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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Rational-representation Group\n\n## Contents\n\n- 1 Definition\n - 1.1 Equivalence of Definitions\n- 2 Examples\n- 3 Facts\n- 4 Relation with Other Properties\n - 4.1 Weaker Properties\n\n**BEWARE!** This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page.\n\nThis article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism.\n\n## Definition\n\nA rational-representation group is a finite group satisfying the following properties:\n1. The field of rational numbers is a splitting field, i.e., every irreducible representation in characteristic zero is realizable over the rational numbers.\n2. Every irreducible representation in characteristic zero can be realized over the field of fractions.\n\n### Equivalence of Definitions\n\nDefinitions (1) and (2) are equivalent because a linear representation is realizable over a principal ideal domain if and only if it is realizable over the field of fractions.\n\n## Examples\n\nHere are some small examples, with relevant contextual information:\n\n| Group | Order | List of Degrees of Irreducible Representations | List of Entries of Matrices Arising Across All Irreducible Representations When Chosen with Simplest Entries |\n|--------------------------|-------|------------------------------------------------|--------------------------------------------------------------------------------------------------|\n| Trivial group | 1 | 1 | |\n| Cyclic group: \\(Z_2\\) | 2 | 1, 1 | |\n| Klein four-group | 4 | 1, 1, 1, 1 | |\n| Symmetric group: \\(S_3\\) | 6 | 1, 1, 2 | |\n| Elementary abelian group: \\(E_8\\) | 8 | 1, 1, 1, 1, 1, 1, 1, 1 | |\n| Dihedral group: \\(D_8\\) | 8 | 1, 1, 1, 1, 2 | |\n| Direct product of \\(S_3\\) and \\(Z_2\\) | 12 | 1, 1, 1, 1, 2, 2 | |\n| Elementary abelian group: \\(E_{16}\\) | 16 | 1 (16 times) | |\n| Direct product of \\(D_8\\) and \\(Z_2\\) | 16 | 1, 1, 1, 1, 1, 1, 1, 1, 2, 2 | |\n| Symmetric group: \\(S_4\\) | 24 | 1, 1, 2, 3, 3 | |\n| Direct product of \\(S_3\\) and \\(V_4\\) | 24 | 1 (8 times), 2 (4 times) | |\n\n## Facts\n\n- Symmetric groups on finite sets are rational-representation groups. For a full proof, refer: Symmetric groups are rational-representation.\n- Dihedral group: \\(D_8\\) is a rational-representation group.\n- Trivial group, cyclic group: \\(Z_2\\), and dihedral group: \\(D_8\\) are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. For a full proof, refer: Classification of finite groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers.\n\n## Relation with Other Properties\n\n### Weaker Properties\n\n- **Rational group:** A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The quaternion group is an example of a rational group that is not a rational-representation group.\n- **Ambivalent group:** A finite group in which every character is real-valued."
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "\nTo: Chris G. Demetriou <>\nFrom: Robby Griffin <>\nList: netbsd-bugs\nDate: 11/14/1997 19:40:38\n>> \t64-bit assignment statements tend to cause bus errors if misaligned.\n>> Other OS's simply report an alignment error, generate a couple of extra\n>> instructions, and get on with life. I don't know exactly what is causing\n>> this behavior, but it should probably be changed to make porting easier.\n>> If unintentional, then it's a bug and should be fixed :)\n\n>It might be nice to provide an option so that the\n>user can _enable_ fixups, but it shouldn't be mandatory.\n\n>I've yet to see a case where an unaligned access in any of the NetBSD\n>source tree was anything other than a programming bug. From the\n>\"good\" developer's standoint, printing a warning tells you that\n>there's a bug, but gives you no opportunity to debug it. From the\n>\"bad\" developer's standpoint, the only things that are actually bugs\n>are those which cause incorrect operation or which crash the program,\n>so unaligned accesses (which generally do indicate broken code, and\n>which do significantly slow down program operation if they happen\n>often enough) \"aren't really bugs.\"\n\nWell, I first encountered the problem not as a developer but as an end-user\nof third-party software. The software worked everywhere else in the world\nbut just died with a bus error on this machine. Quite frustrating, you\nunderstand. The average user never wants to see their favorite software\ncroak \"Bus error\" and die.\n\nAfter gaining experience with unix and C (no longer the average user), I was\nfinally able to go back and figure out what had made the software behave like\nthat. The author had used a buffer tagging scheme on a pool of dynamically\nallocated memory. For each chunk allocated, he slapped on a header and a\nfooter, and returned a pointer to memory starting at the end of the header.\nDue to a ridiculous combination of type sizes, NetBSD/sparc appears to be the\nonly platform on which the header size and the kinds of data stored in these\nbuffers interact to produce misalignment problems.\n\nHis header struct consists of two ints and three pointers, so I suggested he\nadd something like \n char align[(8-(2*sizeof(int)+3*sizeof(char*))%8)%8];\nto the header.\n\nApparently that's too gross for him, just to support one little platform,\nand an ifdef would make it depend even more heavily on existing machine\nconditions. Sigh. Even with fixups I have a sneaking suspicion that it\nwouldn't operate as desired because it'd be too busy printing out warning"
} |
questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
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"answer_type": "categorical",
"index": 67,
"split": "train",
"text": "\nTo: Chris G. Demetriou <>\nFrom: Robby Griffin <>\nList: netbsd-bugs\nDate: 11/14/1997 19:40:38\n>> \t64-bit assignment statements tend to cause bus errors if misaligned.\n>> Other OS's simply report an alignment error, generate a couple of extra\n>> instructions, and get on with life. I don't know exactly what is causing\n>> this behavior, but it should probably be changed to make porting easier.\n>> If unintentional, then it's a bug and should be fixed :)\n\n>It might be nice to provide an option so that the\n>user can _enable_ fixups, but it shouldn't be mandatory.\n\n>I've yet to see a case where an unaligned access in any of the NetBSD\n>source tree was anything other than a programming bug. From the\n>\"good\" developer's standoint, printing a warning tells you that\n>there's a bug, but gives you no opportunity to debug it. From the\n>\"bad\" developer's standpoint, the only things that are actually bugs\n>are those which cause incorrect operation or which crash the program,\n>so unaligned accesses (which generally do indicate broken code, and\n>which do significantly slow down program operation if they happen\n>often enough) \"aren't really bugs.\"\n\nWell, I first encountered the problem not as a developer but as an end-user\nof third-party software. The software worked everywhere else in the world\nbut just died with a bus error on this machine. Quite frustrating, you\nunderstand. The average user never wants to see their favorite software\ncroak \"Bus error\" and die.\n\nAfter gaining experience with unix and C (no longer the average user), I was\nfinally able to go back and figure out what had made the software behave like\nthat. The author had used a buffer tagging scheme on a pool of dynamically\nallocated memory. For each chunk allocated, he slapped on a header and a\nfooter, and returned a pointer to memory starting at the end of the header.\nDue to a ridiculous combination of type sizes, NetBSD/sparc appears to be the\nonly platform on which the header size and the kinds of data stored in these\nbuffers interact to produce misalignment problems.\n\nHis header struct consists of two ints and three pointers, so I suggested he\nadd something like \n char align[(8-(2*sizeof(int)+3*sizeof(char*))%8)%8];\nto the header.\n\nApparently that's too gross for him, just to support one little platform,\nand an ifdef would make it depend even more heavily on existing machine\nconditions. Sigh. Even with fixups I have a sneaking suspicion that it\nwouldn't operate as desired because it'd be too busy printing out warning"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "categorical",
"index": 67,
"split": "train",
"text": "\nTo: Chris G. Demetriou <>\nFrom: Robby Griffin <>\nList: netbsd-bugs\nDate: 11/14/1997 19:40:38\n>> \t64-bit assignment statements tend to cause bus errors if misaligned.\n>> Other OS's simply report an alignment error, generate a couple of extra\n>> instructions, and get on with life. I don't know exactly what is causing\n>> this behavior, but it should probably be changed to make porting easier.\n>> If unintentional, then it's a bug and should be fixed :)\n\n>It might be nice to provide an option so that the\n>user can _enable_ fixups, but it shouldn't be mandatory.\n\n>I've yet to see a case where an unaligned access in any of the NetBSD\n>source tree was anything other than a programming bug. From the\n>\"good\" developer's standoint, printing a warning tells you that\n>there's a bug, but gives you no opportunity to debug it. From the\n>\"bad\" developer's standpoint, the only things that are actually bugs\n>are those which cause incorrect operation or which crash the program,\n>so unaligned accesses (which generally do indicate broken code, and\n>which do significantly slow down program operation if they happen\n>often enough) \"aren't really bugs.\"\n\nWell, I first encountered the problem not as a developer but as an end-user\nof third-party software. The software worked everywhere else in the world\nbut just died with a bus error on this machine. Quite frustrating, you\nunderstand. The average user never wants to see their favorite software\ncroak \"Bus error\" and die.\n\nAfter gaining experience with unix and C (no longer the average user), I was\nfinally able to go back and figure out what had made the software behave like\nthat. The author had used a buffer tagging scheme on a pool of dynamically\nallocated memory. For each chunk allocated, he slapped on a header and a\nfooter, and returned a pointer to memory starting at the end of the header.\nDue to a ridiculous combination of type sizes, NetBSD/sparc appears to be the\nonly platform on which the header size and the kinds of data stored in these\nbuffers interact to produce misalignment problems.\n\nHis header struct consists of two ints and three pointers, so I suggested he\nadd something like \n char align[(8-(2*sizeof(int)+3*sizeof(char*))%8)%8];\nto the header.\n\nApparently that's too gross for him, just to support one little platform,\nand an ifdef would make it depend even more heavily on existing machine\nconditions. Sigh. Even with fixups I have a sneaking suspicion that it\nwouldn't operate as desired because it'd be too busy printing out warning"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
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"answer_type": "categorical",
"index": 68,
"split": "train",
"text": "Paul Davidson: What Makes Economists So Sure of Themselves, Anyway?\n\nBy Paul Davidson, America’s foremost post-Keynesian economist. Davidson is currently the Holly Professor of Excellence, Emeritus at the University of Tennessee in Knoxville. In 1978 Davidson and Sydney Weintraub founded the Journal for Post-Keynesian Economics. Davidson is the author of numerous books, the most recent of which is an introduction to a post-Keynesian perspective on the recent crisis entitled ‘The Keynes Solution: The Path to Global Prosperity’.\n\nIntroduction by Philip Pilkington\n\nIn a recent interview I asked the US’s leading post-Keynesian economist and founder of the Journal of Post-Keynesian Economics, Paul Davidson to discuss what is known as the ‘ergodic axiom’ in economics. This is a particularly important axiom as it allows mainstream economists (including left-wing Keynesians like Paul Krugman and Joseph Stiglitz) to claim that they can essentially know the future in a very tangible way. It does this by assuming that the future can be known by examining the past.\n\nWithout this axiom the whole edifice of mainstream theory rests on very shaky grounds. Yet, it should be clear to anyone that given that the theory is supposed to explain human behaviour it is unlikely that the future will correlate with the past because people and institutions tend to change and evolve over a given period of time.\n\nYes, often past behaviour will help us understand future behaviour – apply this in a simple psychological way to anyone you know and you will find it to be true – however, it should be quite clear that all future behaviour cannot be wholly explained by past behaviour. Clearly it should be quite obvious that the same should apply when we consider large aggregates of individuals and yet mainstream economics steadfastly refuses to accept this.\n\nWhat follows is an particularly succinct overview of these ideas and a summary of their importance that Professor Davidson has kindly written for us. He also lays out the alternative view that the future is not determined by probabilistic risk but is instead subject to an absolute unknowability or uncertainty as laid out in the work of John Maynard Keynes and the implications of this.\n\nThis is a hugely important debate in that it essentially touches on what a good economic theory would allow for in terms of government policy. The ergodic axiom is possibly one of the key reasons that many economists show a remarkable anxiety when it comes to any human action undertaken outside of their models.\n\n– — Philip Pilkington\n\n\n\nThe economy is a process in historical time. Time is a device that prevents everything from happening at once. The production of commodities takes time; and the consumption of goods, especially durables, takes considerable time. Economics is the study of how households and firms make decision choices regarding production and consumption when the outcome (pay-off) of today’s decision occurs at a significantly later date.\n\nAny study of the behavior of economic decision-makers, therefore, requires the analyst to make some assumption regarding what today’s decision-makers ‘know’ about future outcomes.\n\nThere are two different concepts of the uncertain knowledge regarding the future outcomes of decisions made today. The mainstream concept regarding knowledge about the future and the Keynes General Theory concept. The ability of economists to explain the importance of the role of money, liquidity, and the existence of persistent unemployment in a market economy depends on which concept of knowledge about the future economists use as the basis of their economic analysis.\n\nBecause economists are split into two major theoretical camps about what decision-makers know about the future, these groups provide differing explanations of economic problems and their policy solutions. Understanding the differences in these two concepts of knowledge of future outcomes is essential to understanding the philosophical differences between economists on the role for government and economic policies in the economic system.\n\nAs explained below, mainstream economics assumes that households and entrepreneurs are optimal decision makers, that is, they choose the decision today that optimizes their utility, income and profits over time.\n\n\nRicardo (1817), the father of 19th century classical economics, assumed a world of perfect certainty. All households and businesses were assumed to possess a full and correct knowledge of a presumed programmed external economic reality that governed all past, present, and future economic outcomes. The external economic environment was assumed immutable in the sense that it was not susceptible to change induced by human action. The path of the economy, like the path of the planets under Newton’s celestial mechanics, was determined by timeless natural laws. Economic decision makers had complete knowledge of these laws. Households and firms never made errors in their spending choices. They always spend everything they earned on things with the highest ‘known’ future pay-off in terms of utility for households and profits for businesses. Accordingly, there could never be a lack of demand for the products of industry or for workers who wanted to work. Classical economics justified a laissez-faire philosophy for the economic system. No government action could provide a higher pay-off than the decisions individuals made with complete information about the future in free markets.\n\n\nIn the early 20th century, classical economists tended to substitute the notion of probabilistic risk premiums and “certainty equivalents” for the perfect knowledge assumption of earlier classical theory. Risk premiums was said to provide “uncertainty” allowances where the latter referred to the difference between the estimated value of a future event, held with an objective (frequency distribution) probability of less than unity and the value of a perfectly certain (p = 1) event that evokes the same behavior.\n\nBy the 1970s this classical risk analysis had evolved into what mainstream economists call the New Classical Theory of ‘rational expectations’ where individuals make decisions based on their subjective probability distributions regarding future events where the subjective probabilities are presumed to be equal to immutable objective probability distributions that govern future outcomes [Lucas, 1972]. Today all mainstream economists interpret uncertainty in economics as synonymous with objective probability distributions [Lucas and Sargent, 1981; Machina 1987] that govern future events but are completely known to all persons today.\n\nThis device of labelling statistically reliable estimates of probabilistic risk regarding future outcomes as uncertainty permits mainstream economists to preserve intact most of the laissez faire efficient market analysis that had been developed under the earlier perfect certainty assumption. While rejecting the perfect certainty model, mainstream economists still accept, as a universal truth, the existence of a predetermined external economic reality (similar to Newton’s celestial mechanics which, for example, permits the astronomer to accurately predict the next solar eclipse) that can be fully described by unchanging objective conditional probability functions that are fully known by the decision makers in one’s model… Unlike the perfect certainty model, however, conflating the concept of uncertainty with the probabilitistic risk permits individual decision makers to make an occasional erroneous choice (in the short run) just as a single sample means can differ from the true universe value. In the long run, the assumption that people with rational expectations already “know” the objective probabilities assures correct choices on average for those “fittest” decision makers who survived in the Darwinian world of free markets. In other words, free markets lead to optimal solutions at least in the long run\n\nIn mainstream economics, economic data are typically viewed as part of time series realization generated by an ergodic stochastic processes. In fact, Nobel Prize winner Paul Samuelson (1969) has made the acceptance of the ergodic axiom the sine qua non of the scientific method in economics. What is this ergodic axiom that Samuelson insists is necessary for economics to be a science?\n\n\nLogically, to make statistically reliable probabilistic forecasts about future economic events, today’s decision-makers should obtain and analyze sample data from the future. Since that is impossible, the assumption of ergodic stochastic economic processes permits the analyst to assert that the outcome at any future date is the statistical shadow of past and current market data.\n\nA realization of a stochastic process is a sample value of a multidimensional variable over a period of time, i.e., a single time series. A stochastic process makes a universe of such time series. Time statistics refer to statistical averages (e.g., the mean, standard deviation) calculated from a single fixed realization over an indefinite time space. Space statistics, on the other hand, refer to a fixed point of time and are formed over the universe of realizations (i.e. they are statistics obtained from cross-sectional data).\n\nStatistical theory asserts that if the stochastic process is ergodic then for an infinite realization, the time statistics and the space statistics will coincide. For finite realizations of ergodic processes, time and space statistics coincide except for random errors; they will tend to converge (with the probability of unity) as the number of observations increase. Consequently, if ergodicity is assumed, statistics calculated from past time series or cross-sectional data are statistically reliable estimates of the statistics probabilities that will occur at any future date.\n\nIn simple language, the ergodic presumption assures that economic outcomes on any specific future date can be reliably predicted by a statistical probability analysis of existing market data. By assumption, New Classical economic theory imposes the condition that economic relationships are timeless or ahistoric ‘natural’ laws. The historical dates when observations are collected do not affect the estimates of the statistical time and space averages. Accordingly, the mainstream presumption (utilized by both New Classical economists and New Keynesian economists) that decision-makers possess rational expectations imply that people in one’s model process information embedded in past and present market data to form statistical averages (or decision weights) that reliably forecast the future. Or as 2011 Nobel Prize winner Thomas Sargent [1993, p. 3], one of the leaders of the rational expectations school, states “rational expectations models impute much more knowledge to the agents within the model (who use the equilibrium probability distributions)… than is possessed by an econometrician, who faces estimation and inference problems that the agents in the model have somehow solved”.\n\nBy using probabilistic distributions calculated from past market data, rational expectations theory assumes that, on average, the actions fostered by these expectations are precisely those that would be forthcoming in a perfectly certain world – at least in the long run.\n\nIn recent years, partly in reaction to the rational expectations hypothesis, some mainstream economists have raised questions regarding the use of such stochastic concepts to define uncertainty. For example, Nobel Prize winner R. M. Solow (1985, p. 328) has stated “economics is a social science….much of what we observe cannot be treated as the realization of a stationary stochastic process without straining credulity”. Since stationary is a necessary but not sufficient condition for ergodicity, Solow’s statement implies that only the very gullible would ever believe that most important macroeconomic processes are ergodic.\n\n\nBeginning with Knight’s [1921] seminal work, some economists have drawn a distinction between “true” uncertainty and probabilistic risk, where the latter is calculable based on past frequency distributions and is, therefore, conceptually insurable, while uncertainty is neither calculable nor insurable.\n\nJohn Maynard Keynes (1936) launched a revolution in economics. Keynes explicitly developed an alternative “general theory” to classical theory. Keynes argued that the difference between probabilistic risk and uncertainty had important implications for understanding (a) the operations of a market economy and (b) the role of government in influencing market outcomes through deliberate legislative policies.\n\nIn Keynes’s (1936) analysis, whenever the full consequences of today’s economic decisions occur many days in the future, uncertainty would prevail and economic behavior could not be described as an “outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities”.\n\nUnlike today’s orthodox economists, Keynes did not write in the idiom of stochastic processes in developing his concept of uncertainty. Keynes (1937) simply described uncertainty as occurring when there is no scientific basis to form any calculable probability. Nevertheless, in criticizing Tinbergen’s use of econometric analysis, Keynes (1939) argued that Tinbergen’s ‘method’ was not applicable to economic data because “the economic environment is not homogeneous over a period of time”, a criticism that is equivalent to stating economic time series are not stationary.\n\nWith the development of ergodic theory and stochastic process analysis since Keynes wrote, it is possible to interpret Keynes’s uncertainty concept in terms of this statistical lexicon. Keynes’s theory required decision-makers to recognize that in the market system in which they operate, in some but not necessarily all economic dimensions, the future is uncertain and cannot be reliably predicted on the basis of any statistical analysis of past evidence. The absence of ergodic conditions, therefore, is a sufficient condition for Keynes’s concept of uncertainty. In a nonergodic environment, even if agents have the capacity to obtain and statistically process past and current market data, these observations do not, and cannot, provide a statistically reliable basis for forecasting the probability distributions, if any, that will govern outcomes at any specific date in the future. According to Keynes (1937), “About these [future] matters there is no scientific basis to form any calculable probability whatever. We simply do not know.”\n\nKeynes’s uncertainty concept implies that the future is transmutable or creative in the sense that future economic outcomes may be permanently changed in nature and substance by today’s actions of individuals, groups (e.g., unions, cartels) and/or governments, often in ways not even perceived by the creators of change. (It is also possible that changes that are not predetermined can occur even without any deliberate human economic action.). [George Soros’s concept of ‘reflexivity’ asserts future market outcomes are determined by market participants’ actions today.]\n\nThis nonergodic view of modelling the future out comes as being determined by peoples’ actions rather than a timeless probability distribution has been described by Nobel Prize winner Sir John Hicks (1977) as a situation where people in the model “do not know what is going to happen and know that they do not what is going to happen. As in history!” Hicks (1979) declared that “I am bold enough to conclude from these considerations that the usefulness of ‘statistical’ or ‘stochastic’ methods in economics is a good deal less than is now conventionally supposed.”\n\nAccordingly, mainstream macroeconomics is logically inconsistent with Keynes’ macroeconomic general theory explaining employment, interest, and money. The result has been a continuing debate between the followers of Keynes and mainstream theorists over the relevant policy prescriptions for solving the macroeconomic problems of the real world.\n\nThe first postulate of mainstream economics is the presumption that there exists a finite set of acts and outcomes and that each agent can make a complete and transitive preference ordering of all possible alternative choices. Decision making by agents who know the statistically reliable future can characterize the decision process as “Look before you leap”. This “Look before you leap” analysis, however, is not a general theory of decision making for it does not explicitly deal with uncertainty per se. As the statistical theorist Leonard Savage recognized “a person may not know [all] the consequences of the acts open to him in each state of the world. He might be … ignorant” and hence might want to leave his options open; a decision that Savage characterizes as “You can cross that bridge when you come to it”. Savage admits the latter is often a more accurate description of the human economic predicament. When a decision maker is ‘ignorant’ and wants to wait before making a decision, we can classify the situation as one involving Keynes’ uncertainty concept and therefore the mainstream ergodic axiom is violated.\n\nAs Savage puts it, mainstream economics “attack[s] relatively simple problems of decision by artificially confining attention to so small a world that the `Look before you leap’ principle can be applied”, i.e., where Keynes’ uncertainty concept is not relevant. Savage warns that mainstream theory is “practical [only] in suitably limited domains… At the same time, the behavior of people is often at variance with the theory. The departure is sometimes flagrant … the `Look before you leap’ principle is preposterous if carried to extremes”. Yet when today’s mainstream economic theorists talk about efficient free markets they treat uncertainty in economics as synonymous with a probability measure The behavior they describe flagrantly departs from the behavior that determines employment in a money-using market economy.\n\nIf, as Savage recognizes, in some areas of economic activity the ability of humans to form a complete preference ordering regarding all potential consequences of all actions is not possible, then mainstream theory cannot provide a useful explanation of the behavior of decision-makers in these areas. It is here that Keynes’ uncertainty concept becomes paramount\n\nIn the classical (ergodic) theory, where all outcomes are conceptually calculable, there is never a need to keep options open. People will therefore spend all they earn on the products of industry (Say’s Law) and there can never be a lack of effective demand to prevent the system from reaching full employment.\n\nOn the other hand, when households and firms “know that they do not know” the future and therefore cannot order all future consequences associated with any possible choice today, they may wish to defer forever making “look before they leap” decisions. When people believe the future is uncertain in the sense of Keynes, they prefer to leave their options open, i.e., to cross that bridge when, and if, they come to it.\n\nWhenever households and business managers believe they cannot predict the future with any degree of a priori or statistically reliable probability, then the axiomatic foundation of mainstream economic theory is violated. Hicks (1979) has associates this transgression of mainstream axiomatic ergodic basis with Keynes’ long-term ‘liquidity’ concept. For Keynes, it is the existence of an uncertain future that makes a long-run demand for liquidity (money and other liquid assets traded in well organized markets where prices movements are ‘orderly’) a ubiquitous fact of life. The ability to save one’s income in the form of money and other liquid assets permits households and firms to keep their options open by not having to spend all of their earned income on the products of industry, even in the long-run.\n\nAs long as income-earning decision-makers have this option of demanding liquidity rather than the products of industry, then a laissez-faire market system cannot assure that peoples’ total market demand for goods and services will be sufficient to make it profitable for firms to fully employ all who want to work.\n\nThe notion of a demand for long-term liquidity can only be justified in a world of Keynes’ (nonergodic) uncertainty. This desire for long-term liquidity is incompatible with mainstream’s optimal decision makers in an ergodic environment. Only the Keynes concept of uncertainty in economics provides a logical, statistical explanation of the phenomenon of persistent unemployment that occurs in the market economies in the world we inhabit. Only the Keynes uncertainty concept can justify a role for governmental policies to assure full employment when questions of liquidity are important.\n\n\nDavidson, P. (1991) “Is Probability Theory Relevant For Uncertainty? A Post Keynesian Perspective”, Journal of Economic Perspectives, 5. (Distinguishes between economic decisions where ergodic circumstances might prevail, and situations where nonergodic circumstances are likely. The former are called routine decisions, the latter are crucial decisions.)\n\nHicks, J. R. (1977), Economic Perspectives, Oxford University Press, Oxford.(Argues for economic models where agents ‘know’ that they cannot reliably predict the future.)\n\nHicks, J. R (1979), Causality in Economics, Basic Books, New York. (Argues that economics is embedded in time in a way that the physical sciences are not. Consequently stochastic theory is not applicable to most dynamic economic problems.)\n\nKeynes, J. M. (1936), The General Theory of Employment, Interest and Money Harcourt, New York. (The basis for the ‘Keynesian Revolution’ where the existence of uncertainty explains why market economies have no endogenous forces that assure full employment.)\n\nKeynes, J. M. (1937), “The General Theory of Employment” Quarterly Journal of Economics, 52. (A further extension of what Keynes means by ‘uncertainty’ and why uncertainty is the root cause of unemployment in market economies.)\n\nKeynes, J. M. (1939), “Professor Tinbergen’s Method”, The Economic Journal, 47. (Keynes attacks the statistical method of regression analysis as not applicable to economic time series data.)\n\nKnight, F. N. (1921), Risk, Uncertainty, and Profit, Houghton Mifflin, New York. (Distinguished between probabilistic risk and uncertainty.)\n\nLucas, R. E., (1972) “Expectations and the Neutrality of money”, Journal of Economic Theory,4. (The article that initiated the rational expectations analysis in macroeconomics.)\n\nLucas R. E., and Sargent, T. J. (1981), Rational Expectations and Econometric Practices, Minneapolis, University of Minnesota Press. (Develops the relationship between the rational expectations hypothesis and the axioms underlying econometric analysis for macroeconomic analysis.)\n\nMachina, M. J. “Choice Under Uncertainty; Problems Solved and Unsolved”, Journal of Economic Perspectives, 1. (Attempts to shore up the theory of choice under uncertainty on “solid axiomatic foundations” of probabilistic risk in the face of the famous St. Petersburg paradox and other challenges to expected utility theory).\n\nRicardo, D. (1817), On the Principles of Political Economy and Taxation. (The first economist to formulate the axiom of perfect certainty in economics.)\n\nSavage, L. (1954), The Foundations of Statistics Wiley, New York.(Develops the Expected Utility Theory of economics for making decision with complete subjective probabilistic information.)\n\nSargent, T. J. (1993), Bounded Rationality in Macroeconomics, Oxford, Clarendon Press. (A founder of the rational expectations school who now argues that rational expectations are not applicable to situations where people find themselves in new, i.e., nonergodic, situations.)\n\nSolow, R. M. (1985), “Economic History and Economics”, American Economic Review Papers and Proceedings, 75.\n\nPrint Friendly\nTweet about this on Twitter0Digg thisShare on Reddit0Share on StumbleUpon0Share on Facebook0Share on LinkedIn3Share on Google+2Buffer this pageEmail this to someone\n\n\n 1. craazyman\n\n wow. 1817 to 2012.\n\n one hundred and ninety-five years of total nonsense.\n\n that’s almost a triumph of imagination over reality. hahahah.\n\n 1. Valissa\n\n Since economists lack the creative imagination to create their own deity, I nominate the Wrathful Deity Mahakala in both his Hindu and Japanese aspects.\n\n Mahakala was the personal tutelary deity for the Mongol ruler Kublai Khan. His terrifying imagery ultimately derives from the angry form of the Hindu god Shiva, known as Bhairava. In Tibetan iconography he typically has one head with three bulging eyes. His eyebrows are like small flames, and his beard is made of hook-like shapes. He can have two to six arms.\n\n The essential nature of Mahakala in the Tibetan pantheon can be gauged from the fact that he is worshipped as the Protector of the tent.\n\n In this case the “tent” symbolizes the “big tent” of economic dogma.\n\n In Japan, Mahakala…\n\n enjoys an exalted position as a household deity in Japan, as he is one of the Seven Lucky Gods in Japanese folklore. Mahākāla’s association with wealth and prosperity gave rise to a strange custom known as Fuku-nusubi. This custom started with the belief that one who stole divine figures (gods and goddesses) was assured of good fortune, if not caught in the act of stealing. In the course of time stealing of divine images became so common a practice in Japan that the Toshi-no-ichi or the ‘year-end-market’ held in the Asakusa Kannon temple became the main venue of the sale and disposal of such images by the fortune-seekers.\n\n 1. steelhead23\n\n God I love NC. Where else can I read of the Keynesian perspective on uncertainty and comparative Asian religion in a single thread? Where’s the course syllabus for this seminar?\n\n 2. digi_owl\n\n Actually their deity would be “the invisible hand”. And most mainstreams seems to think of it as being very much benevolent. Never mind that it seems to have ruined just about any community it has touched…\n\n 1. F. Beard\n\n But the collection basket remains and you don’t even have to reach into your own pocket! The banks do it for you while most economists look the other way.\n\n 2. enouf\n\n Think of economics as religion without the music, the art and the cathedrals.–Jake chase says: January 19, 2012 at 9:01 am\n\n Think of everything and everyone whom blatantly denies (and even those “agnostics”) the existence of a Higher Intelligence as a “humanist” or Belief in Humanism.\n\n The denial of reality of a consciousness/awareness of self, the creativity, and innate personal moral fortitude (the sense of Injustice, the feeling of Love, etc) leads to nothing short of a carnival act.\n\n 3. enouf\n\n\n\n\n 1. tyaresun\n\n The father of rational expectations is John Muth. I had the priviledge of working with him in 1985-86. He was working on an errors in expectations model at that time. I don’t think you will be able to find any published research on that model. Also, Muth never received the Nobel Prize even though rational expectations forms the fundamental brick of modern economics. Why? One of the reasons given is that others took over the idea and made more fundamental contributions.\n\n Could it be that Muth did not receive the prize because he had changed his mind on ratioanal expectations?\n\n 2. Tom Dority\n\n I suppose economists are stuck in a closed loop. Describing events but not understanding. Maybe quantum economic theory and understanding of the Heisenberg principle would be a start. Every market participant is defined by their own uncertainty and environment – they are a molecule or atom in the economic world – thus economists have only described what they see and fool themselves when they think they can know why – having limited themselves to old earth centrism quasi science in my view. The butterfly principle alive and well. I am not an economist but, I think looking into math and scientific understanding of our physical universe may lead to an expansion in understanding our economics. Sorry first post and a bit loose.\n\n 1. Birch\n\n Before getting into quantum economic theory, it would be very useful for economists to apply the laws of thermodynamics to their discipline. Frederick Soddy did this in the ’30s to great effect, and the tradition has been continued; most notably by Herman Daly whose economic theories actually relate to the real world.\n\n 1. Fiver\n\n Would like to see NC take an interest and do a piece on Daly – we either develop a sustainable economy or we are headed for a very bad outcome.\n\n 3. jake chase\n\n The only value of economics is derived from the fact that a substantial number of people believe it, at least selectively. Their actions create opportunity for contrarians who ignore the signals cherished by the believers.\n\n 4. Tom Dority\n\n To add, Is there a periodic table of economic elements of a grand theory like E=mc2, maybe something like Economic output is equal to money supply times x too the second power.\n\n 5. René\n\n\n This is true for all models created by the elite and their helpers. Hence, NDAA and the SOPA Act.\n\n 6. Eric Patton\n\n I am not an economist, but the fact that Robin Hahnel and Michael Albert’s 1990 Princeton University Press book Quiet Revolution in Welfare Economics is out of print and persona-non-grata in the profession really tells you all you need to know about the religion that is economics.\n\n 1. Frank Speaking\n\n just as, “War is…an act of force to compel our enemy to do our will” so is economics an act of force to compel members of a society to do the will of others.\n\n Herbert Gintis and an entire school of economic thought has been marginalized be"
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"text": "Paul Davidson: What Makes Economists So Sure of Themselves, Anyway?\n\nBy Paul Davidson, America’s foremost post-Keynesian economist. Davidson is currently the Holly Professor of Excellence, Emeritus at the University of Tennessee in Knoxville. In 1978 Davidson and Sydney Weintraub founded the Journal for Post-Keynesian Economics. Davidson is the author of numerous books, the most recent of which is an introduction to a post-Keynesian perspective on the recent crisis entitled ‘The Keynes Solution: The Path to Global Prosperity’.\n\nIntroduction by Philip Pilkington\n\nIn a recent interview I asked the US’s leading post-Keynesian economist and founder of the Journal of Post-Keynesian Economics, Paul Davidson to discuss what is known as the ‘ergodic axiom’ in economics. This is a particularly important axiom as it allows mainstream economists (including left-wing Keynesians like Paul Krugman and Joseph Stiglitz) to claim that they can essentially know the future in a very tangible way. It does this by assuming that the future can be known by examining the past.\n\nWithout this axiom the whole edifice of mainstream theory rests on very shaky grounds. Yet, it should be clear to anyone that given that the theory is supposed to explain human behaviour it is unlikely that the future will correlate with the past because people and institutions tend to change and evolve over a given period of time.\n\nYes, often past behaviour will help us understand future behaviour – apply this in a simple psychological way to anyone you know and you will find it to be true – however, it should be quite clear that all future behaviour cannot be wholly explained by past behaviour. Clearly it should be quite obvious that the same should apply when we consider large aggregates of individuals and yet mainstream economics steadfastly refuses to accept this.\n\nWhat follows is an particularly succinct overview of these ideas and a summary of their importance that Professor Davidson has kindly written for us. He also lays out the alternative view that the future is not determined by probabilistic risk but is instead subject to an absolute unknowability or uncertainty as laid out in the work of John Maynard Keynes and the implications of this.\n\nThis is a hugely important debate in that it essentially touches on what a good economic theory would allow for in terms of government policy. The ergodic axiom is possibly one of the key reasons that many economists show a remarkable anxiety when it comes to any human action undertaken outside of their models.\n\n– — Philip Pilkington\n\n\n\nThe economy is a process in historical time. Time is a device that prevents everything from happening at once. The production of commodities takes time; and the consumption of goods, especially durables, takes considerable time. Economics is the study of how households and firms make decision choices regarding production and consumption when the outcome (pay-off) of today’s decision occurs at a significantly later date.\n\nAny study of the behavior of economic decision-makers, therefore, requires the analyst to make some assumption regarding what today’s decision-makers ‘know’ about future outcomes.\n\nThere are two different concepts of the uncertain knowledge regarding the future outcomes of decisions made today. The mainstream concept regarding knowledge about the future and the Keynes General Theory concept. The ability of economists to explain the importance of the role of money, liquidity, and the existence of persistent unemployment in a market economy depends on which concept of knowledge about the future economists use as the basis of their economic analysis.\n\nBecause economists are split into two major theoretical camps about what decision-makers know about the future, these groups provide differing explanations of economic problems and their policy solutions. Understanding the differences in these two concepts of knowledge of future outcomes is essential to understanding the philosophical differences between economists on the role for government and economic policies in the economic system.\n\nAs explained below, mainstream economics assumes that households and entrepreneurs are optimal decision makers, that is, they choose the decision today that optimizes their utility, income and profits over time.\n\n\nRicardo (1817), the father of 19th century classical economics, assumed a world of perfect certainty. All households and businesses were assumed to possess a full and correct knowledge of a presumed programmed external economic reality that governed all past, present, and future economic outcomes. The external economic environment was assumed immutable in the sense that it was not susceptible to change induced by human action. The path of the economy, like the path of the planets under Newton’s celestial mechanics, was determined by timeless natural laws. Economic decision makers had complete knowledge of these laws. Households and firms never made errors in their spending choices. They always spend everything they earned on things with the highest ‘known’ future pay-off in terms of utility for households and profits for businesses. Accordingly, there could never be a lack of demand for the products of industry or for workers who wanted to work. Classical economics justified a laissez-faire philosophy for the economic system. No government action could provide a higher pay-off than the decisions individuals made with complete information about the future in free markets.\n\n\nIn the early 20th century, classical economists tended to substitute the notion of probabilistic risk premiums and “certainty equivalents” for the perfect knowledge assumption of earlier classical theory. Risk premiums was said to provide “uncertainty” allowances where the latter referred to the difference between the estimated value of a future event, held with an objective (frequency distribution) probability of less than unity and the value of a perfectly certain (p = 1) event that evokes the same behavior.\n\nBy the 1970s this classical risk analysis had evolved into what mainstream economists call the New Classical Theory of ‘rational expectations’ where individuals make decisions based on their subjective probability distributions regarding future events where the subjective probabilities are presumed to be equal to immutable objective probability distributions that govern future outcomes [Lucas, 1972]. Today all mainstream economists interpret uncertainty in economics as synonymous with objective probability distributions [Lucas and Sargent, 1981; Machina 1987] that govern future events but are completely known to all persons today.\n\nThis device of labelling statistically reliable estimates of probabilistic risk regarding future outcomes as uncertainty permits mainstream economists to preserve intact most of the laissez faire efficient market analysis that had been developed under the earlier perfect certainty assumption. While rejecting the perfect certainty model, mainstream economists still accept, as a universal truth, the existence of a predetermined external economic reality (similar to Newton’s celestial mechanics which, for example, permits the astronomer to accurately predict the next solar eclipse) that can be fully described by unchanging objective conditional probability functions that are fully known by the decision makers in one’s model… Unlike the perfect certainty model, however, conflating the concept of uncertainty with the probabilitistic risk permits individual decision makers to make an occasional erroneous choice (in the short run) just as a single sample means can differ from the true universe value. In the long run, the assumption that people with rational expectations already “know” the objective probabilities assures correct choices on average for those “fittest” decision makers who survived in the Darwinian world of free markets. In other words, free markets lead to optimal solutions at least in the long run\n\nIn mainstream economics, economic data are typically viewed as part of time series realization generated by an ergodic stochastic processes. In fact, Nobel Prize winner Paul Samuelson (1969) has made the acceptance of the ergodic axiom the sine qua non of the scientific method in economics. What is this ergodic axiom that Samuelson insists is necessary for economics to be a science?\n\n\nLogically, to make statistically reliable probabilistic forecasts about future economic events, today’s decision-makers should obtain and analyze sample data from the future. Since that is impossible, the assumption of ergodic stochastic economic processes permits the analyst to assert that the outcome at any future date is the statistical shadow of past and current market data.\n\nA realization of a stochastic process is a sample value of a multidimensional variable over a period of time, i.e., a single time series. A stochastic process makes a universe of such time series. Time statistics refer to statistical averages (e.g., the mean, standard deviation) calculated from a single fixed realization over an indefinite time space. Space statistics, on the other hand, refer to a fixed point of time and are formed over the universe of realizations (i.e. they are statistics obtained from cross-sectional data).\n\nStatistical theory asserts that if the stochastic process is ergodic then for an infinite realization, the time statistics and the space statistics will coincide. For finite realizations of ergodic processes, time and space statistics coincide except for random errors; they will tend to converge (with the probability of unity) as the number of observations increase. Consequently, if ergodicity is assumed, statistics calculated from past time series or cross-sectional data are statistically reliable estimates of the statistics probabilities that will occur at any future date.\n\nIn simple language, the ergodic presumption assures that economic outcomes on any specific future date can be reliably predicted by a statistical probability analysis of existing market data. By assumption, New Classical economic theory imposes the condition that economic relationships are timeless or ahistoric ‘natural’ laws. The historical dates when observations are collected do not affect the estimates of the statistical time and space averages. Accordingly, the mainstream presumption (utilized by both New Classical economists and New Keynesian economists) that decision-makers possess rational expectations imply that people in one’s model process information embedded in past and present market data to form statistical averages (or decision weights) that reliably forecast the future. Or as 2011 Nobel Prize winner Thomas Sargent [1993, p. 3], one of the leaders of the rational expectations school, states “rational expectations models impute much more knowledge to the agents within the model (who use the equilibrium probability distributions)… than is possessed by an econometrician, who faces estimation and inference problems that the agents in the model have somehow solved”.\n\nBy using probabilistic distributions calculated from past market data, rational expectations theory assumes that, on average, the actions fostered by these expectations are precisely those that would be forthcoming in a perfectly certain world – at least in the long run.\n\nIn recent years, partly in reaction to the rational expectations hypothesis, some mainstream economists have raised questions regarding the use of such stochastic concepts to define uncertainty. For example, Nobel Prize winner R. M. Solow (1985, p. 328) has stated “economics is a social science….much of what we observe cannot be treated as the realization of a stationary stochastic process without straining credulity”. Since stationary is a necessary but not sufficient condition for ergodicity, Solow’s statement implies that only the very gullible would ever believe that most important macroeconomic processes are ergodic.\n\n\nBeginning with Knight’s [1921] seminal work, some economists have drawn a distinction between “true” uncertainty and probabilistic risk, where the latter is calculable based on past frequency distributions and is, therefore, conceptually insurable, while uncertainty is neither calculable nor insurable.\n\nJohn Maynard Keynes (1936) launched a revolution in economics. Keynes explicitly developed an alternative “general theory” to classical theory. Keynes argued that the difference between probabilistic risk and uncertainty had important implications for understanding (a) the operations of a market economy and (b) the role of government in influencing market outcomes through deliberate legislative policies.\n\nIn Keynes’s (1936) analysis, whenever the full consequences of today’s economic decisions occur many days in the future, uncertainty would prevail and economic behavior could not be described as an “outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities”.\n\nUnlike today’s orthodox economists, Keynes did not write in the idiom of stochastic processes in developing his concept of uncertainty. Keynes (1937) simply described uncertainty as occurring when there is no scientific basis to form any calculable probability. Nevertheless, in criticizing Tinbergen’s use of econometric analysis, Keynes (1939) argued that Tinbergen’s ‘method’ was not applicable to economic data because “the economic environment is not homogeneous over a period of time”, a criticism that is equivalent to stating economic time series are not stationary.\n\nWith the development of ergodic theory and stochastic process analysis since Keynes wrote, it is possible to interpret Keynes’s uncertainty concept in terms of this statistical lexicon. Keynes’s theory required decision-makers to recognize that in the market system in which they operate, in some but not necessarily all economic dimensions, the future is uncertain and cannot be reliably predicted on the basis of any statistical analysis of past evidence. The absence of ergodic conditions, therefore, is a sufficient condition for Keynes’s concept of uncertainty. In a nonergodic environment, even if agents have the capacity to obtain and statistically process past and current market data, these observations do not, and cannot, provide a statistically reliable basis for forecasting the probability distributions, if any, that will govern outcomes at any specific date in the future. According to Keynes (1937), “About these [future] matters there is no scientific basis to form any calculable probability whatever. We simply do not know.”\n\nKeynes’s uncertainty concept implies that the future is transmutable or creative in the sense that future economic outcomes may be permanently changed in nature and substance by today’s actions of individuals, groups (e.g., unions, cartels) and/or governments, often in ways not even perceived by the creators of change. (It is also possible that changes that are not predetermined can occur even without any deliberate human economic action.). [George Soros’s concept of ‘reflexivity’ asserts future market outcomes are determined by market participants’ actions today.]\n\nThis nonergodic view of modelling the future out comes as being determined by peoples’ actions rather than a timeless probability distribution has been described by Nobel Prize winner Sir John Hicks (1977) as a situation where people in the model “do not know what is going to happen and know that they do not what is going to happen. As in history!” Hicks (1979) declared that “I am bold enough to conclude from these considerations that the usefulness of ‘statistical’ or ‘stochastic’ methods in economics is a good deal less than is now conventionally supposed.”\n\nAccordingly, mainstream macroeconomics is logically inconsistent with Keynes’ macroeconomic general theory explaining employment, interest, and money. The result has been a continuing debate between the followers of Keynes and mainstream theorists over the relevant policy prescriptions for solving the macroeconomic problems of the real world.\n\nThe first postulate of mainstream economics is the presumption that there exists a finite set of acts and outcomes and that each agent can make a complete and transitive preference ordering of all possible alternative choices. Decision making by agents who know the statistically reliable future can characterize the decision process as “Look before you leap”. This “Look before you leap” analysis, however, is not a general theory of decision making for it does not explicitly deal with uncertainty per se. As the statistical theorist Leonard Savage recognized “a person may not know [all] the consequences of the acts open to him in each state of the world. He might be … ignorant” and hence might want to leave his options open; a decision that Savage characterizes as “You can cross that bridge when you come to it”. Savage admits the latter is often a more accurate description of the human economic predicament. When a decision maker is ‘ignorant’ and wants to wait before making a decision, we can classify the situation as one involving Keynes’ uncertainty concept and therefore the mainstream ergodic axiom is violated.\n\nAs Savage puts it, mainstream economics “attack[s] relatively simple problems of decision by artificially confining attention to so small a world that the `Look before you leap’ principle can be applied”, i.e., where Keynes’ uncertainty concept is not relevant. Savage warns that mainstream theory is “practical [only] in suitably limited domains… At the same time, the behavior of people is often at variance with the theory. The departure is sometimes flagrant … the `Look before you leap’ principle is preposterous if carried to extremes”. Yet when today’s mainstream economic theorists talk about efficient free markets they treat uncertainty in economics as synonymous with a probability measure The behavior they describe flagrantly departs from the behavior that determines employment in a money-using market economy.\n\nIf, as Savage recognizes, in some areas of economic activity the ability of humans to form a complete preference ordering regarding all potential consequences of all actions is not possible, then mainstream theory cannot provide a useful explanation of the behavior of decision-makers in these areas. It is here that Keynes’ uncertainty concept becomes paramount\n\nIn the classical (ergodic) theory, where all outcomes are conceptually calculable, there is never a need to keep options open. People will therefore spend all they earn on the products of industry (Say’s Law) and there can never be a lack of effective demand to prevent the system from reaching full employment.\n\nOn the other hand, when households and firms “know that they do not know” the future and therefore cannot order all future consequences associated with any possible choice today, they may wish to defer forever making “look before they leap” decisions. When people believe the future is uncertain in the sense of Keynes, they prefer to leave their options open, i.e., to cross that bridge when, and if, they come to it.\n\nWhenever households and business managers believe they cannot predict the future with any degree of a priori or statistically reliable probability, then the axiomatic foundation of mainstream economic theory is violated. Hicks (1979) has associates this transgression of mainstream axiomatic ergodic basis with Keynes’ long-term ‘liquidity’ concept. For Keynes, it is the existence of an uncertain future that makes a long-run demand for liquidity (money and other liquid assets traded in well organized markets where prices movements are ‘orderly’) a ubiquitous fact of life. The ability to save one’s income in the form of money and other liquid assets permits households and firms to keep their options open by not having to spend all of their earned income on the products of industry, even in the long-run.\n\nAs long as income-earning decision-makers have this option of demanding liquidity rather than the products of industry, then a laissez-faire market system cannot assure that peoples’ total market demand for goods and services will be sufficient to make it profitable for firms to fully employ all who want to work.\n\nThe notion of a demand for long-term liquidity can only be justified in a world of Keynes’ (nonergodic) uncertainty. This desire for long-term liquidity is incompatible with mainstream’s optimal decision makers in an ergodic environment. Only the Keynes concept of uncertainty in economics provides a logical, statistical explanation of the phenomenon of persistent unemployment that occurs in the market economies in the world we inhabit. Only the Keynes uncertainty concept can justify a role for governmental policies to assure full employment when questions of liquidity are important.\n\n\nDavidson, P. (1991) “Is Probability Theory Relevant For Uncertainty? A Post Keynesian Perspective”, Journal of Economic Perspectives, 5. (Distinguishes between economic decisions where ergodic circumstances might prevail, and situations where nonergodic circumstances are likely. The former are called routine decisions, the latter are crucial decisions.)\n\nHicks, J. R. (1977), Economic Perspectives, Oxford University Press, Oxford.(Argues for economic models where agents ‘know’ that they cannot reliably predict the future.)\n\nHicks, J. R (1979), Causality in Economics, Basic Books, New York. (Argues that economics is embedded in time in a way that the physical sciences are not. Consequently stochastic theory is not applicable to most dynamic economic problems.)\n\nKeynes, J. M. (1936), The General Theory of Employment, Interest and Money Harcourt, New York. (The basis for the ‘Keynesian Revolution’ where the existence of uncertainty explains why market economies have no endogenous forces that assure full employment.)\n\nKeynes, J. M. (1937), “The General Theory of Employment” Quarterly Journal of Economics, 52. (A further extension of what Keynes means by ‘uncertainty’ and why uncertainty is the root cause of unemployment in market economies.)\n\nKeynes, J. M. (1939), “Professor Tinbergen’s Method”, The Economic Journal, 47. (Keynes attacks the statistical method of regression analysis as not applicable to economic time series data.)\n\nKnight, F. N. (1921), Risk, Uncertainty, and Profit, Houghton Mifflin, New York. (Distinguished between probabilistic risk and uncertainty.)\n\nLucas, R. E., (1972) “Expectations and the Neutrality of money”, Journal of Economic Theory,4. (The article that initiated the rational expectations analysis in macroeconomics.)\n\nLucas R. E., and Sargent, T. J. (1981), Rational Expectations and Econometric Practices, Minneapolis, University of Minnesota Press. (Develops the relationship between the rational expectations hypothesis and the axioms underlying econometric analysis for macroeconomic analysis.)\n\nMachina, M. J. “Choice Under Uncertainty; Problems Solved and Unsolved”, Journal of Economic Perspectives, 1. (Attempts to shore up the theory of choice under uncertainty on “solid axiomatic foundations” of probabilistic risk in the face of the famous St. Petersburg paradox and other challenges to expected utility theory).\n\nRicardo, D. (1817), On the Principles of Political Economy and Taxation. (The first economist to formulate the axiom of perfect certainty in economics.)\n\nSavage, L. (1954), The Foundations of Statistics Wiley, New York.(Develops the Expected Utility Theory of economics for making decision with complete subjective probabilistic information.)\n\nSargent, T. J. (1993), Bounded Rationality in Macroeconomics, Oxford, Clarendon Press. (A founder of the rational expectations school who now argues that rational expectations are not applicable to situations where people find themselves in new, i.e., nonergodic, situations.)\n\nSolow, R. M. (1985), “Economic History and Economics”, American Economic Review Papers and Proceedings, 75.\n\nPrint Friendly\nTweet about this on Twitter0Digg thisShare on Reddit0Share on StumbleUpon0Share on Facebook0Share on LinkedIn3Share on Google+2Buffer this pageEmail this to someone\n\n\n 1. craazyman\n\n wow. 1817 to 2012.\n\n one hundred and ninety-five years of total nonsense.\n\n that’s almost a triumph of imagination over reality. hahahah.\n\n 1. Valissa\n\n Since economists lack the creative imagination to create their own deity, I nominate the Wrathful Deity Mahakala in both his Hindu and Japanese aspects.\n\n Mahakala was the personal tutelary deity for the Mongol ruler Kublai Khan. His terrifying imagery ultimately derives from the angry form of the Hindu god Shiva, known as Bhairava. In Tibetan iconography he typically has one head with three bulging eyes. His eyebrows are like small flames, and his beard is made of hook-like shapes. He can have two to six arms.\n\n The essential nature of Mahakala in the Tibetan pantheon can be gauged from the fact that he is worshipped as the Protector of the tent.\n\n In this case the “tent” symbolizes the “big tent” of economic dogma.\n\n In Japan, Mahakala…\n\n enjoys an exalted position as a household deity in Japan, as he is one of the Seven Lucky Gods in Japanese folklore. Mahākāla’s association with wealth and prosperity gave rise to a strange custom known as Fuku-nusubi. This custom started with the belief that one who stole divine figures (gods and goddesses) was assured of good fortune, if not caught in the act of stealing. In the course of time stealing of divine images became so common a practice in Japan that the Toshi-no-ichi or the ‘year-end-market’ held in the Asakusa Kannon temple became the main venue of the sale and disposal of such images by the fortune-seekers.\n\n 1. steelhead23\n\n God I love NC. Where else can I read of the Keynesian perspective on uncertainty and comparative Asian religion in a single thread? Where’s the course syllabus for this seminar?\n\n 2. digi_owl\n\n Actually their deity would be “the invisible hand”. And most mainstreams seems to think of it as being very much benevolent. Never mind that it seems to have ruined just about any community it has touched…\n\n 1. F. Beard\n\n But the collection basket remains and you don’t even have to reach into your own pocket! The banks do it for you while most economists look the other way.\n\n 2. enouf\n\n Think of economics as religion without the music, the art and the cathedrals.–Jake chase says: January 19, 2012 at 9:01 am\n\n Think of everything and everyone whom blatantly denies (and even those “agnostics”) the existence of a Higher Intelligence as a “humanist” or Belief in Humanism.\n\n The denial of reality of a consciousness/awareness of self, the creativity, and innate personal moral fortitude (the sense of Injustice, the feeling of Love, etc) leads to nothing short of a carnival act.\n\n 3. enouf\n\n\n\n\n 1. tyaresun\n\n The father of rational expectations is John Muth. I had the priviledge of working with him in 1985-86. He was working on an errors in expectations model at that time. I don’t think you will be able to find any published research on that model. Also, Muth never received the Nobel Prize even though rational expectations forms the fundamental brick of modern economics. Why? One of the reasons given is that others took over the idea and made more fundamental contributions.\n\n Could it be that Muth did not receive the prize because he had changed his mind on ratioanal expectations?\n\n 2. Tom Dority\n\n I suppose economists are stuck in a closed loop. Describing events but not understanding. Maybe quantum economic theory and understanding of the Heisenberg principle would be a start. Every market participant is defined by their own uncertainty and environment – they are a molecule or atom in the economic world – thus economists have only described what they see and fool themselves when they think they can know why – having limited themselves to old earth centrism quasi science in my view. The butterfly principle alive and well. I am not an economist but, I think looking into math and scientific understanding of our physical universe may lead to an expansion in understanding our economics. Sorry first post and a bit loose.\n\n 1. Birch\n\n Before getting into quantum economic theory, it would be very useful for economists to apply the laws of thermodynamics to their discipline. Frederick Soddy did this in the ’30s to great effect, and the tradition has been continued; most notably by Herman Daly whose economic theories actually relate to the real world.\n\n 1. Fiver\n\n Would like to see NC take an interest and do a piece on Daly – we either develop a sustainable economy or we are headed for a very bad outcome.\n\n 3. jake chase\n\n The only value of economics is derived from the fact that a substantial number of people believe it, at least selectively. Their actions create opportunity for contrarians who ignore the signals cherished by the believers.\n\n 4. Tom Dority\n\n To add, Is there a periodic table of economic elements of a grand theory like E=mc2, maybe something like Economic output is equal to money supply times x too the second power.\n\n 5. René\n\n\n This is true for all models created by the elite and their helpers. Hence, NDAA and the SOPA Act.\n\n 6. Eric Patton\n\n I am not an economist, but the fact that Robin Hahnel and Michael Albert’s 1990 Princeton University Press book Quiet Revolution in Welfare Economics is out of print and persona-non-grata in the profession really tells you all you need to know about the religion that is economics.\n\n 1. Frank Speaking\n\n just as, “War is…an act of force to compel our enemy to do our will” so is economics an act of force to compel members of a society to do the will of others.\n\n Herbert Gintis and an entire school of economic thought has been marginalized be"
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"text": "Paul Davidson: What Makes Economists So Sure of Themselves, Anyway?\n\nBy Paul Davidson, America’s foremost post-Keynesian economist. Davidson is currently the Holly Professor of Excellence, Emeritus at the University of Tennessee in Knoxville. In 1978 Davidson and Sydney Weintraub founded the Journal for Post-Keynesian Economics. Davidson is the author of numerous books, the most recent of which is an introduction to a post-Keynesian perspective on the recent crisis entitled ‘The Keynes Solution: The Path to Global Prosperity’.\n\nIntroduction by Philip Pilkington\n\nIn a recent interview I asked the US’s leading post-Keynesian economist and founder of the Journal of Post-Keynesian Economics, Paul Davidson to discuss what is known as the ‘ergodic axiom’ in economics. This is a particularly important axiom as it allows mainstream economists (including left-wing Keynesians like Paul Krugman and Joseph Stiglitz) to claim that they can essentially know the future in a very tangible way. It does this by assuming that the future can be known by examining the past.\n\nWithout this axiom the whole edifice of mainstream theory rests on very shaky grounds. Yet, it should be clear to anyone that given that the theory is supposed to explain human behaviour it is unlikely that the future will correlate with the past because people and institutions tend to change and evolve over a given period of time.\n\nYes, often past behaviour will help us understand future behaviour – apply this in a simple psychological way to anyone you know and you will find it to be true – however, it should be quite clear that all future behaviour cannot be wholly explained by past behaviour. Clearly it should be quite obvious that the same should apply when we consider large aggregates of individuals and yet mainstream economics steadfastly refuses to accept this.\n\nWhat follows is an particularly succinct overview of these ideas and a summary of their importance that Professor Davidson has kindly written for us. He also lays out the alternative view that the future is not determined by probabilistic risk but is instead subject to an absolute unknowability or uncertainty as laid out in the work of John Maynard Keynes and the implications of this.\n\nThis is a hugely important debate in that it essentially touches on what a good economic theory would allow for in terms of government policy. The ergodic axiom is possibly one of the key reasons that many economists show a remarkable anxiety when it comes to any human action undertaken outside of their models.\n\n– — Philip Pilkington\n\n\n\nThe economy is a process in historical time. Time is a device that prevents everything from happening at once. The production of commodities takes time; and the consumption of goods, especially durables, takes considerable time. Economics is the study of how households and firms make decision choices regarding production and consumption when the outcome (pay-off) of today’s decision occurs at a significantly later date.\n\nAny study of the behavior of economic decision-makers, therefore, requires the analyst to make some assumption regarding what today’s decision-makers ‘know’ about future outcomes.\n\nThere are two different concepts of the uncertain knowledge regarding the future outcomes of decisions made today. The mainstream concept regarding knowledge about the future and the Keynes General Theory concept. The ability of economists to explain the importance of the role of money, liquidity, and the existence of persistent unemployment in a market economy depends on which concept of knowledge about the future economists use as the basis of their economic analysis.\n\nBecause economists are split into two major theoretical camps about what decision-makers know about the future, these groups provide differing explanations of economic problems and their policy solutions. Understanding the differences in these two concepts of knowledge of future outcomes is essential to understanding the philosophical differences between economists on the role for government and economic policies in the economic system.\n\nAs explained below, mainstream economics assumes that households and entrepreneurs are optimal decision makers, that is, they choose the decision today that optimizes their utility, income and profits over time.\n\n\nRicardo (1817), the father of 19th century classical economics, assumed a world of perfect certainty. All households and businesses were assumed to possess a full and correct knowledge of a presumed programmed external economic reality that governed all past, present, and future economic outcomes. The external economic environment was assumed immutable in the sense that it was not susceptible to change induced by human action. The path of the economy, like the path of the planets under Newton’s celestial mechanics, was determined by timeless natural laws. Economic decision makers had complete knowledge of these laws. Households and firms never made errors in their spending choices. They always spend everything they earned on things with the highest ‘known’ future pay-off in terms of utility for households and profits for businesses. Accordingly, there could never be a lack of demand for the products of industry or for workers who wanted to work. Classical economics justified a laissez-faire philosophy for the economic system. No government action could provide a higher pay-off than the decisions individuals made with complete information about the future in free markets.\n\n\nIn the early 20th century, classical economists tended to substitute the notion of probabilistic risk premiums and “certainty equivalents” for the perfect knowledge assumption of earlier classical theory. Risk premiums was said to provide “uncertainty” allowances where the latter referred to the difference between the estimated value of a future event, held with an objective (frequency distribution) probability of less than unity and the value of a perfectly certain (p = 1) event that evokes the same behavior.\n\nBy the 1970s this classical risk analysis had evolved into what mainstream economists call the New Classical Theory of ‘rational expectations’ where individuals make decisions based on their subjective probability distributions regarding future events where the subjective probabilities are presumed to be equal to immutable objective probability distributions that govern future outcomes [Lucas, 1972]. Today all mainstream economists interpret uncertainty in economics as synonymous with objective probability distributions [Lucas and Sargent, 1981; Machina 1987] that govern future events but are completely known to all persons today.\n\nThis device of labelling statistically reliable estimates of probabilistic risk regarding future outcomes as uncertainty permits mainstream economists to preserve intact most of the laissez faire efficient market analysis that had been developed under the earlier perfect certainty assumption. While rejecting the perfect certainty model, mainstream economists still accept, as a universal truth, the existence of a predetermined external economic reality (similar to Newton’s celestial mechanics which, for example, permits the astronomer to accurately predict the next solar eclipse) that can be fully described by unchanging objective conditional probability functions that are fully known by the decision makers in one’s model… Unlike the perfect certainty model, however, conflating the concept of uncertainty with the probabilitistic risk permits individual decision makers to make an occasional erroneous choice (in the short run) just as a single sample means can differ from the true universe value. In the long run, the assumption that people with rational expectations already “know” the objective probabilities assures correct choices on average for those “fittest” decision makers who survived in the Darwinian world of free markets. In other words, free markets lead to optimal solutions at least in the long run\n\nIn mainstream economics, economic data are typically viewed as part of time series realization generated by an ergodic stochastic processes. In fact, Nobel Prize winner Paul Samuelson (1969) has made the acceptance of the ergodic axiom the sine qua non of the scientific method in economics. What is this ergodic axiom that Samuelson insists is necessary for economics to be a science?\n\n\nLogically, to make statistically reliable probabilistic forecasts about future economic events, today’s decision-makers should obtain and analyze sample data from the future. Since that is impossible, the assumption of ergodic stochastic economic processes permits the analyst to assert that the outcome at any future date is the statistical shadow of past and current market data.\n\nA realization of a stochastic process is a sample value of a multidimensional variable over a period of time, i.e., a single time series. A stochastic process makes a universe of such time series. Time statistics refer to statistical averages (e.g., the mean, standard deviation) calculated from a single fixed realization over an indefinite time space. Space statistics, on the other hand, refer to a fixed point of time and are formed over the universe of realizations (i.e. they are statistics obtained from cross-sectional data).\n\nStatistical theory asserts that if the stochastic process is ergodic then for an infinite realization, the time statistics and the space statistics will coincide. For finite realizations of ergodic processes, time and space statistics coincide except for random errors; they will tend to converge (with the probability of unity) as the number of observations increase. Consequently, if ergodicity is assumed, statistics calculated from past time series or cross-sectional data are statistically reliable estimates of the statistics probabilities that will occur at any future date.\n\nIn simple language, the ergodic presumption assures that economic outcomes on any specific future date can be reliably predicted by a statistical probability analysis of existing market data. By assumption, New Classical economic theory imposes the condition that economic relationships are timeless or ahistoric ‘natural’ laws. The historical dates when observations are collected do not affect the estimates of the statistical time and space averages. Accordingly, the mainstream presumption (utilized by both New Classical economists and New Keynesian economists) that decision-makers possess rational expectations imply that people in one’s model process information embedded in past and present market data to form statistical averages (or decision weights) that reliably forecast the future. Or as 2011 Nobel Prize winner Thomas Sargent [1993, p. 3], one of the leaders of the rational expectations school, states “rational expectations models impute much more knowledge to the agents within the model (who use the equilibrium probability distributions)… than is possessed by an econometrician, who faces estimation and inference problems that the agents in the model have somehow solved”.\n\nBy using probabilistic distributions calculated from past market data, rational expectations theory assumes that, on average, the actions fostered by these expectations are precisely those that would be forthcoming in a perfectly certain world – at least in the long run.\n\nIn recent years, partly in reaction to the rational expectations hypothesis, some mainstream economists have raised questions regarding the use of such stochastic concepts to define uncertainty. For example, Nobel Prize winner R. M. Solow (1985, p. 328) has stated “economics is a social science….much of what we observe cannot be treated as the realization of a stationary stochastic process without straining credulity”. Since stationary is a necessary but not sufficient condition for ergodicity, Solow’s statement implies that only the very gullible would ever believe that most important macroeconomic processes are ergodic.\n\n\nBeginning with Knight’s [1921] seminal work, some economists have drawn a distinction between “true” uncertainty and probabilistic risk, where the latter is calculable based on past frequency distributions and is, therefore, conceptually insurable, while uncertainty is neither calculable nor insurable.\n\nJohn Maynard Keynes (1936) launched a revolution in economics. Keynes explicitly developed an alternative “general theory” to classical theory. Keynes argued that the difference between probabilistic risk and uncertainty had important implications for understanding (a) the operations of a market economy and (b) the role of government in influencing market outcomes through deliberate legislative policies.\n\nIn Keynes’s (1936) analysis, whenever the full consequences of today’s economic decisions occur many days in the future, uncertainty would prevail and economic behavior could not be described as an “outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities”.\n\nUnlike today’s orthodox economists, Keynes did not write in the idiom of stochastic processes in developing his concept of uncertainty. Keynes (1937) simply described uncertainty as occurring when there is no scientific basis to form any calculable probability. Nevertheless, in criticizing Tinbergen’s use of econometric analysis, Keynes (1939) argued that Tinbergen’s ‘method’ was not applicable to economic data because “the economic environment is not homogeneous over a period of time”, a criticism that is equivalent to stating economic time series are not stationary.\n\nWith the development of ergodic theory and stochastic process analysis since Keynes wrote, it is possible to interpret Keynes’s uncertainty concept in terms of this statistical lexicon. Keynes’s theory required decision-makers to recognize that in the market system in which they operate, in some but not necessarily all economic dimensions, the future is uncertain and cannot be reliably predicted on the basis of any statistical analysis of past evidence. The absence of ergodic conditions, therefore, is a sufficient condition for Keynes’s concept of uncertainty. In a nonergodic environment, even if agents have the capacity to obtain and statistically process past and current market data, these observations do not, and cannot, provide a statistically reliable basis for forecasting the probability distributions, if any, that will govern outcomes at any specific date in the future. According to Keynes (1937), “About these [future] matters there is no scientific basis to form any calculable probability whatever. We simply do not know.”\n\nKeynes’s uncertainty concept implies that the future is transmutable or creative in the sense that future economic outcomes may be permanently changed in nature and substance by today’s actions of individuals, groups (e.g., unions, cartels) and/or governments, often in ways not even perceived by the creators of change. (It is also possible that changes that are not predetermined can occur even without any deliberate human economic action.). [George Soros’s concept of ‘reflexivity’ asserts future market outcomes are determined by market participants’ actions today.]\n\nThis nonergodic view of modelling the future out comes as being determined by peoples’ actions rather than a timeless probability distribution has been described by Nobel Prize winner Sir John Hicks (1977) as a situation where people in the model “do not know what is going to happen and know that they do not what is going to happen. As in history!” Hicks (1979) declared that “I am bold enough to conclude from these considerations that the usefulness of ‘statistical’ or ‘stochastic’ methods in economics is a good deal less than is now conventionally supposed.”\n\nAccordingly, mainstream macroeconomics is logically inconsistent with Keynes’ macroeconomic general theory explaining employment, interest, and money. The result has been a continuing debate between the followers of Keynes and mainstream theorists over the relevant policy prescriptions for solving the macroeconomic problems of the real world.\n\nThe first postulate of mainstream economics is the presumption that there exists a finite set of acts and outcomes and that each agent can make a complete and transitive preference ordering of all possible alternative choices. Decision making by agents who know the statistically reliable future can characterize the decision process as “Look before you leap”. This “Look before you leap” analysis, however, is not a general theory of decision making for it does not explicitly deal with uncertainty per se. As the statistical theorist Leonard Savage recognized “a person may not know [all] the consequences of the acts open to him in each state of the world. He might be … ignorant” and hence might want to leave his options open; a decision that Savage characterizes as “You can cross that bridge when you come to it”. Savage admits the latter is often a more accurate description of the human economic predicament. When a decision maker is ‘ignorant’ and wants to wait before making a decision, we can classify the situation as one involving Keynes’ uncertainty concept and therefore the mainstream ergodic axiom is violated.\n\nAs Savage puts it, mainstream economics “attack[s] relatively simple problems of decision by artificially confining attention to so small a world that the `Look before you leap’ principle can be applied”, i.e., where Keynes’ uncertainty concept is not relevant. Savage warns that mainstream theory is “practical [only] in suitably limited domains… At the same time, the behavior of people is often at variance with the theory. The departure is sometimes flagrant … the `Look before you leap’ principle is preposterous if carried to extremes”. Yet when today’s mainstream economic theorists talk about efficient free markets they treat uncertainty in economics as synonymous with a probability measure The behavior they describe flagrantly departs from the behavior that determines employment in a money-using market economy.\n\nIf, as Savage recognizes, in some areas of economic activity the ability of humans to form a complete preference ordering regarding all potential consequences of all actions is not possible, then mainstream theory cannot provide a useful explanation of the behavior of decision-makers in these areas. It is here that Keynes’ uncertainty concept becomes paramount\n\nIn the classical (ergodic) theory, where all outcomes are conceptually calculable, there is never a need to keep options open. People will therefore spend all they earn on the products of industry (Say’s Law) and there can never be a lack of effective demand to prevent the system from reaching full employment.\n\nOn the other hand, when households and firms “know that they do not know” the future and therefore cannot order all future consequences associated with any possible choice today, they may wish to defer forever making “look before they leap” decisions. When people believe the future is uncertain in the sense of Keynes, they prefer to leave their options open, i.e., to cross that bridge when, and if, they come to it.\n\nWhenever households and business managers believe they cannot predict the future with any degree of a priori or statistically reliable probability, then the axiomatic foundation of mainstream economic theory is violated. Hicks (1979) has associates this transgression of mainstream axiomatic ergodic basis with Keynes’ long-term ‘liquidity’ concept. For Keynes, it is the existence of an uncertain future that makes a long-run demand for liquidity (money and other liquid assets traded in well organized markets where prices movements are ‘orderly’) a ubiquitous fact of life. The ability to save one’s income in the form of money and other liquid assets permits households and firms to keep their options open by not having to spend all of their earned income on the products of industry, even in the long-run.\n\nAs long as income-earning decision-makers have this option of demanding liquidity rather than the products of industry, then a laissez-faire market system cannot assure that peoples’ total market demand for goods and services will be sufficient to make it profitable for firms to fully employ all who want to work.\n\nThe notion of a demand for long-term liquidity can only be justified in a world of Keynes’ (nonergodic) uncertainty. This desire for long-term liquidity is incompatible with mainstream’s optimal decision makers in an ergodic environment. Only the Keynes concept of uncertainty in economics provides a logical, statistical explanation of the phenomenon of persistent unemployment that occurs in the market economies in the world we inhabit. Only the Keynes uncertainty concept can justify a role for governmental policies to assure full employment when questions of liquidity are important.\n\n\nDavidson, P. (1991) “Is Probability Theory Relevant For Uncertainty? A Post Keynesian Perspective”, Journal of Economic Perspectives, 5. (Distinguishes between economic decisions where ergodic circumstances might prevail, and situations where nonergodic circumstances are likely. The former are called routine decisions, the latter are crucial decisions.)\n\nHicks, J. R. (1977), Economic Perspectives, Oxford University Press, Oxford.(Argues for economic models where agents ‘know’ that they cannot reliably predict the future.)\n\nHicks, J. R (1979), Causality in Economics, Basic Books, New York. (Argues that economics is embedded in time in a way that the physical sciences are not. Consequently stochastic theory is not applicable to most dynamic economic problems.)\n\nKeynes, J. M. (1936), The General Theory of Employment, Interest and Money Harcourt, New York. (The basis for the ‘Keynesian Revolution’ where the existence of uncertainty explains why market economies have no endogenous forces that assure full employment.)\n\nKeynes, J. M. (1937), “The General Theory of Employment” Quarterly Journal of Economics, 52. (A further extension of what Keynes means by ‘uncertainty’ and why uncertainty is the root cause of unemployment in market economies.)\n\nKeynes, J. M. (1939), “Professor Tinbergen’s Method”, The Economic Journal, 47. (Keynes attacks the statistical method of regression analysis as not applicable to economic time series data.)\n\nKnight, F. N. (1921), Risk, Uncertainty, and Profit, Houghton Mifflin, New York. (Distinguished between probabilistic risk and uncertainty.)\n\nLucas, R. E., (1972) “Expectations and the Neutrality of money”, Journal of Economic Theory,4. (The article that initiated the rational expectations analysis in macroeconomics.)\n\nLucas R. E., and Sargent, T. J. (1981), Rational Expectations and Econometric Practices, Minneapolis, University of Minnesota Press. (Develops the relationship between the rational expectations hypothesis and the axioms underlying econometric analysis for macroeconomic analysis.)\n\nMachina, M. J. “Choice Under Uncertainty; Problems Solved and Unsolved”, Journal of Economic Perspectives, 1. (Attempts to shore up the theory of choice under uncertainty on “solid axiomatic foundations” of probabilistic risk in the face of the famous St. Petersburg paradox and other challenges to expected utility theory).\n\nRicardo, D. (1817), On the Principles of Political Economy and Taxation. (The first economist to formulate the axiom of perfect certainty in economics.)\n\nSavage, L. (1954), The Foundations of Statistics Wiley, New York.(Develops the Expected Utility Theory of economics for making decision with complete subjective probabilistic information.)\n\nSargent, T. J. (1993), Bounded Rationality in Macroeconomics, Oxford, Clarendon Press. (A founder of the rational expectations school who now argues that rational expectations are not applicable to situations where people find themselves in new, i.e., nonergodic, situations.)\n\nSolow, R. M. (1985), “Economic History and Economics”, American Economic Review Papers and Proceedings, 75.\n\nPrint Friendly\nTweet about this on Twitter0Digg thisShare on Reddit0Share on StumbleUpon0Share on Facebook0Share on LinkedIn3Share on Google+2Buffer this pageEmail this to someone\n\n\n 1. craazyman\n\n wow. 1817 to 2012.\n\n one hundred and ninety-five years of total nonsense.\n\n that’s almost a triumph of imagination over reality. hahahah.\n\n 1. Valissa\n\n Since economists lack the creative imagination to create their own deity, I nominate the Wrathful Deity Mahakala in both his Hindu and Japanese aspects.\n\n Mahakala was the personal tutelary deity for the Mongol ruler Kublai Khan. His terrifying imagery ultimately derives from the angry form of the Hindu god Shiva, known as Bhairava. In Tibetan iconography he typically has one head with three bulging eyes. His eyebrows are like small flames, and his beard is made of hook-like shapes. He can have two to six arms.\n\n The essential nature of Mahakala in the Tibetan pantheon can be gauged from the fact that he is worshipped as the Protector of the tent.\n\n In this case the “tent” symbolizes the “big tent” of economic dogma.\n\n In Japan, Mahakala…\n\n enjoys an exalted position as a household deity in Japan, as he is one of the Seven Lucky Gods in Japanese folklore. Mahākāla’s association with wealth and prosperity gave rise to a strange custom known as Fuku-nusubi. This custom started with the belief that one who stole divine figures (gods and goddesses) was assured of good fortune, if not caught in the act of stealing. In the course of time stealing of divine images became so common a practice in Japan that the Toshi-no-ichi or the ‘year-end-market’ held in the Asakusa Kannon temple became the main venue of the sale and disposal of such images by the fortune-seekers.\n\n 1. steelhead23\n\n God I love NC. Where else can I read of the Keynesian perspective on uncertainty and comparative Asian religion in a single thread? Where’s the course syllabus for this seminar?\n\n 2. digi_owl\n\n Actually their deity would be “the invisible hand”. And most mainstreams seems to think of it as being very much benevolent. Never mind that it seems to have ruined just about any community it has touched…\n\n 1. F. Beard\n\n But the collection basket remains and you don’t even have to reach into your own pocket! The banks do it for you while most economists look the other way.\n\n 2. enouf\n\n Think of economics as religion without the music, the art and the cathedrals.–Jake chase says: January 19, 2012 at 9:01 am\n\n Think of everything and everyone whom blatantly denies (and even those “agnostics”) the existence of a Higher Intelligence as a “humanist” or Belief in Humanism.\n\n The denial of reality of a consciousness/awareness of self, the creativity, and innate personal moral fortitude (the sense of Injustice, the feeling of Love, etc) leads to nothing short of a carnival act.\n\n 3. enouf\n\n\n\n\n 1. tyaresun\n\n The father of rational expectations is John Muth. I had the priviledge of working with him in 1985-86. He was working on an errors in expectations model at that time. I don’t think you will be able to find any published research on that model. Also, Muth never received the Nobel Prize even though rational expectations forms the fundamental brick of modern economics. Why? One of the reasons given is that others took over the idea and made more fundamental contributions.\n\n Could it be that Muth did not receive the prize because he had changed his mind on ratioanal expectations?\n\n 2. Tom Dority\n\n I suppose economists are stuck in a closed loop. Describing events but not understanding. Maybe quantum economic theory and understanding of the Heisenberg principle would be a start. Every market participant is defined by their own uncertainty and environment – they are a molecule or atom in the economic world – thus economists have only described what they see and fool themselves when they think they can know why – having limited themselves to old earth centrism quasi science in my view. The butterfly principle alive and well. I am not an economist but, I think looking into math and scientific understanding of our physical universe may lead to an expansion in understanding our economics. Sorry first post and a bit loose.\n\n 1. Birch\n\n Before getting into quantum economic theory, it would be very useful for economists to apply the laws of thermodynamics to their discipline. Frederick Soddy did this in the ’30s to great effect, and the tradition has been continued; most notably by Herman Daly whose economic theories actually relate to the real world.\n\n 1. Fiver\n\n Would like to see NC take an interest and do a piece on Daly – we either develop a sustainable economy or we are headed for a very bad outcome.\n\n 3. jake chase\n\n The only value of economics is derived from the fact that a substantial number of people believe it, at least selectively. Their actions create opportunity for contrarians who ignore the signals cherished by the believers.\n\n 4. Tom Dority\n\n To add, Is there a periodic table of economic elements of a grand theory like E=mc2, maybe something like Economic output is equal to money supply times x too the second power.\n\n 5. René\n\n\n This is true for all models created by the elite and their helpers. Hence, NDAA and the SOPA Act.\n\n 6. Eric Patton\n\n I am not an economist, but the fact that Robin Hahnel and Michael Albert’s 1990 Princeton University Press book Quiet Revolution in Welfare Economics is out of print and persona-non-grata in the profession really tells you all you need to know about the religion that is economics.\n\n 1. Frank Speaking\n\n just as, “War is…an act of force to compel our enemy to do our will” so is economics an act of force to compel members of a society to do the will of others.\n\n Herbert Gintis and an entire school of economic thought has been marginalized be"
