Given a square, nonsingular matrix of univariate polynomials F ∈ K[x] over a field K, we give a fast, deterministic algorithm for finding the Hermite normal form of F with complexity O (nωd) where d is the degree of F. Here soft-O notation is Big-O with log factors removed and ω is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries ...

Asymptotic expressions for the reflected, transmitted, and internal scattered radiation field in optically thick, vertically homogeneous, plane-parallel media are derived from first principles by using the discrete ordinates method of radiative transfer. Compact matrix equations are derived for computing the escape function, diffusion pattern, diffusion exponent, and the reflection function of ...

Linear systems with structures such as Toeplitz, Vandermonde or Cauchy-likeness can be solved in O (̃α2n) operations, where n is the matrix size, α is its displacement rank, and O ̃ denotes the omission of logarithmic factors. We show that for such matrices, this cost can be reduced to O (̃αω−1n), where ω is a feasible exponent for matrix multiplication over the base field. The best known estimate...

The Shoshan-Zwick algorithm solves the all pairs shortest paths problem in undirected graphs with integer edge costs in the range {1, 2, . . . ,M}. It runs in Õ(M · n) time, where n is the number of vertices, M is the largest integer edge cost, and ω < 2.3727 is the exponent of matrix multiplication. It is the fastest known algorithm for this problem. This paper points out the erroneous behavio...

We adapt White’s density matrix renormalisation group (DMRG) to the direct study of critical phenomena. We use the DMRG to generate transformations in the space of coupling constants. We postulate that a study of density matrix eigenvalues leads to a natural identification of renormalised blocks, operators and Hamiltonians. We apply the scheme to the phase transition in the anisotropic spin-1/2...

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication. In 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2 in this framework. In this note we rule out obtaining ω = 2 in this framework from the groups Fp , using the breakthro...

This paper constructs tridiagonal random matrix models for general (β > 0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β = 1, 2, 4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix mod...

We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show nontrivial lower bounds for the rest of the group algebras. These lower bounds are built on the top of Bläser’s results for semisimple algebras and algebras with ...

We compute the pseudo complexity of purification corresponding to reduced transition matrices for free scalar field theories with an arbitrary dynamical exponent. plot behaviour various parameters theory under study and compare it density two states $|\psi_1\rangle$ $|\psi_2\rangle$ that constitute matrix. first find matrix by reducing a small number ($1$ $2$) degrees freedom in lattice from sy...

The one-dimensional wave equation describes many phenomena in physics [1]. In a periodic medium, the solution of the wave equation has a Floquet–Bloch structure. Very long waves propagate without attenuation and the medium can be viewed as homogeneous. However, for higher frequencies there are certain intervals (so-called gaps) where waves cannot propagate even though the system is perfectly co...