A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissa of convergence. In this article, we relax the condition on the coefficients while still arriving at the same conclusion. Specifically, we write $a_n$ as $|a_n| e^{i \theta _n}$ and we consider the sequences $\{ |a_n| \}$ and $\{ \cos{\theta _n} \}$. Let $M \in \mathbb{N}$ be given. The condition on $\{ |a_n| \}$ is that, dividing the sequence sequentially into vectors of length $M$, each vector lies in a certain convex cone $B \subset [0,\infty)^M$. The condition on $\{ \cos{\theta _n} \}$ is (roughly) that, again dividing the sequence sequentially into vectors of length $M$, each vector lies in the negative of the polar cone of $B$. We demonstrate the additional freedom allowed in choosing the $\theta _n$, compared to Landau's Theorem. We also obtain sharpness results.

Integrals and sums involving special functions are in constant demand in applied mathematics. Rather than refer to a handbook of integrals or to a computer algebra system, we present a do-it-yourself systematic approach that shows how the evaluation of such integrals and sums can be made as simple as possible. Illustrating our method, we present several examples of integrals of Poisson type, Fourier transform, as well as integrals involving product of Bessel functions. We also obtain a new identity involving the sums of $_{2}F_{1}$.

In a tournament, each of $n$ teams wins or loses against each of the other $n-1$ teams $c$ times. If team $i$ wins a total of $w_i$ games, then the sequence $( w_1, w_2, \ldots , w_n )$ is called the score sequence of the tournament. In this paper we give necessary and sufficient conditions on a sequence in order that it be a score sequence for a tournament.

By representing a set of $N$ data points as a vector $\mathbf{x}$ in $\mathbb{R}^{N}$, we show that certain data analysis concepts, in particular regression and quantile regression, can be interpreted as vectors that minimize the distance to the vector $\mathbf{x}$, with respect to an appropriate metric or quasimetric.

In this paper we introduce a degree of primality for natural numbers, and hence, a measure of primality for intervals of consecutive numbers. We characterize maximally prime intervals of length $\le 3$ and their primalities. Maximally prime intervals of length 2 are those that contain Mersenne or Fermat primes; maximally prime intervals of length 3 are, but for a few exceptions, those whose midpoints are Dan numbers. There are relatively few maxprimes for larger lengths. We present a heuristic argument for an asymptotic form describing the distribution of maximal primalities. Finally, we mention open problems and directions for further research.

Let $SE(\mathcal{C})$ (respectively, $SM(\mathcal{C})$) be the subcategory of a category $\mathcal{C}$ with the same objects and whose morphisms are strong epimorphisms (respectively, strong monomorphisms) of $\mathcal{C}$. In this paper, we give conditions in some categories $\mathcal{C}$ for an object $X$ of pro-$\mathcal{C}$ to be isomorphic to an object of pro-$SE(\mathcal{C})$ (respectively, pro-$SM(\mathcal{C})$). As an application, we give conditions under which objects of pro-categories are stable.

Panjer's recursion formula is used for the evaluation of compound distributions. The use of this algorithm has become a widespread standard technique for life and general insurance problems. This study reviews and extends Panjer's recursion formula for evaluation of compound negative binomial distributions. For gamma and a mixture of gamma distributions, the theory has been developed so that the software R can be directly applicable. The accuracy of the method used in this study is better and the computation time is faster.

In this note, we establish some new interesting inequalities in Ptolemaic spaces.