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.     

Spatially localized free vibrations for a non-linear string model

We prove the existence of infinitely many non-zero time-periodic solutions (breathers) to the dispersive wave equation of the form which are localized in the spatial variable, that is The main tool employed is the concentration compactness principle of P. L. Lions.     

Bounds for the ramsey number of a disconnected graph versus any graph

Bounds are determined for the Ramsey number of the union of graphs versus a fixed graph H, based on the Ramsey number of the components versus H. For certain unions of graphs, the exact Ramsey number is determined. From these formulas, some new Ramsey numbers are indicated. In particular, if . Where k_i is the number of components of order i and t₁ (H) is the minimum order of a color class over all critical colorings of the vertices of H, then .     

On the riemann problem for a prototype of a mixed type conservation law

We study the system of conservation laws given by With initial value The system is elliptic when u² + v² < ρ² and hyperbolic when u² + v² ≧ ρ². Following Liu's construction it is found that the system always has a weak solution which however is not necessarily unique.     

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.     

Numerical solution of a time-like Cauchy problem for the wave equation

Let D ? ?ⁿ be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.     

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 .     

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      

Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

An ordinary differential equation of the type with parameterξ ? IRⁿ and smooth coefficients a_j,a ? C^∞([-T,T]) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t ≠ 0 they are real and distinct. The constructions of real-valued phase functions ?pH_kl (k,l = 1., m) and of amplitude functions A_jkl such that for a given s ? [-T, T] every solution u(t, ξ) of the equation can be represented as where Ψ_j(s, ξ)= D^j_tu(s,ξ), j = 0,m-1 are given.     

Forced lattice vibrations: Part II

This is the second part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1, is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling-wave solutions of the doubly infinite system exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions exist for all k, γ_k > γ > γ_k+1. By a general result proved in Part I, this implies that there exist time-periodic solutions of the driven system (i) with k-phase wave asymptotics in n of the type with k = 0 or 1 for general F and k arbitrary for F(x) = e^x (when k = 0, take γ₀ = ∞ and X₀ ≡ 0).     

Stability and Oscillations of Solutions of Integro-Differential Equations of Pendulum—like Systems

A system of nonlinear differential equations of the type on a domain of ?ⁿ is studied. Functional relations between the f_j's, j = 1, …, n, and other necessary conditions are deduced when at each point of the domain the system has a manifold of local solutions. A structure theorem, that makes possible to reduce the problems of the system, e.g. the global solvability of it, to the corresponding questions for a connection of the type ?_z?w = g(z, w) in a fibre bundle over a Riemann surface is proved, and through this reduction we obtain theorems of identity, extension, global factorization, and so on, for the solutions of the system. As an example, a system of nonlinear differential equations of the type is studied and its global solutions are constructed.     

A central limit theorem on gln(fq)

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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.     

Existence for a characteristic boundary value problem for an equation of mixed type in three-dimensional space

The equation of mixed type With k(x₃) = sign x₃|x₃|^m, m > 0, d?C¹(?), x = (x₁, x₂, x₃), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ₀ lying in the half-space x₃ > 0 which intersects the plane x₃ = 0 in the unit circle, and for x₃ < 0 by the characteristic surfaces We prove existence of a generalized solution for the characteristic boundary value problem: Lu = fin G, u_Γ0∪Γ1 = 0. The result is obtained by using a variant of the energy-integral method.     

Hamilton cycles in claw-free graphs

Bondy conjectured that if G is a k-connected graph of order n such that for any (k + 1)-independent set / of G, then the subgraph outside any longest cycle contains no path of length k ? 1. In this paper, we are going to prove that, if G is a k-connected claw-free (K_1,3-free) graph of order n such that for any (k + 1)-independent set /, then G contains a Hamilton cycle. The theorem in this paper implies Bondy's conjecture in the case of claw-free graphs.     

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.     

Weighted (LF)-spaces of Continuous Functions

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The Forwarding Indices of Random Graphs

A routing R of a graph G is a set of n(n ? 1) elementary paths R(u, v) specified for all ordered pairs (u, v) of vertices of G. The vertex-forwarding index ξ(G) of G, is defined by Where ξ(G, R) is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G. Let G_p denote the random graph on n vertices with edge probability p and let m = np. It is proved among other things that, under natural growth conditions on the function p = p(n), the ratio Tends to 1 in probability as n tends to infinity.     

On a neutral Lotka–Volterra system

Sufficient conditions are obtained for the linear stability of the positive equilibrium of the neutral system in terms of the parameters of the system. The case n=2 is considered in detail and the general case is discussed briefly.     

Compatibility Conditions and the Manifold of Solutions of some Nonlinear Cauchy-Riemann System in Several Complex Variables

The system of differential equations Is studied. On the assumption that the system admits certain class of solutions, necessary conditions of compatibility are deduced, and by introducing some changes of variables the system is reduced to a single differential equation, then conversely by solving it, the formula giving all local solutions of the system is infered. The validity of the factorization theorem and the identity theorem for solutions in the class is also treated.