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 a divisor of the central binomial coefficient

The Ramanujan Journal - It is well known that for all $$n\ge 1$$ the number $$n+1$$ is a divisor of the central binomial coefficient $${2n\atopwithdelims ()n}$$ . Since the nth central binomial...     

A note on polydegree (n, 1) rational inner functions,slice matrices,and singularities

Archiv der Mathematik - We analyze certain compositions of rational inner functions in the unit polydisk $$\mathbb {D}^{d}$$ with polydegree (n, 1), $$n\in \mathbb {N}^{d-1}$$ , and...     

Foundations of Quasiconformal Analysis of a Two-Index Scale of Spatial Mappings

Doklady Mathematics - A scale of mappings that depends on two real parameters $$p,q$$ ( $$n - 1 \leqslant q \leqslant p < \infty $$ ) and a weight function $$\theta $$ is defined. In the...     

A Remark on the Steklov–Poincaré Inequality

Mathematical Notes - In an $$n$$ -dimensional bounded domain $$\Omega_n$$ , $$n\ge 2$$ , we prove the Steklov–Poincaré inequality with the best constant in the case where $$\Omega_n$$ is...     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions

Mathematical Notes - The paper deals with sharp estimates of derivatives of intermediate order $$k\le n-1$$ in the Sobolev space $$\mathring W^n_2[0;1]$$ , $$n\in\mathbb N$$ . The functions...     

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Semigroup Forum - Let $$f_1(n), \ldots , f_k(n)$$ be polynomial functions of n. For fixed $$n\in \mathbb {N}$$ , let $$S_n\subseteq \mathbb {N}$$ be the numerical semigroup generated by...     

Inequalities for the overpartition function

The Ramanujan Journal - Let $$\overline{p}(n)$$ denote the overpartition function. Engel showed that for $$n\ge 2$$ , $$\overline{p}(n)$$ satisfy the Turán inequalities, that is,...     

Polynomials in the Differentiation Operator and Formulas for the Sums of Certain Convergent Series

Functional Analysis and Its Applications - Let $$P_n(x)$$ be any polynomial of degree $$n\geq 2$$ with real coefficients such that $$P_n(k)\ne 0$$ for $$k\in\mathbb{Z}$$ . In the paper, in...     

Universal Relations in Asymptotic Formulas for Orthogonal Polynomials

Functional Analysis and Its Applications - Orthogonal polynomials $$P_{n}(\lambda)$$ are oscillating functions of $$n$$ as $$n\to\infty$$ for $$\lambda$$ in the absolutely continuous spectrum of...     

On generalized square-full numbers in an arithmetic progression

Czechoslovak Mathematical Journal - Let a and $$b \in \mathbb{N}$$ . Denote by Ra,b the set of all integers n > 1 whose canonical prime representation $$n = p_1^{{\alpha _1}}p_2^{{\alpha...     

Critical Values of Finite Blaschke Products

Doklady Mathematics - For finite Blaschke products B of degree $$n \geqslant 2,$$ $$B(0) = 0,$$ $$B'(0) \ne 0$$ a sharp upper bound for the least critical values and a sharp lower bound for the...     

Exact expansions of Hankel transforms and related integrals

The Ramanujan Journal - The Hankel transform $$\mathcal {H}_n [f(x)](q)=\int _0^{\infty } \!\! \, x f(x) J_n(q x) \mathrm{d}x$$ is studied for integer $$n\geqslant -1$$ and positive parameter q. It...     

Hamming weight distributions of multi-twisted codes over finite fields

Designs, Codes and Cryptography - Let $${\mathbb {F}}_q$$ denote the finite field of order q,  and let $$n = m_1+m_2+\cdots +m_\ell ,$$ where $$m_1,m_2,\ldots ,m_\ell $$ are arbitrary...     

Smooth Nonprojective Equivariant Completions of Affine Space

Mathematical Notes - An open translation-equivariant embedding of the affine space $$\mathbb A^n$$ into a complete nonprojective algebraic variety $$X$$ is constructed for any $$n\ge 3$$ . The main...     

Rogosinsky–Bernstein Polynomial Method of Summation of Trigonometric Fourier Series

Mathematical Notes - General Rogosinsky–Bernstein linear polynomial means $$R_n(f)$$ of Fourier series are introduced and three convergence criteria as $$n\to \infty$$ are obtained: for...     

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.     

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...     

On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity

Differential Equations - For the Laplace equation in a bounded domain of the space $$\mathbb {R}^n $$ , $$n\geq 3$$ , with a boundary that contains a piece $$\Gamma $$ of a hyperplane $$L $$ , we...