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"text": "Take the 2-minute tour ×\n\nPell equations can be solved using continued fractions. I have heard that some elliptic curves can be \"solved\" using continued fractions. Is this true?\n\nWhich Diophantine equations other than Pell equations can be solved for rational or integer points using continued fractions? If there are others, what are some good references?\n\n\nProfessor Elkies has given an excellent response as to the role of continued fractions in solving general Diophantine equations including elliptic curves. What are some other methods to solve the Diophantine equations $$X^2 - \\Delta Y^2 = 4 Z^3$$ and $$18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2 ?$$\n\nshare|improve this question\nSamuel, could you be a little more specific about your comment about solving $x^2 - D y^2 = 4 z^3,$ especially why you are not satisfied with your parametrized solution? I printed out your paper with Franz, \"Arithmetic of Pell Surfaces,\" where I think equation (1.1) with $n=3$ is what you are discussing below. Also Buell's book, especially pages 147-157, discussing Nagell 1922 and Yamamoto 1970. – Will Jagy Oct 17 '11 at 0:03\nThank you Dr. Jagy for comment, interest in the question, and for your help with question 71727 ! I've also seen that you're interested in ternary quadratic forms, which I would like to learn about too. My response will require quite the maximum characters, so I'll stop chatting and get to it. – Samuel Hambleton Oct 19 '11 at 2:13\nLet $\\Delta = 229$ and $K = \\mathbb{Q}(\\sqrt{\\Delta })$. Then $\\text{Cl}^+(K)[3] \\simeq (\\mathbb{Z}/ \\3 )$. Using the \"parametrization\" in Mathematica5, d = 229; S = Union[DeleteCases[Partition[Flatten[Table[P = {(t^3 + 3 d t u^2)/4,(3 t^2 u + d u^3)/4,(t^2 - d u^2)/4};If[IntegerQ[P[[1]]] && IntegerQ[P[[2]]] && IntegerQ[P[[3]]] && GCD[P[[1]], P[[3]]] == 1, P, {w, w, w}], {t, -100, 100}, {u, -100, 100}]], 3], {w, w, w}]]; gives the Points of the \"Pell surface\" $x^2 - d y^2 = 4 z^3$ from the \"parametrization\". On the other hand, a brute force search for points satisfying this Eqn. can be ... – Samuel Hambleton Oct 19 '11 at 2:27\n... done as: T = Union[DeleteCases[Partition[Flatten[Table[If[IntegerQ[Sqrt[d y^2 + 4 z^3]] && GCD[Sqrt[d y^2 + 4 z^3], z] == 1, {Sqrt[d y^2 + 4 z^3], y, z}, {w, w, w}], {y, -100, 100}, {z, -100, 100}]], 3], {w, w, w}]]; There are points of the set $T$ not in $S$, for example : $(11, 1, 3)$. The other form of the Pell surface is $B^2 + B C - 57 C^2 = A^3$. With point $(11, 1, 3)$ corresponding to $(A, B, C) = (3, 5, 1)$, which should map to the ideal $(3, 2 + (1 + \\sqrt{\\Delta })/2)$. I suspect that the \"parametrization\" leads to principal ideals. – Samuel Hambleton Oct 19 '11 at 2:39\nThe points should read $(11, 1, -3)$, $(-3, 5, 1)$, and ideal $(-3, 2 + (1 + \\sqrt{229})/2)$. I would like to know of more methods for solving Diophantine equations, especially surfaces. Professor Elkies'methods look promising. Joro's question 70913, and particularly Schoof's article linked there shows that there may be some good reasons to want an easy method for finding non-principal ideals. I am particularly keen to learn methods for solving $18 x y +x^2 y^2 -4 x^3 -4 y^3 -27 = D z^2$. – Samuel Hambleton Oct 19 '11 at 2:50\nshow 3 more comments\n\n6 Answers\n\nup vote 25 down vote accepted\n\n[edited to insert paragraph on Cornacchia and point-counting]\n\nContinued fractions, or (more-or-less) equivalently the Euclidean algorithm, can be used to find small integer solutions of linear Diophantine equations $ax+by=c$, and integer solutions of quadratic equations such as $x^2-Dy^2=\\pm1$ (\"Pell\"). Continued fractions in themselves won't find rational points on elliptic curves, but there's a technique using Heegner points that calculates a close real approximation to a rational point, which is then recovered from a continued fraction — this is possible because the recovery problem amounts to finding a small integer solution of a linear Diophantine equation. My paper\n\nNoam D. Elkies: Heegner point computations, Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/94; L.M. Adleman and M.-D. Huang, eds.), 122-133.\n\nmight have been the first to describe this approach.\n\nAnother application of continued fractions is Cornacchia's algorithm to solve $x^2+Dy^2=m$ for large $m>0$ coprime to $D$, given $x/y \\bmod m$ which is a square root of $-D \\bmod m$. This has an application to counting points on elliptic curves $E\\bmod p$ for $E$ such as $y^2 = x^3 + b$ or $y^2 = x^3 + ax$ for which the CM field ${\\bf Q}(\\sqrt{-D})$ is known: the count (including the point at infinity) is $p+1-t$ where $t^2+Du^2=4p$ for some integers $t$ and $u$, and this determines $t$ up to an ambiguity of at most $6$ possibilities that in practice is readily resolved. The necessary square root mod $p$ is readily found in random polynomial time, though it is a persistent embarrassment that we cannot extract square roots modulo a large prime in deterministic polynomial time without assuming something like the extended Riemann hypothesis. Indeed the application that Schoof gave to motivate his polynomial-time algorithm to compute $t$ for any elliptic curve mod $p$ was to recover a square root of $-D \\bmod p$ for small $D$ ! (Though this would never be done in practice because the exponent in Schoof's algorithm is much larger than for the randomized algorithm.) The reference for Schoof's paper is\n\nRené Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots $\\bmod p$, Math. Comp. 44 (#170, April 1985), 483-494.\n\nA natural generalization of the Euclidean algorithm to higher dimensions is the LLL algorithm and other techniques for lattice basis reduction (LBR), which have found various other Diophantine uses, including some other techniques for finding rational points on elliptic curves; another of my papers describes some of these Diophantine applications of LBR.\n\nshare|improve this answer\nAwesome. Thank you Professor Elkies. That should get me started. I can't remember who told me about continued fractions with respect to elliptic curves but I thought they mentioned Artin. I'm not sure. – Samuel Hambleton Oct 13 '11 at 2:51\nThis is a slightly different topic now but I've seen a cool paper of Professor Elkies' : \"Pythagorean triples and Hilbert's Theorem 90\", which I tried to apply to $x^2 - D y^2 = 4 z^3$. It partially worked but I couldn't seem to get the points I was interested in, and so I was wondering about continued fractions. – Samuel Hambleton Oct 13 '11 at 3:40\n@R.Thornburn: 1a) You're welcome! 1b) Perhaps Artin was thinking about point-counting on elliptic curves modulo a prime; see the paragraph I inserted. 2) Looks like this equation $x^2-Dy^2=4z^3$ will involve the $3$-torsion in the class group of ${\\bf Z}[(D+\\sqrt{D})/2]$, which may be accessible via the continued fraction for $(D+\\sqrt{D})/2$ but I suspect that this is not the most efficient method for large $D$. – Noam D. Elkies Oct 14 '11 at 20:02\nWith a change of variables, one can handle all of the equations $A x^2 + B x y + C y^2 + D x + E y + F =0$ in much the same way as Pell's Equation. – Kevin O'Bryant Oct 14 '11 at 23:37\nThank you both very much, and sorry about my identity crisis. I initially wanted to vote the answer to Question 70913 to a non-negative number. I don't want to be deceitful. The incomplete \"parametrization\" I found for $x^2 - D y^2 = 4 z^3$ is $((t^3+3 D t u^2)/4, (3 t^2 u + D u^3)/4 , (t^2 - D u^2)/4)$ but I can't seem to get elements of to narrow class group of exact order $3$ with this \"parametrization\". I am also interested in solving $18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2$. – Samuel Hambleton Oct 16 '11 at 4:32\nadd comment\n\nFor your sample problem, I get two flavors of identity, principal and non-principal.\n\nFor discriminant 229, I take the identity form as $$ f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$ f(x^3 + 3 x y^2 + 5 y^3, \\; 3 x^2 y + 45 x y^2 + 226 y^3 ) \\; = \\; f^3(x,y).$$ As an automorph of $f$ is $$ \\left( \\begin{array}{rr} 1 & 15 \\\\\\ 15 & 226 \\end{array} \\right) , $$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.\n\nFor the other two classes, take $$ g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$ A second family is $$ f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \\; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \\; = \\; g^3(x,y).$$\n$$ $$ $$ $$ The following cycles of reduced forms are as in Buell's book, pages 21-30.\n\nInput three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 \n1 15 -1\n\n0 form 1 15 -1 delta -15\n1 form -1 15 1 delta 15\n2 form 1 15 -1\nminimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229\nAutomorph, written on right of Gram matrix: \n-1 -15\n-15 -226\n Trace: -227 gcd(a21, a22 - a11, a12) : 15\n\n3 13 -5\n\n0 form 3 13 -5 delta -2\n1 form -5 7 9 delta 1\n2 form 9 11 -3 delta -4\n3 form -3 13 5 delta 2\n4 form 5 7 -9 delta -1\n5 form -9 11 3 delta 4\n6 form 3 13 -5\nminimum was 3rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 25.44444\nAutomorph, written on right of Gram matrix: \n-16 -75\n-45 -211\nshare|improve this answer\nThanks. Are you saying that the binary quadratic form $3 x^2 + 13 x y -5 y^2$ comes from the parametrization? From Yamamoto's work, which appears in Buell's book, the binary quadratic form $(z, x, z^2)$ is discussed, where $x^2 - \\Delta y^2 = 4 z^3$ and $gcd(x, z) = 1$. Franz Lemmermeyer and I looked at $z T^2 + x T U + z^2 U^2 = \\Delta y^2$. If rel. prime integers $T, U$ exist satisfying this, then $(z, x, z^2)$ is principle. I think there should be polynomials $T(t,u)$ and $U(t, u)$ : $$((t^2 - Δ u^2)/4)T^2 + ((t^3 + 3 Δ t u^2)/4) T U + ((t^2 - Δ u^2)/4)^2U^2 = Δ ((3t^2 u + Δ u^3)/4)^2 ,$$ – Samuel Hambleton Oct 19 '11 at 10:57\nIf there are such polynomials $T(t, u)$ and $U(t, u)$, then points from the \"parametrization\" yield principal forms and principal ideals. I haven't found any such polynomials. – Samuel Hambleton Oct 19 '11 at 11:02\nadd comment\n\nIn 1993, Tzanakis ( http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6435.pdf ) showed that solving a quartic Thue equation, which correponding quartic field is the compositum of two real quadratic fields, reduces to solving a system of Pellian equations.\n\nEven if the system of Pellian equations cannot be solved completely, the information on solutions obtained from the theory of continued fractions and Diophantine approximations might be sufficient to show that the Thue equation (or Thue inequality) has no solutions or has only trivial solutions. For that purpose, very useful tool is Worley's result characterizing all rational approximations satifying $|\\alpha - \\frac{p}{q}| < \\frac{c}{q^2}$ in terms of convergents of continued fraction of $\\alpha$. You may consult the paper \"Solving a family of quartic Thue inequalities using continued fractions\" ( http://web.math.pmf.unizg.hr/~duje/pdf/dij.pdf ) and the references given there.\n\nshare|improve this answer\nadd comment\n\nI should have stuck with your preferred notation, as in your $B^2 + B C - 57 C^2 = A^3$ in a comment. So the form of interest will be $x^2 + x y - 57 y^2.$The other classes with this discriminant of indefinite integral binary quadratic forms would then be given by $ 3 x^2 \\pm xy - 19 y^2.$\n\nTherefore, take $$ \\phi(x,y) = x^2 + x y - 57 y^2.$$ The identity you need to deal with your $A= \\pm 3$ is $$ \\phi( 15 x^3 - 99 x^2 y + 252 x y^2 - 181 y^3 , \\; 2 x^3 - 15 x^2 y + 33 x y^2 - 28 y^3 ) \\; = \\; ( 3 x^2 + xy - 19 y^2 )^3 $$\n\nThis leads most directly to $\\phi(15,2) = 27.$ Using $ 3 x^2 + x y - 19 y^2 = -3$ when $x=7, y=3,$ this leads directly to $ \\phi(1581, -196) = -27.$\n\nHowever, we have an automorph of $\\phi,$ $$ W \\; = \\; \\left( \\begin{array}{rr} 106 & 855 \\\\\\ 15 & 121 \\end{array} \\right) , $$ and $ W \\cdot (1581,-196)^T = (6, -1)^T,$ so $\\phi(6,-1) = -27.$\n\nFinally, any principal form of odd discriminant, call it $x^2 + x y + k y^2,$ (you have $k=-57$) has the improper automorph $$ Z \\; = \\; \\left( \\begin{array}{rr} 1 & 1 \\\\\\ 0 & -1 \\end{array} \\right) , $$ while $ Z \\cdot (6,-1)^T = (5, 1)^T,$ so $\\phi(5,1) = -27.$\n\nEDIT: a single formula cannot be visually obvious for all desired outcomes. There are an infinite number of integral solutions to $3 x^2 + x y - 19y^2 = -3.$ It is an excellent bet that one of these leads, through the identity I give, to at least one of the desired $\\phi(5,1) = -27$ or $\\phi(6,-1) = -27,$ but not necessarily both, largely because $3 x^2 + x y - 19y^2$ and $3 x^2 - x y - 19y^2$ are not properly equivalent. Worth investigating, I should think.\n\nshare|improve this answer\nI think Dr. Jagy is saying that one can obtain all solutions of X^2 - D y^2 = 4 Z^3 by using representative binary quadratic forms of each of the classes of forms. For D = 229, there are three. This is correct, we can solve X^2 - D y^2 = 4 Z^3 by looking at classes of forms. Part of the proof is that the map from points of ( X^2 - D y^2 = 4 Z^3 ) to Cl^+(D)[3] is surjective. I am curious to know whether it can get computationally easier than this. Possibly not? – Samuel Hambleton Oct 23 '11 at 6:47\nadd comment\n\nI was about to mention H J S Smith's algorithm for finding integer solutions to $x^2 + y^2 = p$ for $p \\equiv 1$ mod 4; but this is referred to in a related thread at Applications of finite continued fractions\n\n(Apologies if that thread is easily found from this one; but I wouldn't have noticed it without doing a Google search, and perhaps some other readers are equally inexperienced in StackOverflow ways or unobservant!)\n\nAlso, what about higher-dimensional continued fractions, expressed as matrix recurrence relations? I seem to recall that these can be used to find rational solutions of equations involving some kinds of cubic forms.\n\nshare|improve this answer\nadd comment\n\nHello, I want to necromance this thread back to life because I was working on a very similar question (and therefore it would only make sense to post it here).\n\nIf I have the diophantine equation\n\nX^2 - Y^2 = C\n\nAssuming we have already proven that this equation has solutions and a finite number of them, how do I efficiently find the smallest value of X and Y, both greater than zero that satisfy this equation.\n\n\nx^2 - y^2 = 2684\n\nA solution to this equation is x = 672, y = 670. But the smallest solution to this equation is x = 72, y = 50. If i'm most interested in finding the latter how would I go about doing so without just trial and error. Can continued fractions play a role here and if so then how? Can secant and tangent lines to the graph of X^2 - Y^2 = C play a role in finding rational points? If so then how? Thank you very much for your time!\n\nshare|improve this answer\nIf you want to ask a new question that may be related to an existing one, better to just ask the question and give the reference. In the present case, though, your question seems to be almost equivalent to a question that was already asked here earlier this month (#98637: \"find the minimum difference between the factors of a number\"), and I think was also seen here before that. So you might just consult that thread. – Noam D. Elkies Jun 20 '12 at 0:58\nThank you for your response! I did look at the thread and our threads are quite different. My question is basically attempting to maximize the value of abs(x - y) whereas that thread was concerned with minimizing the value. In other words the asker on the other post would be most interested in what I am least interested in. Additionally I am interested in the absolute minimal solution to x^2 - y^2 = 2684 (where x and y > 0) regardless of whether x and y do multiply out to create 2684. That being said would you suggest that I post this question in altogether new thread or keep it here for now? – Sid Jun 20 '12 at 1:18\nSince $X^2 - Y^2 = (X+Y) (X-Y)$, you can take $(x,y) = (X+Y,X-Y)$. Then $xy=C$, and $|x-y|=2Y$, so the questions are almost the same except that we must impose the condition $x \\equiv y \\bmod 2$ to ensure the integrality of $X$ and $Y$ [which are $(x\\pm y)/2$]. – Noam D. Elkies Jun 20 '12 at 2:16\nadd comment\n\nYour Answer\n\n\n"
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"text": "Take the 2-minute tour ×\n\nPell equations can be solved using continued fractions. I have heard that some elliptic curves can be \"solved\" using continued fractions. Is this true?\n\nWhich Diophantine equations other than Pell equations can be solved for rational or integer points using continued fractions? If there are others, what are some good references?\n\n\nProfessor Elkies has given an excellent response as to the role of continued fractions in solving general Diophantine equations including elliptic curves. What are some other methods to solve the Diophantine equations $$X^2 - \\Delta Y^2 = 4 Z^3$$ and $$18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2 ?$$\n\nshare|improve this question\nSamuel, could you be a little more specific about your comment about solving $x^2 - D y^2 = 4 z^3,$ especially why you are not satisfied with your parametrized solution? I printed out your paper with Franz, \"Arithmetic of Pell Surfaces,\" where I think equation (1.1) with $n=3$ is what you are discussing below. Also Buell's book, especially pages 147-157, discussing Nagell 1922 and Yamamoto 1970. – Will Jagy Oct 17 '11 at 0:03\nThank you Dr. Jagy for comment, interest in the question, and for your help with question 71727 ! I've also seen that you're interested in ternary quadratic forms, which I would like to learn about too. My response will require quite the maximum characters, so I'll stop chatting and get to it. – Samuel Hambleton Oct 19 '11 at 2:13\nLet $\\Delta = 229$ and $K = \\mathbb{Q}(\\sqrt{\\Delta })$. Then $\\text{Cl}^+(K)[3] \\simeq (\\mathbb{Z}/ \\3 )$. Using the \"parametrization\" in Mathematica5, d = 229; S = Union[DeleteCases[Partition[Flatten[Table[P = {(t^3 + 3 d t u^2)/4,(3 t^2 u + d u^3)/4,(t^2 - d u^2)/4};If[IntegerQ[P[[1]]] && IntegerQ[P[[2]]] && IntegerQ[P[[3]]] && GCD[P[[1]], P[[3]]] == 1, P, {w, w, w}], {t, -100, 100}, {u, -100, 100}]], 3], {w, w, w}]]; gives the Points of the \"Pell surface\" $x^2 - d y^2 = 4 z^3$ from the \"parametrization\". On the other hand, a brute force search for points satisfying this Eqn. can be ... – Samuel Hambleton Oct 19 '11 at 2:27\n... done as: T = Union[DeleteCases[Partition[Flatten[Table[If[IntegerQ[Sqrt[d y^2 + 4 z^3]] && GCD[Sqrt[d y^2 + 4 z^3], z] == 1, {Sqrt[d y^2 + 4 z^3], y, z}, {w, w, w}], {y, -100, 100}, {z, -100, 100}]], 3], {w, w, w}]]; There are points of the set $T$ not in $S$, for example : $(11, 1, 3)$. The other form of the Pell surface is $B^2 + B C - 57 C^2 = A^3$. With point $(11, 1, 3)$ corresponding to $(A, B, C) = (3, 5, 1)$, which should map to the ideal $(3, 2 + (1 + \\sqrt{\\Delta })/2)$. I suspect that the \"parametrization\" leads to principal ideals. – Samuel Hambleton Oct 19 '11 at 2:39\nThe points should read $(11, 1, -3)$, $(-3, 5, 1)$, and ideal $(-3, 2 + (1 + \\sqrt{229})/2)$. I would like to know of more methods for solving Diophantine equations, especially surfaces. Professor Elkies'methods look promising. Joro's question 70913, and particularly Schoof's article linked there shows that there may be some good reasons to want an easy method for finding non-principal ideals. I am particularly keen to learn methods for solving $18 x y +x^2 y^2 -4 x^3 -4 y^3 -27 = D z^2$. – Samuel Hambleton Oct 19 '11 at 2:50\nshow 3 more comments\n\n6 Answers\n\nup vote 25 down vote accepted\n\n[edited to insert paragraph on Cornacchia and point-counting]\n\nContinued fractions, or (more-or-less) equivalently the Euclidean algorithm, can be used to find small integer solutions of linear Diophantine equations $ax+by=c$, and integer solutions of quadratic equations such as $x^2-Dy^2=\\pm1$ (\"Pell\"). Continued fractions in themselves won't find rational points on elliptic curves, but there's a technique using Heegner points that calculates a close real approximation to a rational point, which is then recovered from a continued fraction — this is possible because the recovery problem amounts to finding a small integer solution of a linear Diophantine equation. My paper\n\nNoam D. Elkies: Heegner point computations, Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/94; L.M. Adleman and M.-D. Huang, eds.), 122-133.\n\nmight have been the first to describe this approach.\n\nAnother application of continued fractions is Cornacchia's algorithm to solve $x^2+Dy^2=m$ for large $m>0$ coprime to $D$, given $x/y \\bmod m$ which is a square root of $-D \\bmod m$. This has an application to counting points on elliptic curves $E\\bmod p$ for $E$ such as $y^2 = x^3 + b$ or $y^2 = x^3 + ax$ for which the CM field ${\\bf Q}(\\sqrt{-D})$ is known: the count (including the point at infinity) is $p+1-t$ where $t^2+Du^2=4p$ for some integers $t$ and $u$, and this determines $t$ up to an ambiguity of at most $6$ possibilities that in practice is readily resolved. The necessary square root mod $p$ is readily found in random polynomial time, though it is a persistent embarrassment that we cannot extract square roots modulo a large prime in deterministic polynomial time without assuming something like the extended Riemann hypothesis. Indeed the application that Schoof gave to motivate his polynomial-time algorithm to compute $t$ for any elliptic curve mod $p$ was to recover a square root of $-D \\bmod p$ for small $D$ ! (Though this would never be done in practice because the exponent in Schoof's algorithm is much larger than for the randomized algorithm.) The reference for Schoof's paper is\n\nRené Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots $\\bmod p$, Math. Comp. 44 (#170, April 1985), 483-494.\n\nA natural generalization of the Euclidean algorithm to higher dimensions is the LLL algorithm and other techniques for lattice basis reduction (LBR), which have found various other Diophantine uses, including some other techniques for finding rational points on elliptic curves; another of my papers describes some of these Diophantine applications of LBR.\n\nshare|improve this answer\nAwesome. Thank you Professor Elkies. That should get me started. I can't remember who told me about continued fractions with respect to elliptic curves but I thought they mentioned Artin. I'm not sure. – Samuel Hambleton Oct 13 '11 at 2:51\nThis is a slightly different topic now but I've seen a cool paper of Professor Elkies' : \"Pythagorean triples and Hilbert's Theorem 90\", which I tried to apply to $x^2 - D y^2 = 4 z^3$. It partially worked but I couldn't seem to get the points I was interested in, and so I was wondering about continued fractions. – Samuel Hambleton Oct 13 '11 at 3:40\n@R.Thornburn: 1a) You're welcome! 1b) Perhaps Artin was thinking about point-counting on elliptic curves modulo a prime; see the paragraph I inserted. 2) Looks like this equation $x^2-Dy^2=4z^3$ will involve the $3$-torsion in the class group of ${\\bf Z}[(D+\\sqrt{D})/2]$, which may be accessible via the continued fraction for $(D+\\sqrt{D})/2$ but I suspect that this is not the most efficient method for large $D$. – Noam D. Elkies Oct 14 '11 at 20:02\nWith a change of variables, one can handle all of the equations $A x^2 + B x y + C y^2 + D x + E y + F =0$ in much the same way as Pell's Equation. – Kevin O'Bryant Oct 14 '11 at 23:37\nThank you both very much, and sorry about my identity crisis. I initially wanted to vote the answer to Question 70913 to a non-negative number. I don't want to be deceitful. The incomplete \"parametrization\" I found for $x^2 - D y^2 = 4 z^3$ is $((t^3+3 D t u^2)/4, (3 t^2 u + D u^3)/4 , (t^2 - D u^2)/4)$ but I can't seem to get elements of to narrow class group of exact order $3$ with this \"parametrization\". I am also interested in solving $18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2$. – Samuel Hambleton Oct 16 '11 at 4:32\nadd comment\n\nFor your sample problem, I get two flavors of identity, principal and non-principal.\n\nFor discriminant 229, I take the identity form as $$ f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$ f(x^3 + 3 x y^2 + 5 y^3, \\; 3 x^2 y + 45 x y^2 + 226 y^3 ) \\; = \\; f^3(x,y).$$ As an automorph of $f$ is $$ \\left( \\begin{array}{rr} 1 & 15 \\\\\\ 15 & 226 \\end{array} \\right) , $$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.\n\nFor the other two classes, take $$ g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$ A second family is $$ f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \\; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \\; = \\; g^3(x,y).$$\n$$ $$ $$ $$ The following cycles of reduced forms are as in Buell's book, pages 21-30.\n\nInput three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 \n1 15 -1\n\n0 form 1 15 -1 delta -15\n1 form -1 15 1 delta 15\n2 form 1 15 -1\nminimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229\nAutomorph, written on right of Gram matrix: \n-1 -15\n-15 -226\n Trace: -227 gcd(a21, a22 - a11, a12) : 15\n\n3 13 -5\n\n0 form 3 13 -5 delta -2\n1 form -5 7 9 delta 1\n2 form 9 11 -3 delta -4\n3 form -3 13 5 delta 2\n4 form 5 7 -9 delta -1\n5 form -9 11 3 delta 4\n6 form 3 13 -5\nminimum was 3rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 25.44444\nAutomorph, written on right of Gram matrix: \n-16 -75\n-45 -211\nshare|improve this answer\nThanks. Are you saying that the binary quadratic form $3 x^2 + 13 x y -5 y^2$ comes from the parametrization? From Yamamoto's work, which appears in Buell's book, the binary quadratic form $(z, x, z^2)$ is discussed, where $x^2 - \\Delta y^2 = 4 z^3$ and $gcd(x, z) = 1$. Franz Lemmermeyer and I looked at $z T^2 + x T U + z^2 U^2 = \\Delta y^2$. If rel. prime integers $T, U$ exist satisfying this, then $(z, x, z^2)$ is principle. I think there should be polynomials $T(t,u)$ and $U(t, u)$ : $$((t^2 - Δ u^2)/4)T^2 + ((t^3 + 3 Δ t u^2)/4) T U + ((t^2 - Δ u^2)/4)^2U^2 = Δ ((3t^2 u + Δ u^3)/4)^2 ,$$ – Samuel Hambleton Oct 19 '11 at 10:57\nIf there are such polynomials $T(t, u)$ and $U(t, u)$, then points from the \"parametrization\" yield principal forms and principal ideals. I haven't found any such polynomials. – Samuel Hambleton Oct 19 '11 at 11:02\nadd comment\n\nIn 1993, Tzanakis ( http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6435.pdf ) showed that solving a quartic Thue equation, which correponding quartic field is the compositum of two real quadratic fields, reduces to solving a system of Pellian equations.\n\nEven if the system of Pellian equations cannot be solved completely, the information on solutions obtained from the theory of continued fractions and Diophantine approximations might be sufficient to show that the Thue equation (or Thue inequality) has no solutions or has only trivial solutions. For that purpose, very useful tool is Worley's result characterizing all rational approximations satifying $|\\alpha - \\frac{p}{q}| < \\frac{c}{q^2}$ in terms of convergents of continued fraction of $\\alpha$. You may consult the paper \"Solving a family of quartic Thue inequalities using continued fractions\" ( http://web.math.pmf.unizg.hr/~duje/pdf/dij.pdf ) and the references given there.\n\nshare|improve this answer\nadd comment\n\nI should have stuck with your preferred notation, as in your $B^2 + B C - 57 C^2 = A^3$ in a comment. So the form of interest will be $x^2 + x y - 57 y^2.$The other classes with this discriminant of indefinite integral binary quadratic forms would then be given by $ 3 x^2 \\pm xy - 19 y^2.$\n\nTherefore, take $$ \\phi(x,y) = x^2 + x y - 57 y^2.$$ The identity you need to deal with your $A= \\pm 3$ is $$ \\phi( 15 x^3 - 99 x^2 y + 252 x y^2 - 181 y^3 , \\; 2 x^3 - 15 x^2 y + 33 x y^2 - 28 y^3 ) \\; = \\; ( 3 x^2 + xy - 19 y^2 )^3 $$\n\nThis leads most directly to $\\phi(15,2) = 27.$ Using $ 3 x^2 + x y - 19 y^2 = -3$ when $x=7, y=3,$ this leads directly to $ \\phi(1581, -196) = -27.$\n\nHowever, we have an automorph of $\\phi,$ $$ W \\; = \\; \\left( \\begin{array}{rr} 106 & 855 \\\\\\ 15 & 121 \\end{array} \\right) , $$ and $ W \\cdot (1581,-196)^T = (6, -1)^T,$ so $\\phi(6,-1) = -27.$\n\nFinally, any principal form of odd discriminant, call it $x^2 + x y + k y^2,$ (you have $k=-57$) has the improper automorph $$ Z \\; = \\; \\left( \\begin{array}{rr} 1 & 1 \\\\\\ 0 & -1 \\end{array} \\right) , $$ while $ Z \\cdot (6,-1)^T = (5, 1)^T,$ so $\\phi(5,1) = -27.$\n\nEDIT: a single formula cannot be visually obvious for all desired outcomes. There are an infinite number of integral solutions to $3 x^2 + x y - 19y^2 = -3.$ It is an excellent bet that one of these leads, through the identity I give, to at least one of the desired $\\phi(5,1) = -27$ or $\\phi(6,-1) = -27,$ but not necessarily both, largely because $3 x^2 + x y - 19y^2$ and $3 x^2 - x y - 19y^2$ are not properly equivalent. Worth investigating, I should think.\n\nshare|improve this answer\nI think Dr. Jagy is saying that one can obtain all solutions of X^2 - D y^2 = 4 Z^3 by using representative binary quadratic forms of each of the classes of forms. For D = 229, there are three. This is correct, we can solve X^2 - D y^2 = 4 Z^3 by looking at classes of forms. Part of the proof is that the map from points of ( X^2 - D y^2 = 4 Z^3 ) to Cl^+(D)[3] is surjective. I am curious to know whether it can get computationally easier than this. Possibly not? – Samuel Hambleton Oct 23 '11 at 6:47\nadd comment\n\nI was about to mention H J S Smith's algorithm for finding integer solutions to $x^2 + y^2 = p$ for $p \\equiv 1$ mod 4; but this is referred to in a related thread at Applications of finite continued fractions\n\n(Apologies if that thread is easily found from this one; but I wouldn't have noticed it without doing a Google search, and perhaps some other readers are equally inexperienced in StackOverflow ways or unobservant!)\n\nAlso, what about higher-dimensional continued fractions, expressed as matrix recurrence relations? I seem to recall that these can be used to find rational solutions of equations involving some kinds of cubic forms.\n\nshare|improve this answer\nadd comment\n\nHello, I want to necromance this thread back to life because I was working on a very similar question (and therefore it would only make sense to post it here).\n\nIf I have the diophantine equation\n\nX^2 - Y^2 = C\n\nAssuming we have already proven that this equation has solutions and a finite number of them, how do I efficiently find the smallest value of X and Y, both greater than zero that satisfy this equation.\n\n\nx^2 - y^2 = 2684\n\nA solution to this equation is x = 672, y = 670. But the smallest solution to this equation is x = 72, y = 50. If i'm most interested in finding the latter how would I go about doing so without just trial and error. Can continued fractions play a role here and if so then how? Can secant and tangent lines to the graph of X^2 - Y^2 = C play a role in finding rational points? If so then how? Thank you very much for your time!\n\nshare|improve this answer\nIf you want to ask a new question that may be related to an existing one, better to just ask the question and give the reference. In the present case, though, your question seems to be almost equivalent to a question that was already asked here earlier this month (#98637: \"find the minimum difference between the factors of a number\"), and I think was also seen here before that. So you might just consult that thread. – Noam D. Elkies Jun 20 '12 at 0:58\nThank you for your response! I did look at the thread and our threads are quite different. My question is basically attempting to maximize the value of abs(x - y) whereas that thread was concerned with minimizing the value. In other words the asker on the other post would be most interested in what I am least interested in. Additionally I am interested in the absolute minimal solution to x^2 - y^2 = 2684 (where x and y > 0) regardless of whether x and y do multiply out to create 2684. That being said would you suggest that I post this question in altogether new thread or keep it here for now? – Sid Jun 20 '12 at 1:18\nSince $X^2 - Y^2 = (X+Y) (X-Y)$, you can take $(x,y) = (X+Y,X-Y)$. Then $xy=C$, and $|x-y|=2Y$, so the questions are almost the same except that we must impose the condition $x \\equiv y \\bmod 2$ to ensure the integrality of $X$ and $Y$ [which are $(x\\pm y)/2$]. – Noam D. Elkies Jun 20 '12 at 2:16\nadd comment\n\nYour Answer\n\n\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Take the 2-minute tour ×\n\nPell equations can be solved using continued fractions. I have heard that some elliptic curves can be \"solved\" using continued fractions. Is this true?\n\nWhich Diophantine equations other than Pell equations can be solved for rational or integer points using continued fractions? If there are others, what are some good references?\n\n\nProfessor Elkies has given an excellent response as to the role of continued fractions in solving general Diophantine equations including elliptic curves. What are some other methods to solve the Diophantine equations $$X^2 - \\Delta Y^2 = 4 Z^3$$ and $$18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2 ?$$\n\nshare|improve this question\nSamuel, could you be a little more specific about your comment about solving $x^2 - D y^2 = 4 z^3,$ especially why you are not satisfied with your parametrized solution? I printed out your paper with Franz, \"Arithmetic of Pell Surfaces,\" where I think equation (1.1) with $n=3$ is what you are discussing below. Also Buell's book, especially pages 147-157, discussing Nagell 1922 and Yamamoto 1970. – Will Jagy Oct 17 '11 at 0:03\nThank you Dr. Jagy for comment, interest in the question, and for your help with question 71727 ! I've also seen that you're interested in ternary quadratic forms, which I would like to learn about too. My response will require quite the maximum characters, so I'll stop chatting and get to it. – Samuel Hambleton Oct 19 '11 at 2:13\nLet $\\Delta = 229$ and $K = \\mathbb{Q}(\\sqrt{\\Delta })$. Then $\\text{Cl}^+(K)[3] \\simeq (\\mathbb{Z}/ \\3 )$. Using the \"parametrization\" in Mathematica5, d = 229; S = Union[DeleteCases[Partition[Flatten[Table[P = {(t^3 + 3 d t u^2)/4,(3 t^2 u + d u^3)/4,(t^2 - d u^2)/4};If[IntegerQ[P[[1]]] && IntegerQ[P[[2]]] && IntegerQ[P[[3]]] && GCD[P[[1]], P[[3]]] == 1, P, {w, w, w}], {t, -100, 100}, {u, -100, 100}]], 3], {w, w, w}]]; gives the Points of the \"Pell surface\" $x^2 - d y^2 = 4 z^3$ from the \"parametrization\". On the other hand, a brute force search for points satisfying this Eqn. can be ... – Samuel Hambleton Oct 19 '11 at 2:27\n... done as: T = Union[DeleteCases[Partition[Flatten[Table[If[IntegerQ[Sqrt[d y^2 + 4 z^3]] && GCD[Sqrt[d y^2 + 4 z^3], z] == 1, {Sqrt[d y^2 + 4 z^3], y, z}, {w, w, w}], {y, -100, 100}, {z, -100, 100}]], 3], {w, w, w}]]; There are points of the set $T$ not in $S$, for example : $(11, 1, 3)$. The other form of the Pell surface is $B^2 + B C - 57 C^2 = A^3$. With point $(11, 1, 3)$ corresponding to $(A, B, C) = (3, 5, 1)$, which should map to the ideal $(3, 2 + (1 + \\sqrt{\\Delta })/2)$. I suspect that the \"parametrization\" leads to principal ideals. – Samuel Hambleton Oct 19 '11 at 2:39\nThe points should read $(11, 1, -3)$, $(-3, 5, 1)$, and ideal $(-3, 2 + (1 + \\sqrt{229})/2)$. I would like to know of more methods for solving Diophantine equations, especially surfaces. Professor Elkies'methods look promising. Joro's question 70913, and particularly Schoof's article linked there shows that there may be some good reasons to want an easy method for finding non-principal ideals. I am particularly keen to learn methods for solving $18 x y +x^2 y^2 -4 x^3 -4 y^3 -27 = D z^2$. – Samuel Hambleton Oct 19 '11 at 2:50\nshow 3 more comments\n\n6 Answers\n\nup vote 25 down vote accepted\n\n[edited to insert paragraph on Cornacchia and point-counting]\n\nContinued fractions, or (more-or-less) equivalently the Euclidean algorithm, can be used to find small integer solutions of linear Diophantine equations $ax+by=c$, and integer solutions of quadratic equations such as $x^2-Dy^2=\\pm1$ (\"Pell\"). Continued fractions in themselves won't find rational points on elliptic curves, but there's a technique using Heegner points that calculates a close real approximation to a rational point, which is then recovered from a continued fraction — this is possible because the recovery problem amounts to finding a small integer solution of a linear Diophantine equation. My paper\n\nNoam D. Elkies: Heegner point computations, Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/94; L.M. Adleman and M.-D. Huang, eds.), 122-133.\n\nmight have been the first to describe this approach.\n\nAnother application of continued fractions is Cornacchia's algorithm to solve $x^2+Dy^2=m$ for large $m>0$ coprime to $D$, given $x/y \\bmod m$ which is a square root of $-D \\bmod m$. This has an application to counting points on elliptic curves $E\\bmod p$ for $E$ such as $y^2 = x^3 + b$ or $y^2 = x^3 + ax$ for which the CM field ${\\bf Q}(\\sqrt{-D})$ is known: the count (including the point at infinity) is $p+1-t$ where $t^2+Du^2=4p$ for some integers $t$ and $u$, and this determines $t$ up to an ambiguity of at most $6$ possibilities that in practice is readily resolved. The necessary square root mod $p$ is readily found in random polynomial time, though it is a persistent embarrassment that we cannot extract square roots modulo a large prime in deterministic polynomial time without assuming something like the extended Riemann hypothesis. Indeed the application that Schoof gave to motivate his polynomial-time algorithm to compute $t$ for any elliptic curve mod $p$ was to recover a square root of $-D \\bmod p$ for small $D$ ! (Though this would never be done in practice because the exponent in Schoof's algorithm is much larger than for the randomized algorithm.) The reference for Schoof's paper is\n\nRené Schoof: Elliptic Curves over Finite Fields and the Computation of Square Roots $\\bmod p$, Math. Comp. 44 (#170, April 1985), 483-494.\n\nA natural generalization of the Euclidean algorithm to higher dimensions is the LLL algorithm and other techniques for lattice basis reduction (LBR), which have found various other Diophantine uses, including some other techniques for finding rational points on elliptic curves; another of my papers describes some of these Diophantine applications of LBR.\n\nshare|improve this answer\nAwesome. Thank you Professor Elkies. That should get me started. I can't remember who told me about continued fractions with respect to elliptic curves but I thought they mentioned Artin. I'm not sure. – Samuel Hambleton Oct 13 '11 at 2:51\nThis is a slightly different topic now but I've seen a cool paper of Professor Elkies' : \"Pythagorean triples and Hilbert's Theorem 90\", which I tried to apply to $x^2 - D y^2 = 4 z^3$. It partially worked but I couldn't seem to get the points I was interested in, and so I was wondering about continued fractions. – Samuel Hambleton Oct 13 '11 at 3:40\n@R.Thornburn: 1a) You're welcome! 1b) Perhaps Artin was thinking about point-counting on elliptic curves modulo a prime; see the paragraph I inserted. 2) Looks like this equation $x^2-Dy^2=4z^3$ will involve the $3$-torsion in the class group of ${\\bf Z}[(D+\\sqrt{D})/2]$, which may be accessible via the continued fraction for $(D+\\sqrt{D})/2$ but I suspect that this is not the most efficient method for large $D$. – Noam D. Elkies Oct 14 '11 at 20:02\nWith a change of variables, one can handle all of the equations $A x^2 + B x y + C y^2 + D x + E y + F =0$ in much the same way as Pell's Equation. – Kevin O'Bryant Oct 14 '11 at 23:37\nThank you both very much, and sorry about my identity crisis. I initially wanted to vote the answer to Question 70913 to a non-negative number. I don't want to be deceitful. The incomplete \"parametrization\" I found for $x^2 - D y^2 = 4 z^3$ is $((t^3+3 D t u^2)/4, (3 t^2 u + D u^3)/4 , (t^2 - D u^2)/4)$ but I can't seem to get elements of to narrow class group of exact order $3$ with this \"parametrization\". I am also interested in solving $18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2$. – Samuel Hambleton Oct 16 '11 at 4:32\nadd comment\n\nFor your sample problem, I get two flavors of identity, principal and non-principal.\n\nFor discriminant 229, I take the identity form as $$ f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$ f(x^3 + 3 x y^2 + 5 y^3, \\; 3 x^2 y + 45 x y^2 + 226 y^3 ) \\; = \\; f^3(x,y).$$ As an automorph of $f$ is $$ \\left( \\begin{array}{rr} 1 & 15 \\\\\\ 15 & 226 \\end{array} \\right) , $$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.\n\nFor the other two classes, take $$ g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$ A second family is $$ f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \\; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \\; = \\; g^3(x,y).$$\n$$ $$ $$ $$ The following cycles of reduced forms are as in Buell's book, pages 21-30.\n\nInput three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 \n1 15 -1\n\n0 form 1 15 -1 delta -15\n1 form -1 15 1 delta 15\n2 form 1 15 -1\nminimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229\nAutomorph, written on right of Gram matrix: \n-1 -15\n-15 -226\n Trace: -227 gcd(a21, a22 - a11, a12) : 15\n\n3 13 -5\n\n0 form 3 13 -5 delta -2\n1 form -5 7 9 delta 1\n2 form 9 11 -3 delta -4\n3 form -3 13 5 delta 2\n4 form 5 7 -9 delta -1\n5 form -9 11 3 delta 4\n6 form 3 13 -5\nminimum was 3rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 25.44444\nAutomorph, written on right of Gram matrix: \n-16 -75\n-45 -211\nshare|improve this answer\nThanks. Are you saying that the binary quadratic form $3 x^2 + 13 x y -5 y^2$ comes from the parametrization? From Yamamoto's work, which appears in Buell's book, the binary quadratic form $(z, x, z^2)$ is discussed, where $x^2 - \\Delta y^2 = 4 z^3$ and $gcd(x, z) = 1$. Franz Lemmermeyer and I looked at $z T^2 + x T U + z^2 U^2 = \\Delta y^2$. If rel. prime integers $T, U$ exist satisfying this, then $(z, x, z^2)$ is principle. I think there should be polynomials $T(t,u)$ and $U(t, u)$ : $$((t^2 - Δ u^2)/4)T^2 + ((t^3 + 3 Δ t u^2)/4) T U + ((t^2 - Δ u^2)/4)^2U^2 = Δ ((3t^2 u + Δ u^3)/4)^2 ,$$ – Samuel Hambleton Oct 19 '11 at 10:57\nIf there are such polynomials $T(t, u)$ and $U(t, u)$, then points from the \"parametrization\" yield principal forms and principal ideals. I haven't found any such polynomials. – Samuel Hambleton Oct 19 '11 at 11:02\nadd comment\n\nIn 1993, Tzanakis ( http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6435.pdf ) showed that solving a quartic Thue equation, which correponding quartic field is the compositum of two real quadratic fields, reduces to solving a system of Pellian equations.\n\nEven if the system of Pellian equations cannot be solved completely, the information on solutions obtained from the theory of continued fractions and Diophantine approximations might be sufficient to show that the Thue equation (or Thue inequality) has no solutions or has only trivial solutions. For that purpose, very useful tool is Worley's result characterizing all rational approximations satifying $|\\alpha - \\frac{p}{q}| < \\frac{c}{q^2}$ in terms of convergents of continued fraction of $\\alpha$. You may consult the paper \"Solving a family of quartic Thue inequalities using continued fractions\" ( http://web.math.pmf.unizg.hr/~duje/pdf/dij.pdf ) and the references given there.\n\nshare|improve this answer\nadd comment\n\nI should have stuck with your preferred notation, as in your $B^2 + B C - 57 C^2 = A^3$ in a comment. So the form of interest will be $x^2 + x y - 57 y^2.$The other classes with this discriminant of indefinite integral binary quadratic forms would then be given by $ 3 x^2 \\pm xy - 19 y^2.$\n\nTherefore, take $$ \\phi(x,y) = x^2 + x y - 57 y^2.$$ The identity you need to deal with your $A= \\pm 3$ is $$ \\phi( 15 x^3 - 99 x^2 y + 252 x y^2 - 181 y^3 , \\; 2 x^3 - 15 x^2 y + 33 x y^2 - 28 y^3 ) \\; = \\; ( 3 x^2 + xy - 19 y^2 )^3 $$\n\nThis leads most directly to $\\phi(15,2) = 27.$ Using $ 3 x^2 + x y - 19 y^2 = -3$ when $x=7, y=3,$ this leads directly to $ \\phi(1581, -196) = -27.$\n\nHowever, we have an automorph of $\\phi,$ $$ W \\; = \\; \\left( \\begin{array}{rr} 106 & 855 \\\\\\ 15 & 121 \\end{array} \\right) , $$ and $ W \\cdot (1581,-196)^T = (6, -1)^T,$ so $\\phi(6,-1) = -27.$\n\nFinally, any principal form of odd discriminant, call it $x^2 + x y + k y^2,$ (you have $k=-57$) has the improper automorph $$ Z \\; = \\; \\left( \\begin{array}{rr} 1 & 1 \\\\\\ 0 & -1 \\end{array} \\right) , $$ while $ Z \\cdot (6,-1)^T = (5, 1)^T,$ so $\\phi(5,1) = -27.$\n\nEDIT: a single formula cannot be visually obvious for all desired outcomes. There are an infinite number of integral solutions to $3 x^2 + x y - 19y^2 = -3.$ It is an excellent bet that one of these leads, through the identity I give, to at least one of the desired $\\phi(5,1) = -27$ or $\\phi(6,-1) = -27,$ but not necessarily both, largely because $3 x^2 + x y - 19y^2$ and $3 x^2 - x y - 19y^2$ are not properly equivalent. Worth investigating, I should think.\n\nshare|improve this answer\nI think Dr. Jagy is saying that one can obtain all solutions of X^2 - D y^2 = 4 Z^3 by using representative binary quadratic forms of each of the classes of forms. For D = 229, there are three. This is correct, we can solve X^2 - D y^2 = 4 Z^3 by looking at classes of forms. Part of the proof is that the map from points of ( X^2 - D y^2 = 4 Z^3 ) to Cl^+(D)[3] is surjective. I am curious to know whether it can get computationally easier than this. Possibly not? – Samuel Hambleton Oct 23 '11 at 6:47\nadd comment\n\nI was about to mention H J S Smith's algorithm for finding integer solutions to $x^2 + y^2 = p$ for $p \\equiv 1$ mod 4; but this is referred to in a related thread at Applications of finite continued fractions\n\n(Apologies if that thread is easily found from this one; but I wouldn't have noticed it without doing a Google search, and perhaps some other readers are equally inexperienced in StackOverflow ways or unobservant!)\n\nAlso, what about higher-dimensional continued fractions, expressed as matrix recurrence relations? I seem to recall that these can be used to find rational solutions of equations involving some kinds of cubic forms.\n\nshare|improve this answer\nadd comment\n\nHello, I want to necromance this thread back to life because I was working on a very similar question (and therefore it would only make sense to post it here).\n\nIf I have the diophantine equation\n\nX^2 - Y^2 = C\n\nAssuming we have already proven that this equation has solutions and a finite number of them, how do I efficiently find the smallest value of X and Y, both greater than zero that satisfy this equation.\n\n\nx^2 - y^2 = 2684\n\nA solution to this equation is x = 672, y = 670. But the smallest solution to this equation is x = 72, y = 50. If i'm most interested in finding the latter how would I go about doing so without just trial and error. Can continued fractions play a role here and if so then how? Can secant and tangent lines to the graph of X^2 - Y^2 = C play a role in finding rational points? If so then how? Thank you very much for your time!\n\nshare|improve this answer\nIf you want to ask a new question that may be related to an existing one, better to just ask the question and give the reference. In the present case, though, your question seems to be almost equivalent to a question that was already asked here earlier this month (#98637: \"find the minimum difference between the factors of a number\"), and I think was also seen here before that. So you might just consult that thread. – Noam D. Elkies Jun 20 '12 at 0:58\nThank you for your response! I did look at the thread and our threads are quite different. My question is basically attempting to maximize the value of abs(x - y) whereas that thread was concerned with minimizing the value. In other words the asker on the other post would be most interested in what I am least interested in. Additionally I am interested in the absolute minimal solution to x^2 - y^2 = 2684 (where x and y > 0) regardless of whether x and y do multiply out to create 2684. That being said would you suggest that I post this question in altogether new thread or keep it here for now? – Sid Jun 20 '12 at 1:18\nSince $X^2 - Y^2 = (X+Y) (X-Y)$, you can take $(x,y) = (X+Y,X-Y)$. Then $xy=C$, and $|x-y|=2Y$, so the questions are almost the same except that we must impose the condition $x \\equiv y \\bmod 2$ to ensure the integrality of $X$ and $Y$ [which are $(x\\pm y)/2$]. – Noam D. Elkies Jun 20 '12 at 2:16\nadd comment\n\nYour Answer\n\n\n"
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Precalculus: Introduction to Limits\n\n**By:** Ann Morris \n**Primary Subject:** Math \n**Grade Level:** 11-12\n\n## Objective:\nBy the end of this lesson, the student will be able to compute a basic limit of a function using limit notation.\n\n## Pre-Class Assignment:\nStudents will create a poster titled “There is a Limit to……” or “There is No Limit to…….”. They will draw or cut out pictures from magazines. Examples include: “There is No Limit to the Number of Pairs of Shoes One Can Have” or “There is a Limit to the Amount you can Spend on a Credit Card”. These posters will be hung up in the room or in the hall.\n\n## Resources:\n- Precalculus Textbook\n\n## Time Required:\n90 minutes\n\n## Equipment:\n- Overhead projector or Interwrite Pad\n- Graphics calculator (TI-84 recommended)\n- Method of posting a graph on a large screen (TI-Presenter/TI-84 calculator, Epson projector/document camera, etc.)\n\n## Outline:\n\n1. **Introduction:**\n - Have students show their posters and discuss the concept of limits related to their topics.\n\n2. **Discussion:**\n - Explain that many mathematical problems involve the behavior of a function at a particular number.\n - Present the question: “What is the value of the function \\( f(x) \\) when \\( x = c \\)?”\n - Clarify that the concept of a limit involves the behavior of a function (y-value) as the x-value approaches a particular number, rather than when \\( x \\) equals that number.\n\n3. **Introduce the Concept of \"Approach\":**\n - Example: Walk to a desk and stand close, then sit down. Discuss the difference between approaching and reaching.\n - Use the x-axis to illustrate approaching a number versus reaching it.\n\n4. **Examples:**\n - **A.** \\( y = x^2 \\)\n - Determine what \\( x \\) approaches when \\( y = 4 \\).\n\n - **B.** \\( y = x^2 - x + 2 \\)\n - Investigate behavior as \\( x \\) nears 2 from the left:\n - Table of Values:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 1.0 & 2.000000 \\\\\n 1.5 & 2.750000 \\\\\n 1.8 & 3.440000 \\\\\n 1.9 & 3.710000 \\\\\n 1.95 & 3.852500 \\\\\n 1.995 & 3.985025 \\\\\n 1.999 & 3.997001 \\\\\n \\end{array}\n \\]\n - As \\( x \\) approaches 2 from the right:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 3.0 & 8.000000 \\\\\n 2.5 & 5.750000 \\\\\n 2.2 & 4.640000 \\\\\n 2.05 & 4.152000 \\\\\n 2.01 & 4.030100 \\\\\n 2.005 & 4.015025 \\\\\n 2.001 & 4.003001 \\\\\n \\end{array}\n \\]\n\n - **C.** \\( y = \\sin x \\)\n - Discuss how as \\( x \\) approaches \\( \\pi \\), \\( y \\) approaches 0.\n\n5. **Formal Definition of Limit:**\n - Introduce the formal definition from the textbook.\n\n6. **Calculator Exploration:**\n - Create Table of Values for:\n - \\( y = \\frac{\\sin x}{x} \\)\n - \\( y = \\frac{0.1x^4 - 0.8x^3 - 2x}{x-4} \\)\n - Discuss behavior as \\( x \\) approaches 0.\n\n7. **Limit Notation:**\n - Introduce notation for limit as \\( x \\) approaches a specific number.\n - Re-introduce the infinity symbol and discuss behavior as \\( x \\) approaches infinity.\n\n8. **Cooperative Learning Activity:**\n - Number students from 1-7, form groups, and work on limit notation problems from the textbook.\n - Each group creates examples to present in the next class.\n\n## Assignment:\nComplete an assignment from the textbook on limits.\n\n## Prerequisite to:\nNonexistence of Limits (oscillating functions, etc.)"
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"text": "# Precalculus: Introduction to Limits\n\n**By:** Ann Morris \n**Primary Subject:** Math \n**Grade Level:** 11-12\n\n## Objective:\nBy the end of this lesson, the student will be able to compute a basic limit of a function using limit notation.\n\n## Pre-Class Assignment:\nStudents will create a poster titled “There is a Limit to……” or “There is No Limit to…….”. They will draw or cut out pictures from magazines. Examples include: “There is No Limit to the Number of Pairs of Shoes One Can Have” or “There is a Limit to the Amount you can Spend on a Credit Card”. These posters will be hung up in the room or in the hall.\n\n## Resources:\n- Precalculus Textbook\n\n## Time Required:\n90 minutes\n\n## Equipment:\n- Overhead projector or Interwrite Pad\n- Graphics calculator (TI-84 recommended)\n- Method of posting a graph on a large screen (TI-Presenter/TI-84 calculator, Epson projector/document camera, etc.)\n\n## Outline:\n\n1. **Introduction:**\n - Have students show their posters and discuss the concept of limits related to their topics.\n\n2. **Discussion:**\n - Explain that many mathematical problems involve the behavior of a function at a particular number.\n - Present the question: “What is the value of the function \\( f(x) \\) when \\( x = c \\)?”\n - Clarify that the concept of a limit involves the behavior of a function (y-value) as the x-value approaches a particular number, rather than when \\( x \\) equals that number.\n\n3. **Introduce the Concept of \"Approach\":**\n - Example: Walk to a desk and stand close, then sit down. Discuss the difference between approaching and reaching.\n - Use the x-axis to illustrate approaching a number versus reaching it.\n\n4. **Examples:**\n - **A.** \\( y = x^2 \\)\n - Determine what \\( x \\) approaches when \\( y = 4 \\).\n\n - **B.** \\( y = x^2 - x + 2 \\)\n - Investigate behavior as \\( x \\) nears 2 from the left:\n - Table of Values:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 1.0 & 2.000000 \\\\\n 1.5 & 2.750000 \\\\\n 1.8 & 3.440000 \\\\\n 1.9 & 3.710000 \\\\\n 1.95 & 3.852500 \\\\\n 1.995 & 3.985025 \\\\\n 1.999 & 3.997001 \\\\\n \\end{array}\n \\]\n - As \\( x \\) approaches 2 from the right:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 3.0 & 8.000000 \\\\\n 2.5 & 5.750000 \\\\\n 2.2 & 4.640000 \\\\\n 2.05 & 4.152000 \\\\\n 2.01 & 4.030100 \\\\\n 2.005 & 4.015025 \\\\\n 2.001 & 4.003001 \\\\\n \\end{array}\n \\]\n\n - **C.** \\( y = \\sin x \\)\n - Discuss how as \\( x \\) approaches \\( \\pi \\), \\( y \\) approaches 0.\n\n5. **Formal Definition of Limit:**\n - Introduce the formal definition from the textbook.\n\n6. **Calculator Exploration:**\n - Create Table of Values for:\n - \\( y = \\frac{\\sin x}{x} \\)\n - \\( y = \\frac{0.1x^4 - 0.8x^3 - 2x}{x-4} \\)\n - Discuss behavior as \\( x \\) approaches 0.\n\n7. **Limit Notation:**\n - Introduce notation for limit as \\( x \\) approaches a specific number.\n - Re-introduce the infinity symbol and discuss behavior as \\( x \\) approaches infinity.\n\n8. **Cooperative Learning Activity:**\n - Number students from 1-7, form groups, and work on limit notation problems from the textbook.\n - Each group creates examples to present in the next class.\n\n## Assignment:\nComplete an assignment from the textbook on limits.\n\n## Prerequisite to:\nNonexistence of Limits (oscillating functions, etc.)"
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"text": "# Precalculus: Introduction to Limits\n\n**By:** Ann Morris \n**Primary Subject:** Math \n**Grade Level:** 11-12\n\n## Objective:\nBy the end of this lesson, the student will be able to compute a basic limit of a function using limit notation.\n\n## Pre-Class Assignment:\nStudents will create a poster titled “There is a Limit to……” or “There is No Limit to…….”. They will draw or cut out pictures from magazines. Examples include: “There is No Limit to the Number of Pairs of Shoes One Can Have” or “There is a Limit to the Amount you can Spend on a Credit Card”. These posters will be hung up in the room or in the hall.\n\n## Resources:\n- Precalculus Textbook\n\n## Time Required:\n90 minutes\n\n## Equipment:\n- Overhead projector or Interwrite Pad\n- Graphics calculator (TI-84 recommended)\n- Method of posting a graph on a large screen (TI-Presenter/TI-84 calculator, Epson projector/document camera, etc.)\n\n## Outline:\n\n1. **Introduction:**\n - Have students show their posters and discuss the concept of limits related to their topics.\n\n2. **Discussion:**\n - Explain that many mathematical problems involve the behavior of a function at a particular number.\n - Present the question: “What is the value of the function \\( f(x) \\) when \\( x = c \\)?”\n - Clarify that the concept of a limit involves the behavior of a function (y-value) as the x-value approaches a particular number, rather than when \\( x \\) equals that number.\n\n3. **Introduce the Concept of \"Approach\":**\n - Example: Walk to a desk and stand close, then sit down. Discuss the difference between approaching and reaching.\n - Use the x-axis to illustrate approaching a number versus reaching it.\n\n4. **Examples:**\n - **A.** \\( y = x^2 \\)\n - Determine what \\( x \\) approaches when \\( y = 4 \\).\n\n - **B.** \\( y = x^2 - x + 2 \\)\n - Investigate behavior as \\( x \\) nears 2 from the left:\n - Table of Values:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 1.0 & 2.000000 \\\\\n 1.5 & 2.750000 \\\\\n 1.8 & 3.440000 \\\\\n 1.9 & 3.710000 \\\\\n 1.95 & 3.852500 \\\\\n 1.995 & 3.985025 \\\\\n 1.999 & 3.997001 \\\\\n \\end{array}\n \\]\n - As \\( x \\) approaches 2 from the right:\n \\[\n \\begin{array}{c|c}\n x & f(x) \\\\\n \\hline\n 3.0 & 8.000000 \\\\\n 2.5 & 5.750000 \\\\\n 2.2 & 4.640000 \\\\\n 2.05 & 4.152000 \\\\\n 2.01 & 4.030100 \\\\\n 2.005 & 4.015025 \\\\\n 2.001 & 4.003001 \\\\\n \\end{array}\n \\]\n\n - **C.** \\( y = \\sin x \\)\n - Discuss how as \\( x \\) approaches \\( \\pi \\), \\( y \\) approaches 0.\n\n5. **Formal Definition of Limit:**\n - Introduce the formal definition from the textbook.\n\n6. **Calculator Exploration:**\n - Create Table of Values for:\n - \\( y = \\frac{\\sin x}{x} \\)\n - \\( y = \\frac{0.1x^4 - 0.8x^3 - 2x}{x-4} \\)\n - Discuss behavior as \\( x \\) approaches 0.\n\n7. **Limit Notation:**\n - Introduce notation for limit as \\( x \\) approaches a specific number.\n - Re-introduce the infinity symbol and discuss behavior as \\( x \\) approaches infinity.\n\n8. **Cooperative Learning Activity:**\n - Number students from 1-7, form groups, and work on limit notation problems from the textbook.\n - Each group creates examples to present in the next class.\n\n## Assignment:\nComplete an assignment from the textbook on limits.\n\n## Prerequisite to:\nNonexistence of Limits (oscillating functions, etc.)"
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"text": "Take the 2-minute tour ×\n\nI want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.\n\nWhat I am looking for are references for classical enumerative geometry, back before Hilbert's 15th Problem asked people to actually make it work as rigorous mathematics. Are there good references for the original (flawed!) arguments? I'd prefer perhaps something more recent than the original papers and books (many are hard to find, and even when I can, I tend to be a bit uncomfortable just handling 150 year old books if there's another option.)\n\nMore specifically, are there modern expositions of the original arguments by Schubert, Zeuthen and their contemporaries? And if not, are there translations or modern (20th century, say...) reprints of their work available, or are scanned copies available online (I couldn't find much, though I admit my German is awful enough that I might have missed them by not having the right search terms, so I'm hoping for English review papers or the like, though I'll deal with it if I need to.)\n\nshare|improve this question\nI wish you (sincerely!) good luck with this. Some famous mathematician (Rota?) once lamented that there are too few people who do the work of reading old mathematics, putting it context and reinterpreting it for a contemporary audience, because this requires skill and interest in both mathematics and history. Most historians of mathematics do not have the mathematical skills, and most mathematicians know and care much more about contemporary work (especially their own) than past mathematics, even if this causes each generation to independently rediscover many results in its own language. – Pete L. Clark Jan 8 '10 at 6:17\nThanks, I've had a hobby in history of math since I was an undergrad and I think that it's something I'd like to pursue as a sideline. Specifically, enumerative geometry has interested me, but I found out last May that I just didn't have the stomach for the kinds of arguments going on in GW theory, and switched subfields. Probably partly the influence of Anders Buch, one of my ugrad profs. And the big reason I'm asking is that I feel kind of weird holding an actual copy of Schubert's \"Kalkul der Abzahlenden Geometrie\" in my hands without damaging it. – Charles Siegel Jan 8 '10 at 6:40\nadd comment\n\n5 Answers\n\nI do know of one article taking a historical approach to Schubert calculus:\n\nKleiman, Steven L. Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445--482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.\n\nI am much less of a Schubertist than the average Berkeley/Harvard-educated mathematician with research interests in algebraic geometry, but nevertheless I found this article to be fascinating reading.\n\nshare|improve this answer\nadd comment\nup vote 3 down vote accepted\n\nAnd actually, as a partial answer to my own question, I just stumbled across Schubert's \"Kalkul\" on Google Books, and it looks complete, which makes me rather happy, though other portions of the question still stand.\n\nEDIT: A friend of mine has informed me that Zeuthen's \"Lehrbuch\" is also there, and now it's linked.\n\nshare|improve this answer\nYeah, it's complete, you can download the pdf, and it even appears to have a good bibliography, index, and endnotes. – Jonas Meyer Jan 8 '10 at 6:54\nadd comment\n\nHere is a link to a more than 3 page list of works on enumerative geometry from the second half of the 19th century, including those of Schubert, Zeuthen, and many others. (Perhaps it will be a useful guide for searching for scanned copies of the originals, e.g. through UPenn fulltext subscriptions.)\n\nshare|improve this answer\nThanks, I'll definitely be sorting through some of that. – Charles Siegel Jan 8 '10 at 6:41\nadd comment\n\nYou could also take a look at Schubert Calculus by Kleiman and Laksov in the Monthly, Vol. 79, No. 10, pp 1061-1082 or the monograph Geometry of Coxeter Groups by H. Hiller.\n\nshare|improve this answer\nI've seen Kleiman/Laksov, though not the Hiller monograph, but generally seems rather more modern than I was looking for, though thanks for pointing out the Hiller. – Charles Siegel May 28 '10 at 1:54\nadd comment\n\nHave you looked at Semple and Roth, Introduction to Algebraic Geometry? It was published in 1949 and contains a wealth of classical results (there is a chapter devoted to enumerative geometry). Going back a bit further, both German and French Encyclopaedias of Mathematical Sciences published in the early 20th century had surveys of algebraic geometry. Moving in the opposite direction, Fulton's \"Intersection theory\" discusses applications of his theory to classical enumerative geometry problems where excessive intersections play crucial role (such as finding the number of conics touching 5 given ones).\n\nI know you said you've decided to move away from GW theory, but I thought I'd just throw it in here: Sheldon Katz's book \"Enumerative geometry and string theory\" (Student Mathematical Library, vol 32) is actually very readable.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
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questioner_given_difficulty_id | [
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"text": "Take the 2-minute tour ×\n\nI want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.\n\nWhat I am looking for are references for classical enumerative geometry, back before Hilbert's 15th Problem asked people to actually make it work as rigorous mathematics. Are there good references for the original (flawed!) arguments? I'd prefer perhaps something more recent than the original papers and books (many are hard to find, and even when I can, I tend to be a bit uncomfortable just handling 150 year old books if there's another option.)\n\nMore specifically, are there modern expositions of the original arguments by Schubert, Zeuthen and their contemporaries? And if not, are there translations or modern (20th century, say...) reprints of their work available, or are scanned copies available online (I couldn't find much, though I admit my German is awful enough that I might have missed them by not having the right search terms, so I'm hoping for English review papers or the like, though I'll deal with it if I need to.)\n\nshare|improve this question\nI wish you (sincerely!) good luck with this. Some famous mathematician (Rota?) once lamented that there are too few people who do the work of reading old mathematics, putting it context and reinterpreting it for a contemporary audience, because this requires skill and interest in both mathematics and history. Most historians of mathematics do not have the mathematical skills, and most mathematicians know and care much more about contemporary work (especially their own) than past mathematics, even if this causes each generation to independently rediscover many results in its own language. – Pete L. Clark Jan 8 '10 at 6:17\nThanks, I've had a hobby in history of math since I was an undergrad and I think that it's something I'd like to pursue as a sideline. Specifically, enumerative geometry has interested me, but I found out last May that I just didn't have the stomach for the kinds of arguments going on in GW theory, and switched subfields. Probably partly the influence of Anders Buch, one of my ugrad profs. And the big reason I'm asking is that I feel kind of weird holding an actual copy of Schubert's \"Kalkul der Abzahlenden Geometrie\" in my hands without damaging it. – Charles Siegel Jan 8 '10 at 6:40\nadd comment\n\n5 Answers\n\nI do know of one article taking a historical approach to Schubert calculus:\n\nKleiman, Steven L. Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445--482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.\n\nI am much less of a Schubertist than the average Berkeley/Harvard-educated mathematician with research interests in algebraic geometry, but nevertheless I found this article to be fascinating reading.\n\nshare|improve this answer\nadd comment\nup vote 3 down vote accepted\n\nAnd actually, as a partial answer to my own question, I just stumbled across Schubert's \"Kalkul\" on Google Books, and it looks complete, which makes me rather happy, though other portions of the question still stand.\n\nEDIT: A friend of mine has informed me that Zeuthen's \"Lehrbuch\" is also there, and now it's linked.\n\nshare|improve this answer\nYeah, it's complete, you can download the pdf, and it even appears to have a good bibliography, index, and endnotes. – Jonas Meyer Jan 8 '10 at 6:54\nadd comment\n\nHere is a link to a more than 3 page list of works on enumerative geometry from the second half of the 19th century, including those of Schubert, Zeuthen, and many others. (Perhaps it will be a useful guide for searching for scanned copies of the originals, e.g. through UPenn fulltext subscriptions.)\n\nshare|improve this answer\nThanks, I'll definitely be sorting through some of that. – Charles Siegel Jan 8 '10 at 6:41\nadd comment\n\nYou could also take a look at Schubert Calculus by Kleiman and Laksov in the Monthly, Vol. 79, No. 10, pp 1061-1082 or the monograph Geometry of Coxeter Groups by H. Hiller.\n\nshare|improve this answer\nI've seen Kleiman/Laksov, though not the Hiller monograph, but generally seems rather more modern than I was looking for, though thanks for pointing out the Hiller. – Charles Siegel May 28 '10 at 1:54\nadd comment\n\nHave you looked at Semple and Roth, Introduction to Algebraic Geometry? It was published in 1949 and contains a wealth of classical results (there is a chapter devoted to enumerative geometry). Going back a bit further, both German and French Encyclopaedias of Mathematical Sciences published in the early 20th century had surveys of algebraic geometry. Moving in the opposite direction, Fulton's \"Intersection theory\" discusses applications of his theory to classical enumerative geometry problems where excessive intersections play crucial role (such as finding the number of conics touching 5 given ones).\n\nI know you said you've decided to move away from GW theory, but I thought I'd just throw it in here: Sheldon Katz's book \"Enumerative geometry and string theory\" (Student Mathematical Library, vol 32) is actually very readable.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
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"text": "Take the 2-minute tour ×\n\nI want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.\n\nWhat I am looking for are references for classical enumerative geometry, back before Hilbert's 15th Problem asked people to actually make it work as rigorous mathematics. Are there good references for the original (flawed!) arguments? I'd prefer perhaps something more recent than the original papers and books (many are hard to find, and even when I can, I tend to be a bit uncomfortable just handling 150 year old books if there's another option.)\n\nMore specifically, are there modern expositions of the original arguments by Schubert, Zeuthen and their contemporaries? And if not, are there translations or modern (20th century, say...) reprints of their work available, or are scanned copies available online (I couldn't find much, though I admit my German is awful enough that I might have missed them by not having the right search terms, so I'm hoping for English review papers or the like, though I'll deal with it if I need to.)\n\nshare|improve this question\nI wish you (sincerely!) good luck with this. Some famous mathematician (Rota?) once lamented that there are too few people who do the work of reading old mathematics, putting it context and reinterpreting it for a contemporary audience, because this requires skill and interest in both mathematics and history. Most historians of mathematics do not have the mathematical skills, and most mathematicians know and care much more about contemporary work (especially their own) than past mathematics, even if this causes each generation to independently rediscover many results in its own language. – Pete L. Clark Jan 8 '10 at 6:17\nThanks, I've had a hobby in history of math since I was an undergrad and I think that it's something I'd like to pursue as a sideline. Specifically, enumerative geometry has interested me, but I found out last May that I just didn't have the stomach for the kinds of arguments going on in GW theory, and switched subfields. Probably partly the influence of Anders Buch, one of my ugrad profs. And the big reason I'm asking is that I feel kind of weird holding an actual copy of Schubert's \"Kalkul der Abzahlenden Geometrie\" in my hands without damaging it. – Charles Siegel Jan 8 '10 at 6:40\nadd comment\n\n5 Answers\n\nI do know of one article taking a historical approach to Schubert calculus:\n\nKleiman, Steven L. Problem 15: rigorous foundation of Schubert's enumerative calculus. Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pp. 445--482. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.\n\nI am much less of a Schubertist than the average Berkeley/Harvard-educated mathematician with research interests in algebraic geometry, but nevertheless I found this article to be fascinating reading.\n\nshare|improve this answer\nadd comment\nup vote 3 down vote accepted\n\nAnd actually, as a partial answer to my own question, I just stumbled across Schubert's \"Kalkul\" on Google Books, and it looks complete, which makes me rather happy, though other portions of the question still stand.\n\nEDIT: A friend of mine has informed me that Zeuthen's \"Lehrbuch\" is also there, and now it's linked.\n\nshare|improve this answer\nYeah, it's complete, you can download the pdf, and it even appears to have a good bibliography, index, and endnotes. – Jonas Meyer Jan 8 '10 at 6:54\nadd comment\n\nHere is a link to a more than 3 page list of works on enumerative geometry from the second half of the 19th century, including those of Schubert, Zeuthen, and many others. (Perhaps it will be a useful guide for searching for scanned copies of the originals, e.g. through UPenn fulltext subscriptions.)\n\nshare|improve this answer\nThanks, I'll definitely be sorting through some of that. – Charles Siegel Jan 8 '10 at 6:41\nadd comment\n\nYou could also take a look at Schubert Calculus by Kleiman and Laksov in the Monthly, Vol. 79, No. 10, pp 1061-1082 or the monograph Geometry of Coxeter Groups by H. Hiller.\n\nshare|improve this answer\nI've seen Kleiman/Laksov, though not the Hiller monograph, but generally seems rather more modern than I was looking for, though thanks for pointing out the Hiller. – Charles Siegel May 28 '10 at 1:54\nadd comment\n\nHave you looked at Semple and Roth, Introduction to Algebraic Geometry? It was published in 1949 and contains a wealth of classical results (there is a chapter devoted to enumerative geometry). Going back a bit further, both German and French Encyclopaedias of Mathematical Sciences published in the early 20th century had surveys of algebraic geometry. Moving in the opposite direction, Fulton's \"Intersection theory\" discusses applications of his theory to classical enumerative geometry problems where excessive intersections play crucial role (such as finding the number of conics touching 5 given ones).\n\nI know you said you've decided to move away from GW theory, but I thought I'd just throw it in here: Sheldon Katz's book \"Enumerative geometry and string theory\" (Student Mathematical Library, vol 32) is actually very readable.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
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"text": "# Optimal Solution to the Knapsack Problem with Linear Programming in R\n\n**Posted on 7. February 2021 by Fabian Leuthold**\n\nIn a previous post, I demonstrated how to develop a heuristic to solve the knapsack problem. We managed to solve the problem quite well and had an optimality gap of about 1%, meaning our solution was 1% away from the optimal solution. Here, I’d like to demonstrate how simple it is to solve the knapsack problem using linear programming (LP) in R.\n\n### Introduction\n\nThe knapsack problem is a classic optimization problem. Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.\n\n### Mathematical Formulation\n\nThe problem can be formally stated as follows:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ x_i \\in \\{0, 1\\}, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\nWhere:\n- \\( n \\) is the number of items,\n- \\( v_i \\) is the value of item \\( i \\),\n- \\( w_i \\) is the weight of item \\( i \\),\n- \\( x_i \\) is a binary variable indicating whether item \\( i \\) is included in the knapsack,\n- \\( W \\) is the maximum weight capacity of the knapsack.\n\n### Linear Programming Approach\n\nTo solve this using linear programming, we relax the integer constraint on \\( x_i \\) to allow fractional values, solving:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ 0 \\leq x_i \\leq 1, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\n### Implementation in R\n\nHere is how you can implement this approach in R using the `lpSolve` package:\n\n```r\n# Install and load the lpSolve package\ninstall.packages(\"lpSolve\")\nlibrary(lpSolve)\n\n# Define the parameters\nvalues <- c(60, 100, 120) # Example values\nweights <- c(10, 20, 30) # Example weights\ncapacity <- 50 # Example capacity\n\n# Create the LP model\nlp_model <- lp(\"max\", \n obj = values, \n const.mat = matrix(weights, nrow = 1), \n const.dir = \"<=\", \n const.rhs = capacity, \n all.int = FALSE)\n\n# Extract the solution\nsolution <- lp_model$solution\noptimal_value <- lp_model$objval\n\n# Print results\ncat(\"Optimal solution value:\", optimal_value, \"\\n\")\ncat(\"Items included:\", solution, \"\\n\")\n```\n\n### Discussion\n\nThe LP relaxation provides an upper bound on the optimal solution of the integer knapsack problem. The solution obtained from the LP model may include fractional items, which are not feasible in the original problem. However, this approach gives a quick and efficient way to estimate the optimal value.\n\n### Conclusion\n\nUsing linear programming to solve the knapsack problem in R is a straightforward method that provides valuable insights into the problem's structure. While it may not always yield an integer solution, it is a useful tool for understanding the bounds and characteristics of the knapsack problem.\n\n### References\n\n- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). *Introduction to Algorithms*. MIT Press.\n- Grötschel, M., Lovász, L., & Schrijver, A. (1981). *The ellipsoid method and its consequences in combinatorial optimization*. Combinatorica, 1(2), 153-158.\n\nThis article provides a foundational understanding of solving the knapsack problem using linear programming in R. For further exploration, consider applying heuristics or integer programming methods to obtain exact solutions."
