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).     

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.     

Generalized herglotz domains

Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound-soft cylinder of cross section D. Suppose F has the Fourier expansion where a_n = a_n(k, . Then if ?² is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |b_n| where the b_n are independent of such that for all directions Domains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ?D or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions.     

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.     

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.     

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.     

Common root functions of two digraphs

Let D₁ and D₂ be finite digraphs, both with vertex set V, let a and b be given functions from V to Z+, and let k be a positive integer. In this paper we give a necessary and sufficient condition for the existence of k arc-disjoint arborescences in each of D₁ and D₂ satisfying the condition that for each v in V a(v) <r₁(v) = r₂(v) < b(v). Where r_i(v) denotes the number of the arborescences in D_i rooted at v, i = 1, 2.     

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 .     

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 .     

Growth of a PSL2R Manifold Group

The purpose of this paper is to present characterizations of the inhomogeneous Hardy-Bessel potential spaces F^p_α (K) over the 2-series field K defined by Littlewood-Paley type function, where Δ₀(x) = 1(|x|≥1), = 0(other), Δ_j(x) = 2^j ≥, = 0(other) (j ≥ 1). These characterizations are given by difference of functions, ball means of difference and atoms. As applications of these results we shall determine when F^p_α(K) is a multiplication algebra, and prove the lower majorant property, the uniform localization property and the equivalence of Fourier multipliers.     

Global existence and blow-up for a system of heat equations with non-linear boundary conditions

In this paper, we consider non-negative solutions of We prove that if pq ? 1 every solution remains bounded, while all solutions blow up in a finite time if pq > 1. We also show that if p,q > 1, then blow-up can only occur on the boundary.     

On Hardy - Littlewood Maximal Functions in Orlicz Spaces

Let Φ(t) and Ψ(t) be the functions having the following representations Φ(t) = ∫a(s)ds and Ψ(t) = ∫b(s) ds, where a(s) is a positive continuous function such that ∫a(s)/s ds = + ∞ and b(s) is an increasing function such that lim_s→ ∞ b(s) = + ∞. Then the following statements for the Hardy - Littlewood maximal function M f (x) are equivalent:

• 1 (i) there exist positive constants c₁ and s₀ such that
• 1 (ii) there exist positive constant c₂ and c₃ such that

.     

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.     

On regular rings and annihilators

Let F_n stand for the distribution of a normalized sum of n independent random variables with common distribution H. In [6] we assumed the restricted convergence. and obtain an analogous result. The method of proof is considerably different, in particular a very recent continuation theorem (lemma 3.2) for infinitely divisible distributions is needed.     

Sojourns above a high level for a gaussian process with a point of maximum variance

Let X(t), 0≦t≦ 1, be a measurable Gaussian process with mean 0 and variance σ²(t) = EX²(t). Suppose that σ²(t) assumes a unique maximum value σ² at a point τ [0,1]. Define L_u = mes(t: 0≦t≦1, X(t)>)Under appropriate conditions, there exists a function fσ(u) such that fσ(u) → ∞ for u ∞, and The function fσ and the constant δ > 0 are determined by the behavior of the function R(x) = mes(t: 0 ≦ t ≦ 1, σ² - σ²(t) ≦ x) for x > 0.     

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.     

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.     

Hamiltonian cycles in 2-connected claw-free-graphs

M. Matthews and D. Sumner have proved that of G is a 2-connected claw-free graph of order n such that δ ≧ (n ? 2)/3, then G is hamiltonian. We prove that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in F, where F is the set of all graphs defined as follows: any graph H in F can be decomposed into three vertex disjoint subgraphs H₁, H₂, H₃ such that , where u_i, v_i ? V(H_i), u_j v_j ? V(H_j) 1 ? i ≦ j ≦ 3. Examples are given to show that the bound n/4 is sharp. © 1995 John Wiley & Sons, Inc.     

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.