This paper provides a careful and accessible exposition of static bifurcation theory for degenerate boundary value problems for semilinear second-order elliptic dierential operators. The purpose of this paper is twofold. The rst purpose is to prove that the rst eigenvalue of the linearized boundary value problem is simple and its associated eigenfunction is positive. The second purpose is to discuss the changes that occur in the structure of the solutions as a parameter varies near the rst eigenvalue of the linearized problem.

Let $G$ be a $(p,q)$-graph with edge domination number $\gamma '$ and edge domatic number $d'$. In this paper we characterize connected graphs for which $\gamma ' = p/2$ and graphs for which $\gamma ' + d' = q + 1$. We also characterize trees and unicyclic graphs for which $\gamma ' = \lfloor p/2 \rfloor $ and $\gamma ' = q - \Delta '$, where $\Delta '$ denotes the maximum degree of an edge in $G$.

B. Liu has recently obtained the maximum value for the $k$th upper multiexponents of primitive, nearly reducible matrices of order $n$ with $1 \leq k \leq n$. In this paper primitive, nearly reducible matrices whose $k$th upper multiexponents attain the maximum value are completely characterized

Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order $(p, 0)$, $(0, p)$, and $(p, p)$ with the weight $(\overline{|j|}\, \overline{|k|})^{p-1}$ for some $p\gt 1$. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted $L^r$-integrability and weighted $L^r$-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], M\'oricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].

This paper is a continuation of [3-6] and it is devoted to a stability analysis of multilayered vibrating viscoelastic spheres, both in vacuo and in an acoustical uid. The analysis is done by investigating the eect of viscoelastic damping on the (continuous) Ladyzenskaya- Babuska-Brezzi (LBB) constants for the related boundary-value problems. The sphere is modeled using both 3-D viscoelasticity and the Kirchho-Love shell theory

To estimate the mean and the variance of a lognormal distribution, Finney (1941) derived the uniformly minimum variance unbiased estimators ($UMVUE$) in the form of infinite series. In this paper, we give an alternative derivation of the $UMVUE$s, and also obtain them in integral forms.

In this paper, we will prove that the solution of stochastic differential equation with a small diffusion coefficient in a nonsmooth domain normally reflected at boundary satisfies a large deviation principle and converges to a deterministic path in $L^p$.