Abstract:
We first investigate the structures of the square roots of a class of circulant matrices and give classifications of the square roots of these circulant matrices. Then, we develop several algorithms for computing their square roots. We show that our algorithms are faster than the standard algorithm which is based on the Schur decomposition.

Abstract:
Holonomy R-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.

Abstract:
Various properties of a class of braid matrices, presented before, are studied considering $N^2 \times N^2 (N=3,4,...)$ vector representations for two subclasses. For $q=1$ the matrices are nontrivial. Triangularity $(\hat R^2 =I)$ corresponds to polynomial equations for $q$, the solutions ranging from roots of unity to hyperelliptic functions. The algebras of $L-$ operators are studied. As a crucial feature one obtains $2N$ central, group-like, homogenous quadratic functions of $L_{ij}$ constrained to equality among themselves by the $RLL$ equations. They are studied in detail for $N =3$ and are proportional to $I$ for the fundamental $3\times3$ representation and hence for all iterated coproducts. The implications are analysed through a detailed study of the $9\times 9$ representation for N=3. The Turaev construction for link invariants is adapted to our class. A skein relation is obtained. Noncommutative spaces associated to our class of $\hat R$ are constructed. The transfer matrix map is implemented, with the N=3 case as example, for an iterated construction of noncommutative coordinates starting from an $(N-1)$ dimensional commutative base space. Further possibilities, such as multistate statistical models, are indicated.

Abstract:
We consider a special symmetric matrix and obtain a similar formula as the one obtained by Weyl's criterion. Some applications of the formula are given, where we give a new way to calculate the integral of $\ln\Gamma(x)$ on $[0,1]$, and we claim that one class of matrices are not Hadamard matrices.

Abstract:
A matrix S is said to be an nth root of a matrix A if Sn = A, where n is a positive integer greater than or equal to 2. If there is no such matrix for any integer n > 2, A is called a rootless matrix. After investigating the properties of these matrices, we conclude that we always find an nth root of a non-singular matrix and a diagonalizable matrix for any positive integer n. On the other hand, we find some matrix having an nth root for some positive integer n. We call it p-nilpotent matrix.

Abstract:
Eventually positive matrices are real matrices whose powers become and remain strictly positive. As such, eventually positive matrices are a fortiori matrix roots of positive matrices, which motivates us to study the matrix roots of primitive matrices. Using classical matrix function theory and Perron-Frobenius theory, we characterize, classify, and describe in terms of the real Jordan canonical form the $p$th-roots of eventually positive matrices.

Abstract:
The purpose of this paper is the study of the roots in the mapping class groups. Let $\Sigma$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $\Sigma$, and let $\MM (\Sigma, \PP)$ denote the mapping class group of $(\Sigma, \PP)$. We prove that, if $\Sigma$ is of genus 0, then each $f \in \MM (\Sigma)$ has at most one $m$-root for all $m \ge 1$. We prove that, if $\Sigma$ is of genus 1 and has non-empty boundary, then each $f \in \MM (\Sigma)$ has at most one $m$-root up to conjugation for all $m \ge 1$. We prove that, however, if $\Sigma$ is of genus $\ge 2$, then there exist $f,g \in \MM (\Sigma, \PP)$ such that $f^2=g^2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Afterwards, we focus our study on the roots of the pseudo-Anosov elements. We prove that, if $\partial \Sigma \neq \emptyset$, then each pseudo-Anosov element $f \in \MM(\Sigma, \PP)$ has at most one $m$-root for all $m \ge 1$. We prove that, however, if $\partial \Sigma = \emptyset$ and the genus of $\Sigma$ is $\ge 2$, then there exist two pseudo-Anosov elements $f,g \in \MM (\Sigma)$ (explicitely constructed) such that $f^m=g^m$ for some $m\ge 2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Furthermore, if the genus of $\Sigma$ is $\equiv 0 (\mod 4)$, then we can take $m=2$. Finally, we show that, if $\Gamma$ is a pure subgroup of $\MM (\Sigma, \PP)$ and $f \in \Gamma$, then $f$ has at most one $m$-root in $\Gamma$ for all $m \ge 1$. Note that there are finite index pure subgroups in $\MM (\Sigma, \PP)$.

Abstract:
Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well.

Abstract:
The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.

Abstract:
The Pauli matrices are a set of three 2x2 complex Hermitian, unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum circuits. Techniques for simplifying such circuits are given. In particular, we show how those techniques can be used to find a circuit of Clifford+T gates starting from a circuit composed of gates from the well studied NCV library.