Let $f$ be a modular form that is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated with $f$. In the case where ${{a}_{p}}\,=\,0$, he showed that, as in the one-variable case, Pollack’s plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where ${{a}_{p}}\ne 0$ using Sprung’s logarithmic matrix.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs, first introduced in 1978 by A. R. Camina. Camina’s work was inspired by the study of Frobenius groups. We show that if $(G,\,N,\,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

We give a short proof of the Brascamp–Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prèkopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

Given a group automorphism $\phi :\,\Gamma \,\to \,\Gamma $, one has an action of $\Gamma $ on itself by $\phi $-twisted conjugacy, namely, $g.x\,=\,gx\phi ({{g}^{-1}})$. The orbits of this action are called $\phi $-twisted conjugacy classes. One says that $\Gamma $ has the ${{R}_{\infty }}$-property if there are infinitely many $\phi $-twisted conjugacy classes for every automorphism $\phi $ of $\Gamma $. In this paper we show that $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ and its congruence subgroups have the ${{R}_{\infty }}$-property. Further we show that any (countable) abelian extension of $\Gamma $ has the ${{R}_{\infty }}$-property where $\Gamma $ is a torsion free non-elementary hyperbolic group, or $\text{SL(}n\text{,}\mathbb{Z}\text{)},\text{Sp(2}n\text{,}\mathbb{Z}\text{)}$ or a principal congruence subgroup of $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Let ${{\Gamma }_{1}}$ and ${{\Gamma }_{2}}$ be Bieberbach groups contained in the full isometry group $G$ of ${{\mathbb{R}}^{n}}$. We prove that if the compact flat manifolds ${{\Gamma }_{1}}\backslash {{\mathbb{R}}^{n}}$ and ${{\Gamma }_{2}}\backslash {{\mathbb{R}}^{n}}$ are strongly isospectral, then the Bieberbach groups ${{\Gamma }_{1}}$ and ${{\Gamma }_{2}}$ are representation equivalent; that is, the right regular representations ${{L}^{2}}\left( {{\Gamma }_{1}}\backslash G \right)$ and ${{L}^{2}}\left( {{\Gamma }_{2}}\backslash G \right)$ are unitarily equivalent.

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ${{\mathbb{R}}^{n}}$.

Let $X$ be a compact Hausdorff space. In this paper, we give an example to show that there is $u\,\in \,C\left( X \right)\,\otimes \,{{M}_{n}}$ with $\det \left( u\left( x \right) \right)\,=\,1$ for all $x\,\in \,X$ and $u{{\tilde{\ }}_{h}}1$ such that the ${{C}^{*}}$ exponential length of $u$ (denoted by $\text{cel}\left( u \right)$) cannot be controlled by $\pi$. Moreover, in simple inductive limit ${{C}^{*}}$-algebras, similar examples also exist.

Let $X$ and $Y$ be metric spaces and $E$, $F$ be Banach spaces. Suppose that both $X$ and $Y$ are realcompact, or both $E$, $F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z\left( f \right)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if

$$z\left( f \right)\,\subseteq \,z\left( g \right)\,\,\,\,\Leftrightarrow \,\,\,\,z\left( Tf \right)\,\subseteq \,z\left( Tg \right),\,\,\,\,\,\text{or}\,\,\,\,z\left( f \right)\,=\,\varnothing \,\,\,\Leftrightarrow \,\,\,z\left( Tf \right)\,=\,\varnothing ,$$

respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim\,E\,=\,\dim\,F\,<\,+\infty$, is a weighted composition operator $\left( Tf \right)\left( y \right)\,=\,{{J}_{y}}\left( f\left( \tau \left( y \right) \right) \right)$. We show that the map $\tau \,:\,Y\,\to \,X$ is a locally (little) Lipschitz homeomorphism.

We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied, and the best upper bounds to date are linear in genus, due to a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound that slightly improves the best known bound.

We give a sharp upper bound on the size of a triangle-free graph of a given order and connected domination. Our bound, apart from strengthening an old classical theorem of Mantel and of Turán improves on a theorem of Sanchis. Further, as corollaries, we settle a long standing conjecture of Graffiti on the leaf number and local independence for triangle-free graphs and answer a question of Griggs, Kleitman, and Shastri on a lower bound of the leaf number in triangle-free graphs.

