Tensor Transpose and Its Properties

Tensor transpose is a higher order generalization of matrix transpose. In this paper, we use permutations and symmetry group to define? the tensor transpose. Then we discuss the classification and composition of tensor transposes. Properties of tensor transpose are studied in relation to tensor multiplication, tensor eigenvalues, tensor decompositions and tensor rank

On the typical rank of real binary forms

We determine the rank of a general real binary form of degree d=4 and d=5. In the case d=5, the possible values of the rank of such general forms are 3,4,5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d.Comment: 12 pages, 2 figure

STORM: a nonlinear model order reduction method via symmetric tensor decomposition

Nonlinear model order reduction has always been a challenging but important task in various science and engineering fields. In this paper, a novel symmetric tensor-based orderreduction method (STORM) is presented for simulating largescale nonlinear systems. The multidimensional data structure of symmetric tensors, as the higher order generalization of symmetric matrices, is utilized for the effective capture of highorder nonlinearities and efficient generation of compact models. Compared to the recent tensor-based nonlinear model order reduction (TNMOR) algorithm [1], STORM shows advantages in two aspects. First, STORM avoids the assumption of the existence of a low-rank tensor approximation. Second, with the use of the symmetric tensor decomposition, STORM allows significantly faster computation and less storage complexity than TNMOR. Numerical experiments demonstrate the superior computational efficiency and accuracy of STORM against existing nonlinear model order reduction methods.postprin

Continuous-Discrete Path Integral Filtering

A summary of the relationship between the Langevin equation, Fokker-Planck-Kolmogorov forward equation (FPKfe) and the Feynman path integral descriptions of stochastic processes relevant for the solution of the continuous-discrete filtering problem is provided in this paper. The practical utility of the path integral formula is demonstrated via some nontrivial examples. Specifically, it is shown that the simplest approximation of the path integral formula for the fundamental solution of the FPKfe can be applied to solve nonlinear continuous-discrete filtering problems quite accurately. The Dirac-Feynman path integral filtering algorithm is quite simple, and is suitable for real-time implementation.Comment: 35 pages, 18 figures, JHEP3 clas