code: def transition(measure, N):
      """A, B transition matrices for different measures."""
      # Legendre (translated)
      if measure == "legt":
          Q = np.arange(N, dtype=np.float64)
          R = (2 * Q + 1) ** 0.5
          j, i = np.meshgrid(Q, Q)
          A = (
              R[:, None]
              * np.where(i < j, (-1.0) ** (i - j), 1)
              * R[None, :]
          )
          B = R[:, None]
          A = -A
  
          # Halve again for timescale correctness
          A *= 0.5
          B *= 0.5
      # Legendre (scaled)
      elif measure == "legs":
          q = np.arange(N, dtype=np.float64)
          col, row = np.meshgrid(q, q)
          r = 2 * q + 1
          M = -(np.where(row >= col, r, 0) - np.diag(q))
          T = np.sqrt(np.diag(2 * q + 1))
          A = T @ M @ np.linalg.inv(T)
          B = np.diag(T)[:, None]
          B = (
              B.copy()
          )  # Otherwise "UserWarning: given NumPY array is not writeable..."
          # after torch.as_tensor(B)
      elif measure == "legsd":
          # Essentially equivalent to S4D-LegS
          q = np.arange(N, dtype=np.float64)
          col, row = np.meshgrid(q, q)
          r = 2 * q + 1
          M = -(np.where(row >= col, r, 0) - np.diag(q))
          T = np.sqrt(np.diag(2 * q + 1))
          A = T @ M @ np.linalg.inv(T)
          B = np.diag(T)[:, None]
          B = (
              B.copy()
          )  # Otherwise "UserWarning: given NumPY array is not writeable..."
          # after torch.as_tensor(B)
          A += 0.5 * B * B[None, :, 0]
          B = B / 2.0
      elif measure in ["fourier_diag", "foud"]:
          # Essentially equivalent to S4D-Lin
          freqs = np.arange(N // 2)
          d = np.stack([freqs, np.zeros(N // 2)], axis=-1).reshape(-1)[
              :-1
          ]
          A = 2 * np.pi * (-np.diag(d, 1) + np.diag(d, -1))
          A = A - 0.5 * np.eye(N)
          B = np.zeros(N)
          B[0::2] = 2**0.5
          B[0] = 1
          B = B[:, None]
      elif measure in ["fourier", "fout"]:
          freqs = np.arange(N // 2)
          d = np.stack([np.zeros(N // 2), freqs], axis=-1).reshape(-1)[
              1:
          ]
          A = np.pi * (-np.diag(d, 1) + np.diag(d, -1))
          B = np.zeros(N)
          B[0::2] = 2**0.5
          B[0] = 1
  
          # Subtract off rank correction - this corresponds
          # to the other endpoint u(t-1) in this case
          A = A - B[:, None] * B[None, :]
          B = B[:, None]
      else:
          raise NotImplementedError
  
      return A, B
  
imports: import logging
  import math
  import os
  from functools import wraps
  from typing import Any, Callable, Optional
  import numpy
  import opt_einsum
  import torch
  import torch.nn
  import torch.nn.functional
  from einops import rearrange, repeat
  from espnet2.asr.state_spaces.components import Activation, DropoutNd, LinearActivation
label_desc: Use image node_id 6 for calling a global function with example usage: transition(measure, N) and returns: A,  B