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"text": "# Optimal Solution to the Knapsack Problem with Linear Programming in R\n\n**Posted on 7. February 2021 by Fabian Leuthold**\n\nIn a previous post, I demonstrated how to develop a heuristic to solve the knapsack problem. We managed to solve the problem quite well and had an optimality gap of about 1%, meaning our solution was 1% away from the optimal solution. Here, I’d like to demonstrate how simple it is to solve the knapsack problem using linear programming (LP) in R.\n\n### Introduction\n\nThe knapsack problem is a classic optimization problem. Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.\n\n### Mathematical Formulation\n\nThe problem can be formally stated as follows:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ x_i \\in \\{0, 1\\}, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\nWhere:\n- \\( n \\) is the number of items,\n- \\( v_i \\) is the value of item \\( i \\),\n- \\( w_i \\) is the weight of item \\( i \\),\n- \\( x_i \\) is a binary variable indicating whether item \\( i \\) is included in the knapsack,\n- \\( W \\) is the maximum weight capacity of the knapsack.\n\n### Linear Programming Approach\n\nTo solve this using linear programming, we relax the integer constraint on \\( x_i \\) to allow fractional values, solving:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ 0 \\leq x_i \\leq 1, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\n### Implementation in R\n\nHere is how you can implement this approach in R using the `lpSolve` package:\n\n```r\n# Install and load the lpSolve package\ninstall.packages(\"lpSolve\")\nlibrary(lpSolve)\n\n# Define the parameters\nvalues <- c(60, 100, 120) # Example values\nweights <- c(10, 20, 30) # Example weights\ncapacity <- 50 # Example capacity\n\n# Create the LP model\nlp_model <- lp(\"max\", \n obj = values, \n const.mat = matrix(weights, nrow = 1), \n const.dir = \"<=\", \n const.rhs = capacity, \n all.int = FALSE)\n\n# Extract the solution\nsolution <- lp_model$solution\noptimal_value <- lp_model$objval\n\n# Print results\ncat(\"Optimal solution value:\", optimal_value, \"\\n\")\ncat(\"Items included:\", solution, \"\\n\")\n```\n\n### Discussion\n\nThe LP relaxation provides an upper bound on the optimal solution of the integer knapsack problem. The solution obtained from the LP model may include fractional items, which are not feasible in the original problem. However, this approach gives a quick and efficient way to estimate the optimal value.\n\n### Conclusion\n\nUsing linear programming to solve the knapsack problem in R is a straightforward method that provides valuable insights into the problem's structure. While it may not always yield an integer solution, it is a useful tool for understanding the bounds and characteristics of the knapsack problem.\n\n### References\n\n- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). *Introduction to Algorithms*. MIT Press.\n- Grötschel, M., Lovász, L., & Schrijver, A. (1981). *The ellipsoid method and its consequences in combinatorial optimization*. Combinatorica, 1(2), 153-158.\n\nThis article provides a foundational understanding of solving the knapsack problem using linear programming in R. For further exploration, consider applying heuristics or integer programming methods to obtain exact solutions."
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"text": "# Optimal Solution to the Knapsack Problem with Linear Programming in R\n\n**Posted on 7. February 2021 by Fabian Leuthold**\n\nIn a previous post, I demonstrated how to develop a heuristic to solve the knapsack problem. We managed to solve the problem quite well and had an optimality gap of about 1%, meaning our solution was 1% away from the optimal solution. Here, I’d like to demonstrate how simple it is to solve the knapsack problem using linear programming (LP) in R.\n\n### Introduction\n\nThe knapsack problem is a classic optimization problem. Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.\n\n### Mathematical Formulation\n\nThe problem can be formally stated as follows:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ x_i \\in \\{0, 1\\}, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\nWhere:\n- \\( n \\) is the number of items,\n- \\( v_i \\) is the value of item \\( i \\),\n- \\( w_i \\) is the weight of item \\( i \\),\n- \\( x_i \\) is a binary variable indicating whether item \\( i \\) is included in the knapsack,\n- \\( W \\) is the maximum weight capacity of the knapsack.\n\n### Linear Programming Approach\n\nTo solve this using linear programming, we relax the integer constraint on \\( x_i \\) to allow fractional values, solving:\n\nMaximize: \n\\[ \\sum_{i=1}^{n} v_i x_i \\]\n\nSubject to: \n\\[ \\sum_{i=1}^{n} w_i x_i \\leq W \\] \n\\[ 0 \\leq x_i \\leq 1, \\quad \\forall i \\in \\{1, \\ldots, n\\} \\]\n\n### Implementation in R\n\nHere is how you can implement this approach in R using the `lpSolve` package:\n\n```r\n# Install and load the lpSolve package\ninstall.packages(\"lpSolve\")\nlibrary(lpSolve)\n\n# Define the parameters\nvalues <- c(60, 100, 120) # Example values\nweights <- c(10, 20, 30) # Example weights\ncapacity <- 50 # Example capacity\n\n# Create the LP model\nlp_model <- lp(\"max\", \n obj = values, \n const.mat = matrix(weights, nrow = 1), \n const.dir = \"<=\", \n const.rhs = capacity, \n all.int = FALSE)\n\n# Extract the solution\nsolution <- lp_model$solution\noptimal_value <- lp_model$objval\n\n# Print results\ncat(\"Optimal solution value:\", optimal_value, \"\\n\")\ncat(\"Items included:\", solution, \"\\n\")\n```\n\n### Discussion\n\nThe LP relaxation provides an upper bound on the optimal solution of the integer knapsack problem. The solution obtained from the LP model may include fractional items, which are not feasible in the original problem. However, this approach gives a quick and efficient way to estimate the optimal value.\n\n### Conclusion\n\nUsing linear programming to solve the knapsack problem in R is a straightforward method that provides valuable insights into the problem's structure. While it may not always yield an integer solution, it is a useful tool for understanding the bounds and characteristics of the knapsack problem.\n\n### References\n\n- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). *Introduction to Algorithms*. MIT Press.\n- Grötschel, M., Lovász, L., & Schrijver, A. (1981). *The ellipsoid method and its consequences in combinatorial optimization*. Combinatorica, 1(2), 153-158.\n\nThis article provides a foundational understanding of solving the knapsack problem using linear programming in R. For further exploration, consider applying heuristics or integer programming methods to obtain exact solutions."
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"text": "Take the 2-minute tour ×\n\nIf I understand correctly, there was a huge birth of programming languages during the early decades of computing, but then things have stabilized.\n\nBasically, why are many universities and industries still using languages such as C and C++ (which were created decades ago), when there have been plenty of decades since then to improve upon and create more effective (and human-friendly) languages?\n\nshare|improve this question\n\nclosed as not constructive by Walter, Tom Squires, StuperUser, Jonas, Jeremy Oct 28 '11 at 18:28\n\n\nHey Jacob, your question is off topic for programmers.stackexchange, you should only ask practical, answerable questions based on actual problems, as its stated on the faq. As for your question, C and C++ keep evolving and are still extremely efficient if used correctly, although many seem to think that they are not so friendly. But human friendliness is not always what you look for in a programming language. – Yannis Rizos Oct 28 '11 at 2:18\nI think the premise is entirely incorrect. My impression is that there are a lot more new programming languages these days than at any time before. – Kyralessa Oct 28 '11 at 2:26\nI wish more universities would use C++ for teaching programming. – sbi Oct 28 '11 at 8:06\n@sbi - I wish more universities would use anything but C++ for teaching programming. Pen&paper if possible, but will settle for anything else. – ldigas Oct 28 '11 at 17:22\n@sbi - No, you see that is exactly what I don't agree with: \"in order to learn concepts, you need tools that support those concepts\". I believe concepts should be learned without the tools, so once \"one gets the tools\" one already knows the concepts. I'm of the school that it is terrible to go learning the concept of pointers by programming in C++. You should understand both pointers, memory management and ... before writing your first line of code. You don't go learning drawing by working in Autocad, you learn to draw and then you learn ACAD. One has not much to do with another. – ldigas Oct 28 '11 at 23:22\nshow 8 more comments\n\n8 Answers\n\nup vote 11 down vote accepted\n\nOne can extend this question to many things: why has the progress of chess slowed down? Why has the progress of aviation slowed down? Why was I learning math so fast in grade school but in grad school and thereafter I am publishing papers VERY SLOWLY. Why was Facebook progressing so fast as start-up, and now with 1200+ employees it does not really change all that much?\n\nThe progress has not stopped; it goes on, but, as George Carlin put it, \"Now you've got shit all over the world. The supply lines are getting harder to maintain\".\n\nThe success of languages depends partly on having very large API base and user base, and the success of airplanes depends on being able to provide a million of conveniences to the customer. And the work of an academic is more complicated than that of a student.\n\nSo, having to deal with backwards compatibility, having to test a new language feature in conjunction with all of the existing ones, having to worry about breaking libraries that worked with Python 2.7 (for instance), just concentrating on improving the API, dealing with new computer architectures that need to be supported, dealing with inter-operability with some other popular technologies, being invited to conferences as the father of a language, having to deal with lots of fan/hate mail, general time waste due to the inreased size of code base / organization, finding time to write a book about the language you have just made ... the responsibilities can suddenly pile up exponentially, but the time ticks away at the same speed. So, as complexity of things grows, the progress appears to slow down. The only way to deal with that is to throw more people at the problem, but not every organization can afford to do that. They would have to be benefiting financially from the product at least proportionally to its complexity. The for-profit model could work better in this case, although Sun & Oracle had cash but have/are not making revolutionary changes to Java now days.\n\nThese are just some things that I thought of ...\n\nshare|improve this answer\nDoes this also explain why there are fewer successful start-ups these days than decades ago? – Dark Templar Oct 28 '11 at 3:32\n@Jacob Hayden, maybe. One can find holes in my answer for sure. On one hand you have better tools; on the other customers are expecting more. When it comes to \"winner takes all situation\", such as Facebook, it has grown like crazy when social media was a new thing. It is becoming a commodity now. I predict the next big thing will be widespread genetic testing. Today people are willing to pay to find out how black / Asian / white they are. In the future they might settle for nothing less than a thorough 10-page report. First start-ups in new space can have good margins while having ok products. – Job Oct 28 '11 at 3:36\nThanks. Do you know what \"commodity\" is supposed to mean in this context? – Dark Templar Oct 28 '11 at 3:53\nI don't know about aviation, but the progress of chess has definitely not slowed down in recent decades; quite the contrary. – Konrad Morawski Oct 28 '11 at 8:54\n@Morawski, I think there is also a perceived component to this. Chess is a well-studied problem, and ever since Kasparov lost to the computer, the software and algorithms are getting better, but they do not generate buzz anymore. We take for granted that a computer can beat a person, just as we take for granted that one can get a modern cell phone that weighs less than a wallet and can call anyone for under $100 per month. This kind of stuff used to generate headlines; now it is does not because non-sensation is non-news. I should have written about the 'perception is reality' aspect of this. – Job Oct 28 '11 at 15:14\nadd comment\n\nSimply put, because there are no new successful languages in similar problem spaces. There are no languages which perform the same job as C++, but better enough to justify throwing away all of the experience, tools, existing codebases, etc. The only language that I can name that's even made a half-assed attempt is D, and it resembles C# more closely.\n\nUnlike C. C exists for two reasons: C++ is an impractical language to implement, so there are some (very small) problem spaces in which C++ won't fit, and because people are too ignorant to understand how incredibly poor it is. C is not suitable for teaching as a general-purpose language any more.\n\nOne of the reasons that there are fewer new languages is because successful languages are much larger and more complex to implement competitive toolchains.\n\nI would suggest that another reason is because people hardly know the languages they have. Especially a language like C++. You find many more people with incredible misconceptions than you do people who actually know how to use the language. People who were only taught \"C with Classes\". You're asking people who can't grasp that std::vector<T> is better than malloc and free to learn something? Good luck.\n\nshare|improve this answer\nadd comment\n\nYou've got the wrong impression. There's a lot of \"new\" languages out there. Google alone has released Dart and Go within little more than a year.\n\nAs for what is used at the university. If you take MIT, they went from LISP to Scala to Python within the last 2 decades or so. Some universities try hard to pick the right languages for their courses and some just use whatever they think is established and stick to that.\n\nshare|improve this answer\nadd comment\n\nWikipedia has a chronological list of important programming languages. It's not a complete list of every language ever invented, but I'm not sure that would help. Counting by hand, I see the following:\n\n • 1950's: 47 languages\n • 1960's: 46 languages\n • 1970's: 45 languages\n • 1980's: 50 languages\n • 1990's: 57 languages\n • 2000's: 36 languages\n\nSo there may have been a bit of a drop-off in \"important\" languages (although some of the languages invented in the last few years may yet prove important enough to be added later). Other than that, the rate of language creation has been remarkably flat.\n\nDo you have any alternative data that supports your assertion?\n\nAs for universities: Do you know that they also still teach Latin? Nobody speaks Latin anymore, but you'll find plenty of people who will tell you that it's important. Learning Latin helps you understand a lot about modern languages, and Latin is also the source of a great deal of great literature. C is similar (although people still \"speak\" it): it's the root of many more modern programming languages, and a great deal of useful code is written in C.\n\nshare|improve this answer\nPeople still use Latin. How would you name newly discovered species without it? – SK-logic Oct 28 '11 at 7:53\nCopy-paste, of course, just like you maintain COBOL ;) – MSalters Oct 28 '11 at 10:12\n@SK-logic Of course people still use Latin -- they still teach Latin because it's so useful. Virtually nobody speaks Latin, though -- modern languages are more convenient. – Caleb Oct 28 '11 at 12:25\nThese lists look pretty laughable, at least to me. Somebody may think all these languages are \"important\", but there's a pretty fair number there I've never heard of before. At least half of which I was previously aware, I'd still say were more \"obscure\" than \"important\". – Jerry Coffin Oct 28 '11 at 14:57\n@JerryCoffin, I feel the same way, but I figure that a community-reviewed list is a better source than the entirely subjective list of languages that any one person considers \"important.\" Perhaps I should use quotes instead of italics for that word above. Nevertheless, the languages on the list are at least important or well-known enough that somebody bothered to add them to the list, and I think that speaks to the premise that \"rate of language popularization has slowed down in recent decades.\" – Caleb Oct 28 '11 at 15:36\nadd comment\n\nWhy do Universities still teach old languages? Because those old languages got it right. Sure they have rough edges, C's strings could be better and C++ could certainly be decrufted a bit. But they work. And there's no denying that. What counts more than anything else isn't efficacy of language, it's environment.\n\nA million years ago when Mainframes were hot stuff and Dr. Knuth still had hair, we had the likes of Cobol, APL and FORTRAN (then spelled in all caps because we hadn't figured out that it wasn't cool to do that). Lisp was there too and so was Algol and so forth (FORTH wasn't quite there yet though, we're still in the 60s). However, most of these languages were fairly limited unless they were supported by the Government (FORTRAN) or Business (COBOL). Remember that the Internet was 4 computers in 1968 so getting information on a language would involve hearing about it from a colleague or some such and then going to a library, looking through a card catalog (a literal file system) and seeing if they had it. If you were lucky you could go to the source directly at whatever university or company was working on it. So knowledge ecosystems were limited.\n\nNow let's talk about hardware. I'm just old enough to have been raised on a 386. A 386 was a God Machine compared to what they had back then. We're talking about transistors the size of cockroaches and wired collections of magnetic cores (Giving us the term \"core dump\") that held kilobytes of memory. These computers cost hundreds of thousands or millions of dollars. With machines being so expensive and with relatively few users, you had a lot of variation. Word and bit collection sizes that would make most of us go insane with frustration if we had to deal with them today. Bytes have spoiled us all. So what did you program these things in? Assembly on Punched cards. Sometimes you might have a machine specific higher level language to work with like PL/I or similar.\n\nBy the end of the decade, integrated circuits had started to come down in price and had begun to displace large scale transistors while memory also got cheaper. You still ran into the issue of all kinds of architectures but things were progressing. Then some bored programmer wanted to play Space War. And it changed the world. While languages like COBOL and Lisp were portable, C was the first language that was portable at a low level. Combined with UNIX (Like a really nerdy Voltron), it battled Pascal for dominion.\n\nBy this time, computers at universities were fairly common and Hacker culture was beginning to form. This led to the formation of things like the GNU project and other such initiatives as well as jokes like INTERCAL but I'm getting ahead of myself. There was a parallel stream here, the screaming birth of Software Engineering. Object Oriented Programming was not new in the early 80s but by then computers were common enough that large scale code bases were becoming hard to manage due to greater demand so it spurred development there.\n\nOther developments ensued: the widespread use of home computers, the spread of the internet, the rise of windowing operating systems, the rise of the video game as a popular art form, and the rise of the parallel core machine amongst others.\n\nThe most important of these developments for the proliferation of languages are the introduction of the x86 and its various competitors that were folded into the IBM PC and its competitors and then the rise of the internet to facilitate language spread. So you've solved the problem of architecture portability and 90% of the market uses the same operating system. What does this do to the need for languages? It flattens them.\n\nWhy? Fundamentally language development and propagation are driven by need to solve problems. If you've got a language that works, you don't need to change it. Perhaps you add a framework here or there to streamline development or a library for task automation or the methodology of the month but ultimately you're crufting onto the language. This cruft though, it's important because there's where you get the environment from. You get culture that way and propagation follows the waves of culture.\n\nLanguages adapt and C family languages are incredibly adaptable due to being both widely deployed and extremely powerful. They also have something else, tremendous momentum. Say what you like about COBOL, there is a lot of wrapped COBOL running around out there; some of that is mutating into Java and other things but it will still being going along none the less. Some languages never really take off because they weren't designed to solve a widely scoped problem. C and C++ are widely scoped problem solvers. APL is not, at all. APL was designed to use as few keystrokes as possible in an era when that really mattered. It took it too far compared to other languages at the time and thus balance was not achieved, momentum was not gained and culture rampancy didn't take hold.\n\nRampancy is another key. The very best languages have something viral about them. They get inside your head and they stay there. That something viral might be monetary like a company saying \"This language is win\" and giving out T Shirts but I think it's more than that. Java may be a funky, crufty language but it solves a problem (several actually) and has a very powerful, meme-plague ridden ecosystem. So it keeps on propagating. And Propagation really is the key, your professors teach you old languages because they know there languages and these languages are still used out there. C is so close to the metal and so flexible that it is useful conceptually even if you never really use it. C++ is a thornier sell perhaps but I'd argue that as a much \"harder\" Object Oriented language than most, it serves as an excellent teaching tool.\n\nComputer Science departments are out to teach concepts, not memetic fads. So they pick the best tool that solves that problem. That's why MIT used Scheme for so long. The modern language pantheon offers a lot of choice and therein perhaps lies the final answer. The internet has made it very easy to design, build and market a language. It can provide an ecosystem for you but it also provide a host of competitors. Which do you choose if you are teaching CS? The new hotness that may burn out soon? Which one? Why or why not? Most \"fixer\" languages are aimed at programmer productivity, not academic learning. And even with industrial usage, would you rather spend 6 months retraining your guys to Newfangled? I'd suspect you'd rather spend 6 months using the C++ you already have a team for and make money. Because that's the problem set in business. How do I make money faster than my rivals? I use proven technology run by smart people. So I can hired more smart people versed in proven technologies to write more code that makes me more money so I can hire even more smart people and pretty soon I have an R&D budget so I can indulge in things like exotic paradigms in Newfangled. But I'm still running in an \"old\" ecosystem. Because it works.\n\nIt just works.\n\nshare|improve this answer\nadd comment\n\nYes, but now a days most them are using open source programming languages like PHP, Ruby on rails etc. Past days they are used Visual basic for stand lone application. Now days that was changed into VB.Net. And machine languages like c and C++ are replaced with Java for machine level coding..........!!\n\nshare|improve this answer\nadd comment\n\nA lot of \"modern\" programming languages aren't so different to C++.\n\nJava has classes and function calls. So does C++. Java has block structures such as if and for - which are copied from C.\n\nA major thing that Java has, but which C++ doesn't, is garbage collection. That's no so modern - it was invented for Lisp in the 1950s. The resource management system used in C++ (based on RAII and timely reliable destructors) is actually much newer than GC.\n\nFunctional languages like Haskell and the ML family are much more interestingly modern - and have mostly been considered academic because you're likely to find them in universities, but not so much in industry - though even that is changing. Even if you don't end up using Haskell in industry, Microsofts F# is a variant of Objective CAML - ie in the ML family - and C# and Python steal quite a few ideas from functional languages.\n\nFrom what I've seen, the best universities cover quite a few languages, making sure students get a good idea of the various different paradigms that go with them. Take a look at some of the Stanford University courses with freely downloadable materials (inc. lecture videos), for example...\n\n\nTo the extent that there's a real problem, though, the answer is simple enough...\n\n 1. The new languages mostly don't include that many new ideas.\n 2. The new ideas that do exist tend to be controversial - a lot of people familiar with older ways of working are resistant to them. The greater the paradigm shift, the bigger this problem gets - though circumstances sometimes force change, as the increasing need for concurrency is doing right now.\n 3. You don't understand the new ideas (like GC) unless you know how to build them yourself, or at least work without them.\n 4. A lot of the benefit in a lot of modern languages come from the \"batteries included\" standard libraries. But in some ways, it doesn't matter if a library is included as official standards or not - there are plenty of de-facto standard libraries out there, and once you've set up your build environment, it's a bit of a non-issue. This is particularly the case on Linux, where your repository manager can pretty much set up your build environment for you.\n\nIf you really believe that old languages never get discarded, give some thought to COBOL, Fortran and Algol. OK, they're still out there, but mostly for maintenance of old code - and the \"living death\" of old code in mostly forgotten languages is another reason to not jump on the latest programming language bandwagon. It takes a long time to become a true expert in any particular language. A jack of all programming languages is easily replaced, and isn't all that valuable.\n\nshare|improve this answer\nC++ and Java are extremely different. The need for a VM, enforced reference and GC semantics in Java, performance, etc. C++ has GC too - it's called Boehm. – DeadMG Oct 28 '11 at 16:23\n@DeadMG - I know about Boehm - it's not standard, and it's not compatible with reliable destructors so it violates a key rule of the C++ language - the circular references issue. As for Java being extremely different to C++ - the idioms are mainly different because the libraries are different, but the languages are mostly very similar give or take a tweak here and there. C++ templates are quite different to Java generics, Java has no member pointers but C++ needs a library to provide delegates yada yada, but considering how different two languages can be that's not a big deal. – Steve314 Oct 28 '11 at 17:39\nadd comment\n\nLanguages such as C/C++ are far closer to what the metal actually executes. This means they can teach you what the machine is really doing in tandem with teaching you the fundamentals of algorithms and data structures.\n\nDon't forget, even though C & C++ were \"created decades ago\", they are still fundamental to a lot of new development (see: Linux Kernel).\n\nAlso, don't forget that it actually takes time to create, review, polish and get approved new course content. The academic world is on general, more conservative in vetting changes than the non-academic world.\n\nHaving said this, while I studied Computer Science, I still had classes on .NET, Java, PHP and more.\n\nI direct your eyes to this study of Australian University introductory programming courses and the languages they used. You will note that the highest was Java at almost 44% and the second highest was VB at almost 19%.\n\nshare|improve this answer\nSo how much farther from the \"metal\" is C++ compared with C? – Dark Templar Oct 28 '11 at 2:39\n@Jacob - basically, as far as you need it to be. Libraries can do pretty much everything for you - due to templates and overloading, they're a fair bit more flexible than in C. Even memory (and other resources) can have automatic cleanup, though (normally) via smart pointers and RAII rather than the more popular garbage collection. But when you need to do low level stuff, you can do anything that you could in C. – Steve314 Oct 28 '11 at 4:21\n@Jacob: C++ isn't any farther from the metal than C. It's main difference is that it also reaches further upwards. – sbi Oct 28 '11 at 8:08\nadd comment\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Take the 2-minute tour ×\n\nIf I understand correctly, there was a huge birth of programming languages during the early decades of computing, but then things have stabilized.\n\nBasically, why are many universities and industries still using languages such as C and C++ (which were created decades ago), when there have been plenty of decades since then to improve upon and create more effective (and human-friendly) languages?\n\nshare|improve this question\n\nclosed as not constructive by Walter, Tom Squires, StuperUser, Jonas, Jeremy Oct 28 '11 at 18:28\n\n\nHey Jacob, your question is off topic for programmers.stackexchange, you should only ask practical, answerable questions based on actual problems, as its stated on the faq. As for your question, C and C++ keep evolving and are still extremely efficient if used correctly, although many seem to think that they are not so friendly. But human friendliness is not always what you look for in a programming language. – Yannis Rizos Oct 28 '11 at 2:18\nI think the premise is entirely incorrect. My impression is that there are a lot more new programming languages these days than at any time before. – Kyralessa Oct 28 '11 at 2:26\nI wish more universities would use C++ for teaching programming. – sbi Oct 28 '11 at 8:06\n@sbi - I wish more universities would use anything but C++ for teaching programming. Pen&paper if possible, but will settle for anything else. – ldigas Oct 28 '11 at 17:22\n@sbi - No, you see that is exactly what I don't agree with: \"in order to learn concepts, you need tools that support those concepts\". I believe concepts should be learned without the tools, so once \"one gets the tools\" one already knows the concepts. I'm of the school that it is terrible to go learning the concept of pointers by programming in C++. You should understand both pointers, memory management and ... before writing your first line of code. You don't go learning drawing by working in Autocad, you learn to draw and then you learn ACAD. One has not much to do with another. – ldigas Oct 28 '11 at 23:22\nshow 8 more comments\n\n8 Answers\n\nup vote 11 down vote accepted\n\nOne can extend this question to many things: why has the progress of chess slowed down? Why has the progress of aviation slowed down? Why was I learning math so fast in grade school but in grad school and thereafter I am publishing papers VERY SLOWLY. Why was Facebook progressing so fast as start-up, and now with 1200+ employees it does not really change all that much?\n\nThe progress has not stopped; it goes on, but, as George Carlin put it, \"Now you've got shit all over the world. The supply lines are getting harder to maintain\".\n\nThe success of languages depends partly on having very large API base and user base, and the success of airplanes depends on being able to provide a million of conveniences to the customer. And the work of an academic is more complicated than that of a student.\n\nSo, having to deal with backwards compatibility, having to test a new language feature in conjunction with all of the existing ones, having to worry about breaking libraries that worked with Python 2.7 (for instance), just concentrating on improving the API, dealing with new computer architectures that need to be supported, dealing with inter-operability with some other popular technologies, being invited to conferences as the father of a language, having to deal with lots of fan/hate mail, general time waste due to the inreased size of code base / organization, finding time to write a book about the language you have just made ... the responsibilities can suddenly pile up exponentially, but the time ticks away at the same speed. So, as complexity of things grows, the progress appears to slow down. The only way to deal with that is to throw more people at the problem, but not every organization can afford to do that. They would have to be benefiting financially from the product at least proportionally to its complexity. The for-profit model could work better in this case, although Sun & Oracle had cash but have/are not making revolutionary changes to Java now days.\n\nThese are just some things that I thought of ...\n\nshare|improve this answer\nDoes this also explain why there are fewer successful start-ups these days than decades ago? – Dark Templar Oct 28 '11 at 3:32\n@Jacob Hayden, maybe. One can find holes in my answer for sure. On one hand you have better tools; on the other customers are expecting more. When it comes to \"winner takes all situation\", such as Facebook, it has grown like crazy when social media was a new thing. It is becoming a commodity now. I predict the next big thing will be widespread genetic testing. Today people are willing to pay to find out how black / Asian / white they are. In the future they might settle for nothing less than a thorough 10-page report. First start-ups in new space can have good margins while having ok products. – Job Oct 28 '11 at 3:36\nThanks. Do you know what \"commodity\" is supposed to mean in this context? – Dark Templar Oct 28 '11 at 3:53\nI don't know about aviation, but the progress of chess has definitely not slowed down in recent decades; quite the contrary. – Konrad Morawski Oct 28 '11 at 8:54\n@Morawski, I think there is also a perceived component to this. Chess is a well-studied problem, and ever since Kasparov lost to the computer, the software and algorithms are getting better, but they do not generate buzz anymore. We take for granted that a computer can beat a person, just as we take for granted that one can get a modern cell phone that weighs less than a wallet and can call anyone for under $100 per month. This kind of stuff used to generate headlines; now it is does not because non-sensation is non-news. I should have written about the 'perception is reality' aspect of this. – Job Oct 28 '11 at 15:14\nadd comment\n\nSimply put, because there are no new successful languages in similar problem spaces. There are no languages which perform the same job as C++, but better enough to justify throwing away all of the experience, tools, existing codebases, etc. The only language that I can name that's even made a half-assed attempt is D, and it resembles C# more closely.\n\nUnlike C. C exists for two reasons: C++ is an impractical language to implement, so there are some (very small) problem spaces in which C++ won't fit, and because people are too ignorant to understand how incredibly poor it is. C is not suitable for teaching as a general-purpose language any more.\n\nOne of the reasons that there are fewer new languages is because successful languages are much larger and more complex to implement competitive toolchains.\n\nI would suggest that another reason is because people hardly know the languages they have. Especially a language like C++. You find many more people with incredible misconceptions than you do people who actually know how to use the language. People who were only taught \"C with Classes\". You're asking people who can't grasp that std::vector<T> is better than malloc and free to learn something? Good luck.\n\nshare|improve this answer\nadd comment\n\nYou've got the wrong impression. There's a lot of \"new\" languages out there. Google alone has released Dart and Go within little more than a year.\n\nAs for what is used at the university. If you take MIT, they went from LISP to Scala to Python within the last 2 decades or so. Some universities try hard to pick the right languages for their courses and some just use whatever they think is established and stick to that.\n\nshare|improve this answer\nadd comment\n\nWikipedia has a chronological list of important programming languages. It's not a complete list of every language ever invented, but I'm not sure that would help. Counting by hand, I see the following:\n\n • 1950's: 47 languages\n • 1960's: 46 languages\n • 1970's: 45 languages\n • 1980's: 50 languages\n • 1990's: 57 languages\n • 2000's: 36 languages\n\nSo there may have been a bit of a drop-off in \"important\" languages (although some of the languages invented in the last few years may yet prove important enough to be added later). Other than that, the rate of language creation has been remarkably flat.\n\nDo you have any alternative data that supports your assertion?\n\nAs for universities: Do you know that they also still teach Latin? Nobody speaks Latin anymore, but you'll find plenty of people who will tell you that it's important. Learning Latin helps you understand a lot about modern languages, and Latin is also the source of a great deal of great literature. C is similar (although people still \"speak\" it): it's the root of many more modern programming languages, and a great deal of useful code is written in C.\n\nshare|improve this answer\nPeople still use Latin. How would you name newly discovered species without it? – SK-logic Oct 28 '11 at 7:53\nCopy-paste, of course, just like you maintain COBOL ;) – MSalters Oct 28 '11 at 10:12\n@SK-logic Of course people still use Latin -- they still teach Latin because it's so useful. Virtually nobody speaks Latin, though -- modern languages are more convenient. – Caleb Oct 28 '11 at 12:25\nThese lists look pretty laughable, at least to me. Somebody may think all these languages are \"important\", but there's a pretty fair number there I've never heard of before. At least half of which I was previously aware, I'd still say were more \"obscure\" than \"important\". – Jerry Coffin Oct 28 '11 at 14:57\n@JerryCoffin, I feel the same way, but I figure that a community-reviewed list is a better source than the entirely subjective list of languages that any one person considers \"important.\" Perhaps I should use quotes instead of italics for that word above. Nevertheless, the languages on the list are at least important or well-known enough that somebody bothered to add them to the list, and I think that speaks to the premise that \"rate of language popularization has slowed down in recent decades.\" – Caleb Oct 28 '11 at 15:36\nadd comment\n\nWhy do Universities still teach old languages? Because those old languages got it right. Sure they have rough edges, C's strings could be better and C++ could certainly be decrufted a bit. But they work. And there's no denying that. What counts more than anything else isn't efficacy of language, it's environment.\n\nA million years ago when Mainframes were hot stuff and Dr. Knuth still had hair, we had the likes of Cobol, APL and FORTRAN (then spelled in all caps because we hadn't figured out that it wasn't cool to do that). Lisp was there too and so was Algol and so forth (FORTH wasn't quite there yet though, we're still in the 60s). However, most of these languages were fairly limited unless they were supported by the Government (FORTRAN) or Business (COBOL). Remember that the Internet was 4 computers in 1968 so getting information on a language would involve hearing about it from a colleague or some such and then going to a library, looking through a card catalog (a literal file system) and seeing if they had it. If you were lucky you could go to the source directly at whatever university or company was working on it. So knowledge ecosystems were limited.\n\nNow let's talk about hardware. I'm just old enough to have been raised on a 386. A 386 was a God Machine compared to what they had back then. We're talking about transistors the size of cockroaches and wired collections of magnetic cores (Giving us the term \"core dump\") that held kilobytes of memory. These computers cost hundreds of thousands or millions of dollars. With machines being so expensive and with relatively few users, you had a lot of variation. Word and bit collection sizes that would make most of us go insane with frustration if we had to deal with them today. Bytes have spoiled us all. So what did you program these things in? Assembly on Punched cards. Sometimes you might have a machine specific higher level language to work with like PL/I or similar.\n\nBy the end of the decade, integrated circuits had started to come down in price and had begun to displace large scale transistors while memory also got cheaper. You still ran into the issue of all kinds of architectures but things were progressing. Then some bored programmer wanted to play Space War. And it changed the world. While languages like COBOL and Lisp were portable, C was the first language that was portable at a low level. Combined with UNIX (Like a really nerdy Voltron), it battled Pascal for dominion.\n\nBy this time, computers at universities were fairly common and Hacker culture was beginning to form. This led to the formation of things like the GNU project and other such initiatives as well as jokes like INTERCAL but I'm getting ahead of myself. There was a parallel stream here, the screaming birth of Software Engineering. Object Oriented Programming was not new in the early 80s but by then computers were common enough that large scale code bases were becoming hard to manage due to greater demand so it spurred development there.\n\nOther developments ensued: the widespread use of home computers, the spread of the internet, the rise of windowing operating systems, the rise of the video game as a popular art form, and the rise of the parallel core machine amongst others.\n\nThe most important of these developments for the proliferation of languages are the introduction of the x86 and its various competitors that were folded into the IBM PC and its competitors and then the rise of the internet to facilitate language spread. So you've solved the problem of architecture portability and 90% of the market uses the same operating system. What does this do to the need for languages? It flattens them.\n\nWhy? Fundamentally language development and propagation are driven by need to solve problems. If you've got a language that works, you don't need to change it. Perhaps you add a framework here or there to streamline development or a library for task automation or the methodology of the month but ultimately you're crufting onto the language. This cruft though, it's important because there's where you get the environment from. You get culture that way and propagation follows the waves of culture.\n\nLanguages adapt and C family languages are incredibly adaptable due to being both widely deployed and extremely powerful. They also have something else, tremendous momentum. Say what you like about COBOL, there is a lot of wrapped COBOL running around out there; some of that is mutating into Java and other things but it will still being going along none the less. Some languages never really take off because they weren't designed to solve a widely scoped problem. C and C++ are widely scoped problem solvers. APL is not, at all. APL was designed to use as few keystrokes as possible in an era when that really mattered. It took it too far compared to other languages at the time and thus balance was not achieved, momentum was not gained and culture rampancy didn't take hold.\n\nRampancy is another key. The very best languages have something viral about them. They get inside your head and they stay there. That something viral might be monetary like a company saying \"This language is win\" and giving out T Shirts but I think it's more than that. Java may be a funky, crufty language but it solves a problem (several actually) and has a very powerful, meme-plague ridden ecosystem. So it keeps on propagating. And Propagation really is the key, your professors teach you old languages because they know there languages and these languages are still used out there. C is so close to the metal and so flexible that it is useful conceptually even if you never really use it. C++ is a thornier sell perhaps but I'd argue that as a much \"harder\" Object Oriented language than most, it serves as an excellent teaching tool.\n\nComputer Science departments are out to teach concepts, not memetic fads. So they pick the best tool that solves that problem. That's why MIT used Scheme for so long. The modern language pantheon offers a lot of choice and therein perhaps lies the final answer. The internet has made it very easy to design, build and market a language. It can provide an ecosystem for you but it also provide a host of competitors. Which do you choose if you are teaching CS? The new hotness that may burn out soon? Which one? Why or why not? Most \"fixer\" languages are aimed at programmer productivity, not academic learning. And even with industrial usage, would you rather spend 6 months retraining your guys to Newfangled? I'd suspect you'd rather spend 6 months using the C++ you already have a team for and make money. Because that's the problem set in business. How do I make money faster than my rivals? I use proven technology run by smart people. So I can hired more smart people versed in proven technologies to write more code that makes me more money so I can hire even more smart people and pretty soon I have an R&D budget so I can indulge in things like exotic paradigms in Newfangled. But I'm still running in an \"old\" ecosystem. Because it works.\n\nIt just works.\n\nshare|improve this answer\nadd comment\n\nYes, but now a days most them are using open source programming languages like PHP, Ruby on rails etc. Past days they are used Visual basic for stand lone application. Now days that was changed into VB.Net. And machine languages like c and C++ are replaced with Java for machine level coding..........!!\n\nshare|improve this answer\nadd comment\n\nA lot of \"modern\" programming languages aren't so different to C++.\n\nJava has classes and function calls. So does C++. Java has block structures such as if and for - which are copied from C.\n\nA major thing that Java has, but which C++ doesn't, is garbage collection. That's no so modern - it was invented for Lisp in the 1950s. The resource management system used in C++ (based on RAII and timely reliable destructors) is actually much newer than GC.\n\nFunctional languages like Haskell and the ML family are much more interestingly modern - and have mostly been considered academic because you're likely to find them in universities, but not so much in industry - though even that is changing. Even if you don't end up using Haskell in industry, Microsofts F# is a variant of Objective CAML - ie in the ML family - and C# and Python steal quite a few ideas from functional languages.\n\nFrom what I've seen, the best universities cover quite a few languages, making sure students get a good idea of the various different paradigms that go with them. Take a look at some of the Stanford University courses with freely downloadable materials (inc. lecture videos), for example...\n\n\nTo the extent that there's a real problem, though, the answer is simple enough...\n\n 1. The new languages mostly don't include that many new ideas.\n 2. The new ideas that do exist tend to be controversial - a lot of people familiar with older ways of working are resistant to them. The greater the paradigm shift, the bigger this problem gets - though circumstances sometimes force change, as the increasing need for concurrency is doing right now.\n 3. You don't understand the new ideas (like GC) unless you know how to build them yourself, or at least work without them.\n 4. A lot of the benefit in a lot of modern languages come from the \"batteries included\" standard libraries. But in some ways, it doesn't matter if a library is included as official standards or not - there are plenty of de-facto standard libraries out there, and once you've set up your build environment, it's a bit of a non-issue. This is particularly the case on Linux, where your repository manager can pretty much set up your build environment for you.\n\nIf you really believe that old languages never get discarded, give some thought to COBOL, Fortran and Algol. OK, they're still out there, but mostly for maintenance of old code - and the \"living death\" of old code in mostly forgotten languages is another reason to not jump on the latest programming language bandwagon. It takes a long time to become a true expert in any particular language. A jack of all programming languages is easily replaced, and isn't all that valuable.\n\nshare|improve this answer\nC++ and Java are extremely different. The need for a VM, enforced reference and GC semantics in Java, performance, etc. C++ has GC too - it's called Boehm. – DeadMG Oct 28 '11 at 16:23\n@DeadMG - I know about Boehm - it's not standard, and it's not compatible with reliable destructors so it violates a key rule of the C++ language - the circular references issue. As for Java being extremely different to C++ - the idioms are mainly different because the libraries are different, but the languages are mostly very similar give or take a tweak here and there. C++ templates are quite different to Java generics, Java has no member pointers but C++ needs a library to provide delegates yada yada, but considering how different two languages can be that's not a big deal. – Steve314 Oct 28 '11 at 17:39\nadd comment\n\nLanguages such as C/C++ are far closer to what the metal actually executes. This means they can teach you what the machine is really doing in tandem with teaching you the fundamentals of algorithms and data structures.\n\nDon't forget, even though C & C++ were \"created decades ago\", they are still fundamental to a lot of new development (see: Linux Kernel).\n\nAlso, don't forget that it actually takes time to create, review, polish and get approved new course content. The academic world is on general, more conservative in vetting changes than the non-academic world.\n\nHaving said this, while I studied Computer Science, I still had classes on .NET, Java, PHP and more.\n\nI direct your eyes to this study of Australian University introductory programming courses and the languages they used. You will note that the highest was Java at almost 44% and the second highest was VB at almost 19%.\n\nshare|improve this answer\nSo how much farther from the \"metal\" is C++ compared with C? – Dark Templar Oct 28 '11 at 2:39\n@Jacob - basically, as far as you need it to be. Libraries can do pretty much everything for you - due to templates and overloading, they're a fair bit more flexible than in C. Even memory (and other resources) can have automatic cleanup, though (normally) via smart pointers and RAII rather than the more popular garbage collection. But when you need to do low level stuff, you can do anything that you could in C. – Steve314 Oct 28 '11 at 4:21\n@Jacob: C++ isn't any farther from the metal than C. It's main difference is that it also reaches further upwards. – sbi Oct 28 '11 at 8:08\nadd comment\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "categorical",
"index": 73,
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"text": "Take the 2-minute tour ×\n\nIf I understand correctly, there was a huge birth of programming languages during the early decades of computing, but then things have stabilized.\n\nBasically, why are many universities and industries still using languages such as C and C++ (which were created decades ago), when there have been plenty of decades since then to improve upon and create more effective (and human-friendly) languages?\n\nshare|improve this question\n\nclosed as not constructive by Walter, Tom Squires, StuperUser, Jonas, Jeremy Oct 28 '11 at 18:28\n\n\nHey Jacob, your question is off topic for programmers.stackexchange, you should only ask practical, answerable questions based on actual problems, as its stated on the faq. As for your question, C and C++ keep evolving and are still extremely efficient if used correctly, although many seem to think that they are not so friendly. But human friendliness is not always what you look for in a programming language. – Yannis Rizos Oct 28 '11 at 2:18\nI think the premise is entirely incorrect. My impression is that there are a lot more new programming languages these days than at any time before. – Kyralessa Oct 28 '11 at 2:26\nI wish more universities would use C++ for teaching programming. – sbi Oct 28 '11 at 8:06\n@sbi - I wish more universities would use anything but C++ for teaching programming. Pen&paper if possible, but will settle for anything else. – ldigas Oct 28 '11 at 17:22\n@sbi - No, you see that is exactly what I don't agree with: \"in order to learn concepts, you need tools that support those concepts\". I believe concepts should be learned without the tools, so once \"one gets the tools\" one already knows the concepts. I'm of the school that it is terrible to go learning the concept of pointers by programming in C++. You should understand both pointers, memory management and ... before writing your first line of code. You don't go learning drawing by working in Autocad, you learn to draw and then you learn ACAD. One has not much to do with another. – ldigas Oct 28 '11 at 23:22\nshow 8 more comments\n\n8 Answers\n\nup vote 11 down vote accepted\n\nOne can extend this question to many things: why has the progress of chess slowed down? Why has the progress of aviation slowed down? Why was I learning math so fast in grade school but in grad school and thereafter I am publishing papers VERY SLOWLY. Why was Facebook progressing so fast as start-up, and now with 1200+ employees it does not really change all that much?\n\nThe progress has not stopped; it goes on, but, as George Carlin put it, \"Now you've got shit all over the world. The supply lines are getting harder to maintain\".\n\nThe success of languages depends partly on having very large API base and user base, and the success of airplanes depends on being able to provide a million of conveniences to the customer. And the work of an academic is more complicated than that of a student.\n\nSo, having to deal with backwards compatibility, having to test a new language feature in conjunction with all of the existing ones, having to worry about breaking libraries that worked with Python 2.7 (for instance), just concentrating on improving the API, dealing with new computer architectures that need to be supported, dealing with inter-operability with some other popular technologies, being invited to conferences as the father of a language, having to deal with lots of fan/hate mail, general time waste due to the inreased size of code base / organization, finding time to write a book about the language you have just made ... the responsibilities can suddenly pile up exponentially, but the time ticks away at the same speed. So, as complexity of things grows, the progress appears to slow down. The only way to deal with that is to throw more people at the problem, but not every organization can afford to do that. They would have to be benefiting financially from the product at least proportionally to its complexity. The for-profit model could work better in this case, although Sun & Oracle had cash but have/are not making revolutionary changes to Java now days.\n\nThese are just some things that I thought of ...\n\nshare|improve this answer\nDoes this also explain why there are fewer successful start-ups these days than decades ago? – Dark Templar Oct 28 '11 at 3:32\n@Jacob Hayden, maybe. One can find holes in my answer for sure. On one hand you have better tools; on the other customers are expecting more. When it comes to \"winner takes all situation\", such as Facebook, it has grown like crazy when social media was a new thing. It is becoming a commodity now. I predict the next big thing will be widespread genetic testing. Today people are willing to pay to find out how black / Asian / white they are. In the future they might settle for nothing less than a thorough 10-page report. First start-ups in new space can have good margins while having ok products. – Job Oct 28 '11 at 3:36\nThanks. Do you know what \"commodity\" is supposed to mean in this context? – Dark Templar Oct 28 '11 at 3:53\nI don't know about aviation, but the progress of chess has definitely not slowed down in recent decades; quite the contrary. – Konrad Morawski Oct 28 '11 at 8:54\n@Morawski, I think there is also a perceived component to this. Chess is a well-studied problem, and ever since Kasparov lost to the computer, the software and algorithms are getting better, but they do not generate buzz anymore. We take for granted that a computer can beat a person, just as we take for granted that one can get a modern cell phone that weighs less than a wallet and can call anyone for under $100 per month. This kind of stuff used to generate headlines; now it is does not because non-sensation is non-news. I should have written about the 'perception is reality' aspect of this. – Job Oct 28 '11 at 15:14\nadd comment\n\nSimply put, because there are no new successful languages in similar problem spaces. There are no languages which perform the same job as C++, but better enough to justify throwing away all of the experience, tools, existing codebases, etc. The only language that I can name that's even made a half-assed attempt is D, and it resembles C# more closely.\n\nUnlike C. C exists for two reasons: C++ is an impractical language to implement, so there are some (very small) problem spaces in which C++ won't fit, and because people are too ignorant to understand how incredibly poor it is. C is not suitable for teaching as a general-purpose language any more.\n\nOne of the reasons that there are fewer new languages is because successful languages are much larger and more complex to implement competitive toolchains.\n\nI would suggest that another reason is because people hardly know the languages they have. Especially a language like C++. You find many more people with incredible misconceptions than you do people who actually know how to use the language. People who were only taught \"C with Classes\". You're asking people who can't grasp that std::vector<T> is better than malloc and free to learn something? Good luck.\n\nshare|improve this answer\nadd comment\n\nYou've got the wrong impression. There's a lot of \"new\" languages out there. Google alone has released Dart and Go within little more than a year.\n\nAs for what is used at the university. If you take MIT, they went from LISP to Scala to Python within the last 2 decades or so. Some universities try hard to pick the right languages for their courses and some just use whatever they think is established and stick to that.\n\nshare|improve this answer\nadd comment\n\nWikipedia has a chronological list of important programming languages. It's not a complete list of every language ever invented, but I'm not sure that would help. Counting by hand, I see the following:\n\n • 1950's: 47 languages\n • 1960's: 46 languages\n • 1970's: 45 languages\n • 1980's: 50 languages\n • 1990's: 57 languages\n • 2000's: 36 languages\n\nSo there may have been a bit of a drop-off in \"important\" languages (although some of the languages invented in the last few years may yet prove important enough to be added later). Other than that, the rate of language creation has been remarkably flat.\n\nDo you have any alternative data that supports your assertion?\n\nAs for universities: Do you know that they also still teach Latin? Nobody speaks Latin anymore, but you'll find plenty of people who will tell you that it's important. Learning Latin helps you understand a lot about modern languages, and Latin is also the source of a great deal of great literature. C is similar (although people still \"speak\" it): it's the root of many more modern programming languages, and a great deal of useful code is written in C.\n\nshare|improve this answer\nPeople still use Latin. How would you name newly discovered species without it? – SK-logic Oct 28 '11 at 7:53\nCopy-paste, of course, just like you maintain COBOL ;) – MSalters Oct 28 '11 at 10:12\n@SK-logic Of course people still use Latin -- they still teach Latin because it's so useful. Virtually nobody speaks Latin, though -- modern languages are more convenient. – Caleb Oct 28 '11 at 12:25\nThese lists look pretty laughable, at least to me. Somebody may think all these languages are \"important\", but there's a pretty fair number there I've never heard of before. At least half of which I was previously aware, I'd still say were more \"obscure\" than \"important\". – Jerry Coffin Oct 28 '11 at 14:57\n@JerryCoffin, I feel the same way, but I figure that a community-reviewed list is a better source than the entirely subjective list of languages that any one person considers \"important.\" Perhaps I should use quotes instead of italics for that word above. Nevertheless, the languages on the list are at least important or well-known enough that somebody bothered to add them to the list, and I think that speaks to the premise that \"rate of language popularization has slowed down in recent decades.\" – Caleb Oct 28 '11 at 15:36\nadd comment\n\nWhy do Universities still teach old languages? Because those old languages got it right. Sure they have rough edges, C's strings could be better and C++ could certainly be decrufted a bit. But they work. And there's no denying that. What counts more than anything else isn't efficacy of language, it's environment.\n\nA million years ago when Mainframes were hot stuff and Dr. Knuth still had hair, we had the likes of Cobol, APL and FORTRAN (then spelled in all caps because we hadn't figured out that it wasn't cool to do that). Lisp was there too and so was Algol and so forth (FORTH wasn't quite there yet though, we're still in the 60s). However, most of these languages were fairly limited unless they were supported by the Government (FORTRAN) or Business (COBOL). Remember that the Internet was 4 computers in 1968 so getting information on a language would involve hearing about it from a colleague or some such and then going to a library, looking through a card catalog (a literal file system) and seeing if they had it. If you were lucky you could go to the source directly at whatever university or company was working on it. So knowledge ecosystems were limited.\n\nNow let's talk about hardware. I'm just old enough to have been raised on a 386. A 386 was a God Machine compared to what they had back then. We're talking about transistors the size of cockroaches and wired collections of magnetic cores (Giving us the term \"core dump\") that held kilobytes of memory. These computers cost hundreds of thousands or millions of dollars. With machines being so expensive and with relatively few users, you had a lot of variation. Word and bit collection sizes that would make most of us go insane with frustration if we had to deal with them today. Bytes have spoiled us all. So what did you program these things in? Assembly on Punched cards. Sometimes you might have a machine specific higher level language to work with like PL/I or similar.\n\nBy the end of the decade, integrated circuits had started to come down in price and had begun to displace large scale transistors while memory also got cheaper. You still ran into the issue of all kinds of architectures but things were progressing. Then some bored programmer wanted to play Space War. And it changed the world. While languages like COBOL and Lisp were portable, C was the first language that was portable at a low level. Combined with UNIX (Like a really nerdy Voltron), it battled Pascal for dominion.\n\nBy this time, computers at universities were fairly common and Hacker culture was beginning to form. This led to the formation of things like the GNU project and other such initiatives as well as jokes like INTERCAL but I'm getting ahead of myself. There was a parallel stream here, the screaming birth of Software Engineering. Object Oriented Programming was not new in the early 80s but by then computers were common enough that large scale code bases were becoming hard to manage due to greater demand so it spurred development there.\n\nOther developments ensued: the widespread use of home computers, the spread of the internet, the rise of windowing operating systems, the rise of the video game as a popular art form, and the rise of the parallel core machine amongst others.\n\nThe most important of these developments for the proliferation of languages are the introduction of the x86 and its various competitors that were folded into the IBM PC and its competitors and then the rise of the internet to facilitate language spread. So you've solved the problem of architecture portability and 90% of the market uses the same operating system. What does this do to the need for languages? It flattens them.\n\nWhy? Fundamentally language development and propagation are driven by need to solve problems. If you've got a language that works, you don't need to change it. Perhaps you add a framework here or there to streamline development or a library for task automation or the methodology of the month but ultimately you're crufting onto the language. This cruft though, it's important because there's where you get the environment from. You get culture that way and propagation follows the waves of culture.\n\nLanguages adapt and C family languages are incredibly adaptable due to being both widely deployed and extremely powerful. They also have something else, tremendous momentum. Say what you like about COBOL, there is a lot of wrapped COBOL running around out there; some of that is mutating into Java and other things but it will still being going along none the less. Some languages never really take off because they weren't designed to solve a widely scoped problem. C and C++ are widely scoped problem solvers. APL is not, at all. APL was designed to use as few keystrokes as possible in an era when that really mattered. It took it too far compared to other languages at the time and thus balance was not achieved, momentum was not gained and culture rampancy didn't take hold.\n\nRampancy is another key. The very best languages have something viral about them. They get inside your head and they stay there. That something viral might be monetary like a company saying \"This language is win\" and giving out T Shirts but I think it's more than that. Java may be a funky, crufty language but it solves a problem (several actually) and has a very powerful, meme-plague ridden ecosystem. So it keeps on propagating. And Propagation really is the key, your professors teach you old languages because they know there languages and these languages are still used out there. C is so close to the metal and so flexible that it is useful conceptually even if you never really use it. C++ is a thornier sell perhaps but I'd argue that as a much \"harder\" Object Oriented language than most, it serves as an excellent teaching tool.\n\nComputer Science departments are out to teach concepts, not memetic fads. So they pick the best tool that solves that problem. That's why MIT used Scheme for so long. The modern language pantheon offers a lot of choice and therein perhaps lies the final answer. The internet has made it very easy to design, build and market a language. It can provide an ecosystem for you but it also provide a host of competitors. Which do you choose if you are teaching CS? The new hotness that may burn out soon? Which one? Why or why not? Most \"fixer\" languages are aimed at programmer productivity, not academic learning. And even with industrial usage, would you rather spend 6 months retraining your guys to Newfangled? I'd suspect you'd rather spend 6 months using the C++ you already have a team for and make money. Because that's the problem set in business. How do I make money faster than my rivals? I use proven technology run by smart people. So I can hired more smart people versed in proven technologies to write more code that makes me more money so I can hire even more smart people and pretty soon I have an R&D budget so I can indulge in things like exotic paradigms in Newfangled. But I'm still running in an \"old\" ecosystem. Because it works.\n\nIt just works.\n\nshare|improve this answer\nadd comment\n\nYes, but now a days most them are using open source programming languages like PHP, Ruby on rails etc. Past days they are used Visual basic for stand lone application. Now days that was changed into VB.Net. And machine languages like c and C++ are replaced with Java for machine level coding..........!!\n\nshare|improve this answer\nadd comment\n\nA lot of \"modern\" programming languages aren't so different to C++.\n\nJava has classes and function calls. So does C++. Java has block structures such as if and for - which are copied from C.\n\nA major thing that Java has, but which C++ doesn't, is garbage collection. That's no so modern - it was invented for Lisp in the 1950s. The resource management system used in C++ (based on RAII and timely reliable destructors) is actually much newer than GC.\n\nFunctional languages like Haskell and the ML family are much more interestingly modern - and have mostly been considered academic because you're likely to find them in universities, but not so much in industry - though even that is changing. Even if you don't end up using Haskell in industry, Microsofts F# is a variant of Objective CAML - ie in the ML family - and C# and Python steal quite a few ideas from functional languages.\n\nFrom what I've seen, the best universities cover quite a few languages, making sure students get a good idea of the various different paradigms that go with them. Take a look at some of the Stanford University courses with freely downloadable materials (inc. lecture videos), for example...\n\n\nTo the extent that there's a real problem, though, the answer is simple enough...\n\n 1. The new languages mostly don't include that many new ideas.\n 2. The new ideas that do exist tend to be controversial - a lot of people familiar with older ways of working are resistant to them. The greater the paradigm shift, the bigger this problem gets - though circumstances sometimes force change, as the increasing need for concurrency is doing right now.\n 3. You don't understand the new ideas (like GC) unless you know how to build them yourself, or at least work without them.\n 4. A lot of the benefit in a lot of modern languages come from the \"batteries included\" standard libraries. But in some ways, it doesn't matter if a library is included as official standards or not - there are plenty of de-facto standard libraries out there, and once you've set up your build environment, it's a bit of a non-issue. This is particularly the case on Linux, where your repository manager can pretty much set up your build environment for you.\n\nIf you really believe that old languages never get discarded, give some thought to COBOL, Fortran and Algol. OK, they're still out there, but mostly for maintenance of old code - and the \"living death\" of old code in mostly forgotten languages is another reason to not jump on the latest programming language bandwagon. It takes a long time to become a true expert in any particular language. A jack of all programming languages is easily replaced, and isn't all that valuable.\n\nshare|improve this answer\nC++ and Java are extremely different. The need for a VM, enforced reference and GC semantics in Java, performance, etc. C++ has GC too - it's called Boehm. – DeadMG Oct 28 '11 at 16:23\n@DeadMG - I know about Boehm - it's not standard, and it's not compatible with reliable destructors so it violates a key rule of the C++ language - the circular references issue. As for Java being extremely different to C++ - the idioms are mainly different because the libraries are different, but the languages are mostly very similar give or take a tweak here and there. C++ templates are quite different to Java generics, Java has no member pointers but C++ needs a library to provide delegates yada yada, but considering how different two languages can be that's not a big deal. – Steve314 Oct 28 '11 at 17:39\nadd comment\n\nLanguages such as C/C++ are far closer to what the metal actually executes. This means they can teach you what the machine is really doing in tandem with teaching you the fundamentals of algorithms and data structures.\n\nDon't forget, even though C & C++ were \"created decades ago\", they are still fundamental to a lot of new development (see: Linux Kernel).\n\nAlso, don't forget that it actually takes time to create, review, polish and get approved new course content. The academic world is on general, more conservative in vetting changes than the non-academic world.\n\nHaving said this, while I studied Computer Science, I still had classes on .NET, Java, PHP and more.\n\nI direct your eyes to this study of Australian University introductory programming courses and the languages they used. You will note that the highest was Java at almost 44% and the second highest was VB at almost 19%.\n\nshare|improve this answer\nSo how much farther from the \"metal\" is C++ compared with C? – Dark Templar Oct 28 '11 at 2:39\n@Jacob - basically, as far as you need it to be. Libraries can do pretty much everything for you - due to templates and overloading, they're a fair bit more flexible than in C. Even memory (and other resources) can have automatic cleanup, though (normally) via smart pointers and RAII rather than the more popular garbage collection. But when you need to do low level stuff, you can do anything that you could in C. – Steve314 Oct 28 '11 at 4:21\n@Jacob: C++ isn't any farther from the metal than C. It's main difference is that it also reaches further upwards. – sbi Oct 28 '11 at 8:08\nadd comment\n"
