A note on infinite systems of linear inequalities in Rn

A necessary and sufficient condition for infinite systems of linear inequalities to have solutions is given, using the Kuhn-Fourier Theorem, which deals with finite systems.     

Partitioning the lattice points in Rn

A (linear) lattice point path is the intersection of a connected subset of a straight line in euclidean n-space Rⁿ with the lattice points Zⁿ ? Rⁿ, where Z denotes the integers. Given a subset X ? Zⁿ, a typical problem considered in this paper is the determination of the largest integer k (it turns out that k ? 2n ?1) with the property that given any partition of X into two parts, then at least one of the parts contains as a subset a lattice point path of k points. Also considered are similar questions which arise by placing restrictions on the type of partitions allowed, or by restricting attention to those lattice point paths which are obtainable from lines having preassigned (integral) direction numbers.     

The realization of distances in measurable subsets covering Rn

A method is developed to investgate the minimum number of subsets that n-dimensional Euclidean space must be divided into so that no two points that are at unit distance are in the same subset, in the particular case where the division is into measurable subsets.     

Stationary distributions of the simplest dynamical systems in Rn perturbed by generalized Poisson process

Let z(t) ∈ Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Proof of the impossibility of the fermat equation Xp + Yp = Zp for special values of p and of the more general equation bXn + cYn = dZn

This paper continues the search to determine for what exponents n Fermat's Last Theorem is true. The main theorem and Corollary 1 consider the set of prime exponents p for which mp + 1 is prime for certain even integers m and prove the truth of FLT in Case 1 for such primes p. The remaining theorems prove the inequality of the more general Fermat equation bXⁿ + cYⁿ = dZⁿ.     

On the Laplacian spectrum of an infinite graph

In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.     

On the spectrum of the Dirichlet Laplacian in a narrow strip

We consider the Dirichlet Laplacian Δ_∈ in a family of bounded domains {−a < x < b, 0 < y < εh(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We find the two-term asymptotics in ε → 0 of the eigenvalues and the one-term asymptotics of the corresponding eigenfunctions. The asymptotic formulas obtained involve the eigenvalues and eigenfunctions of an auxiliary ODE on ℝ that depends on the behavior of h(x) as x → 0. The proof is based on a detailed study of the resolvent of the operator Δ_∈.     

On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

We study the Laplacian in deformed thin (bounded or unbounded) tubes in R³, i.e., tubular regions along a curve r(s) whose cross sections are multiplied by an appropriate deformation function h(s)>0. One of the main requirements on h(s) is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.     

On large deviations for sums of random vectors in Rk

The paper deals with limit theorems for probabilities of large deviations for sums of independent identically distributed random vectors. We give more detailed bounds for the remainder in von Bahr's limit theorem. New asymptotic formulas for probabilities of large deviations on the outside of balls are established.