On Mean Convergence of Fourier-Bessel Series of Negative Order

The expansion of f ∈ L^p(0, 1) Fourier series of Bessel functions of order converges to f in L^p whenever Let be the space of p-integrable functions with respect to the measure t dt and where {s_n}, n = 1, 2, …, is the set of positive zeros of J_v. Then, the expansion of in a Fourier series of functions ψ_n, ?1 < ν < ?½, converges to in whenever      

Probabilistic approach to the neumann problem

The basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically: where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

On the Eigenvalues of Half‐Linear Boundary Value Problems

We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .     

The eigenvalues and eigenfunctions of a spherically symmetric anharmonic oscillator

We show that the operator H_s has a complete set of eigenfunctions and eigenvalues , which satisfy [2l(l + 1) - (3n² + 3n + 1)]s + o(s) and lim_s→0 = 0. The functions are given in spherical coordinates as a product of generalized Laguerre functions and spherical harmonics.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.     

On the Nonoscillatory Behavior of Solutions of a Second Order Linear Differential Equation

In this paper, wo improve the Sturm comparison theorem and two nonoscillation criteria of Leighton and Wintner, and establish two variants of a Wintner' s nonoscillatory criterion of the second order linear differential equation where r, c : t₀,∞) →, R > 0 a. e. on t₀,∞) and 1/r, c ε L^l(t₀,b) for each b ∞ (t₀,>) for some t₀ > 0. Using these two criteria, we improve some nonoscillation criteria of Hartman. Hille. Moore. Potter. WintnEr, and Willett. These proofs are more elegant and concise than those of theirs.     

A power law for connectedness of some random graphs at the critical point

We consider P(G is connected) when G is a graph with vertex set Z⁺ = {1,2, …}, and the edge between i and j is present with probability p(i, j) = min(λ h(i, j), 1) for certain functions h(i, j) homogeneous of degree -1. It is known that there is a critical value λ_c of λ such that . We show that the probability, at the critical point λ_c, that n₁, and n₂ are connected satisfies a power law, in the sense that for n₂ ≧ n_t ≧ 1 for any δ > 0 and certain constants c₁ and c₂.     

An upper bound for some ramsey numbers

New upper bounds for the ramsey numbers r(k, l) are obtained. In particular it is shown there is a constant A such that      

The Triebel - Lizorkin Scale of Function Spaces for the Fourier - Helgason Transform

We define and investigate the Triebel - Lizorkin scale of function spaces F, with 1< p < ∞, 1< q ≤ ∞ for the Fourier-Helgason transform on symmetric Riemannian manifolds of the noncompact type.     

Constructions of covering triangulations with folds

Let G be a graph with a known triangular embedding in a surface S, and consider G_(m), the composition of G with an independant set of order m. The purpose of this paper is to construct a triangular embedding of G_(m) into a surface by using a covering triangulation with folds. We make the construction for three cases. One of them is used for proving that G_(m) can be triangularly embedded into a surface if G is an Eulerian graph which can be triangularly embedded into a surface S with the same orientability characteristic as .     

Existence of solutions to some non-linear thermoelastic systems with viscosity

In this paper a non-linear system of equations in a one-dimensional domain is considered, where both the viscosity, v, and R are positive. The existence of solutions in the case of Neumann boundary conditions for θ and a polynomial form of p(θ, ?) is proved. When θ satisfies the Dirichlet boundary condition, the existence of solutions is obtained under additional assumptions limiting the growth of p(θ, ?) with respect to θ. In both cases the uniqueness of the solutions is also established.     

Endpoint regularity criterion for weak solutions of the 3D incompressible liquid crystals system

In this paper, we consider the endpoint case regularity for the 3D liquid crystals system. We prove that if , then weak solution (v,d) is smooth, and our main observation is that the condition is not necessary in this situation. The proof is based on the blow‐up analysis and backward uniqueness for the parabolic operator developed by Escauriaza‐Seregin‐S̆verák.     

A central limit theorem on gln(fq)

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Korovkin‐type results for multivariate functions which are periodic with respect to one variable

Let K be a compact metric space and let denote the real Banach space of all continuous functions which are 2π‐periodic with respect to the second variable. We prove the following Korovkin‐type result: Let be a continuous algebraic separating function such that for all , and let be a sequence of positive linear operators. If uniformly with respect to and uniformly on for all , then uniformly on for every . As a corollary we deduce: If , then uniformly on for every if and only if uniformly on for every , where and .     

Spectral Decomposition of Symmetric Operator Matrices

The authors study symmetric operator matrices in the product of Hilbert spaces H = H₁×H₂, where the entries are not necessarily bounded operators. Under suitable assumptions the closure L_o exists and is a selfadjoint operator in H. With L_o, the closure of the transfer function is considered. Under the assumption that there exists a real number β < inf p(A) such that M(β)<< 0, it follows that β ε p(L_o). Applying a factorization result of A.I. Virozub and V.I. Matsaev [VM] to the holomorphic operator function M(λ, the_spectral subspaces of L_o corresponding to the intervals ] — ∞, β] and [β, ∞[ and the restrictions of L_o to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Krein space.     

Analysis of the thermoelastic model of a plate with non-linear shape memory alloy reinforcements

In this paper we study the questions of the existence and uniqueness of the solutions for a thermoelastic system of equations in a two-dimensional domain, where both the viscosity v and the rigidity D are positive. It seems that such a system has up to now been considered in a one-dimensional setting only. The change of dimensions enforces the growth conditions with respect to θ and the additional regularity of the data. The existence of solutions in the case of the Neumann boundary conditions for θ and some weak regularity of data is proved. Under stronger regularity conditions the uniqueness is also established. The system has an interpretation as a plate reinforced by a shape memory alloy (SMA) wire mesh.     

Long cycles passing through vertices in k‐connected graphs

In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k‐connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values of m and k, with , must G have a cycle of length at least passing through X? Fujisawa and Yamashita solved this problem for the case and in 2008. We provide an affirmative answer to this problem for the case of and .     

Estimating ∇u by div u and curl u

In this paper we obtain necessary and sufficient conditions for the validity of the estimate The constant c does not depend on u. u is a vector field with values in ?³. It is defined on a bounded set G of ?³ or an unbounded one, denoted by ?. The boundary conditions are as follows: either the normal component of u vanishes or the tangential one does. Our conditions are expressed in terms of the Betti numbers of G or ?.     

Periodic solutions of quasilinear non-autonomous systems with impulses

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form where x ? R ⁿ, ε is a small parameter, Sufficient conditions have been found for existence of the periodic solution of the given system in the critical and non-critical cases.