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1.
Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of Nn given by m1?m2???mn?0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in Nn to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m1?m2???mn?0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.  相似文献   

2.
We study the approximation of functions from anisotropic Sobolev classes B(Wrp([0,1]d)) and Hölder-Nikolskii classes B(Hrp([0,1]d)) in the Lq([0,1]d) norm with qp in the quantum model of computation. We determine the quantum query complexity of this problem up to logarithmic factors. It shows that the quantum algorithms are significantly better than the classical deterministic or randomized algorithms.  相似文献   

3.
Out of n i.i.d. random vectors in Rd let X1n be the one closest to the origin. We show that X1n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density estimation.  相似文献   

4.
A review is made of some of the fundamental properties of the sequence of functions {tieλkt}, k = 1,…,s, i = 0,…,mk?1, distinct λ i. In particular it is shown how the Wronskian and Gram matrices of this sequence of functions appear naturally in such fields as spectral matrix theory, controllability, and Lyapunov stability theory.  相似文献   

5.
Given a random measure η and a fixed number p>1, the Lp-intensity 6η6p of ηis defined as the total variation measure of the subadditive set function 6η(·)6p. It is shown that 6η6p can exist (be locally finite) only if the usual intensity measure Eη exists and η 《Eη a.s, and that in this case 6η6pB=?B6dηdEη6pdEη. If η is the conditional intensity of a simple point process ξ, then 6η6p equals the total variation of the subadditive set functions 6P[ξB = 1|Bcξ]6p and 6E[ξB|Bcξ]6p. Some applications to stochastic geometry and particle systems are discussed briefly.  相似文献   

6.
We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L=F+G of a given increasing convex additively slowly varying function L into a sum of two increasing convex functions F and G. We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class OΠ2+ we obtain is related to the well-known class OΠg where, instead of first order differences as in OΠg, we have second order differences.  相似文献   

7.
This paper introduces the use of conjugate transforms in the study of τT semigroups of probability distribution functions. If Δ+ denotes the space of one-dimensional distribution functions concentrated on [0, ∞) and T is a t-norm, i.e., a suitable binary operation on [0, 1], then the operation τT is defined for F, G in Δ+by τT(F, G)(x) = supu+v = xT(F(u), G(v)) for all x. The pair (Δ+, τT) is then a semigroup. For any Archimedean t-norm T, a conjugate transform CT is defined on (Δ+, τT). These transforms are shown to play a role similar to that played by the Laplace transform on the convolution semigroup. Thus a theory of “characteristic functions” for τT semigroups is developed. In addition to establishing their basic algebraic properties, we also use conjugate transforms to study the algebraic questions of the cancellation law, infinitely divisible elements, and solutions of equations in τT semigroups.  相似文献   

8.
We prove the existence of many homographic solutions of the n-body problem in E4 by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E4 central configurations are a proper subset of the relative equilibria for any n ? 3 and for any (mi)?R+n. We compare the existence and classification of homographic solutions of the n-body problem in E3 with the Newtonian potential and that of homographic solutions of the n-body problem in E4. Classifying relative equilibria leads to classifying homographic solutions.  相似文献   

9.
For a Banach space E and for 1 ? p < ∞ let ?p<∞ let LEp(μ) = LEp(S,B,μ) denote all Bochner p-integrable E-valued functions on a measure space (S,B,μ). Under study are convergence theorems for integrals of functions in LEp(μ) with respect to Nemytskii measures. Weak integrals are then denoted to Hammerstein operators, and a study of topologies generated by vector measures leads to a characterization of compact Hammerstein operators.  相似文献   

10.
We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal IP(N) the quotient space l(I)/c0(I) is complete. The second main result states that consistently there is an admissible ideal IP(N) such that the sets W(I), of all real sequences with finite I-variation, and c?(I), of all restrictively I-convergent sequences, are equal.  相似文献   

11.
The (real-time) Green's functions for the P(φ)2 quantum field theory are studied. The main result is that they are C functions of the coupling constant λ, for λ sufficiently small and nonzero. Explicit expressions for the derivatives are obtained.  相似文献   

12.
13.
It is proved that the classical S-operator for relativistic spin-0 and spin-12 particles in time-dependent external fields is gauge invariant, and that S+- and S?+ are entire functions of the coupling constant in the Hilbert-Schmidt norm. As a result the Fock space S-operator exists for any real value of the coupling constant, and is gauge invariant. The external fields and the gauge function are assumed to be real-valued resp. complex-valued functions in S(R4).  相似文献   

14.
The b-clique polytope CPnb is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QPn=CPnn as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over CPnn. The problem of optimizing linear functions over CPnb has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CPnb in further detail and present a new computational approach to the linear optimization problem based on the idea of integrating cutting planes into a Lagrangian relaxation of an integer programming problem that Balas and Christofides had suggested for the traveling salesman problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.  相似文献   

15.
The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L 2-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.  相似文献   

16.
Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H01 (Ω) of all functions ψ0 ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ0 a class Wweak0) of functions which captures robust topological properties, of the level sets of {ψ0} and, moreover, is closed under weak limits.  相似文献   

17.
In this work wome connections are pursued between weak and strong convergence in the spaces Cm (m-times continuously differentiable functions on Rn). Let fn, f?Cm + 1, where n = 1, 2,…, and m is a nonnegative integer. Suppose that the sequence {fn} converges to f relative to the weak topology of Cm + 1. It is shown that this implies the convergence of {fn} to f with respect to the strong topology of Cm. Several corollaries to this theorem are established; among them is a sufficient condition for uniform convergence. A stronger result is shown to exist when the sequence constitutes an output sequence of a linear weakly continuous operator.  相似文献   

18.
We consider an extremal problem for even positive definite entire functions of exponential type with zero mean with power weight on the semiaxis. This problem is related to the multidimensional Jackson-Stechkin theorem in the space L 2(?n).  相似文献   

19.
In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region inR 2. We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error inL 1 is controlled.  相似文献   

20.
A second countable developable T1-space D1 is defined which has the following properties: (1) D1 is an absolute extensor for the class of perfect spaces. (2) D1?0 is a universal space for second countable developable T1-spaces.  相似文献   

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