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "web analytics\n\nDrilling Square Holes with a Reuleaux Triangle\n\nWritten by mike on October 3rd, 2011\n\nCircular motion can be converted into a four-sided square using the Reuleaux triangle. The process relies on the property of Reuleaux triangle’s diameter being consistent across all points, and when rolled and rotated simultaneously, tracing a square area with slightly rounded corners. By slightly altering the shape of the triangle to mimic a drill bit (for cutting and extracting material), a drill with a specialized rotating chuck can create perfect, four-sided holes. These drills were pioneered by Harry Watts, and can still be found today, albeit not at your typical home improvement store.\n\nThe following is a bit of literature that helps explain and illustrate the concepts behind the Reuleaux triangle. Please check out more of Prof. Smith’s work here.\n\nDrilling Square Holes\nby Scott Smith\npublished in The Mathematics Teacher, October 1993 (Volume 86 Number 7)\nA bit that drills square holes … it defies common sense. How can a revolving edge cut anything but a circular hole? Not only do such bits exist (as well as bits for pentagonal, hexagonal and octagonal holes), but they derive their shape from a simple geometric construction known as a Reuleaux triangle (after Franz Reuleaux, 1829-1905).\n\nTo construct a Reuleaux triangle, start with an equilateral triangle of side s (Figure 1). With a radius equal to s and the center at one of the vertices, draw an arc connecting the other two vertices. Similarly, draw arcs connecting the endpoints of the other two sides. The three arcs form the Reuleaux triangle. One of its properties is that of constant width, meaning the figure could be rotated completely around between two parallel lines separated by distance s.\nIt was with this property of constant width that the Reuleaux triangle was introduced in a sidebar of our geometry text (Moise and Downs, Teachers’ Edition, p. 555). “This figure has constant width,” I lectured, “just like a circle.” Without thinking, I volunteered, “Imagine it as wheels on a cart.” “What sort of cart?” “Why, a math cart, to carry my board compass and protractor,” I replied, digging myself in deeper. This was the first of several impulsive misstatements I made about the Reuleaux triangle, only to admit after a little reflection that it wasn’t so. Not in twenty years of teaching had my intuition failed me so completely.\n\nThe constant width property can be used to transport loads, but not by using Reuleaux triangles as wheels. If several poles had congruent Reuleaux triangles as cross sections, bulky items could ride atop them (Figure 2). Movement would occur as poles were transferred from back to front, providing a moveable base of constant height.\n\nBut the Reuleaux triangle cannot be a wheel. The only conceivable point for the axle, at the triangle’s centroid, is not the same distance from the Reuleaux triangle’s “sides” (Figure 3). If the sides of the equilateral triangle are s, then\n\n 2 s sqrt(3)\n(1) AP = - - sqrt(3) = ------- s » 0.577s,\n 3 2 3\n\n\n sqrt(3) sqrt(3)\n PB = s - ------- s = s(1 - -------) » 0.423s.\n 3 3\n\nEven if four Reuleaux triangle wheels were synchronized, the load would rise and fall continuously — you’d need Dramamine to ride this cart!\n\n“And since it has constant width, it would just fit inside a square whose sides are that width,” I continued, trying to regain their attention. I carefully drew a square circumscribing the Reuleaux triangle (Figure 4). The triangle is normally tangent to two sides of the square with two vertices touching the square directly opposite the points of tangency (why?), as in Figure 4a. The exception is Figure 4b, where the Reuleaux triangle has one point of tangency and all three vertices on the square (one directly opposite that point of tangency).\n“If the Reuleaux triangle just fits inside the square, no matter what position it’s in, couldn’t it rotate around the inside of the square?” They needed convincing — a model would have to be built. “But if it did rotate around the inside, doesn’t that mean that a sharp Reuleaux triangle could carve out a square as it rotated?” I had them. “Drill a square hole?”, one countered. “No way!”\n\nThat night I cut a four inch Reuleaux triangle from a manila folder to take to class the next day. With a lot of effort, I was able to show the triangle rotate around the inside of a four inch square. “And if this was metal at the end of a rotating shaft, it would cut out a square”, I continued, racking up two more falsehoods. Firstly, it was implied that the center of the Reuleaux triangle would coincide with the center of a drill’s shaft; it cannot. And secondly, the corners of the holes are not right angles, but slightly rounded.\n\nTrying to show the triangle should be centered at the end of a rotating shaft, I stuck a pen through the triangle’s center which, while a student manually rotated the triangle within the square, traced the center’s path on paper beneath (Figure 5). “It’s definitely not a single point,” I had to admit, holding up the traced curve, “but it sure looks like a circle!” Falsehood #4.\n\nJust what is the path of the centroid of a Reuleaux triangle boring a square hole? Assume the square and the equilateral triangle have sides of length 1. Center the square about the origin and position the Reuleaux triangle so vertex A is at (-1/2,0), as in Figure 6a. Using (1), the triangle’s centroid will be P (-1/2+sqrt(3)/3,0). Now imagine rotating the triangle clockwise through the position in Figure 6b, and ending up in Figure 6c, where the centroid is P”(0,-1/2+sqrt(3)/3). The path from P to P” lies in quadrant I. Let a be angle MA’B', ß the angle formed by A’P’ and a horizontal line through A’, and c the y-coordinate of point A’. We are interested in the coordinates of P’. Note that cosa=1/2+c and that ß=270°+a+30°=300°+a. Also note that during this rotation, a goes from 60° to 30°. Because A’P'=sqrt(3)/3, if we measure from the coordinates of A’(-1/2,c), the x and y coordinates of P’ can be found.\n\n -1 sqrt(3) -3 + sqrt(3) cos(a) + 3 sin(a)\n(2) x = - + ------- cos(300°+a) = ------------------------------,\n 2 3 6\n\n\n(3) y = c + ------- sin(300°+a)\n\n= (cosa – 1/2) + ——- sin(300°+a)\n\n-3 + 3 cos(a) + sqrt(3) sin(a)\n= ——————————\n\nas a goes from 60° to 30°. Finding the path of the triangle’s center in the other three quadrants is similar in procedure and produces equations symmetric to the origin and both axes.\n\n 3 - sqrt(3) cos(a) - 3 sin(a)\n\ny = ——————————\n\n3 – sqrt(3) cos(a) – 3 sin(a)\nQuadrant III: x = —————————–\n\n3 – 3 cos(a) – sqrt(3) sin(a)\ny = —————————–\n\nQuadrant IV: x = ——————————\n\ny = —————————–\n\nBut these equations do not describe a circle. In equations (2) and (3), when a=30°, P is on the x-axis at approximately (0.07735,0). But when a=45°,\n\n -6 + sqrt(6) + 3sqrt(2)\n\nwhich makes the distance from P’ to the origin about 0.08168. This non-circularity is also shown by graphing the four parametric equations above with a circle whose radius is slightly smaller or larger. In Figure 7, the circle is the outer curve. Notice that the centroid’s path is farther from the circle at the axes than mid-quadrant.\n\nSo the Reuleaux triangle’s centroid does not follow a circular path. How then is the Reuleaux bit contained within the square outline it’s to cut? Harry Watts designed a drill in 1914 with a patented “full floating chuck” to accommodate his irregular bits. Bits for square, pentagonal, hexagonal and octagonal holes are still sold by the Watts Brothers Tool Works in Wilmerding PA. The actual drill bit for the square is a Reuleaux triangle made concave in three spots to allow for unobstructed corner-cutting and the discharge of shavings (Figure 8).\n\nEven the modified bit leaves slightly rounded corners. How rounded? Assume the starting position in Figure 9a, in which the Reuleaux triangle is just tangent at point C. As the triangle rotates counterclockwise, C leaves that edge of the square temporarily (labeled C’ in Figure 9b) only to rejoin another at position C” in Figure 9c. In Figure 9b, let a be angle MA’B', ß be the angle formed by A’C’ and the horizontal line through A’, and c the y-coordinate of A’. Then ß = a+60°-90° = a-30° and cosa = 1/2+c. To generate the corner by C, a starts at 30° in Figure 9a and ends up at 60° in Figure 9c. Using A’C’ = 1 and measuring from the coordinates of A’, the coordinates of C’ are described by\n\n -1 -1 + sqrt(3) cos(a) + sin(a)\n x = - + 1 cos(a-30°) = ----------------------------,\n 2 2\n\n\n y = c + 1 sin(a-30°) = (cosa - -) + sin(a-30°)\n\n-1 + cos(a) + sqrt(3) sin(a)\n= —————————-.\n\nThe equations for the other three corners are similar, and when graphed with the rest of the square yield Figure 10.\nNot only does the Reuleaux triangle have practical and interesting applications, and is easy to describe geometrically, but it generates a lot of discussion due to its nonintuitive properties. With this background, you can avoid the blunders I made. Further explorations into the topic might include other figures of constant width (see Gardner and Rademacher/Toeplitz); further identifying the curve of the Reuleaux triangle’s center as it cuts a square; and the shapes of bits for pentagonal, hexagonal and octagonal holes.\n\n\n 1. Gardner, Martin. “Mathematical Games”. Scientific American 208:2 (February 1963): 148-156.\n 2. How to Drill Square, Hexagon, Octagon, Pentagon Holes. Wilmerding, Penn: Watts Brothers Tool Works, 1966.\n 3. Moise, Edwin and Floyd Downs Jr. Geometry. Reading, Mass: Addison-Wesley, 1982.\n 4. Rademacher, Hans and Otto Toeplitz. The Enjoyment of Mathematics. New York: Dover Publications, 1990.\n 5. Smart, James R. “Problem Solving in Geometry – a Sequence of Reuleaux Triangles.” Mathematics Teacher 79 (January 1986): 11-14.\n\nRelated posts:\n\n • http://twitter.com/leetoma/status/129312691834728449 Lee Toma\n\n Fascinating tech: How to drill a square hole! videos http://t.co/aRHiZ5mR\n\nAlso on DOIT:"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "web analytics\n\nDrilling Square Holes with a Reuleaux Triangle\n\nWritten by mike on October 3rd, 2011\n\nCircular motion can be converted into a four-sided square using the Reuleaux triangle. The process relies on the property of Reuleaux triangle’s diameter being consistent across all points, and when rolled and rotated simultaneously, tracing a square area with slightly rounded corners. By slightly altering the shape of the triangle to mimic a drill bit (for cutting and extracting material), a drill with a specialized rotating chuck can create perfect, four-sided holes. These drills were pioneered by Harry Watts, and can still be found today, albeit not at your typical home improvement store.\n\nThe following is a bit of literature that helps explain and illustrate the concepts behind the Reuleaux triangle. Please check out more of Prof. Smith’s work here.\n\nDrilling Square Holes\nby Scott Smith\npublished in The Mathematics Teacher, October 1993 (Volume 86 Number 7)\nA bit that drills square holes … it defies common sense. How can a revolving edge cut anything but a circular hole? Not only do such bits exist (as well as bits for pentagonal, hexagonal and octagonal holes), but they derive their shape from a simple geometric construction known as a Reuleaux triangle (after Franz Reuleaux, 1829-1905).\n\nTo construct a Reuleaux triangle, start with an equilateral triangle of side s (Figure 1). With a radius equal to s and the center at one of the vertices, draw an arc connecting the other two vertices. Similarly, draw arcs connecting the endpoints of the other two sides. The three arcs form the Reuleaux triangle. One of its properties is that of constant width, meaning the figure could be rotated completely around between two parallel lines separated by distance s.\nIt was with this property of constant width that the Reuleaux triangle was introduced in a sidebar of our geometry text (Moise and Downs, Teachers’ Edition, p. 555). “This figure has constant width,” I lectured, “just like a circle.” Without thinking, I volunteered, “Imagine it as wheels on a cart.” “What sort of cart?” “Why, a math cart, to carry my board compass and protractor,” I replied, digging myself in deeper. This was the first of several impulsive misstatements I made about the Reuleaux triangle, only to admit after a little reflection that it wasn’t so. Not in twenty years of teaching had my intuition failed me so completely.\n\nThe constant width property can be used to transport loads, but not by using Reuleaux triangles as wheels. If several poles had congruent Reuleaux triangles as cross sections, bulky items could ride atop them (Figure 2). Movement would occur as poles were transferred from back to front, providing a moveable base of constant height.\n\nBut the Reuleaux triangle cannot be a wheel. The only conceivable point for the axle, at the triangle’s centroid, is not the same distance from the Reuleaux triangle’s “sides” (Figure 3). If the sides of the equilateral triangle are s, then\n\n 2 s sqrt(3)\n(1) AP = - - sqrt(3) = ------- s » 0.577s,\n 3 2 3\n\n\n sqrt(3) sqrt(3)\n PB = s - ------- s = s(1 - -------) » 0.423s.\n 3 3\n\nEven if four Reuleaux triangle wheels were synchronized, the load would rise and fall continuously — you’d need Dramamine to ride this cart!\n\n“And since it has constant width, it would just fit inside a square whose sides are that width,” I continued, trying to regain their attention. I carefully drew a square circumscribing the Reuleaux triangle (Figure 4). The triangle is normally tangent to two sides of the square with two vertices touching the square directly opposite the points of tangency (why?), as in Figure 4a. The exception is Figure 4b, where the Reuleaux triangle has one point of tangency and all three vertices on the square (one directly opposite that point of tangency).\n“If the Reuleaux triangle just fits inside the square, no matter what position it’s in, couldn’t it rotate around the inside of the square?” They needed convincing — a model would have to be built. “But if it did rotate around the inside, doesn’t that mean that a sharp Reuleaux triangle could carve out a square as it rotated?” I had them. “Drill a square hole?”, one countered. “No way!”\n\nThat night I cut a four inch Reuleaux triangle from a manila folder to take to class the next day. With a lot of effort, I was able to show the triangle rotate around the inside of a four inch square. “And if this was metal at the end of a rotating shaft, it would cut out a square”, I continued, racking up two more falsehoods. Firstly, it was implied that the center of the Reuleaux triangle would coincide with the center of a drill’s shaft; it cannot. And secondly, the corners of the holes are not right angles, but slightly rounded.\n\nTrying to show the triangle should be centered at the end of a rotating shaft, I stuck a pen through the triangle’s center which, while a student manually rotated the triangle within the square, traced the center’s path on paper beneath (Figure 5). “It’s definitely not a single point,” I had to admit, holding up the traced curve, “but it sure looks like a circle!” Falsehood #4.\n\nJust what is the path of the centroid of a Reuleaux triangle boring a square hole? Assume the square and the equilateral triangle have sides of length 1. Center the square about the origin and position the Reuleaux triangle so vertex A is at (-1/2,0), as in Figure 6a. Using (1), the triangle’s centroid will be P (-1/2+sqrt(3)/3,0). Now imagine rotating the triangle clockwise through the position in Figure 6b, and ending up in Figure 6c, where the centroid is P”(0,-1/2+sqrt(3)/3). The path from P to P” lies in quadrant I. Let a be angle MA’B', ß the angle formed by A’P’ and a horizontal line through A’, and c the y-coordinate of point A’. We are interested in the coordinates of P’. Note that cosa=1/2+c and that ß=270°+a+30°=300°+a. Also note that during this rotation, a goes from 60° to 30°. Because A’P'=sqrt(3)/3, if we measure from the coordinates of A’(-1/2,c), the x and y coordinates of P’ can be found.\n\n -1 sqrt(3) -3 + sqrt(3) cos(a) + 3 sin(a)\n(2) x = - + ------- cos(300°+a) = ------------------------------,\n 2 3 6\n\n\n(3) y = c + ------- sin(300°+a)\n\n= (cosa – 1/2) + ——- sin(300°+a)\n\n-3 + 3 cos(a) + sqrt(3) sin(a)\n= ——————————\n\nas a goes from 60° to 30°. Finding the path of the triangle’s center in the other three quadrants is similar in procedure and produces equations symmetric to the origin and both axes.\n\n 3 - sqrt(3) cos(a) - 3 sin(a)\n\ny = ——————————\n\n3 – sqrt(3) cos(a) – 3 sin(a)\nQuadrant III: x = —————————–\n\n3 – 3 cos(a) – sqrt(3) sin(a)\ny = —————————–\n\nQuadrant IV: x = ——————————\n\ny = —————————–\n\nBut these equations do not describe a circle. In equations (2) and (3), when a=30°, P is on the x-axis at approximately (0.07735,0). But when a=45°,\n\n -6 + sqrt(6) + 3sqrt(2)\n\nwhich makes the distance from P’ to the origin about 0.08168. This non-circularity is also shown by graphing the four parametric equations above with a circle whose radius is slightly smaller or larger. In Figure 7, the circle is the outer curve. Notice that the centroid’s path is farther from the circle at the axes than mid-quadrant.\n\nSo the Reuleaux triangle’s centroid does not follow a circular path. How then is the Reuleaux bit contained within the square outline it’s to cut? Harry Watts designed a drill in 1914 with a patented “full floating chuck” to accommodate his irregular bits. Bits for square, pentagonal, hexagonal and octagonal holes are still sold by the Watts Brothers Tool Works in Wilmerding PA. The actual drill bit for the square is a Reuleaux triangle made concave in three spots to allow for unobstructed corner-cutting and the discharge of shavings (Figure 8).\n\nEven the modified bit leaves slightly rounded corners. How rounded? Assume the starting position in Figure 9a, in which the Reuleaux triangle is just tangent at point C. As the triangle rotates counterclockwise, C leaves that edge of the square temporarily (labeled C’ in Figure 9b) only to rejoin another at position C” in Figure 9c. In Figure 9b, let a be angle MA’B', ß be the angle formed by A’C’ and the horizontal line through A’, and c the y-coordinate of A’. Then ß = a+60°-90° = a-30° and cosa = 1/2+c. To generate the corner by C, a starts at 30° in Figure 9a and ends up at 60° in Figure 9c. Using A’C’ = 1 and measuring from the coordinates of A’, the coordinates of C’ are described by\n\n -1 -1 + sqrt(3) cos(a) + sin(a)\n x = - + 1 cos(a-30°) = ----------------------------,\n 2 2\n\n\n y = c + 1 sin(a-30°) = (cosa - -) + sin(a-30°)\n\n-1 + cos(a) + sqrt(3) sin(a)\n= —————————-.\n\nThe equations for the other three corners are similar, and when graphed with the rest of the square yield Figure 10.\nNot only does the Reuleaux triangle have practical and interesting applications, and is easy to describe geometrically, but it generates a lot of discussion due to its nonintuitive properties. With this background, you can avoid the blunders I made. Further explorations into the topic might include other figures of constant width (see Gardner and Rademacher/Toeplitz); further identifying the curve of the Reuleaux triangle’s center as it cuts a square; and the shapes of bits for pentagonal, hexagonal and octagonal holes.\n\n\n 1. Gardner, Martin. “Mathematical Games”. Scientific American 208:2 (February 1963): 148-156.\n 2. How to Drill Square, Hexagon, Octagon, Pentagon Holes. Wilmerding, Penn: Watts Brothers Tool Works, 1966.\n 3. Moise, Edwin and Floyd Downs Jr. Geometry. Reading, Mass: Addison-Wesley, 1982.\n 4. Rademacher, Hans and Otto Toeplitz. The Enjoyment of Mathematics. New York: Dover Publications, 1990.\n 5. Smart, James R. “Problem Solving in Geometry – a Sequence of Reuleaux Triangles.” Mathematics Teacher 79 (January 1986): 11-14.\n\nRelated posts:\n\n • http://twitter.com/leetoma/status/129312691834728449 Lee Toma\n\n Fascinating tech: How to drill a square hole! videos http://t.co/aRHiZ5mR\n\nAlso on DOIT:"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "categorical",
"index": 74,
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"text": "web analytics\n\nDrilling Square Holes with a Reuleaux Triangle\n\nWritten by mike on October 3rd, 2011\n\nCircular motion can be converted into a four-sided square using the Reuleaux triangle. The process relies on the property of Reuleaux triangle’s diameter being consistent across all points, and when rolled and rotated simultaneously, tracing a square area with slightly rounded corners. By slightly altering the shape of the triangle to mimic a drill bit (for cutting and extracting material), a drill with a specialized rotating chuck can create perfect, four-sided holes. These drills were pioneered by Harry Watts, and can still be found today, albeit not at your typical home improvement store.\n\nThe following is a bit of literature that helps explain and illustrate the concepts behind the Reuleaux triangle. Please check out more of Prof. Smith’s work here.\n\nDrilling Square Holes\nby Scott Smith\npublished in The Mathematics Teacher, October 1993 (Volume 86 Number 7)\nA bit that drills square holes … it defies common sense. How can a revolving edge cut anything but a circular hole? Not only do such bits exist (as well as bits for pentagonal, hexagonal and octagonal holes), but they derive their shape from a simple geometric construction known as a Reuleaux triangle (after Franz Reuleaux, 1829-1905).\n\nTo construct a Reuleaux triangle, start with an equilateral triangle of side s (Figure 1). With a radius equal to s and the center at one of the vertices, draw an arc connecting the other two vertices. Similarly, draw arcs connecting the endpoints of the other two sides. The three arcs form the Reuleaux triangle. One of its properties is that of constant width, meaning the figure could be rotated completely around between two parallel lines separated by distance s.\nIt was with this property of constant width that the Reuleaux triangle was introduced in a sidebar of our geometry text (Moise and Downs, Teachers’ Edition, p. 555). “This figure has constant width,” I lectured, “just like a circle.” Without thinking, I volunteered, “Imagine it as wheels on a cart.” “What sort of cart?” “Why, a math cart, to carry my board compass and protractor,” I replied, digging myself in deeper. This was the first of several impulsive misstatements I made about the Reuleaux triangle, only to admit after a little reflection that it wasn’t so. Not in twenty years of teaching had my intuition failed me so completely.\n\nThe constant width property can be used to transport loads, but not by using Reuleaux triangles as wheels. If several poles had congruent Reuleaux triangles as cross sections, bulky items could ride atop them (Figure 2). Movement would occur as poles were transferred from back to front, providing a moveable base of constant height.\n\nBut the Reuleaux triangle cannot be a wheel. The only conceivable point for the axle, at the triangle’s centroid, is not the same distance from the Reuleaux triangle’s “sides” (Figure 3). If the sides of the equilateral triangle are s, then\n\n 2 s sqrt(3)\n(1) AP = - - sqrt(3) = ------- s » 0.577s,\n 3 2 3\n\n\n sqrt(3) sqrt(3)\n PB = s - ------- s = s(1 - -------) » 0.423s.\n 3 3\n\nEven if four Reuleaux triangle wheels were synchronized, the load would rise and fall continuously — you’d need Dramamine to ride this cart!\n\n“And since it has constant width, it would just fit inside a square whose sides are that width,” I continued, trying to regain their attention. I carefully drew a square circumscribing the Reuleaux triangle (Figure 4). The triangle is normally tangent to two sides of the square with two vertices touching the square directly opposite the points of tangency (why?), as in Figure 4a. The exception is Figure 4b, where the Reuleaux triangle has one point of tangency and all three vertices on the square (one directly opposite that point of tangency).\n“If the Reuleaux triangle just fits inside the square, no matter what position it’s in, couldn’t it rotate around the inside of the square?” They needed convincing — a model would have to be built. “But if it did rotate around the inside, doesn’t that mean that a sharp Reuleaux triangle could carve out a square as it rotated?” I had them. “Drill a square hole?”, one countered. “No way!”\n\nThat night I cut a four inch Reuleaux triangle from a manila folder to take to class the next day. With a lot of effort, I was able to show the triangle rotate around the inside of a four inch square. “And if this was metal at the end of a rotating shaft, it would cut out a square”, I continued, racking up two more falsehoods. Firstly, it was implied that the center of the Reuleaux triangle would coincide with the center of a drill’s shaft; it cannot. And secondly, the corners of the holes are not right angles, but slightly rounded.\n\nTrying to show the triangle should be centered at the end of a rotating shaft, I stuck a pen through the triangle’s center which, while a student manually rotated the triangle within the square, traced the center’s path on paper beneath (Figure 5). “It’s definitely not a single point,” I had to admit, holding up the traced curve, “but it sure looks like a circle!” Falsehood #4.\n\nJust what is the path of the centroid of a Reuleaux triangle boring a square hole? Assume the square and the equilateral triangle have sides of length 1. Center the square about the origin and position the Reuleaux triangle so vertex A is at (-1/2,0), as in Figure 6a. Using (1), the triangle’s centroid will be P (-1/2+sqrt(3)/3,0). Now imagine rotating the triangle clockwise through the position in Figure 6b, and ending up in Figure 6c, where the centroid is P”(0,-1/2+sqrt(3)/3). The path from P to P” lies in quadrant I. Let a be angle MA’B', ß the angle formed by A’P’ and a horizontal line through A’, and c the y-coordinate of point A’. We are interested in the coordinates of P’. Note that cosa=1/2+c and that ß=270°+a+30°=300°+a. Also note that during this rotation, a goes from 60° to 30°. Because A’P'=sqrt(3)/3, if we measure from the coordinates of A’(-1/2,c), the x and y coordinates of P’ can be found.\n\n -1 sqrt(3) -3 + sqrt(3) cos(a) + 3 sin(a)\n(2) x = - + ------- cos(300°+a) = ------------------------------,\n 2 3 6\n\n\n(3) y = c + ------- sin(300°+a)\n\n= (cosa – 1/2) + ——- sin(300°+a)\n\n-3 + 3 cos(a) + sqrt(3) sin(a)\n= ——————————\n\nas a goes from 60° to 30°. Finding the path of the triangle’s center in the other three quadrants is similar in procedure and produces equations symmetric to the origin and both axes.\n\n 3 - sqrt(3) cos(a) - 3 sin(a)\n\ny = ——————————\n\n3 – sqrt(3) cos(a) – 3 sin(a)\nQuadrant III: x = —————————–\n\n3 – 3 cos(a) – sqrt(3) sin(a)\ny = —————————–\n\nQuadrant IV: x = ——————————\n\ny = —————————–\n\nBut these equations do not describe a circle. In equations (2) and (3), when a=30°, P is on the x-axis at approximately (0.07735,0). But when a=45°,\n\n -6 + sqrt(6) + 3sqrt(2)\n\nwhich makes the distance from P’ to the origin about 0.08168. This non-circularity is also shown by graphing the four parametric equations above with a circle whose radius is slightly smaller or larger. In Figure 7, the circle is the outer curve. Notice that the centroid’s path is farther from the circle at the axes than mid-quadrant.\n\nSo the Reuleaux triangle’s centroid does not follow a circular path. How then is the Reuleaux bit contained within the square outline it’s to cut? Harry Watts designed a drill in 1914 with a patented “full floating chuck” to accommodate his irregular bits. Bits for square, pentagonal, hexagonal and octagonal holes are still sold by the Watts Brothers Tool Works in Wilmerding PA. The actual drill bit for the square is a Reuleaux triangle made concave in three spots to allow for unobstructed corner-cutting and the discharge of shavings (Figure 8).\n\nEven the modified bit leaves slightly rounded corners. How rounded? Assume the starting position in Figure 9a, in which the Reuleaux triangle is just tangent at point C. As the triangle rotates counterclockwise, C leaves that edge of the square temporarily (labeled C’ in Figure 9b) only to rejoin another at position C” in Figure 9c. In Figure 9b, let a be angle MA’B', ß be the angle formed by A’C’ and the horizontal line through A’, and c the y-coordinate of A’. Then ß = a+60°-90° = a-30° and cosa = 1/2+c. To generate the corner by C, a starts at 30° in Figure 9a and ends up at 60° in Figure 9c. Using A’C’ = 1 and measuring from the coordinates of A’, the coordinates of C’ are described by\n\n -1 -1 + sqrt(3) cos(a) + sin(a)\n x = - + 1 cos(a-30°) = ----------------------------,\n 2 2\n\n\n y = c + 1 sin(a-30°) = (cosa - -) + sin(a-30°)\n\n-1 + cos(a) + sqrt(3) sin(a)\n= —————————-.\n\nThe equations for the other three corners are similar, and when graphed with the rest of the square yield Figure 10.\nNot only does the Reuleaux triangle have practical and interesting applications, and is easy to describe geometrically, but it generates a lot of discussion due to its nonintuitive properties. With this background, you can avoid the blunders I made. Further explorations into the topic might include other figures of constant width (see Gardner and Rademacher/Toeplitz); further identifying the curve of the Reuleaux triangle’s center as it cuts a square; and the shapes of bits for pentagonal, hexagonal and octagonal holes.\n\n\n 1. Gardner, Martin. “Mathematical Games”. Scientific American 208:2 (February 1963): 148-156.\n 2. How to Drill Square, Hexagon, Octagon, Pentagon Holes. Wilmerding, Penn: Watts Brothers Tool Works, 1966.\n 3. Moise, Edwin and Floyd Downs Jr. Geometry. Reading, Mass: Addison-Wesley, 1982.\n 4. Rademacher, Hans and Otto Toeplitz. The Enjoyment of Mathematics. New York: Dover Publications, 1990.\n 5. Smart, James R. “Problem Solving in Geometry – a Sequence of Reuleaux Triangles.” Mathematics Teacher 79 (January 1986): 11-14.\n\nRelated posts:\n\n • http://twitter.com/leetoma/status/129312691834728449 Lee Toma\n\n Fascinating tech: How to drill a square hole! videos http://t.co/aRHiZ5mR\n\nAlso on DOIT:"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
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"answer_type": "categorical",
"index": 75,
"split": "train",
"text": "Tell me more ×\n\nFirst question here, so please go easy:\n\nI have a debian linux 5.0 server with two public interfaces. I would like to route outbound traffic from one instance of an application via one interface and the second instance through the second interface. There are some challenges:\n\n • both instances of the application use the same protocol\n • both instances of the application can access the entire internet (can't route based on dest network)\n • I can't change the code of the application\n • I don't think a typical approach to load balancing all traffic is going to work well, because there are relatively few destination servers being accessed in the outbound traffic, and all traffic would really need to be distributed pretty evenly across these relatively few servers.\n\nI could probably run two virtualized servers on the box and bind each of them to a different interface, but I'm looking for a simpler solution, maybe using iproute or iptables?\n\nAny ideas for me? Thanks in advance - and I'm happy to answer any questions.\n\nshare|improve this question\nHi Warner, I assure you that I'm not trolling - if this is a dumb question, I can handle that. – UnicornsAndRainbows Apr 1 '10 at 6:04\nMy mistake, I apologize. – Warner Apr 1 '10 at 18:34\nadd comment\n\n3 Answers\n\nup vote 3 down vote accepted\n\nHave you looked at the owner module within iptables, possibly combined with a source NAT rule?\n\nThis lets you setup OUTPUT chain rules based on UID, GID, PID, SID and command name.\n\nshare|improve this answer\nThat's a great idea, but I don't think I can count on the VPSes I'm using having the owner iptables module. At least I think that's what this means (just using the DROP for a test): (edit: eek, yucky formatting!) ct-1592:~# iptables -A OUTPUT -o venet0 -m owner --uid-owner ip1 -j DROP FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory iptables: No chain/target/match by that name ct-1592:~# – UnicornsAndRainbows Apr 3 '10 at 1:02\nI chose this answer because it would be the best option for my needs if my host supported the owner module. It's best for me because it doesn't require application support for binding to a specific interface. – UnicornsAndRainbows May 6 '10 at 23:53\nonly uid and gid currently (unfortunately) – akostadinov Apr 8 at 19:44\nadd comment\n\nI believe the best option is to bind each application to a different IP, and use source-based routing to change which interface packets from that source IP go out via.\n\nThe general idea is that you can create multiple parallel routing tables ('ip route' has an optional 'table' parameter). Then you add a rule saying \"if it's from IP [x.x.x.x] then use table [foo]\".\n\n\n • Create routing tables [foo1, foo2] in /etc/iproute2/rt_tables\n • Populate routing tables (something like \"ip route add default gateway dev eth0 table foo1\" nd likewise for foo2)\n • Create rules to say which table to use based on source IP - \"ip rule add from table foo1\", \"ip rule add from table foo2\".\n\nI haven't tested those examples specifically, but I've used similar in the past succesfully.\n\nshare|improve this answer\nadd comment\n\nThere is support in linux for binding an application to a specific IP (used for example by Apache). If your application does not support this you are out of luck.\n\nI am not aware of any linux kernel modules (because that is what you would need) to bind a specific application to an interface even when it tries to bind to *. Using a virtual machine is one possibility.\n\nThe closest thing I can come up with to your requirements if there is no application support is Ethernet interface bonding ( But then you would only be running one instance of the application on multiple interfaces with the same ip.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 75,
"split": "train",
"text": "Tell me more ×\n\nFirst question here, so please go easy:\n\nI have a debian linux 5.0 server with two public interfaces. I would like to route outbound traffic from one instance of an application via one interface and the second instance through the second interface. There are some challenges:\n\n • both instances of the application use the same protocol\n • both instances of the application can access the entire internet (can't route based on dest network)\n • I can't change the code of the application\n • I don't think a typical approach to load balancing all traffic is going to work well, because there are relatively few destination servers being accessed in the outbound traffic, and all traffic would really need to be distributed pretty evenly across these relatively few servers.\n\nI could probably run two virtualized servers on the box and bind each of them to a different interface, but I'm looking for a simpler solution, maybe using iproute or iptables?\n\nAny ideas for me? Thanks in advance - and I'm happy to answer any questions.\n\nshare|improve this question\nHi Warner, I assure you that I'm not trolling - if this is a dumb question, I can handle that. – UnicornsAndRainbows Apr 1 '10 at 6:04\nMy mistake, I apologize. – Warner Apr 1 '10 at 18:34\nadd comment\n\n3 Answers\n\nup vote 3 down vote accepted\n\nHave you looked at the owner module within iptables, possibly combined with a source NAT rule?\n\nThis lets you setup OUTPUT chain rules based on UID, GID, PID, SID and command name.\n\nshare|improve this answer\nThat's a great idea, but I don't think I can count on the VPSes I'm using having the owner iptables module. At least I think that's what this means (just using the DROP for a test): (edit: eek, yucky formatting!) ct-1592:~# iptables -A OUTPUT -o venet0 -m owner --uid-owner ip1 -j DROP FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory iptables: No chain/target/match by that name ct-1592:~# – UnicornsAndRainbows Apr 3 '10 at 1:02\nI chose this answer because it would be the best option for my needs if my host supported the owner module. It's best for me because it doesn't require application support for binding to a specific interface. – UnicornsAndRainbows May 6 '10 at 23:53\nonly uid and gid currently (unfortunately) – akostadinov Apr 8 at 19:44\nadd comment\n\nI believe the best option is to bind each application to a different IP, and use source-based routing to change which interface packets from that source IP go out via.\n\nThe general idea is that you can create multiple parallel routing tables ('ip route' has an optional 'table' parameter). Then you add a rule saying \"if it's from IP [x.x.x.x] then use table [foo]\".\n\n\n • Create routing tables [foo1, foo2] in /etc/iproute2/rt_tables\n • Populate routing tables (something like \"ip route add default gateway dev eth0 table foo1\" nd likewise for foo2)\n • Create rules to say which table to use based on source IP - \"ip rule add from table foo1\", \"ip rule add from table foo2\".\n\nI haven't tested those examples specifically, but I've used similar in the past succesfully.\n\nshare|improve this answer\nadd comment\n\nThere is support in linux for binding an application to a specific IP (used for example by Apache). If your application does not support this you are out of luck.\n\nI am not aware of any linux kernel modules (because that is what you would need) to bind a specific application to an interface even when it tries to bind to *. Using a virtual machine is one possibility.\n\nThe closest thing I can come up with to your requirements if there is no application support is Ethernet interface bonding ( But then you would only be running one instance of the application on multiple interfaces with the same ip.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 75,
"split": "train",
"text": "Tell me more ×\n\nFirst question here, so please go easy:\n\nI have a debian linux 5.0 server with two public interfaces. I would like to route outbound traffic from one instance of an application via one interface and the second instance through the second interface. There are some challenges:\n\n • both instances of the application use the same protocol\n • both instances of the application can access the entire internet (can't route based on dest network)\n • I can't change the code of the application\n • I don't think a typical approach to load balancing all traffic is going to work well, because there are relatively few destination servers being accessed in the outbound traffic, and all traffic would really need to be distributed pretty evenly across these relatively few servers.\n\nI could probably run two virtualized servers on the box and bind each of them to a different interface, but I'm looking for a simpler solution, maybe using iproute or iptables?\n\nAny ideas for me? Thanks in advance - and I'm happy to answer any questions.\n\nshare|improve this question\nHi Warner, I assure you that I'm not trolling - if this is a dumb question, I can handle that. – UnicornsAndRainbows Apr 1 '10 at 6:04\nMy mistake, I apologize. – Warner Apr 1 '10 at 18:34\nadd comment\n\n3 Answers\n\nup vote 3 down vote accepted\n\nHave you looked at the owner module within iptables, possibly combined with a source NAT rule?\n\nThis lets you setup OUTPUT chain rules based on UID, GID, PID, SID and command name.\n\nshare|improve this answer\nThat's a great idea, but I don't think I can count on the VPSes I'm using having the owner iptables module. At least I think that's what this means (just using the DROP for a test): (edit: eek, yucky formatting!) ct-1592:~# iptables -A OUTPUT -o venet0 -m owner --uid-owner ip1 -j DROP FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory FATAL: Could not load /lib/modules/2.6.18-028stab068.9/modules.dep: No such file or directory iptables: No chain/target/match by that name ct-1592:~# – UnicornsAndRainbows Apr 3 '10 at 1:02\nI chose this answer because it would be the best option for my needs if my host supported the owner module. It's best for me because it doesn't require application support for binding to a specific interface. – UnicornsAndRainbows May 6 '10 at 23:53\nonly uid and gid currently (unfortunately) – akostadinov Apr 8 at 19:44\nadd comment\n\nI believe the best option is to bind each application to a different IP, and use source-based routing to change which interface packets from that source IP go out via.\n\nThe general idea is that you can create multiple parallel routing tables ('ip route' has an optional 'table' parameter). Then you add a rule saying \"if it's from IP [x.x.x.x] then use table [foo]\".\n\n\n • Create routing tables [foo1, foo2] in /etc/iproute2/rt_tables\n • Populate routing tables (something like \"ip route add default gateway dev eth0 table foo1\" nd likewise for foo2)\n • Create rules to say which table to use based on source IP - \"ip rule add from table foo1\", \"ip rule add from table foo2\".\n\nI haven't tested those examples specifically, but I've used similar in the past succesfully.\n\nshare|improve this answer\nadd comment\n\nThere is support in linux for binding an application to a specific IP (used for example by Apache). If your application does not support this you are out of luck.\n\nI am not aware of any linux kernel modules (because that is what you would need) to bind a specific application to an interface even when it tries to bind to *. Using a virtual machine is one possibility.\n\nThe closest thing I can come up with to your requirements if there is no application support is Ethernet interface bonding ( But then you would only be running one instance of the application on multiple interfaces with the same ip.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"index": 76,
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"text": "### MythBusters - Testing Bullet Proof and Bullet Speeds\n\n**By rallain on November 7, 2008**\n\nOn MythBusters, Adam and Jamie tested the bullet-proofness of various objects. Notably, they examined an iPod. They referenced a report of a soldier being shot by an AK-47 but surviving because the bullet hit his iPod. To test this, Adam shot an AK-47 at an iPod, and it penetrated. Their conclusion was that the soldier was likely also wearing body armor. This raises questions about the reliability of such a report.\n\nI considered that the bullet might have penetrated the iPod due to being only 10 feet away. This brings up an interesting point regarding numerical calculations involving air resistance.\n\nFor this calculation, I'll focus on horizontal motion and ignore vertical motion, simplifying the analysis. Once the bullet leaves the barrel, the following (modified) free body diagram is considered:\n\nIn this one-dimensional model, the only force acting on the bullet is air resistance, opposing the velocity. The model for air resistance is given by:\n\n\\[ F_{\\text{drag}} = \\frac{1}{2} \\rho A C v^2 \\]\n\nwhere \\( \\rho \\) is the air density, \\( A \\) is the cross-sectional area of the object, \\( C \\) is the drag coefficient, and \\( v \\) is the velocity of the bullet. The force of air resistance is proportional to the square of the velocity. You can experience this force by sticking your hand out of a moving car window—the faster the car, the greater the force.\n\nTo apply the momentum principle (or Newton's second law), consider the work-energy theorem. The force is not constant because it depends on velocity. The average velocity can be expressed as:\n\n\\[ v_{\\text{avg}} = \\frac{v_{\\text{final}} + v_{\\text{initial}}}{2} \\]\n\nThis leads to the kinematic equations. To examine this, consider a numerical approach:\n\n1. Break the motion into tiny time-steps.\n2. Calculate the air resistance force.\n3. Calculate the acceleration (assuming constant force).\n4. Calculate the change in position and velocity (assuming constant acceleration).\n5. Move to the next time step.\n6. Repeat.\n\nI used Easy Java Simulations for this, which simplifies model building. The interactive graph shows the speed of the bullet as a function of distance, using data for the AK-47 from Wikipedia. The applet allows users to enter the initial velocity and bullet mass.\n\nAt 3 meters (about 10 feet), the bullet's speed is around 715 m/s, showing minimal slowing. At 50 meters, it's around 675 m/s, and at 100 meters, 634 m/s. At 500 meters, the speed drops to 390 m/s. This affects the kinetic energy, which is crucial for understanding impact damage. The kinetic energy at various distances can be calculated as:\n\n\\[ KE = \\frac{1}{2} mv^2 \\]\n\nwhere \\( m \\) is the mass of the bullet and \\( v \\) is its velocity. At close range (less than 100 meters), there's little difference in speed or kinetic energy. At longer ranges (500 meters), there's a significant difference, affecting both speed and kinetic energy.\n\n### Tags\n\n- acceleration\n- air resistance\n- analysis\n- energy\n- kinematics\n- kinetic energy\n- momentum\n- mythbusters\n- Physics\n- EJS\n- forces\n- numerical model\n- analysis\n\n### Comments\n\n1. **Adam was a poor shot.** By hitting the periphery of the iPod, he missed the circuit board and hard drive. The materials (polycarbonate, metal disk, glass fiber/epoxy composite, and metal backing) effectively degrade a bullet. The bullet's jacket, if any, also significantly affects impact.\n\n2. **Calculate deceleration of the bullet with flight.** At 715 m/s (Mach 1.43), there can be a significant break in slope as the bullet slows from transonic to subsonic if it is not streamlined.\n\n3. **Numerical integral in Excel.** Using Excel for a numerical integral can demonstrate that students have the necessary knowledge to solve problems. A plot of kinetic energy against distance would be appropriate.\n\n4. **Difference between shooting something alone and against a body.** Body armor fails more easily when shot alone because it dissipates energy over a larger area. When shot alone, energy remains concentrated, making penetration easier. This should be considered in the soldier's case.\n\n5. **Watermelon Collisions - ouch.** A clip from The Amazing Race shows a girl launching watermelons with a slingshot. The setup suggests potential injury, though it appears she is not seriously harmed."
} |
questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
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"answer_type": "integer",
"index": 76,
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"text": "### MythBusters - Testing Bullet Proof and Bullet Speeds\n\n**By rallain on November 7, 2008**\n\nOn MythBusters, Adam and Jamie tested the bullet-proofness of various objects. Notably, they examined an iPod. They referenced a report of a soldier being shot by an AK-47 but surviving because the bullet hit his iPod. To test this, Adam shot an AK-47 at an iPod, and it penetrated. Their conclusion was that the soldier was likely also wearing body armor. This raises questions about the reliability of such a report.\n\nI considered that the bullet might have penetrated the iPod due to being only 10 feet away. This brings up an interesting point regarding numerical calculations involving air resistance.\n\nFor this calculation, I'll focus on horizontal motion and ignore vertical motion, simplifying the analysis. Once the bullet leaves the barrel, the following (modified) free body diagram is considered:\n\nIn this one-dimensional model, the only force acting on the bullet is air resistance, opposing the velocity. The model for air resistance is given by:\n\n\\[ F_{\\text{drag}} = \\frac{1}{2} \\rho A C v^2 \\]\n\nwhere \\( \\rho \\) is the air density, \\( A \\) is the cross-sectional area of the object, \\( C \\) is the drag coefficient, and \\( v \\) is the velocity of the bullet. The force of air resistance is proportional to the square of the velocity. You can experience this force by sticking your hand out of a moving car window—the faster the car, the greater the force.\n\nTo apply the momentum principle (or Newton's second law), consider the work-energy theorem. The force is not constant because it depends on velocity. The average velocity can be expressed as:\n\n\\[ v_{\\text{avg}} = \\frac{v_{\\text{final}} + v_{\\text{initial}}}{2} \\]\n\nThis leads to the kinematic equations. To examine this, consider a numerical approach:\n\n1. Break the motion into tiny time-steps.\n2. Calculate the air resistance force.\n3. Calculate the acceleration (assuming constant force).\n4. Calculate the change in position and velocity (assuming constant acceleration).\n5. Move to the next time step.\n6. Repeat.\n\nI used Easy Java Simulations for this, which simplifies model building. The interactive graph shows the speed of the bullet as a function of distance, using data for the AK-47 from Wikipedia. The applet allows users to enter the initial velocity and bullet mass.\n\nAt 3 meters (about 10 feet), the bullet's speed is around 715 m/s, showing minimal slowing. At 50 meters, it's around 675 m/s, and at 100 meters, 634 m/s. At 500 meters, the speed drops to 390 m/s. This affects the kinetic energy, which is crucial for understanding impact damage. The kinetic energy at various distances can be calculated as:\n\n\\[ KE = \\frac{1}{2} mv^2 \\]\n\nwhere \\( m \\) is the mass of the bullet and \\( v \\) is its velocity. At close range (less than 100 meters), there's little difference in speed or kinetic energy. At longer ranges (500 meters), there's a significant difference, affecting both speed and kinetic energy.\n\n### Tags\n\n- acceleration\n- air resistance\n- analysis\n- energy\n- kinematics\n- kinetic energy\n- momentum\n- mythbusters\n- Physics\n- EJS\n- forces\n- numerical model\n- analysis\n\n### Comments\n\n1. **Adam was a poor shot.** By hitting the periphery of the iPod, he missed the circuit board and hard drive. The materials (polycarbonate, metal disk, glass fiber/epoxy composite, and metal backing) effectively degrade a bullet. The bullet's jacket, if any, also significantly affects impact.\n\n2. **Calculate deceleration of the bullet with flight.** At 715 m/s (Mach 1.43), there can be a significant break in slope as the bullet slows from transonic to subsonic if it is not streamlined.\n\n3. **Numerical integral in Excel.** Using Excel for a numerical integral can demonstrate that students have the necessary knowledge to solve problems. A plot of kinetic energy against distance would be appropriate.\n\n4. **Difference between shooting something alone and against a body.** Body armor fails more easily when shot alone because it dissipates energy over a larger area. When shot alone, energy remains concentrated, making penetration easier. This should be considered in the soldier's case.\n\n5. **Watermelon Collisions - ouch.** A clip from The Amazing Race shows a girl launching watermelons with a slingshot. The setup suggests potential injury, though it appears she is not seriously harmed."
} |
questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "integer",
"index": 76,
"split": "train",
"text": "### MythBusters - Testing Bullet Proof and Bullet Speeds\n\n**By rallain on November 7, 2008**\n\nOn MythBusters, Adam and Jamie tested the bullet-proofness of various objects. Notably, they examined an iPod. They referenced a report of a soldier being shot by an AK-47 but surviving because the bullet hit his iPod. To test this, Adam shot an AK-47 at an iPod, and it penetrated. Their conclusion was that the soldier was likely also wearing body armor. This raises questions about the reliability of such a report.\n\nI considered that the bullet might have penetrated the iPod due to being only 10 feet away. This brings up an interesting point regarding numerical calculations involving air resistance.\n\nFor this calculation, I'll focus on horizontal motion and ignore vertical motion, simplifying the analysis. Once the bullet leaves the barrel, the following (modified) free body diagram is considered:\n\nIn this one-dimensional model, the only force acting on the bullet is air resistance, opposing the velocity. The model for air resistance is given by:\n\n\\[ F_{\\text{drag}} = \\frac{1}{2} \\rho A C v^2 \\]\n\nwhere \\( \\rho \\) is the air density, \\( A \\) is the cross-sectional area of the object, \\( C \\) is the drag coefficient, and \\( v \\) is the velocity of the bullet. The force of air resistance is proportional to the square of the velocity. You can experience this force by sticking your hand out of a moving car window—the faster the car, the greater the force.\n\nTo apply the momentum principle (or Newton's second law), consider the work-energy theorem. The force is not constant because it depends on velocity. The average velocity can be expressed as:\n\n\\[ v_{\\text{avg}} = \\frac{v_{\\text{final}} + v_{\\text{initial}}}{2} \\]\n\nThis leads to the kinematic equations. To examine this, consider a numerical approach:\n\n1. Break the motion into tiny time-steps.\n2. Calculate the air resistance force.\n3. Calculate the acceleration (assuming constant force).\n4. Calculate the change in position and velocity (assuming constant acceleration).\n5. Move to the next time step.\n6. Repeat.\n\nI used Easy Java Simulations for this, which simplifies model building. The interactive graph shows the speed of the bullet as a function of distance, using data for the AK-47 from Wikipedia. The applet allows users to enter the initial velocity and bullet mass.\n\nAt 3 meters (about 10 feet), the bullet's speed is around 715 m/s, showing minimal slowing. At 50 meters, it's around 675 m/s, and at 100 meters, 634 m/s. At 500 meters, the speed drops to 390 m/s. This affects the kinetic energy, which is crucial for understanding impact damage. The kinetic energy at various distances can be calculated as:\n\n\\[ KE = \\frac{1}{2} mv^2 \\]\n\nwhere \\( m \\) is the mass of the bullet and \\( v \\) is its velocity. At close range (less than 100 meters), there's little difference in speed or kinetic energy. At longer ranges (500 meters), there's a significant difference, affecting both speed and kinetic energy.\n\n### Tags\n\n- acceleration\n- air resistance\n- analysis\n- energy\n- kinematics\n- kinetic energy\n- momentum\n- mythbusters\n- Physics\n- EJS\n- forces\n- numerical model\n- analysis\n\n### Comments\n\n1. **Adam was a poor shot.** By hitting the periphery of the iPod, he missed the circuit board and hard drive. The materials (polycarbonate, metal disk, glass fiber/epoxy composite, and metal backing) effectively degrade a bullet. The bullet's jacket, if any, also significantly affects impact.\n\n2. **Calculate deceleration of the bullet with flight.** At 715 m/s (Mach 1.43), there can be a significant break in slope as the bullet slows from transonic to subsonic if it is not streamlined.\n\n3. **Numerical integral in Excel.** Using Excel for a numerical integral can demonstrate that students have the necessary knowledge to solve problems. A plot of kinetic energy against distance would be appropriate.\n\n4. **Difference between shooting something alone and against a body.** Body armor fails more easily when shot alone because it dissipates energy over a larger area. When shot alone, energy remains concentrated, making penetration easier. This should be considered in the soldier's case.\n\n5. **Watermelon Collisions - ouch.** A clip from The Amazing Race shows a girl launching watermelons with a slingshot. The setup suggests potential injury, though it appears she is not seriously harmed."
} |
questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Let's Make Robots!\n\nJuice Box Robot Plan, Needs review\n\n***UPDATE MARCH 11: After reading the comments here I have made a couple of changes to my plans. I have replaced the PICAXE 18 with the PICAXE 28 and appropriate board. I have switched to 3 AAA cells as a power source and I have changed to servos to power the \"arms\". When I wrote the original post used the word \"arms\" as a placeholder, but while I always intended to have actuation it was never actually going to be a set of arms. One servo will power a number of separate actions that allow the robot components to fold out of the box (for example the gearboxes descend so the wheels stick through slots in the bottom of the box). The other servo will be used to actuate the tabs of the juice box like \"ears\". I also have a few new questions with my updated plan. Wil the projecct board have the correct outputs to run the motors forward/backwards and also the servos? Or what else do I need? And if anybody has experience with this board will it be able to fit in the 4\"x2\"x1\" space of a juice box (the website doesn't list a size).***\n\nI want to create a mobile robot project that hides inside of a juice box. While I have done robotics before in the FIRST robotics competition but never somethiing working with smaller components like those I will need here. I have talked to some people from a local robot shop in Calgary, AB, Canada \"Solarbotics\" and together we have come up with a list of compenents to make this project a reality:\n\nPICAXE 18 Projecct Board: https://solarbotics.com/product/28460/\n\n***PICAXE 28x1 Chip: https://solarbotics.com/product/28480/\n\n***PICAXE 28 Project Board: https://solarbotics.com/product/28510/\n\nPICaxe Programming Cable: https://solarbotics.com/product/28405/\n\nA Remote Control IR Receiver: https://solarbotics.com/product/TSOP4038/\n\nGM3 Motors for Drivetrain: https://solarbotics.com/product/gm3/\n\n***Tiny Gear Motor Mounts for Wheels: https://solarbotics.com/product/gmw/\n\nPager motors to actuate arms: https://solarbotics.com/product/gm15a/https://solarbotics.com/product/gm15/\n\n***Micro servos for actuating features: https://solarbotics.com/product/25500/\n\nLithium Ion Batery: https://solarbotics.com/product/battr15/\n\nLi-ion Charger: https://solarbotics.com/product/50834/\n\n***3 AAA Cell Holders for power: https://solarbotics.com/product/bholdaaa_1_cell/\n\nI will CAD and 3d print the chassis itself. Does this seem like a set of components that will work effectively together or do you have some suggestions how to improve it. Also, where can I find a good tutorial on how to program the IR receiver. \n\nThanks in Advance, Alex\n\nComment viewing options\n\n\nI can tell its a good one because you have successfully re-invented the Start Here Robot.\n\nI would order the Start Here kit (and save yourself a few bucks over buying everything stand-alone) then add the few extra parts you show (extra servos and the like).\n\nThanks for all your advice! I have updated the original post with the updated plan, take a look.\n\nBased on your BOM, you have just about rebuilt the Start Here robot with a few additions and of course, a juice box.\n\nIf you are looking at making a robot in general, it might not be worth reinventing the wheel here. Not to mention, you could get a better selection of parts (ones that would work for this application) for less money.\n\n • 28x board instead of the 18x --The 18 does not have a motor driver, the 28x does\n • IR reciever --If the 28x board is used, it has provisions (and an available kit) for the addition of IR\n • Pretty much all of the GMx motors would be fine, a little math is needed on gear ratios, wheel dia, etc.\n • If you go with a li-ion battery, you will need 2. Each is a little over 3v, with 2 of them adding up to 7.2v or so. This would need to be then regulated down to 5v for the picaxe. Option 2 is to use the M2 versions of the Picaxe chips that allow for 3.3v operation. \n\n***NOTE*** You are NOT showing Li-Ion battery in your BOM. You are showing a Li-po battery (lithium polymer). There is a substantial difference between the two and no newbie should ever play with Li-po's until they have a firm understanding of the proper handling, discharge and charging. As an example, in the above text, I told you to use 2 of these to get 7.2v. This is just fine with a Li-ion, but li-po's on the other hand, need to be \"balanced\" and \"matched\" before being assembeled into a pack. Simply wiring 2 of these together is a really bad idea. Heck, the pack you show does not even seem to have under/over volt protection.\n\n**I don't mean to harp on you here about these batteries, but this is sorta a big one. You just can't screw around with Li-po batteries. If you short them, they explode. If you try to suck too much current, they explode. If you charge them improperly, they explode. Its just not something to play around with until one knows what they are doing. \n\na motor driver. Actually, you will need a pair of motor drivers since you are planning on using 4 motors. Second, you won't need to \"learn\" how to interact with the IR receiver. PICAXE chips come preprogrammed being able to understand the Sony protocol. You just need to use the proper command, which I believe is irin or infrain. I don't rightly recall. The geared pager motors say they only have .5 oz/in of torque, but, their speed is 920rpm or just over 15 rev/sec. I am not sure, without seeing your plans for an arm, they will really do what you are after.\n\nPS: I just recalled that the PICAXE 18 board you are looking at has a darlington bridge(?) onboard. I am guessing that is what is being planned for use to drive the motors.\n\nThe page won't load for me quite right, but one thing I see now that bird has pointed out the board in question is that you'll actually need to buy your 18 pin PicAxe chip separately according to the link you provided. Otherwise it's a Wizard-of-Oz-Scarecrow situation:no brains.\n\nIf these guys are local to you then you're in luck (Dave Hrynkiw is the owner, I think, and he's a really decent guy from all the interactions I've had with him.). In spit of the fact that he is well known for his \"BEAM\" robotics he is equally knowledgable and enthusiastic about microcontroller units and he won't steer you wrong (no puns intended.)\n\nAs far as IR tutorials go, this is the gold standard: http://learn.adafruit.com/ir-sensor It is geared towards Arduino but you should be able to apply it to Picaxe.\n\nMy only questions are what do you mean by \"...utilizes hides inside of a juicebox.\" And what kind of \"arms\" are you planning on making with pager motors?"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Let's Make Robots!\n\nJuice Box Robot Plan, Needs review\n\n***UPDATE MARCH 11: After reading the comments here I have made a couple of changes to my plans. I have replaced the PICAXE 18 with the PICAXE 28 and appropriate board. I have switched to 3 AAA cells as a power source and I have changed to servos to power the \"arms\". When I wrote the original post used the word \"arms\" as a placeholder, but while I always intended to have actuation it was never actually going to be a set of arms. One servo will power a number of separate actions that allow the robot components to fold out of the box (for example the gearboxes descend so the wheels stick through slots in the bottom of the box). The other servo will be used to actuate the tabs of the juice box like \"ears\". I also have a few new questions with my updated plan. Wil the projecct board have the correct outputs to run the motors forward/backwards and also the servos? Or what else do I need? And if anybody has experience with this board will it be able to fit in the 4\"x2\"x1\" space of a juice box (the website doesn't list a size).***\n\nI want to create a mobile robot project that hides inside of a juice box. While I have done robotics before in the FIRST robotics competition but never somethiing working with smaller components like those I will need here. I have talked to some people from a local robot shop in Calgary, AB, Canada \"Solarbotics\" and together we have come up with a list of compenents to make this project a reality:\n\nPICAXE 18 Projecct Board: https://solarbotics.com/product/28460/\n\n***PICAXE 28x1 Chip: https://solarbotics.com/product/28480/\n\n***PICAXE 28 Project Board: https://solarbotics.com/product/28510/\n\nPICaxe Programming Cable: https://solarbotics.com/product/28405/\n\nA Remote Control IR Receiver: https://solarbotics.com/product/TSOP4038/\n\nGM3 Motors for Drivetrain: https://solarbotics.com/product/gm3/\n\n***Tiny Gear Motor Mounts for Wheels: https://solarbotics.com/product/gmw/\n\nPager motors to actuate arms: https://solarbotics.com/product/gm15a/https://solarbotics.com/product/gm15/\n\n***Micro servos for actuating features: https://solarbotics.com/product/25500/\n\nLithium Ion Batery: https://solarbotics.com/product/battr15/\n\nLi-ion Charger: https://solarbotics.com/product/50834/\n\n***3 AAA Cell Holders for power: https://solarbotics.com/product/bholdaaa_1_cell/\n\nI will CAD and 3d print the chassis itself. Does this seem like a set of components that will work effectively together or do you have some suggestions how to improve it. Also, where can I find a good tutorial on how to program the IR receiver. \n\nThanks in Advance, Alex\n\nComment viewing options\n\n\nI can tell its a good one because you have successfully re-invented the Start Here Robot.\n\nI would order the Start Here kit (and save yourself a few bucks over buying everything stand-alone) then add the few extra parts you show (extra servos and the like).\n\nThanks for all your advice! I have updated the original post with the updated plan, take a look.\n\nBased on your BOM, you have just about rebuilt the Start Here robot with a few additions and of course, a juice box.\n\nIf you are looking at making a robot in general, it might not be worth reinventing the wheel here. Not to mention, you could get a better selection of parts (ones that would work for this application) for less money.\n\n • 28x board instead of the 18x --The 18 does not have a motor driver, the 28x does\n • IR reciever --If the 28x board is used, it has provisions (and an available kit) for the addition of IR\n • Pretty much all of the GMx motors would be fine, a little math is needed on gear ratios, wheel dia, etc.\n • If you go with a li-ion battery, you will need 2. Each is a little over 3v, with 2 of them adding up to 7.2v or so. This would need to be then regulated down to 5v for the picaxe. Option 2 is to use the M2 versions of the Picaxe chips that allow for 3.3v operation. \n\n***NOTE*** You are NOT showing Li-Ion battery in your BOM. You are showing a Li-po battery (lithium polymer). There is a substantial difference between the two and no newbie should ever play with Li-po's until they have a firm understanding of the proper handling, discharge and charging. As an example, in the above text, I told you to use 2 of these to get 7.2v. This is just fine with a Li-ion, but li-po's on the other hand, need to be \"balanced\" and \"matched\" before being assembeled into a pack. Simply wiring 2 of these together is a really bad idea. Heck, the pack you show does not even seem to have under/over volt protection.\n\n**I don't mean to harp on you here about these batteries, but this is sorta a big one. You just can't screw around with Li-po batteries. If you short them, they explode. If you try to suck too much current, they explode. If you charge them improperly, they explode. Its just not something to play around with until one knows what they are doing. \n\na motor driver. Actually, you will need a pair of motor drivers since you are planning on using 4 motors. Second, you won't need to \"learn\" how to interact with the IR receiver. PICAXE chips come preprogrammed being able to understand the Sony protocol. You just need to use the proper command, which I believe is irin or infrain. I don't rightly recall. The geared pager motors say they only have .5 oz/in of torque, but, their speed is 920rpm or just over 15 rev/sec. I am not sure, without seeing your plans for an arm, they will really do what you are after.\n\nPS: I just recalled that the PICAXE 18 board you are looking at has a darlington bridge(?) onboard. I am guessing that is what is being planned for use to drive the motors.\n\nThe page won't load for me quite right, but one thing I see now that bird has pointed out the board in question is that you'll actually need to buy your 18 pin PicAxe chip separately according to the link you provided. Otherwise it's a Wizard-of-Oz-Scarecrow situation:no brains.\n\nIf these guys are local to you then you're in luck (Dave Hrynkiw is the owner, I think, and he's a really decent guy from all the interactions I've had with him.). In spit of the fact that he is well known for his \"BEAM\" robotics he is equally knowledgable and enthusiastic about microcontroller units and he won't steer you wrong (no puns intended.)\n\nAs far as IR tutorials go, this is the gold standard: http://learn.adafruit.com/ir-sensor It is geared towards Arduino but you should be able to apply it to Picaxe.\n\nMy only questions are what do you mean by \"...utilizes hides inside of a juicebox.\" And what kind of \"arms\" are you planning on making with pager motors?"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "categorical",
"index": 77,
"split": "train",
"text": "Let's Make Robots!\n\nJuice Box Robot Plan, Needs review\n\n***UPDATE MARCH 11: After reading the comments here I have made a couple of changes to my plans. I have replaced the PICAXE 18 with the PICAXE 28 and appropriate board. I have switched to 3 AAA cells as a power source and I have changed to servos to power the \"arms\". When I wrote the original post used the word \"arms\" as a placeholder, but while I always intended to have actuation it was never actually going to be a set of arms. One servo will power a number of separate actions that allow the robot components to fold out of the box (for example the gearboxes descend so the wheels stick through slots in the bottom of the box). The other servo will be used to actuate the tabs of the juice box like \"ears\". I also have a few new questions with my updated plan. Wil the projecct board have the correct outputs to run the motors forward/backwards and also the servos? Or what else do I need? And if anybody has experience with this board will it be able to fit in the 4\"x2\"x1\" space of a juice box (the website doesn't list a size).***\n\nI want to create a mobile robot project that hides inside of a juice box. While I have done robotics before in the FIRST robotics competition but never somethiing working with smaller components like those I will need here. I have talked to some people from a local robot shop in Calgary, AB, Canada \"Solarbotics\" and together we have come up with a list of compenents to make this project a reality:\n\nPICAXE 18 Projecct Board: https://solarbotics.com/product/28460/\n\n***PICAXE 28x1 Chip: https://solarbotics.com/product/28480/\n\n***PICAXE 28 Project Board: https://solarbotics.com/product/28510/\n\nPICaxe Programming Cable: https://solarbotics.com/product/28405/\n\nA Remote Control IR Receiver: https://solarbotics.com/product/TSOP4038/\n\nGM3 Motors for Drivetrain: https://solarbotics.com/product/gm3/\n\n***Tiny Gear Motor Mounts for Wheels: https://solarbotics.com/product/gmw/\n\nPager motors to actuate arms: https://solarbotics.com/product/gm15a/https://solarbotics.com/product/gm15/\n\n***Micro servos for actuating features: https://solarbotics.com/product/25500/\n\nLithium Ion Batery: https://solarbotics.com/product/battr15/\n\nLi-ion Charger: https://solarbotics.com/product/50834/\n\n***3 AAA Cell Holders for power: https://solarbotics.com/product/bholdaaa_1_cell/\n\nI will CAD and 3d print the chassis itself. Does this seem like a set of components that will work effectively together or do you have some suggestions how to improve it. Also, where can I find a good tutorial on how to program the IR receiver. \n\nThanks in Advance, Alex\n\nComment viewing options\n\n\nI can tell its a good one because you have successfully re-invented the Start Here Robot.\n\nI would order the Start Here kit (and save yourself a few bucks over buying everything stand-alone) then add the few extra parts you show (extra servos and the like).\n\nThanks for all your advice! I have updated the original post with the updated plan, take a look.\n\nBased on your BOM, you have just about rebuilt the Start Here robot with a few additions and of course, a juice box.\n\nIf you are looking at making a robot in general, it might not be worth reinventing the wheel here. Not to mention, you could get a better selection of parts (ones that would work for this application) for less money.\n\n • 28x board instead of the 18x --The 18 does not have a motor driver, the 28x does\n • IR reciever --If the 28x board is used, it has provisions (and an available kit) for the addition of IR\n • Pretty much all of the GMx motors would be fine, a little math is needed on gear ratios, wheel dia, etc.\n • If you go with a li-ion battery, you will need 2. Each is a little over 3v, with 2 of them adding up to 7.2v or so. This would need to be then regulated down to 5v for the picaxe. Option 2 is to use the M2 versions of the Picaxe chips that allow for 3.3v operation. \n\n***NOTE*** You are NOT showing Li-Ion battery in your BOM. You are showing a Li-po battery (lithium polymer). There is a substantial difference between the two and no newbie should ever play with Li-po's until they have a firm understanding of the proper handling, discharge and charging. As an example, in the above text, I told you to use 2 of these to get 7.2v. This is just fine with a Li-ion, but li-po's on the other hand, need to be \"balanced\" and \"matched\" before being assembeled into a pack. Simply wiring 2 of these together is a really bad idea. Heck, the pack you show does not even seem to have under/over volt protection.\n\n**I don't mean to harp on you here about these batteries, but this is sorta a big one. You just can't screw around with Li-po batteries. If you short them, they explode. If you try to suck too much current, they explode. If you charge them improperly, they explode. Its just not something to play around with until one knows what they are doing. \n\na motor driver. Actually, you will need a pair of motor drivers since you are planning on using 4 motors. Second, you won't need to \"learn\" how to interact with the IR receiver. PICAXE chips come preprogrammed being able to understand the Sony protocol. You just need to use the proper command, which I believe is irin or infrain. I don't rightly recall. The geared pager motors say they only have .5 oz/in of torque, but, their speed is 920rpm or just over 15 rev/sec. I am not sure, without seeing your plans for an arm, they will really do what you are after.\n\nPS: I just recalled that the PICAXE 18 board you are looking at has a darlington bridge(?) onboard. I am guessing that is what is being planned for use to drive the motors.\n\nThe page won't load for me quite right, but one thing I see now that bird has pointed out the board in question is that you'll actually need to buy your 18 pin PicAxe chip separately according to the link you provided. Otherwise it's a Wizard-of-Oz-Scarecrow situation:no brains.\n\nIf these guys are local to you then you're in luck (Dave Hrynkiw is the owner, I think, and he's a really decent guy from all the interactions I've had with him.). In spit of the fact that he is well known for his \"BEAM\" robotics he is equally knowledgable and enthusiastic about microcontroller units and he won't steer you wrong (no puns intended.)\n\nAs far as IR tutorials go, this is the gold standard: http://learn.adafruit.com/ir-sensor It is geared towards Arduino but you should be able to apply it to Picaxe.\n\nMy only questions are what do you mean by \"...utilizes hides inside of a juicebox.\" And what kind of \"arms\" are you planning on making with pager motors?"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "integer",
"index": 78,
"split": "train",
"text": "# Linear Programming Piecewise Linear Objective\n\n## Problem Statement\n\nI am fairly new to linear programming/optimization and am currently working on implementing a linear program that is stated as follows:\n\nMaximize $\\sum_{i=1}^{k}{p(\\vec \\alpha \\cdot \\vec c_i)}$\n\nSubject to:\n\n$|\\alpha_j| \\le 1$\n\nWhere \\(p(x) = 2x\\) if \\(x < 0\\), \\(x\\) otherwise, and \\(\\vec c\\) is a constant.\n\nThe \\(p(x)\\) function is what's troubling me, since one can only determine \\(x\\)'s sign after an assignment of \\(\\vec \\alpha\\). How can I remove the function \\(p\\) from the objective and express this objective equivalently as a linear combination of the variables?\n\n## Comments and Corrections\n\n- The original post was full of mistakes, sorry about that. It should be $|\\alpha_i| \\le 1$. – AFJ\n- The last question: do you want to maximize the sum or to minimize it? – fedja\n- Maximize, although I don't think it matters in answering my question, but you're right that it should have been included. – AFJ\n\n## Solution\n\nThe problem is equivalent to maximizing the linear expression $\\sum_i y_i$ under the linear restrictions $\\alpha_j \\ge -1$, $\\alpha_j \\le 1$, $y_i \\le c_i \\cdot \\alpha$, $y_i \\le 2c_i \\cdot \\alpha$. It is as simple as that but it is crucial that your \\(p\\) is concave and that you maximize.\n\n## Explanation\n\nYour function \\(p\\) is actually \\(p(x) = \\operatorname{min}(x, 2x)\\). By setting \\(y_i := p(\\alpha \\cdot c_i) = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\), your problem becomes: maximize \\(\\sum y_i\\) subject to \\(|\\alpha_j| \\leq 1\\) and \\(y_i = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\). But clearly, you will get the same maximum by relaxing \\(y_i = \\cdots\\) to \\(y_i \\leq \\cdots\\). Now use the fact that \\(y_i \\leq \\operatorname{min}(a, b)\\) if and only if \\(y_i \\leq a\\) and \\(y_i \\leq b\\) to get the conclusion.\n\n## Additional Considerations\n\n- It seems to me that it can't be split into the maximum of just two problems, since it is a sum over \\(p(x_i)\\) where the \\(x_i\\) can be different, resulting in \\(p(x_i)\\) being \\(2x_i\\) or \\(x_i\\) depending on the sign of \\(x_i\\), for a total of \\(2^k\\) ways that \\(p\\) can play a role in the objective. An exponentially growing number of separate problems to solve won't be doable, since \\(k\\) can be 25 or more. – AFJ\n\n## Alternative Approach\n\nI think that the best way of dealing with this is to solve the dual problem. I'll have to think about the details, but that should get the number of vertices in the polytope down to a manageable level. – Victor Miller\n\n## Further Inquiry\n\nI am wondering if something similar to the solution works if \\(p(x) = 2x\\) for \\(x < 5\\) and \\(p(x) = x\\) for \\(x \\ge 5\\). In this case, \\(p(x)\\) is discontinuous at 5 whereas in the given problem \\(p(x)\\) is continuous at 0. Now you no longer have the case that \\(y = \\operatorname{min}(2x, x)\\). – Jeff"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
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"answer_type": "integer",
"index": 78,
"split": "train",
"text": "# Linear Programming Piecewise Linear Objective\n\n## Problem Statement\n\nI am fairly new to linear programming/optimization and am currently working on implementing a linear program that is stated as follows:\n\nMaximize $\\sum_{i=1}^{k}{p(\\vec \\alpha \\cdot \\vec c_i)}$\n\nSubject to:\n\n$|\\alpha_j| \\le 1$\n\nWhere \\(p(x) = 2x\\) if \\(x < 0\\), \\(x\\) otherwise, and \\(\\vec c\\) is a constant.\n\nThe \\(p(x)\\) function is what's troubling me, since one can only determine \\(x\\)'s sign after an assignment of \\(\\vec \\alpha\\). How can I remove the function \\(p\\) from the objective and express this objective equivalently as a linear combination of the variables?\n\n## Comments and Corrections\n\n- The original post was full of mistakes, sorry about that. It should be $|\\alpha_i| \\le 1$. – AFJ\n- The last question: do you want to maximize the sum or to minimize it? – fedja\n- Maximize, although I don't think it matters in answering my question, but you're right that it should have been included. – AFJ\n\n## Solution\n\nThe problem is equivalent to maximizing the linear expression $\\sum_i y_i$ under the linear restrictions $\\alpha_j \\ge -1$, $\\alpha_j \\le 1$, $y_i \\le c_i \\cdot \\alpha$, $y_i \\le 2c_i \\cdot \\alpha$. It is as simple as that but it is crucial that your \\(p\\) is concave and that you maximize.\n\n## Explanation\n\nYour function \\(p\\) is actually \\(p(x) = \\operatorname{min}(x, 2x)\\). By setting \\(y_i := p(\\alpha \\cdot c_i) = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\), your problem becomes: maximize \\(\\sum y_i\\) subject to \\(|\\alpha_j| \\leq 1\\) and \\(y_i = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\). But clearly, you will get the same maximum by relaxing \\(y_i = \\cdots\\) to \\(y_i \\leq \\cdots\\). Now use the fact that \\(y_i \\leq \\operatorname{min}(a, b)\\) if and only if \\(y_i \\leq a\\) and \\(y_i \\leq b\\) to get the conclusion.\n\n## Additional Considerations\n\n- It seems to me that it can't be split into the maximum of just two problems, since it is a sum over \\(p(x_i)\\) where the \\(x_i\\) can be different, resulting in \\(p(x_i)\\) being \\(2x_i\\) or \\(x_i\\) depending on the sign of \\(x_i\\), for a total of \\(2^k\\) ways that \\(p\\) can play a role in the objective. An exponentially growing number of separate problems to solve won't be doable, since \\(k\\) can be 25 or more. – AFJ\n\n## Alternative Approach\n\nI think that the best way of dealing with this is to solve the dual problem. I'll have to think about the details, but that should get the number of vertices in the polytope down to a manageable level. – Victor Miller\n\n## Further Inquiry\n\nI am wondering if something similar to the solution works if \\(p(x) = 2x\\) for \\(x < 5\\) and \\(p(x) = x\\) for \\(x \\ge 5\\). In this case, \\(p(x)\\) is discontinuous at 5 whereas in the given problem \\(p(x)\\) is continuous at 0. Now you no longer have the case that \\(y = \\operatorname{min}(2x, x)\\). – Jeff"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "integer",
"index": 78,
"split": "train",
"text": "# Linear Programming Piecewise Linear Objective\n\n## Problem Statement\n\nI am fairly new to linear programming/optimization and am currently working on implementing a linear program that is stated as follows:\n\nMaximize $\\sum_{i=1}^{k}{p(\\vec \\alpha \\cdot \\vec c_i)}$\n\nSubject to:\n\n$|\\alpha_j| \\le 1$\n\nWhere \\(p(x) = 2x\\) if \\(x < 0\\), \\(x\\) otherwise, and \\(\\vec c\\) is a constant.\n\nThe \\(p(x)\\) function is what's troubling me, since one can only determine \\(x\\)'s sign after an assignment of \\(\\vec \\alpha\\). How can I remove the function \\(p\\) from the objective and express this objective equivalently as a linear combination of the variables?\n\n## Comments and Corrections\n\n- The original post was full of mistakes, sorry about that. It should be $|\\alpha_i| \\le 1$. – AFJ\n- The last question: do you want to maximize the sum or to minimize it? – fedja\n- Maximize, although I don't think it matters in answering my question, but you're right that it should have been included. – AFJ\n\n## Solution\n\nThe problem is equivalent to maximizing the linear expression $\\sum_i y_i$ under the linear restrictions $\\alpha_j \\ge -1$, $\\alpha_j \\le 1$, $y_i \\le c_i \\cdot \\alpha$, $y_i \\le 2c_i \\cdot \\alpha$. It is as simple as that but it is crucial that your \\(p\\) is concave and that you maximize.\n\n## Explanation\n\nYour function \\(p\\) is actually \\(p(x) = \\operatorname{min}(x, 2x)\\). By setting \\(y_i := p(\\alpha \\cdot c_i) = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\), your problem becomes: maximize \\(\\sum y_i\\) subject to \\(|\\alpha_j| \\leq 1\\) and \\(y_i = \\operatorname{min}(\\alpha \\cdot c_i, 2\\alpha \\cdot c_i)\\). But clearly, you will get the same maximum by relaxing \\(y_i = \\cdots\\) to \\(y_i \\leq \\cdots\\). Now use the fact that \\(y_i \\leq \\operatorname{min}(a, b)\\) if and only if \\(y_i \\leq a\\) and \\(y_i \\leq b\\) to get the conclusion.\n\n## Additional Considerations\n\n- It seems to me that it can't be split into the maximum of just two problems, since it is a sum over \\(p(x_i)\\) where the \\(x_i\\) can be different, resulting in \\(p(x_i)\\) being \\(2x_i\\) or \\(x_i\\) depending on the sign of \\(x_i\\), for a total of \\(2^k\\) ways that \\(p\\) can play a role in the objective. An exponentially growing number of separate problems to solve won't be doable, since \\(k\\) can be 25 or more. – AFJ\n\n## Alternative Approach\n\nI think that the best way of dealing with this is to solve the dual problem. I'll have to think about the details, but that should get the number of vertices in the polytope down to a manageable level. – Victor Miller\n\n## Further Inquiry\n\nI am wondering if something similar to the solution works if \\(p(x) = 2x\\) for \\(x < 5\\) and \\(p(x) = x\\) for \\(x \\ge 5\\). In this case, \\(p(x)\\) is discontinuous at 5 whereas in the given problem \\(p(x)\\) is continuous at 0. Now you no longer have the case that \\(y = \\operatorname{min}(2x, x)\\). – Jeff"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
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"answer_type": "integer",
"index": 79,
"split": "train",
"text": "# Set Theory\n\nSet theory was developed by George Cantor between 1845 and 1918. Today, it is used in almost every branch of mathematics and serves as a fundamental part of present-day mathematics.\n\n## Overview\n\nIn set theory, we will learn about representation in roster form and set builder form, types of sets (empty set, singleton set, finite and infinite sets, equal and equivalent sets), cardinal number of a set, subsets (proper subset, superset, power set), number of proper subsets, universal set, and operations on sets (union, intersection, difference, and complement of sets). In everyday life, we often talk of the collection of objects such as a bunch of keys, a flock of birds, or a pack of cards. In mathematics, we come across collections like natural numbers, whole numbers, prime and composite numbers.\n\n## Examples of Collections\n\nLet us examine the following collections:\n\n- Even natural numbers less than 20, i.e., $2, 4, 6, 8, 10, 12, 14, 16, 18$.\n- Vowels in the English alphabet, i.e., $a, e, i, o, u$.\n- Prime factors of 30, i.e., $2, 3, 5$.\n- Triangles based on sides, i.e., equilateral, isosceles, and scalene.\n\nWe observe that these examples are well-defined collections of objects. Let us examine some more collections:\n\n- Five most renowned scientists of the world.\n- Seven most beautiful girls in a society.\n- Three best surgeons in America.\n\nThese examples are not well-defined collections of objects because the criterion for determining as most renowned, most beautiful, best, varies from person to person.\n\n## Sets\n\nA set is a well-defined collection of distinct objects. We assume that:\n\n- The word set is synonymous with the word collection, aggregate, class, and comprises elements.\n- Objects, elements, and members of a set are synonymous terms.\n- Sets are usually denoted by capital letters $A, B, C, \\ldots$ etc.\n- Elements of the set are represented by small letters $a, b, c, \\ldots$ etc.\n\nIf ‘$a$’ is an element of set $A$, then we say that ‘$a$’ belongs to $A$. We denote the phrase ‘belongs to’ by the Greek symbol ‘$\\in$’ (epsilon). Thus, we say that $a \\in A$. If ‘$b$’ is an element which does not belong to $A$, we represent this as $b \\notin A$.\n\n## Important Sets in Mathematics\n\nSome important sets used in mathematics are:\n\n- $N$: the set of all natural numbers $= \\{1, 2, 3, 4, \\ldots\\}$\n- $Z$: the set of all integers $= \\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots\\}$\n- $Q$: the set of all rational numbers\n- $R$: the set of all real numbers\n- $Z^+$: the set of all positive integers\n- $W$: the set of all whole numbers\n\n## Key Topics in Set Theory\n\n- Representation of a Set\n- Types of Sets\n- Finite Sets and Infinite Sets\n- Power Set\n- Problems on Union of Sets\n- Problems on Intersection of Sets\n- Difference of Two Sets\n- Complement of a Set\n- Problems on Complement of a Set\n- Problems on Operation on Sets\n- Word Problems on Sets\n- Venn Diagrams in Different Situations\n- Relationship in Sets using Venn Diagram\n- Union of Sets using Venn Diagram\n- Intersection of Sets using Venn Diagram\n- Disjoint of Sets using Venn Diagram\n- Difference of Sets using Venn Diagram\n- Examples on Venn Diagram\n\n## 8th Grade Math Practice\n\nFrom Set Theory to HOME PAGE."
