A generalization of gamma functions and Kronecker's limit formulas

We obtain a “Kronecker limit formula” for the Epstein zeta function. This is done by introducing a generalized gamma function attached to the Epstein zeta function. The methods involve generalizing ideas of Shintani and Stark. We first show that a generalized gamma function appears as the value at s=0 of the first derivative of the associated Epstein zeta function. Then this is used to yield Kronecker's limit formula and its “s=0”-version.     

Biases in the prime number race of function fields

We derive a formula for the density of positive integers satisfying a certain system of inequality, often referred as prime number races, in the case of the polynomial rings over finite fields. This is a function field analog of the work of Feuerverger and Martin, who established such formula in the number field case, building up on the fundamental work of Rubinstein and Sarnak.     

Rectangularity of products and completions of their subsets

The approach to the problem of the distribution of the functors of the Stone-?ech compactification, the Hewitt realcompactification or the Dieudonné completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given.     

Bad normed spaces, convexity properties, separated sets

Some years ago, a parameter-denoted by A₁(X)-was defined in real Banach spaces. In the same setting, several years before, a notion called Q-convexity had been defined. Studying these two notions seems to be rather awkward and up until now this has not been done in deep.Here we indicate some properties and connections between these two parameters and some other related ones, in infinite-dimensional Banach spaces. We also consider another notion, a natural extension of Q-convexity, and we discuss the case when A₁(X) attains its maximum value. The spaces where this happens can be considered as ”bad” since they cannot have several properties which are usually considered as nice (like uniform non-squareness or P-convexity).     

Sharkovsky’s program for the classification of triangular maps is almost completed

For a continuous map of the interval, there are more than 50 conditions characterizing zero topological entropy. Some are applicable to the class of triangular maps (x,y)?(f(x),g_x(y)) of the square, but only a few of them are equivalent in this more general setting. In 1989, A.N. Sharkovsky posed the problem of proving or disproving all possible implications between them. During last 20 years, 32 conditions were considered, and most of the work was done. Only 45 relations out of 992 remained not clear. In this paper we give a survey of known results, provide two new examples disproving another 26 possible implications, and spell out the remaining 19 open problems; all but one concern distributional chaos.     

Comparison between the shifted-Laplacian preconditioning and the controllability methods for computational acoustics

Processes that can be modelled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmholtz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.     

Nonlinear age-dependent diffusive equations: A bifurcation approach

The main goal of this paper is the study of the existence and uniqueness of positive solutions of some nonlinear age-dependent diffusive models, arising from dynamic populations. We use a bifurcation method, for which it has been necessary to study in detail the linear and eigenvalue problems associated to the nonlinear problem in an appropriate space.     

Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics

The dynamics of gaseous stars is often described by magnetic fields coupled to self-gravitation and radiation effects. In this paper we consider an initial-boundary value problem for nonlinear planar magnetohydrodynamics (MHD) in the case that the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes are taken into account. Based on the fundamental local existence results and global-in-time a priori estimates, we establish the global existence of a unique classical solution with large initial data to the initial-boundary value problem under quite general assumptions on the heat conductivity.     

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (Md,g), where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.