In this paper we consider the almost sure convergence of a series of contractions (of an arbitrary Hilbert space) with random weights. The paper is a continuation of a previous work [PSW], in which only convergence in operator norm was investigated. We obtain conditions ensuring the existence of universal sets on which these series are converging almost everywhere, for any contraction. The paper is also a continuation of the paper [SW], in which an analogous problem concerning ergodic averages was considered, as well as the paper [S], which deals with a variant of the problem. The proofs of our results rely on uniform estimates of random polynomials which were established in a recent paper by the second author and proved by means of metric entropy methods.

We improve the existing upper bound for the mean-square of the absolute value of the Rankin-Selberg zeta-function (attached to a holomorphic cusp form) defined for the full modular group in the critical strip.

In this paper we show that neither the Weyl algebra $A_n(K)$ nor the derivative algebra $DA_n(K)$ has infinite irreducible representations in the case when the ground field $K$ has characteristic $p \gt 0$. We also give a complete classification of irreducible representations of the first derivative algebra $DA_1$ when $K$ is algebraically closed. Finally, we present an algorithm that determines, in finitely many steps, whether $DA_1/L$ is a simple $DA_1$-module, where $L$ is any left ideal of $DA_1$.

This paper addresses the problem of determining the best results one can expect using the Thue-Siegel method as developed by Bombieri in his equivariant approach to effective irrationality measures to roots of high order of algebraic numbers, in the non-archimedean setting. As an application, we show that this method, under a non-vanishing assumption for the auxiliary polynomial which replaces the appeal to Dyson's Lemma type arguments and together with a version of Siegel's Lemma due to Struppeck and Vaaler, yields a result comparable to the best results obtained to date by transcendence methods.

$K$-theory and $\mathrm{Ext}$ are computed for the $C^*$-algebra $C^*(E)$ of any countable directed graph $E$. The results generalize the $K$-theory computations of Raeburn and Szymański and the $\mathrm{Ext}$ computations of Tomforde for row-finite graphs. As a consequence, it is shown that if $A$ is a countable $\{0,1\}$ matrix and $E_A$ is the graph obtained by viewing $A$ as a vertex matrix, then $C^*(E_A)$ is not necessarily Morita equivalent to the Exel-Laca algebra $\mathcal{O}_A$.

We prove that if a function $f \in L^p(\mathbb{R}^d)$ has vanishing periodizations then $\|f\|_{p'} \lesssim \|f\|_{p}$, provided $1 \le p < {2d}/{(d + 2)}$ and $d \ge 3$.

We prove that, in any o-minimal structure, definable $C^r$ mappings can always be approximated by definable $C^{r+1}$ mappings. As an application we obtain definable triviality for pairs of definable proper submersions.

Liaison theory has been extensively studied during the past decades. In codimension 2, the theory has reached a very satisfactory state, but in higher codimensions there are still many open problems. In this paper we prove that two unions $V= \bigcup_{i=1}^k L_i$ and $V'= \bigcup_{i=1}^{k'} L'_i$ of independent linear varieties of dimension $d \geq 1$ in $\mathbb{P}^n$ are in the same G-liaison class if and only if $k=k'$ or, equivalently, if $V$ and $V'$ have isomorphic deficiency modules $M^i(V) \cong M^i (V')$, $i=1, \dots, d$. We also describe the G-liaison classes of arithmetically Buchsbaum divisors on rational normal scrolls.

Let $C$ be a smooth geometrically integral real algebraic curve in projective $n$-space $\mathbb{P}^n$. Let $c$ be its degree and let $g$ be its genus. Let $d$, $s$ and $m$ be nonzero natural integers. Let $\nu$ be the number of real hypersurfaces of degree $d$ that are tangent to at least $s$ real branches of $C$ with order of tangency at least $m$. We show that $\nu$ is finite if $s=g$, $gm=cd$ and the restriction map $H^0(\mathbb{P}^n,\mathcal{O}(d))\rightarrow H^0(C,\mathcal{O}(d))$ is an isomorphism. Moreover, we determine explicitly the value of $\nu$ in that case.

Let $T_\theta$ be the first exit time of an $n$-dimensional symmetric stable process with parameter $\alpha \in (0,2)$ from a cone of angle $\theta$, $0 \lt \theta \lt \pi$. Then there exists a constant $ p(\theta,\alpha,n)$ such that for $x$ in the cone, $E^x T_\theta^p \lt \infty$ if $p \lt p(\theta,\alpha,n)$, and $E^x T_\theta^p = \infty$ if $ p \gt p(\theta,\alpha,n)$. We characterize $p(\theta,\alpha,n)$ in terms of the principal eigenvalue of an operator and give upper and lower bounds for it. We also present a generalization of this result to more general cones in $\mathbf{R}^n$. These results extend the two-dimensional results of R. D. DeBlassie to $n$ dimensions and more general cones.

We show that a generator of the cohomology group $H^n(S^n)$ cannot be realized as the Euler class of a vector bundle over the $n$-sphere unless $n$ equals 2, 4, or 8.