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questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
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"index": 79,
"split": "train",
"text": "# Set Theory\n\nSet theory was developed by George Cantor between 1845 and 1918. Today, it is used in almost every branch of mathematics and serves as a fundamental part of present-day mathematics.\n\n## Overview\n\nIn set theory, we will learn about representation in roster form and set builder form, types of sets (empty set, singleton set, finite and infinite sets, equal and equivalent sets), cardinal number of a set, subsets (proper subset, superset, power set), number of proper subsets, universal set, and operations on sets (union, intersection, difference, and complement of sets). In everyday life, we often talk of the collection of objects such as a bunch of keys, a flock of birds, or a pack of cards. In mathematics, we come across collections like natural numbers, whole numbers, prime and composite numbers.\n\n## Examples of Collections\n\nLet us examine the following collections:\n\n- Even natural numbers less than 20, i.e., $2, 4, 6, 8, 10, 12, 14, 16, 18$.\n- Vowels in the English alphabet, i.e., $a, e, i, o, u$.\n- Prime factors of 30, i.e., $2, 3, 5$.\n- Triangles based on sides, i.e., equilateral, isosceles, and scalene.\n\nWe observe that these examples are well-defined collections of objects. Let us examine some more collections:\n\n- Five most renowned scientists of the world.\n- Seven most beautiful girls in a society.\n- Three best surgeons in America.\n\nThese examples are not well-defined collections of objects because the criterion for determining as most renowned, most beautiful, best, varies from person to person.\n\n## Sets\n\nA set is a well-defined collection of distinct objects. We assume that:\n\n- The word set is synonymous with the word collection, aggregate, class, and comprises elements.\n- Objects, elements, and members of a set are synonymous terms.\n- Sets are usually denoted by capital letters $A, B, C, \\ldots$ etc.\n- Elements of the set are represented by small letters $a, b, c, \\ldots$ etc.\n\nIf ‘$a$’ is an element of set $A$, then we say that ‘$a$’ belongs to $A$. We denote the phrase ‘belongs to’ by the Greek symbol ‘$\\in$’ (epsilon). Thus, we say that $a \\in A$. If ‘$b$’ is an element which does not belong to $A$, we represent this as $b \\notin A$.\n\n## Important Sets in Mathematics\n\nSome important sets used in mathematics are:\n\n- $N$: the set of all natural numbers $= \\{1, 2, 3, 4, \\ldots\\}$\n- $Z$: the set of all integers $= \\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots\\}$\n- $Q$: the set of all rational numbers\n- $R$: the set of all real numbers\n- $Z^+$: the set of all positive integers\n- $W$: the set of all whole numbers\n\n## Key Topics in Set Theory\n\n- Representation of a Set\n- Types of Sets\n- Finite Sets and Infinite Sets\n- Power Set\n- Problems on Union of Sets\n- Problems on Intersection of Sets\n- Difference of Two Sets\n- Complement of a Set\n- Problems on Complement of a Set\n- Problems on Operation on Sets\n- Word Problems on Sets\n- Venn Diagrams in Different Situations\n- Relationship in Sets using Venn Diagram\n- Union of Sets using Venn Diagram\n- Intersection of Sets using Venn Diagram\n- Disjoint of Sets using Venn Diagram\n- Difference of Sets using Venn Diagram\n- Examples on Venn Diagram\n\n## 8th Grade Math Practice\n\nFrom Set Theory to HOME PAGE."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 79,
"split": "train",
"text": "# Set Theory\n\nSet theory was developed by George Cantor between 1845 and 1918. Today, it is used in almost every branch of mathematics and serves as a fundamental part of present-day mathematics.\n\n## Overview\n\nIn set theory, we will learn about representation in roster form and set builder form, types of sets (empty set, singleton set, finite and infinite sets, equal and equivalent sets), cardinal number of a set, subsets (proper subset, superset, power set), number of proper subsets, universal set, and operations on sets (union, intersection, difference, and complement of sets). In everyday life, we often talk of the collection of objects such as a bunch of keys, a flock of birds, or a pack of cards. In mathematics, we come across collections like natural numbers, whole numbers, prime and composite numbers.\n\n## Examples of Collections\n\nLet us examine the following collections:\n\n- Even natural numbers less than 20, i.e., $2, 4, 6, 8, 10, 12, 14, 16, 18$.\n- Vowels in the English alphabet, i.e., $a, e, i, o, u$.\n- Prime factors of 30, i.e., $2, 3, 5$.\n- Triangles based on sides, i.e., equilateral, isosceles, and scalene.\n\nWe observe that these examples are well-defined collections of objects. Let us examine some more collections:\n\n- Five most renowned scientists of the world.\n- Seven most beautiful girls in a society.\n- Three best surgeons in America.\n\nThese examples are not well-defined collections of objects because the criterion for determining as most renowned, most beautiful, best, varies from person to person.\n\n## Sets\n\nA set is a well-defined collection of distinct objects. We assume that:\n\n- The word set is synonymous with the word collection, aggregate, class, and comprises elements.\n- Objects, elements, and members of a set are synonymous terms.\n- Sets are usually denoted by capital letters $A, B, C, \\ldots$ etc.\n- Elements of the set are represented by small letters $a, b, c, \\ldots$ etc.\n\nIf ‘$a$’ is an element of set $A$, then we say that ‘$a$’ belongs to $A$. We denote the phrase ‘belongs to’ by the Greek symbol ‘$\\in$’ (epsilon). Thus, we say that $a \\in A$. If ‘$b$’ is an element which does not belong to $A$, we represent this as $b \\notin A$.\n\n## Important Sets in Mathematics\n\nSome important sets used in mathematics are:\n\n- $N$: the set of all natural numbers $= \\{1, 2, 3, 4, \\ldots\\}$\n- $Z$: the set of all integers $= \\{\\ldots, -3, -2, -1, 0, 1, 2, 3, \\ldots\\}$\n- $Q$: the set of all rational numbers\n- $R$: the set of all real numbers\n- $Z^+$: the set of all positive integers\n- $W$: the set of all whole numbers\n\n## Key Topics in Set Theory\n\n- Representation of a Set\n- Types of Sets\n- Finite Sets and Infinite Sets\n- Power Set\n- Problems on Union of Sets\n- Problems on Intersection of Sets\n- Difference of Two Sets\n- Complement of a Set\n- Problems on Complement of a Set\n- Problems on Operation on Sets\n- Word Problems on Sets\n- Venn Diagrams in Different Situations\n- Relationship in Sets using Venn Diagram\n- Union of Sets using Venn Diagram\n- Intersection of Sets using Venn Diagram\n- Disjoint of Sets using Venn Diagram\n- Difference of Sets using Venn Diagram\n- Examples on Venn Diagram\n\n## 8th Grade Math Practice\n\nFrom Set Theory to HOME PAGE."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 80,
"split": "train",
"text": "Take the 2-minute tour ×\n\nPossible Duplicate:\nSizeof array passed as parameter\n\nI am being stupid with this sizeof operator in c++, do you have any idea why it is 4 and 12 ?\n\n void function (int arg[]) {\n cout<<sizeof(arg)<<endl; // 4\n\n int main ()\n cout<<sizeof array<<endl; // 12\n function (array);\n return 0;\nshare|improve this question\n\nmarked as duplicate by Pablo, arasmussen, Robᵩ, iammilind, Graviton Mar 1 '12 at 4:31\n\n\n4 Answers 4\n\nup vote 18 down vote accepted\n\nIn main, the name array is an array so you get the size in bytes of the array with sizeof. However, an array decays to a pointer when passed to a function, so you get sizeof(int*) inside the function.\n\nBe aware that taking an argument in the form of T arg[] is exactly the same as taking the argument as T* arg. So your function is the exact equivalent of\n\nvoid function(int* arg) {\n cout << sizeof(arg) << endl;\nshare|improve this answer\n void function (int arg[]) // or void function (int arg[N])\n\nis equivalent to\n\n void function (int *arg)\n\n\nsizeof(arg) == sizeof(int*)\n\nIf you intend to pass array itself, then C++ offers you to pass it by reference:\n\nvoid function (int (&arg)[3])\n // ^^^ pass by reference\n\n\nsizeof(arg) == sizeof(int[3])\nshare|improve this answer\n\nArrays are simply pointers to an arbitrary amount of memory. If you do sizeof(array) it will return the size of a pointer - 4 bytes on 32 bit systems, and 8 bytes on 64 bit systems (if the program is compiled as 64 bit).\n\nThis is the same reason that you have to null-terminate your strings in c/c++ - to denote the end of the array.\n\nSimply put, you have the keep track of the size of your arrays yourself. If you allocate an array of 40 bytes, you have to make sure you never access the array above the 40th index (ie. array[39]).\n\nHope this helps.\n\nshare|improve this answer\nIf arrays are simply pointers, how come inside main it prints 12? – FredOverflow Mar 1 '12 at 7:33\nI can't be 100% sure, but my guess is because your array is static. It's size is determined at compile-time, unlike dynamic memory. Sizeof() may work on static arrays, but i can assure you it won't work on arrays that are allocated dynamically during runtime. – Gogeta70 Mar 1 '12 at 14:08\n\nYour program below is similar to the next one.\n\nvoid function (int arg[]) {\n\nProgram below prints the size of pointer.\n\nvoid function (int *arg) {\nshare|improve this answer\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 80,
"split": "train",
"text": "Take the 2-minute tour ×\n\nPossible Duplicate:\nSizeof array passed as parameter\n\nI am being stupid with this sizeof operator in c++, do you have any idea why it is 4 and 12 ?\n\n void function (int arg[]) {\n cout<<sizeof(arg)<<endl; // 4\n\n int main ()\n cout<<sizeof array<<endl; // 12\n function (array);\n return 0;\nshare|improve this question\n\nmarked as duplicate by Pablo, arasmussen, Robᵩ, iammilind, Graviton Mar 1 '12 at 4:31\n\n\n4 Answers 4\n\nup vote 18 down vote accepted\n\nIn main, the name array is an array so you get the size in bytes of the array with sizeof. However, an array decays to a pointer when passed to a function, so you get sizeof(int*) inside the function.\n\nBe aware that taking an argument in the form of T arg[] is exactly the same as taking the argument as T* arg. So your function is the exact equivalent of\n\nvoid function(int* arg) {\n cout << sizeof(arg) << endl;\nshare|improve this answer\n void function (int arg[]) // or void function (int arg[N])\n\nis equivalent to\n\n void function (int *arg)\n\n\nsizeof(arg) == sizeof(int*)\n\nIf you intend to pass array itself, then C++ offers you to pass it by reference:\n\nvoid function (int (&arg)[3])\n // ^^^ pass by reference\n\n\nsizeof(arg) == sizeof(int[3])\nshare|improve this answer\n\nArrays are simply pointers to an arbitrary amount of memory. If you do sizeof(array) it will return the size of a pointer - 4 bytes on 32 bit systems, and 8 bytes on 64 bit systems (if the program is compiled as 64 bit).\n\nThis is the same reason that you have to null-terminate your strings in c/c++ - to denote the end of the array.\n\nSimply put, you have the keep track of the size of your arrays yourself. If you allocate an array of 40 bytes, you have to make sure you never access the array above the 40th index (ie. array[39]).\n\nHope this helps.\n\nshare|improve this answer\nIf arrays are simply pointers, how come inside main it prints 12? – FredOverflow Mar 1 '12 at 7:33\nI can't be 100% sure, but my guess is because your array is static. It's size is determined at compile-time, unlike dynamic memory. Sizeof() may work on static arrays, but i can assure you it won't work on arrays that are allocated dynamically during runtime. – Gogeta70 Mar 1 '12 at 14:08\n\nYour program below is similar to the next one.\n\nvoid function (int arg[]) {\n\nProgram below prints the size of pointer.\n\nvoid function (int *arg) {\nshare|improve this answer\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 80,
"split": "train",
"text": "Take the 2-minute tour ×\n\nPossible Duplicate:\nSizeof array passed as parameter\n\nI am being stupid with this sizeof operator in c++, do you have any idea why it is 4 and 12 ?\n\n void function (int arg[]) {\n cout<<sizeof(arg)<<endl; // 4\n\n int main ()\n cout<<sizeof array<<endl; // 12\n function (array);\n return 0;\nshare|improve this question\n\nmarked as duplicate by Pablo, arasmussen, Robᵩ, iammilind, Graviton Mar 1 '12 at 4:31\n\n\n4 Answers 4\n\nup vote 18 down vote accepted\n\nIn main, the name array is an array so you get the size in bytes of the array with sizeof. However, an array decays to a pointer when passed to a function, so you get sizeof(int*) inside the function.\n\nBe aware that taking an argument in the form of T arg[] is exactly the same as taking the argument as T* arg. So your function is the exact equivalent of\n\nvoid function(int* arg) {\n cout << sizeof(arg) << endl;\nshare|improve this answer\n void function (int arg[]) // or void function (int arg[N])\n\nis equivalent to\n\n void function (int *arg)\n\n\nsizeof(arg) == sizeof(int*)\n\nIf you intend to pass array itself, then C++ offers you to pass it by reference:\n\nvoid function (int (&arg)[3])\n // ^^^ pass by reference\n\n\nsizeof(arg) == sizeof(int[3])\nshare|improve this answer\n\nArrays are simply pointers to an arbitrary amount of memory. If you do sizeof(array) it will return the size of a pointer - 4 bytes on 32 bit systems, and 8 bytes on 64 bit systems (if the program is compiled as 64 bit).\n\nThis is the same reason that you have to null-terminate your strings in c/c++ - to denote the end of the array.\n\nSimply put, you have the keep track of the size of your arrays yourself. If you allocate an array of 40 bytes, you have to make sure you never access the array above the 40th index (ie. array[39]).\n\nHope this helps.\n\nshare|improve this answer\nIf arrays are simply pointers, how come inside main it prints 12? – FredOverflow Mar 1 '12 at 7:33\nI can't be 100% sure, but my guess is because your array is static. It's size is determined at compile-time, unlike dynamic memory. Sizeof() may work on static arrays, but i can assure you it won't work on arrays that are allocated dynamically during runtime. – Gogeta70 Mar 1 '12 at 14:08\n\nYour program below is similar to the next one.\n\nvoid function (int arg[]) {\n\nProgram below prints the size of pointer.\n\nvoid function (int *arg) {\nshare|improve this answer\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "integer",
"index": 81,
"split": "train",
"text": "## Basic Algebra\n\n### Introduction to Basic Algebra\n\nBasic algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a powerful tool for solving equations and understanding relationships between variables. In this section, we will explore the fundamental concepts of basic algebra, including variables, constants, coefficients, and algebraic expressions.\n\n#### Variables and Constants\n\nIn algebra, a **variable** is a symbol, often a letter, that represents an unknown value. For example, in the equation \\(x + 5 = 12\\), \\(x\\) is the variable. A **constant** is a value that does not change. In the same equation, 5 and 12 are constants.\n\n#### Coefficients\n\nA **coefficient** is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression. For example, in the term \\(3x\\), 3 is the coefficient of the variable \\(x\\).\n\n#### Algebraic Expressions\n\nAn **algebraic expression** is a combination of variables, constants, and coefficients, connected by arithmetic operations. For example, \\(2x + 3\\) and \\(4y^2 - 7y + 1\\) are algebraic expressions.\n\n### Solving Linear Equations\n\nA **linear equation** is an equation that forms a straight line when graphed. It has the general form \\(ax + b = 0\\), where \\(a\\) and \\(b\\) are constants.\n\n#### Example\n\nSolve the equation \\(2x + 3 = 11\\).\n\n**Solution:**\n\n1. Subtract 3 from both sides:\n \\[\n 2x + 3 - 3 = 11 - 3\n \\]\n \\[\n 2x = 8\n \\]\n\n2. Divide both sides by 2:\n \\[\n \\frac{2x}{2} = \\frac{8}{2}\n \\]\n \\[\n x = 4\n \\]\n\n### Quadratic Equations\n\nA **quadratic equation** is an equation of the form \\(ax^2 + bx + c = 0\\), where \\(a\\), \\(b\\), and \\(c\\) are constants, and \\(a \\neq 0\\).\n\n#### Example\n\nSolve the quadratic equation \\(x^2 - 5x + 6 = 0\\).\n\n**Solution:**\n\n1. Factor the quadratic:\n \\[\n (x - 2)(x - 3) = 0\n \\]\n\n2. Set each factor to zero:\n \\[\n x - 2 = 0 \\quad \\text{or} \\quad x - 3 = 0\n \\]\n\n3. Solve for \\(x\\):\n \\[\n x = 2 \\quad \\text{or} \\quad x = 3\n \\]\n\n### Systems of Equations\n\nA **system of equations** consists of two or more equations with the same set of variables. The solution to a system of equations is the set of values that satisfy all equations simultaneously.\n\n#### Example\n\nSolve the system of equations:\n\\[\n\\begin{align*}\nx + y &= 5 \\\\\nx - y &= 1\n\\end{align*}\n\\]\n\n**Solution:**\n\n1. Add the two equations:\n \\[\n (x + y) + (x - y) = 5 + 1\n \\]\n \\[\n 2x = 6\n \\]\n \\[\n x = 3\n \\]\n\n2. Substitute \\(x = 3\\) into the first equation:\n \\[\n 3 + y = 5\n \\]\n \\[\n y = 2\n \\]\n\nThe solution is \\(x = 3\\) and \\(y = 2\\).\n\n### Conclusion\n\nBasic algebra provides the tools necessary to solve equations and understand relationships between variables. By mastering the concepts of variables, constants, coefficients, and algebraic expressions, one can tackle a wide range of mathematical problems. Solving linear equations, quadratic equations, and systems of equations are fundamental skills that form the foundation for more advanced mathematical studies."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "float",
"index": 81,
"split": "train",
"text": "## Basic Algebra\n\n### Introduction to Basic Algebra\n\nBasic algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a powerful tool for solving equations and understanding relationships between variables. In this section, we will explore the fundamental concepts of basic algebra, including variables, constants, coefficients, and algebraic expressions.\n\n#### Variables and Constants\n\nIn algebra, a **variable** is a symbol, often a letter, that represents an unknown value. For example, in the equation \\(x + 5 = 12\\), \\(x\\) is the variable. A **constant** is a value that does not change. In the same equation, 5 and 12 are constants.\n\n#### Coefficients\n\nA **coefficient** is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression. For example, in the term \\(3x\\), 3 is the coefficient of the variable \\(x\\).\n\n#### Algebraic Expressions\n\nAn **algebraic expression** is a combination of variables, constants, and coefficients, connected by arithmetic operations. For example, \\(2x + 3\\) and \\(4y^2 - 7y + 1\\) are algebraic expressions.\n\n### Solving Linear Equations\n\nA **linear equation** is an equation that forms a straight line when graphed. It has the general form \\(ax + b = 0\\), where \\(a\\) and \\(b\\) are constants.\n\n#### Example\n\nSolve the equation \\(2x + 3 = 11\\).\n\n**Solution:**\n\n1. Subtract 3 from both sides:\n \\[\n 2x + 3 - 3 = 11 - 3\n \\]\n \\[\n 2x = 8\n \\]\n\n2. Divide both sides by 2:\n \\[\n \\frac{2x}{2} = \\frac{8}{2}\n \\]\n \\[\n x = 4\n \\]\n\n### Quadratic Equations\n\nA **quadratic equation** is an equation of the form \\(ax^2 + bx + c = 0\\), where \\(a\\), \\(b\\), and \\(c\\) are constants, and \\(a \\neq 0\\).\n\n#### Example\n\nSolve the quadratic equation \\(x^2 - 5x + 6 = 0\\).\n\n**Solution:**\n\n1. Factor the quadratic:\n \\[\n (x - 2)(x - 3) = 0\n \\]\n\n2. Set each factor to zero:\n \\[\n x - 2 = 0 \\quad \\text{or} \\quad x - 3 = 0\n \\]\n\n3. Solve for \\(x\\):\n \\[\n x = 2 \\quad \\text{or} \\quad x = 3\n \\]\n\n### Systems of Equations\n\nA **system of equations** consists of two or more equations with the same set of variables. The solution to a system of equations is the set of values that satisfy all equations simultaneously.\n\n#### Example\n\nSolve the system of equations:\n\\[\n\\begin{align*}\nx + y &= 5 \\\\\nx - y &= 1\n\\end{align*}\n\\]\n\n**Solution:**\n\n1. Add the two equations:\n \\[\n (x + y) + (x - y) = 5 + 1\n \\]\n \\[\n 2x = 6\n \\]\n \\[\n x = 3\n \\]\n\n2. Substitute \\(x = 3\\) into the first equation:\n \\[\n 3 + y = 5\n \\]\n \\[\n y = 2\n \\]\n\nThe solution is \\(x = 3\\) and \\(y = 2\\).\n\n### Conclusion\n\nBasic algebra provides the tools necessary to solve equations and understand relationships between variables. By mastering the concepts of variables, constants, coefficients, and algebraic expressions, one can tackle a wide range of mathematical problems. Solving linear equations, quadratic equations, and systems of equations are fundamental skills that form the foundation for more advanced mathematical studies."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 81,
"split": "train",
"text": "## Basic Algebra\n\n### Introduction to Basic Algebra\n\nBasic algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a powerful tool for solving equations and understanding relationships between variables. In this section, we will explore the fundamental concepts of basic algebra, including variables, constants, coefficients, and algebraic expressions.\n\n#### Variables and Constants\n\nIn algebra, a **variable** is a symbol, often a letter, that represents an unknown value. For example, in the equation \\(x + 5 = 12\\), \\(x\\) is the variable. A **constant** is a value that does not change. In the same equation, 5 and 12 are constants.\n\n#### Coefficients\n\nA **coefficient** is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression. For example, in the term \\(3x\\), 3 is the coefficient of the variable \\(x\\).\n\n#### Algebraic Expressions\n\nAn **algebraic expression** is a combination of variables, constants, and coefficients, connected by arithmetic operations. For example, \\(2x + 3\\) and \\(4y^2 - 7y + 1\\) are algebraic expressions.\n\n### Solving Linear Equations\n\nA **linear equation** is an equation that forms a straight line when graphed. It has the general form \\(ax + b = 0\\), where \\(a\\) and \\(b\\) are constants.\n\n#### Example\n\nSolve the equation \\(2x + 3 = 11\\).\n\n**Solution:**\n\n1. Subtract 3 from both sides:\n \\[\n 2x + 3 - 3 = 11 - 3\n \\]\n \\[\n 2x = 8\n \\]\n\n2. Divide both sides by 2:\n \\[\n \\frac{2x}{2} = \\frac{8}{2}\n \\]\n \\[\n x = 4\n \\]\n\n### Quadratic Equations\n\nA **quadratic equation** is an equation of the form \\(ax^2 + bx + c = 0\\), where \\(a\\), \\(b\\), and \\(c\\) are constants, and \\(a \\neq 0\\).\n\n#### Example\n\nSolve the quadratic equation \\(x^2 - 5x + 6 = 0\\).\n\n**Solution:**\n\n1. Factor the quadratic:\n \\[\n (x - 2)(x - 3) = 0\n \\]\n\n2. Set each factor to zero:\n \\[\n x - 2 = 0 \\quad \\text{or} \\quad x - 3 = 0\n \\]\n\n3. Solve for \\(x\\):\n \\[\n x = 2 \\quad \\text{or} \\quad x = 3\n \\]\n\n### Systems of Equations\n\nA **system of equations** consists of two or more equations with the same set of variables. The solution to a system of equations is the set of values that satisfy all equations simultaneously.\n\n#### Example\n\nSolve the system of equations:\n\\[\n\\begin{align*}\nx + y &= 5 \\\\\nx - y &= 1\n\\end{align*}\n\\]\n\n**Solution:**\n\n1. Add the two equations:\n \\[\n (x + y) + (x - y) = 5 + 1\n \\]\n \\[\n 2x = 6\n \\]\n \\[\n x = 3\n \\]\n\n2. Substitute \\(x = 3\\) into the first equation:\n \\[\n 3 + y = 5\n \\]\n \\[\n y = 2\n \\]\n\nThe solution is \\(x = 3\\) and \\(y = 2\\).\n\n### Conclusion\n\nBasic algebra provides the tools necessary to solve equations and understand relationships between variables. By mastering the concepts of variables, constants, coefficients, and algebraic expressions, one can tackle a wide range of mathematical problems. Solving linear equations, quadratic equations, and systems of equations are fundamental skills that form the foundation for more advanced mathematical studies."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 82,
"split": "train",
"text": "Written by Solomon D. Erulkar\n\nnervous system\n\nArticle Free Pass\nWritten by Solomon D. Erulkar\n\nThe sequence of sodium activation–sodium inactivation–potassium activation creates a nerve impulse that is brief in duration, lasting only a few milliseconds, and that travels down the nerve fibre like a wave, the membrane depolarizing in front of the current and repolarizing behind. Because nerve impulses are not graded in amplitude, it is not the size of the action potential that is important in processing information within the nervous system; rather, it is the number and frequency with which the impulses are fired.\n\nAs stated above, the action potential is propagated along the axon without any decrease in amplitude with distance. However, the velocity of conduction along the nerve fibre is dependent upon several factors. The first factor is the outside diameter of the nerve fibre. The fastest conduction velocity occurs in the largest diameter nerve fibres. This phenomenon has formed the basis for classifying mammalian nerve fibres into groups in order of decreasing diameter and decreasing conduction velocity. Another factor is the temperature of the nerve fibre. Conduction velocity increases at high temperatures and decreases at low temperatures. Indeed, nerve conduction can be blocked by the local application of cold to a nerve fibre. Conduction velocity is also affected by myelination of the nerve fibre. Since ions cannot cross the lipid content of the myelin sheath, they spread passively down the nerve fibre until reaching the unmyelinated nodes of Ranvier. The nodes of Ranvier are packed with a high concentration of ion channels, which, upon stimulation, propagate the nerve impulse to the next node. In this manner the action potential jumps quickly from node to node along the fibre in a process called saltatory conduction (from Latin saltare, “to jump”; see the figure).\n\nTransmission at the synapse\n\nOnce an action potential has been generated at the axon hillock, it is conducted along the length of the axon until it reaches the terminals, the fingerlike extensions of the neuron that are next to other neurons and muscle cells (see the section The nerve cell: The neuron). At this point there exist two methods for transmitting the action potential from one cell to the other. In electrical transmission, the ionic current flows directly through channels that couple the cells. In chemical transmission, a chemical substance called the neurotransmitter passes from one cell to the other, stimulating the second cell to generate its own action potential.\n\nElectrical transmission\n\nThis method of transmitting nerve impulses, while far less common than chemical transmission, occurs in the nervous systems of invertebrates and lower vertebrates, as well as in the central nervous systems of some mammals. Transmission takes place through gap junctions, which are protein channels that link the cellular contents of adjacent neurons. Direct diffusion of ions through these junctions allows the action potential to be transmitted with little or no delay or distortion, in effect synchronizing the response of an entire group of neurons. The channels often allow ions to diffuse in both directions, but some gated channels restrict transmission to only one direction.\n\nChemical transmission\n\nThere are two classic preparations for the study of chemical transmission at the synapse. One is the vertebrate neuromuscular junction, and the other is the giant synapse of the squid Loligo. These sites have the advantage of being readily accessible for recording by electrodes—especially the squid synapse, which is large enough that electrodes can be inserted directly into the presynaptic terminal and postsynaptic fibre. In addition, only a single synapse is involved at these sites, whereas a single neuron of the central nervous system may have many synapses with many other neurons, each with a different neurotransmitter.\n\nNeurotransmitter release\n\nTwo factors are essential for the release of the neurotransmitter from the presynaptic terminal: (1) depolarization of the terminal and (2) the presence of calcium ions (Ca2+) in the extracellular fluid. The membrane of the presynaptic terminal contains voltage-dependent calcium channels that open when the membrane is depolarized by a nerve impulse, allowing Ca2+ to diffuse into the terminal along its concentration gradient. (See the figure.) Following the entrance of Ca2+ is the release of neurotransmitter.\n\nIt is uncertain what happens in the time between Ca2+ entry and transmitter release. Ca2+ is known to be sequestered by certain organelles within the terminal, including the endoplasmic reticulum. The ions may attach to the membranes of synaptic vesicles, in some way facilitating their fusion with the nerve terminal membrane. They may also be removed from the terminal by exchange with extracellular Na+—a mechanism known to occur at some neuronal membranes. What is certain is that when the concentration of Ca2+ is increased within the terminal, the probability of transmitter release is also increased.\n\nNeurotransmitters are packed into small, membrane-bound synaptic vesicles. Each vesicle contains thousands of neurotransmitter molecules, and there are thousands of vesicles in each axon terminal. Once stimulated by Ca2+, the vesicles move through the cytoplasm and fuse their membranes with the plasma membrane of the terminal. The transmitter molecules are then expelled from the vesicles into the synaptic cleft. This expulsion process is called exocytosis. Vesicle membranes are then recovered from the plasma membrane through endocytosis. In this process the membranes are surrounded by a protein coat at the lateral margins of the synapse and are then transferred to cisternae, which form in the terminal during nerve stimulation. There the vesicles lose their coats, are probably refilled with neurotransmitter, and pinch off from the cisternae to become synaptic vesicles once more.\n\nBecause the neurotransmitter chemicals are packed into separate, almost identically sized vesicles, their release into the synaptic cleft is said to be quantal—that is, they are expelled in parcels, each vesicle adding its contents incrementally to the contents released from other parcels. This quantal release of neurotransmitter has a critical influence on the electrical potential created in the postsynaptic membrane.\n\nTake Quiz Add To This Article\nShare Stories, photos and video Surprise Me!\n\n\nPlease select the sections you want to print\nSelect All\nMLA style:\n\"nervous system\". Encyclopædia Britannica. Encyclopædia Britannica Online.\nAPA style:\nHarvard style:\nnervous system. 2014. Encyclopædia Britannica Online. Retrieved 11 July, 2014, from http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction\nChicago Manual of Style:\nEncyclopædia Britannica Online, s. v. \"nervous system\", accessed July 11, 2014, http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction.\n\n\nWe welcome suggested improvements to any of our articles.\n(Please limit to 900 characters)\n\nOr click Continue to submit anonymously:"
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{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
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"answer_type": "categorical",
"index": 82,
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"text": "Written by Solomon D. Erulkar\n\nnervous system\n\nArticle Free Pass\nWritten by Solomon D. Erulkar\n\nThe sequence of sodium activation–sodium inactivation–potassium activation creates a nerve impulse that is brief in duration, lasting only a few milliseconds, and that travels down the nerve fibre like a wave, the membrane depolarizing in front of the current and repolarizing behind. Because nerve impulses are not graded in amplitude, it is not the size of the action potential that is important in processing information within the nervous system; rather, it is the number and frequency with which the impulses are fired.\n\nAs stated above, the action potential is propagated along the axon without any decrease in amplitude with distance. However, the velocity of conduction along the nerve fibre is dependent upon several factors. The first factor is the outside diameter of the nerve fibre. The fastest conduction velocity occurs in the largest diameter nerve fibres. This phenomenon has formed the basis for classifying mammalian nerve fibres into groups in order of decreasing diameter and decreasing conduction velocity. Another factor is the temperature of the nerve fibre. Conduction velocity increases at high temperatures and decreases at low temperatures. Indeed, nerve conduction can be blocked by the local application of cold to a nerve fibre. Conduction velocity is also affected by myelination of the nerve fibre. Since ions cannot cross the lipid content of the myelin sheath, they spread passively down the nerve fibre until reaching the unmyelinated nodes of Ranvier. The nodes of Ranvier are packed with a high concentration of ion channels, which, upon stimulation, propagate the nerve impulse to the next node. In this manner the action potential jumps quickly from node to node along the fibre in a process called saltatory conduction (from Latin saltare, “to jump”; see the figure).\n\nTransmission at the synapse\n\nOnce an action potential has been generated at the axon hillock, it is conducted along the length of the axon until it reaches the terminals, the fingerlike extensions of the neuron that are next to other neurons and muscle cells (see the section The nerve cell: The neuron). At this point there exist two methods for transmitting the action potential from one cell to the other. In electrical transmission, the ionic current flows directly through channels that couple the cells. In chemical transmission, a chemical substance called the neurotransmitter passes from one cell to the other, stimulating the second cell to generate its own action potential.\n\nElectrical transmission\n\nThis method of transmitting nerve impulses, while far less common than chemical transmission, occurs in the nervous systems of invertebrates and lower vertebrates, as well as in the central nervous systems of some mammals. Transmission takes place through gap junctions, which are protein channels that link the cellular contents of adjacent neurons. Direct diffusion of ions through these junctions allows the action potential to be transmitted with little or no delay or distortion, in effect synchronizing the response of an entire group of neurons. The channels often allow ions to diffuse in both directions, but some gated channels restrict transmission to only one direction.\n\nChemical transmission\n\nThere are two classic preparations for the study of chemical transmission at the synapse. One is the vertebrate neuromuscular junction, and the other is the giant synapse of the squid Loligo. These sites have the advantage of being readily accessible for recording by electrodes—especially the squid synapse, which is large enough that electrodes can be inserted directly into the presynaptic terminal and postsynaptic fibre. In addition, only a single synapse is involved at these sites, whereas a single neuron of the central nervous system may have many synapses with many other neurons, each with a different neurotransmitter.\n\nNeurotransmitter release\n\nTwo factors are essential for the release of the neurotransmitter from the presynaptic terminal: (1) depolarization of the terminal and (2) the presence of calcium ions (Ca2+) in the extracellular fluid. The membrane of the presynaptic terminal contains voltage-dependent calcium channels that open when the membrane is depolarized by a nerve impulse, allowing Ca2+ to diffuse into the terminal along its concentration gradient. (See the figure.) Following the entrance of Ca2+ is the release of neurotransmitter.\n\nIt is uncertain what happens in the time between Ca2+ entry and transmitter release. Ca2+ is known to be sequestered by certain organelles within the terminal, including the endoplasmic reticulum. The ions may attach to the membranes of synaptic vesicles, in some way facilitating their fusion with the nerve terminal membrane. They may also be removed from the terminal by exchange with extracellular Na+—a mechanism known to occur at some neuronal membranes. What is certain is that when the concentration of Ca2+ is increased within the terminal, the probability of transmitter release is also increased.\n\nNeurotransmitters are packed into small, membrane-bound synaptic vesicles. Each vesicle contains thousands of neurotransmitter molecules, and there are thousands of vesicles in each axon terminal. Once stimulated by Ca2+, the vesicles move through the cytoplasm and fuse their membranes with the plasma membrane of the terminal. The transmitter molecules are then expelled from the vesicles into the synaptic cleft. This expulsion process is called exocytosis. Vesicle membranes are then recovered from the plasma membrane through endocytosis. In this process the membranes are surrounded by a protein coat at the lateral margins of the synapse and are then transferred to cisternae, which form in the terminal during nerve stimulation. There the vesicles lose their coats, are probably refilled with neurotransmitter, and pinch off from the cisternae to become synaptic vesicles once more.\n\nBecause the neurotransmitter chemicals are packed into separate, almost identically sized vesicles, their release into the synaptic cleft is said to be quantal—that is, they are expelled in parcels, each vesicle adding its contents incrementally to the contents released from other parcels. This quantal release of neurotransmitter has a critical influence on the electrical potential created in the postsynaptic membrane.\n\nTake Quiz Add To This Article\nShare Stories, photos and video Surprise Me!\n\n\nPlease select the sections you want to print\nSelect All\nMLA style:\n\"nervous system\". Encyclopædia Britannica. Encyclopædia Britannica Online.\nAPA style:\nHarvard style:\nnervous system. 2014. Encyclopædia Britannica Online. Retrieved 11 July, 2014, from http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction\nChicago Manual of Style:\nEncyclopædia Britannica Online, s. v. \"nervous system\", accessed July 11, 2014, http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction.\n\n\nWe welcome suggested improvements to any of our articles.\n(Please limit to 900 characters)\n\nOr click Continue to submit anonymously:"
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questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
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"answer_type": "categorical",
"index": 82,
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"text": "Written by Solomon D. Erulkar\n\nnervous system\n\nArticle Free Pass\nWritten by Solomon D. Erulkar\n\nThe sequence of sodium activation–sodium inactivation–potassium activation creates a nerve impulse that is brief in duration, lasting only a few milliseconds, and that travels down the nerve fibre like a wave, the membrane depolarizing in front of the current and repolarizing behind. Because nerve impulses are not graded in amplitude, it is not the size of the action potential that is important in processing information within the nervous system; rather, it is the number and frequency with which the impulses are fired.\n\nAs stated above, the action potential is propagated along the axon without any decrease in amplitude with distance. However, the velocity of conduction along the nerve fibre is dependent upon several factors. The first factor is the outside diameter of the nerve fibre. The fastest conduction velocity occurs in the largest diameter nerve fibres. This phenomenon has formed the basis for classifying mammalian nerve fibres into groups in order of decreasing diameter and decreasing conduction velocity. Another factor is the temperature of the nerve fibre. Conduction velocity increases at high temperatures and decreases at low temperatures. Indeed, nerve conduction can be blocked by the local application of cold to a nerve fibre. Conduction velocity is also affected by myelination of the nerve fibre. Since ions cannot cross the lipid content of the myelin sheath, they spread passively down the nerve fibre until reaching the unmyelinated nodes of Ranvier. The nodes of Ranvier are packed with a high concentration of ion channels, which, upon stimulation, propagate the nerve impulse to the next node. In this manner the action potential jumps quickly from node to node along the fibre in a process called saltatory conduction (from Latin saltare, “to jump”; see the figure).\n\nTransmission at the synapse\n\nOnce an action potential has been generated at the axon hillock, it is conducted along the length of the axon until it reaches the terminals, the fingerlike extensions of the neuron that are next to other neurons and muscle cells (see the section The nerve cell: The neuron). At this point there exist two methods for transmitting the action potential from one cell to the other. In electrical transmission, the ionic current flows directly through channels that couple the cells. In chemical transmission, a chemical substance called the neurotransmitter passes from one cell to the other, stimulating the second cell to generate its own action potential.\n\nElectrical transmission\n\nThis method of transmitting nerve impulses, while far less common than chemical transmission, occurs in the nervous systems of invertebrates and lower vertebrates, as well as in the central nervous systems of some mammals. Transmission takes place through gap junctions, which are protein channels that link the cellular contents of adjacent neurons. Direct diffusion of ions through these junctions allows the action potential to be transmitted with little or no delay or distortion, in effect synchronizing the response of an entire group of neurons. The channels often allow ions to diffuse in both directions, but some gated channels restrict transmission to only one direction.\n\nChemical transmission\n\nThere are two classic preparations for the study of chemical transmission at the synapse. One is the vertebrate neuromuscular junction, and the other is the giant synapse of the squid Loligo. These sites have the advantage of being readily accessible for recording by electrodes—especially the squid synapse, which is large enough that electrodes can be inserted directly into the presynaptic terminal and postsynaptic fibre. In addition, only a single synapse is involved at these sites, whereas a single neuron of the central nervous system may have many synapses with many other neurons, each with a different neurotransmitter.\n\nNeurotransmitter release\n\nTwo factors are essential for the release of the neurotransmitter from the presynaptic terminal: (1) depolarization of the terminal and (2) the presence of calcium ions (Ca2+) in the extracellular fluid. The membrane of the presynaptic terminal contains voltage-dependent calcium channels that open when the membrane is depolarized by a nerve impulse, allowing Ca2+ to diffuse into the terminal along its concentration gradient. (See the figure.) Following the entrance of Ca2+ is the release of neurotransmitter.\n\nIt is uncertain what happens in the time between Ca2+ entry and transmitter release. Ca2+ is known to be sequestered by certain organelles within the terminal, including the endoplasmic reticulum. The ions may attach to the membranes of synaptic vesicles, in some way facilitating their fusion with the nerve terminal membrane. They may also be removed from the terminal by exchange with extracellular Na+—a mechanism known to occur at some neuronal membranes. What is certain is that when the concentration of Ca2+ is increased within the terminal, the probability of transmitter release is also increased.\n\nNeurotransmitters are packed into small, membrane-bound synaptic vesicles. Each vesicle contains thousands of neurotransmitter molecules, and there are thousands of vesicles in each axon terminal. Once stimulated by Ca2+, the vesicles move through the cytoplasm and fuse their membranes with the plasma membrane of the terminal. The transmitter molecules are then expelled from the vesicles into the synaptic cleft. This expulsion process is called exocytosis. Vesicle membranes are then recovered from the plasma membrane through endocytosis. In this process the membranes are surrounded by a protein coat at the lateral margins of the synapse and are then transferred to cisternae, which form in the terminal during nerve stimulation. There the vesicles lose their coats, are probably refilled with neurotransmitter, and pinch off from the cisternae to become synaptic vesicles once more.\n\nBecause the neurotransmitter chemicals are packed into separate, almost identically sized vesicles, their release into the synaptic cleft is said to be quantal—that is, they are expelled in parcels, each vesicle adding its contents incrementally to the contents released from other parcels. This quantal release of neurotransmitter has a critical influence on the electrical potential created in the postsynaptic membrane.\n\nTake Quiz Add To This Article\nShare Stories, photos and video Surprise Me!\n\n\nPlease select the sections you want to print\nSelect All\nMLA style:\n\"nervous system\". Encyclopædia Britannica. Encyclopædia Britannica Online.\nAPA style:\nHarvard style:\nnervous system. 2014. Encyclopædia Britannica Online. Retrieved 11 July, 2014, from http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction\nChicago Manual of Style:\nEncyclopædia Britannica Online, s. v. \"nervous system\", accessed July 11, 2014, http://www.britannica.com/EBchecked/topic/409665/nervous-system/75841/Conduction.\n\n\nWe welcome suggested improvements to any of our articles.\n(Please limit to 900 characters)\n\nOr click Continue to submit anonymously:"
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questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 83,
"split": "train",
"text": "European Space Agency\n\nTowards Reusable Launchers - A Widening Perspective\n\nH. Pfeffer\n\nFuture Launchers Office, Directorate of Launchers, ESA, Paris\n\nOne of the prime tasks of the Agency, identified in the Report of ESA's Long-Term Space Policy Committee (LSPC) endorsed by the Ministers at their Toulouse Meeting in October last year, is to search for ways to reduce the cost of access to space and thereby open up new markets. One possible approach is that of launcher reuse. Because all such concepts envisaged so far require advances in technology before possibly becoming of potential economic interest, the Agency has begun a programme called FESTIP (Future European Space Transportation Investigations Programme), in which reusable launcher concepts are being studied and the corresponding technology work is being done.\nThis article presents some possible reusable launcher design options in the context of a scenario that might lead to a technological convergence between space vehicles and high-speed global aviation.\n\nThe ideal reusable launcher\n\nThe ideal reusable launcher would be analogous to a normal aircraft in that it would be able to take off from many possible bases on Earth, enter the desired orbital plane, accelerate to orbital velocity, release its payload, de-orbit, dissipate its kinetic energy and land at the take-off base to be readied for its next flight after a quick turn-around. The vehicle would maintain physical integrity during a mission, would be flown 'very often', and its cost per flight would be compatible with the value of the missions undertaken. The vehicle would also be able to abort its mission any time in the case of mishap and land intact with its payload. This ideal vehicle is the 'single-stage-to-orbit reusable rocket launcher', or SSTO-RRL.\n\nThe value of the 'single stage' concept is that the vehicle maintains its integrity throughout its lifetime. This integrity, which results in the minimisation of the inherent interfaces, is a major factor in the vehicle's dependability. Indeed, the majority of today's practical transport systems are also 'single- stage' vehicles, which include aircraft, ships, cars, etc.\n\nSo far, however, all space launchers are staged to orbit and expendable (with the exception of the US Space Shuttle System) and there is no SSTO-RRL as yet, despite its apparent desirability. Clearly, something must be standing in the way of achieving this 'ideal' vehicle.\n\nThe problems posed by SSTO-RRLs\n\nBecause rocket propulsion is mandatory to accelerate to orbital speed in vacuum, the most logical design option is to use rocket propulsion from take-off until orbit insertion. Both gravity and drag losses must be overcome on the trajectory to orbit. The ideal velocity increment, Delta V, required from an SSTO-RRL is then about 9000 m/s in order to reach a Low Earth Orbit (LEO). All further considerations concentrate on reaching LEO, because this is the most difficult part of gaining access to space and the major hurdle to be mastered in terms of reusability.\n\nThe mass that can be accelerated into orbit using rocket propulsion is given by the equation: M 1 /M 0 =exp ( -Delta V /V E ) where M 0 is the mass at take-off, M 1 is the mass which has received the ideal velocity increment Delta V, and V E is the ejection velocity of the rocket engine.\n\nFor a given Delta V, which is mission-imposed, the mass ratio M 1 /M 0 increases with increasing V E . The highest practical rocket ejection velocities are achieved by burning hydrogen with oxygen in a combustion chamber and ejecting the produced gases through a convergent/divergent nozzle. When averaged over the trajectory, the exhaust velocity V E is in the order of 4000 m/s. The corresponding mass ratio to reach LEO is: M 1 /M 0 =exp(Delta V /V E ) = exp ( 9000/4000) = 0.1054 = 10.54%\n\nThis means that 100 10.54 = 89.46% of the take-off mass must be made up of propellants, and that just 10.54% of the trade-off mass remains available for the tankage to contain the propellants, the engines, the structures, the equipment, and last but not least the payload. Assuming that the payload is 1% of the take-off mass, the mass fraction left available to build the complete launcher is 10.54 1 = 9.54%. This is not very much, and we have already assumed the highest performing propulsion system possible (cryogenic propulsion).\n\nMass ratios of the order of 10% are achieved today with expendable rocket stages, but are a nearly impossible requirement for a reusable vehicle, which is more stressed during its lifetime and which must carry the additional provisions for recovery and reuse. Because today's rocket propulsion is already quite near to its theoretical maximum, the whole burden of achieving such small mass ratios falls on the materials from which the launcher is built. Such launchers are, however, still studied within the FESTIP initiative (Fig. 1) in order to define the areas in which technology advances would be most effective.\n\nMeans for increasing the mass ratio\n\nIf the mass ratio allocated to the SSTO-RRL could be larger, the vehicle might become more feasible or more robust to use. The options offered by the rocket equation for increasing the mass ratio are to increase V E ,or to decrease Delta V, or a combination of the two.\n\nFigure 1. A possible single-stage-to-orbit launcher studied within ESA's FESTIP initiative\n\nIf V E were increased by 10% from 4000 m/s to 4400 m/s, the mass ratio would become\n\nM 11 /M 10 = exp ( 9000 / 4400) = 0.1293\n\nAgain assuming 1% for the payload, the mass available for the vehicle becomes\n\n12.93% 1% = 11.93%\n\nwhich represents a percentage increase of\n\n(11.93 9.54) / 9.54 = 25.1%\n\nwith respect to the nominal SSTO-RRL, which is quite a significant gain.\n\nIncreasing V E is equivalent to finding a propulsion system which produces the required thrust while consuming less propellant. For example, the apparent V E of propellants carried aboard a rocket increases when some of the products ejected are taken from the ambient air instead of having to be carried onboard from the start. In general, however, increasing the V E of a rocket motor with the help of the atmosphere is a complex problem to which Europe cannot expect an early solution. A typical configuration of this type, shown in Figure 2, is the SKYLON (UK), derived from HOTOL.\n\nrocket propulsion\nFigure 2. A possible single-stage-to-orbit launcher with combined air-breathing and rocket propulsion\n\nTurning now to the effect of a reduction in velocity increment, if the required Delta V were reduced by 10% from 9000 to 8100 m/s, the mass ratio would become\n\nM 1 /M 0 = exp ( 8100 / 4000) = 0.1320\n\nIf we still want a 1% payload in orbit, the payload MP0, ejected at 9000 8100= 900 m/s short of orbital velocity, must carry an additional propulsion system. Assuming a storable bipropellant system having an effective exhaust velocity of V E2 = 3000 m/s, the mass fraction for the payload is\n\nMP 1 /MP 0 = exp ( 900 / 3000) = 0.7408\n\nThe mass ejected 900 m/s short of orbital velocity must therefore be 1 / 0.7408=1.350% of the launcher take-off mass to again have a final payload mass in orbit of 1%. The net mass ratio remaining for the launcher itself becomes\n\n13.20% 1.350% = 11.85%\n\nwhich represents a gain of\n\n(11.85 9.54) / 9.54 = 24.2%\n\nwith respect to that required for the pure SSTO-RRL.\n\nTherefore, reducing Delta V is as effective in creating a mass margin as is increasing V E. In more general terms, reducing Delta V leads to having more than one stage to reach orbit. As each stage has to deliver only a fraction of the total Delta V, its mass fraction can increase and a more robust design becomes possible.\n\nStaging options for RRLs\n\n(a) The SSTO-AL(air- launched)-RRL\nPrinciple: A carrier aircraft takes the SSTO-RRL to a certain altitude and speed, thereby reducing the remaining DV required to reach orbit. The Interim HOTOL carried on-top of an Antonov-225 aircraft is one such example (Fig. 3).\n\nHOTOL launched\nFigure 3. The Interim HOTOL launched from the Antonov 225 aircraft\n\nDrawbacks: The carrier aircraft has a limited payload capability, which constrains the SSTO-RRL's mass at separation. As a result, an air-launched SSTO-RRL becomes as difficult to build as a ground-launched version for which no take-off mass limitations apply. This comment is applicable to any upper stage of any Two Stage To Orbit (TSTO) system, because large vehicles are more effective and have a lower overall mass-ratio capability. Air-launching also involves a dangerous separation in the presence of aerodynamic flows.\n\nAdvantages: Air-launching gives launch azimuth flexibility and provides a self-contained flying launch base.\n\nConclusion: Not a promising solution for Europe.\n\n(b) The TSTO-RTL (return to launch site) and DRL(down- 0range-landing)-RRL\n\nPrinciple: Here, a reusable rocket first stage carries a reusable (or expendable) rocket second stage. In the RTL mode, the first stage returns to the launch site after separation. RTL is practical as long as the first stage does not re-enter too far down range, which corresponds to up to 1/4 of the total Delta V required to reach LEO. The FLS as studied by Aerospatiale for ESA in the early 1980s (Fig. 4), the RRL studied in the early 1990s (Fig. 5) and the FSS 9 as studied within FESTIP (Fig. 6) are vehicle's of this type.\n\nsemi-reusable 1980\nFigure 4. The semi-reusable future launching system studied by Aerospatiale in the early 1980s\n\nsemi-reusable 1990\nFigure 5. A semi-reusable rocket launcher as studied by Aerospatiale in the 1990s\n\nStudied FESTIP\nFigure 6. A fully reusable two-stage-to-orbit rocket launcher as studied within FESTIP\n\nIn the DRL mode, the first stage provides a larger share of the total Delta V and is allowed to land down-range of the launch site after separation and re-entry. The first stage returns to the launch site at a later time. The TARANIS, studied by Aerospatiale for ESA in the early 1990s (Fig. 7) was this type of vehicle.\n\nFigure 7. TAKANIS: a fully reusable rocket launcher with down- range landing of the first stage\n\nDrawbacks: There are now three aerodynamic configurations to be controlled - the first stage, the second stage when reusable, and the composite - two vehicles to be developed and qualified, and the amount of operations is increased. Staging is optimised for a given technology level and prevents one from deriving full benefit from any subsequent technology improvements. Because of the mass limitations on the second stage, its reusability represents a penalty rather than a cost advantage. A more effective solution is that of an expendable second stage, which could be considered an interesting interim solution.\n\nWhen implemented with today's technology, the TSTO-RTL/DRL-RRL is not much cheaper in use than an expendable launcher, but it can already offer higher reliability and safety. Launch/in-flight abortability remains limited.\n\nAdvantages: Feasible in the near term, if needed.\n\nConclusion: Not an interesting solution for Europe, in that the expendable Ariane-5 is already a modern launcher with significant evolution potential.\n\n(c) The SOSS (Sub-Orbital Single Stage) OA (Once- Around) RRL\n\nPrinciple: The down-range capability of the first stage is now extended to the point where it can complete one trip around the Earth so as to land back at the launch base (Fig. 8). The first stage therefore provides almost the total Delta V needed to reach orbit and the payload, ejected in vacuum but still at sub-orbital speed, achieves orbital velocity with its own propulsion system. This principle is comparable to that of the SSTO-AL-RRL, except that now the Delta V complement provided by the carrier aircraft (a large and expensive item) is provided by the payload itself (a small system) and the large sub-orbital stage itself retains the freedom of unconstrained mass at take-off.\n\nonce-around Earth\nFigure 8. The trajectory of a once-around Earth sub-orbital single-stage reusable launcher\n\nDrawbacks: The payload must provide its propulsion into orbit (from 300 to 3000 m/s) to compensate for the Delta V deficit of the sub-orbital stage. Not all payloads might welcome this, but this requirement could be taken into account when designing them.\n\nAdvantages: For the payload: For the majority of payloads, LEO is not the ultimate destination and many of them already carry an integrated propulsion stage to acquire higher energy orbits. Adding the need to provide the initial impulse to reach orbit therefore does not represent an excessive penalty.\n\nFor the launcher: The launcher is now back at the launch site after one revolution. The Delta V needed to circularise and to de-orbit is saved, there is no launcher in- orbit phase with the associated functions and mass, and a high launcher efficiency is therefore possible.\n\nConclusion: The author believes that the SOSS-OA-RRL (sub-orbital single-stage once-around reusable rocket launcher) can be a very promising solution, the particular features of which are discussed below.\n\nOperating mode of the SOSS-OA-RRL\n\nLet us start by defining a large reusable stage with a given take-off mass, of say 500 tons. Since we do not yet have the technology to achieve the mass ratio necessary for true SSTO operation, we must design the launcher to provide the largest Delta V it can, and leave it to the payload to provide the complement needed to achieve orbital conditions. Keeping the take-off mass constant allows us to freeze the launcher's external shape, so that the aero-dynamic database remains valid once established, and the installed thrust can be fixed. The launcher tankage is designed for the full volume of propellants ultimately required for SSTO, and the payload bay is given the volume needed by the largest possible combination of payload with its upper-stage propulsion.\n\nWe can now trade-off propellant masses between the launcher and the payload whilst keeping the total take-off mass constant. Depending on the sharing of Delta V between the launcher and the payload, we can achieve a range of payload-in-orbit capability. This approach therefore combines the flexibility and performance robustness of a pure TSTO system with the operational benefits of a single-stage vehicle. We refer to this approach as 'internal staging'.\n\nThe SOSS-OA-RRL, when already dimensioned for ultimate SSTO capability, can therefore become an interim single-stage vehicle with which practical experience can be accumulated early in a revenue-generating utilisation environment, whilst still preserving the possibility to upgrade it as technology progresses, without affecting its overall layout. With improving technology, the launcher dry-mass or the propellant mass needed will decrease, the launcher will reach higher speeds, and the propulsion requirements on the payload will decrease accordingly, until a pure SSTO capability becomes possible, should this indeed prove to be an economically viable option.\n\nIn a first approximation, the cost of using an SOSS-RRL is constant and independent of the mass of the payload with propulsion it carries, because its take-off mass is constant. The same SOSS-OA-RRL is used for each payload, irrespective of its mass, the only adaptation being the correct loading of propellants and the calculation of the corresponding trajectory. This repeated utilisation should lead to significant cost savings as experience accrues.\n\nOur SOSS-OA-RRL now resembles a truck which has a maximum payload, but can also operate profitably with a partial load.\n\nThe value of good high-speed aerodynamics\n\nThe velocity deficit of the SOSS-OA-RRL with respect to orbital velocity cannot be too large if we want the reusable launcher stage to complete one turn of the Earth and return to the launch base in a non-propulsive manner. The permissible deficit will depend on the vehicle's aerodynamic qualities during the high- speed re-entry phase: a greater lift-to-drag ratio at hypersonic speed permits a greater passive down-range capability, which permits a larger allowable velocity deficit with respect to orbital speed, which in turn permits a greater launcher dry-mass (at the expense of increased aerothermal problems). Suitable shapes for the hypersonic glide-back flight are lifting bodies (Fig. 9) or waveriders (Fig. 10).\n\nFigure 9. A lifting-body sub-orbital, once-around reusable rocket launcher studied within FESTIP\n\nFigure 10. A generic 'wave-rider' shape as studied within FESTIP\n\nAn SOSS-OA-RRL which needs a down-range capability to return to its base therefore makes constructive use of the kinetic energy it received during ascent, instead of simply destroying it as is the case for re-entry from orbital speed.\n\nIf the velocity deficit is such that even good aerodynamics do not allow a complete once-around glide, the stage can land after a partial orbit, but then there is the problem of returning it to the launch base, or we can equip it with an auxiliary propulsion system for a powered fly-back to complete the once- around. This powered fly-back could either be ensured by a scramjet for high-altitude, high-speed flight (although there might be thermal problems), or as a subsonic cruise on turbojet power (aerial refuelling with kerosene is a possibility). Again, the better the aerodynamics, the less penalising the cruise-back becomes.\n\nVertical versus horizontal take-off\n\nNow that we have a reason to require good aerodynamics for the return flight of the SOSS-OA-RRL, the next step is to use the installed aerodynamics for horizontal take-off (HTO) and lifting ascent on rocket power only. Indeed, the integrated velocity losses with HTO are a little less than with vertical take-off (VTO) on rocket power. A conventional SSTO-RRL must, however, rely on VTO because its required low mass fraction does not permit the implementation of lifting surfaces for ascent.\n\nFor a reusable vehicle, however, VTO has unpleasant implications. With VTO, the installed thrust which translates into number, mass and cost of engines must lift the vehicle off the pad, must provide the thrust needed to accelerate it, and there must also be a thrust margin for some engine-out capability. The total installed thrust then corresponds to about 1.5 times the take-off weight of the vehicle, only about a third of which does useful work in accelerating the vehicle during the initial phase of the mission. This installed thrust becomes excessive towards the end of the ascent and engines have either to be throttled back (cost) or shut-off (dead mass into orbit).\n\nBecause rocket engines must be located at the base of the vehicle, a high installed thrust shifts the centre of mass of the empty vehicle to the rear and compromises must be made in the aerodynamic shape in order to maintain a controllable vehicle at re-entry.\n\nEven when planning for engine-out situations, VTO remains a dangerous phase and while this is normal for an expendable launcher (if it has to fail, it does not matter where it fails), it is far from ideal for a reusable vehicle. Indeed, until there is enough dynamic pressure to provide aerodynamic control authority, the reusable launcher behaves like an expendable vehicle and the range-safety and vehicle destruct may have to be retained.\n\nFinally, VTO requires a heavy ground infrastructure which is both costly and limits the number of launch bases that one can build.\n\nThe SOSS-OA-RRL launcher, which needs aerodynamics for its return flight (and which can afford it because of its higher permissible mass fraction) allows one to exploit the Delta V savings of HTO. However, HTO also has more fundamental advantages. With HTO followed by lifting ascent, the weight of the vehicle is carried by the lift due to the forward speed and only the drag generated by that lift has to be compensated for by rocket thrust. Depending on the aerodynamic shape, the drag generated by the lift is only one half to one quarter of the vehicle's weight. Rocket thrust to accelerate the vehicle at the rate of 0.5 g at take-off is sufficient, so that the total rocket thrust needed by an HTO vehicle is about 0.5 + 1/2 to 1/4 =1.0 to 0.75 times the weight of the vehicle at take-off. This reduced thrust budget leads to associated reductions in engine mass (which improves the aerodynamic centering) and cost.\n\nHTO also provides an increased engine-out capability and therefore greater safety, at least when implemented as follows. The vehicle can rest on a passive trolley, on a linear ground track a few kilometres long. The vehicle accelerates under its own rocket power until the appropriate velocity is reached, at which point its nose is pushed upwards mechanically causing the vehicle to lift off from the trolley and continue its ascent. Operational advantages of this approach are:\n\nWith this scenario, rocket-powered HTO could offer the same thrust and safety margins as are now customary for multi-engined aviation.\n\nFinally, the HTO lifting ascent phase prolongs the time spent in the Earth's atmosphere by several minutes, which might allow some form of air-breathing rocket propulsion to be added to the vehicle at a later date, thereby improving the apparent specific impulse of the propulsion system and further increasing the allowable mass ratio.\n\nAn eventual boost-glide vehicle for global Earth travel?\n\nBy now, our proposed SOSS-OA-RRL has quite striking parallels with conventional aviation: horizontal processing, horizontal take-off, horizontal ascent, gliding or powered hyper-sonic flight, horizontal landing, and similar engine-out tolerances. The vehicle can cover all speed ranges from low subsonic to nearly orbital, and can evolve through the whole depth of the atmosphere. But such a vehicle might have other applications too, as it is not obliged to make a full revolution of the Earth: it could then serve as a high-speed global transport vehicle.\n\nIndeed, high-speed commercial transport (HSCT) programmes in the USA, Japan and Europe are already devoting significant funds to solving the fundamental technology problems, and the approach being pursued so far envisages cruising at higher speeds (up to Mach 5.5), as a direct extrapolation of today's lower speed aviation. At such speeds, however, global range trip times remain long, the aircraft cruises in contamination-sensitive layers of the upper atmosphere where its engines deposit emissions, the aircraft generates a sonic-boom carpet on the ground, which makes supersonic overflight of land masses unlikely, the aircraft faces severe thermal soaking problems, the propulsion still has to be developed to civilian standards, etc.\n\nThe boost-glide approach of the SOSS-OA-RRL is a logical alternative for such global travel: the vehicle is accelerated to high speed (Mach 10?,15?) by rocket (not necessarily O 2 /H 2 ) or by rocket plus air-breathing propulsion, follows a ballistic arc in near-vacuum, re-enters and dissipates its kinetic energy during an equilibrium glide at hypersonic speed and at high altitude. Once this energy has been dissipated, the vehicle can make a conventional landing like a normal subsonic aircraft.\n\nThe advantages of boost-glide for global travel are:\n\n\nAssembling the features described, one can conclude that, with today's emerging technologies, launcher reusability offers the opportunity to serve both the space-launch and global-travel markets with one type of vehicle: the sub- orbital single-stage reusable rocket launcher. The vehicles themselves would differ in detail, depending on their exact roles, but they would rely on a common technological base.\n\nSuch synergy would be to the benefit of both communities: it would enlarge the technology base available for space activities, it would spread the resulting development costs, it would amortise more rapidly all investments made, and it would stimulate a new approach to travelling around our planet, which in turn could prove a strong motivator for the younger generation to pursue a scientific/technical career.\n\nSpace has the potential to become a very large industrial park, stimulated by more routine and less costly access to space in the same way that global markets have been facilitated by the growth and falling cost of today's Earth-bound transport systems. Assessment of the possibilities for making this happen deserves commensurate spending for exploratory work. ESA's FESTIP programme will hopefully contribute to this goal.\n\nAbout | Search | Feedback\n\nRight Left Up Home ESA Bulletin Nr. 87.\nPublished August 1996.\nDeveloped by ESA-ESRIN ID/D."
