We show that the function \[ V_q(x)=\frac{2e^{x^2}}{\Gamma(q+1)}\int_{x}^{\infty}e^{-t^2}(t^2-x^2)^qdt\quad{(-1<q\in\mathbf{R}; 0<x\in \mathbf{R})}, \] which has applications in the study of atoms in magnetic fields, satisfies certain monotonicity and convexity properties as well as inequalities. In particular, we prove that $1/V_q$ is convex on $(0,\infty)$ if and only if $q\geq 0$. This extends a recent result of M. B. Ruskai and E. Werner, who established the convexity for all integers $q\geq 0$.

In this paper, general existence theorems are presented for the singular equation \[\left\{\begin{array}{@{}l}-(\varphi_p(u^{\prime}))^{\prime}=f(t,u,u^{\prime}),\;0<t<1\\[3pt]u(0)=u(1)=0.\end{array}\right.\] Throughout, our nonlinearity is allowed to change sign. The singularity may occur at $u=0,$$t=0$ and $t=1$.

In [11] the authors obtained an operator matrix with two variables that distinguishes the classes of $p$-hyponormal operators, $w$-hyponormal, absolute-$p$-paranormal, and normaloid operators on Hilbert spaces. We establish the general model for $n$ variables, which provides many more examples to show that such classes are distinct.

Given a sequence $\{A_n\}_{n\in\mathbb{Z}_+}$ of bounded linear operators between complex Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ we characterize the existence of a contraction (resp. isometry, unitary operator, shift) $T$ on $\mathcal{K}$ such that \[A_n=T^nA_0,\quad n\in\mathbb{Z}_+.\] Such moment problems are motivated by their connection with the dilatability of positive operator measures having applications in the theory of stochastic processes.

The solutions, based on the fact that a certain operator function attached to $T$ is positive definite on $\mathbb{Z}$, extend the ones given by Sebestyén in [18], [19] or, recently, by Jabłoński and Stochel in [8]. Some applications, containing new characterizations for isometric, unitary operators, orthogonal projections or commuting pairs having regular dilation, conclude the paper.

Let X and Y be Banach spaces such that each one is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on X, Y, the supplemented subspaces of Y in X and of X in Y to yield that X is isomorphic to Y. In other words, we obtain generalizations of Pełczyński's decomposition method via supplemented subspaces. In order to determine all the possible generalizations, we introduce the notion of Mixed Schroeder-Bernstein Quadruples for Banach spaces. Then, we use some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997 to characterize them.

A ring is called clean if every element is the sum of an idempotent and a unit. It is an open question whether the tensor products of two clean algebras over a field is clean. In this note we study the tensor product of clean algebras over a field and we provide some examples to show that the tensor product of two clean algebras over a field need not be clean.

The discrete universality of the derivative and logarithmic derivative of zeta-functions of normalized eigenforms is obtained. This is used to estimate the number of zeros of the derivatives in the critical strip. For the proof the method of functional limit theorems in the sense of weak convergence of probability measures is applied.

Let $\varphi(\cdot)$ denote the Euler function, and let $a>1$ be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as $a^n \equiv 1\!\!\!\!\pmod{\varphi(n)}$, and we estimate the number of positive integers $n\le x$ satisfying these conditions.

In this paper we establish a uniform distribution result for the time spent by a billiard particle in the unit square having vertices (0,0), (1,0), (1,1), (0,1), with small triangular pockets of size $\epsilon$ removed from its corners.

We use Arkhipov's twisting functors to show that the universal enveloping algebra of a semi-simple complex finite-dimensional Lie algebra surjects onto the space of ad-finite endomorphisms of the simple highest weight module $L(\lambda)$, whose highest weight is associated (in the natural way) with a subset of simple roots and a simple root in this subset. This is a new step towards a complete answer to a classical question of Kostant. We also show how one can use the twisting functors to reprove the classical results related to this question.

A basis $\{x_n\}$ for a Hilbert space H is called a Riesz basis if it has the property that $\sum a_nx_n$ converges in H if and only if $\sum|a_n|^2<\infty$, and hence if and only if $\{x_n\}$ is the isomorphic image of some orthonormal basis for H. A consequence of a classical result of Bary [1] is that any basis for H that is quadratically near an orthonormal basis must be a Riesz basis. Motivated by this result, we study in this paper the class of normalized bases in a Hilbert space that are quadratically near some orthonormal basis, bases we call almost-orthonormal bases. In particular, we prove that any such basis must be quadratically near its Gram-Schmidt orthonormalization, and derive an internal characterization of these bases that indicates how restrictive the property of being almost-orthonormal is.