A Riemannian manifold $\left( M,\,\rho \right)$ is called Einstein if the metric $\rho $ satisfies the condition $\text{Ric}\left( \rho \right)\,=\,c\,\cdot \,\rho $ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO\left( n \right)/SO\left( l \right)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$$SO\left( n \right)$-invariant Einstein metrics.

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $t{{R}^{1/3}}$ on a circle of radius $R$, for any given $t\,>\,0$. In particular we prove that any arc of length ${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$ on a circle of radius $R$, with $R\,>\,\sqrt{65}$, contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, $\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$, each of which lies on an arc of length ${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$ on a circle of radius ${{R}_{n}}$.

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)$. The periodic ${{\text{C}}^{*}}$-algebras are characterized, and it is shown that ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)\,\simeq \,\text{C}\left( \mathbb{T} \right)\,\otimes \,\mathfrak{U}$ where $\mathfrak{A}$ is a simple ${{\text{C}}^{*}}$-algebra.

In this paper, we mainly study operator spaces which have the locally lifting property $\left( \text{LLP} \right)$. The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the $\text{LLP}$. Several homological characterizations of the $\text{LLP}$ and weak expectation property are given. We also prove that for any operator space $V$, ${{V}^{**}}$ has the $\text{LLP}$ if and only if $V$ has the $\text{LLP}$ and ${{V}^{*}}$ is exact.

Let $\Xi $ be a discrete set in ${{\mathbb{R}}^{d}}$. Call the elements of $\Xi $centers. The well-known Voronoi tessellation partitions ${{\mathbb{R}}^{d}}$ into polyhedral regions (of varying volumes) by allocating each site of ${{\mathbb{R}}^{d}}$ to the closest center. Here we study allocations of ${{\mathbb{R}}^{d}}$ to $\Xi $ in which each center attempts to claim a region of equal volume $\alpha $.

We focus on the case where $\Xi $ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation.

The model exhibits a phase transition in the appetite $\alpha $. In the critical case $\alpha \,=\,1$ we prove a power law upper bound on $X$ in dimension $d\,=\,1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha <1$ and $\alpha \,>1$we prove exponential upper bounds on $X$.

For every polytope $\mathcal{P}$ there is the universal regular polytope of the same rank as $\mathcal{P}$ corresponding to the Coxeter group $\mathcal{C}\,=\,\left[ \infty ,\,.\,.\,.\,,\,\infty \right]$. For a given automorphism $d$ of $\mathcal{C}$, using monodromy groups, we construct a combinatorial structure ${{P}^{d}}$. When ${{P}^{d}}$ is a polytope isomorphic to $\mathcal{P}$ we say that $\mathcal{P}$ is self-invariant with respect to $d$, or $d$-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a $d$-automorphism of a given order. As an application, we analyze properties of self-dual edge-transitive polyhedra and polyhedra with two flag-orbits. We investigate properties of medials of such polyhedra. Furthermore, we give an example of a self-dual equivelar polyhedron which contains no polarity (duality of order 2). We also extend the concept of Petrie dual to higher dimensions, and we show how it can be dealt with using self-invariance.

Let ${{\mathbb{F}}_{q}}$ be a finite field of $q$ elements, $\mathcal{F}$ a $p$-adic field, and $D$ a quaternion division algebra over $\mathcal{F}$. This paper proves uniqueness of Shalika models for $\text{G}{{\text{L}}_{2n}}\left( {{\mathbb{F}}_{q}} \right)$ and $\text{G}{{\text{L}}_{2n}}\left( D \right)$, and re-obtains uniqueness of Shalika models for $\text{G}{{\text{L}}_{2n}}\left( \mathcal{F} \right)$ for any $n\,\in \,\mathbb{N}$.

We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé approximants either of the polylogarithms or of Hurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Prévost. Finally, we make a detailed comparison of our work with Beukers’. Such constructions are useful in the arithmetical study of the values of the Riemann zeta function at integer points and of the Kubota–Leopold $p$ -adic zeta function.

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We show that a regular Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, $\phi \,=\,\phi \left( s \right)$ , for which there are a Riemannian metric $\alpha $ and a 1-form $\beta $ on a manifold $M$ such that the scalar function $F\,=\,\alpha \phi \left( \beta /\alpha \right)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.

Write ${{\Theta }^{E}}$ for the stable discrete series character associated with an irreducible finite-dimensional representation $E$ of a connected real reductive group $G$. Let $M$ be the centralizer of the split component of a maximal torus $T$, and denote by ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ Arthur’s extension of $|D_{M}^{G}\,\left( \gamma \right)|{{\,}^{1/2}}\,{{\Theta }^{E}}\,\left( \gamma \right)$ to $T\left( \mathbb{R} \right)$. In this paper we give a simple explicit expression for ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ when $\gamma $ is elliptic in $G$. We do not assume $\gamma $ is regular.

Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin–Selberg type for groups $G\,\times \,G{{L}_{n}}$, where $G$ is of split rank $n$. Here we show that their method can equally well be applied to the product ${{U}_{3}}\,\times \,G{{L}_{2}}$, where ${{U}_{3}}$ denotes the quasisplit unitary group in three variables. As an application, we describe which cuspidal automorphic representations of ${{U}_{3}}$ occur in the Siegel induced residual spectrum of the quasisplit ${{U}_{4}}$.

In this work, we investigate how to decompose a pair $\left( A,\,B \right)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf{H}_{\mathbb{C}}^{2}$ under the form $A\,=\,{{I}_{1}}{{I}_{2}}$ and $B\,=\,{{I}_{3}}{{I}_{2}}$, where the ${{I}_{k}}$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\left\langle A,\,B \right\rangle $.