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"text": "European Space Agency\n\nTowards Reusable Launchers - A Widening Perspective\n\nH. Pfeffer\n\nFuture Launchers Office, Directorate of Launchers, ESA, Paris\n\nOne of the prime tasks of the Agency, identified in the Report of ESA's Long-Term Space Policy Committee (LSPC) endorsed by the Ministers at their Toulouse Meeting in October last year, is to search for ways to reduce the cost of access to space and thereby open up new markets. One possible approach is that of launcher reuse. Because all such concepts envisaged so far require advances in technology before possibly becoming of potential economic interest, the Agency has begun a programme called FESTIP (Future European Space Transportation Investigations Programme), in which reusable launcher concepts are being studied and the corresponding technology work is being done.\nThis article presents some possible reusable launcher design options in the context of a scenario that might lead to a technological convergence between space vehicles and high-speed global aviation.\n\nThe ideal reusable launcher\n\nThe ideal reusable launcher would be analogous to a normal aircraft in that it would be able to take off from many possible bases on Earth, enter the desired orbital plane, accelerate to orbital velocity, release its payload, de-orbit, dissipate its kinetic energy and land at the take-off base to be readied for its next flight after a quick turn-around. The vehicle would maintain physical integrity during a mission, would be flown 'very often', and its cost per flight would be compatible with the value of the missions undertaken. The vehicle would also be able to abort its mission any time in the case of mishap and land intact with its payload. This ideal vehicle is the 'single-stage-to-orbit reusable rocket launcher', or SSTO-RRL.\n\nThe value of the 'single stage' concept is that the vehicle maintains its integrity throughout its lifetime. This integrity, which results in the minimisation of the inherent interfaces, is a major factor in the vehicle's dependability. Indeed, the majority of today's practical transport systems are also 'single- stage' vehicles, which include aircraft, ships, cars, etc.\n\nSo far, however, all space launchers are staged to orbit and expendable (with the exception of the US Space Shuttle System) and there is no SSTO-RRL as yet, despite its apparent desirability. Clearly, something must be standing in the way of achieving this 'ideal' vehicle.\n\nThe problems posed by SSTO-RRLs\n\nBecause rocket propulsion is mandatory to accelerate to orbital speed in vacuum, the most logical design option is to use rocket propulsion from take-off until orbit insertion. Both gravity and drag losses must be overcome on the trajectory to orbit. The ideal velocity increment, Delta V, required from an SSTO-RRL is then about 9000 m/s in order to reach a Low Earth Orbit (LEO). All further considerations concentrate on reaching LEO, because this is the most difficult part of gaining access to space and the major hurdle to be mastered in terms of reusability.\n\nThe mass that can be accelerated into orbit using rocket propulsion is given by the equation: M 1 /M 0 =exp ( -Delta V /V E ) where M 0 is the mass at take-off, M 1 is the mass which has received the ideal velocity increment Delta V, and V E is the ejection velocity of the rocket engine.\n\nFor a given Delta V, which is mission-imposed, the mass ratio M 1 /M 0 increases with increasing V E . The highest practical rocket ejection velocities are achieved by burning hydrogen with oxygen in a combustion chamber and ejecting the produced gases through a convergent/divergent nozzle. When averaged over the trajectory, the exhaust velocity V E is in the order of 4000 m/s. The corresponding mass ratio to reach LEO is: M 1 /M 0 =exp(Delta V /V E ) = exp ( 9000/4000) = 0.1054 = 10.54%\n\nThis means that 100 10.54 = 89.46% of the take-off mass must be made up of propellants, and that just 10.54% of the trade-off mass remains available for the tankage to contain the propellants, the engines, the structures, the equipment, and last but not least the payload. Assuming that the payload is 1% of the take-off mass, the mass fraction left available to build the complete launcher is 10.54 1 = 9.54%. This is not very much, and we have already assumed the highest performing propulsion system possible (cryogenic propulsion).\n\nMass ratios of the order of 10% are achieved today with expendable rocket stages, but are a nearly impossible requirement for a reusable vehicle, which is more stressed during its lifetime and which must carry the additional provisions for recovery and reuse. Because today's rocket propulsion is already quite near to its theoretical maximum, the whole burden of achieving such small mass ratios falls on the materials from which the launcher is built. Such launchers are, however, still studied within the FESTIP initiative (Fig. 1) in order to define the areas in which technology advances would be most effective.\n\nMeans for increasing the mass ratio\n\nIf the mass ratio allocated to the SSTO-RRL could be larger, the vehicle might become more feasible or more robust to use. The options offered by the rocket equation for increasing the mass ratio are to increase V E ,or to decrease Delta V, or a combination of the two.\n\nFigure 1. A possible single-stage-to-orbit launcher studied within ESA's FESTIP initiative\n\nIf V E were increased by 10% from 4000 m/s to 4400 m/s, the mass ratio would become\n\nM 11 /M 10 = exp ( 9000 / 4400) = 0.1293\n\nAgain assuming 1% for the payload, the mass available for the vehicle becomes\n\n12.93% 1% = 11.93%\n\nwhich represents a percentage increase of\n\n(11.93 9.54) / 9.54 = 25.1%\n\nwith respect to the nominal SSTO-RRL, which is quite a significant gain.\n\nIncreasing V E is equivalent to finding a propulsion system which produces the required thrust while consuming less propellant. For example, the apparent V E of propellants carried aboard a rocket increases when some of the products ejected are taken from the ambient air instead of having to be carried onboard from the start. In general, however, increasing the V E of a rocket motor with the help of the atmosphere is a complex problem to which Europe cannot expect an early solution. A typical configuration of this type, shown in Figure 2, is the SKYLON (UK), derived from HOTOL.\n\nrocket propulsion\nFigure 2. A possible single-stage-to-orbit launcher with combined air-breathing and rocket propulsion\n\nTurning now to the effect of a reduction in velocity increment, if the required Delta V were reduced by 10% from 9000 to 8100 m/s, the mass ratio would become\n\nM 1 /M 0 = exp ( 8100 / 4000) = 0.1320\n\nIf we still want a 1% payload in orbit, the payload MP0, ejected at 9000 8100= 900 m/s short of orbital velocity, must carry an additional propulsion system. Assuming a storable bipropellant system having an effective exhaust velocity of V E2 = 3000 m/s, the mass fraction for the payload is\n\nMP 1 /MP 0 = exp ( 900 / 3000) = 0.7408\n\nThe mass ejected 900 m/s short of orbital velocity must therefore be 1 / 0.7408=1.350% of the launcher take-off mass to again have a final payload mass in orbit of 1%. The net mass ratio remaining for the launcher itself becomes\n\n13.20% 1.350% = 11.85%\n\nwhich represents a gain of\n\n(11.85 9.54) / 9.54 = 24.2%\n\nwith respect to that required for the pure SSTO-RRL.\n\nTherefore, reducing Delta V is as effective in creating a mass margin as is increasing V E. In more general terms, reducing Delta V leads to having more than one stage to reach orbit. As each stage has to deliver only a fraction of the total Delta V, its mass fraction can increase and a more robust design becomes possible.\n\nStaging options for RRLs\n\n(a) The SSTO-AL(air- launched)-RRL\nPrinciple: A carrier aircraft takes the SSTO-RRL to a certain altitude and speed, thereby reducing the remaining DV required to reach orbit. The Interim HOTOL carried on-top of an Antonov-225 aircraft is one such example (Fig. 3).\n\nHOTOL launched\nFigure 3. The Interim HOTOL launched from the Antonov 225 aircraft\n\nDrawbacks: The carrier aircraft has a limited payload capability, which constrains the SSTO-RRL's mass at separation. As a result, an air-launched SSTO-RRL becomes as difficult to build as a ground-launched version for which no take-off mass limitations apply. This comment is applicable to any upper stage of any Two Stage To Orbit (TSTO) system, because large vehicles are more effective and have a lower overall mass-ratio capability. Air-launching also involves a dangerous separation in the presence of aerodynamic flows.\n\nAdvantages: Air-launching gives launch azimuth flexibility and provides a self-contained flying launch base.\n\nConclusion: Not a promising solution for Europe.\n\n(b) The TSTO-RTL (return to launch site) and DRL(down- 0range-landing)-RRL\n\nPrinciple: Here, a reusable rocket first stage carries a reusable (or expendable) rocket second stage. In the RTL mode, the first stage returns to the launch site after separation. RTL is practical as long as the first stage does not re-enter too far down range, which corresponds to up to 1/4 of the total Delta V required to reach LEO. The FLS as studied by Aerospatiale for ESA in the early 1980s (Fig. 4), the RRL studied in the early 1990s (Fig. 5) and the FSS 9 as studied within FESTIP (Fig. 6) are vehicle's of this type.\n\nsemi-reusable 1980\nFigure 4. The semi-reusable future launching system studied by Aerospatiale in the early 1980s\n\nsemi-reusable 1990\nFigure 5. A semi-reusable rocket launcher as studied by Aerospatiale in the 1990s\n\nStudied FESTIP\nFigure 6. A fully reusable two-stage-to-orbit rocket launcher as studied within FESTIP\n\nIn the DRL mode, the first stage provides a larger share of the total Delta V and is allowed to land down-range of the launch site after separation and re-entry. The first stage returns to the launch site at a later time. The TARANIS, studied by Aerospatiale for ESA in the early 1990s (Fig. 7) was this type of vehicle.\n\nFigure 7. TAKANIS: a fully reusable rocket launcher with down- range landing of the first stage\n\nDrawbacks: There are now three aerodynamic configurations to be controlled - the first stage, the second stage when reusable, and the composite - two vehicles to be developed and qualified, and the amount of operations is increased. Staging is optimised for a given technology level and prevents one from deriving full benefit from any subsequent technology improvements. Because of the mass limitations on the second stage, its reusability represents a penalty rather than a cost advantage. A more effective solution is that of an expendable second stage, which could be considered an interesting interim solution.\n\nWhen implemented with today's technology, the TSTO-RTL/DRL-RRL is not much cheaper in use than an expendable launcher, but it can already offer higher reliability and safety. Launch/in-flight abortability remains limited.\n\nAdvantages: Feasible in the near term, if needed.\n\nConclusion: Not an interesting solution for Europe, in that the expendable Ariane-5 is already a modern launcher with significant evolution potential.\n\n(c) The SOSS (Sub-Orbital Single Stage) OA (Once- Around) RRL\n\nPrinciple: The down-range capability of the first stage is now extended to the point where it can complete one trip around the Earth so as to land back at the launch base (Fig. 8). The first stage therefore provides almost the total Delta V needed to reach orbit and the payload, ejected in vacuum but still at sub-orbital speed, achieves orbital velocity with its own propulsion system. This principle is comparable to that of the SSTO-AL-RRL, except that now the Delta V complement provided by the carrier aircraft (a large and expensive item) is provided by the payload itself (a small system) and the large sub-orbital stage itself retains the freedom of unconstrained mass at take-off.\n\nonce-around Earth\nFigure 8. The trajectory of a once-around Earth sub-orbital single-stage reusable launcher\n\nDrawbacks: The payload must provide its propulsion into orbit (from 300 to 3000 m/s) to compensate for the Delta V deficit of the sub-orbital stage. Not all payloads might welcome this, but this requirement could be taken into account when designing them.\n\nAdvantages: For the payload: For the majority of payloads, LEO is not the ultimate destination and many of them already carry an integrated propulsion stage to acquire higher energy orbits. Adding the need to provide the initial impulse to reach orbit therefore does not represent an excessive penalty.\n\nFor the launcher: The launcher is now back at the launch site after one revolution. The Delta V needed to circularise and to de-orbit is saved, there is no launcher in- orbit phase with the associated functions and mass, and a high launcher efficiency is therefore possible.\n\nConclusion: The author believes that the SOSS-OA-RRL (sub-orbital single-stage once-around reusable rocket launcher) can be a very promising solution, the particular features of which are discussed below.\n\nOperating mode of the SOSS-OA-RRL\n\nLet us start by defining a large reusable stage with a given take-off mass, of say 500 tons. Since we do not yet have the technology to achieve the mass ratio necessary for true SSTO operation, we must design the launcher to provide the largest Delta V it can, and leave it to the payload to provide the complement needed to achieve orbital conditions. Keeping the take-off mass constant allows us to freeze the launcher's external shape, so that the aero-dynamic database remains valid once established, and the installed thrust can be fixed. The launcher tankage is designed for the full volume of propellants ultimately required for SSTO, and the payload bay is given the volume needed by the largest possible combination of payload with its upper-stage propulsion.\n\nWe can now trade-off propellant masses between the launcher and the payload whilst keeping the total take-off mass constant. Depending on the sharing of Delta V between the launcher and the payload, we can achieve a range of payload-in-orbit capability. This approach therefore combines the flexibility and performance robustness of a pure TSTO system with the operational benefits of a single-stage vehicle. We refer to this approach as 'internal staging'.\n\nThe SOSS-OA-RRL, when already dimensioned for ultimate SSTO capability, can therefore become an interim single-stage vehicle with which practical experience can be accumulated early in a revenue-generating utilisation environment, whilst still preserving the possibility to upgrade it as technology progresses, without affecting its overall layout. With improving technology, the launcher dry-mass or the propellant mass needed will decrease, the launcher will reach higher speeds, and the propulsion requirements on the payload will decrease accordingly, until a pure SSTO capability becomes possible, should this indeed prove to be an economically viable option.\n\nIn a first approximation, the cost of using an SOSS-RRL is constant and independent of the mass of the payload with propulsion it carries, because its take-off mass is constant. The same SOSS-OA-RRL is used for each payload, irrespective of its mass, the only adaptation being the correct loading of propellants and the calculation of the corresponding trajectory. This repeated utilisation should lead to significant cost savings as experience accrues.\n\nOur SOSS-OA-RRL now resembles a truck which has a maximum payload, but can also operate profitably with a partial load.\n\nThe value of good high-speed aerodynamics\n\nThe velocity deficit of the SOSS-OA-RRL with respect to orbital velocity cannot be too large if we want the reusable launcher stage to complete one turn of the Earth and return to the launch base in a non-propulsive manner. The permissible deficit will depend on the vehicle's aerodynamic qualities during the high- speed re-entry phase: a greater lift-to-drag ratio at hypersonic speed permits a greater passive down-range capability, which permits a larger allowable velocity deficit with respect to orbital speed, which in turn permits a greater launcher dry-mass (at the expense of increased aerothermal problems). Suitable shapes for the hypersonic glide-back flight are lifting bodies (Fig. 9) or waveriders (Fig. 10).\n\nFigure 9. A lifting-body sub-orbital, once-around reusable rocket launcher studied within FESTIP\n\nFigure 10. A generic 'wave-rider' shape as studied within FESTIP\n\nAn SOSS-OA-RRL which needs a down-range capability to return to its base therefore makes constructive use of the kinetic energy it received during ascent, instead of simply destroying it as is the case for re-entry from orbital speed.\n\nIf the velocity deficit is such that even good aerodynamics do not allow a complete once-around glide, the stage can land after a partial orbit, but then there is the problem of returning it to the launch base, or we can equip it with an auxiliary propulsion system for a powered fly-back to complete the once- around. This powered fly-back could either be ensured by a scramjet for high-altitude, high-speed flight (although there might be thermal problems), or as a subsonic cruise on turbojet power (aerial refuelling with kerosene is a possibility). Again, the better the aerodynamics, the less penalising the cruise-back becomes.\n\nVertical versus horizontal take-off\n\nNow that we have a reason to require good aerodynamics for the return flight of the SOSS-OA-RRL, the next step is to use the installed aerodynamics for horizontal take-off (HTO) and lifting ascent on rocket power only. Indeed, the integrated velocity losses with HTO are a little less than with vertical take-off (VTO) on rocket power. A conventional SSTO-RRL must, however, rely on VTO because its required low mass fraction does not permit the implementation of lifting surfaces for ascent.\n\nFor a reusable vehicle, however, VTO has unpleasant implications. With VTO, the installed thrust which translates into number, mass and cost of engines must lift the vehicle off the pad, must provide the thrust needed to accelerate it, and there must also be a thrust margin for some engine-out capability. The total installed thrust then corresponds to about 1.5 times the take-off weight of the vehicle, only about a third of which does useful work in accelerating the vehicle during the initial phase of the mission. This installed thrust becomes excessive towards the end of the ascent and engines have either to be throttled back (cost) or shut-off (dead mass into orbit).\n\nBecause rocket engines must be located at the base of the vehicle, a high installed thrust shifts the centre of mass of the empty vehicle to the rear and compromises must be made in the aerodynamic shape in order to maintain a controllable vehicle at re-entry.\n\nEven when planning for engine-out situations, VTO remains a dangerous phase and while this is normal for an expendable launcher (if it has to fail, it does not matter where it fails), it is far from ideal for a reusable vehicle. Indeed, until there is enough dynamic pressure to provide aerodynamic control authority, the reusable launcher behaves like an expendable vehicle and the range-safety and vehicle destruct may have to be retained.\n\nFinally, VTO requires a heavy ground infrastructure which is both costly and limits the number of launch bases that one can build.\n\nThe SOSS-OA-RRL launcher, which needs aerodynamics for its return flight (and which can afford it because of its higher permissible mass fraction) allows one to exploit the Delta V savings of HTO. However, HTO also has more fundamental advantages. With HTO followed by lifting ascent, the weight of the vehicle is carried by the lift due to the forward speed and only the drag generated by that lift has to be compensated for by rocket thrust. Depending on the aerodynamic shape, the drag generated by the lift is only one half to one quarter of the vehicle's weight. Rocket thrust to accelerate the vehicle at the rate of 0.5 g at take-off is sufficient, so that the total rocket thrust needed by an HTO vehicle is about 0.5 + 1/2 to 1/4 =1.0 to 0.75 times the weight of the vehicle at take-off. This reduced thrust budget leads to associated reductions in engine mass (which improves the aerodynamic centering) and cost.\n\nHTO also provides an increased engine-out capability and therefore greater safety, at least when implemented as follows. The vehicle can rest on a passive trolley, on a linear ground track a few kilometres long. The vehicle accelerates under its own rocket power until the appropriate velocity is reached, at which point its nose is pushed upwards mechanically causing the vehicle to lift off from the trolley and continue its ascent. Operational advantages of this approach are:\n\nWith this scenario, rocket-powered HTO could offer the same thrust and safety margins as are now customary for multi-engined aviation.\n\nFinally, the HTO lifting ascent phase prolongs the time spent in the Earth's atmosphere by several minutes, which might allow some form of air-breathing rocket propulsion to be added to the vehicle at a later date, thereby improving the apparent specific impulse of the propulsion system and further increasing the allowable mass ratio.\n\nAn eventual boost-glide vehicle for global Earth travel?\n\nBy now, our proposed SOSS-OA-RRL has quite striking parallels with conventional aviation: horizontal processing, horizontal take-off, horizontal ascent, gliding or powered hyper-sonic flight, horizontal landing, and similar engine-out tolerances. The vehicle can cover all speed ranges from low subsonic to nearly orbital, and can evolve through the whole depth of the atmosphere. But such a vehicle might have other applications too, as it is not obliged to make a full revolution of the Earth: it could then serve as a high-speed global transport vehicle.\n\nIndeed, high-speed commercial transport (HSCT) programmes in the USA, Japan and Europe are already devoting significant funds to solving the fundamental technology problems, and the approach being pursued so far envisages cruising at higher speeds (up to Mach 5.5), as a direct extrapolation of today's lower speed aviation. At such speeds, however, global range trip times remain long, the aircraft cruises in contamination-sensitive layers of the upper atmosphere where its engines deposit emissions, the aircraft generates a sonic-boom carpet on the ground, which makes supersonic overflight of land masses unlikely, the aircraft faces severe thermal soaking problems, the propulsion still has to be developed to civilian standards, etc.\n\nThe boost-glide approach of the SOSS-OA-RRL is a logical alternative for such global travel: the vehicle is accelerated to high speed (Mach 10?,15?) by rocket (not necessarily O 2 /H 2 ) or by rocket plus air-breathing propulsion, follows a ballistic arc in near-vacuum, re-enters and dissipates its kinetic energy during an equilibrium glide at hypersonic speed and at high altitude. Once this energy has been dissipated, the vehicle can make a conventional landing like a normal subsonic aircraft.\n\nThe advantages of boost-glide for global travel are:\n\n\nAssembling the features described, one can conclude that, with today's emerging technologies, launcher reusability offers the opportunity to serve both the space-launch and global-travel markets with one type of vehicle: the sub- orbital single-stage reusable rocket launcher. The vehicles themselves would differ in detail, depending on their exact roles, but they would rely on a common technological base.\n\nSuch synergy would be to the benefit of both communities: it would enlarge the technology base available for space activities, it would spread the resulting development costs, it would amortise more rapidly all investments made, and it would stimulate a new approach to travelling around our planet, which in turn could prove a strong motivator for the younger generation to pursue a scientific/technical career.\n\nSpace has the potential to become a very large industrial park, stimulated by more routine and less costly access to space in the same way that global markets have been facilitated by the growth and falling cost of today's Earth-bound transport systems. Assessment of the possibilities for making this happen deserves commensurate spending for exploratory work. ESA's FESTIP programme will hopefully contribute to this goal.\n\nAbout | Search | Feedback\n\nRight Left Up Home ESA Bulletin Nr. 87.\nPublished August 1996.\nDeveloped by ESA-ESRIN ID/D."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "European Space Agency\n\nTowards Reusable Launchers - A Widening Perspective\n\nH. Pfeffer\n\nFuture Launchers Office, Directorate of Launchers, ESA, Paris\n\nOne of the prime tasks of the Agency, identified in the Report of ESA's Long-Term Space Policy Committee (LSPC) endorsed by the Ministers at their Toulouse Meeting in October last year, is to search for ways to reduce the cost of access to space and thereby open up new markets. One possible approach is that of launcher reuse. Because all such concepts envisaged so far require advances in technology before possibly becoming of potential economic interest, the Agency has begun a programme called FESTIP (Future European Space Transportation Investigations Programme), in which reusable launcher concepts are being studied and the corresponding technology work is being done.\nThis article presents some possible reusable launcher design options in the context of a scenario that might lead to a technological convergence between space vehicles and high-speed global aviation.\n\nThe ideal reusable launcher\n\nThe ideal reusable launcher would be analogous to a normal aircraft in that it would be able to take off from many possible bases on Earth, enter the desired orbital plane, accelerate to orbital velocity, release its payload, de-orbit, dissipate its kinetic energy and land at the take-off base to be readied for its next flight after a quick turn-around. The vehicle would maintain physical integrity during a mission, would be flown 'very often', and its cost per flight would be compatible with the value of the missions undertaken. The vehicle would also be able to abort its mission any time in the case of mishap and land intact with its payload. This ideal vehicle is the 'single-stage-to-orbit reusable rocket launcher', or SSTO-RRL.\n\nThe value of the 'single stage' concept is that the vehicle maintains its integrity throughout its lifetime. This integrity, which results in the minimisation of the inherent interfaces, is a major factor in the vehicle's dependability. Indeed, the majority of today's practical transport systems are also 'single- stage' vehicles, which include aircraft, ships, cars, etc.\n\nSo far, however, all space launchers are staged to orbit and expendable (with the exception of the US Space Shuttle System) and there is no SSTO-RRL as yet, despite its apparent desirability. Clearly, something must be standing in the way of achieving this 'ideal' vehicle.\n\nThe problems posed by SSTO-RRLs\n\nBecause rocket propulsion is mandatory to accelerate to orbital speed in vacuum, the most logical design option is to use rocket propulsion from take-off until orbit insertion. Both gravity and drag losses must be overcome on the trajectory to orbit. The ideal velocity increment, Delta V, required from an SSTO-RRL is then about 9000 m/s in order to reach a Low Earth Orbit (LEO). All further considerations concentrate on reaching LEO, because this is the most difficult part of gaining access to space and the major hurdle to be mastered in terms of reusability.\n\nThe mass that can be accelerated into orbit using rocket propulsion is given by the equation: M 1 /M 0 =exp ( -Delta V /V E ) where M 0 is the mass at take-off, M 1 is the mass which has received the ideal velocity increment Delta V, and V E is the ejection velocity of the rocket engine.\n\nFor a given Delta V, which is mission-imposed, the mass ratio M 1 /M 0 increases with increasing V E . The highest practical rocket ejection velocities are achieved by burning hydrogen with oxygen in a combustion chamber and ejecting the produced gases through a convergent/divergent nozzle. When averaged over the trajectory, the exhaust velocity V E is in the order of 4000 m/s. The corresponding mass ratio to reach LEO is: M 1 /M 0 =exp(Delta V /V E ) = exp ( 9000/4000) = 0.1054 = 10.54%\n\nThis means that 100 10.54 = 89.46% of the take-off mass must be made up of propellants, and that just 10.54% of the trade-off mass remains available for the tankage to contain the propellants, the engines, the structures, the equipment, and last but not least the payload. Assuming that the payload is 1% of the take-off mass, the mass fraction left available to build the complete launcher is 10.54 1 = 9.54%. This is not very much, and we have already assumed the highest performing propulsion system possible (cryogenic propulsion).\n\nMass ratios of the order of 10% are achieved today with expendable rocket stages, but are a nearly impossible requirement for a reusable vehicle, which is more stressed during its lifetime and which must carry the additional provisions for recovery and reuse. Because today's rocket propulsion is already quite near to its theoretical maximum, the whole burden of achieving such small mass ratios falls on the materials from which the launcher is built. Such launchers are, however, still studied within the FESTIP initiative (Fig. 1) in order to define the areas in which technology advances would be most effective.\n\nMeans for increasing the mass ratio\n\nIf the mass ratio allocated to the SSTO-RRL could be larger, the vehicle might become more feasible or more robust to use. The options offered by the rocket equation for increasing the mass ratio are to increase V E ,or to decrease Delta V, or a combination of the two.\n\nFigure 1. A possible single-stage-to-orbit launcher studied within ESA's FESTIP initiative\n\nIf V E were increased by 10% from 4000 m/s to 4400 m/s, the mass ratio would become\n\nM 11 /M 10 = exp ( 9000 / 4400) = 0.1293\n\nAgain assuming 1% for the payload, the mass available for the vehicle becomes\n\n12.93% 1% = 11.93%\n\nwhich represents a percentage increase of\n\n(11.93 9.54) / 9.54 = 25.1%\n\nwith respect to the nominal SSTO-RRL, which is quite a significant gain.\n\nIncreasing V E is equivalent to finding a propulsion system which produces the required thrust while consuming less propellant. For example, the apparent V E of propellants carried aboard a rocket increases when some of the products ejected are taken from the ambient air instead of having to be carried onboard from the start. In general, however, increasing the V E of a rocket motor with the help of the atmosphere is a complex problem to which Europe cannot expect an early solution. A typical configuration of this type, shown in Figure 2, is the SKYLON (UK), derived from HOTOL.\n\nrocket propulsion\nFigure 2. A possible single-stage-to-orbit launcher with combined air-breathing and rocket propulsion\n\nTurning now to the effect of a reduction in velocity increment, if the required Delta V were reduced by 10% from 9000 to 8100 m/s, the mass ratio would become\n\nM 1 /M 0 = exp ( 8100 / 4000) = 0.1320\n\nIf we still want a 1% payload in orbit, the payload MP0, ejected at 9000 8100= 900 m/s short of orbital velocity, must carry an additional propulsion system. Assuming a storable bipropellant system having an effective exhaust velocity of V E2 = 3000 m/s, the mass fraction for the payload is\n\nMP 1 /MP 0 = exp ( 900 / 3000) = 0.7408\n\nThe mass ejected 900 m/s short of orbital velocity must therefore be 1 / 0.7408=1.350% of the launcher take-off mass to again have a final payload mass in orbit of 1%. The net mass ratio remaining for the launcher itself becomes\n\n13.20% 1.350% = 11.85%\n\nwhich represents a gain of\n\n(11.85 9.54) / 9.54 = 24.2%\n\nwith respect to that required for the pure SSTO-RRL.\n\nTherefore, reducing Delta V is as effective in creating a mass margin as is increasing V E. In more general terms, reducing Delta V leads to having more than one stage to reach orbit. As each stage has to deliver only a fraction of the total Delta V, its mass fraction can increase and a more robust design becomes possible.\n\nStaging options for RRLs\n\n(a) The SSTO-AL(air- launched)-RRL\nPrinciple: A carrier aircraft takes the SSTO-RRL to a certain altitude and speed, thereby reducing the remaining DV required to reach orbit. The Interim HOTOL carried on-top of an Antonov-225 aircraft is one such example (Fig. 3).\n\nHOTOL launched\nFigure 3. The Interim HOTOL launched from the Antonov 225 aircraft\n\nDrawbacks: The carrier aircraft has a limited payload capability, which constrains the SSTO-RRL's mass at separation. As a result, an air-launched SSTO-RRL becomes as difficult to build as a ground-launched version for which no take-off mass limitations apply. This comment is applicable to any upper stage of any Two Stage To Orbit (TSTO) system, because large vehicles are more effective and have a lower overall mass-ratio capability. Air-launching also involves a dangerous separation in the presence of aerodynamic flows.\n\nAdvantages: Air-launching gives launch azimuth flexibility and provides a self-contained flying launch base.\n\nConclusion: Not a promising solution for Europe.\n\n(b) The TSTO-RTL (return to launch site) and DRL(down- 0range-landing)-RRL\n\nPrinciple: Here, a reusable rocket first stage carries a reusable (or expendable) rocket second stage. In the RTL mode, the first stage returns to the launch site after separation. RTL is practical as long as the first stage does not re-enter too far down range, which corresponds to up to 1/4 of the total Delta V required to reach LEO. The FLS as studied by Aerospatiale for ESA in the early 1980s (Fig. 4), the RRL studied in the early 1990s (Fig. 5) and the FSS 9 as studied within FESTIP (Fig. 6) are vehicle's of this type.\n\nsemi-reusable 1980\nFigure 4. The semi-reusable future launching system studied by Aerospatiale in the early 1980s\n\nsemi-reusable 1990\nFigure 5. A semi-reusable rocket launcher as studied by Aerospatiale in the 1990s\n\nStudied FESTIP\nFigure 6. A fully reusable two-stage-to-orbit rocket launcher as studied within FESTIP\n\nIn the DRL mode, the first stage provides a larger share of the total Delta V and is allowed to land down-range of the launch site after separation and re-entry. The first stage returns to the launch site at a later time. The TARANIS, studied by Aerospatiale for ESA in the early 1990s (Fig. 7) was this type of vehicle.\n\nFigure 7. TAKANIS: a fully reusable rocket launcher with down- range landing of the first stage\n\nDrawbacks: There are now three aerodynamic configurations to be controlled - the first stage, the second stage when reusable, and the composite - two vehicles to be developed and qualified, and the amount of operations is increased. Staging is optimised for a given technology level and prevents one from deriving full benefit from any subsequent technology improvements. Because of the mass limitations on the second stage, its reusability represents a penalty rather than a cost advantage. A more effective solution is that of an expendable second stage, which could be considered an interesting interim solution.\n\nWhen implemented with today's technology, the TSTO-RTL/DRL-RRL is not much cheaper in use than an expendable launcher, but it can already offer higher reliability and safety. Launch/in-flight abortability remains limited.\n\nAdvantages: Feasible in the near term, if needed.\n\nConclusion: Not an interesting solution for Europe, in that the expendable Ariane-5 is already a modern launcher with significant evolution potential.\n\n(c) The SOSS (Sub-Orbital Single Stage) OA (Once- Around) RRL\n\nPrinciple: The down-range capability of the first stage is now extended to the point where it can complete one trip around the Earth so as to land back at the launch base (Fig. 8). The first stage therefore provides almost the total Delta V needed to reach orbit and the payload, ejected in vacuum but still at sub-orbital speed, achieves orbital velocity with its own propulsion system. This principle is comparable to that of the SSTO-AL-RRL, except that now the Delta V complement provided by the carrier aircraft (a large and expensive item) is provided by the payload itself (a small system) and the large sub-orbital stage itself retains the freedom of unconstrained mass at take-off.\n\nonce-around Earth\nFigure 8. The trajectory of a once-around Earth sub-orbital single-stage reusable launcher\n\nDrawbacks: The payload must provide its propulsion into orbit (from 300 to 3000 m/s) to compensate for the Delta V deficit of the sub-orbital stage. Not all payloads might welcome this, but this requirement could be taken into account when designing them.\n\nAdvantages: For the payload: For the majority of payloads, LEO is not the ultimate destination and many of them already carry an integrated propulsion stage to acquire higher energy orbits. Adding the need to provide the initial impulse to reach orbit therefore does not represent an excessive penalty.\n\nFor the launcher: The launcher is now back at the launch site after one revolution. The Delta V needed to circularise and to de-orbit is saved, there is no launcher in- orbit phase with the associated functions and mass, and a high launcher efficiency is therefore possible.\n\nConclusion: The author believes that the SOSS-OA-RRL (sub-orbital single-stage once-around reusable rocket launcher) can be a very promising solution, the particular features of which are discussed below.\n\nOperating mode of the SOSS-OA-RRL\n\nLet us start by defining a large reusable stage with a given take-off mass, of say 500 tons. Since we do not yet have the technology to achieve the mass ratio necessary for true SSTO operation, we must design the launcher to provide the largest Delta V it can, and leave it to the payload to provide the complement needed to achieve orbital conditions. Keeping the take-off mass constant allows us to freeze the launcher's external shape, so that the aero-dynamic database remains valid once established, and the installed thrust can be fixed. The launcher tankage is designed for the full volume of propellants ultimately required for SSTO, and the payload bay is given the volume needed by the largest possible combination of payload with its upper-stage propulsion.\n\nWe can now trade-off propellant masses between the launcher and the payload whilst keeping the total take-off mass constant. Depending on the sharing of Delta V between the launcher and the payload, we can achieve a range of payload-in-orbit capability. This approach therefore combines the flexibility and performance robustness of a pure TSTO system with the operational benefits of a single-stage vehicle. We refer to this approach as 'internal staging'.\n\nThe SOSS-OA-RRL, when already dimensioned for ultimate SSTO capability, can therefore become an interim single-stage vehicle with which practical experience can be accumulated early in a revenue-generating utilisation environment, whilst still preserving the possibility to upgrade it as technology progresses, without affecting its overall layout. With improving technology, the launcher dry-mass or the propellant mass needed will decrease, the launcher will reach higher speeds, and the propulsion requirements on the payload will decrease accordingly, until a pure SSTO capability becomes possible, should this indeed prove to be an economically viable option.\n\nIn a first approximation, the cost of using an SOSS-RRL is constant and independent of the mass of the payload with propulsion it carries, because its take-off mass is constant. The same SOSS-OA-RRL is used for each payload, irrespective of its mass, the only adaptation being the correct loading of propellants and the calculation of the corresponding trajectory. This repeated utilisation should lead to significant cost savings as experience accrues.\n\nOur SOSS-OA-RRL now resembles a truck which has a maximum payload, but can also operate profitably with a partial load.\n\nThe value of good high-speed aerodynamics\n\nThe velocity deficit of the SOSS-OA-RRL with respect to orbital velocity cannot be too large if we want the reusable launcher stage to complete one turn of the Earth and return to the launch base in a non-propulsive manner. The permissible deficit will depend on the vehicle's aerodynamic qualities during the high- speed re-entry phase: a greater lift-to-drag ratio at hypersonic speed permits a greater passive down-range capability, which permits a larger allowable velocity deficit with respect to orbital speed, which in turn permits a greater launcher dry-mass (at the expense of increased aerothermal problems). Suitable shapes for the hypersonic glide-back flight are lifting bodies (Fig. 9) or waveriders (Fig. 10).\n\nFigure 9. A lifting-body sub-orbital, once-around reusable rocket launcher studied within FESTIP\n\nFigure 10. A generic 'wave-rider' shape as studied within FESTIP\n\nAn SOSS-OA-RRL which needs a down-range capability to return to its base therefore makes constructive use of the kinetic energy it received during ascent, instead of simply destroying it as is the case for re-entry from orbital speed.\n\nIf the velocity deficit is such that even good aerodynamics do not allow a complete once-around glide, the stage can land after a partial orbit, but then there is the problem of returning it to the launch base, or we can equip it with an auxiliary propulsion system for a powered fly-back to complete the once- around. This powered fly-back could either be ensured by a scramjet for high-altitude, high-speed flight (although there might be thermal problems), or as a subsonic cruise on turbojet power (aerial refuelling with kerosene is a possibility). Again, the better the aerodynamics, the less penalising the cruise-back becomes.\n\nVertical versus horizontal take-off\n\nNow that we have a reason to require good aerodynamics for the return flight of the SOSS-OA-RRL, the next step is to use the installed aerodynamics for horizontal take-off (HTO) and lifting ascent on rocket power only. Indeed, the integrated velocity losses with HTO are a little less than with vertical take-off (VTO) on rocket power. A conventional SSTO-RRL must, however, rely on VTO because its required low mass fraction does not permit the implementation of lifting surfaces for ascent.\n\nFor a reusable vehicle, however, VTO has unpleasant implications. With VTO, the installed thrust which translates into number, mass and cost of engines must lift the vehicle off the pad, must provide the thrust needed to accelerate it, and there must also be a thrust margin for some engine-out capability. The total installed thrust then corresponds to about 1.5 times the take-off weight of the vehicle, only about a third of which does useful work in accelerating the vehicle during the initial phase of the mission. This installed thrust becomes excessive towards the end of the ascent and engines have either to be throttled back (cost) or shut-off (dead mass into orbit).\n\nBecause rocket engines must be located at the base of the vehicle, a high installed thrust shifts the centre of mass of the empty vehicle to the rear and compromises must be made in the aerodynamic shape in order to maintain a controllable vehicle at re-entry.\n\nEven when planning for engine-out situations, VTO remains a dangerous phase and while this is normal for an expendable launcher (if it has to fail, it does not matter where it fails), it is far from ideal for a reusable vehicle. Indeed, until there is enough dynamic pressure to provide aerodynamic control authority, the reusable launcher behaves like an expendable vehicle and the range-safety and vehicle destruct may have to be retained.\n\nFinally, VTO requires a heavy ground infrastructure which is both costly and limits the number of launch bases that one can build.\n\nThe SOSS-OA-RRL launcher, which needs aerodynamics for its return flight (and which can afford it because of its higher permissible mass fraction) allows one to exploit the Delta V savings of HTO. However, HTO also has more fundamental advantages. With HTO followed by lifting ascent, the weight of the vehicle is carried by the lift due to the forward speed and only the drag generated by that lift has to be compensated for by rocket thrust. Depending on the aerodynamic shape, the drag generated by the lift is only one half to one quarter of the vehicle's weight. Rocket thrust to accelerate the vehicle at the rate of 0.5 g at take-off is sufficient, so that the total rocket thrust needed by an HTO vehicle is about 0.5 + 1/2 to 1/4 =1.0 to 0.75 times the weight of the vehicle at take-off. This reduced thrust budget leads to associated reductions in engine mass (which improves the aerodynamic centering) and cost.\n\nHTO also provides an increased engine-out capability and therefore greater safety, at least when implemented as follows. The vehicle can rest on a passive trolley, on a linear ground track a few kilometres long. The vehicle accelerates under its own rocket power until the appropriate velocity is reached, at which point its nose is pushed upwards mechanically causing the vehicle to lift off from the trolley and continue its ascent. Operational advantages of this approach are:\n\nWith this scenario, rocket-powered HTO could offer the same thrust and safety margins as are now customary for multi-engined aviation.\n\nFinally, the HTO lifting ascent phase prolongs the time spent in the Earth's atmosphere by several minutes, which might allow some form of air-breathing rocket propulsion to be added to the vehicle at a later date, thereby improving the apparent specific impulse of the propulsion system and further increasing the allowable mass ratio.\n\nAn eventual boost-glide vehicle for global Earth travel?\n\nBy now, our proposed SOSS-OA-RRL has quite striking parallels with conventional aviation: horizontal processing, horizontal take-off, horizontal ascent, gliding or powered hyper-sonic flight, horizontal landing, and similar engine-out tolerances. The vehicle can cover all speed ranges from low subsonic to nearly orbital, and can evolve through the whole depth of the atmosphere. But such a vehicle might have other applications too, as it is not obliged to make a full revolution of the Earth: it could then serve as a high-speed global transport vehicle.\n\nIndeed, high-speed commercial transport (HSCT) programmes in the USA, Japan and Europe are already devoting significant funds to solving the fundamental technology problems, and the approach being pursued so far envisages cruising at higher speeds (up to Mach 5.5), as a direct extrapolation of today's lower speed aviation. At such speeds, however, global range trip times remain long, the aircraft cruises in contamination-sensitive layers of the upper atmosphere where its engines deposit emissions, the aircraft generates a sonic-boom carpet on the ground, which makes supersonic overflight of land masses unlikely, the aircraft faces severe thermal soaking problems, the propulsion still has to be developed to civilian standards, etc.\n\nThe boost-glide approach of the SOSS-OA-RRL is a logical alternative for such global travel: the vehicle is accelerated to high speed (Mach 10?,15?) by rocket (not necessarily O 2 /H 2 ) or by rocket plus air-breathing propulsion, follows a ballistic arc in near-vacuum, re-enters and dissipates its kinetic energy during an equilibrium glide at hypersonic speed and at high altitude. Once this energy has been dissipated, the vehicle can make a conventional landing like a normal subsonic aircraft.\n\nThe advantages of boost-glide for global travel are:\n\n\nAssembling the features described, one can conclude that, with today's emerging technologies, launcher reusability offers the opportunity to serve both the space-launch and global-travel markets with one type of vehicle: the sub- orbital single-stage reusable rocket launcher. The vehicles themselves would differ in detail, depending on their exact roles, but they would rely on a common technological base.\n\nSuch synergy would be to the benefit of both communities: it would enlarge the technology base available for space activities, it would spread the resulting development costs, it would amortise more rapidly all investments made, and it would stimulate a new approach to travelling around our planet, which in turn could prove a strong motivator for the younger generation to pursue a scientific/technical career.\n\nSpace has the potential to become a very large industrial park, stimulated by more routine and less costly access to space in the same way that global markets have been facilitated by the growth and falling cost of today's Earth-bound transport systems. Assessment of the possibilities for making this happen deserves commensurate spending for exploratory work. ESA's FESTIP programme will hopefully contribute to this goal.\n\nAbout | Search | Feedback\n\nRight Left Up Home ESA Bulletin Nr. 87.\nPublished August 1996.\nDeveloped by ESA-ESRIN ID/D."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "From Wikipedia, the free encyclopedia\nJump to: navigation, search\nFor the business-related fill rate, see Service rate.\n\nThe term fillrate usually refers to the number of pixels a video card can render and write to video memory in a second. In this case, fillrates are given in megapixels per second or in gigapixels per second (in the case of newer cards), and they are obtained by multiplying the number of raster operations (ROPs) by the clock frequency of the graphics processor unit (GPU) of a video card. However, there is no agreement on how to calculate and report fillrates. Other possible methods are: to multiply the number of texture units by the clock frequency, or to multiply the number of pixel pipelines by the clock frequency. [1] The results of these multiplications correspond to a theoretical number. The actual fillrate depends on many other factors. In the past, the fillrate has been used as an indicator of performance by video card manufacturers such as ATI and NVIDIA, however, the importance of the fillrate as a measurement of performance has declined as the bottleneck in graphics applications has shifted. For example, today, the number and speed of pixel shader units has gained attention.\n\nScene complexity can be increased by overdrawing, which happens when \"an object is drawn to the frame buffer, and then another object (such as a wall) is drawn on top of it, covering it up. The time spent drawing the first object was wasted because it isn't visible.\" When a sequence of scenes is extremely complex (many pixels have to be drawn for each scene), the frame rate for the sequence may drop. When designing graphics intensive applications, one can determine whether the application is fillrate-limited by seeing if the frame rate increases dramatically when the application runs at a lower resolution or in a smaller window. [2]\n\nSee also[edit]\n\n\n 1. ^ Don Woligroski (July 31, 2006). Graphics Beginner's Guide, Part 2: Graphics Technology. Tom's Hardware. \n 2. ^ \"Fill rate\". DmWiki."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"index": 84,
"split": "train",
"text": "From Wikipedia, the free encyclopedia\nJump to: navigation, search\nFor the business-related fill rate, see Service rate.\n\nThe term fillrate usually refers to the number of pixels a video card can render and write to video memory in a second. In this case, fillrates are given in megapixels per second or in gigapixels per second (in the case of newer cards), and they are obtained by multiplying the number of raster operations (ROPs) by the clock frequency of the graphics processor unit (GPU) of a video card. However, there is no agreement on how to calculate and report fillrates. Other possible methods are: to multiply the number of texture units by the clock frequency, or to multiply the number of pixel pipelines by the clock frequency. [1] The results of these multiplications correspond to a theoretical number. The actual fillrate depends on many other factors. In the past, the fillrate has been used as an indicator of performance by video card manufacturers such as ATI and NVIDIA, however, the importance of the fillrate as a measurement of performance has declined as the bottleneck in graphics applications has shifted. For example, today, the number and speed of pixel shader units has gained attention.\n\nScene complexity can be increased by overdrawing, which happens when \"an object is drawn to the frame buffer, and then another object (such as a wall) is drawn on top of it, covering it up. The time spent drawing the first object was wasted because it isn't visible.\" When a sequence of scenes is extremely complex (many pixels have to be drawn for each scene), the frame rate for the sequence may drop. When designing graphics intensive applications, one can determine whether the application is fillrate-limited by seeing if the frame rate increases dramatically when the application runs at a lower resolution or in a smaller window. [2]\n\nSee also[edit]\n\n\n 1. ^ Don Woligroski (July 31, 2006). Graphics Beginner's Guide, Part 2: Graphics Technology. Tom's Hardware. \n 2. ^ \"Fill rate\". DmWiki."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 84,
"split": "train",
"text": "From Wikipedia, the free encyclopedia\nJump to: navigation, search\nFor the business-related fill rate, see Service rate.\n\nThe term fillrate usually refers to the number of pixels a video card can render and write to video memory in a second. In this case, fillrates are given in megapixels per second or in gigapixels per second (in the case of newer cards), and they are obtained by multiplying the number of raster operations (ROPs) by the clock frequency of the graphics processor unit (GPU) of a video card. However, there is no agreement on how to calculate and report fillrates. Other possible methods are: to multiply the number of texture units by the clock frequency, or to multiply the number of pixel pipelines by the clock frequency. [1] The results of these multiplications correspond to a theoretical number. The actual fillrate depends on many other factors. In the past, the fillrate has been used as an indicator of performance by video card manufacturers such as ATI and NVIDIA, however, the importance of the fillrate as a measurement of performance has declined as the bottleneck in graphics applications has shifted. For example, today, the number and speed of pixel shader units has gained attention.\n\nScene complexity can be increased by overdrawing, which happens when \"an object is drawn to the frame buffer, and then another object (such as a wall) is drawn on top of it, covering it up. The time spent drawing the first object was wasted because it isn't visible.\" When a sequence of scenes is extremely complex (many pixels have to be drawn for each scene), the frame rate for the sequence may drop. When designing graphics intensive applications, one can determine whether the application is fillrate-limited by seeing if the frame rate increases dramatically when the application runs at a lower resolution or in a smaller window. [2]\n\nSee also[edit]\n\n\n 1. ^ Don Woligroski (July 31, 2006). Graphics Beginner's Guide, Part 2: Graphics Technology. Tom's Hardware. \n 2. ^ \"Fill rate\". DmWiki."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "integer",
"index": 85,
"split": "train",
"text": "# Math Help Forum\n\n## Discrete Math\n### Combinatorics and Coins\n\n#### Posts\n\n1. **April 8th, 2012, 02:09 PM #1** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Question: If you toss 1000 fair coins 10 times each, what is the probability that *some* coin will get 10 heads?\n\n2. **April 8th, 2012, 02:15 PM #2** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Answer: Approximately 63%. And my question is, why?\n\n3. **April 8th, 2012, 02:16 PM #3** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Explanation: The use of *some* coin getting 10 heads suggests that you are interested in the probability that at least one coin gets ten heads. If we let \\( N \\) be the number of coins getting 10 heads, then \\( \\Pr(N \\geq 1) = 1 - \\Pr(N = 0) \\). Therefore, if we can find the probability that none of the 1000 coins get 10 heads, we can calculate your needed answer.\n\n4. **April 8th, 2012, 02:18 PM #4** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: I can't see if this problem is easier to get solved.\n\n5. **April 8th, 2012, 02:29 PM #5** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Detailed Solution: If we calculate the probability that one of the 1000 coins does not get 10 heads, then we raise this to the power of 1000 to get the probability that none of the 1000 coins get 10 heads as the events are independent. So, one way to find this would be to find the probabilities of getting 1 head, 2 heads, ..., 9 heads out of 10 and adding these together: \n \\[\n \\left(\\frac{1}{2}\\right) + \\left(\\frac{1}{2}\\right)^2 + \\ldots + \\left(\\frac{1}{2}\\right)^9 = 0.999023438\n \\] \n Or, we could calculate \\( 1 - \\) (Probability of getting 10 heads) = \\( 1 - \\left(\\frac{1}{2}\\right)^{10} = 0.999023438 \\). Now, we raise this to the power of 1000 as we have 1000 coins, and find the probability that none of the 1000 coins get 10 heads = \\( 0.999023438^{1000} = 0.376423986 \\), then take this away from 1 to find the probability that at least one gets 10 heads: \n \\[\n 1 - 0.376423986 = 0.623 = 62.3\\%\n \\] \n This is one of my first replies, so please feel free to ask questions if this isn't clear.\n\n6. **April 8th, 2012, 02:46 PM #6** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: It's a very clear solution and explanation! Thank you.\n\n7. **April 8th, 2012, 03:45 PM #7** \n **User:** Soroban \n **Status:** Super Member \n **Joined:** May 2006 \n **From:** Lexington, MA (USA) \n **Posts:** 11,546 \n **Thanks:** 539 \n\n Rehash Solution: Hello, mathquest! \n If you toss 1000 fair coins 10 times each, what is the probability that some coin will get 10 heads? \n The probability of one coin getting 10 Heads: \n \\[\n \\left(\\frac{1}{2}\\right)^{10} = \\frac{1}{1024}\n \\] \n The probability of a coin not getting 10 Heads: \n \\[\n 1 - \\frac{1}{1024} = \\frac{1023}{1024}\n \\] \n The probability of 1000 coins, not getting 10 Heads: \n \\[\n \\left(\\frac{1023}{1024}\\right)^{1000}\n \\] \n The probability of 1000 coins, some getting 10 Heads: \n \\[\n 1 - \\left(\\frac{1023}{1024}\\right)^{1000} = 0.623576202 \\approx 62.3\\%\n \\]\n\n8. **April 20th, 2012, 06:54 AM #8** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: Thank you, Soroban! Good rehash!"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "integer",
"index": 85,
"split": "train",
"text": "# Math Help Forum\n\n## Discrete Math\n### Combinatorics and Coins\n\n#### Posts\n\n1. **April 8th, 2012, 02:09 PM #1** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Question: If you toss 1000 fair coins 10 times each, what is the probability that *some* coin will get 10 heads?\n\n2. **April 8th, 2012, 02:15 PM #2** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Answer: Approximately 63%. And my question is, why?\n\n3. **April 8th, 2012, 02:16 PM #3** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Explanation: The use of *some* coin getting 10 heads suggests that you are interested in the probability that at least one coin gets ten heads. If we let \\( N \\) be the number of coins getting 10 heads, then \\( \\Pr(N \\geq 1) = 1 - \\Pr(N = 0) \\). Therefore, if we can find the probability that none of the 1000 coins get 10 heads, we can calculate your needed answer.\n\n4. **April 8th, 2012, 02:18 PM #4** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: I can't see if this problem is easier to get solved.\n\n5. **April 8th, 2012, 02:29 PM #5** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Detailed Solution: If we calculate the probability that one of the 1000 coins does not get 10 heads, then we raise this to the power of 1000 to get the probability that none of the 1000 coins get 10 heads as the events are independent. So, one way to find this would be to find the probabilities of getting 1 head, 2 heads, ..., 9 heads out of 10 and adding these together: \n \\[\n \\left(\\frac{1}{2}\\right) + \\left(\\frac{1}{2}\\right)^2 + \\ldots + \\left(\\frac{1}{2}\\right)^9 = 0.999023438\n \\] \n Or, we could calculate \\( 1 - \\) (Probability of getting 10 heads) = \\( 1 - \\left(\\frac{1}{2}\\right)^{10} = 0.999023438 \\). Now, we raise this to the power of 1000 as we have 1000 coins, and find the probability that none of the 1000 coins get 10 heads = \\( 0.999023438^{1000} = 0.376423986 \\), then take this away from 1 to find the probability that at least one gets 10 heads: \n \\[\n 1 - 0.376423986 = 0.623 = 62.3\\%\n \\] \n This is one of my first replies, so please feel free to ask questions if this isn't clear.\n\n6. **April 8th, 2012, 02:46 PM #6** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: It's a very clear solution and explanation! Thank you.\n\n7. **April 8th, 2012, 03:45 PM #7** \n **User:** Soroban \n **Status:** Super Member \n **Joined:** May 2006 \n **From:** Lexington, MA (USA) \n **Posts:** 11,546 \n **Thanks:** 539 \n\n Rehash Solution: Hello, mathquest! \n If you toss 1000 fair coins 10 times each, what is the probability that some coin will get 10 heads? \n The probability of one coin getting 10 Heads: \n \\[\n \\left(\\frac{1}{2}\\right)^{10} = \\frac{1}{1024}\n \\] \n The probability of a coin not getting 10 Heads: \n \\[\n 1 - \\frac{1}{1024} = \\frac{1023}{1024}\n \\] \n The probability of 1000 coins, not getting 10 Heads: \n \\[\n \\left(\\frac{1023}{1024}\\right)^{1000}\n \\] \n The probability of 1000 coins, some getting 10 Heads: \n \\[\n 1 - \\left(\\frac{1023}{1024}\\right)^{1000} = 0.623576202 \\approx 62.3\\%\n \\]\n\n8. **April 20th, 2012, 06:54 AM #8** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: Thank you, Soroban! Good rehash!"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 85,
"split": "train",
"text": "# Math Help Forum\n\n## Discrete Math\n### Combinatorics and Coins\n\n#### Posts\n\n1. **April 8th, 2012, 02:09 PM #1** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Question: If you toss 1000 fair coins 10 times each, what is the probability that *some* coin will get 10 heads?\n\n2. **April 8th, 2012, 02:15 PM #2** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Answer: Approximately 63%. And my question is, why?\n\n3. **April 8th, 2012, 02:16 PM #3** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Explanation: The use of *some* coin getting 10 heads suggests that you are interested in the probability that at least one coin gets ten heads. If we let \\( N \\) be the number of coins getting 10 heads, then \\( \\Pr(N \\geq 1) = 1 - \\Pr(N = 0) \\). Therefore, if we can find the probability that none of the 1000 coins get 10 heads, we can calculate your needed answer.\n\n4. **April 8th, 2012, 02:18 PM #4** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: I can't see if this problem is easier to get solved.\n\n5. **April 8th, 2012, 02:29 PM #5** \n **User:** Milokerr90 \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** UK \n **Posts:** 2 \n **Thanks:** 1 \n\n Detailed Solution: If we calculate the probability that one of the 1000 coins does not get 10 heads, then we raise this to the power of 1000 to get the probability that none of the 1000 coins get 10 heads as the events are independent. So, one way to find this would be to find the probabilities of getting 1 head, 2 heads, ..., 9 heads out of 10 and adding these together: \n \\[\n \\left(\\frac{1}{2}\\right) + \\left(\\frac{1}{2}\\right)^2 + \\ldots + \\left(\\frac{1}{2}\\right)^9 = 0.999023438\n \\] \n Or, we could calculate \\( 1 - \\) (Probability of getting 10 heads) = \\( 1 - \\left(\\frac{1}{2}\\right)^{10} = 0.999023438 \\). Now, we raise this to the power of 1000 as we have 1000 coins, and find the probability that none of the 1000 coins get 10 heads = \\( 0.999023438^{1000} = 0.376423986 \\), then take this away from 1 to find the probability that at least one gets 10 heads: \n \\[\n 1 - 0.376423986 = 0.623 = 62.3\\%\n \\] \n This is one of my first replies, so please feel free to ask questions if this isn't clear.\n\n6. **April 8th, 2012, 02:46 PM #6** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: It's a very clear solution and explanation! Thank you.\n\n7. **April 8th, 2012, 03:45 PM #7** \n **User:** Soroban \n **Status:** Super Member \n **Joined:** May 2006 \n **From:** Lexington, MA (USA) \n **Posts:** 11,546 \n **Thanks:** 539 \n\n Rehash Solution: Hello, mathquest! \n If you toss 1000 fair coins 10 times each, what is the probability that some coin will get 10 heads? \n The probability of one coin getting 10 Heads: \n \\[\n \\left(\\frac{1}{2}\\right)^{10} = \\frac{1}{1024}\n \\] \n The probability of a coin not getting 10 Heads: \n \\[\n 1 - \\frac{1}{1024} = \\frac{1023}{1024}\n \\] \n The probability of 1000 coins, not getting 10 Heads: \n \\[\n \\left(\\frac{1023}{1024}\\right)^{1000}\n \\] \n The probability of 1000 coins, some getting 10 Heads: \n \\[\n 1 - \\left(\\frac{1023}{1024}\\right)^{1000} = 0.623576202 \\approx 62.3\\%\n \\]\n\n8. **April 20th, 2012, 06:54 AM #8** \n **User:** mathquest \n **Status:** Newbie \n **Joined:** Apr 2012 \n **From:** Greece \n **Posts:** 8 \n\n Comment: Thank you, Soroban! Good rehash!"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "integer",
"index": 86,
"split": "train",
"text": "# Show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots\n\n**Problem Statement:**\n\nShow that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\).\n\n**Discussion:**\n\nOriginally Posted by fardeen_gen:\n\nTo show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\), consider the following steps:\n\nStarting with the equation:\n\\[\n\\frac{1}{\\sin \\theta} + \\frac{1}{\\cos \\theta} = c\n\\]\n\nSquare the equation:\n\\[\n\\left(\\frac{1}{\\sin^2 \\theta} + \\frac{1}{\\cos^2 \\theta}\\right) + \\frac{2}{\\sin \\theta \\cos \\theta} = c^2\n\\]\n\nUsing the identity \\(\\sin(2\\theta) = 2 \\cos(\\theta) \\sin(\\theta)\\), we have:\n\\[\n\\cos^2(\\theta) \\sin^2(\\theta) = \\frac{\\sin^2(2\\theta)}{4}\n\\]\n\nThus:\n\\[\nA = \\frac{\\sin^2 \\theta + \\cos^2 \\theta}{\\sin^2 \\theta \\cos^2 \\theta} = \\frac{1}{\\frac{\\sin^2(2\\theta)}{4}} = \\frac{4}{\\sin^2(2\\theta)}\n\\]\n\nAnd:\n\\[\nB = \\frac{2}{\\sin \\theta \\cos \\theta} = \\frac{2}{\\frac{\\sin(2\\theta)}{2}} = \\frac{4}{\\sin(2\\theta)}\n\\]\n\nThe equation becomes:\n\\[\n\\frac{4}{\\sin^2(2\\theta)} + \\frac{4}{\\sin(2\\theta)} = c^2\n\\]\n\nMultiply both sides by \\(\\sin^2(2\\theta)\\):\n\\[\n4 + 4 \\sin(2\\theta) = c^2 \\cdot \\sin^2(2\\theta)\n\\]\n\nLet \\( X = \\sin(2\\theta) \\). The equation becomes:\n\\[\nc^2 X^2 - 4X - 4 = 0\n\\]\n\nThe discriminant is:\n\\[\n\\Delta = 16 + 4 \\cdot 4 \\cdot c^2 = 16(1 + c^2)\n\\]\n\nSince \\(\\Delta\\) is positive, the solutions are:\n\\[\nX = \\frac{4 \\pm 4 \\sqrt{1+c^2}}{2c^2} = \\frac{2 \\pm 2 \\sqrt{1+c^2}}{c^2}\n\\]\n\nWe need \\( -1 \\leq X \\leq 1 \\).\n\n**Solution:**\n\nFor \\( 2 < c^2 < 8 \\), the inequality:\n\\[\n\\left|2 \\pm 2 \\sqrt{1+c^2}\\right| \\leq c^2\n\\]\n\n1. \\( 2 + 2 \\sqrt{1+c^2} \\leq c^2 \\)\n - \\( 2 + 2 \\sqrt{1+c^2} - c^2 \\leq 0 \\)\n - \\( 2 < c^2 < 8 \\) implies \\( \\sqrt{1+c^2} < 3 \\)\n - \\( 2 + 2\\sqrt{1+c^2} < 8 \\)\n - \\( 2 + 2\\sqrt{1+c^2} - c^2 \\leq 6 \\)\n\n2. \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| \\leq c^2 \\)\n - \\( \\sqrt{1+c^2} \\geq 1 \\) implies \\( 2 - 2\\sqrt{1+c^2} \\leq 0 \\)\n - \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| = 2 \\sqrt{1+c^2} - 2 \\)\n - \\( 2 \\sqrt{1+c^2} - 2 \\leq c^2 \\)\n - \\( 2 \\sqrt{1+c^2} - c^2 \\leq 2 \\)\n\n**Alternative Proof:**\n\nUsing the graph, for \\( |c| < 2\\sqrt{2} \\), the equation:\n\\[\n\\frac{1}{\\sin x} + \\frac{1}{\\cos x} = c\n\\]\nhas two solutions. Define \\( f(x) = \\frac{1}{\\sin x} + \\frac{1}{\\cos x} \\).\n\nThe derivative:\n\\[\nf'(x) = \\frac{\\sin^3 x - \\cos^3 x}{\\sin^2 x \\cos^2 x} = \\frac{(\\sin x - \\cos x)(1 + \\sin x \\cos x)}{\\sin^2 x \\cos^2 x}\n\\]\n\n\\( f'(x) = 0 \\) when \\( \\sin x = \\cos x \\), i.e., \\( x = \\frac{\\pi}{4} \\) or \\( x = \\frac{5\\pi}{4} \\).\n\n\\( f\\left(\\frac{\\pi}{4}\\right) = 2\\sqrt{2} \\) is the minimum, and \\( f\\left(\\frac{5\\pi}{4}\\right) = -2\\sqrt{2} \\) is the maximum.\n\nFor \\( x \\in \\left(\\frac{\\pi}{2}, \\pi\\right) \\), \\( f'(x) > 0 \\), so \\( f \\) is increasing. By the intermediate value theorem, \\( f(x) = c \\) has exactly one solution.\n\nSimilarly, \\( f(x) = c \\) has one solution in \\( \\left(\\frac{3\\pi}{2}, 2\\pi\\right) \\).\n\nThus, if \\( |c| < 2\\sqrt{2} \\), the equation has two solutions, satisfying \\( 2 < c^2 < 8 \\)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
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"answer_type": "integer",
"index": 86,
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"text": "# Show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots\n\n**Problem Statement:**\n\nShow that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\).\n\n**Discussion:**\n\nOriginally Posted by fardeen_gen:\n\nTo show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\), consider the following steps:\n\nStarting with the equation:\n\\[\n\\frac{1}{\\sin \\theta} + \\frac{1}{\\cos \\theta} = c\n\\]\n\nSquare the equation:\n\\[\n\\left(\\frac{1}{\\sin^2 \\theta} + \\frac{1}{\\cos^2 \\theta}\\right) + \\frac{2}{\\sin \\theta \\cos \\theta} = c^2\n\\]\n\nUsing the identity \\(\\sin(2\\theta) = 2 \\cos(\\theta) \\sin(\\theta)\\), we have:\n\\[\n\\cos^2(\\theta) \\sin^2(\\theta) = \\frac{\\sin^2(2\\theta)}{4}\n\\]\n\nThus:\n\\[\nA = \\frac{\\sin^2 \\theta + \\cos^2 \\theta}{\\sin^2 \\theta \\cos^2 \\theta} = \\frac{1}{\\frac{\\sin^2(2\\theta)}{4}} = \\frac{4}{\\sin^2(2\\theta)}\n\\]\n\nAnd:\n\\[\nB = \\frac{2}{\\sin \\theta \\cos \\theta} = \\frac{2}{\\frac{\\sin(2\\theta)}{2}} = \\frac{4}{\\sin(2\\theta)}\n\\]\n\nThe equation becomes:\n\\[\n\\frac{4}{\\sin^2(2\\theta)} + \\frac{4}{\\sin(2\\theta)} = c^2\n\\]\n\nMultiply both sides by \\(\\sin^2(2\\theta)\\):\n\\[\n4 + 4 \\sin(2\\theta) = c^2 \\cdot \\sin^2(2\\theta)\n\\]\n\nLet \\( X = \\sin(2\\theta) \\). The equation becomes:\n\\[\nc^2 X^2 - 4X - 4 = 0\n\\]\n\nThe discriminant is:\n\\[\n\\Delta = 16 + 4 \\cdot 4 \\cdot c^2 = 16(1 + c^2)\n\\]\n\nSince \\(\\Delta\\) is positive, the solutions are:\n\\[\nX = \\frac{4 \\pm 4 \\sqrt{1+c^2}}{2c^2} = \\frac{2 \\pm 2 \\sqrt{1+c^2}}{c^2}\n\\]\n\nWe need \\( -1 \\leq X \\leq 1 \\).\n\n**Solution:**\n\nFor \\( 2 < c^2 < 8 \\), the inequality:\n\\[\n\\left|2 \\pm 2 \\sqrt{1+c^2}\\right| \\leq c^2\n\\]\n\n1. \\( 2 + 2 \\sqrt{1+c^2} \\leq c^2 \\)\n - \\( 2 + 2 \\sqrt{1+c^2} - c^2 \\leq 0 \\)\n - \\( 2 < c^2 < 8 \\) implies \\( \\sqrt{1+c^2} < 3 \\)\n - \\( 2 + 2\\sqrt{1+c^2} < 8 \\)\n - \\( 2 + 2\\sqrt{1+c^2} - c^2 \\leq 6 \\)\n\n2. \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| \\leq c^2 \\)\n - \\( \\sqrt{1+c^2} \\geq 1 \\) implies \\( 2 - 2\\sqrt{1+c^2} \\leq 0 \\)\n - \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| = 2 \\sqrt{1+c^2} - 2 \\)\n - \\( 2 \\sqrt{1+c^2} - 2 \\leq c^2 \\)\n - \\( 2 \\sqrt{1+c^2} - c^2 \\leq 2 \\)\n\n**Alternative Proof:**\n\nUsing the graph, for \\( |c| < 2\\sqrt{2} \\), the equation:\n\\[\n\\frac{1}{\\sin x} + \\frac{1}{\\cos x} = c\n\\]\nhas two solutions. Define \\( f(x) = \\frac{1}{\\sin x} + \\frac{1}{\\cos x} \\).\n\nThe derivative:\n\\[\nf'(x) = \\frac{\\sin^3 x - \\cos^3 x}{\\sin^2 x \\cos^2 x} = \\frac{(\\sin x - \\cos x)(1 + \\sin x \\cos x)}{\\sin^2 x \\cos^2 x}\n\\]\n\n\\( f'(x) = 0 \\) when \\( \\sin x = \\cos x \\), i.e., \\( x = \\frac{\\pi}{4} \\) or \\( x = \\frac{5\\pi}{4} \\).\n\n\\( f\\left(\\frac{\\pi}{4}\\right) = 2\\sqrt{2} \\) is the minimum, and \\( f\\left(\\frac{5\\pi}{4}\\right) = -2\\sqrt{2} \\) is the maximum.\n\nFor \\( x \\in \\left(\\frac{\\pi}{2}, \\pi\\right) \\), \\( f'(x) > 0 \\), so \\( f \\) is increasing. By the intermediate value theorem, \\( f(x) = c \\) has exactly one solution.\n\nSimilarly, \\( f(x) = c \\) has one solution in \\( \\left(\\frac{3\\pi}{2}, 2\\pi\\right) \\).\n\nThus, if \\( |c| < 2\\sqrt{2} \\), the equation has two solutions, satisfying \\( 2 < c^2 < 8 \\)."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots\n\n**Problem Statement:**\n\nShow that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\).\n\n**Discussion:**\n\nOriginally Posted by fardeen_gen:\n\nTo show that the equation \\( \\sec \\theta + \\csc \\theta = c \\) has two roots between 0 and \\( 2\\pi \\) if \\( 2 < c^2 < 8 \\), consider the following steps:\n\nStarting with the equation:\n\\[\n\\frac{1}{\\sin \\theta} + \\frac{1}{\\cos \\theta} = c\n\\]\n\nSquare the equation:\n\\[\n\\left(\\frac{1}{\\sin^2 \\theta} + \\frac{1}{\\cos^2 \\theta}\\right) + \\frac{2}{\\sin \\theta \\cos \\theta} = c^2\n\\]\n\nUsing the identity \\(\\sin(2\\theta) = 2 \\cos(\\theta) \\sin(\\theta)\\), we have:\n\\[\n\\cos^2(\\theta) \\sin^2(\\theta) = \\frac{\\sin^2(2\\theta)}{4}\n\\]\n\nThus:\n\\[\nA = \\frac{\\sin^2 \\theta + \\cos^2 \\theta}{\\sin^2 \\theta \\cos^2 \\theta} = \\frac{1}{\\frac{\\sin^2(2\\theta)}{4}} = \\frac{4}{\\sin^2(2\\theta)}\n\\]\n\nAnd:\n\\[\nB = \\frac{2}{\\sin \\theta \\cos \\theta} = \\frac{2}{\\frac{\\sin(2\\theta)}{2}} = \\frac{4}{\\sin(2\\theta)}\n\\]\n\nThe equation becomes:\n\\[\n\\frac{4}{\\sin^2(2\\theta)} + \\frac{4}{\\sin(2\\theta)} = c^2\n\\]\n\nMultiply both sides by \\(\\sin^2(2\\theta)\\):\n\\[\n4 + 4 \\sin(2\\theta) = c^2 \\cdot \\sin^2(2\\theta)\n\\]\n\nLet \\( X = \\sin(2\\theta) \\). The equation becomes:\n\\[\nc^2 X^2 - 4X - 4 = 0\n\\]\n\nThe discriminant is:\n\\[\n\\Delta = 16 + 4 \\cdot 4 \\cdot c^2 = 16(1 + c^2)\n\\]\n\nSince \\(\\Delta\\) is positive, the solutions are:\n\\[\nX = \\frac{4 \\pm 4 \\sqrt{1+c^2}}{2c^2} = \\frac{2 \\pm 2 \\sqrt{1+c^2}}{c^2}\n\\]\n\nWe need \\( -1 \\leq X \\leq 1 \\).\n\n**Solution:**\n\nFor \\( 2 < c^2 < 8 \\), the inequality:\n\\[\n\\left|2 \\pm 2 \\sqrt{1+c^2}\\right| \\leq c^2\n\\]\n\n1. \\( 2 + 2 \\sqrt{1+c^2} \\leq c^2 \\)\n - \\( 2 + 2 \\sqrt{1+c^2} - c^2 \\leq 0 \\)\n - \\( 2 < c^2 < 8 \\) implies \\( \\sqrt{1+c^2} < 3 \\)\n - \\( 2 + 2\\sqrt{1+c^2} < 8 \\)\n - \\( 2 + 2\\sqrt{1+c^2} - c^2 \\leq 6 \\)\n\n2. \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| \\leq c^2 \\)\n - \\( \\sqrt{1+c^2} \\geq 1 \\) implies \\( 2 - 2\\sqrt{1+c^2} \\leq 0 \\)\n - \\( \\left|2 - 2 \\sqrt{1+c^2}\\right| = 2 \\sqrt{1+c^2} - 2 \\)\n - \\( 2 \\sqrt{1+c^2} - 2 \\leq c^2 \\)\n - \\( 2 \\sqrt{1+c^2} - c^2 \\leq 2 \\)\n\n**Alternative Proof:**\n\nUsing the graph, for \\( |c| < 2\\sqrt{2} \\), the equation:\n\\[\n\\frac{1}{\\sin x} + \\frac{1}{\\cos x} = c\n\\]\nhas two solutions. Define \\( f(x) = \\frac{1}{\\sin x} + \\frac{1}{\\cos x} \\).\n\nThe derivative:\n\\[\nf'(x) = \\frac{\\sin^3 x - \\cos^3 x}{\\sin^2 x \\cos^2 x} = \\frac{(\\sin x - \\cos x)(1 + \\sin x \\cos x)}{\\sin^2 x \\cos^2 x}\n\\]\n\n\\( f'(x) = 0 \\) when \\( \\sin x = \\cos x \\), i.e., \\( x = \\frac{\\pi}{4} \\) or \\( x = \\frac{5\\pi}{4} \\).\n\n\\( f\\left(\\frac{\\pi}{4}\\right) = 2\\sqrt{2} \\) is the minimum, and \\( f\\left(\\frac{5\\pi}{4}\\right) = -2\\sqrt{2} \\) is the maximum.\n\nFor \\( x \\in \\left(\\frac{\\pi}{2}, \\pi\\right) \\), \\( f'(x) > 0 \\), so \\( f \\) is increasing. By the intermediate value theorem, \\( f(x) = c \\) has exactly one solution.\n\nSimilarly, \\( f(x) = c \\) has one solution in \\( \\left(\\frac{3\\pi}{2}, 2\\pi\\right) \\).\n\nThus, if \\( |c| < 2\\sqrt{2} \\), the equation has two solutions, satisfying \\( 2 < c^2 < 8 \\)."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# The Fundamentals of Calculus\n\n## Introduction to Calculus\n\nCalculus is a branch of mathematics that studies changes and motion. It is divided into two main areas: differential calculus and integral calculus. Differential calculus concerns the concept of a derivative, which represents an instantaneous rate of change. Integral calculus, on the other hand, deals with the accumulation of quantities and the areas under and between curves.\n\n## The Derivative\n\nThe derivative of a function at a certain point is the slope of the tangent line to the function at that point. This can be mathematically expressed as:\n\n\\[ f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\]\n\nA common example is the derivative of the function \\( f(x) = x^2 \\), which is:\n\n\\[ f'(x) = 2x \\]\n\n### Applications of Derivatives\n\n1. **Motion**: Derivatives can describe velocity and acceleration.\n2. **Optimization**: Used to find maximum and minimum values of functions.\n3. **Curve Sketching**: Helps in understanding the behavior of graphs.\n\n## The Integral\n\nThe integral is essentially the inverse operation of the derivative. While the derivative represents a rate of change, the integral represents a cumulative total. The definite integral of a function \\( f(x) \\) from \\( a \\) to \\( b \\) is given by:\n\n\\[ \\int_a^b f(x) \\, dx \\]\n\nFor example, the integral of \\( f(x) = x^2 \\) from 0 to 1 is:\n\n\\[ \\int_0^1 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{3} \\]\n\n### Applications of Integrals\n\n1. **Area Under a Curve**: Calculating the area between a curve and the x-axis.\n2. **Physics**: Used in calculating quantities like displacement and work.\n3. **Probability**: Integral calculus is used to find probabilities in continuous distributions.\n\n## The Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus links the concept of differentiation and integration. It has two parts:\n\n1. **First Part**: If \\( F \\) is an antiderivative of \\( f \\) on an interval \\([a, b]\\), then:\n\n \\[ \\int_a^b f(x) \\, dx = F(b) - F(a) \\]\n\n2. **Second Part**: If \\( f \\) is continuous on \\([a, b]\\) and \\( F(x) = \\int_a^x f(t) \\, dt \\), then \\( F \\) is differentiable on \\((a, b)\\) and \\( F'(x) = f(x) \\).\n\n## Example: Calculating the Area Between Two Curves\n\nTo find the area between the curves \\( y = x^2 \\) and \\( y = x \\) from \\( x = 0 \\) to \\( x = 1 \\), we set up the integral:\n\n\\[ \\text{Area} = \\int_0^1 (x - x^2) \\, dx = \\left[ \\frac{x^2}{2} - \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{2} - \\frac{1}{3} = \\frac{1}{6} \\]\n\n## Conclusion\n\nCalculus is a powerful mathematical tool with a wide range of applications. Understanding its fundamental concepts of derivatives and integrals allows us to model and solve real-world problems involving change and accumulation.\n\n## References\n\n1. Stewart, J. (2008). *Calculus: Early Transcendentals*. Cengage Learning.\n2. Thomas, G. B., Weir, M. D., & Hass, J. (2014). *Thomas' Calculus*. Pearson.\n\n### Author Note\n\nThe content above is a simplified overview of calculus intended for educational purposes. For a deeper understanding, it is recommended to refer to comprehensive calculus textbooks and resources."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# The Fundamentals of Calculus\n\n## Introduction to Calculus\n\nCalculus is a branch of mathematics that studies changes and motion. It is divided into two main areas: differential calculus and integral calculus. Differential calculus concerns the concept of a derivative, which represents an instantaneous rate of change. Integral calculus, on the other hand, deals with the accumulation of quantities and the areas under and between curves.\n\n## The Derivative\n\nThe derivative of a function at a certain point is the slope of the tangent line to the function at that point. This can be mathematically expressed as:\n\n\\[ f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\]\n\nA common example is the derivative of the function \\( f(x) = x^2 \\), which is:\n\n\\[ f'(x) = 2x \\]\n\n### Applications of Derivatives\n\n1. **Motion**: Derivatives can describe velocity and acceleration.\n2. **Optimization**: Used to find maximum and minimum values of functions.\n3. **Curve Sketching**: Helps in understanding the behavior of graphs.\n\n## The Integral\n\nThe integral is essentially the inverse operation of the derivative. While the derivative represents a rate of change, the integral represents a cumulative total. The definite integral of a function \\( f(x) \\) from \\( a \\) to \\( b \\) is given by:\n\n\\[ \\int_a^b f(x) \\, dx \\]\n\nFor example, the integral of \\( f(x) = x^2 \\) from 0 to 1 is:\n\n\\[ \\int_0^1 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{3} \\]\n\n### Applications of Integrals\n\n1. **Area Under a Curve**: Calculating the area between a curve and the x-axis.\n2. **Physics**: Used in calculating quantities like displacement and work.\n3. **Probability**: Integral calculus is used to find probabilities in continuous distributions.\n\n## The Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus links the concept of differentiation and integration. It has two parts:\n\n1. **First Part**: If \\( F \\) is an antiderivative of \\( f \\) on an interval \\([a, b]\\), then:\n\n \\[ \\int_a^b f(x) \\, dx = F(b) - F(a) \\]\n\n2. **Second Part**: If \\( f \\) is continuous on \\([a, b]\\) and \\( F(x) = \\int_a^x f(t) \\, dt \\), then \\( F \\) is differentiable on \\((a, b)\\) and \\( F'(x) = f(x) \\).\n\n## Example: Calculating the Area Between Two Curves\n\nTo find the area between the curves \\( y = x^2 \\) and \\( y = x \\) from \\( x = 0 \\) to \\( x = 1 \\), we set up the integral:\n\n\\[ \\text{Area} = \\int_0^1 (x - x^2) \\, dx = \\left[ \\frac{x^2}{2} - \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{2} - \\frac{1}{3} = \\frac{1}{6} \\]\n\n## Conclusion\n\nCalculus is a powerful mathematical tool with a wide range of applications. Understanding its fundamental concepts of derivatives and integrals allows us to model and solve real-world problems involving change and accumulation.\n\n## References\n\n1. Stewart, J. (2008). *Calculus: Early Transcendentals*. Cengage Learning.\n2. Thomas, G. B., Weir, M. D., & Hass, J. (2014). *Thomas' Calculus*. Pearson.\n\n### Author Note\n\nThe content above is a simplified overview of calculus intended for educational purposes. For a deeper understanding, it is recommended to refer to comprehensive calculus textbooks and resources."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# The Fundamentals of Calculus\n\n## Introduction to Calculus\n\nCalculus is a branch of mathematics that studies changes and motion. It is divided into two main areas: differential calculus and integral calculus. Differential calculus concerns the concept of a derivative, which represents an instantaneous rate of change. Integral calculus, on the other hand, deals with the accumulation of quantities and the areas under and between curves.\n\n## The Derivative\n\nThe derivative of a function at a certain point is the slope of the tangent line to the function at that point. This can be mathematically expressed as:\n\n\\[ f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\]\n\nA common example is the derivative of the function \\( f(x) = x^2 \\), which is:\n\n\\[ f'(x) = 2x \\]\n\n### Applications of Derivatives\n\n1. **Motion**: Derivatives can describe velocity and acceleration.\n2. **Optimization**: Used to find maximum and minimum values of functions.\n3. **Curve Sketching**: Helps in understanding the behavior of graphs.\n\n## The Integral\n\nThe integral is essentially the inverse operation of the derivative. While the derivative represents a rate of change, the integral represents a cumulative total. The definite integral of a function \\( f(x) \\) from \\( a \\) to \\( b \\) is given by:\n\n\\[ \\int_a^b f(x) \\, dx \\]\n\nFor example, the integral of \\( f(x) = x^2 \\) from 0 to 1 is:\n\n\\[ \\int_0^1 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{3} \\]\n\n### Applications of Integrals\n\n1. **Area Under a Curve**: Calculating the area between a curve and the x-axis.\n2. **Physics**: Used in calculating quantities like displacement and work.\n3. **Probability**: Integral calculus is used to find probabilities in continuous distributions.\n\n## The Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus links the concept of differentiation and integration. It has two parts:\n\n1. **First Part**: If \\( F \\) is an antiderivative of \\( f \\) on an interval \\([a, b]\\), then:\n\n \\[ \\int_a^b f(x) \\, dx = F(b) - F(a) \\]\n\n2. **Second Part**: If \\( f \\) is continuous on \\([a, b]\\) and \\( F(x) = \\int_a^x f(t) \\, dt \\), then \\( F \\) is differentiable on \\((a, b)\\) and \\( F'(x) = f(x) \\).\n\n## Example: Calculating the Area Between Two Curves\n\nTo find the area between the curves \\( y = x^2 \\) and \\( y = x \\) from \\( x = 0 \\) to \\( x = 1 \\), we set up the integral:\n\n\\[ \\text{Area} = \\int_0^1 (x - x^2) \\, dx = \\left[ \\frac{x^2}{2} - \\frac{x^3}{3} \\right]_0^1 = \\frac{1}{2} - \\frac{1}{3} = \\frac{1}{6} \\]\n\n## Conclusion\n\nCalculus is a powerful mathematical tool with a wide range of applications. Understanding its fundamental concepts of derivatives and integrals allows us to model and solve real-world problems involving change and accumulation.\n\n## References\n\n1. Stewart, J. (2008). *Calculus: Early Transcendentals*. Cengage Learning.\n2. Thomas, G. B., Weir, M. D., & Hass, J. (2014). *Thomas' Calculus*. Pearson.\n\n### Author Note\n\nThe content above is a simplified overview of calculus intended for educational purposes. For a deeper understanding, it is recommended to refer to comprehensive calculus textbooks and resources."
