This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of $\text{G}{{\text{L}}_{5}}\left( \mathcal{F} \right)$, where $\mathcal{F}$ is a $p$-adic field. It is an extension of the work of Heumos and Rallis on $\text{G}{{\text{L}}_{4}}\left( \mathcal{F} \right)$.

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 $.

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.

We give a constructive proof, in the special case of $\text{G}{{\text{L}}_{3}}$, of a theorem of Ash and Stevens which compares overconvergent cohomology to classical cohomology. Namely, we show that every ordinary classical Hecke-eigenclass can be lifted uniquely to a rigid analytic eigenclass. Our basic method builds on the ideas of M. Greenberg; we first form an arbitrary lift of the classical eigenclass to a distribution-valued cochain. Then, by appropriately iterating the ${{U}_{p}}$-operator, we produce a cocycle whose image in cohomology is the desired eigenclass. The constructive nature of this proof makes it possible to perform computer computations to approximate these interesting overconvergent eigenclasses.

Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$ . In this paper we determine all the cases in which Lie$(N)$ is prehomogeneous under $\text{Ad}\left( m \right)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.

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.

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following.

1. (a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets.

2. (b) If $G$ is a countably infinite Abelian group with finitely many elements of order 2 and $\beta G$ is the Stone–Čech compactification of $G$ as a discrete semigroup, then for every idempotent $p\,\,\in \,\,\beta G\backslash \{0\}$, the subset $\{p,-p\}\subset \beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and $\{-p\}$.