We define periodic functions on infinite blow-ups of the Sierpinski gasket as lifts of functions defined on certain compact fractafolds via covering maps. This is analogous to defining periodic functions on the line as lifts of functions on the circle via covering maps. In our setting there is only a countable set of covering maps. We give two different characterizations of periodic functions in terms of repeating patterns. However, there is no discrete group action that can be used to characterize periodic functions. We also give a Fourier series type description in terms of periodic eigenfunctions of the Laplacian. We define almost periodic functions as uniform limits of periodic functions.

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 $n\,\in \,\mathbb{N}$.

Let $\Theta \,=\,\left( \alpha ,\,\beta \right)$ be a point in ${{\text{R}}^{2}}$, with $1,\,\alpha ,\,\beta $ linearly independent over $\mathrm{Q}$. We attach to $\Theta $ a quadruple $\Omega \left( \Theta \right)$ of exponents that measure the quality of approximation to $\Theta $ both by rational points and by rational lines. The two “uniform” components of $\Omega \left( \Theta \right)$ are related by an equation due to Jarník, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega $ fulfilling these necessary conditions, there exists a point $\Theta \,\in \,{{\text{R}}^{2}}$ for which $\Omega \left( \Theta \right)\,=\,\Omega $.

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.

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

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.

Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight ${{\aleph }_{1}}$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

In this paper we study square integrable representations and $L$ -functions for quasisplit general spin groups over a $p$-adic field. In the first part, the holomorphy of $L$ -functions in a half plane is proved by using a variant form of Casselman's square integrability criterion and the Langlands–Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Muić's idea via the Langlands–Shahidi method towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

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.

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

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 ${{\mathcal{A}}^{**}}$.

For an isotropic submanifold ${{M}^{n}}(n\underline{\underline{>}}3)$ of a space form ${{\tilde{M}}^{n+p}}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of ${{M}^{n}}$ is parallel and the sectional curvature $K$ of ${{M}^{n}}$ satisfies some inequality, then the second fundamental form of ${{M}^{n}}$ in ${{\tilde{M}}^{n+p}}$ is parallel and our manifold ${{M}^{n}}$ is a space form.

Let $G$ be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let $\mathfrak{g}$ be its Lie algebra. First we extend a well-known result about the Picard group of a semi-simple group to reductive groups. Then we prove that if the derived group is simply connected and $\mathfrak{g}$ satisfies a mild condition, the algebra $K{{[G]}^{\mathfrak{g}}}$ of regular functions on $G$ that are invariant under the action of $\mathfrak{g}$ derived from the conjugation action is a unique factorisation domain.

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 $\Pi$.

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.

Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

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 ${{U}_{4}}$.

An unsettled conjecture of V. Bergelson and I. Håland proposes that if $(X,\,\mathcal{A},\,\mu ,\,T)$ is an invertible weak mixing measure preserving system, where $\mu (X)<\infty $, and if ${{p}_{1}},{{p}_{2}},...,{{p}_{k}}$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no ${{p}_{i}}$, nor any ${{p}_{i}}-{{p}_{j,}}i\ne j$, is constant on a set of positive density, then for any measurable sets ${{A}_{0}},{{A}_{1}},...,{{A}_{K}}$, there exists a zero-density set $E\subset Z$ such that

1

$$\underset{n\notin E}{\mathop{\underset{n\to \infty }{\mathop{\lim }}\,}}\,\mu ({{A}_{0}}\cap {{T}^{p1(n)}}{{A}_{1}}\cap \ldots \cap {{T}^{pk(n)}}{{A}_{k}})=\prod\limits_{i=0}^{k}{\mu ({{A}_{i}}).}$$

We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\left\{ {{p}_{1}},{{p}_{2}},\,.\,.\,.\,,\,{{p}_{k}} \right\}$ satisfying the hypotheses of this theorem.

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.