Here's example two:
[
{
"type": "text",
"text": "Command started with PID 139\nInitial output:\n\n=== 5th RANK SYMBOLIC TENSOR OPERATIONS ===\n\nCreating symbolic tensors T and U of rank 5...\n\nSample elements from tensor T:\nT[0,0,0,0,0] = T_0_0_0_0_0\nT[1,0,1,0,1] = T_1_0_1_0_1\nT[1,1,1,1,1] = T_1_1_1_1_1\n\n=== TENSOR ADDITION ===\nT + U for selected elements:\n(T+U)[0,0,0,0,0] = (T_0_0_0_0_0) + (U_0_0_0_0_0)\n(T+U)[1,1,0,0,1] = (T_1_1_0_0_1) + (U_1_1_0_0_1)\n\n=== TENSOR CONTRACTION ===\nContracting the last two dimensions of tensor T...\nResult of contraction (now a rank-3 tensor):\nContracted T[0,0,0] = T_0_0_0_0_0 + T_0_0_0_0_1 + T_0_0_0_1_0 + T_0_0_0_1_1\nContracted T[1,1,1] = T_1_1_1_0_0 + T_1_1_1_0_1 + T_1_1_1_1_0 + T_1_1_1_1_1\n\n=== TENSOR PRODUCT ===\nComputing symbolic outer product for select elements...\nT[0,0,0,0,0] ⊗ U[1,1,1,1,1] = (T_0_0_0_0_0) * (U_1_1_1_1_1)\nT[1,0,1,0,1] ⊗ U[0,1,0,1,0] = (T_1_0_1_0_1) * (U_0_1_0_1_0)\n\n=== ADVANCED SYMBOLIC TENSOR OPERATIONS ===\nExpression with T[0,1,0,1,0]: (T_0_1_0_1_0) ^ 2 + 3 * (T_0_1_0_1_0) + 5\n\n=== EIGENVALUE CONCEPT FOR HIGHER-ORDER TENSORS ===\nHigher-order tensors have generalizations of eigenvalues\nEigenvalue-like equation: lambda * (T_0_0_0_0_0) - (T_1_1_1_1_1) = 0\nThis is an example of a symbolic representation for a\nHigher-order tensor eigenvalue problem\n\n=== TENSOR INVARIANTS ===\nTensor invariants are scalar values that remain unchanged\nunder certain transformations. For our 5th rank tensor, we can\ncreate an example invariant based on contraction.\nSymbolic tensor invariant: T_0_0_0_0_0 + T_1_1_1_1_1 + T_0_1_0_1_0 + T_1_0_1_0_1\nThis invariant represents a quantity that would remain unchanged\nunder specific tensor transformations.\n\n=== MULTI-LINEAR MAPPING REPRESENTATION ===\nA 5th rank tensor can be viewed as a multi-linear mapping\nfrom four vector spaces to a fifth vector space.\nMulti-linear mapping (first component):\nresult_0 = (T_0_0_0_0_0) * v1_0 * v2_0 * v3_0 * v4_0 + (T_0_0_0_1_0) * v1_0 * v2_0 * v3_0 * v4_1 + (T_0_0_1_0_0) * v1_0 * v2_0 * v3_1 * v4_0 + ... (and 27 more terms)\n\n=== TENSOR DECOMPOSITION ===\nHigher-order tensors can be decomposed into simpler components.\nOne approach is the CP decomposition (CANDECOMP/PARAFAC),\nwhich expresses a tensor as a sum of rank-one tensors:\nSymbolic CP decomposition with rank 3:\nT_{ijklm} ≈ ∑ λ_p * a_i^p * b_j^p * c_k^p * d_l^p * e_m^p\nfor p = 1 to 3\n\nThis decomposition can be useful for finding\nlow-dimensional representations of the tensor data.\n"
}
]