相关论文: Exponential sums and rank of double persymmetric m…

Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x…

A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…

In this paper, we introduce symmetric diagram matrices $A_{s+r,s}$ of size ${_{(s+r)}}C_s$ whose entries are $\{x_i\}_{min\{s,r\}}$. We compute the eigenvalues of symmetric diagram matrices using elementary row and column operations…

Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de La Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the…

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.

A square matrix of order $n$ with $n\geq 2$ is called a \textit{permutative matrix} or permutative when all its rows (up to the first one) are permutations of precisely its first row. In this paper, the spectra of a class of permutative…

We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…

We prove that symmetry group of the pfaffian polynomial of a symmetric matrix is a dihedral group. We calculate pfaffians of symmetric matrices with components $(x_i-x_j)^2$ and $\cos(x_i-x_j)$ for $i<j.$

We count with a smooth weight the number of $2 \times 2$ integer matrices with a fixed characteristic polynomial with a main term and an error term using bounds for sums of Weyl sums for quadratic roots.

Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like…

We study characteristic polynomials of symmetric matrices with entries ${i+j\choose i}$ the binomial coefficients, over finite fields.

We give a complete description of all solutions to the equation $f_1^3 + f_2^3 = f_3^3 + f_4^3$ for quadratic forms $f_j \in \mathbb C[x,y]$ and show how Ramanujan's example can be extended to three equal sums of pairs of cubes. We also…

We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of…

We say that a symmetric noncommutative polynomial in the noncommutative free variables (x_1, x_2, ..., x_g) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative complex hessian that is positive…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are…

In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…

How to calculate the exponential of matrices in an explicit manner is one of fundamental problems in almost all subjects in Science. Especially in Mathematical Physics or Quantum Optics many problems are reduced to this calculation by…

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$…

Consider the regular representation of the sum over all permutations weighted by the sum of their descent, inversion, and fixed point multinomials. We compute the spectrum and the multiplicities of its elements of that matrix. Note that…