Related papers: Supplement to the article "irreducible polynomials…

This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial. We first generalized a global-local theorem of Vaserstein. Then we proved these…

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the…

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is…

We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute.

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…

Based on a less-known result, we prove a recent conjecture concerning the determinant of a certain Sylvester-Kac type matrix and consider an extension of it.

In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A…

Pick a random matrix $\gamma$ in $\Gamma={\rm SL}(2,\mathbb{Z})$. Denote by $\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let $\Delta_K$, $\Delta_\gamma$ and $\Delta = {\rm Tr}(\gamma)^2-4$ be the respective…

The probability of the small deviations of the matrix $AA^T$ determinant is estimated, where $A$ is an $n\times\infty$ random matrix with centered entries having joint Gaussian distribution. The inequality obtained is sharp in a sence.

It is proved that for the top Lyapunov exponent of a random matrix system of the form $\{A D(\omega)\}$, where $A$ is a nonnegative matrix and $D(\omega)$ is a diagonal matrix with positive diagonal entries, is bounded from below by the top…

Let $\FF$ be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of $\FF$ is sufficiently large. We also give an example of non-bijective map when…

It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord…

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability.

We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping…

Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from…

We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their…

Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} = 1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n \geq…

We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart of this statement is that in the case of unbounded combinatorics renormalized mappings become almost…

In this note we describe the singular locus of diagonally-dominant Hermitian matrices with nonnegative diagonal entries over the reals, the complex numbers, and the quaternions. This yields explicit expressions for the probability that such…