A local compactness theorem for Maxwell's equations

The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {_n} be a sequence of vector-valued functions on G where the L₂-norms of {_n}, {curl _n}, and {div(β _n)} are bounded, and where all _n either satisfy x _n = 0 or (β F_n) = 0 at the boundary ?G of G ( = normal to ?G): then {_n} has a L₂-convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ?G if β is interpreted as electric dielectricity ? or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations.     

Über eine Verallgemeinerung des Begriffs „Charakteristik”︁ bei partiellen Differentialgleichungen erster Ordnung

As, in general, the projections of characteristics into the x-space intersect for finite values of t, the global solution of a conservation law cannot be determined from the characteristic system of the equation, is considered. Only in the linear case, this equation coincides with the equation of the projections of characteristics. For convex h and all x₀ this equation has a solution almost everywhere, and the properties of this solution permit to construct a global solution of the conservation law using strips, in the same way as this is done for linear problems by the method of characteristics.     

Determinants of Regular Singular Sturm - Liouville Operators

We consider a regular singular Sturm-Liouville operator on the line segment (0,1]. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known (cf. Theorem 1.1 below) that the ζ-function of this operator has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to [RS] the ζ - regularized determinant of L is defined by In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation L_y = 0 generalizing earlier work of S. Levit and U. Smilansky [LS], T. Dreyfus and H. Dym [DD], and D. Burghelea, L. Friedlander and T. Kappeler [BFK1, BFK2). More precisely we prove the formula Here ? ψ is a certain fundamental system of solutions for the homogeneous equation L_y = 0, W(? ψ), denotes their Wronski determinant, and v₀, v₁ are numbers related to the characteristic roots of the regular singular points 0, 1.     

A note on backward errors for structured linear systems

Let x? be a computed solution to a linear system Ax=b with , where is a proper subclass of matrices in . A structured backward error (SBE) of x? is defined by a measure of the minimal perturbations and such that (1) and that the SBE can be used to distinguish the structured backward stability of the computed solution x?. For simplicity, we may define a partial SBE of x? by a measure of the minimal perturbation such that (2) Can one use the partial SBE to distinguish the structured backward stability of x?? In this note we show that the partial SBE may be much larger than the SBE for certain structured linear systems such as symmetric Toeplitz systems, KKT systems, and dual Vandermonde systems. Besides, certain backward errors for linear least squares are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem

We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Q_t = Ω × (0, t), where T[u] = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. In [10] we proved the existence of a generalized solution of problem (1) and the existence of a generalized solution of the corresponding stationary boundary value problem (non-linear Tricomi problem) In this paper we give sufficient conditions for the uniqueness of these solutions.     

Scattering for Schrödinger Operators with Magnetic Fields

We study the existence and completeness of the wave operators W_ω(A(b),-Δ) for general Schrodinger operators of the form is a magnetic potential.     

Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH¹(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A¹(X) ? CH¹(X) the subgroup of cycles algebraically equivalent to zero. We say that A¹(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH¹(Γ × X) such that z_*:A¹(Γ)-A¹(X) is surjective. There is the well known Abel-Jacobi map λ₁:A¹(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A¹(X)→J(X) A¹(X) finite dimensional. Now set with corresponding map A^*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A^*(X) finite dimensional ?? Level (H^*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.     

The Khinchin Inequality for Generalized Rademacher Functions

We give a new proof of the Khinchin inequality for the sequence of k-Rademacher functions: We obtain constants which are independent of k. Although the constants are not best possible, they improve estimates of Floret and Matos [4] and they do have optimal dependence on p as p → ∞.     

