In this paper, we investigate the reduced form of circulant matrices and we show that the problem of computing the q-th roots of a nonsingular circulant matrix A can be reduced to that of computing the q-th roots of two half size matrices B - C and B + C.

Floquet phases of matter have attracted great attention due to their dynamical and topological nature that are unique nonequilibrium settings. In this work, we introduce a generic way taking any integer q q th-root the evolution operator U display="inline">U describe...

We present an r-th root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive r-th root of unity ξ where s is the largest positive integer satisfying r|q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms.

Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $A(G)$ the adjacency matrix of $G$. The  signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of  graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

let $g = (v, e)$ be a simple graph. denote by $d(g)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $a(g)$ the adjacency matrix of $g$. the  signless laplacianmatrix of $g$ is $q(g) = d(g) + a(g)$ and the $k-$th signless laplacian spectral moment of  graph $g$ is defined as $t_k(g)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2-th root of unity ξ where s is the largest positive integer satisfying 2|q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square roo...

Let m > 2, ζm an m-th primitive root of 1, q ≡ 1 mod 2m a prime number, s = sq a primitive root modulo q and f = fq = (q − 1)/m. We study the Jacobi sums Ja,b = − ∑q−1 k=2 ζ a inds(k)+b inds(1−k) m , 0 ≤ a, b ≤ m−1, where inds(k) is the least nonnegative integer such that s inds(k) ≡ k mod q. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE...

Explicit formulae for τ ′ r(L(p, q)) and τ(L(p, q)) are obtained for all L(p, q). There are three systems of invariants of Witten-type for closed oriented 3manifolds: 1. {τr(M), r ≥ 2; τ ′ r(M), r odd ≥ 3}, where τr was defined by Reshetikhin and Turaev [1], and τ ′ r was defined by Kirby-Melvin [2] 2. {Θr(M,A), r ≥ 1,whereA is a 2r-th primitive root of unity} defined by Blanchet, Habegger, Mas...