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"index": 88,
"split": "train",
"text": "# The Fundamental Theorems of Calculus\n\nThe Fundamental Theorems of Calculus establish a connection between differentiation and integration, two of the principal operations in calculus. They serve as a bridge between the concept of the derivative of a function and the concept of the integral of a function.\n\n## The First Fundamental Theorem of Calculus\n\nThe First Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_a^x f(t) \\, dt, \\]\n\nfor every \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor every \\( x \\) in \\((a, b)\\). In other words, the derivative of the integral of \\( f \\) is \\( f \\) itself.\n\n### Proof\n\nTo prove this theorem, we start by considering the definition of the derivative:\n\n\\[ F'(x) = \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h}. \\]\n\nBy the definition of \\( F \\), we have\n\n\\[ F(x+h) = \\int_a^{x+h} f(t) \\, dt \\quad \\text{and} \\quad F(x) = \\int_a^x f(t) \\, dt. \\]\n\nThus,\n\n\\[ F(x+h) - F(x) = \\int_a^{x+h} f(t) \\, dt - \\int_a^x f(t) \\, dt. \\]\n\nUsing the properties of definite integrals, we can rewrite this as\n\n\\[ F(x+h) - F(x) = \\int_x^{x+h} f(t) \\, dt. \\]\n\nTherefore,\n\n\\[ \\frac{F(x+h) - F(x)}{h} = \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt. \\]\n\nSince \\( f \\) is continuous at \\( x \\), for any \\( \\epsilon > 0 \\), there exists a \\( \\delta > 0 \\) such that if \\( |t - x| < \\delta \\), then \\( |f(t) - f(x)| < \\epsilon \\). For \\( h \\) sufficiently small, \\( t \\) in the interval \\([x, x+h]\\) satisfies \\( |t - x| < \\delta \\), and thus \\( |f(t) - f(x)| < \\epsilon \\).\n\nWe can then write\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| = \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right|. \\]\n\nBy the Mean Value Theorem for integrals, there exists a \\( c \\) in \\([x, x+h]\\) such that\n\n\\[ \\int_x^{x+h} (f(t) - f(x)) \\, dt = (f(c) - f(x)) \\cdot h. \\]\n\nThus,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right| = |f(c) - f(x)|. \\]\n\nSince \\( c \\) is in \\([x, x+h]\\) and \\( |c - x| < \\delta \\) for sufficiently small \\( h \\), we have \\( |f(c) - f(x)| < \\epsilon \\). Therefore,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| < \\epsilon. \\]\n\nTaking the limit as \\( h \\to 0 \\), we get\n\n\\[ \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h} = f(x). \\]\n\nHence, \\( F'(x) = f(x) \\), completing the proof of the First Fundamental Theorem of Calculus.\n\n## The Second Fundamental Theorem of Calculus\n\nThe Second Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\), and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_a^b f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis theorem provides a practical way to evaluate definite integrals. Instead of computing the integral directly, we can find an antiderivative of the integrand and use it to compute the integral's value.\n\n### Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{x^3}{3} \\). According to the Second Fundamental Theorem of Calculus,\n\n\\[ \\int_1^2 x^2 \\, dx = F(2) - F(1) = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThis result can be verified by directly computing the integral:\n\n\\[ \\int_1^2 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_1^2 = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThus, the Second Fundamental Theorem of Calculus confirms the correctness of our antiderivative-based calculation.\n\n## Conclusion\n\nThe Fundamental Theorems of Calculus are central to understanding the relationship between differentiation and integration. They provide powerful tools for solving problems involving areas under curves, accumulation of quantities, and more. By understanding these theorems, we gain deeper insights into the behavior of functions and the geometric interpretation of calculus."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "integer",
"index": 88,
"split": "train",
"text": "# The Fundamental Theorems of Calculus\n\nThe Fundamental Theorems of Calculus establish a connection between differentiation and integration, two of the principal operations in calculus. They serve as a bridge between the concept of the derivative of a function and the concept of the integral of a function.\n\n## The First Fundamental Theorem of Calculus\n\nThe First Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_a^x f(t) \\, dt, \\]\n\nfor every \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor every \\( x \\) in \\((a, b)\\). In other words, the derivative of the integral of \\( f \\) is \\( f \\) itself.\n\n### Proof\n\nTo prove this theorem, we start by considering the definition of the derivative:\n\n\\[ F'(x) = \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h}. \\]\n\nBy the definition of \\( F \\), we have\n\n\\[ F(x+h) = \\int_a^{x+h} f(t) \\, dt \\quad \\text{and} \\quad F(x) = \\int_a^x f(t) \\, dt. \\]\n\nThus,\n\n\\[ F(x+h) - F(x) = \\int_a^{x+h} f(t) \\, dt - \\int_a^x f(t) \\, dt. \\]\n\nUsing the properties of definite integrals, we can rewrite this as\n\n\\[ F(x+h) - F(x) = \\int_x^{x+h} f(t) \\, dt. \\]\n\nTherefore,\n\n\\[ \\frac{F(x+h) - F(x)}{h} = \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt. \\]\n\nSince \\( f \\) is continuous at \\( x \\), for any \\( \\epsilon > 0 \\), there exists a \\( \\delta > 0 \\) such that if \\( |t - x| < \\delta \\), then \\( |f(t) - f(x)| < \\epsilon \\). For \\( h \\) sufficiently small, \\( t \\) in the interval \\([x, x+h]\\) satisfies \\( |t - x| < \\delta \\), and thus \\( |f(t) - f(x)| < \\epsilon \\).\n\nWe can then write\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| = \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right|. \\]\n\nBy the Mean Value Theorem for integrals, there exists a \\( c \\) in \\([x, x+h]\\) such that\n\n\\[ \\int_x^{x+h} (f(t) - f(x)) \\, dt = (f(c) - f(x)) \\cdot h. \\]\n\nThus,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right| = |f(c) - f(x)|. \\]\n\nSince \\( c \\) is in \\([x, x+h]\\) and \\( |c - x| < \\delta \\) for sufficiently small \\( h \\), we have \\( |f(c) - f(x)| < \\epsilon \\). Therefore,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| < \\epsilon. \\]\n\nTaking the limit as \\( h \\to 0 \\), we get\n\n\\[ \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h} = f(x). \\]\n\nHence, \\( F'(x) = f(x) \\), completing the proof of the First Fundamental Theorem of Calculus.\n\n## The Second Fundamental Theorem of Calculus\n\nThe Second Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\), and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_a^b f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis theorem provides a practical way to evaluate definite integrals. Instead of computing the integral directly, we can find an antiderivative of the integrand and use it to compute the integral's value.\n\n### Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{x^3}{3} \\). According to the Second Fundamental Theorem of Calculus,\n\n\\[ \\int_1^2 x^2 \\, dx = F(2) - F(1) = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThis result can be verified by directly computing the integral:\n\n\\[ \\int_1^2 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_1^2 = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThus, the Second Fundamental Theorem of Calculus confirms the correctness of our antiderivative-based calculation.\n\n## Conclusion\n\nThe Fundamental Theorems of Calculus are central to understanding the relationship between differentiation and integration. They provide powerful tools for solving problems involving areas under curves, accumulation of quantities, and more. By understanding these theorems, we gain deeper insights into the behavior of functions and the geometric interpretation of calculus."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 88,
"split": "train",
"text": "# The Fundamental Theorems of Calculus\n\nThe Fundamental Theorems of Calculus establish a connection between differentiation and integration, two of the principal operations in calculus. They serve as a bridge between the concept of the derivative of a function and the concept of the integral of a function.\n\n## The First Fundamental Theorem of Calculus\n\nThe First Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_a^x f(t) \\, dt, \\]\n\nfor every \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor every \\( x \\) in \\((a, b)\\). In other words, the derivative of the integral of \\( f \\) is \\( f \\) itself.\n\n### Proof\n\nTo prove this theorem, we start by considering the definition of the derivative:\n\n\\[ F'(x) = \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h}. \\]\n\nBy the definition of \\( F \\), we have\n\n\\[ F(x+h) = \\int_a^{x+h} f(t) \\, dt \\quad \\text{and} \\quad F(x) = \\int_a^x f(t) \\, dt. \\]\n\nThus,\n\n\\[ F(x+h) - F(x) = \\int_a^{x+h} f(t) \\, dt - \\int_a^x f(t) \\, dt. \\]\n\nUsing the properties of definite integrals, we can rewrite this as\n\n\\[ F(x+h) - F(x) = \\int_x^{x+h} f(t) \\, dt. \\]\n\nTherefore,\n\n\\[ \\frac{F(x+h) - F(x)}{h} = \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt. \\]\n\nSince \\( f \\) is continuous at \\( x \\), for any \\( \\epsilon > 0 \\), there exists a \\( \\delta > 0 \\) such that if \\( |t - x| < \\delta \\), then \\( |f(t) - f(x)| < \\epsilon \\). For \\( h \\) sufficiently small, \\( t \\) in the interval \\([x, x+h]\\) satisfies \\( |t - x| < \\delta \\), and thus \\( |f(t) - f(x)| < \\epsilon \\).\n\nWe can then write\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| = \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right|. \\]\n\nBy the Mean Value Theorem for integrals, there exists a \\( c \\) in \\([x, x+h]\\) such that\n\n\\[ \\int_x^{x+h} (f(t) - f(x)) \\, dt = (f(c) - f(x)) \\cdot h. \\]\n\nThus,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} (f(t) - f(x)) \\, dt \\right| = |f(c) - f(x)|. \\]\n\nSince \\( c \\) is in \\([x, x+h]\\) and \\( |c - x| < \\delta \\) for sufficiently small \\( h \\), we have \\( |f(c) - f(x)| < \\epsilon \\). Therefore,\n\n\\[ \\left| \\frac{1}{h} \\int_x^{x+h} f(t) \\, dt - f(x) \\right| < \\epsilon. \\]\n\nTaking the limit as \\( h \\to 0 \\), we get\n\n\\[ \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h} = f(x). \\]\n\nHence, \\( F'(x) = f(x) \\), completing the proof of the First Fundamental Theorem of Calculus.\n\n## The Second Fundamental Theorem of Calculus\n\nThe Second Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\), and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_a^b f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis theorem provides a practical way to evaluate definite integrals. Instead of computing the integral directly, we can find an antiderivative of the integrand and use it to compute the integral's value.\n\n### Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{x^3}{3} \\). According to the Second Fundamental Theorem of Calculus,\n\n\\[ \\int_1^2 x^2 \\, dx = F(2) - F(1) = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThis result can be verified by directly computing the integral:\n\n\\[ \\int_1^2 x^2 \\, dx = \\left[ \\frac{x^3}{3} \\right]_1^2 = \\frac{2^3}{3} - \\frac{1^3}{3} = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\nThus, the Second Fundamental Theorem of Calculus confirms the correctness of our antiderivative-based calculation.\n\n## Conclusion\n\nThe Fundamental Theorems of Calculus are central to understanding the relationship between differentiation and integration. They provide powerful tools for solving problems involving areas under curves, accumulation of quantities, and more. By understanding these theorems, we gain deeper insights into the behavior of functions and the geometric interpretation of calculus."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "float",
"index": 89,
"split": "train",
"text": "# OEIS\n\n## A054765\n\n$a(n+2) = (2n+3) \\cdot a(n+1) + (n+1)^2 \\cdot a(n)$, $a(0) = 0$, $a(1) = 1$\n\nThe sequence is as follows: $0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600$.\n\n### COMMENTS\n\nThe denominators of the convergents of $[1/3, 4/5, 9/7, 16/9, \\ldots]$. To produce $\\pi$, the above continued fraction is used. It is formed by $n^2/(2n+1)$, which starts at $n=1$. Most numerators of continued fractions are $1$ and thus are left out of the brackets. In the case of $\\pi$ they vary. Therefore, here both numerators and denominators are given. The first 4 convergents are $1/3, 5/19, 44/160, 476/1744$. The value of this continued fraction is $0.273239\\ldots$. $4 \\cdot \\text{INV}(1+0.273239\\ldots)$ is $\\pi$. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008\n\nStarting with offset $1 = \\text{row sums of triangle A155729}$. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]\n\n### FORMULA\n\n$a(n) \\sim \\pi \\cdot (1+\\sqrt{2})^{n + 1/2} \\cdot \\frac{n^n}{2^{9/4} \\cdot \\exp(n)}$. - Vaclav Kotesovec, Feb 18 2017\n\n### MAPLE\n\n```maple\nA054765 := proc(n)\n option remember;\n if n <= 1 then\n n;\n else\n (2*n-1)*procname(n-1)+(n-1)^2*procname(n-2);\n end if;\nend proc;\n```\n# R. J. Mathar, Jul 13 2013\n\n### MATHEMATICA\n\n```mathematica\nRecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],\n a[0] == 0, a[1] == 1}, a, {n, 0, 50}]\n```\n# G. C. Greubel, Feb 18 2017\n\n### CROSSREFS\n\nCf. A155729, A012244, A054766. \nSequence in context: A232607, A320352, A301921, A232691, A057719, A289258 \nAdjacent sequences: A054762, A054763, A054764, A054766, A054767, A054768\n\n### KEYWORD\n\nnonn, easy\n\n### AUTHOR\n\nN. J. A. Sloane, May 26 2000\n\n### EXTENSIONS\n\nMore terms from James A. Sellers, May 27 2000\n\n### STATUS\n\napproved"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "float",
"index": 89,
"split": "train",
"text": "# OEIS\n\n## A054765\n\n$a(n+2) = (2n+3) \\cdot a(n+1) + (n+1)^2 \\cdot a(n)$, $a(0) = 0$, $a(1) = 1$\n\nThe sequence is as follows: $0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600$.\n\n### COMMENTS\n\nThe denominators of the convergents of $[1/3, 4/5, 9/7, 16/9, \\ldots]$. To produce $\\pi$, the above continued fraction is used. It is formed by $n^2/(2n+1)$, which starts at $n=1$. Most numerators of continued fractions are $1$ and thus are left out of the brackets. In the case of $\\pi$ they vary. Therefore, here both numerators and denominators are given. The first 4 convergents are $1/3, 5/19, 44/160, 476/1744$. The value of this continued fraction is $0.273239\\ldots$. $4 \\cdot \\text{INV}(1+0.273239\\ldots)$ is $\\pi$. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008\n\nStarting with offset $1 = \\text{row sums of triangle A155729}$. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]\n\n### FORMULA\n\n$a(n) \\sim \\pi \\cdot (1+\\sqrt{2})^{n + 1/2} \\cdot \\frac{n^n}{2^{9/4} \\cdot \\exp(n)}$. - Vaclav Kotesovec, Feb 18 2017\n\n### MAPLE\n\n```maple\nA054765 := proc(n)\n option remember;\n if n <= 1 then\n n;\n else\n (2*n-1)*procname(n-1)+(n-1)^2*procname(n-2);\n end if;\nend proc;\n```\n# R. J. Mathar, Jul 13 2013\n\n### MATHEMATICA\n\n```mathematica\nRecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],\n a[0] == 0, a[1] == 1}, a, {n, 0, 50}]\n```\n# G. C. Greubel, Feb 18 2017\n\n### CROSSREFS\n\nCf. A155729, A012244, A054766. \nSequence in context: A232607, A320352, A301921, A232691, A057719, A289258 \nAdjacent sequences: A054762, A054763, A054764, A054766, A054767, A054768\n\n### KEYWORD\n\nnonn, easy\n\n### AUTHOR\n\nN. J. A. Sloane, May 26 2000\n\n### EXTENSIONS\n\nMore terms from James A. Sellers, May 27 2000\n\n### STATUS\n\napproved"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 89,
"split": "train",
"text": "# OEIS\n\n## A054765\n\n$a(n+2) = (2n+3) \\cdot a(n+1) + (n+1)^2 \\cdot a(n)$, $a(0) = 0$, $a(1) = 1$\n\nThe sequence is as follows: $0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600$.\n\n### COMMENTS\n\nThe denominators of the convergents of $[1/3, 4/5, 9/7, 16/9, \\ldots]$. To produce $\\pi$, the above continued fraction is used. It is formed by $n^2/(2n+1)$, which starts at $n=1$. Most numerators of continued fractions are $1$ and thus are left out of the brackets. In the case of $\\pi$ they vary. Therefore, here both numerators and denominators are given. The first 4 convergents are $1/3, 5/19, 44/160, 476/1744$. The value of this continued fraction is $0.273239\\ldots$. $4 \\cdot \\text{INV}(1+0.273239\\ldots)$ is $\\pi$. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008\n\nStarting with offset $1 = \\text{row sums of triangle A155729}$. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]\n\n### FORMULA\n\n$a(n) \\sim \\pi \\cdot (1+\\sqrt{2})^{n + 1/2} \\cdot \\frac{n^n}{2^{9/4} \\cdot \\exp(n)}$. - Vaclav Kotesovec, Feb 18 2017\n\n### MAPLE\n\n```maple\nA054765 := proc(n)\n option remember;\n if n <= 1 then\n n;\n else\n (2*n-1)*procname(n-1)+(n-1)^2*procname(n-2);\n end if;\nend proc;\n```\n# R. J. Mathar, Jul 13 2013\n\n### MATHEMATICA\n\n```mathematica\nRecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],\n a[0] == 0, a[1] == 1}, a, {n, 0, 50}]\n```\n# G. C. Greubel, Feb 18 2017\n\n### CROSSREFS\n\nCf. A155729, A012244, A054766. \nSequence in context: A232607, A320352, A301921, A232691, A057719, A289258 \nAdjacent sequences: A054762, A054763, A054764, A054766, A054767, A054768\n\n### KEYWORD\n\nnonn, easy\n\n### AUTHOR\n\nN. J. A. Sloane, May 26 2000\n\n### EXTENSIONS\n\nMore terms from James A. Sellers, May 27 2000\n\n### STATUS\n\napproved"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 90,
"split": "train",
"text": "Editor's note: This is Part 1 in a two-part series. See Rethinking Central Utility Plants, Part 2\n\nTimes have changed and so have the buildings in which we live, work, and teach. They have become more energy-efficient because of better facades and more sophisticated HVAC, electrical, and automation systems. But what about the central plants that produce the heating and cooling delivered to these buildings? How have they evolved over the years? Are they serving us in an efficient, sustainable manner?\n\nMany universities are making a big push to reduce their facilities' carbon footprints but relegating the assignment to those who operate and maintain their central plants. Although those individuals most likely have implemented capital-improvement projects that have reduced operating and maintenance costs, the greater question is whether the plants still are relevant in a world increasingly concerned with global warming, fossil-fuel dependence, and economic competitiveness. (The use of cogeneration and thermal-energy storage are exceptions, but will not be addressed in this column.)\n\n\nCertain HVAC trends, such as decoupling ventilation systems from space-heating and cooling systems, eliminating reheat for comfort-cooling applications, minimizing fan energy, and using warmer chilled water and cooler heating hot water, impact how central-plant equipment relates to the buildings it serves. These trends are necessary to achieve the building energy savings our society and government desire.\n\nCampus waste is most evident at a central heating plant, where anachronistic boilers burn large quantities of fossil fuel at a modest thermal efficiency. Some energy goes up the stack, some goes down the drain, and some is radiated and distributed in an inefficient manner. Many heating plants are located in cold-weather climates where coal once was or still is used. Saturated steam is generated at 80 psig or more and distributed via tunnels to buildings, where the pressure is reduced. The steam then is used in heating coils or radiators or, more commonly, converted to hot water for space heating. Problems associated with these types of systems involve:\n\n • Thermal efficiency\n\n Efficiency suffers partly because a central boiler plant must generate a heating medium, such as steam, at a higher temperature or pressure than a smaller distributed plant. High-delivery temperatures and pressures compensate for thermal and friction losses that occur en route to the heating load. But by generating a heating medium at a higher temperature, the central plant misses out on potential thermal efficiencies, such as those that can be realized with a condensing boiler.\n\n • Radiation and convective losses\n\n Steam boilers have large surface areas that constantly radiate energy when operating or idle. Simple math demonstrates that a boiler estimated to have convective and radiation losses of 1 percent at full load will have 2-percent losses at half load and 4-percent losses at one-quarter load.\n\n • Distribution losses\n\n\n Campus piping distribution systems experience thermal and friction losses. Central steam plants require large networks of tunnels, shallow trenches, and direct-buried piping systems to deliver energy to remote points throughout a campus. Friction losses require boilers to produce higher steam pressures than typically needed at remote buildings. Higher pressures create higher distribution temperatures and thermal losses as well as lower boiler thermal efficiencies.\n\n • Condensate losses\n\n Steam boilers require high-quality water for longevity. Consequently, operating personnel go to great lengths to ensure feedwater quality by treating it expensively. Boilers turn the water to steam and send it to buildings, where it is used for space heating, domestic water heating, and processes, such as humidification. In an ideal steam-boiler system, all of a boiler plant's condensate returns make another trip through the system. Condensate then can be lost at faulty steam traps, via aging leaky piping systems, and through poor practices, such as allowing receivers to vent to the atmosphere.\n\n • Blowdown losses\n\n Because steam systems lose condensate, they require makeup water. The water leaves behind impurities when it evaporates. Over time, the impurities must be “blown down” a drain, taking energy with them. Often, this blowdown is mixed with cold water before being sent to the sewer system, wasting more resources.\n\nIf central heating plants are so inefficient, why are they utilized? Once regarded as inexpensive, coal used to be a popular heat source. Systems that utilized coal were major pollutants, creating black soot that covered the landscape. Therefore, it was imperative to locate campus heat sources away from occupied areas, such as classrooms, libraries, dormitories, administrative offices, and patient rooms.\n\nThe Clean Air Act of 1963 and its amendments in 1970, 1977, and 1990 caused the U.S. Environmental Protection Agency to tighten restrictions on large campus boiler plants. Costly emissions control and monitoring equipment was installed on coal plants, while other boiler plants were converted to fuel oil and natural gas. Many steam radiators still are operating in campus buildings, and, unfortunately, windows are kept open for much of the year to compensate for a lack of temperature control. All of this has increased the cost of owning, operating, and maintaining central heating plants.\n\nCentral cooling plants also have proliferated over time. In the past, chillers were large and inefficient and tended to surge when unloaded at less than 80 percent of full-load capacity. As chillers aged, owners were forced to replace them directly or construct a central chilled-water plant that could serve a number of buildings. Central plants were able to have smaller-than-required installed capacities by replacing every building's chillers because not all buildings would have simultaneously peaking cooling loads.\n\nBecause large centrifugal chillers operate at a higher efficiency when fully loaded, central plants could operate more efficiently by virtue of their enhanced opportunity to keep their chillers fully loaded. Couple that with the fact many campuses already had steam tunnels in which chilled-water pipe could be installed, and the economics of the central plant were favorable. However, central chilled-water plants have their drawbacks, such as:\n\n • Distribution losses\n\n Distribution losses come in two varieties: friction and thermal. It is not uncommon to see cold pipes in extremely hot steam tunnels or buried in soil warmed by the summer sun. I know of campuses at which direct-buried chilled-water return lines are uninsulated and bell-and-spigot connections that have a catalogued fluid leakage rate are used. Some campuses extend expensive steam-tunnel systems just to accommodate chilled-water-plant expansion. A central plant may produce chilled water at much colder temperatures than required by the majority of end users just to satisfy the needs of a small load, such as a surgical suite.\n\n • Central-plant distribution and other problems\n\n Low-delta-T syndrome has been a plague to many campuses that have been connecting buildings to central chilled-water loops without first upgrading the buildings to conserve chilled water. Campus planners also make the mistake of not thinking through the hydraulic decoupling of buildings from central loops. A common approach is to use a heat exchanger for isolation, which requires a chilled-water plant to produce water a couple of degrees colder than otherwise required, thereby decreasing chiller-plant efficiency and increasing thermal losses.\n\nThe refrigeration industry has made great technological strides to advance chiller efficiency. Chillers truly can unload to match building loads with favorable coefficients of performance. Smaller machines now have excellent efficiencies approaching that of the largest centrifugal chillers. Economical and efficient chillers in smaller sizes were not available until water-cooled screw chillers were adapted and refined for HVAC applications.\n\nChiller manufacturers also have made great strides in accommodating colder condensing temperatures and fitting chillers with variable-frequency drives. Another development is the use of magnetic bearings on small centrifugal compressors, which offer excellent part-load efficiencies. Now, small distributed plants can operate as efficiently as large central plants.\n\nNext month, the author will discuss the use of renewable energy sources for heating and cooling.\n\nA vice president with Advanced Engineering Consultants, Carl C. Schultz, PE, LEED AP, has 20 years of experience designing mechanical systems for hospitals, laboratories, universities, prisons, data centers, and large office complexes. Additionally, he has extensive experience designing central steam, high-temperature-hot-water, and chilled-water plants. He can be contacted at carls@aecmep.com."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 90,
"split": "train",
"text": "Editor's note: This is Part 1 in a two-part series. See Rethinking Central Utility Plants, Part 2\n\nTimes have changed and so have the buildings in which we live, work, and teach. They have become more energy-efficient because of better facades and more sophisticated HVAC, electrical, and automation systems. But what about the central plants that produce the heating and cooling delivered to these buildings? How have they evolved over the years? Are they serving us in an efficient, sustainable manner?\n\nMany universities are making a big push to reduce their facilities' carbon footprints but relegating the assignment to those who operate and maintain their central plants. Although those individuals most likely have implemented capital-improvement projects that have reduced operating and maintenance costs, the greater question is whether the plants still are relevant in a world increasingly concerned with global warming, fossil-fuel dependence, and economic competitiveness. (The use of cogeneration and thermal-energy storage are exceptions, but will not be addressed in this column.)\n\n\nCertain HVAC trends, such as decoupling ventilation systems from space-heating and cooling systems, eliminating reheat for comfort-cooling applications, minimizing fan energy, and using warmer chilled water and cooler heating hot water, impact how central-plant equipment relates to the buildings it serves. These trends are necessary to achieve the building energy savings our society and government desire.\n\nCampus waste is most evident at a central heating plant, where anachronistic boilers burn large quantities of fossil fuel at a modest thermal efficiency. Some energy goes up the stack, some goes down the drain, and some is radiated and distributed in an inefficient manner. Many heating plants are located in cold-weather climates where coal once was or still is used. Saturated steam is generated at 80 psig or more and distributed via tunnels to buildings, where the pressure is reduced. The steam then is used in heating coils or radiators or, more commonly, converted to hot water for space heating. Problems associated with these types of systems involve:\n\n • Thermal efficiency\n\n Efficiency suffers partly because a central boiler plant must generate a heating medium, such as steam, at a higher temperature or pressure than a smaller distributed plant. High-delivery temperatures and pressures compensate for thermal and friction losses that occur en route to the heating load. But by generating a heating medium at a higher temperature, the central plant misses out on potential thermal efficiencies, such as those that can be realized with a condensing boiler.\n\n • Radiation and convective losses\n\n Steam boilers have large surface areas that constantly radiate energy when operating or idle. Simple math demonstrates that a boiler estimated to have convective and radiation losses of 1 percent at full load will have 2-percent losses at half load and 4-percent losses at one-quarter load.\n\n • Distribution losses\n\n\n Campus piping distribution systems experience thermal and friction losses. Central steam plants require large networks of tunnels, shallow trenches, and direct-buried piping systems to deliver energy to remote points throughout a campus. Friction losses require boilers to produce higher steam pressures than typically needed at remote buildings. Higher pressures create higher distribution temperatures and thermal losses as well as lower boiler thermal efficiencies.\n\n • Condensate losses\n\n Steam boilers require high-quality water for longevity. Consequently, operating personnel go to great lengths to ensure feedwater quality by treating it expensively. Boilers turn the water to steam and send it to buildings, where it is used for space heating, domestic water heating, and processes, such as humidification. In an ideal steam-boiler system, all of a boiler plant's condensate returns make another trip through the system. Condensate then can be lost at faulty steam traps, via aging leaky piping systems, and through poor practices, such as allowing receivers to vent to the atmosphere.\n\n • Blowdown losses\n\n Because steam systems lose condensate, they require makeup water. The water leaves behind impurities when it evaporates. Over time, the impurities must be “blown down” a drain, taking energy with them. Often, this blowdown is mixed with cold water before being sent to the sewer system, wasting more resources.\n\nIf central heating plants are so inefficient, why are they utilized? Once regarded as inexpensive, coal used to be a popular heat source. Systems that utilized coal were major pollutants, creating black soot that covered the landscape. Therefore, it was imperative to locate campus heat sources away from occupied areas, such as classrooms, libraries, dormitories, administrative offices, and patient rooms.\n\nThe Clean Air Act of 1963 and its amendments in 1970, 1977, and 1990 caused the U.S. Environmental Protection Agency to tighten restrictions on large campus boiler plants. Costly emissions control and monitoring equipment was installed on coal plants, while other boiler plants were converted to fuel oil and natural gas. Many steam radiators still are operating in campus buildings, and, unfortunately, windows are kept open for much of the year to compensate for a lack of temperature control. All of this has increased the cost of owning, operating, and maintaining central heating plants.\n\nCentral cooling plants also have proliferated over time. In the past, chillers were large and inefficient and tended to surge when unloaded at less than 80 percent of full-load capacity. As chillers aged, owners were forced to replace them directly or construct a central chilled-water plant that could serve a number of buildings. Central plants were able to have smaller-than-required installed capacities by replacing every building's chillers because not all buildings would have simultaneously peaking cooling loads.\n\nBecause large centrifugal chillers operate at a higher efficiency when fully loaded, central plants could operate more efficiently by virtue of their enhanced opportunity to keep their chillers fully loaded. Couple that with the fact many campuses already had steam tunnels in which chilled-water pipe could be installed, and the economics of the central plant were favorable. However, central chilled-water plants have their drawbacks, such as:\n\n • Distribution losses\n\n Distribution losses come in two varieties: friction and thermal. It is not uncommon to see cold pipes in extremely hot steam tunnels or buried in soil warmed by the summer sun. I know of campuses at which direct-buried chilled-water return lines are uninsulated and bell-and-spigot connections that have a catalogued fluid leakage rate are used. Some campuses extend expensive steam-tunnel systems just to accommodate chilled-water-plant expansion. A central plant may produce chilled water at much colder temperatures than required by the majority of end users just to satisfy the needs of a small load, such as a surgical suite.\n\n • Central-plant distribution and other problems\n\n Low-delta-T syndrome has been a plague to many campuses that have been connecting buildings to central chilled-water loops without first upgrading the buildings to conserve chilled water. Campus planners also make the mistake of not thinking through the hydraulic decoupling of buildings from central loops. A common approach is to use a heat exchanger for isolation, which requires a chilled-water plant to produce water a couple of degrees colder than otherwise required, thereby decreasing chiller-plant efficiency and increasing thermal losses.\n\nThe refrigeration industry has made great technological strides to advance chiller efficiency. Chillers truly can unload to match building loads with favorable coefficients of performance. Smaller machines now have excellent efficiencies approaching that of the largest centrifugal chillers. Economical and efficient chillers in smaller sizes were not available until water-cooled screw chillers were adapted and refined for HVAC applications.\n\nChiller manufacturers also have made great strides in accommodating colder condensing temperatures and fitting chillers with variable-frequency drives. Another development is the use of magnetic bearings on small centrifugal compressors, which offer excellent part-load efficiencies. Now, small distributed plants can operate as efficiently as large central plants.\n\nNext month, the author will discuss the use of renewable energy sources for heating and cooling.\n\nA vice president with Advanced Engineering Consultants, Carl C. Schultz, PE, LEED AP, has 20 years of experience designing mechanical systems for hospitals, laboratories, universities, prisons, data centers, and large office complexes. Additionally, he has extensive experience designing central steam, high-temperature-hot-water, and chilled-water plants. He can be contacted at carls@aecmep.com."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 90,
"split": "train",
"text": "Editor's note: This is Part 1 in a two-part series. See Rethinking Central Utility Plants, Part 2\n\nTimes have changed and so have the buildings in which we live, work, and teach. They have become more energy-efficient because of better facades and more sophisticated HVAC, electrical, and automation systems. But what about the central plants that produce the heating and cooling delivered to these buildings? How have they evolved over the years? Are they serving us in an efficient, sustainable manner?\n\nMany universities are making a big push to reduce their facilities' carbon footprints but relegating the assignment to those who operate and maintain their central plants. Although those individuals most likely have implemented capital-improvement projects that have reduced operating and maintenance costs, the greater question is whether the plants still are relevant in a world increasingly concerned with global warming, fossil-fuel dependence, and economic competitiveness. (The use of cogeneration and thermal-energy storage are exceptions, but will not be addressed in this column.)\n\n\nCertain HVAC trends, such as decoupling ventilation systems from space-heating and cooling systems, eliminating reheat for comfort-cooling applications, minimizing fan energy, and using warmer chilled water and cooler heating hot water, impact how central-plant equipment relates to the buildings it serves. These trends are necessary to achieve the building energy savings our society and government desire.\n\nCampus waste is most evident at a central heating plant, where anachronistic boilers burn large quantities of fossil fuel at a modest thermal efficiency. Some energy goes up the stack, some goes down the drain, and some is radiated and distributed in an inefficient manner. Many heating plants are located in cold-weather climates where coal once was or still is used. Saturated steam is generated at 80 psig or more and distributed via tunnels to buildings, where the pressure is reduced. The steam then is used in heating coils or radiators or, more commonly, converted to hot water for space heating. Problems associated with these types of systems involve:\n\n • Thermal efficiency\n\n Efficiency suffers partly because a central boiler plant must generate a heating medium, such as steam, at a higher temperature or pressure than a smaller distributed plant. High-delivery temperatures and pressures compensate for thermal and friction losses that occur en route to the heating load. But by generating a heating medium at a higher temperature, the central plant misses out on potential thermal efficiencies, such as those that can be realized with a condensing boiler.\n\n • Radiation and convective losses\n\n Steam boilers have large surface areas that constantly radiate energy when operating or idle. Simple math demonstrates that a boiler estimated to have convective and radiation losses of 1 percent at full load will have 2-percent losses at half load and 4-percent losses at one-quarter load.\n\n • Distribution losses\n\n\n Campus piping distribution systems experience thermal and friction losses. Central steam plants require large networks of tunnels, shallow trenches, and direct-buried piping systems to deliver energy to remote points throughout a campus. Friction losses require boilers to produce higher steam pressures than typically needed at remote buildings. Higher pressures create higher distribution temperatures and thermal losses as well as lower boiler thermal efficiencies.\n\n • Condensate losses\n\n Steam boilers require high-quality water for longevity. Consequently, operating personnel go to great lengths to ensure feedwater quality by treating it expensively. Boilers turn the water to steam and send it to buildings, where it is used for space heating, domestic water heating, and processes, such as humidification. In an ideal steam-boiler system, all of a boiler plant's condensate returns make another trip through the system. Condensate then can be lost at faulty steam traps, via aging leaky piping systems, and through poor practices, such as allowing receivers to vent to the atmosphere.\n\n • Blowdown losses\n\n Because steam systems lose condensate, they require makeup water. The water leaves behind impurities when it evaporates. Over time, the impurities must be “blown down” a drain, taking energy with them. Often, this blowdown is mixed with cold water before being sent to the sewer system, wasting more resources.\n\nIf central heating plants are so inefficient, why are they utilized? Once regarded as inexpensive, coal used to be a popular heat source. Systems that utilized coal were major pollutants, creating black soot that covered the landscape. Therefore, it was imperative to locate campus heat sources away from occupied areas, such as classrooms, libraries, dormitories, administrative offices, and patient rooms.\n\nThe Clean Air Act of 1963 and its amendments in 1970, 1977, and 1990 caused the U.S. Environmental Protection Agency to tighten restrictions on large campus boiler plants. Costly emissions control and monitoring equipment was installed on coal plants, while other boiler plants were converted to fuel oil and natural gas. Many steam radiators still are operating in campus buildings, and, unfortunately, windows are kept open for much of the year to compensate for a lack of temperature control. All of this has increased the cost of owning, operating, and maintaining central heating plants.\n\nCentral cooling plants also have proliferated over time. In the past, chillers were large and inefficient and tended to surge when unloaded at less than 80 percent of full-load capacity. As chillers aged, owners were forced to replace them directly or construct a central chilled-water plant that could serve a number of buildings. Central plants were able to have smaller-than-required installed capacities by replacing every building's chillers because not all buildings would have simultaneously peaking cooling loads.\n\nBecause large centrifugal chillers operate at a higher efficiency when fully loaded, central plants could operate more efficiently by virtue of their enhanced opportunity to keep their chillers fully loaded. Couple that with the fact many campuses already had steam tunnels in which chilled-water pipe could be installed, and the economics of the central plant were favorable. However, central chilled-water plants have their drawbacks, such as:\n\n • Distribution losses\n\n Distribution losses come in two varieties: friction and thermal. It is not uncommon to see cold pipes in extremely hot steam tunnels or buried in soil warmed by the summer sun. I know of campuses at which direct-buried chilled-water return lines are uninsulated and bell-and-spigot connections that have a catalogued fluid leakage rate are used. Some campuses extend expensive steam-tunnel systems just to accommodate chilled-water-plant expansion. A central plant may produce chilled water at much colder temperatures than required by the majority of end users just to satisfy the needs of a small load, such as a surgical suite.\n\n • Central-plant distribution and other problems\n\n Low-delta-T syndrome has been a plague to many campuses that have been connecting buildings to central chilled-water loops without first upgrading the buildings to conserve chilled water. Campus planners also make the mistake of not thinking through the hydraulic decoupling of buildings from central loops. A common approach is to use a heat exchanger for isolation, which requires a chilled-water plant to produce water a couple of degrees colder than otherwise required, thereby decreasing chiller-plant efficiency and increasing thermal losses.\n\nThe refrigeration industry has made great technological strides to advance chiller efficiency. Chillers truly can unload to match building loads with favorable coefficients of performance. Smaller machines now have excellent efficiencies approaching that of the largest centrifugal chillers. Economical and efficient chillers in smaller sizes were not available until water-cooled screw chillers were adapted and refined for HVAC applications.\n\nChiller manufacturers also have made great strides in accommodating colder condensing temperatures and fitting chillers with variable-frequency drives. Another development is the use of magnetic bearings on small centrifugal compressors, which offer excellent part-load efficiencies. Now, small distributed plants can operate as efficiently as large central plants.\n\nNext month, the author will discuss the use of renewable energy sources for heating and cooling.\n\nA vice president with Advanced Engineering Consultants, Carl C. Schultz, PE, LEED AP, has 20 years of experience designing mechanical systems for hospitals, laboratories, universities, prisons, data centers, and large office complexes. Additionally, he has extensive experience designing central steam, high-temperature-hot-water, and chilled-water plants. He can be contacted at carls@aecmep.com."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 91,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI've recently noticed something interesting when looking at Python 3.3 grammar specification:\n\nfuncdef: 'def' NAME parameters ['->' test] ':' suite\n\nThe optional 'arrow' block was absent in Python 2 and I couldn't find any information regarding its meaning in Python 3. It turns out this is correct Python and it's accepted by the interpreter:\n\ndef f(x) -> 123:\n return x\n\nI thought that this might be some kind of a precondition syntax, but:\n\n • I cannot test x here, at it is still undefined,\n • No matter what I put after the arrow (e.g. 2 < 1), it doesn't affect the function behaviour.\n\nCould anyone accustomed with this syntax explain it?\n\nshare|improve this question\n\n2 Answers 2\n\nup vote 73 down vote accepted\n\nIt's a function annotation.\n\nIn more detail, Python 2.x has docstrings, which allow you to attach a metadata string to various types of object. This is amazingly handy, so Python 3 extends the feature by allowing you to attach metadata to functions describing their parameters and return values.\n\nThere's no preconceived use case, but the PEP suggests several. One very handy one is to allow you to annotate parameters with their expected types; it would then be easy to write a decorator that verifies the annotations or coerces the arguments to the right type. Another is to allow parameter-specific documentation instead of encoding it into the docstring.\n\nshare|improve this answer\nAnd the information is available as a .__annotations__ attribute. – Martijn Pieters Jan 17 '13 at 13:06\nWow, I missed quite a broad area of knowledge - not only return value annotations, but also parameter annotations. Thank you very much :). – Krotton Jan 17 '13 at 13:16\n@Krotton Can't blame you for missing it, it's practically unused. I only ever met a single library using them, and it's quite obscure. – delnan Jan 17 '13 at 13:19\nAnd the __annotations__ attribute is a dictionary. The key return is the one used to retrieve the value after the arrow. – Keith Jan 17 '13 at 13:20\n@delnan -- probably the reason that it's mostly unused is because most python libraries still aim to be compatible with python2.x. As python3.x begins to become more standard, we might see more of these things popping up here and there... – mgilson Jan 17 '13 at 14:15\n\nThese are function annotations covered in PEP 3107. Specifically, the -> marks the return function annotation.\n\n\n>>> def kinetic_energy(m:'in KG', v:'in M/S')->'Joules': \n... return 1/2*m*v**2\n>>> kinetic_energy.__annotations__\n{'return': 'Joules', 'v': 'in M/S', 'm': 'in KG'}\n\nAnnotations are dictionaries, so you can do this:\n\n>>> '{:,} {}'.format(kinetic_energy(20,3000),\n'90,000,000.0 Joules'\n\nYou can also have a python data structure rather than just a string:\n\n>>> rd={'type':float,'units':'Joules','docstring':'Given mass and velocity returns kinetic energy in Joules'}\n>>> def f()->rd:\n... pass\n>>> f.__annotations__['return']['type']\n<class 'float'>\n>>> f.__annotations__['return']['units']\n\nOr, you can use function attributes to validate called values:\n\ndef validate(func, locals):\n for var, test in func.__annotations__.items():\n value = locals[var]\n pr=test.__name__+': '+test.__docstring__\n except AttributeError:\n msg = '{}=={}; Test: {}'.format(var, value, pr)\n assert test(value), msg\n\ndef between(lo, hi):\n def _between(x):\n return lo <= x <= hi\n _between.__docstring__='must be between {} and {}'.format(lo,hi) \n return _between\n\ndef f(x: between(3,10), y:lambda _y: isinstance(_y,int)):\n validate(f, locals())\n\n\n>>> f(2,2) \nAssertionError: x==2; Test: _between: must be between 3 and 10\n>>> f(3,2.1)\nAssertionError: y==2.1; Test: <lambda>\nshare|improve this answer\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 91,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI've recently noticed something interesting when looking at Python 3.3 grammar specification:\n\nfuncdef: 'def' NAME parameters ['->' test] ':' suite\n\nThe optional 'arrow' block was absent in Python 2 and I couldn't find any information regarding its meaning in Python 3. It turns out this is correct Python and it's accepted by the interpreter:\n\ndef f(x) -> 123:\n return x\n\nI thought that this might be some kind of a precondition syntax, but:\n\n • I cannot test x here, at it is still undefined,\n • No matter what I put after the arrow (e.g. 2 < 1), it doesn't affect the function behaviour.\n\nCould anyone accustomed with this syntax explain it?\n\nshare|improve this question\n\n2 Answers 2\n\nup vote 73 down vote accepted\n\nIt's a function annotation.\n\nIn more detail, Python 2.x has docstrings, which allow you to attach a metadata string to various types of object. This is amazingly handy, so Python 3 extends the feature by allowing you to attach metadata to functions describing their parameters and return values.\n\nThere's no preconceived use case, but the PEP suggests several. One very handy one is to allow you to annotate parameters with their expected types; it would then be easy to write a decorator that verifies the annotations or coerces the arguments to the right type. Another is to allow parameter-specific documentation instead of encoding it into the docstring.\n\nshare|improve this answer\nAnd the information is available as a .__annotations__ attribute. – Martijn Pieters Jan 17 '13 at 13:06\nWow, I missed quite a broad area of knowledge - not only return value annotations, but also parameter annotations. Thank you very much :). – Krotton Jan 17 '13 at 13:16\n@Krotton Can't blame you for missing it, it's practically unused. I only ever met a single library using them, and it's quite obscure. – delnan Jan 17 '13 at 13:19\nAnd the __annotations__ attribute is a dictionary. The key return is the one used to retrieve the value after the arrow. – Keith Jan 17 '13 at 13:20\n@delnan -- probably the reason that it's mostly unused is because most python libraries still aim to be compatible with python2.x. As python3.x begins to become more standard, we might see more of these things popping up here and there... – mgilson Jan 17 '13 at 14:15\n\nThese are function annotations covered in PEP 3107. Specifically, the -> marks the return function annotation.\n\n\n>>> def kinetic_energy(m:'in KG', v:'in M/S')->'Joules': \n... return 1/2*m*v**2\n>>> kinetic_energy.__annotations__\n{'return': 'Joules', 'v': 'in M/S', 'm': 'in KG'}\n\nAnnotations are dictionaries, so you can do this:\n\n>>> '{:,} {}'.format(kinetic_energy(20,3000),\n'90,000,000.0 Joules'\n\nYou can also have a python data structure rather than just a string:\n\n>>> rd={'type':float,'units':'Joules','docstring':'Given mass and velocity returns kinetic energy in Joules'}\n>>> def f()->rd:\n... pass\n>>> f.__annotations__['return']['type']\n<class 'float'>\n>>> f.__annotations__['return']['units']\n\nOr, you can use function attributes to validate called values:\n\ndef validate(func, locals):\n for var, test in func.__annotations__.items():\n value = locals[var]\n pr=test.__name__+': '+test.__docstring__\n except AttributeError:\n msg = '{}=={}; Test: {}'.format(var, value, pr)\n assert test(value), msg\n\ndef between(lo, hi):\n def _between(x):\n return lo <= x <= hi\n _between.__docstring__='must be between {} and {}'.format(lo,hi) \n return _between\n\ndef f(x: between(3,10), y:lambda _y: isinstance(_y,int)):\n validate(f, locals())\n\n\n>>> f(2,2) \nAssertionError: x==2; Test: _between: must be between 3 and 10\n>>> f(3,2.1)\nAssertionError: y==2.1; Test: <lambda>\nshare|improve this answer\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 91,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI've recently noticed something interesting when looking at Python 3.3 grammar specification:\n\nfuncdef: 'def' NAME parameters ['->' test] ':' suite\n\nThe optional 'arrow' block was absent in Python 2 and I couldn't find any information regarding its meaning in Python 3. It turns out this is correct Python and it's accepted by the interpreter:\n\ndef f(x) -> 123:\n return x\n\nI thought that this might be some kind of a precondition syntax, but:\n\n • I cannot test x here, at it is still undefined,\n • No matter what I put after the arrow (e.g. 2 < 1), it doesn't affect the function behaviour.\n\nCould anyone accustomed with this syntax explain it?\n\nshare|improve this question\n\n2 Answers 2\n\nup vote 73 down vote accepted\n\nIt's a function annotation.\n\nIn more detail, Python 2.x has docstrings, which allow you to attach a metadata string to various types of object. This is amazingly handy, so Python 3 extends the feature by allowing you to attach metadata to functions describing their parameters and return values.\n\nThere's no preconceived use case, but the PEP suggests several. One very handy one is to allow you to annotate parameters with their expected types; it would then be easy to write a decorator that verifies the annotations or coerces the arguments to the right type. Another is to allow parameter-specific documentation instead of encoding it into the docstring.\n\nshare|improve this answer\nAnd the information is available as a .__annotations__ attribute. – Martijn Pieters Jan 17 '13 at 13:06\nWow, I missed quite a broad area of knowledge - not only return value annotations, but also parameter annotations. Thank you very much :). – Krotton Jan 17 '13 at 13:16\n@Krotton Can't blame you for missing it, it's practically unused. I only ever met a single library using them, and it's quite obscure. – delnan Jan 17 '13 at 13:19\nAnd the __annotations__ attribute is a dictionary. The key return is the one used to retrieve the value after the arrow. – Keith Jan 17 '13 at 13:20\n@delnan -- probably the reason that it's mostly unused is because most python libraries still aim to be compatible with python2.x. As python3.x begins to become more standard, we might see more of these things popping up here and there... – mgilson Jan 17 '13 at 14:15\n\nThese are function annotations covered in PEP 3107. Specifically, the -> marks the return function annotation.\n\n\n>>> def kinetic_energy(m:'in KG', v:'in M/S')->'Joules': \n... return 1/2*m*v**2\n>>> kinetic_energy.__annotations__\n{'return': 'Joules', 'v': 'in M/S', 'm': 'in KG'}\n\nAnnotations are dictionaries, so you can do this:\n\n>>> '{:,} {}'.format(kinetic_energy(20,3000),\n'90,000,000.0 Joules'\n\nYou can also have a python data structure rather than just a string:\n\n>>> rd={'type':float,'units':'Joules','docstring':'Given mass and velocity returns kinetic energy in Joules'}\n>>> def f()->rd:\n... pass\n>>> f.__annotations__['return']['type']\n<class 'float'>\n>>> f.__annotations__['return']['units']\n\nOr, you can use function attributes to validate called values:\n\ndef validate(func, locals):\n for var, test in func.__annotations__.items():\n value = locals[var]\n pr=test.__name__+': '+test.__docstring__\n except AttributeError:\n msg = '{}=={}; Test: {}'.format(var, value, pr)\n assert test(value), msg\n\ndef between(lo, hi):\n def _between(x):\n return lo <= x <= hi\n _between.__docstring__='must be between {} and {}'.format(lo,hi) \n return _between\n\ndef f(x: between(3,10), y:lambda _y: isinstance(_y,int)):\n validate(f, locals())\n\n\n>>> f(2,2) \nAssertionError: x==2; Test: _between: must be between 3 and 10\n>>> f(3,2.1)\nAssertionError: y==2.1; Test: <lambda>\nshare|improve this answer\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "integer",
"index": 92,
"split": "train",
"text": "# Linear Equations\n\n## Quick Definition:\nAn equation that forms a straight line on a graph.\n\nMore precisely, a linear equation is one that is dependent only on constants and a variable raised to the first power. For example, \\( y = 6x + 2 \\) is linear because it has no squares, cubes, square roots, sines, etc. Linear equations can always be manipulated to take this form:\n\n## Formula of a Linear Equation\nYou won't always see linear equations written exactly like that, but keep in mind that we can manipulate equations to put them in a particular form if necessary.\n\nLinear equations are often written with more than one variable, often \\( x \\) and \\( y \\). Such equations will have many possible combinations of \\( x \\) and \\( y \\) that work. When those points (known as coordinates) are plotted on an \\( x-y \\) axis, they will form a straight line. Consider the following examples graphically:\n\n- \\( y = 3 \\) (dependent on just a constant, 3)\n- \\( y = 0.75x - 0.5 \\) (a linear term and a constant)\n\n## How to Determine If an Equation Is Linear\n\nTo determine if an equation is linear, check for the following:\n- Does the equation (or function) include any squared terms? How about other terms with exponents other than 1 (or technically, zero)?\n- If the function has no terms with an order higher than 1, then it is linear!\n\nWhat if it has a log or trig function? These are not linear terms. They aren't constants (regular numbers) or variables with an exponent of 1, so the function isn't linear. If we could write \\( \\sin(x) \\) or \\( \\log(x) \\) as something linear like \"2x+3,\" we'd do that instead of using complicated non-linear functions like sine and log!\n\n## Solving a Linear Equation\n\nSome linear equations are really simple to solve. For example, consider:\n\n- An equation already solved for \\( y \\) is straightforward. The goal is to rewrite the equation so that the variable you are solving for is on one side, and everything else is on the other side.\n\nSteps to solve a linear equation:\n1. **Collect Like Terms**: Gather all \\( x \\)'s together, all \\( y \\)'s together, and all constants. For example, \\( 4x + 2y + 3x - 5 + 10 \\) becomes \\( 7x + 2y + 5 \\).\n2. **Isolate the Variable**: Get the variable you wish to solve for on one side of the equals sign. \n3. **Remove Coefficients**: If your answer after step 2 looks like \\( 5y = 7x - 10 \\), divide both sides by 5 to get \\( y = \\frac{7x}{5} - 2 \\).\n4. **Check Your Answer**: Substitute your answer back into the original equation to verify.\n\n## Examples of Solving Linear Equations\n\nIt's important to remember that you can't always solve the equation to something definite like \\( y = 5 \\). It's perfectly ok to have \\( y = x + 5 \\), which means \\( y \\) depends on \\( x \\). There's exactly one value of \\( y \\) for each value of \\( x \\), all of which make points that lie on a straight line.\n\n### Example:\nSolve for \\( y \\):\n\nIf you substitute 2 for \\( y \\) back in the original problem, you get \\( 9 = 9 \\), so it's correct!\n\n## Summary\n\nRemember that linear equations are inherently simple. They consist only of linear terms (like \\( 3x \\), \\( 2y \\), \\( y/2 \\), etc.) and constants. If you get stuck, take it one step at a time: collect like terms, isolate the variable you wish to solve for, and perform any additional math required to solve for \\( y \\) or \\( x \\)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "integer",
"index": 92,
"split": "train",
"text": "# Linear Equations\n\n## Quick Definition:\nAn equation that forms a straight line on a graph.\n\nMore precisely, a linear equation is one that is dependent only on constants and a variable raised to the first power. For example, \\( y = 6x + 2 \\) is linear because it has no squares, cubes, square roots, sines, etc. Linear equations can always be manipulated to take this form:\n\n## Formula of a Linear Equation\nYou won't always see linear equations written exactly like that, but keep in mind that we can manipulate equations to put them in a particular form if necessary.\n\nLinear equations are often written with more than one variable, often \\( x \\) and \\( y \\). Such equations will have many possible combinations of \\( x \\) and \\( y \\) that work. When those points (known as coordinates) are plotted on an \\( x-y \\) axis, they will form a straight line. Consider the following examples graphically:\n\n- \\( y = 3 \\) (dependent on just a constant, 3)\n- \\( y = 0.75x - 0.5 \\) (a linear term and a constant)\n\n## How to Determine If an Equation Is Linear\n\nTo determine if an equation is linear, check for the following:\n- Does the equation (or function) include any squared terms? How about other terms with exponents other than 1 (or technically, zero)?\n- If the function has no terms with an order higher than 1, then it is linear!\n\nWhat if it has a log or trig function? These are not linear terms. They aren't constants (regular numbers) or variables with an exponent of 1, so the function isn't linear. If we could write \\( \\sin(x) \\) or \\( \\log(x) \\) as something linear like \"2x+3,\" we'd do that instead of using complicated non-linear functions like sine and log!\n\n## Solving a Linear Equation\n\nSome linear equations are really simple to solve. For example, consider:\n\n- An equation already solved for \\( y \\) is straightforward. The goal is to rewrite the equation so that the variable you are solving for is on one side, and everything else is on the other side.\n\nSteps to solve a linear equation:\n1. **Collect Like Terms**: Gather all \\( x \\)'s together, all \\( y \\)'s together, and all constants. For example, \\( 4x + 2y + 3x - 5 + 10 \\) becomes \\( 7x + 2y + 5 \\).\n2. **Isolate the Variable**: Get the variable you wish to solve for on one side of the equals sign. \n3. **Remove Coefficients**: If your answer after step 2 looks like \\( 5y = 7x - 10 \\), divide both sides by 5 to get \\( y = \\frac{7x}{5} - 2 \\).\n4. **Check Your Answer**: Substitute your answer back into the original equation to verify.\n\n## Examples of Solving Linear Equations\n\nIt's important to remember that you can't always solve the equation to something definite like \\( y = 5 \\). It's perfectly ok to have \\( y = x + 5 \\), which means \\( y \\) depends on \\( x \\). There's exactly one value of \\( y \\) for each value of \\( x \\), all of which make points that lie on a straight line.\n\n### Example:\nSolve for \\( y \\):\n\nIf you substitute 2 for \\( y \\) back in the original problem, you get \\( 9 = 9 \\), so it's correct!\n\n## Summary\n\nRemember that linear equations are inherently simple. They consist only of linear terms (like \\( 3x \\), \\( 2y \\), \\( y/2 \\), etc.) and constants. If you get stuck, take it one step at a time: collect like terms, isolate the variable you wish to solve for, and perform any additional math required to solve for \\( y \\) or \\( x \\)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 92,
"split": "train",
"text": "# Linear Equations\n\n## Quick Definition:\nAn equation that forms a straight line on a graph.\n\nMore precisely, a linear equation is one that is dependent only on constants and a variable raised to the first power. For example, \\( y = 6x + 2 \\) is linear because it has no squares, cubes, square roots, sines, etc. Linear equations can always be manipulated to take this form:\n\n## Formula of a Linear Equation\nYou won't always see linear equations written exactly like that, but keep in mind that we can manipulate equations to put them in a particular form if necessary.\n\nLinear equations are often written with more than one variable, often \\( x \\) and \\( y \\). Such equations will have many possible combinations of \\( x \\) and \\( y \\) that work. When those points (known as coordinates) are plotted on an \\( x-y \\) axis, they will form a straight line. Consider the following examples graphically:\n\n- \\( y = 3 \\) (dependent on just a constant, 3)\n- \\( y = 0.75x - 0.5 \\) (a linear term and a constant)\n\n## How to Determine If an Equation Is Linear\n\nTo determine if an equation is linear, check for the following:\n- Does the equation (or function) include any squared terms? How about other terms with exponents other than 1 (or technically, zero)?\n- If the function has no terms with an order higher than 1, then it is linear!\n\nWhat if it has a log or trig function? These are not linear terms. They aren't constants (regular numbers) or variables with an exponent of 1, so the function isn't linear. If we could write \\( \\sin(x) \\) or \\( \\log(x) \\) as something linear like \"2x+3,\" we'd do that instead of using complicated non-linear functions like sine and log!\n\n## Solving a Linear Equation\n\nSome linear equations are really simple to solve. For example, consider:\n\n- An equation already solved for \\( y \\) is straightforward. The goal is to rewrite the equation so that the variable you are solving for is on one side, and everything else is on the other side.\n\nSteps to solve a linear equation:\n1. **Collect Like Terms**: Gather all \\( x \\)'s together, all \\( y \\)'s together, and all constants. For example, \\( 4x + 2y + 3x - 5 + 10 \\) becomes \\( 7x + 2y + 5 \\).\n2. **Isolate the Variable**: Get the variable you wish to solve for on one side of the equals sign. \n3. **Remove Coefficients**: If your answer after step 2 looks like \\( 5y = 7x - 10 \\), divide both sides by 5 to get \\( y = \\frac{7x}{5} - 2 \\).\n4. **Check Your Answer**: Substitute your answer back into the original equation to verify.\n\n## Examples of Solving Linear Equations\n\nIt's important to remember that you can't always solve the equation to something definite like \\( y = 5 \\). It's perfectly ok to have \\( y = x + 5 \\), which means \\( y \\) depends on \\( x \\). There's exactly one value of \\( y \\) for each value of \\( x \\), all of which make points that lie on a straight line.\n\n### Example:\nSolve for \\( y \\):\n\nIf you substitute 2 for \\( y \\) back in the original problem, you get \\( 9 = 9 \\), so it's correct!\n\n## Summary\n\nRemember that linear equations are inherently simple. They consist only of linear terms (like \\( 3x \\), \\( 2y \\), \\( y/2 \\), etc.) and constants. If you get stuck, take it one step at a time: collect like terms, isolate the variable you wish to solve for, and perform any additional math required to solve for \\( y \\) or \\( x \\)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 93,
"split": "train",
"text": "\n\nOxygen Radicals & Autism\n\nOverloaded?: Oxygen Radicals & Autism\n\nDecember 13, 2004\n\nPart 2\n\nOxygen radicals damage the brain and nervous system\n\nAutistic children's inability to combat oxidative stress can lead to many health problems. Oxidative stress is caused by oxygen radicals — highly unstable chemicals that react with and destroy healthy cells. These free oxygen radicals are produced by the body in manageable amounts as byproducts of normal body metabolism, but their prevalence can be exacerbated by exposure to environmental chemicals. Oxygen radicals damage cells by reacting with proteins, DNA, carbohydrates, and fats, setting off chain reactions that can only be stopped by a cell's antioxidant defense system. In the process they disrupt cell functions and interfere with signals sent between cells in the body, which can lead to auto-immunity (Klein 2003).\n\nOxidative damage is counteracted by the body's antioxidant systems, which convert oxygen radicals into harmless byproducts. Oxidative stress occurs when oxygen radicals overwhelm the capacity of the body's antioxidant systems. Oxidative stress affects many body systems. It damages cell membrane structure (lipids), the cell machinery that performs the essential work of the cells (proteins), and the body's ability to regulate cell growth and protein synthesis (DNA and RNA). Oxidative stress is associated with premature aging of cells, and can lead to tissue inflammation, damaged cell membranes, autoimmunity and cell death (Klein 2003). Glutathione is the most important antioxidant for metals detoxification and excretion.\n\nThe brain and nervous system are particularly vulnerable to oxidative stress due to limited antioxidant capacity. The brain makes up about two percent of a person's mass but consumes 20 percent of their metabolic oxygen. The vast majority of this energy is used by the neurons (Shulman 2004). Some brain cells, like neurons, cannot make glutathione, but instead rely on surrounding astrocyte cells to provide useable glutathione precursors. Because the brain has limited access to the bulk of antioxidants produced by the body, neurons are the first cells to be affected by a shortage of antioxidants, and are most susceptible to oxidative stress. Researchers studying antioxidant protection of neurons are finding short windows during development of high vulnerability to oxidative stress (Perry 2004).\n\nChildren are more vulnerable than adults to oxidative stress due to their naturally low glutathione levels from conception through infancy (Erden-Inal 2003, Ono 2001). Risks created by this natural deficit in detoxification capacity in infants are compounded by the fact that mercury and other environmental chemicals that invoke oxidative stress are found at higher concentrations in the developing infant than in their mothers and appear to accumulate in the placenta.\n\nIn addition to this natural variability in antioxidant status with age, a person's genes play a strong role in their ability to make antioxidants in response to oxidative stress. A host of genes determine the speed and responsiveness of antioxidant production and recycling. Some genes common in one quarter to one half of the U.S. population reduce glutathione activity and are linked with increased odds of several cancers (Hallier 1994, Engel 2002). People with gene deletions for two types of glutathione genes (GST M1 and T1) are more likely to have allergic reactions to the mercury-based preservative thimerosal (Westphal 2000)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 93,
"split": "train",
"text": "\n\nOxygen Radicals & Autism\n\nOverloaded?: Oxygen Radicals & Autism\n\nDecember 13, 2004\n\nPart 2\n\nOxygen radicals damage the brain and nervous system\n\nAutistic children's inability to combat oxidative stress can lead to many health problems. Oxidative stress is caused by oxygen radicals — highly unstable chemicals that react with and destroy healthy cells. These free oxygen radicals are produced by the body in manageable amounts as byproducts of normal body metabolism, but their prevalence can be exacerbated by exposure to environmental chemicals. Oxygen radicals damage cells by reacting with proteins, DNA, carbohydrates, and fats, setting off chain reactions that can only be stopped by a cell's antioxidant defense system. In the process they disrupt cell functions and interfere with signals sent between cells in the body, which can lead to auto-immunity (Klein 2003).\n\nOxidative damage is counteracted by the body's antioxidant systems, which convert oxygen radicals into harmless byproducts. Oxidative stress occurs when oxygen radicals overwhelm the capacity of the body's antioxidant systems. Oxidative stress affects many body systems. It damages cell membrane structure (lipids), the cell machinery that performs the essential work of the cells (proteins), and the body's ability to regulate cell growth and protein synthesis (DNA and RNA). Oxidative stress is associated with premature aging of cells, and can lead to tissue inflammation, damaged cell membranes, autoimmunity and cell death (Klein 2003). Glutathione is the most important antioxidant for metals detoxification and excretion.\n\nThe brain and nervous system are particularly vulnerable to oxidative stress due to limited antioxidant capacity. The brain makes up about two percent of a person's mass but consumes 20 percent of their metabolic oxygen. The vast majority of this energy is used by the neurons (Shulman 2004). Some brain cells, like neurons, cannot make glutathione, but instead rely on surrounding astrocyte cells to provide useable glutathione precursors. Because the brain has limited access to the bulk of antioxidants produced by the body, neurons are the first cells to be affected by a shortage of antioxidants, and are most susceptible to oxidative stress. Researchers studying antioxidant protection of neurons are finding short windows during development of high vulnerability to oxidative stress (Perry 2004).\n\nChildren are more vulnerable than adults to oxidative stress due to their naturally low glutathione levels from conception through infancy (Erden-Inal 2003, Ono 2001). Risks created by this natural deficit in detoxification capacity in infants are compounded by the fact that mercury and other environmental chemicals that invoke oxidative stress are found at higher concentrations in the developing infant than in their mothers and appear to accumulate in the placenta.\n\nIn addition to this natural variability in antioxidant status with age, a person's genes play a strong role in their ability to make antioxidants in response to oxidative stress. A host of genes determine the speed and responsiveness of antioxidant production and recycling. Some genes common in one quarter to one half of the U.S. population reduce glutathione activity and are linked with increased odds of several cancers (Hallier 1994, Engel 2002). People with gene deletions for two types of glutathione genes (GST M1 and T1) are more likely to have allergic reactions to the mercury-based preservative thimerosal (Westphal 2000)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 93,
"split": "train",
"text": "\n\nOxygen Radicals & Autism\n\nOverloaded?: Oxygen Radicals & Autism\n\nDecember 13, 2004\n\nPart 2\n\nOxygen radicals damage the brain and nervous system\n\nAutistic children's inability to combat oxidative stress can lead to many health problems. Oxidative stress is caused by oxygen radicals — highly unstable chemicals that react with and destroy healthy cells. These free oxygen radicals are produced by the body in manageable amounts as byproducts of normal body metabolism, but their prevalence can be exacerbated by exposure to environmental chemicals. Oxygen radicals damage cells by reacting with proteins, DNA, carbohydrates, and fats, setting off chain reactions that can only be stopped by a cell's antioxidant defense system. In the process they disrupt cell functions and interfere with signals sent between cells in the body, which can lead to auto-immunity (Klein 2003).\n\nOxidative damage is counteracted by the body's antioxidant systems, which convert oxygen radicals into harmless byproducts. Oxidative stress occurs when oxygen radicals overwhelm the capacity of the body's antioxidant systems. Oxidative stress affects many body systems. It damages cell membrane structure (lipids), the cell machinery that performs the essential work of the cells (proteins), and the body's ability to regulate cell growth and protein synthesis (DNA and RNA). Oxidative stress is associated with premature aging of cells, and can lead to tissue inflammation, damaged cell membranes, autoimmunity and cell death (Klein 2003). Glutathione is the most important antioxidant for metals detoxification and excretion.\n\nThe brain and nervous system are particularly vulnerable to oxidative stress due to limited antioxidant capacity. The brain makes up about two percent of a person's mass but consumes 20 percent of their metabolic oxygen. The vast majority of this energy is used by the neurons (Shulman 2004). Some brain cells, like neurons, cannot make glutathione, but instead rely on surrounding astrocyte cells to provide useable glutathione precursors. Because the brain has limited access to the bulk of antioxidants produced by the body, neurons are the first cells to be affected by a shortage of antioxidants, and are most susceptible to oxidative stress. Researchers studying antioxidant protection of neurons are finding short windows during development of high vulnerability to oxidative stress (Perry 2004).\n\nChildren are more vulnerable than adults to oxidative stress due to their naturally low glutathione levels from conception through infancy (Erden-Inal 2003, Ono 2001). Risks created by this natural deficit in detoxification capacity in infants are compounded by the fact that mercury and other environmental chemicals that invoke oxidative stress are found at higher concentrations in the developing infant than in their mothers and appear to accumulate in the placenta.\n\nIn addition to this natural variability in antioxidant status with age, a person's genes play a strong role in their ability to make antioxidants in response to oxidative stress. A host of genes determine the speed and responsiveness of antioxidant production and recycling. Some genes common in one quarter to one half of the U.S. population reduce glutathione activity and are linked with increased odds of several cancers (Hallier 1994, Engel 2002). People with gene deletions for two types of glutathione genes (GST M1 and T1) are more likely to have allergic reactions to the mercury-based preservative thimerosal (Westphal 2000)."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "float",
"index": 94,
"split": "train",
"text": "# The Laws of Biot-Savart & Ampere\n\n## Overview of Lecture\n\n### Fundamental Laws for Calculating Magnetic Field\n\n#### Biot-Savart Law\nThe Biot-Savart Law is a fundamental equation that describes the magnetic field generated by a steady electric current. It is often referred to as the \"brute force\" method due to its direct computation of the magnetic field at a point in space.\n\nThe Biot-Savart Law is expressed as:\n$$\n\\mathbf{B} = \\frac{\\mu_0}{4\\pi} \\int \\frac{I \\, d\\mathbf{l} \\times \\mathbf{\\hat{r}}}{r^2}\n$$\nwhere:\n- \\(\\mathbf{B}\\) is the magnetic field,\n- \\(\\mu_0\\) is the permeability of free space,\n- \\(I\\) is the current,\n- \\(d\\mathbf{l}\\) is a differential element of the wire,\n- \\(\\mathbf{\\hat{r}}\\) is the unit vector from the element to the point of interest,\n- \\(r\\) is the distance from the element to the point of interest.\n\n#### Ampere’s Law\nAmpere’s Law provides a \"high symmetry\" approach to calculating the magnetic field, particularly useful for configurations with high symmetry.\n\nAmpere’s Law is given by:\n$$\n\\oint \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 I_{\\text{enc}}\n$$\nwhere:\n- \\(\\oint \\mathbf{B} \\cdot d\\mathbf{l}\\) is the line integral of the magnetic field around a closed loop,\n- \\(I_{\\text{enc}}\\) is the total current enclosed by the loop.\n\n### Examples\n\n**Example: Calculate Magnetic Field of a Straight Wire**\n\n1. **Using Biot-Savart Law:**\n For a long, straight wire carrying a current \\(I\\), the magnetic field at a distance \\(r\\) from the wire is:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n2. **Using Ampere’s Law:**\n For the same wire, Ampere’s Law can be applied by choosing a circular path of radius \\(r\\) centered on the wire. The symmetry simplifies the integral, yielding the same result:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n### Relevant Comments and References\n\n- **Jean-Baptiste Biot and Felix Savart** were pioneers in the study of magnetism, leading to the formulation of the Biot-Savart Law.\n- **André-Marie Ampère** developed Ampere’s Law, which is crucial for understanding magnetic fields in systems with high symmetry.\n\n### Related Presentations\n\n- **Magnetic Field Calculations**: Various presentations delve into calculating magnetic fields using both the Biot-Savart Law and Ampere’s Law.\n- **Applications in Physics**: These laws are foundational in courses on electromagnetism and are applied in numerous practical scenarios, such as in the design of electromagnets and electrical machines.\n\n### Code Example\n\n```python\n# Example code for calculating magnetic field using Biot-Savart Law\nimport numpy as np\n\ndef biot_savart(I, dl, r, r_hat):\n mu_0 = 4 * np.pi * 1e-7\n B = (mu_0 / (4 * np.pi)) * (I * np.cross(dl, r_hat) / r**2)\n return B\n\n# Parameters\nI = 10 # Current in amperes\ndl = np.array([0, 0, 0.01]) # Differential length element in meters\nr = 0.1 # Distance in meters\nr_hat = np.array([0, 0, 1]) # Unit vector\n\n# Calculate magnetic field\nB = biot_savart(I, dl, r, r_hat)\nprint(\"Magnetic Field:\", B)\n```\n\nThis presentation provides a comprehensive overview of the Biot-Savart and Ampere’s Laws, essential for understanding magnetic fields in various physical contexts."
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "integer",
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"text": "# The Laws of Biot-Savart & Ampere\n\n## Overview of Lecture\n\n### Fundamental Laws for Calculating Magnetic Field\n\n#### Biot-Savart Law\nThe Biot-Savart Law is a fundamental equation that describes the magnetic field generated by a steady electric current. It is often referred to as the \"brute force\" method due to its direct computation of the magnetic field at a point in space.\n\nThe Biot-Savart Law is expressed as:\n$$\n\\mathbf{B} = \\frac{\\mu_0}{4\\pi} \\int \\frac{I \\, d\\mathbf{l} \\times \\mathbf{\\hat{r}}}{r^2}\n$$\nwhere:\n- \\(\\mathbf{B}\\) is the magnetic field,\n- \\(\\mu_0\\) is the permeability of free space,\n- \\(I\\) is the current,\n- \\(d\\mathbf{l}\\) is a differential element of the wire,\n- \\(\\mathbf{\\hat{r}}\\) is the unit vector from the element to the point of interest,\n- \\(r\\) is the distance from the element to the point of interest.\n\n#### Ampere’s Law\nAmpere’s Law provides a \"high symmetry\" approach to calculating the magnetic field, particularly useful for configurations with high symmetry.\n\nAmpere’s Law is given by:\n$$\n\\oint \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 I_{\\text{enc}}\n$$\nwhere:\n- \\(\\oint \\mathbf{B} \\cdot d\\mathbf{l}\\) is the line integral of the magnetic field around a closed loop,\n- \\(I_{\\text{enc}}\\) is the total current enclosed by the loop.\n\n### Examples\n\n**Example: Calculate Magnetic Field of a Straight Wire**\n\n1. **Using Biot-Savart Law:**\n For a long, straight wire carrying a current \\(I\\), the magnetic field at a distance \\(r\\) from the wire is:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n2. **Using Ampere’s Law:**\n For the same wire, Ampere’s Law can be applied by choosing a circular path of radius \\(r\\) centered on the wire. The symmetry simplifies the integral, yielding the same result:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n### Relevant Comments and References\n\n- **Jean-Baptiste Biot and Felix Savart** were pioneers in the study of magnetism, leading to the formulation of the Biot-Savart Law.\n- **André-Marie Ampère** developed Ampere’s Law, which is crucial for understanding magnetic fields in systems with high symmetry.\n\n### Related Presentations\n\n- **Magnetic Field Calculations**: Various presentations delve into calculating magnetic fields using both the Biot-Savart Law and Ampere’s Law.\n- **Applications in Physics**: These laws are foundational in courses on electromagnetism and are applied in numerous practical scenarios, such as in the design of electromagnets and electrical machines.\n\n### Code Example\n\n```python\n# Example code for calculating magnetic field using Biot-Savart Law\nimport numpy as np\n\ndef biot_savart(I, dl, r, r_hat):\n mu_0 = 4 * np.pi * 1e-7\n B = (mu_0 / (4 * np.pi)) * (I * np.cross(dl, r_hat) / r**2)\n return B\n\n# Parameters\nI = 10 # Current in amperes\ndl = np.array([0, 0, 0.01]) # Differential length element in meters\nr = 0.1 # Distance in meters\nr_hat = np.array([0, 0, 1]) # Unit vector\n\n# Calculate magnetic field\nB = biot_savart(I, dl, r, r_hat)\nprint(\"Magnetic Field:\", B)\n```\n\nThis presentation provides a comprehensive overview of the Biot-Savart and Ampere’s Laws, essential for understanding magnetic fields in various physical contexts."
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"text": "# The Laws of Biot-Savart & Ampere\n\n## Overview of Lecture\n\n### Fundamental Laws for Calculating Magnetic Field\n\n#### Biot-Savart Law\nThe Biot-Savart Law is a fundamental equation that describes the magnetic field generated by a steady electric current. It is often referred to as the \"brute force\" method due to its direct computation of the magnetic field at a point in space.\n\nThe Biot-Savart Law is expressed as:\n$$\n\\mathbf{B} = \\frac{\\mu_0}{4\\pi} \\int \\frac{I \\, d\\mathbf{l} \\times \\mathbf{\\hat{r}}}{r^2}\n$$\nwhere:\n- \\(\\mathbf{B}\\) is the magnetic field,\n- \\(\\mu_0\\) is the permeability of free space,\n- \\(I\\) is the current,\n- \\(d\\mathbf{l}\\) is a differential element of the wire,\n- \\(\\mathbf{\\hat{r}}\\) is the unit vector from the element to the point of interest,\n- \\(r\\) is the distance from the element to the point of interest.\n\n#### Ampere’s Law\nAmpere’s Law provides a \"high symmetry\" approach to calculating the magnetic field, particularly useful for configurations with high symmetry.\n\nAmpere’s Law is given by:\n$$\n\\oint \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 I_{\\text{enc}}\n$$\nwhere:\n- \\(\\oint \\mathbf{B} \\cdot d\\mathbf{l}\\) is the line integral of the magnetic field around a closed loop,\n- \\(I_{\\text{enc}}\\) is the total current enclosed by the loop.\n\n### Examples\n\n**Example: Calculate Magnetic Field of a Straight Wire**\n\n1. **Using Biot-Savart Law:**\n For a long, straight wire carrying a current \\(I\\), the magnetic field at a distance \\(r\\) from the wire is:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n2. **Using Ampere’s Law:**\n For the same wire, Ampere’s Law can be applied by choosing a circular path of radius \\(r\\) centered on the wire. The symmetry simplifies the integral, yielding the same result:\n $$\n B = \\frac{\\mu_0 I}{2\\pi r}\n $$\n\n### Relevant Comments and References\n\n- **Jean-Baptiste Biot and Felix Savart** were pioneers in the study of magnetism, leading to the formulation of the Biot-Savart Law.\n- **André-Marie Ampère** developed Ampere’s Law, which is crucial for understanding magnetic fields in systems with high symmetry.\n\n### Related Presentations\n\n- **Magnetic Field Calculations**: Various presentations delve into calculating magnetic fields using both the Biot-Savart Law and Ampere’s Law.\n- **Applications in Physics**: These laws are foundational in courses on electromagnetism and are applied in numerous practical scenarios, such as in the design of electromagnets and electrical machines.\n\n### Code Example\n\n```python\n# Example code for calculating magnetic field using Biot-Savart Law\nimport numpy as np\n\ndef biot_savart(I, dl, r, r_hat):\n mu_0 = 4 * np.pi * 1e-7\n B = (mu_0 / (4 * np.pi)) * (I * np.cross(dl, r_hat) / r**2)\n return B\n\n# Parameters\nI = 10 # Current in amperes\ndl = np.array([0, 0, 0.01]) # Differential length element in meters\nr = 0.1 # Distance in meters\nr_hat = np.array([0, 0, 1]) # Unit vector\n\n# Calculate magnetic field\nB = biot_savart(I, dl, r, r_hat)\nprint(\"Magnetic Field:\", B)\n```\n\nThis presentation provides a comprehensive overview of the Biot-Savart and Ampere’s Laws, essential for understanding magnetic fields in various physical contexts."
