Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Short Note on Kronecker Square Roots
88
0
0.0
(
0
)
Authors
Yorick Hardy
Ask ChatGPT about the research
No Arabic abstract
The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the characterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.
rate research
Read More
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a solution in terms of these functions is to rationalize all occurring square roots by a suitable variable change. In this paper, we give a rigorous definition of rationalizability for square roots of ratios of polynomials. We show that the problem of deciding whether a single square root is rationalizable can be reformulated in geometrical terms. Using this approach, we give easy criteria to decide rationalizability in most cases of square roots in one and two variables. We also give partial results and strategies to prove or disprove rationalizability of sets of square roots. We apply the results to many examples from actual computations in high energy particle physics.
Let $det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $det(det_2(A)) = det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary field. Further insisting that $det_2(AB)=det_2(A)det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $text{Det}$ which satisfies $text{Det}(text{Det}_2(A)) = text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $text{Det}$ is defined.
In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.
We provide a formula for commputing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras.
The aim of this paper is to study linear preservers of the trace of Kronecker sums and their connection with preservers of determinants of Kronecker products. The partial trace and partial determinant play a fundamental role in characterizing the preservers of the trace of Kronecker sums and preservers of the determinant of Kronecker products respectively.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
