Representations of finite number of quadratic forms with same rank

The Ramanujan Journal - Let m, n be positive integers with $$m\le n$$ . Let $$\kappa (m,n)$$ be the largest integer k such that for any (positive definite and integral) quadratic forms...     

An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples

Periodica Mathematica Hungarica - Let m, n be positive integers such that $$m>n$$, $$\gcd (m,n)=1$$ and $$m \not \equiv n \pmod {2}$$. In 1956, L. Je?manowicz conjectured that...     

Methods for Nonnegative Matrix Factorization Based on Low-Rank Cross Approximations

Computational Mathematics and Mathematical Physics - Available methods for nonnegative matrix factorization make use of all elements of the original $$m \times n$$ matrix, and their complexity is...     

The bilinear assignment problem: complexity and polynomially solvable special cases

Mathematical Programming - In this paper we study the bilinear assignment problem (BAP) with size parameters m and n, $$m\le n$$ . BAP is a generalization of the well known quadratic assignment...     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Diagonal groups and arcs over groups

Designs, Codes and Cryptography - In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ , a set of $$m+1$$ partitions of a set $$\Omega $$ , any...     

Menon-type identities with respect to sets of units

The Ramanujan Journal - Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots...     

Existence of primitive normal pairs with one prescribed trace over finite fields

Designs, Codes and Cryptography - Given $$m, n, q\in \mathbb {N}$$ such that q is a prime power and $$m\ge 3$$ , $$a\in \mathbb {F}_q$$ , we establish a sufficient condition for the existence of a...     

Completely positive semidefinite rank

Mathematical Programming - An $$n\times n$$ matrix X is called completely positive semidefinite (cpsd) if there exist $$d\times d$$ Hermitian positive semidefinite matrices $$\{P_i\}_{i=1}^n$$ (for...     

Solutions of Super-Linear Elliptic Equations and Their Morse Indices

Mathematical Notes - We investigate the degenerate bi-harmonic equation $$\Delta_{m}^2 u=f(x,u)\quad \text{in} \ \ \Omega, \qquad u = \Delta u = 0\quad \text{on}\ \ \partial\Omega,$$ with $$m\ge...     

SPECTRAL PROPERTY OF CERTAIN MORAN MEASURES IN $\mathbb{R}^{n}$ R n

Acta Mathematica Hungarica - Let $$R_k = {\rm diag}(r_{k,1}, r_{k,2}, \ldots, r_{k,n})$$ be the $$n \times n$$ expanding integer diagonal matrix, and let $$D_k\subset \mathbb Z^{n}$$ . We show that...     

Moderate Deviations for Extreme Eigenvalues of Real-Valued Sample Covariance Matrices

Journal of Theoretical Probability - Consider the sample covariance matrices of form $$W=n^{-1}C C^{\top }$$ , where C is a $$k\times n$$ matrix with real-valued, independent and identically...     

Semilinear elliptic systems with measure data

Annali di Matematica Pura ed Applicata (1923 -) - We study the Dirichlet problem for systems of the form $$-\varDelta u^k=f^k(x,u)+\mu ^k,\,x\in \varOmega ,\,k=1,\ldots ,n$$ , where $$\varOmega...     

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.     

On the distribution of square-full and cube-full primitive roots

Periodica Mathematica Hungarica - A positive integer n is called an r-full integer if for all primes $$p\mid n$$ we have $$p^r\mid n.$$ Let p be an odd prime. For $$\gcd (n,p)=1$$, the smallest...     

The extension of the $$D(-k)$$ D ( - k ) -pair $$\{k,k+1\}$$ { k,k + 1 } to a quadruple

Periodica Mathematica Hungarica - Let $$n\ne 0$$ be an integer. A set of m distinct positive integers $$\{a_1,a_2,\ldots ,a_m\}$$ is called a D(n)-m-tuple if $$a_ia_j + n$$ is a perfect square for...     

Convex functions and Fourier coefficients

Positivity - The aim of this paper is to prove that the cosine Fourier coefficients $$a_{mn}$$ (with $$m,n\ge 1)$$ of a Popoviciu convex function of two variables are nonnegative.     

Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices

Doklady Mathematics - Two results concerning the number $$P(2,n)$$ of threshold functions and the singularity probability $${{\mathbb{P}}_{n}}$$ of random ( $$n \times n$$ ) $${\text{\{ }} \pm...     

m-Dissections of some infinite products and related identities

The Ramanujan Journal - For integers t and m with $$m\ge 5$$ relatively prime to 6 such that $$1 \le t n_0$$ , the signs of the $$c_n$$ are periodic with period m. In addition, a formula is given...     

Enumeration of Matchings in Complete $$q$$ -ary Trees

Mathematical Notes - We study the asymptotic behavior of the parameters $$m(T_{q,n})$$ and $$im(T_{q,n})$$ , that equal the number of matchings and independent matchings in a complete $$q$$ -ary...