We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

Let $q$ be an algebraic integer of degree $d\ge 2$. Consider the rank of the multiplicative subgroup of ${{\mathbb{C}}^{*}}$ generated by the conjugates of $q$. We say $q$ is of full rank if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\mathbb{Z}[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results.

1. (1) If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\text{disc}\left( \mathbb{Z}\left[ {{q}^{m}} \right] \right)=\text{disc}\left( \mathbb{Z}\left[ {{q}^{n}} \right] \right)$.

2. (2) If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\mathbb{Z}[{{q}^{n}}]=\mathbb{Z}[{{r}^{n}}]$ for infinitely many $n$, then either $q=\omega {r}'$ or $q=\text{Norm}{{(r)}^{2/d}}\omega /r'$ , where $r'$ is some conjugate of $r$ and $\omega $ is some root of unity.

3. (3) Let $r$ be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers $q$ such that $\mathbb{Z}[q]=\mathbb{Z}[r]$.

4. (4) There are only finitely many Pisot-cyclotomic numbers of any fixed order.

We obtain a complete description of closed ideals of the algebra $\mathcal{D}\cap \text{li}{{\text{p}}_{\alpha }},0<\alpha \le \frac{1}{2}$, where $\mathcal{D}$ is the Dirichlet space and $\text{li}{{\text{p}}_{\alpha }}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha $.

A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over ${{\text{R}}^{n}}$, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets.

We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

In this article we study injective representations of infinite quivers. We classify the indecomposable injective representations of trees and describe Gorenstein injective and projective representations of barren trees.

Let $\lambda $ be a fixed integer exceeding 1 and ${{s}_{n}}$ any strictly increasing sequence of positive integers satisfying ${{s}_{n}}\le {{n}^{15/14+o(1)}}$. In this paper we give a version of the large sieve inequality for the sequence ${{\lambda }^{{{s}_{n}}}}$. In particular, we obtain nontrivial estimates of the associated trigonometric sums “on average” and establish equidistribution properties of the numbers ${{\lambda }^{{{s}_{n}}}},n\le p{{(\log p)}^{2+\varepsilon }}$, modulo $p$ for most primes $p$.

Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda $ denotes the isomorphism class of an irreducible representation of $K$, let $\rho \lambda :K\to GL({{V}_{\lambda }})$ denote the corresponding irreducible representation and ${{S}_{\lambda }}$ the $\lambda $-isotypic component of $S$. Write ${{S}_{\lambda }}.{{S}_{\mu }}$ for the subspace of $S$ spanned by products of ${{S}_{\lambda }}$ and ${{S}_{\mu }}$. If ${{V}_{v}}$ occurs as an irreducible constituent of ${{V}_{\lambda }}\otimes {{V}_{\mu }}$, is it true that ${{S}_{v}}\subseteq {{S}_{\lambda }}.{{S}_{\mu }}?$ In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood–Richardson rule.

In this paper we construct a flat smooth section of an induced space $I(s,\eta )$ of $S{{L}_{2}}\left( \mathbb{R} \right)$ so that the attached Whittaker function is not of finite order. An asymptotic method of classical analysis is used.

Let $\mathcal{A}$ be a Banach algebra with a bounded right approximate identity and let $\mathcal{B}$ be a closed ideal of $\mathcal{A}$. We study the relationship between the right identities of the double duals ${{\mathcal{B}}^{*}}^{*}$ and ${{\mathcal{A}}^{**}}$ under the Arens product. We show that every right identity of ${{\mathcal{B}}^{*}}^{*}$ can be extended to a right identity of ${{\mathcal{A}}^{**}}$ in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra $A\left( G \right)$ of a locally compact group $G$, an element $\phi \in A{{\left( G \right)}^{**}}$ is in $A\left( G \right)$ if and only if $A\left( G \right)\phi \subseteq A\left( G \right)$ and $E\phi =\phi$ for all right identities $E$ of $A{{\left( G \right)}^{**}}$. We also prove some results about the topological centers of ${{\mathcal{B}}^{**}}$ and ${{\mathcal{A}}^{**}}$.

Let $\Pi$ be a generic cuspidal automorphic representation of $\text{GSp}\left( 2 \right)$ defined over a totally real algebraic number field $\text{k}$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$-functions $L\left( s,\Pi \times \sigma \right)$ for an arbitrary cuspidal automorphic representation $\sigma $ of $\text{GL}\left( 2 \right)$. We also give an application to the spinor $L$-function of $\Pi$.

The aim of this paper is to prove two general results on parabolic induction of classical $p$-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

We prove that if a finite algebra $\mathbf{A}$ generates a congruence distributive variety, then the subalgebras of the powers of $\mathbf{A}$ satisfy a certain kind of intersection property that fails for finite idempotent algebras that locally exhibit affine or unary behaviour. We demonstrate a connection between this property and the constraint satisfaction problem.

In this paper, we extend the approach first outlined by Hardy and Ramanujan for calculating the asymptotic formulae for the number of partitions into $r$-th powers of primes, ${{p}_{\mathbb{P}\left( r \right)}}\left( n \right)$, to include their difference functions. In doing so, we rectify an oversight of said authors, namely that the first difference function is perforce positive for all values of $n$, and include the magnitude of the error term.