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.     

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.     

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 .     

A central limit theorem on gln(fq)

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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).     

Definitizability of Certain Functions and the Existence of Eigenvectors of Unitary Operators in Pontrjagin Spaces

Let P_k^c(G) denote the set of continuous functions with k negative squares on a locally compact commutative group G. Every function f ? P_k^c(G) is definitizable in the sense that is positive definite for certain complex measures ω on G with finite support [9]. The proof of this fact was base on a result of M. A. Naimark about common nonpositive eigenvectors of commuting unitary operators in a Pontrjagin space. It is the aim of this note to prove without any use of the theory of Pontrjagin spaces the definitizability of functions f ? P_k^c(G) which are of polynomial growth. In Section 3 we show, how the definitizability of functions f ? P_k^c(G) can be used to prove the existence of common non-positive eigenvectors of commuting unitary operators in a Pontrjagin space.     

On the distance matrix of a directed graph

In this note, we show how the determinant of the distance matrix D(G) of a weighted, directed graph G can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks G_i of G. In particular, when cof D(G), the sum of the cofactors of D(G), does not vanish, we have the very attractive formula .     

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 .     

Stabilization and regularization for solutions of an ill posed problem for the wave equation

In this paper we study the problem of finding u: [O, T] → D(A), D(A) ? H, H a Hibert space such that: Is a linear positive, self-adjoint operator with a compact inverse. The problem is well known to be illposed because uniqueness and existence generally fail. We restore the stability with an a priori bound on ∥ du(0)/dt ∥ for some particular values of T.     

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.     

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.     

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.     

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.     

On critical exponents for some quasilinear parabolic equations

We study the Cauchy problem for the quasilinear parabolic equation where p > 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with ? ? sψ′(s)/ψ(s) ? θ for some θ > 0. If 1 < p < 1 + 2/N, there are no global positive solutions, whereas if p > 1 + 2/N, there are global, positive solutions for small initial data.     

Maximum Bandwidth Under Edge Addition

We determine how much the bandwidth B(G) of a graph G can increase when a single edge is added. Let g(b,n) be the maximum possible value of B(G + e) when G has n vertices and bandwidth b. The problem of studying when B(G + e) ≦ B (G) + 1 was originally possed by Erdos. We determine © 1996 John Wiley & Sons, Inc.     

A finite element method and stabilization method for a non-linear tricomi problem

We prove using the Faedo-Galerkin method the existence of a generalized solution of an initial-boundary value problem for the non-linear evolution equation 0 ? Q ? 2, in a cylinder Q_T = Ω × (0, T), where ?? u = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. We then show that the approximate generalized solution of problem (*) converges to the approximate generalized solution of the corresponding stationary boundary value problem as t → ∞.     

Hamiltonian graphs with neighborhood intersections

In this paper, k + 1 real numbers c₁, c₂, ?, c_k+1 are found such that the following condition is sufficient for a k-connected graph of order n to be hamiltonian: for each independent vertex set of k + 1 vertices in G. where S_i = {v ? V:|N(v) ∩ S| = i} for 0 ≦ i ≦ k + 1. Such a set of k + 1 numbers is called an Hk-sequence. A sufficient condition for the existence of Hk-sequences is obtained that generalizes many known results involving sum of degrees, neighborhood unions, and/or neighborhood intersections.     

A list analogue of equitable coloring

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003     

Dominating sets inducing large components in maximal outerplanar graphs

For a maximal outerplanar graph G of order n at least three, Matheson and Tarjan showed that G has domination number at most . Similarly, for a maximal outerplanar graph G of order n at least five, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that G has total domination number at most unless G is isomorphic to one of two exceptional graphs of order 12. We present a unified proof of a common generalization of these two results. For every positive integer k, we specify a set of graphs of order at least and at most such that every maximal outerplanar graph G of order n at least that does not belong to has a dominating set D of order at most such that every component of the subgraph of G induced by D has order at least k.     

Hamiltonicity for K1, r-free graphs

In this paper, we investigate the Hamiltonicity of K_1,r-free graphs with some degree conditions. In particular, let G be a k-connected grph of order n≧3 which is K_1,4-free. If for every independent set {v₀, v₁, …, v_k} then G is hamiltonian. We use an upper bound for the independence number of K_1,r-free graphs to extent the above result to K_1,r-free graphs. Hamiltonian connected and, more generally, q-edge hamiltonian properties are studied here as well. © 1995 John Wiley & Sons, Inc.