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.     

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 .     

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.     

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.     

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.     

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.     

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.     

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 .     

Generalizations of the neumann system a curve-theoretical approach-part I

The following observation, due to E. Trubowitz, illustrates an intimate relationship between spectral theory and Hamiltonian mechanics in the presence of constraints. Let q(s) be a real periodic function such that the Hill operator, has only a finite number g^R of simple eigenvalues. There exist g^R + 1 periodic eigenfunctions x₁,…, x and corresponding eigenvalues a₁,…, a of L such that where y_r = dx_r/ds. The equations Lx_r = a_rx_r, r = 1,…,g_R+1, make up the classical Neumann system, a system of harmonic oscillators constrained to the unit sphere. H. Flaschka obtained similar results about the Neumann system from a more general point of view. His assumption, that there exists an operator of odd order that commutes with L, leads to algebraic curve theory by the method of Krichever and from there to the Neumann formulas above. The familiar Lax pairs, the constants of motion and the quadrics of the Neumann system emerge as consequences of the Riemann-Roch theoreni. The existence of isospectral deformations of L, the Korteweg de-Vries hierarchy of the soliton equations, underlies the complete integrability of the Neumann system. This paper extends Flaschka's techniques, replacing L by an operator of order n 2 2. Higher Neumann systems are defined in a way that leads naturally to interesting symplectic manifolds, Lax pairs and integrals of motion. C. Tomei, using scattering theory, obtained some of our n = 3 formulas.     

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.     

Forced lattice vibrations: Part I

This is the First 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) = _at + _h(_yt), where a and γ are constants and h(·) has period 2π. For a wide variety of restoring forces F (i.e., F′ > 0), numerical calculations indicate the existence of a sequence of thresholds γ₁ = γ₁(a, h, F) > γ₂ = γ₂(a,h,F) > … > γ_k = γ_k(a,h,F) > …, γ_k → 0, as k → ∞. If γ_k > γ > γ_k+1, a k-phase wave that is well described by the wave form, emerges and travels through the lattice. The goal of this series is to describe the emergence and calculate some properties of these wave forms. In Part I the authors first consider the case where F(x) = e^x (i.e., Toda forces) but h is arbitrary, and show how to compute a basic diagnostic (see J(λ), formula (1.26)) for the system in terms of the solution of an associated scalar Riemann-Hilbert problem, once a certain finite set of numbers is known. In another direction, the authors consider the case where F(x) is restoring but arbitrary, and h is small. Here the authors prove a general result, asserting that if there exists a sufficiently ample family of traveling-wave solutions of the doubly infinite lattice, then it is possible to construct time-periodic k-phase wave solutions with asymptotics in n of type (iii) for the driven system (i). In Part II, the authors prove that sufficiently ample families of traveling-wave solutions of the system (iv) 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.     

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).     

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.     

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.     

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.     

Scattering theory for hyperbolic equations of order 2m

Given self-adjoint operators H_j, on Hilbert spaces ??_j, j = 0,l, and J ∈ ?? (??₀, ??₁) (where ?? (??₀ ??₁) denotes the set of bounded linear operators from ??₀to ??₁), define the wave operators where P₀ is the projection onto the subspace for absolute continuity for H₀. We use (i) to study the scattering problem associated with a pair of equations each of the form where L is a positive, self-adjoint operator on a Hilbert space X, m is a positive integer and the α_j are distinct positive constants. Methods patterned after those of Kato are used to study two equations (that is for L = L₀ and L = L_l) each of the form (ii). We show that they are equivalent to equations of the form where each ?_k is a self-adjoint operator on an associated Hilbert space ??_k. Now suppose～he-wave operators W_±,(L₁ L₀) exist and are complete. Then we can find a J ∈ ??(H₁ H₀) such that W₊(?_l, ?₀,J) exists. In the case where L_o and L₁ have the same domain, ??₁ and ??₀ are equal as vector spaces, and under certain conditions (on L_i, i = 0, 1) ??₀ and ??₁ have equivalent norms. Assuming these conditions, let J'∈ ??(??₁' ??₀) be the identity map. We show that (with an additional assumption on L₀ and L₁) W₊(?₁?₀,J) exists andisequal to W₊(?_l,?₀, J).     

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.