We study the non-vanishing on the line $\operatorname{Re}\left( s \right)=1$ of the convolution series associated to two Dirichlet series in a certain class of Dirichlet series. The non-vanishing of various $L$-functions on the line $\operatorname{Re}\left( s \right)=1$ will be simple corollaries of our general theorems.

Let

$$\zeta \left( s,x \right)=\sum\limits_{n=0}^{\infty }{\frac{1}{{{\left( n+x \right)}^{s}}}}\left( s>1,x>0 \right)$$

be the Hurwitz zeta function and let

$$Q\left( x \right)=Q\left( x;\alpha ,\beta ;a,b \right)=\frac{{{\left( \zeta \left( \alpha ,x \right) \right)}^{a}}}{{{\left( \zeta \left( \beta ,x \right) \right)}^{{{b}'}}}}$$

where $\alpha ,\beta >1$ and $a,b>0$ are real numbers. We prove: (i) The function $Q$ is decreasing on $\left( 0,\infty \right)$ iff $\alpha a-\beta b\ge \max \left( a-b,0 \right)$. (ii) $Q$ is increasing on $\left( 0,\infty \right)$ iff $\alpha a-\beta b\le \min \left( a-b,0 \right)$. An application of part (i) reveals that for all $x>0$ the function $s\mapsto {{\left[ \left( s-1 \right)\zeta \left( s,x \right) \right]}^{1/\left( s-1 \right)}}$ is decreasing on $\left( 1,\infty \right)$. This settles a conjecture of Bastien and Rogalski.

Let $\mathcal{M}$ be a type $\text{I}{{\text{I}}_{1}}$ von Neumann algebra, $\tau$ a trace in $\mathcal{M}$, and ${{L}^{2}}\left( \mathcal{M},\tau \right)$ the GNS Hilbert space of $\tau$. We regard the unitary group ${{U}_{\mathcal{M}}}$ as a subset of ${{L}^{2}}\left( \mathcal{M},\tau \right)$ and characterize the shortest smooth curves joining two fixed unitaries in the ${{L}^{2}}$ metric. As a consequence of this we obtain that ${{U}_{\mathcal{M}}}$, though a complete (metric) topological group, is not an embedded riemannian submanifold of ${{L}^{2}}\left( \mathcal{M},\tau \right)$

Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

In this paper we consider multipliers on the real Hardy space ${{H}_{2\pi }}$. It is known that the Marcinkiewicz and the Hörmander–Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on $L_{2\pi }^{p},1<p<\infty$. We show among others that the Hörmander– Mihlin condition extends to ${{H}_{2\pi }}$ but the Marcinkiewicz condition does not.

In this paper we prove the sharp inequality

$$\left| P_{n}^{\left( s \right)}\left( x \right) \right|\le P_{n}^{\left( s \right)}\left( 1 \right)\left( {{\left| x \right|}^{n}}+\frac{n-1}{2s+1}\left( 1-{{\left| x \right|}^{n}} \right) \right)$$

where $P_{n}^{\left( s \right)}\left( x \right)$ is the classical ultraspherical polynomial of degree $n$ and order $s\ge n\frac{1+\sqrt{5}}{4}$. This inequality can be refined in $\left[ 0,z_{n}^{s} \right]$ and $\left[ z_{n}^{s},1 \right]$, where $z_{n}^{s}$ denotes the largest zero of $P_{n}^{\left( s \right)}\left( x \right)$.

Let $n=2m$ be even and denote by $\text{S}{{\text{p}}_{n}}\left( F \right)$ the symplectic group of rank $m$ over an infinite field $F$ of characteristic different from 2. We show that any $n\times n$ symmetric matrix $A$ is equivalent under symplectic congruence transformations to the direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal and $C$ tridiagonal. Since the $\text{S}{{\text{p}}_{n}}\left( F \right)$-module of symmetric $n\times n$ matrices over $F$ is isomorphic to the adjoint module $\mathfrak{s}{{\mathfrak{p}}_{n}}\left( F \right)$, we infer that any adjoint orbit of $\text{S}{{\text{p}}_{n}}\left( F \right)$ in $\mathfrak{s}{{\mathfrak{p}}_{n}}\left( F \right)$ has a representative in the sum of $3m-1$ root spaces, which we explicitly determine.

We present a very short proof of Liouville's theorem for solutions to a non-uniformly elliptic radially symmetric equation. The proof uses the Ricatti equation satisfied by the Dirichlet to Neumann map.

It is shown that there exists an inner function $I$ defined on the unit ball ${{\text{B}}^{n}}$ of ${{\mathbb{C}}^{n}}$ such that each function holomorphic on ${{\text{B}}^{n}}$ and bounded by 1 can be approximated by “non-Euclidean translates” of $I$.

Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$. From the results of Carrell and Lieberman there exists a filtration ${{F}_{0}}\subset {{F}_{1}}\subset \cdot \cdot \cdot$ of $A\left( Z \right)$, the ring of $\mathbb{C}$-valued functions on $Z$, such that $\text{Gr }A\left( Z \right)\cong {{H}^{*}}\left( X,\mathbb{C} \right)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$.

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group ${{S}_{n}}$, in each homogenous degree $n$. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of ${{S}_{n-1}}$. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in ${{S}_{n}}$.

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every ${{x}_{0}}\in X$ and every weakly compact convex symmetric set $K$ such that $\overline{\text{span}}K=X$, there is a point of Gâteaux differentiability of $f$ in ${{x}_{0}}+K$. This extends a Klee's result for separable spaces.

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q\left( z \right)$ and the topology of the complement of the hypersurface defined by $Q\left( z \right)=0$.