We prove that complex harmonic functions in the Lie ball can be represented in Dirichlet series by showing the equivalent fact that it can be constructed explicitly a discrete weakly sufficient set for the space of entire functions of exponential type on the complex light cone.

Let $X_1,X_2,\ldots$ be a sequence of independent replicates of a random variable $X$ and let $\{N_t\}_{t\geq 0}$ be a non-negative integer valued random process and assume that $\{N_t\}_{t\geq 0}$ and $X$ are independent. Then, under some conditions it is shown that the probability $P(\sum_{i=1}^{N_t} X_i\geq 0)$ decays exponentially fast as $t\to\infty$. Moreover, we consider a testing problem in a compound Poisson process, and we study the exact slope of a test statistic based on the sum of random number of independent and exponentially distributed random variables.

Let $n\geq 2$ be an integer. We show that if $G$ is a graph such that every component of $G$ has order at least 3, and $|V(G)|\leq 2^n$ and $|V(G)|\neq 2^n-2$, then there exists an injective mapping $\varphi$ from $V(G)$ to an elementary abelian 2-group of order $2^n$ such that for every component $C$ of $G$, the sum of $\varphi(x)$ as $x$ ranges over $V(C)$ is $o$.

The present note deals with convolution operators $\mathscr{L}_\mu$ on the class of holomorphic multiple Dirichlet series in bounded convex domains in $\mathbf{C}^n$. A surjectivity criterion for this operator is obtained. Moreover, the explicit formula for a particular solution $c$ of the equation $\mathscr{L}_{\mu}c=d$ for a given right-hand side $d$ is given.

Let $k$ be an integer such that $k\geq 2$, and let $G$ be a connected graph of order $n$ such that $n\geq 9k-1-4\sqrt{2(k-1)^2+2}$, $kn$ is even, and the minimum degree is at least $k$. We prove that if $|N_G(u)\cup N_G(v)|\geq\frac{1}{2}(n+k-2)$ for each pair of nonadjacent vertices $u$, $v$ of $G$, then $G$ has a $k$-factor.

In this paper, we study the totally real pseudo-umbilical submanifolds in the complex projective space, and obtain two integral inequalities and some conditions under which the submanifold has parallel second fundamental form.

We determine the arithmetic Fuchsian groups $\Gamma$ with signature $(0;e_1,e_2,e_3,e_4)$ which are the subgroups of normalizer $\Gamma^*(A;O)$ of maximal orders $O$ in quaternion algebras $A$ over the rational number field $\mathbf{Q}$.

For certain algebras of continuous functions, the relationship between the greatest regular subalgebras, the algebras which consist of functions of which corresponding multiplication operators are decomposable, and the sets of functions with natural spectra are studied. In particular, spectral properties of certain Fourier multipliers are considered.

We give a criterion of the vanishing of $\lambda_p$-invariants for relative cyclic extensions of totally real number fields with degree $p$ using Iwasawa's result of Riemann-Hurwitz type which is an analogue to Kida's formula.