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.     

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.     

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.     

Disjoint systems

A disjoint system of type (?, ?, k, n) is a collection ?? = {??₁,…, ??_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??_i and ??_j of distinct members of ?? and for every A ? ??_i there exists a B ? ??_j that does not intersect A. Let D_n (?, ?, k) denote the maximum possible cardinality of a disjoint system of type (?, ?, k, n). It is shown that for every fixed k ? 2,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.     

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 remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem

Recently N. Korevaar developed a method of proving that solutions to elliptic and parabolic boundary value problems on convex domains ω ? R ⁿ are convex functions. He introduced a concavity function and used the classical maximum principle to prove that C ? 0 on ω × ω, i.e. that u is convex. Both he and independently L. Caffarelli and J. Spruck applied this method successfully to various boundary value problems. In this note we weaken the assumptions of their theorems and obtain some interesting new applications which are not covered by their previous results [CS, Ko].     

A new approach to queer differential equations

We study the variational problem Where Ψ* is the increasing rearrangement of Ψ. An approximate problem is introduced which involves a variational problem with n free boundaries (n → ∞). Various estimates are established. In particular, when Ω is convex we show that the solution to the approximate problem is superharmonic and has bounded gradient.     

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 .     

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.     

On Global Existence of Solutions of SDE's with Singular Drift

Let b be a Borel measurable IR^d - valued function, defined on some Borel subset of IR^d. Consider the d- dimensional SDE with singular drift b. A local solution (up to σ) is a tuple (X, W, Q,σ) where X is a stochastic process, W is a Brownian motion under the probability measure Q, and σ is a strictly optional time (i.e., stopping time) such that the above equation is satisfied for all t < σ. Such a local solution was constructed by the author in an earlier paper under very mild conditions on b. In this paper we give criteria for the global existence of the solution, i. e., for Q(σ = ∞) = 1.     

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).     

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.     

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.     

The number of self-complementary achiral necklaces

Necklaces with beads of two colors which are left unchanged both by a reflection as well as by the interchange of the two colors are characterized in terms of their axes of symmetry. This characterization is then used to enumerate them. For n = 2^rm with r ≥ 1 and m odd, the number of self-complementary achiral necklaces is for large even n, this sum is asymptotic to 2^{n/4}-1.     

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.     

Inversion of the exponential X-ray transform. I: Analysis

The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on ?ⁿ by , where μ is a constant. Approximate inversion, and inversion formulae of filtered back-projection type are derived for this operator in all dimensions. In particular, explicit formulae are given for convolution kernels (filters) K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered back-projection algorithm. The results extend and refine work of Tretiak and Metz¹⁷.     

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 linear vertex-arboricity of complementary graphs

The linear vertex-arboricity ρ(G) of a graph G is defined to be the minimum number of subsets into which the vertex set of G can be partitioned such that each subset induces a linear forest. In this paper, we give the sharp upper and lower bounds for the sum and product of linear vertex-arboricities of a graph and its complement. Specifically, we prove that for any graph G of order p. and for any graph G of order p = (2n + 1)², where n ? Z⁺, 2n + 2 ≦ ρ(G) + ρ(G ).     

A central limit theorem on gln(fq)

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