We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts to recent results, and allowing for a simple comparison-at-a-glance between different constructions.

We survey the theory of ordinary self-adjoint differential operators in Hilbert space and their spectrum. Such an operator is generated by a symmetric differential expression and a boundary condition. We discuss the very general modern theory of these symmetric expressions which enlarges the class of these expressions by many dimensions and eliminates the smoothness assumptions required in the classical case as given, e.g., in the celebrated books by Coddington and Levinson and Dunford and Schwartz. The boundary conditions are characterized in terms of square-integrable solutions for a real value of the spectral parameter, and this characterization is used to obtain information about the spectrum. Many of these characterizations are quite recent and widely scattered in the literature, some are new. A comprehensive review of the deficiency index (which determines the number of independent boundary conditions required in the singular case) is also given for an expression $M$ and for its powers. Using the modern theory mentioned above, these powers can be constructed without any smoothness conditions on the coefficients.

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We describe new zero-free regions for the derivatives $\zetak(s)$ of the Riemann zeta function, which take the form of vertical strips in the right half-plane. We show that the zeros located in the narrow complements of these zero-free regions are simple and exhibit vertical periodicities that enable one to give exact formulas for their number.

Analytic functions of two non-commuting bounded operators in a Banach space are considered. Sharp norm estimates are established. Applications to operator equations and differential equations in a Banach space are discussed.

This paper demonstrates equivalence amongst the topological structures of path connected components in the spaces of weighted composition operators from $L^{\infty}$, $H^{\infty}$ and the disk algebra into $L^{\infty}$ on the unit circle.

A congruence relation satisfied by Igusa's cusp form of weight~35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree~2 is included.

With the launch of second line anti-retroviral therapy for HIV infected individuals, there has been an increased expectation of survival %period for people with HIV. We consider previously well-known models in HIV epidemiology where the parameter for the incubation period is used as one of the important components to explain the dynamics of the variables. Such models are extended here to explain the dynamics with respect to a given therapy that prolongs the life of an HIV infected individual. A deconvolution method is demonstrated for estimation of parameters in the situations when no-therapy and multiple therapies are given to the infected population. The models and deconvolution method are extended in order to study the impact of therapy in age-structured populations. A generalization for a situation when $n$-types of therapies are available is given. Models are demonstrated using hypothetical data, and sensitivity of the parameters is also computed.

Let $G$ be the compact group of all characters of the additive group of rational numbers, and let $H_G^\infty$ be the Banach algebra of so-called bounded hyper-analytic functions on the big-disk $\Delta_G$. We characterize the pseudo-hyperbolic distance of the algebra $H_G^\infty$ in terms of the pseudo-hyperbolic distance of the algebra $H^\infty$ and establish relationships between Gleason parts in $M(H_G^\infty)$ and $M(H^\infty)$.

In this article, we show that, under suitable assumptions, the generalized $p$-Laplacian boundary value problem has at least one solution.