Consider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if the Weierstrass Preparation Theoremholds in such a ring, then all elements of it are germs of analytic functions.

In 1970, Kotzig and Rosa defined the concept of edge-magic labelings as follows. Let $G$ be a simple $\left( p,\,q \right)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:\,V\left( G \right)\cup E\left( G \right)\,\to \,\left\{ 1,\,2,\,.\,.\,.\,,\,p\,+\,q \right\}$ is an edge-magic labeling of $G$ if $f\left( u \right)\,+\,f\left( uv \right)\,+f\left( v \right)\,=\,k$, for all $uv\,\in \,E\left( G \right)$. A graph that admits an edge-magic labeling is called an edge-magic graph, and $k$ is called the magic sum of the labeling. An old conjecture of Godbold and Slater states that all possible theoretical magic sums are attained for each cycle of order $n\,\ge \,7$. Motivated by this conjecture, we prove that for all ${{n}_{0}}\,\in \,\mathbb{N}$, there exists $n\,\in \,\mathbb{N}$ such that the cycle ${{C}_{n}}$ admits at least ${{n}_{0}}$ edge-magic labelings with at least ${{n}_{0}}$ mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle ${{C}_{n}}$, depending on the sum of the exponents of the odd primes appearing in the prime factorization of $n$.

Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma (T)\,(\text{resp}\text{.}\,{{\sigma }_{e}}(T))$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator; that is, ${{I}_{H}}\,-\,T*T$ is compact. We show that if $D\backslash \sigma (T)\,\ne \,\varnothing $, then for every $f$ from the disc-algebra

$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( {{\sigma }_{e}}\left( T \right) \right),$$

where $D$ is the open unit disc. In addition, if $T$ lies in the class ${{C}_{0}}.\,\bigcup \,C{{.}_{0}}$, then

$${{\sigma }_{e}}\left( f\left( T \right) \right)\,=\,f\left( \sigma \left( T \right)\,\bigcap \,\Gamma \right),$$

where $\Gamma $ is the unit circle. Some related problems are also discussed.

We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$, all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$, are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$.

In a multiplication module, prime submodules have the following property: if a prime submodule contains a finite intersection of submodules, then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodules. A multiplication module in which every prime submodule is strongly prime will be called a strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, $Q$-modules and strongly 0-dimensional modules.

An inverse semigroup $S$ is combinatorially factorizable if $S\,=\,TG$ where $T$ is a combinatorial (i.e., $\mathcal{H}$ is the equality relation) inverse subsemigroup of $S$ and $G$ is a subgroup of $S$. This concept was introduced and studied by Mills, especially in the case when $S$ is cryptic (i.e., $\mathcal{H}$ is a congruence on $S$). Her approach is mainly analytical considering subsemigroups of a cryptic inverse semigroup.

We start with a combinatorial inverse monoid and a factorizable Clifford monoid and from an action of the former on the latter construct the semigroups in the title. As a special case, we consider semigroups that are direct products of a combinatorial inverse monoid and a group.

We study analytic properties function $m\left( z,\,E \right)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m\left( z,\,E \right)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m\left( z,\,E \right)$ in the strip $\left| \Im z \right|\,<\,2\pi$.

Let $A$ be a subgroup of a finite group $G$ and $\sum \,=\,\{{{G}_{0}}\,\le \,{{G}_{1}}\,\le \,.\,.\,.\,\le \,{{G}_{n}}\}$ some subgroup series of $G$. Suppose that for each pair $\left( K,\,H \right)$ such that $K$ is a maximal subgroup of $H$ and ${{G}_{i-1}}\,\le \,K\,<\,H\,\le \,{{G}_{i}}$, for some i, either $A\,\cap \,H\,=\,A\,\cap \,K\,\text{or}\,\text{AH}\,\text{=}\,\text{AK}$. Then $A$ is said to be $\sum$-embedded in $G$. And $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and $a\,\{1\,\le \,G\}$-embedded subgroup $C$ in $G$ such that $G\,=\,AT$ and $T\cap A\,\le \,C\,\le \,A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order ${{p}^{k}}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.