We show that the image cone of a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be considered as a version of the Atiyah-Guillemin-Sternberg convexity theorem for torus actions on symplectic cones and as a direct generalization of the convexity theorem of Banyaga and Molino for completely integrable torus actions on contact manifolds.

We estimate the well-distribution measure and correlation of order 2 of the binary sequence $E_N=\{e_1,\ldots,e_N\}$ defined by $e_n=+1$ if $0\leqslant\{n^c\} \lt {1}/{2}$ and $e_n=-1$ if ${1}/{2}\leqslant\{n^c\} \lt 1$, where $c$ is a real, non-integral number greater than $1$ and $\{x\}$ denotes the fractional part of $x$. We also prove an upper bound for the well-distribution measure of an arbitrary binary sequence in terms of its generating function and show that there exists no upper bound of this type for the correlation. The proof is based on the Erdős-Turán inequality, which we establish with an improved constant.

We first prove a local $A_r$-weighted Poincaré inequality for solutions to $A$-harmonic equations of the form $d^{\star} A(x, d \omega) =B(x,d\omega)$. Then, as an application of this local result, we prove a global $A_r$-weighted Poincaré inequality for functions that are solutions to such equations in John domains.

In this paper we study the holomorphic Hardy space $\mathcal{H}^p(\Omega)$, where $\Omega$ is a smoothly bounded convex domain of finite type in $\mathbb{C}^n$. We show that for $0<p\le1$, $\mathcal{H}^p(\Omega)$ admits an atomic decomposition. More precisely, we prove that each $f\in\mathcal{H}^p(\Omega)$ can be written as $f=P_S(\sum_{j=0}^{\infty}\nu_j a_j)=\sum_{j=0}^{\infty}\nu_j P_S(a_j)$, where $P_S$ is the Szegö projection, the $a_j$'s are real variable $p$-atoms on the boundary $\partial\Omega$, and the coefficients $\nu_j$ satisfy the condition $\sum_{j=0}^{\infty}|\nu_j|^p \lesssim\|f\|_{\mathcal{H}^p(\Omega)}^p$. Moreover, we prove the following factorization theorem. Each $f\in\mathcal{H}^p(\Omega)$ can be written as $f=\sum_{j=0}^{\infty}f_j g_j$, where $f_j\in\mathcal{H}^{2p}$, $g_j\in\mathcal{H}^{2p}$, and $\sum_{j=0}^{\infty}\|f_j\|_{\mathcal{H}^{2p}} \|g_j\|_{\mathcal{H}^{2p}}$ $\lesssim\|f\|_{\mathcal{H}^p(\Omega)}$. Finally, we extend these theorems to a class of domains of finite type that includes the strongly pseudoconvex domains and the convex domains of finite type.

An automorphism $\alpha$ of an integral group ring $\mathbb{Z} G$, where $G$ is a finite group, is said to have a Zassenhaus factorization if it is the composition of an automorphism of $G$ (extended to a ring automorphism) and a central automorphism. In 1988, Roggenkamp and Scott constructed a group $G$ (of order $2880$) such that $\mathbb{Z} G$ has a normalized (i.e., augmentation preserving) automorphism $\alpha$ which has no Zassenhaus factorization. In this paper, short proofs of the following two results are given. (1) For a group $G$ of order $144$, there is a normalized automorphism $\alpha$ of $\mathbb{Z} G$ which has no Zassenhaus factorization. Moreover, $\alpha$ can be chosen to have finite order. (2) There is a group $G$ of order $1200$, with abelian Sylow subgroups and Sylow tower, such that $\mathbb{Z} G$ has a normalized automorphism which has no Zassenhaus factorization.

Hypersurfaces of constant $2$-mean curvature in spaces of constant sectional curvature are known to be solutions to a variational problem. We extend this characterization to ambient spaces which are Einstein. We then estimate the $2$-mean curvature of certain hypersurfaces in Einstein manifolds. A consequence of our estimates is a generalization of a result, first proved by Chern, showing that there are no complete graphs in the Euclidean space with positive constant $2$-mean curvature.

Mazur's question on the evolutionary stability of reduced local algebras in characteristic zero and a problem of Huneke about derivations are related to the vanishing of cohomology groups.

The notion of a topological vector space of Bochner measurable functions is introduced and studied. Among the main results obtained are characterizations of completeness and of containment of copies of $c_0$ or $\ell_\infty$.

The aim of this paper is to investigate systematically the relationship between the two different types of product probability spaces based on the Loeb space construction. For any two atomless Loeb spaces, it is shown that for fixed $r \lt s$ in $[0,1]$ there exists an increasing sequence $(A_t)_{r \lt t \lt s,t\in [0,1]}$ of in the new sense product measurable sets such that $A_t$ has measure $t$ and, with respect to the usual product, the inner and outer measures are equal to $r$ and $s$, respectively. By constructing a continuum of increasing Loeb product null sets with large gaps, the Loeb product is shown to be much richer than the usual product even on null sets. General results in terms of outer and inner measures with respect to the usual product are also obtained for Loeb product measurable sets that are composed of almost independent events.