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 Ly = 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 Ly = 0, W(? ψ), denotes their Wronski determinant, and v0, v1 are numbers related to the characteristic roots of the regular singular points 0, 1. 相似文献
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 → ∞. 相似文献
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. 相似文献
Let X1, …, Xn 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 an> 0 and bn ∈ ? such that . Let l(?∞,x)(·) denote the indicator function of the set (?∞,x) and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1(?∞,x)((Mn?bn)/an) does not converge almost surely for any x ∈ S(G). But we shall prove . 相似文献
We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??3 with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L2, ?µ(ΩT), initial velocity v0∈H(Ω), µ∈?+\? there exist velocity v∈H(ΩT) and the pressure p, ?p∈L2, ?µ(ΩT) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:
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 an = an(k, . Then if ?2 is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |bn| where the bn 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. 相似文献
We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Qt = Ω × (0, t), where T[u] = yuxx + uyy 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. 相似文献
We consider a boundary value problem where f(x) ∈ Lp(R), p ∈ [1,∞] (L∞(R) ≔ C(R) and 0 ≤ q(x) ∈ Lloc1( R). Boundary value problem (0.1) is called correctly solvable in the given space Lp(R) if for any f(x) ∈ Lp(R) there is a unique solution y(x) ∞ Lp(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in Lp(R). 相似文献
McKean's caricature of the nerve equation: is considered. The H in (1) is the Heaviside function. We prove the existence of multiple impulse solutions consisting of any finite number of pulses. These solutions are referred to as n-ple impulse solutions, where n is an arbitrary positive integer. 相似文献
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 Fn) = 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. 相似文献
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. 相似文献
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. 相似文献
We study the following initial and boundary value problem: In section 1, with u0 in L2(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L2(0, T; H ∩ H2) ∩ L∞(0, T; H) if we assume u0 in H and f in C1(?,?). Numerical results are given for these two cases. 相似文献