Homogenization of elliptic equations with principal part not in divergence form and hamiltonian with quadratic growth

In this paper, we consider the following problem: Here the coefficients a_ij and b_i are smooth, periodic with respect to the second variable, and the matrix (a_ij)_ij is uniformly elliptic. The Hamiltonian H is locally Lipschitz continuous with respect to u^? and Du^?, and has quadratic growth with respect to Du^?. The Hamilton-Jacobi-Beliman equations of some stochastic control problems are of this type. Our aim is to pass to the limit in (0_?) as ? tends to zero. We assume the coefficients b_i to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive L^∞, H and W, p₀ > 2, estimates for the solutions of (0_?). We also prove the following corrector's result: This allows us to pass to the limit in (0_?) and to obtain This problem is of the same type as the initial one. When (0_?) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then (0₀) is also a Hamilton-Jacobi-Bellman equation but one corresponding to a modified set of controls.     

Global existence for two-velocity models of the Boltzmann equation

If A is a symmetric 2 × 2-matrix, then the initial value problem describes the evolution in time of a fictive gas whose particles can move only with the velocities u₁ and v₂. It is proved that, for continuous initial values vanishing at infinity, (1) has a global solution if an H-Theorem holds for the gas described by (1). The validity of an H-Theorem is expressed by the properties of A.     

Die Lösung der Prae-Maxwellschen Gleichungen mit Hilfe vollständiger Lösungssysteme

This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R ³. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .     

One sufficient condition for hamiltonian graphs

Let G be a 2-connected graph of order n. We show that if for each pair of nonadjacent vertices x,y ∈ V(G), then G is Hamiltonian.     

Asymptotic Properties of Delay Differential Equations

The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the ordinary differential equation and the delay differential equation by comparing these equations with a set of the first order advanced differential inequalities.     

Hadamard matrices constructed from supplementary difference sets in the class ℋ︁1

In this article we give the definition of the class ??₁ and prove: (1) ??₁(v) ≠ ? for v ∈ ?? = ??₁ ∪ ??₂ ∪ ??₃ where (2) there exists 2 ? {2q²; q² ± q, q²;q² ± q} supplementary difference sets for q² ∈ ??; (3) there exists an Hadamard matrix of order 4v for v ∈ ??; (4) if t is an order of T-matrices, there exists an Hadamard matrix of order 4tv for v ∈ ??. © 1994 John Wiley & Sons, Inc.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

On the domination of the products of graphs II: Trees

For a graph G, a subset of vertices D is a dominating set if for each vertex X not in D, X is adjacent to at least one vertex of D. The domination number, γ(G), is the order of the smallest such set. An outstanding conjecture in the theory of domination is for any two graph G and H, One result presented in this paper settles this question in the case when at least one of G or H is a tree. We show that for all graphs G and any tree T. Furthermore, we supply a partial characterization for which pairs of trees, T₁ and T₂, strict inequality occurs. We show for almost all pairs of trees.     

Globally sparse vertex-ramsey graphs

For graphs G and F we write F → (G)¹_r if every r-coloring of the vertices of F results in a monochromatic copy of G. The global density m(F) of F is the maximum ratio of the number of edges to the number of vertices taken over all subgraphs of F. Let We show that The lower bound is achieved by complete graphs, whereas, for all r ≥ 2 and ? > 0, m_cr(S_k, r) > r - ? for sufficiently large k, where S_k is the star with k arms. In particular, we prove that      

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.     

On the Harnack principle for strongly elliptic systems with nonsmooth coefficients

In this paper we prove the following theorem (for notation and definitions, see the paragraphs below): “Let Ω ⊆ ℝⁿ be a domain, m ∈ ℕ, and λ, q > 0. Then, there exists r (= r(λ, q)) > 1 such that for every 0 < p < q, whenever are weak solutions of a strongly elliptic system with m equations of ellipticity λ satisfying ∈ 𝒫_r a.e. and Ω′ ⊆ Ω subdomain, the following inequalities hold: where C (= C(n,m,λ,q,p,Ω,Ω′)) is a positive constant.” © 1999 John Wiley & Sons, Inc.     

Necessary and Sufficient Conditions for the Oscillation of a Second Order Linear Differential Equation

In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.