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 .     

Almost Sure Convergence in Extreme Value Theory

Let X₁, …, X_n be independent random variables with common distribution function F. Define and let G(x) be one of the extreme-value distributions. Assume F ∈ D(G), i.e., there exist a_n> 0 and b_n ∈ ? such that . Let l(?∞,x)(·) denote the indicator function of the set (?∞,x) and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1(?∞,x)((M_n?b_n)/a_n) does not converge almost surely for any x ∈ S(G). But we shall prove .     

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.     

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.     

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.     

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 → ∞.     

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.     

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.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Radially symmetric Poisson–Boltzmann equation in a domain expanding to infinity

The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies We study the problem when R → ∞.     

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.     

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.     

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.     

Abel-Summability of Eigenfunction Expansions of Three—Point Boundary Value Problems

We study the Abel-summability of the eigenfunction expansions associated with the differential operator L generated by and splitting three-point boundary conditions. It is shown that there is no straightforward analogy between multipoint and twopoint boundary value problems. Counterexamples show that our main results are best possible. Classification. 34L10, 34B10.     

A Nonlinear Dirichlet Problem on the Unit Ball

In the present paper, we consider the nonlinear Dirichlet problem - Δu(x) u^β(x) = 0 is the unit ball and q is a continuous radially symmetric function on B which may be singular on ?B. We state some mild conditions for the function q so that the Dirichlet problem has a positive classical solution.     

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.     

Discontinuous self-excited systems: An asymptotic analysis

In this paper we derive a uniformly valid asymptotic approximation of the periodic solution of a self-excited system given by the differential equation and β₁,β₂, are positive constants. By uniformly valid asymptotic approximation we mean that no secular terms are present. Our procedure makes use of a nonlinear change of independent variable that transforms the problem from one in which the discontinuities are ? dependent to one in which the discontinuities are ? independent. We obtain an asymptotic approximation up to order ? of the periodic solution and an asymptotic approximation up to order ?² of the period. Some comparisons between our asymptotic results and numerically derived results are given. Application of our technique to other examples of self-excited systems is discussed. The equation is investigated in detail.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Supercritical Branching Brownian Motion and K-P-P Equation In the Critical Speed-Area

If R_t is the position of the rightmost particle at time t in a one dimensional branching brownian motion, whore α is the inverse of the mean life time and m is the mean of the reproduction law. If Z_t denotes the random point measure of particles living at time t, we get in the critical area {c = c₀} The function u(t, x) = P(R_t > x) is studied as a solution of the K-P-P equation for some function f. Conditioned on non-extinction of the spatial tree in the c₀-direction, a limit distribution is obtained and characterized.     

Nontrivial periodic solutions of a nonlinear beam equation

We consider the existence of a nontrivial solution of the following equation: where g is a nondecreasing function defined on R¹, satisfies g(O) = O, and some other additional conditions. Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]-[8]: .