An ordinary Dirichlet series has three abscissae of interest, describing the maximal regions where the Dirichlet series converges, converges uniformly, and converges absolutely. The paper of Hille and Bohnenblust in 1931, regarding the region on which a Dirichlet series can converge uniformly but not absolutely, has prompted much investigation into this region, the "Bohr strip." However, a related natural question has apparently gone unanswered: For a Dirichlet series with non-trivial Bohr strip, how far beyond the Bohr strip might the series converge? We investigate this question by explicit construction, creating Dirichlet series which converge beyond their Bohr strip.

Let $a\in \mathbb{Z}\setminus \{0\}$. A positive squarefree integer $N$ is said to be an $a$-Korselt number ($K_{a}$-number, for short) if $N\neq a$ and $p-a$ divides $N-a$ for each prime divisor $p$ of $N$. By an $a$-Williams number ($W_{a}$-number, for short) we mean a positive integer which is both an $a$-Korselt number and $(-a)$-Korselt number. This paper proves that for each $a$ there are only finitely many $W_{a}$-numbers with exactly three prime factors, as conjectured in 2010 by Bouallegue-Echi-Pinch.

$\newcommand{\p}[1]{#1'}$For a given positive real number $v$ smaller than $\sqrt{2}$, we consider the Fermat-like configuration consisting of a circle $k$ and a rectangle ${AB\p B\p A}$. A point ${P}$ is on ${k}$ if and only if the relation ${|AD|^2+|BC|^2=v^2\,|AB|^2}$ holds, where ${C}$ and ${D}$ are the intersections of the line ${AB}$ with the lines ${\p AP}$ and ${\p BP}$, respectively. There are four such rectangles with the side ${A\p A}$ parallel to any given line of symmetry of the circle. This property is shared by all ellipses. When ${v={\sqrt{2}}}$, analogous statements hold for parabolas. Finally, for ${v\gt{\sqrt{2}}}$, this is true for hyperbolas only for its line of symmetry containing the foci. We also show that many geometric properties of this configuration do not depend on a position of a point on the circle. The original Fermat problem corresponds to the case ${v=1}$.

In this paper, we study the problem of generalized simultaneous approximation in terms of the Minkowski functional. We develop a theory of generalized best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces $W$ and $W+M$ and the quotient space $W/M$. Some other results regarding generalized simultaneous approximation in Banach space are presented.

In this work, we give two applications of Karoubi's fundamental theorem of hermitian K-theory. We prove some isomorphisms in L-theory of monoid rings and the ring of integers in a finite extension of $\mathbb{Q}_{2}$.

An approximate solution of the heat equation on $p$-forms on an $n$-dimensional manifold is constructed. This is used to create a fundamental solution of the heat operator. It is shown there is a link between this solution and the generalized Gauss-Bonnet Theorem on manifolds.

We define the notion of the bias of a Bernoulli random variable and demonstrate its relationship to the property that the mod-2 sum of independent variables converges to a fair coin-toss. We then explore generalizations of these ideas to random walks on a finite cyclic group.

In a prior paper [1] we showed that there is a nonlinear, continuous, dense bijection from $\ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. We now show that there is a linear, continuous, dense bijection whose inverse is everywhere unboundedly discontinuous.

It has been found that a particular sequence in a symmetric Bernoulli process can be squarely linked to a sequence of Fibonacci numbers.

In this note, we introduce objects called prime whales and use them to represent a Boolean algebra as an algebra of sets in a way that is analogous to Stone's Representation Theorem. We also characterize the existence of prime whales in terms of the existence of prime ideals.