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"text": "Take the 2-minute tour ×\n\nIn brief my question is if you have a tcp server that will only ever have one active connection can you just create one server side data socket. In all the tutorials, I see that a new socket is created and I don't understand why this has to be the case. Why not create one server side socket and then open, close, reset it (which I originally hoped async_accept somehow did)?\n\nIn more detail:\n\nI've gone through the boost asio tutorial for an asynchronous day time server and can get it to compile and work. I can even modify it for my application and get it to run as I desire :) . However, my original approach didn't work and I don't understand why which is where I was hoping for your help.\n\nBasically I wanted to create a TCP server that would only accept one TCP client and ignore all others unless that first client disconnected. I did this using acceptor.close() and acceptor.open() so that when the first connection was accepted I just closed the acceptor, then whenever I picked up an eof error I reopened the acceptor to listen for new connections. I naively thought that because I only ever wanted one active connection I only needed to create a single:\n\n\nAs I only ever have one data socket receiving data from a client it seems overkill to create a whole tcp_connection class, as per the tutorial, that as far as I could tell only returned a new socket constructed with the io_service. (In the tutorial I think every time the server accepts a new connection a new socket is created using this code):\n\nclass tcp_connection\n : public boost::enable_shared_from_this<tcp_connection>\n typedef boost::shared_ptr<tcp_connection> pointer;\n\n static pointer create(boost::asio::io_service& io_service)\n return pointer(new tcp_connection(io_service));\n\n tcp::socket& socket()\n return socket_;\n\n tcp_connection(boost::asio::io_service& io_service)\n : socket_(io_service)\n\n tcp::socket socket_;\n std::string message_;\n\nI therefore tried to use just one boost::asio::ip::tcp::socket. I initialised this in the constructor of my server class using the io_service in a similar way to the tutorial above. My testing showed that my first client would connect and only after the first had disconnected would the second connect. However, regardless of what I did the data buffer on the async_accept(socket_,.....) call would never get filled. Initially I just kept getting eof errors, then I tried closing the socked and reopening which removed the eof error and gave a transport end not connected error. Obviously I'm doing something very stupid here but I can't see what is wrong philosophically speaking with what I'm trying to do. When I create a new socket using the tutorial technique everything works as expected.\n\nSo my question is can I just have a single socket and do something like cancel, close, open? Am I meant to bind or something else but isn't that async_accept job?\n\nI've only used boost asio for a week now and this is my first ever post to a forum like this so go easy on me ;) .\n\nshare|improve this question\n\n1 Answer 1\n\nYou could, but you can't.\n\nThere is no reason why TCP could not be implemented as you state but no operating system I'm aware of does so.\n\nThe reason is quite simple in that the OS must support multiple concurrent connections and so that is the design requirement that dictates the implementation. To achieve the \"there can only be one\" case, simply close the listener socket after successfully accepting the new connection and re-open it later. It's not usually worth the effort to do so.\n\nThe overhead you describe is minimal and not worth trying to optimizing out.\n\nshare|improve this answer\nThanks for your answer. I now understand from your answer that the OS must support multiple concurrent connections and agree that the effort to optimise out creating a new data sockets for each connection is probably not worth any gain. It still vexes me though why it just didn't work when I tried to use a single data socket (the acceptor did but not the socket). Did I just mess up when implementing it or are you saying there are fundamental reasons why it won't? – user1759872 Oct 22 '12 at 19:20\nIt's difficult to say because I don't know the internals of how the boost library does its bookkeeping. Perhaps there is some internal state not being properly set when copying a connection socket over a listening socket. – Brian White Oct 22 '12 at 20:01\n\nYour Answer\n\n\n"
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"text": "Take the 2-minute tour ×\n\nIn brief my question is if you have a tcp server that will only ever have one active connection can you just create one server side data socket. In all the tutorials, I see that a new socket is created and I don't understand why this has to be the case. Why not create one server side socket and then open, close, reset it (which I originally hoped async_accept somehow did)?\n\nIn more detail:\n\nI've gone through the boost asio tutorial for an asynchronous day time server and can get it to compile and work. I can even modify it for my application and get it to run as I desire :) . However, my original approach didn't work and I don't understand why which is where I was hoping for your help.\n\nBasically I wanted to create a TCP server that would only accept one TCP client and ignore all others unless that first client disconnected. I did this using acceptor.close() and acceptor.open() so that when the first connection was accepted I just closed the acceptor, then whenever I picked up an eof error I reopened the acceptor to listen for new connections. I naively thought that because I only ever wanted one active connection I only needed to create a single:\n\n\nAs I only ever have one data socket receiving data from a client it seems overkill to create a whole tcp_connection class, as per the tutorial, that as far as I could tell only returned a new socket constructed with the io_service. (In the tutorial I think every time the server accepts a new connection a new socket is created using this code):\n\nclass tcp_connection\n : public boost::enable_shared_from_this<tcp_connection>\n typedef boost::shared_ptr<tcp_connection> pointer;\n\n static pointer create(boost::asio::io_service& io_service)\n return pointer(new tcp_connection(io_service));\n\n tcp::socket& socket()\n return socket_;\n\n tcp_connection(boost::asio::io_service& io_service)\n : socket_(io_service)\n\n tcp::socket socket_;\n std::string message_;\n\nI therefore tried to use just one boost::asio::ip::tcp::socket. I initialised this in the constructor of my server class using the io_service in a similar way to the tutorial above. My testing showed that my first client would connect and only after the first had disconnected would the second connect. However, regardless of what I did the data buffer on the async_accept(socket_,.....) call would never get filled. Initially I just kept getting eof errors, then I tried closing the socked and reopening which removed the eof error and gave a transport end not connected error. Obviously I'm doing something very stupid here but I can't see what is wrong philosophically speaking with what I'm trying to do. When I create a new socket using the tutorial technique everything works as expected.\n\nSo my question is can I just have a single socket and do something like cancel, close, open? Am I meant to bind or something else but isn't that async_accept job?\n\nI've only used boost asio for a week now and this is my first ever post to a forum like this so go easy on me ;) .\n\nshare|improve this question\n\n1 Answer 1\n\nYou could, but you can't.\n\nThere is no reason why TCP could not be implemented as you state but no operating system I'm aware of does so.\n\nThe reason is quite simple in that the OS must support multiple concurrent connections and so that is the design requirement that dictates the implementation. To achieve the \"there can only be one\" case, simply close the listener socket after successfully accepting the new connection and re-open it later. It's not usually worth the effort to do so.\n\nThe overhead you describe is minimal and not worth trying to optimizing out.\n\nshare|improve this answer\nThanks for your answer. I now understand from your answer that the OS must support multiple concurrent connections and agree that the effort to optimise out creating a new data sockets for each connection is probably not worth any gain. It still vexes me though why it just didn't work when I tried to use a single data socket (the acceptor did but not the socket). Did I just mess up when implementing it or are you saying there are fundamental reasons why it won't? – user1759872 Oct 22 '12 at 19:20\nIt's difficult to say because I don't know the internals of how the boost library does its bookkeeping. Perhaps there is some internal state not being properly set when copying a connection socket over a listening socket. – Brian White Oct 22 '12 at 20:01\n\nYour Answer\n\n\n"
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"text": "Take the 2-minute tour ×\n\nIn brief my question is if you have a tcp server that will only ever have one active connection can you just create one server side data socket. In all the tutorials, I see that a new socket is created and I don't understand why this has to be the case. Why not create one server side socket and then open, close, reset it (which I originally hoped async_accept somehow did)?\n\nIn more detail:\n\nI've gone through the boost asio tutorial for an asynchronous day time server and can get it to compile and work. I can even modify it for my application and get it to run as I desire :) . However, my original approach didn't work and I don't understand why which is where I was hoping for your help.\n\nBasically I wanted to create a TCP server that would only accept one TCP client and ignore all others unless that first client disconnected. I did this using acceptor.close() and acceptor.open() so that when the first connection was accepted I just closed the acceptor, then whenever I picked up an eof error I reopened the acceptor to listen for new connections. I naively thought that because I only ever wanted one active connection I only needed to create a single:\n\n\nAs I only ever have one data socket receiving data from a client it seems overkill to create a whole tcp_connection class, as per the tutorial, that as far as I could tell only returned a new socket constructed with the io_service. (In the tutorial I think every time the server accepts a new connection a new socket is created using this code):\n\nclass tcp_connection\n : public boost::enable_shared_from_this<tcp_connection>\n typedef boost::shared_ptr<tcp_connection> pointer;\n\n static pointer create(boost::asio::io_service& io_service)\n return pointer(new tcp_connection(io_service));\n\n tcp::socket& socket()\n return socket_;\n\n tcp_connection(boost::asio::io_service& io_service)\n : socket_(io_service)\n\n tcp::socket socket_;\n std::string message_;\n\nI therefore tried to use just one boost::asio::ip::tcp::socket. I initialised this in the constructor of my server class using the io_service in a similar way to the tutorial above. My testing showed that my first client would connect and only after the first had disconnected would the second connect. However, regardless of what I did the data buffer on the async_accept(socket_,.....) call would never get filled. Initially I just kept getting eof errors, then I tried closing the socked and reopening which removed the eof error and gave a transport end not connected error. Obviously I'm doing something very stupid here but I can't see what is wrong philosophically speaking with what I'm trying to do. When I create a new socket using the tutorial technique everything works as expected.\n\nSo my question is can I just have a single socket and do something like cancel, close, open? Am I meant to bind or something else but isn't that async_accept job?\n\nI've only used boost asio for a week now and this is my first ever post to a forum like this so go easy on me ;) .\n\nshare|improve this question\n\n1 Answer 1\n\nYou could, but you can't.\n\nThere is no reason why TCP could not be implemented as you state but no operating system I'm aware of does so.\n\nThe reason is quite simple in that the OS must support multiple concurrent connections and so that is the design requirement that dictates the implementation. To achieve the \"there can only be one\" case, simply close the listener socket after successfully accepting the new connection and re-open it later. It's not usually worth the effort to do so.\n\nThe overhead you describe is minimal and not worth trying to optimizing out.\n\nshare|improve this answer\nThanks for your answer. I now understand from your answer that the OS must support multiple concurrent connections and agree that the effort to optimise out creating a new data sockets for each connection is probably not worth any gain. It still vexes me though why it just didn't work when I tried to use a single data socket (the acceptor did but not the socket). Did I just mess up when implementing it or are you saying there are fundamental reasons why it won't? – user1759872 Oct 22 '12 at 19:20\nIt's difficult to say because I don't know the internals of how the boost library does its bookkeeping. Perhaps there is some internal state not being properly set when copying a connection socket over a listening socket. – Brian White Oct 22 '12 at 20:01\n\nYour Answer\n\n\n"
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"text": "# Beginning Algebra Videos\n\n## Marvin's Underground Lectures\n\nIn this series, we delve into key concepts of algebra to provide a solid foundation for further study. The content is structured to enhance understanding through clear explanations and examples.\n\n### Introduction to Algebra\n\nAlgebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.\n\n#### Basic Algebraic Expressions\n\nAn algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$), and operators (like add, subtract, multiply, and divide). For example, $3x + 2$ is an algebraic expression.\n\n#### Solving Simple Equations\n\nTo solve an equation means to find the value of the variable that makes the equation true. For instance, consider the equation $2x + 3 = 7$. Solving for $x$ involves the following steps:\n\n1. Subtract 3 from both sides: \n $$2x + 3 - 3 = 7 - 3$$ \n $$2x = 4$$\n\n2. Divide both sides by 2: \n $$\\frac{2x}{2} = \\frac{4}{2}$$ \n $$x = 2$$\n\nThus, $x = 2$ is the solution.\n\n#### The Distributive Property\n\nThe distributive property is a useful tool in algebra. It states that for any numbers $a$, $b$, and $c$: \n$$a(b + c) = ab + ac$$\n\n**Example**: \nApply the distributive property to $3(x + 4)$: \n$$3(x + 4) = 3x + 12$$\n\n#### Factoring Quadratic Expressions\n\nA quadratic expression is of the form $ax^2 + bx + c$. Factoring involves rewriting it as a product of two binomials. Consider $x^2 + 5x + 6$:\n\n1. Find two numbers that multiply to 6 (constant term) and add to 5 (coefficient of $x$).\n2. These numbers are 2 and 3.\n\nThus, the factored form is: \n$$(x + 2)(x + 3)$$\n\n#### Practice Problems\n\n1. Solve $5y - 3 = 2y + 9$.\n2. Factor the expression $x^2 - 7x + 12$.\n3. Use the distributive property to simplify $4(2x - 3)$.\n\n### Conclusion\n\nAlgebra is fundamental to the study of mathematics, providing the tools to model and solve problems. With practice, the concepts introduced here will become second nature.\n\n**Author's Note**: This content is designed to be accessible to beginners, with a focus on clarity and engagement. Feedback and questions are welcome to improve future lessons.\nGiven the provided text, there is no specific main content or meaningful content to extract, as the text consists entirely of navigation menus, categories, and subcategories from an HTML page. No titles, mathematical content, explanations, or any significant narrative or informational content are present. Therefore, the output is:\n\n```\n\n```\n# Beginning Algebra (Math 70)\n\n## Lecture 01: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 116 Views / 0 Likes)*\n\nDr. James Curl of Modesto Junior College teaches beginning algebra. This first lesson reviews the course objectives and schedule, homework procedures, tests, grading, and similar protocols. The lecture covers numbers, why we use them, and some common forms of numbers.\n\n## Lecture 02: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 114 Views / 0 Likes)*\n\nTopics include properties of real numbers, real numbers and the number line, and fractions.\n\n## Lecture 03: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 139 Views / 0 Likes)*\n\nThis lecture explores properties of real numbers, real numbers and the number line, addition and subtraction of real numbers, multiplication and division of real numbers, and simplifying expressions.\n\n## Lecture 04: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nTopics include simplifying expressions, variables, expressions and equations, exponents, order of operations and inequality, real numbers and the number line, and the addition and multiplication properties of equality.\n\n## Lecture 05: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 86 Views / 0 Likes)*\n\nFocus on the addition and multiplication properties of equality.\n\n## Lecture 06: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nSolving equations is the primary focus.\n\n## Lecture 07: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include the addition and multiplication properties of equality, solving equations, and word problems.\n\n## Lecture 08: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 94 Views / 0 Likes)*\n\nThis lecture covers solving equations, word problems, and formulas.\n\n## Lecture 09: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 88 Views / 0 Likes)*\n\nFocus on formulas, ratio, and proportion.\n\n## Lecture 10: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 133 Views / 0 Likes)*\n\nTopics include ratio and proportion, mixture and motion problems, and money rates.\n\n## Lecture 11: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 132 Views / 0 Likes)*\n\nDiscussion on ratios and proportions, and inequalities.\n\n## Lecture 12: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include inequalities, linear equations in two variables, and graphing linear equations in two variables.\n\n## Lecture 13: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 120 Views / 0 Likes)*\n\nFocus on graphing in mathematics and slope.\n\n## Lecture 14: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 99 Views / 0 Likes)*\n\nTopics include the slope of a line, slope and intercept, and graphing linear equations in two variables.\n\n## Lecture 15: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nExam review of linear equations in two variables, graphing linear equations in two variables, and the slope of a line.\n\n## Lecture 16: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 87 Views / 0 Likes)*\n\nFocus on polynomials and parabolas.\n\n## Lecture 17: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 80 Views / 0 Likes)*\n\nTopics include polynomials and parabolas, and rules for exponents.\n\n## Lecture 18: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 110 Views / 0 Likes)*\n\nDiscussion on polynomials and parabolas, rules for exponents, review for exams, the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material, and multiplication of polynomials.\n\n## Lecture 19: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 90 Views / 0 Likes)*\n\nDiscussing the exam of the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material on multiplication of polynomials, and special products.\n\n## Lecture 20: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 78 Views / 0 Likes)*\n\nFocus on multiplication of polynomials and special products.\n\n## Lecture 21: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nDiscussion on special products and integer exponents and the quotient rule.\n\n## Lecture 22: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 109 Views / 0 Likes)*\n\nTopics include integer exponents and quotient rule, and polynomial division.\n\n## Lecture 23: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 57 Views / 0 Likes)*\n\nFocus on polynomial division and an application of exponents: scientific notation.\n\n## Lecture 24: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nDiscussion on the greatest common factor and factoring by grouping.\nThis text appears to be a list of course titles and categories from a website, likely an online education platform. It does not contain any main content, mathematical expressions, comments, references, or code that needs processing. Therefore, the appropriate response is:\n\n```\nNO USEFUL CONTENT\n```\n# Understanding the Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus is a cornerstone of mathematical analysis, connecting differentiation and integration in a profound way. This theorem is divided into two parts, each providing critical insights into how calculus operates.\n\n## Part 1: The Integral of a Derivative\n\nThe first part of the Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_{a}^{x} f(t) \\, dt, \\]\n\nfor all \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor all \\( x \\) in \\((a, b)\\).\n\n### Explanation\n\nThis part of the theorem essentially tells us that the process of integration can be undone by differentiation. In other words, if you start with a function, integrate it, and then differentiate the result, you end up back where you started, assuming the function is continuous.\n\n## Part 2: Evaluation of a Definite Integral\n\nThe second part of the Fundamental Theorem of Calculus provides a way to evaluate definite integrals. It states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\) and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_{a}^{b} f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis part of the theorem allows us to compute the definite integral of a function by finding any antiderivative of the function and then evaluating the difference between the values of this antiderivative at the endpoints of the interval.\n\n## Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{1}{3}x^3 \\). According to the second part of the Fundamental Theorem of Calculus, the definite integral of \\( f \\) from 1 to 2 is\n\n\\[ \\int_{1}^{2} x^2 \\, dx = F(2) - F(1) = \\frac{1}{3}(2^3) - \\frac{1}{3}(1^3) = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\n## Comments and References\n\n- **Clarification**: The continuity of \\( f \\) is crucial for the first part of the theorem. Without it, the derivative \\( F'(x) \\) may not equal \\( f(x) \\).\n- **Reference**: For a deeper understanding, see \"Calculus\" by Michael Spivak, which provides a rigorous treatment of the Fundamental Theorem of Calculus.\n\n### Code Example\n\nHere is a Python code snippet to compute the definite integral using the second part of the theorem:\n\n```python\ndef antiderivative(x):\n return (1/3) * x**3\n\ndef definite_integral(a, b):\n return antiderivative(b) - antiderivative(a)\n\n# Example usage\nresult = definite_integral(1, 2)\nprint(f\"The definite integral of x^2 from 1 to 2 is {result}\")\n```\n\nThis code defines an antiderivative of \\( x^2 \\) and uses it to compute the definite integral from 1 to 2, yielding the expected result of \\( \\frac{7}{3} \\)."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Beginning Algebra Videos\n\n## Marvin's Underground Lectures\n\nIn this series, we delve into key concepts of algebra to provide a solid foundation for further study. The content is structured to enhance understanding through clear explanations and examples.\n\n### Introduction to Algebra\n\nAlgebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.\n\n#### Basic Algebraic Expressions\n\nAn algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$), and operators (like add, subtract, multiply, and divide). For example, $3x + 2$ is an algebraic expression.\n\n#### Solving Simple Equations\n\nTo solve an equation means to find the value of the variable that makes the equation true. For instance, consider the equation $2x + 3 = 7$. Solving for $x$ involves the following steps:\n\n1. Subtract 3 from both sides: \n $$2x + 3 - 3 = 7 - 3$$ \n $$2x = 4$$\n\n2. Divide both sides by 2: \n $$\\frac{2x}{2} = \\frac{4}{2}$$ \n $$x = 2$$\n\nThus, $x = 2$ is the solution.\n\n#### The Distributive Property\n\nThe distributive property is a useful tool in algebra. It states that for any numbers $a$, $b$, and $c$: \n$$a(b + c) = ab + ac$$\n\n**Example**: \nApply the distributive property to $3(x + 4)$: \n$$3(x + 4) = 3x + 12$$\n\n#### Factoring Quadratic Expressions\n\nA quadratic expression is of the form $ax^2 + bx + c$. Factoring involves rewriting it as a product of two binomials. Consider $x^2 + 5x + 6$:\n\n1. Find two numbers that multiply to 6 (constant term) and add to 5 (coefficient of $x$).\n2. These numbers are 2 and 3.\n\nThus, the factored form is: \n$$(x + 2)(x + 3)$$\n\n#### Practice Problems\n\n1. Solve $5y - 3 = 2y + 9$.\n2. Factor the expression $x^2 - 7x + 12$.\n3. Use the distributive property to simplify $4(2x - 3)$.\n\n### Conclusion\n\nAlgebra is fundamental to the study of mathematics, providing the tools to model and solve problems. With practice, the concepts introduced here will become second nature.\n\n**Author's Note**: This content is designed to be accessible to beginners, with a focus on clarity and engagement. Feedback and questions are welcome to improve future lessons.\nGiven the provided text, there is no specific main content or meaningful content to extract, as the text consists entirely of navigation menus, categories, and subcategories from an HTML page. No titles, mathematical content, explanations, or any significant narrative or informational content are present. Therefore, the output is:\n\n```\n\n```\n# Beginning Algebra (Math 70)\n\n## Lecture 01: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 116 Views / 0 Likes)*\n\nDr. James Curl of Modesto Junior College teaches beginning algebra. This first lesson reviews the course objectives and schedule, homework procedures, tests, grading, and similar protocols. The lecture covers numbers, why we use them, and some common forms of numbers.\n\n## Lecture 02: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 114 Views / 0 Likes)*\n\nTopics include properties of real numbers, real numbers and the number line, and fractions.\n\n## Lecture 03: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 139 Views / 0 Likes)*\n\nThis lecture explores properties of real numbers, real numbers and the number line, addition and subtraction of real numbers, multiplication and division of real numbers, and simplifying expressions.\n\n## Lecture 04: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nTopics include simplifying expressions, variables, expressions and equations, exponents, order of operations and inequality, real numbers and the number line, and the addition and multiplication properties of equality.\n\n## Lecture 05: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 86 Views / 0 Likes)*\n\nFocus on the addition and multiplication properties of equality.\n\n## Lecture 06: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nSolving equations is the primary focus.\n\n## Lecture 07: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include the addition and multiplication properties of equality, solving equations, and word problems.\n\n## Lecture 08: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 94 Views / 0 Likes)*\n\nThis lecture covers solving equations, word problems, and formulas.\n\n## Lecture 09: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 88 Views / 0 Likes)*\n\nFocus on formulas, ratio, and proportion.\n\n## Lecture 10: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 133 Views / 0 Likes)*\n\nTopics include ratio and proportion, mixture and motion problems, and money rates.\n\n## Lecture 11: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 132 Views / 0 Likes)*\n\nDiscussion on ratios and proportions, and inequalities.\n\n## Lecture 12: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include inequalities, linear equations in two variables, and graphing linear equations in two variables.\n\n## Lecture 13: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 120 Views / 0 Likes)*\n\nFocus on graphing in mathematics and slope.\n\n## Lecture 14: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 99 Views / 0 Likes)*\n\nTopics include the slope of a line, slope and intercept, and graphing linear equations in two variables.\n\n## Lecture 15: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nExam review of linear equations in two variables, graphing linear equations in two variables, and the slope of a line.\n\n## Lecture 16: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 87 Views / 0 Likes)*\n\nFocus on polynomials and parabolas.\n\n## Lecture 17: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 80 Views / 0 Likes)*\n\nTopics include polynomials and parabolas, and rules for exponents.\n\n## Lecture 18: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 110 Views / 0 Likes)*\n\nDiscussion on polynomials and parabolas, rules for exponents, review for exams, the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material, and multiplication of polynomials.\n\n## Lecture 19: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 90 Views / 0 Likes)*\n\nDiscussing the exam of the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material on multiplication of polynomials, and special products.\n\n## Lecture 20: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 78 Views / 0 Likes)*\n\nFocus on multiplication of polynomials and special products.\n\n## Lecture 21: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nDiscussion on special products and integer exponents and the quotient rule.\n\n## Lecture 22: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 109 Views / 0 Likes)*\n\nTopics include integer exponents and quotient rule, and polynomial division.\n\n## Lecture 23: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 57 Views / 0 Likes)*\n\nFocus on polynomial division and an application of exponents: scientific notation.\n\n## Lecture 24: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nDiscussion on the greatest common factor and factoring by grouping.\nThis text appears to be a list of course titles and categories from a website, likely an online education platform. It does not contain any main content, mathematical expressions, comments, references, or code that needs processing. Therefore, the appropriate response is:\n\n```\nNO USEFUL CONTENT\n```\n# Understanding the Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus is a cornerstone of mathematical analysis, connecting differentiation and integration in a profound way. This theorem is divided into two parts, each providing critical insights into how calculus operates.\n\n## Part 1: The Integral of a Derivative\n\nThe first part of the Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_{a}^{x} f(t) \\, dt, \\]\n\nfor all \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor all \\( x \\) in \\((a, b)\\).\n\n### Explanation\n\nThis part of the theorem essentially tells us that the process of integration can be undone by differentiation. In other words, if you start with a function, integrate it, and then differentiate the result, you end up back where you started, assuming the function is continuous.\n\n## Part 2: Evaluation of a Definite Integral\n\nThe second part of the Fundamental Theorem of Calculus provides a way to evaluate definite integrals. It states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\) and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_{a}^{b} f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis part of the theorem allows us to compute the definite integral of a function by finding any antiderivative of the function and then evaluating the difference between the values of this antiderivative at the endpoints of the interval.\n\n## Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{1}{3}x^3 \\). According to the second part of the Fundamental Theorem of Calculus, the definite integral of \\( f \\) from 1 to 2 is\n\n\\[ \\int_{1}^{2} x^2 \\, dx = F(2) - F(1) = \\frac{1}{3}(2^3) - \\frac{1}{3}(1^3) = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\n## Comments and References\n\n- **Clarification**: The continuity of \\( f \\) is crucial for the first part of the theorem. Without it, the derivative \\( F'(x) \\) may not equal \\( f(x) \\).\n- **Reference**: For a deeper understanding, see \"Calculus\" by Michael Spivak, which provides a rigorous treatment of the Fundamental Theorem of Calculus.\n\n### Code Example\n\nHere is a Python code snippet to compute the definite integral using the second part of the theorem:\n\n```python\ndef antiderivative(x):\n return (1/3) * x**3\n\ndef definite_integral(a, b):\n return antiderivative(b) - antiderivative(a)\n\n# Example usage\nresult = definite_integral(1, 2)\nprint(f\"The definite integral of x^2 from 1 to 2 is {result}\")\n```\n\nThis code defines an antiderivative of \\( x^2 \\) and uses it to compute the definite integral from 1 to 2, yielding the expected result of \\( \\frac{7}{3} \\)."
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Beginning Algebra Videos\n\n## Marvin's Underground Lectures\n\nIn this series, we delve into key concepts of algebra to provide a solid foundation for further study. The content is structured to enhance understanding through clear explanations and examples.\n\n### Introduction to Algebra\n\nAlgebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.\n\n#### Basic Algebraic Expressions\n\nAn algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$), and operators (like add, subtract, multiply, and divide). For example, $3x + 2$ is an algebraic expression.\n\n#### Solving Simple Equations\n\nTo solve an equation means to find the value of the variable that makes the equation true. For instance, consider the equation $2x + 3 = 7$. Solving for $x$ involves the following steps:\n\n1. Subtract 3 from both sides: \n $$2x + 3 - 3 = 7 - 3$$ \n $$2x = 4$$\n\n2. Divide both sides by 2: \n $$\\frac{2x}{2} = \\frac{4}{2}$$ \n $$x = 2$$\n\nThus, $x = 2$ is the solution.\n\n#### The Distributive Property\n\nThe distributive property is a useful tool in algebra. It states that for any numbers $a$, $b$, and $c$: \n$$a(b + c) = ab + ac$$\n\n**Example**: \nApply the distributive property to $3(x + 4)$: \n$$3(x + 4) = 3x + 12$$\n\n#### Factoring Quadratic Expressions\n\nA quadratic expression is of the form $ax^2 + bx + c$. Factoring involves rewriting it as a product of two binomials. Consider $x^2 + 5x + 6$:\n\n1. Find two numbers that multiply to 6 (constant term) and add to 5 (coefficient of $x$).\n2. These numbers are 2 and 3.\n\nThus, the factored form is: \n$$(x + 2)(x + 3)$$\n\n#### Practice Problems\n\n1. Solve $5y - 3 = 2y + 9$.\n2. Factor the expression $x^2 - 7x + 12$.\n3. Use the distributive property to simplify $4(2x - 3)$.\n\n### Conclusion\n\nAlgebra is fundamental to the study of mathematics, providing the tools to model and solve problems. With practice, the concepts introduced here will become second nature.\n\n**Author's Note**: This content is designed to be accessible to beginners, with a focus on clarity and engagement. Feedback and questions are welcome to improve future lessons.\nGiven the provided text, there is no specific main content or meaningful content to extract, as the text consists entirely of navigation menus, categories, and subcategories from an HTML page. No titles, mathematical content, explanations, or any significant narrative or informational content are present. Therefore, the output is:\n\n```\n\n```\n# Beginning Algebra (Math 70)\n\n## Lecture 01: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 116 Views / 0 Likes)*\n\nDr. James Curl of Modesto Junior College teaches beginning algebra. This first lesson reviews the course objectives and schedule, homework procedures, tests, grading, and similar protocols. The lecture covers numbers, why we use them, and some common forms of numbers.\n\n## Lecture 02: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 114 Views / 0 Likes)*\n\nTopics include properties of real numbers, real numbers and the number line, and fractions.\n\n## Lecture 03: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 139 Views / 0 Likes)*\n\nThis lecture explores properties of real numbers, real numbers and the number line, addition and subtraction of real numbers, multiplication and division of real numbers, and simplifying expressions.\n\n## Lecture 04: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nTopics include simplifying expressions, variables, expressions and equations, exponents, order of operations and inequality, real numbers and the number line, and the addition and multiplication properties of equality.\n\n## Lecture 05: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 86 Views / 0 Likes)*\n\nFocus on the addition and multiplication properties of equality.\n\n## Lecture 06: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nSolving equations is the primary focus.\n\n## Lecture 07: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include the addition and multiplication properties of equality, solving equations, and word problems.\n\n## Lecture 08: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 94 Views / 0 Likes)*\n\nThis lecture covers solving equations, word problems, and formulas.\n\n## Lecture 09: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 88 Views / 0 Likes)*\n\nFocus on formulas, ratio, and proportion.\n\n## Lecture 10: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 133 Views / 0 Likes)*\n\nTopics include ratio and proportion, mixture and motion problems, and money rates.\n\n## Lecture 11: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 132 Views / 0 Likes)*\n\nDiscussion on ratios and proportions, and inequalities.\n\n## Lecture 12: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nTopics include inequalities, linear equations in two variables, and graphing linear equations in two variables.\n\n## Lecture 13: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 120 Views / 0 Likes)*\n\nFocus on graphing in mathematics and slope.\n\n## Lecture 14: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 99 Views / 0 Likes)*\n\nTopics include the slope of a line, slope and intercept, and graphing linear equations in two variables.\n\n## Lecture 15: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 118 Views / 0 Likes)*\n\nExam review of linear equations in two variables, graphing linear equations in two variables, and the slope of a line.\n\n## Lecture 16: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 87 Views / 0 Likes)*\n\nFocus on polynomials and parabolas.\n\n## Lecture 17: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 80 Views / 0 Likes)*\n\nTopics include polynomials and parabolas, and rules for exponents.\n\n## Lecture 18: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 110 Views / 0 Likes)*\n\nDiscussion on polynomials and parabolas, rules for exponents, review for exams, the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material, and multiplication of polynomials.\n\n## Lecture 19: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 90 Views / 0 Likes)*\n\nDiscussing the exam of the real number system, linear equations and inequalities in one variable, linear expressions in two variables, polynomials and exponents, new material on multiplication of polynomials, and special products.\n\n## Lecture 20: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 78 Views / 0 Likes)*\n\nFocus on multiplication of polynomials and special products.\n\n## Lecture 21: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 108 Views / 0 Likes)*\n\nDiscussion on special products and integer exponents and the quotient rule.\n\n## Lecture 22: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 109 Views / 0 Likes)*\n\nTopics include integer exponents and quotient rule, and polynomial division.\n\n## Lecture 23: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 57 Views / 0 Likes)*\n\nFocus on polynomial division and an application of exponents: scientific notation.\n\n## Lecture 24: Beginning Algebra (Math 70)\n*(by Admin, Added 4 years ago, 107 Views / 0 Likes)*\n\nDiscussion on the greatest common factor and factoring by grouping.\nThis text appears to be a list of course titles and categories from a website, likely an online education platform. It does not contain any main content, mathematical expressions, comments, references, or code that needs processing. Therefore, the appropriate response is:\n\n```\nNO USEFUL CONTENT\n```\n# Understanding the Fundamental Theorem of Calculus\n\nThe Fundamental Theorem of Calculus is a cornerstone of mathematical analysis, connecting differentiation and integration in a profound way. This theorem is divided into two parts, each providing critical insights into how calculus operates.\n\n## Part 1: The Integral of a Derivative\n\nThe first part of the Fundamental Theorem of Calculus states that if \\( f \\) is a continuous real-valued function defined on a closed interval \\([a, b]\\), and \\( F \\) is the function defined by\n\n\\[ F(x) = \\int_{a}^{x} f(t) \\, dt, \\]\n\nfor all \\( x \\) in \\([a, b]\\), then \\( F \\) is continuous on \\([a, b]\\), differentiable on the open interval \\((a, b)\\), and\n\n\\[ F'(x) = f(x) \\]\n\nfor all \\( x \\) in \\((a, b)\\).\n\n### Explanation\n\nThis part of the theorem essentially tells us that the process of integration can be undone by differentiation. In other words, if you start with a function, integrate it, and then differentiate the result, you end up back where you started, assuming the function is continuous.\n\n## Part 2: Evaluation of a Definite Integral\n\nThe second part of the Fundamental Theorem of Calculus provides a way to evaluate definite integrals. It states that if \\( f \\) is a continuous real-valued function on \\([a, b]\\) and \\( F \\) is any antiderivative of \\( f \\) on \\([a, b]\\), then\n\n\\[ \\int_{a}^{b} f(x) \\, dx = F(b) - F(a). \\]\n\n### Explanation\n\nThis part of the theorem allows us to compute the definite integral of a function by finding any antiderivative of the function and then evaluating the difference between the values of this antiderivative at the endpoints of the interval.\n\n## Example\n\nConsider the function \\( f(x) = x^2 \\). An antiderivative of \\( f \\) is \\( F(x) = \\frac{1}{3}x^3 \\). According to the second part of the Fundamental Theorem of Calculus, the definite integral of \\( f \\) from 1 to 2 is\n\n\\[ \\int_{1}^{2} x^2 \\, dx = F(2) - F(1) = \\frac{1}{3}(2^3) - \\frac{1}{3}(1^3) = \\frac{8}{3} - \\frac{1}{3} = \\frac{7}{3}. \\]\n\n## Comments and References\n\n- **Clarification**: The continuity of \\( f \\) is crucial for the first part of the theorem. Without it, the derivative \\( F'(x) \\) may not equal \\( f(x) \\).\n- **Reference**: For a deeper understanding, see \"Calculus\" by Michael Spivak, which provides a rigorous treatment of the Fundamental Theorem of Calculus.\n\n### Code Example\n\nHere is a Python code snippet to compute the definite integral using the second part of the theorem:\n\n```python\ndef antiderivative(x):\n return (1/3) * x**3\n\ndef definite_integral(a, b):\n return antiderivative(b) - antiderivative(a)\n\n# Example usage\nresult = definite_integral(1, 2)\nprint(f\"The definite integral of x^2 from 1 to 2 is {result}\")\n```\n\nThis code defines an antiderivative of \\( x^2 \\) and uses it to compute the definite integral from 1 to 2, yielding the expected result of \\( \\frac{7}{3} \\)."
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Take the 2-minute tour ×\n\nGiven is a locally finite countable connected poset which satisfies further the following properties:\n\n 1. Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is covered by both the sets ${\\rm Past}(x)$ and ${\\rm Future}(x)$ for $x$ running over $C$ i.e. $A \\subset \\bigcup_{x \\in C} ({\\rm Past}(x) \\cup {\\rm Future}(x))$ and where ${\\rm Past}(x):=\\lbrace y\\mid y \\leq x\\rbrace$ and ${\\rm Future}(x):=\\lbrace y \\mid x \\leq y\\rbrace$.\n\n 2. For any $x$, the intersection of ${\\rm Past}(x)$ with any antichain is finite. Similarily for ${\\rm Future}(x)$.\n\n\n 1. Is the orbit of any point $x$ by an automorphism $f$ of the poset finite?\n 2. Is the group of automorphisms of this poset countable?\n 3. As a polish group, is the group of automorphisms of the poset locally compact?\n\n Thank you\n\nshare|improve this question\nI dont understand the title of the question: what are we quantizing? – André Henriques Sep 28 '11 at 18:53\nIt is related to a quantization procedure on a family of isomorphic causal sets, which are locally finite countable connected posets. – user16974 Sep 28 '11 at 19:21\nThe quantifiers in your hypothesis are not clear. Do you mean to assume that your partial order, in addition to being locally finite, countable and connected, has the covering property for every pair (A,C) of a maximal chain and maximal antichain? And when you say A is covered by Past(x) and Future(x), do you mean that the family of pasts alone covers A and also the family of futures alone covers A? Or do you mean that one might possibly need both futures and pasts from C to cover A? – Joel David Hamkins Sep 29 '11 at 0:46\nJoel: I edited the question. – user16974 Sep 29 '11 at 4:50\nI still do not quite understand it. Every singleton is an antichain, so is condition 1 simply saying that every maximal chain is upwards and downwards cofinal in the poset? Since the poset is countable, isn’t this equivalent to the poset being upwards and downwards directed? – Emil Jeřábek Sep 29 '11 at 9:25\nshow 1 more comment\n\n2 Answers\n\nup vote 1 down vote accepted\n\nIf I've understood the hypotheses correctly, the covering relation gives a connected locally finite directed graph: starting from a point $x$, the maximal elements underneath it form a finite antichain, as do the elements covering it. The automorphism group is therefore a second-countable totally disconnected locally compact group under the topology given by declaring the stabiliser of each point to be open. The automorphism group as a whole will be countable if and only if every point stabiliser is finite and compact if and only if every orbit of the group is finite.\n\nThere's no reason for orbits of even individual automorphisms to be finite: for instance, your poset could be $\\mathbb{Z}$ and the automorphism could be $x \\mapsto x+1$.\n\nEdit: Here's a simple example of how the automorphism group can be large. Start with $\\mathbb{Z}$, and replace every odd number $n$ with $|n|$ incomparable copies of itself. The automorphism group is $H \\rtimes \\mathbb{Z}/2\\mathbb{Z}$, where $H$ is a Cartesian product of symmetric groups of unbounded degree. In particular the automorphism group is uncountable, and contains a copy of every countably-based profinite group.\n\nshare|improve this answer\nIf it is possible, please let me know of any example where the automorphism group is not countable. For the moment, I am satisfied with the result that the group is locally compact. – user16974 Sep 29 '11 at 13:01\nIf we ask whether the orbit of an automorphism which sends $x$ to an element incomparable with $x$ that is $x \\mapsto y$ and not $y \\leq x$ or $x \\leq y$ would the orbit of $x$ be finite? – user16974 Sep 29 '11 at 13:24\nColin: where can I find proofs of what you said? Thank you – user16974 Sep 29 '11 at 19:49\nStandard fact (can be found in a textbook on profinite groups): an inverse limit of finite groups is compact and totally disconnected. Everything else should become clear as soon as you try to prove it. – Colin Reid Sep 30 '11 at 21:24\nadd comment\n\nClaim: The poset is a chain.\n\nProof: Consider an unrelated pair a, b in the poset. Let A = {a}. Then a \\in Past(b) by condition 1, which leads to a contradiction.\n\nI imagine that your definitions are unclear. Please carefully define 'covered by', 'maximal chain', poset?, etc.\n\nshare|improve this answer\nJudging by previous questions, I think the OP is feeling around for axioms that will ensure a reasonably tame automorphism group. My suggestion would be to change 1. to 'every finite set lies in an interval' or something like that. – Colin Reid Sep 29 '11 at 15:30\nDavid, while the OP’s terminology is not quite clear, I think you are a bit overdoing it. Poset and maximal chain are standard terms. It seems that the intended reading of 1 is $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Past}(x)$ and $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Future}(x)$. – Emil Jeřábek Sep 29 '11 at 15:48\nI appologize for not being clear. My intention is $A \\subset \\bigcup_{x \\in C} {\\rm Past}(x) \\cup \\bigcup_{x \\in C} {\\rm Future}(x)$. – user16974 Sep 29 '11 at 17:21\nadd comment\n\nYour Answer\n\n"
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
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"text": "Take the 2-minute tour ×\n\nGiven is a locally finite countable connected poset which satisfies further the following properties:\n\n 1. Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is covered by both the sets ${\\rm Past}(x)$ and ${\\rm Future}(x)$ for $x$ running over $C$ i.e. $A \\subset \\bigcup_{x \\in C} ({\\rm Past}(x) \\cup {\\rm Future}(x))$ and where ${\\rm Past}(x):=\\lbrace y\\mid y \\leq x\\rbrace$ and ${\\rm Future}(x):=\\lbrace y \\mid x \\leq y\\rbrace$.\n\n 2. For any $x$, the intersection of ${\\rm Past}(x)$ with any antichain is finite. Similarily for ${\\rm Future}(x)$.\n\n\n 1. Is the orbit of any point $x$ by an automorphism $f$ of the poset finite?\n 2. Is the group of automorphisms of this poset countable?\n 3. As a polish group, is the group of automorphisms of the poset locally compact?\n\n Thank you\n\nshare|improve this question\nI dont understand the title of the question: what are we quantizing? – André Henriques Sep 28 '11 at 18:53\nIt is related to a quantization procedure on a family of isomorphic causal sets, which are locally finite countable connected posets. – user16974 Sep 28 '11 at 19:21\nThe quantifiers in your hypothesis are not clear. Do you mean to assume that your partial order, in addition to being locally finite, countable and connected, has the covering property for every pair (A,C) of a maximal chain and maximal antichain? And when you say A is covered by Past(x) and Future(x), do you mean that the family of pasts alone covers A and also the family of futures alone covers A? Or do you mean that one might possibly need both futures and pasts from C to cover A? – Joel David Hamkins Sep 29 '11 at 0:46\nJoel: I edited the question. – user16974 Sep 29 '11 at 4:50\nI still do not quite understand it. Every singleton is an antichain, so is condition 1 simply saying that every maximal chain is upwards and downwards cofinal in the poset? Since the poset is countable, isn’t this equivalent to the poset being upwards and downwards directed? – Emil Jeřábek Sep 29 '11 at 9:25\nshow 1 more comment\n\n2 Answers\n\nup vote 1 down vote accepted\n\nIf I've understood the hypotheses correctly, the covering relation gives a connected locally finite directed graph: starting from a point $x$, the maximal elements underneath it form a finite antichain, as do the elements covering it. The automorphism group is therefore a second-countable totally disconnected locally compact group under the topology given by declaring the stabiliser of each point to be open. The automorphism group as a whole will be countable if and only if every point stabiliser is finite and compact if and only if every orbit of the group is finite.\n\nThere's no reason for orbits of even individual automorphisms to be finite: for instance, your poset could be $\\mathbb{Z}$ and the automorphism could be $x \\mapsto x+1$.\n\nEdit: Here's a simple example of how the automorphism group can be large. Start with $\\mathbb{Z}$, and replace every odd number $n$ with $|n|$ incomparable copies of itself. The automorphism group is $H \\rtimes \\mathbb{Z}/2\\mathbb{Z}$, where $H$ is a Cartesian product of symmetric groups of unbounded degree. In particular the automorphism group is uncountable, and contains a copy of every countably-based profinite group.\n\nshare|improve this answer\nIf it is possible, please let me know of any example where the automorphism group is not countable. For the moment, I am satisfied with the result that the group is locally compact. – user16974 Sep 29 '11 at 13:01\nIf we ask whether the orbit of an automorphism which sends $x$ to an element incomparable with $x$ that is $x \\mapsto y$ and not $y \\leq x$ or $x \\leq y$ would the orbit of $x$ be finite? – user16974 Sep 29 '11 at 13:24\nColin: where can I find proofs of what you said? Thank you – user16974 Sep 29 '11 at 19:49\nStandard fact (can be found in a textbook on profinite groups): an inverse limit of finite groups is compact and totally disconnected. Everything else should become clear as soon as you try to prove it. – Colin Reid Sep 30 '11 at 21:24\nadd comment\n\nClaim: The poset is a chain.\n\nProof: Consider an unrelated pair a, b in the poset. Let A = {a}. Then a \\in Past(b) by condition 1, which leads to a contradiction.\n\nI imagine that your definitions are unclear. Please carefully define 'covered by', 'maximal chain', poset?, etc.\n\nshare|improve this answer\nJudging by previous questions, I think the OP is feeling around for axioms that will ensure a reasonably tame automorphism group. My suggestion would be to change 1. to 'every finite set lies in an interval' or something like that. – Colin Reid Sep 29 '11 at 15:30\nDavid, while the OP’s terminology is not quite clear, I think you are a bit overdoing it. Poset and maximal chain are standard terms. It seems that the intended reading of 1 is $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Past}(x)$ and $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Future}(x)$. – Emil Jeřábek Sep 29 '11 at 15:48\nI appologize for not being clear. My intention is $A \\subset \\bigcup_{x \\in C} {\\rm Past}(x) \\cup \\bigcup_{x \\in C} {\\rm Future}(x)$. – user16974 Sep 29 '11 at 17:21\nadd comment\n\nYour Answer\n\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
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"answer_type": "categorical",
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"text": "Take the 2-minute tour ×\n\nGiven is a locally finite countable connected poset which satisfies further the following properties:\n\n 1. Let $C$ be any maximal chain ( i.e. inextendible chain) and $A$ be any antichain. Then $A$ is covered by both the sets ${\\rm Past}(x)$ and ${\\rm Future}(x)$ for $x$ running over $C$ i.e. $A \\subset \\bigcup_{x \\in C} ({\\rm Past}(x) \\cup {\\rm Future}(x))$ and where ${\\rm Past}(x):=\\lbrace y\\mid y \\leq x\\rbrace$ and ${\\rm Future}(x):=\\lbrace y \\mid x \\leq y\\rbrace$.\n\n 2. For any $x$, the intersection of ${\\rm Past}(x)$ with any antichain is finite. Similarily for ${\\rm Future}(x)$.\n\n\n 1. Is the orbit of any point $x$ by an automorphism $f$ of the poset finite?\n 2. Is the group of automorphisms of this poset countable?\n 3. As a polish group, is the group of automorphisms of the poset locally compact?\n\n Thank you\n\nshare|improve this question\nI dont understand the title of the question: what are we quantizing? – André Henriques Sep 28 '11 at 18:53\nIt is related to a quantization procedure on a family of isomorphic causal sets, which are locally finite countable connected posets. – user16974 Sep 28 '11 at 19:21\nThe quantifiers in your hypothesis are not clear. Do you mean to assume that your partial order, in addition to being locally finite, countable and connected, has the covering property for every pair (A,C) of a maximal chain and maximal antichain? And when you say A is covered by Past(x) and Future(x), do you mean that the family of pasts alone covers A and also the family of futures alone covers A? Or do you mean that one might possibly need both futures and pasts from C to cover A? – Joel David Hamkins Sep 29 '11 at 0:46\nJoel: I edited the question. – user16974 Sep 29 '11 at 4:50\nI still do not quite understand it. Every singleton is an antichain, so is condition 1 simply saying that every maximal chain is upwards and downwards cofinal in the poset? Since the poset is countable, isn’t this equivalent to the poset being upwards and downwards directed? – Emil Jeřábek Sep 29 '11 at 9:25\nshow 1 more comment\n\n2 Answers\n\nup vote 1 down vote accepted\n\nIf I've understood the hypotheses correctly, the covering relation gives a connected locally finite directed graph: starting from a point $x$, the maximal elements underneath it form a finite antichain, as do the elements covering it. The automorphism group is therefore a second-countable totally disconnected locally compact group under the topology given by declaring the stabiliser of each point to be open. The automorphism group as a whole will be countable if and only if every point stabiliser is finite and compact if and only if every orbit of the group is finite.\n\nThere's no reason for orbits of even individual automorphisms to be finite: for instance, your poset could be $\\mathbb{Z}$ and the automorphism could be $x \\mapsto x+1$.\n\nEdit: Here's a simple example of how the automorphism group can be large. Start with $\\mathbb{Z}$, and replace every odd number $n$ with $|n|$ incomparable copies of itself. The automorphism group is $H \\rtimes \\mathbb{Z}/2\\mathbb{Z}$, where $H$ is a Cartesian product of symmetric groups of unbounded degree. In particular the automorphism group is uncountable, and contains a copy of every countably-based profinite group.\n\nshare|improve this answer\nIf it is possible, please let me know of any example where the automorphism group is not countable. For the moment, I am satisfied with the result that the group is locally compact. – user16974 Sep 29 '11 at 13:01\nIf we ask whether the orbit of an automorphism which sends $x$ to an element incomparable with $x$ that is $x \\mapsto y$ and not $y \\leq x$ or $x \\leq y$ would the orbit of $x$ be finite? – user16974 Sep 29 '11 at 13:24\nColin: where can I find proofs of what you said? Thank you – user16974 Sep 29 '11 at 19:49\nStandard fact (can be found in a textbook on profinite groups): an inverse limit of finite groups is compact and totally disconnected. Everything else should become clear as soon as you try to prove it. – Colin Reid Sep 30 '11 at 21:24\nadd comment\n\nClaim: The poset is a chain.\n\nProof: Consider an unrelated pair a, b in the poset. Let A = {a}. Then a \\in Past(b) by condition 1, which leads to a contradiction.\n\nI imagine that your definitions are unclear. Please carefully define 'covered by', 'maximal chain', poset?, etc.\n\nshare|improve this answer\nJudging by previous questions, I think the OP is feeling around for axioms that will ensure a reasonably tame automorphism group. My suggestion would be to change 1. to 'every finite set lies in an interval' or something like that. – Colin Reid Sep 29 '11 at 15:30\nDavid, while the OP’s terminology is not quite clear, I think you are a bit overdoing it. Poset and maximal chain are standard terms. It seems that the intended reading of 1 is $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Past}(x)$ and $A\\subseteq\\bigcup_{x\\in C}\\mathrm{Future}(x)$. – Emil Jeřábek Sep 29 '11 at 15:48\nI appologize for not being clear. My intention is $A \\subset \\bigcup_{x \\in C} {\\rm Past}(x) \\cup \\bigcup_{x \\in C} {\\rm Future}(x)$. – user16974 Sep 29 '11 at 17:21\nadd comment\n\nYour Answer\n\n"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"text": "# Harmonic Series is Divergent\n\n## Theorem\nThe harmonic series:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n}$$\n\ndiverges.\n\n## Proof 1\nThe harmonic series can be expressed as:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{1}{2} + \\frac{1}{3}}_{s_1} + \\underbrace{\\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7}}_{s_2} + \\cdots$$\n\nwhere \n\n$$s_k = \\sum_{i = 2^k}^{2^{k + 1} - 1} \\frac{1}{i}$$\n\nFrom the ordering of reciprocals, we have:\n\n$$\\forall m, n \\in \\mathbb{N}_{>0}: m < n \\implies \\frac{1}{m} > \\frac{1}{n}$$\n\nThus, each of the summands in a given $s_k$ is greater than $\\frac{1}{2^{k + 1}}$. The number of summands in $s_k$ is $2^k$, so:\n\n$$s_k > \\frac{2^k}{2^{k + 1}} = \\frac{1}{2}$$\n\nHence, the harmonic sum $H_{2^m}$ satisfies:\n\n$$H_{2^m} = \\sum_{n = 1}^{2^m} \\frac{1}{n} > 1 + \\sum_{a = 0}^m \\frac{1}{2} = 1 + \\frac{m}{2}$$\n\nThe right-hand side diverges by the $n$th term test. The result follows from the Comparison Test for Divergence.\n\n$\\blacksquare$\n\n## Proof 2\nAll terms of the harmonic series are strictly positive and decreasing. By the Cauchy Condensation Test, we examine the convergence of:\n\n$$\\sum_{n = 1}^\\infty 2^n \\frac{1}{2^n} = \\sum_{n = 1}^\\infty 1$$\n\nThis series diverges by the $n$th term test, so $\\sum \\frac{1}{n}$ also diverges.\n\n$\\blacksquare$\n\n## Proof 3\nThe integral of the reciprocal function is divergent. Hence, by the Integral Test, the harmonic series diverges.\n\n$\\blacksquare$\n\n## Proof 4\nFor all $N \\in \\mathbb{N}$:\n\n$$\\frac{1}{N} + \\frac{1}{N + 1} + \\cdots + \\frac{1}{2N} > N \\cdot \\frac{1}{2N} = \\frac{1}{2}$$\n\nBy Cauchy's Convergence Criterion for Series, the harmonic series is divergent.\n\n$\\blacksquare$\n\n## Proof 5\nAssume for $G \\ge 4$, $\\sum_{n = G}^\\infty \\frac{1}{n} = L < \\infty$. Let $s_N := \\sum_{n = G}^N \\frac{1}{n}$ be the partial sum of the series. The sequence $\\{ s_N \\}$ converges if and only if $\\sum_{n = G}^\\infty \\frac{1}{n}$ converges.\n\nConsider the constant sequence $\\{ G \\}$ with limit $G$. The product sequence $\\{ Gs_N \\}$ has limit $GL$. We have:\n\n$$GL = \\sum_{n = G}^\\infty \\frac{G}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{G}{G+1} + \\ldots + \\frac{G}{G+4}}_{s_1} + \\underbrace{\\frac{G}{G+5} + \\ldots + \\frac{G}{G+12}}_{s_2} + \\ldots $$\n\nFor $k \\ge 2$, $s_k \\ge \\frac{2^{k+1}G}{G + 2^{k+2}}$. \n\nClaim: For $k \\ge 1, G \\ge 4$, we have $\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1$. \n\nProof: \n\n$$\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1 \\iff k + 1 \\ge \\log_2\\frac{G}{G-2}.$$ \n\nIf $G = 4$, then $\\log_2(2) = 1$. If $G > 4$, $\\frac{G}{G-2} \\to 1$ as $G \\uparrow$, so $\\log_2\\frac{G}{G-2} \\to 0$.\n\nNow,\n\n$$GL = s_0 + s_1 + s_2 + \\ldots \\ge 1 + 1 + 1 + \\ldots \\to \\infty > GL$$\n\nThis is a contradiction. Therefore, the sequence $\\{s_N\\}$ does not converge, implying the harmonic series diverges.\n\n$\\blacksquare$\n\n## Also see\n\n- Definition: Harmonic Numbers\n- Definition: Riemann Zeta Function\n\n## Historical Note\n\nThe proof that the Harmonic Series is Divergent was discovered by Nicole Oresme, lost for centuries, and rediscovered by Pietro Mengoli in 1647, Johann Bernoulli in 1687, and Jakob II Bernoulli. It is often (erroneously) attributed to Jacob Bernoulli.\n\n## Sources\n\n- 1986: David Wells: Curious and Interesting Numbers\n- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers\n- 1992: George F. Simmons: Calculus Gems\n- 1997: David Wells: Curious and Interesting Numbers (2nd ed.)\n- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.)"
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questioner_given_difficulty_id | [
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"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
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"index": 98,
"split": "train",
"text": "# Harmonic Series is Divergent\n\n## Theorem\nThe harmonic series:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n}$$\n\ndiverges.\n\n## Proof 1\nThe harmonic series can be expressed as:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{1}{2} + \\frac{1}{3}}_{s_1} + \\underbrace{\\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7}}_{s_2} + \\cdots$$\n\nwhere \n\n$$s_k = \\sum_{i = 2^k}^{2^{k + 1} - 1} \\frac{1}{i}$$\n\nFrom the ordering of reciprocals, we have:\n\n$$\\forall m, n \\in \\mathbb{N}_{>0}: m < n \\implies \\frac{1}{m} > \\frac{1}{n}$$\n\nThus, each of the summands in a given $s_k$ is greater than $\\frac{1}{2^{k + 1}}$. The number of summands in $s_k$ is $2^k$, so:\n\n$$s_k > \\frac{2^k}{2^{k + 1}} = \\frac{1}{2}$$\n\nHence, the harmonic sum $H_{2^m}$ satisfies:\n\n$$H_{2^m} = \\sum_{n = 1}^{2^m} \\frac{1}{n} > 1 + \\sum_{a = 0}^m \\frac{1}{2} = 1 + \\frac{m}{2}$$\n\nThe right-hand side diverges by the $n$th term test. The result follows from the Comparison Test for Divergence.\n\n$\\blacksquare$\n\n## Proof 2\nAll terms of the harmonic series are strictly positive and decreasing. By the Cauchy Condensation Test, we examine the convergence of:\n\n$$\\sum_{n = 1}^\\infty 2^n \\frac{1}{2^n} = \\sum_{n = 1}^\\infty 1$$\n\nThis series diverges by the $n$th term test, so $\\sum \\frac{1}{n}$ also diverges.\n\n$\\blacksquare$\n\n## Proof 3\nThe integral of the reciprocal function is divergent. Hence, by the Integral Test, the harmonic series diverges.\n\n$\\blacksquare$\n\n## Proof 4\nFor all $N \\in \\mathbb{N}$:\n\n$$\\frac{1}{N} + \\frac{1}{N + 1} + \\cdots + \\frac{1}{2N} > N \\cdot \\frac{1}{2N} = \\frac{1}{2}$$\n\nBy Cauchy's Convergence Criterion for Series, the harmonic series is divergent.\n\n$\\blacksquare$\n\n## Proof 5\nAssume for $G \\ge 4$, $\\sum_{n = G}^\\infty \\frac{1}{n} = L < \\infty$. Let $s_N := \\sum_{n = G}^N \\frac{1}{n}$ be the partial sum of the series. The sequence $\\{ s_N \\}$ converges if and only if $\\sum_{n = G}^\\infty \\frac{1}{n}$ converges.\n\nConsider the constant sequence $\\{ G \\}$ with limit $G$. The product sequence $\\{ Gs_N \\}$ has limit $GL$. We have:\n\n$$GL = \\sum_{n = G}^\\infty \\frac{G}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{G}{G+1} + \\ldots + \\frac{G}{G+4}}_{s_1} + \\underbrace{\\frac{G}{G+5} + \\ldots + \\frac{G}{G+12}}_{s_2} + \\ldots $$\n\nFor $k \\ge 2$, $s_k \\ge \\frac{2^{k+1}G}{G + 2^{k+2}}$. \n\nClaim: For $k \\ge 1, G \\ge 4$, we have $\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1$. \n\nProof: \n\n$$\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1 \\iff k + 1 \\ge \\log_2\\frac{G}{G-2}.$$ \n\nIf $G = 4$, then $\\log_2(2) = 1$. If $G > 4$, $\\frac{G}{G-2} \\to 1$ as $G \\uparrow$, so $\\log_2\\frac{G}{G-2} \\to 0$.\n\nNow,\n\n$$GL = s_0 + s_1 + s_2 + \\ldots \\ge 1 + 1 + 1 + \\ldots \\to \\infty > GL$$\n\nThis is a contradiction. Therefore, the sequence $\\{s_N\\}$ does not converge, implying the harmonic series diverges.\n\n$\\blacksquare$\n\n## Also see\n\n- Definition: Harmonic Numbers\n- Definition: Riemann Zeta Function\n\n## Historical Note\n\nThe proof that the Harmonic Series is Divergent was discovered by Nicole Oresme, lost for centuries, and rediscovered by Pietro Mengoli in 1647, Johann Bernoulli in 1687, and Jakob II Bernoulli. It is often (erroneously) attributed to Jacob Bernoulli.\n\n## Sources\n\n- 1986: David Wells: Curious and Interesting Numbers\n- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers\n- 1992: George F. Simmons: Calculus Gems\n- 1997: David Wells: Curious and Interesting Numbers (2nd ed.)\n- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.)"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | math | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "integer",
"index": 98,
"split": "train",
"text": "# Harmonic Series is Divergent\n\n## Theorem\nThe harmonic series:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n}$$\n\ndiverges.\n\n## Proof 1\nThe harmonic series can be expressed as:\n\n$$\\displaystyle \\sum_{n = 1}^\\infty \\frac{1}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{1}{2} + \\frac{1}{3}}_{s_1} + \\underbrace{\\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7}}_{s_2} + \\cdots$$\n\nwhere \n\n$$s_k = \\sum_{i = 2^k}^{2^{k + 1} - 1} \\frac{1}{i}$$\n\nFrom the ordering of reciprocals, we have:\n\n$$\\forall m, n \\in \\mathbb{N}_{>0}: m < n \\implies \\frac{1}{m} > \\frac{1}{n}$$\n\nThus, each of the summands in a given $s_k$ is greater than $\\frac{1}{2^{k + 1}}$. The number of summands in $s_k$ is $2^k$, so:\n\n$$s_k > \\frac{2^k}{2^{k + 1}} = \\frac{1}{2}$$\n\nHence, the harmonic sum $H_{2^m}$ satisfies:\n\n$$H_{2^m} = \\sum_{n = 1}^{2^m} \\frac{1}{n} > 1 + \\sum_{a = 0}^m \\frac{1}{2} = 1 + \\frac{m}{2}$$\n\nThe right-hand side diverges by the $n$th term test. The result follows from the Comparison Test for Divergence.\n\n$\\blacksquare$\n\n## Proof 2\nAll terms of the harmonic series are strictly positive and decreasing. By the Cauchy Condensation Test, we examine the convergence of:\n\n$$\\sum_{n = 1}^\\infty 2^n \\frac{1}{2^n} = \\sum_{n = 1}^\\infty 1$$\n\nThis series diverges by the $n$th term test, so $\\sum \\frac{1}{n}$ also diverges.\n\n$\\blacksquare$\n\n## Proof 3\nThe integral of the reciprocal function is divergent. Hence, by the Integral Test, the harmonic series diverges.\n\n$\\blacksquare$\n\n## Proof 4\nFor all $N \\in \\mathbb{N}$:\n\n$$\\frac{1}{N} + \\frac{1}{N + 1} + \\cdots + \\frac{1}{2N} > N \\cdot \\frac{1}{2N} = \\frac{1}{2}$$\n\nBy Cauchy's Convergence Criterion for Series, the harmonic series is divergent.\n\n$\\blacksquare$\n\n## Proof 5\nAssume for $G \\ge 4$, $\\sum_{n = G}^\\infty \\frac{1}{n} = L < \\infty$. Let $s_N := \\sum_{n = G}^N \\frac{1}{n}$ be the partial sum of the series. The sequence $\\{ s_N \\}$ converges if and only if $\\sum_{n = G}^\\infty \\frac{1}{n}$ converges.\n\nConsider the constant sequence $\\{ G \\}$ with limit $G$. The product sequence $\\{ Gs_N \\}$ has limit $GL$. We have:\n\n$$GL = \\sum_{n = G}^\\infty \\frac{G}{n} = \\underbrace{1}_{s_0} + \\underbrace{\\frac{G}{G+1} + \\ldots + \\frac{G}{G+4}}_{s_1} + \\underbrace{\\frac{G}{G+5} + \\ldots + \\frac{G}{G+12}}_{s_2} + \\ldots $$\n\nFor $k \\ge 2$, $s_k \\ge \\frac{2^{k+1}G}{G + 2^{k+2}}$. \n\nClaim: For $k \\ge 1, G \\ge 4$, we have $\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1$. \n\nProof: \n\n$$\\frac{2^{k+1}G}{G+2^{k+2}} \\ge 1 \\iff k + 1 \\ge \\log_2\\frac{G}{G-2}.$$ \n\nIf $G = 4$, then $\\log_2(2) = 1$. If $G > 4$, $\\frac{G}{G-2} \\to 1$ as $G \\uparrow$, so $\\log_2\\frac{G}{G-2} \\to 0$.\n\nNow,\n\n$$GL = s_0 + s_1 + s_2 + \\ldots \\ge 1 + 1 + 1 + \\ldots \\to \\infty > GL$$\n\nThis is a contradiction. Therefore, the sequence $\\{s_N\\}$ does not converge, implying the harmonic series diverges.\n\n$\\blacksquare$\n\n## Also see\n\n- Definition: Harmonic Numbers\n- Definition: Riemann Zeta Function\n\n## Historical Note\n\nThe proof that the Harmonic Series is Divergent was discovered by Nicole Oresme, lost for centuries, and rediscovered by Pietro Mengoli in 1647, Johann Bernoulli in 1687, and Jakob II Bernoulli. It is often (erroneously) attributed to Jacob Bernoulli.\n\n## Sources\n\n- 1986: David Wells: Curious and Interesting Numbers\n- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers\n- 1992: George F. Simmons: Calculus Gems\n- 1997: David Wells: Curious and Interesting Numbers (2nd ed.)\n- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.)"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 1,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 99,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI am currently in the middle of implementing an A* pathfinding for enemies. In order to implement the actual A* logic, I need a navigation mesh for my map.\n\nI am working on a 2D top down rpg map. The world is static, meaning there is no requirement for dynamic runtime mesh generation. My world objects are freely placed (not snapped to grid), rotated and scaled, which means that using a grid based A* is going to be a little complicated.\n\nFor example, how would I know a grid tile contains a collidable object? Lets say that object is actually located on the tile next to this, but is rotated and scaled such that it is still blocking the way?\n\nI could check for the middle pixel of the grid tile to see if it contains any of the nearby objects color data, but it still wouldn't be all that accurate and occasionally enemies could get stuck in between objects.\n\nshare|improve this question\nThis question needs work. 1. What do you mean by pixel-based, not tile-based? Tiles typically are pixel-based. Do you mean they are free-floating, not snapped to the tile grid? 2. You jump from talking about 2D and pixels, to vertex data, I guess you mean for the navmesh? 4. You are not making sense in your third paragraph at all, apologies, but that's the only way I can express this. 3. Your question is about AI, nav meshes, and generation, so why are you talking about 2D, pixels, color? These are render-related and have no bearing on nav meshes. Suggest you try to make the question concise. – Nick Wiggill Oct 28 '12 at 21:35\nadd comment\n\n3 Answers\n\nFrom what I understand of your question, you are trying to figure out how to make a navmesh out of a tile-/grid-based terrain.\n\nWell in fact, tiles/grids are inherently easier to work with, and one way or another you are going to have to apply A*, but you can apply it just as easily to a grid as you can to a navmesh. The point being that A*, like any graph search algorithm, operates on graphs. A navmesh is a graph, and so is a tile grid (although more densely and regularly connected).\n\nI'd suggest that you simply focus on doing pathfinding in a grid, unless you believe there is some substantial gain to be had by using a navmesh (I don't).\n\nAnd if you do, then a possible way is to generate a navmesh first, then rasterise its polygons by filling them in with tiles of different types (you'd use point-in-polygon checks for this). Now you'd have a tile based world, but you'd also have a nice navmesh already present and in agreement with the tiles. It's harder to do this the other way around, IMO (produce navmesh from tiles, which I think you are suggesting). Caveat: The problem here is, well, creating a planar navmesh graph. Not really that easy, I'm afraid, unless you have some ideas of your own. The graph planarity problem is one of the hardest problems in computer science. Hence: Use a grid.\n\nEDIT I am guessing (I have to guess a lot here!) that the reason you might prefer a navmesh over a grid is because you want creatures / objects to be placed or to roam \"pixel-freely\" rather than snapping to grid cells. Well I would still recommend sticking with the grid, but maybe flood-filling certain areas with a code to say, \"this is zone 1\", \"this is zone 2\" etc. That way you can limit your entities to ranging about in a given zone. To remove the aliasing effect of a grid, you can also introduce minor variations to the centre point of each tile, so that it looks less regular when you entities move. Just a thought.\n\nshare|improve this answer\nI have rephrased my initial question a bit. Hopefully its a little more understandable at this point. Introducing variatons to the center of each tile is actually a really good idea and would help in improving 45 or 90 degree turns to look a little bit better. Going the grid based A* would definitely be a lot easier and I can easily get my head around it, but there are still the problems described in my initial question. – J2V Oct 29 '12 at 13:04\nadd comment\n\nIf you want to precalculate the paths (and it fits in memory), you might want to check out some kind of all pairs shortest paths algorithm, like the Johnsons algorithm and stock away all paths in a container where the key is the start and end positions.\n\nReading the edit Nick Wiggill posted, this won't help you make 'pixel perfect' paths though.\n\nIf you want to do a true 2D-Navmesh (and not precalculate the rotation etc and set those overlapped grids to pass or block, which I would suggest) you need to:\n\nRotate, Scale and Transform your blocking rectangles/meshes, to new 'meshes' and then use Boolean unions on all the overlapping ones and there you are, a nav mesh in 2D.\n\nshare|improve this answer\nadd comment\n\nThe key premise to maintain is that navigable regions should not contain collidable objects. In a tile based representation, this means that if a tile contains any collidable object, that tile is impassable to all game entities. Collide against this navigation map, rather than the objects that were inserted into the map, and you avoid your problem.\n\nYou will encounter efficiency problems, as tiles may end up being marked as impassable when only a small part of the tile is obscured by a collidable object. Decreasing the size of your tiles will improve this, at the cost of more tiles, and with that higher memory usage and search times.\n\nDepending on the size of your world, distribution of obstacle sizes and rotations, this may make a purely tile based solution impractical, but remember, you can always adjust your world design to fit your technical constraints.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 2,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 99,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI am currently in the middle of implementing an A* pathfinding for enemies. In order to implement the actual A* logic, I need a navigation mesh for my map.\n\nI am working on a 2D top down rpg map. The world is static, meaning there is no requirement for dynamic runtime mesh generation. My world objects are freely placed (not snapped to grid), rotated and scaled, which means that using a grid based A* is going to be a little complicated.\n\nFor example, how would I know a grid tile contains a collidable object? Lets say that object is actually located on the tile next to this, but is rotated and scaled such that it is still blocking the way?\n\nI could check for the middle pixel of the grid tile to see if it contains any of the nearby objects color data, but it still wouldn't be all that accurate and occasionally enemies could get stuck in between objects.\n\nshare|improve this question\nThis question needs work. 1. What do you mean by pixel-based, not tile-based? Tiles typically are pixel-based. Do you mean they are free-floating, not snapped to the tile grid? 2. You jump from talking about 2D and pixels, to vertex data, I guess you mean for the navmesh? 4. You are not making sense in your third paragraph at all, apologies, but that's the only way I can express this. 3. Your question is about AI, nav meshes, and generation, so why are you talking about 2D, pixels, color? These are render-related and have no bearing on nav meshes. Suggest you try to make the question concise. – Nick Wiggill Oct 28 '12 at 21:35\nadd comment\n\n3 Answers\n\nFrom what I understand of your question, you are trying to figure out how to make a navmesh out of a tile-/grid-based terrain.\n\nWell in fact, tiles/grids are inherently easier to work with, and one way or another you are going to have to apply A*, but you can apply it just as easily to a grid as you can to a navmesh. The point being that A*, like any graph search algorithm, operates on graphs. A navmesh is a graph, and so is a tile grid (although more densely and regularly connected).\n\nI'd suggest that you simply focus on doing pathfinding in a grid, unless you believe there is some substantial gain to be had by using a navmesh (I don't).\n\nAnd if you do, then a possible way is to generate a navmesh first, then rasterise its polygons by filling them in with tiles of different types (you'd use point-in-polygon checks for this). Now you'd have a tile based world, but you'd also have a nice navmesh already present and in agreement with the tiles. It's harder to do this the other way around, IMO (produce navmesh from tiles, which I think you are suggesting). Caveat: The problem here is, well, creating a planar navmesh graph. Not really that easy, I'm afraid, unless you have some ideas of your own. The graph planarity problem is one of the hardest problems in computer science. Hence: Use a grid.\n\nEDIT I am guessing (I have to guess a lot here!) that the reason you might prefer a navmesh over a grid is because you want creatures / objects to be placed or to roam \"pixel-freely\" rather than snapping to grid cells. Well I would still recommend sticking with the grid, but maybe flood-filling certain areas with a code to say, \"this is zone 1\", \"this is zone 2\" etc. That way you can limit your entities to ranging about in a given zone. To remove the aliasing effect of a grid, you can also introduce minor variations to the centre point of each tile, so that it looks less regular when you entities move. Just a thought.\n\nshare|improve this answer\nI have rephrased my initial question a bit. Hopefully its a little more understandable at this point. Introducing variatons to the center of each tile is actually a really good idea and would help in improving 45 or 90 degree turns to look a little bit better. Going the grid based A* would definitely be a lot easier and I can easily get my head around it, but there are still the problems described in my initial question. – J2V Oct 29 '12 at 13:04\nadd comment\n\nIf you want to precalculate the paths (and it fits in memory), you might want to check out some kind of all pairs shortest paths algorithm, like the Johnsons algorithm and stock away all paths in a container where the key is the start and end positions.\n\nReading the edit Nick Wiggill posted, this won't help you make 'pixel perfect' paths though.\n\nIf you want to do a true 2D-Navmesh (and not precalculate the rotation etc and set those overlapped grids to pass or block, which I would suggest) you need to:\n\nRotate, Scale and Transform your blocking rectangles/meshes, to new 'meshes' and then use Boolean unions on all the overlapping ones and there you are, a nav mesh in 2D.\n\nshare|improve this answer\nadd comment\n\nThe key premise to maintain is that navigable regions should not contain collidable objects. In a tile based representation, this means that if a tile contains any collidable object, that tile is impassable to all game entities. Collide against this navigation map, rather than the objects that were inserted into the map, and you avoid your problem.\n\nYou will encounter efficiency problems, as tiles may end up being marked as impassable when only a small part of the tile is obscured by a collidable object. Decreasing the size of your tiles will improve this, at the cost of more tiles, and with that higher memory usage and search times.\n\nDepending on the size of your world, distribution of obstacle sizes and rotations, this may make a purely tile based solution impractical, but remember, you can always adjust your world design to fit your technical constraints.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
} |
questioner_given_difficulty_id | [
{
"content": "Your task is to generate a single self-contained question and its correct answer inspired by the given document.\nThe question must strictly satisfy both the difficulty level and the answer_type constraints.\n\nYou must output exactly one JSON object as specified below. \nAll reasoning MUST be pl... | general | {
"ground_truth": 3,
"style": "rule"
} | {
"answer_type": "categorical",
"index": 99,
"split": "train",
"text": "Take the 2-minute tour ×\n\nI am currently in the middle of implementing an A* pathfinding for enemies. In order to implement the actual A* logic, I need a navigation mesh for my map.\n\nI am working on a 2D top down rpg map. The world is static, meaning there is no requirement for dynamic runtime mesh generation. My world objects are freely placed (not snapped to grid), rotated and scaled, which means that using a grid based A* is going to be a little complicated.\n\nFor example, how would I know a grid tile contains a collidable object? Lets say that object is actually located on the tile next to this, but is rotated and scaled such that it is still blocking the way?\n\nI could check for the middle pixel of the grid tile to see if it contains any of the nearby objects color data, but it still wouldn't be all that accurate and occasionally enemies could get stuck in between objects.\n\nshare|improve this question\nThis question needs work. 1. What do you mean by pixel-based, not tile-based? Tiles typically are pixel-based. Do you mean they are free-floating, not snapped to the tile grid? 2. You jump from talking about 2D and pixels, to vertex data, I guess you mean for the navmesh? 4. You are not making sense in your third paragraph at all, apologies, but that's the only way I can express this. 3. Your question is about AI, nav meshes, and generation, so why are you talking about 2D, pixels, color? These are render-related and have no bearing on nav meshes. Suggest you try to make the question concise. – Nick Wiggill Oct 28 '12 at 21:35\nadd comment\n\n3 Answers\n\nFrom what I understand of your question, you are trying to figure out how to make a navmesh out of a tile-/grid-based terrain.\n\nWell in fact, tiles/grids are inherently easier to work with, and one way or another you are going to have to apply A*, but you can apply it just as easily to a grid as you can to a navmesh. The point being that A*, like any graph search algorithm, operates on graphs. A navmesh is a graph, and so is a tile grid (although more densely and regularly connected).\n\nI'd suggest that you simply focus on doing pathfinding in a grid, unless you believe there is some substantial gain to be had by using a navmesh (I don't).\n\nAnd if you do, then a possible way is to generate a navmesh first, then rasterise its polygons by filling them in with tiles of different types (you'd use point-in-polygon checks for this). Now you'd have a tile based world, but you'd also have a nice navmesh already present and in agreement with the tiles. It's harder to do this the other way around, IMO (produce navmesh from tiles, which I think you are suggesting). Caveat: The problem here is, well, creating a planar navmesh graph. Not really that easy, I'm afraid, unless you have some ideas of your own. The graph planarity problem is one of the hardest problems in computer science. Hence: Use a grid.\n\nEDIT I am guessing (I have to guess a lot here!) that the reason you might prefer a navmesh over a grid is because you want creatures / objects to be placed or to roam \"pixel-freely\" rather than snapping to grid cells. Well I would still recommend sticking with the grid, but maybe flood-filling certain areas with a code to say, \"this is zone 1\", \"this is zone 2\" etc. That way you can limit your entities to ranging about in a given zone. To remove the aliasing effect of a grid, you can also introduce minor variations to the centre point of each tile, so that it looks less regular when you entities move. Just a thought.\n\nshare|improve this answer\nI have rephrased my initial question a bit. Hopefully its a little more understandable at this point. Introducing variatons to the center of each tile is actually a really good idea and would help in improving 45 or 90 degree turns to look a little bit better. Going the grid based A* would definitely be a lot easier and I can easily get my head around it, but there are still the problems described in my initial question. – J2V Oct 29 '12 at 13:04\nadd comment\n\nIf you want to precalculate the paths (and it fits in memory), you might want to check out some kind of all pairs shortest paths algorithm, like the Johnsons algorithm and stock away all paths in a container where the key is the start and end positions.\n\nReading the edit Nick Wiggill posted, this won't help you make 'pixel perfect' paths though.\n\nIf you want to do a true 2D-Navmesh (and not precalculate the rotation etc and set those overlapped grids to pass or block, which I would suggest) you need to:\n\nRotate, Scale and Transform your blocking rectangles/meshes, to new 'meshes' and then use Boolean unions on all the overlapping ones and there you are, a nav mesh in 2D.\n\nshare|improve this answer\nadd comment\n\nThe key premise to maintain is that navigable regions should not contain collidable objects. In a tile based representation, this means that if a tile contains any collidable object, that tile is impassable to all game entities. Collide against this navigation map, rather than the objects that were inserted into the map, and you avoid your problem.\n\nYou will encounter efficiency problems, as tiles may end up being marked as impassable when only a small part of the tile is obscured by a collidable object. Decreasing the size of your tiles will improve this, at the cost of more tiles, and with that higher memory usage and search times.\n\nDepending on the size of your world, distribution of obstacle sizes and rotations, this may make a purely tile based solution impractical, but remember, you can always adjust your world design to fit your technical constraints.\n\nshare|improve this answer\nadd comment\n\nYour Answer\n\n\n"
} |
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