Diestel-Faires定理在局部凸空间中的推广

通过Banach 空间与局部凸空间的对比,将Banach 空间上的Diestel-Faires 定理在局部凸空间上进行推广。进一步给出了局部凸空间上的Orlicz-Pettis定理与推论。     

Loop spaces and the compression theorem

For a smooth, finite-dimensional manifold with a submanifold we study the topology of the straight loop space , the space of loops whose intersections with are subject to a certain transversality condition. Our main tool is Rourke and Sanderson's compression theorem. We prove that the homotopy type of the straight loop space of a link in depends only on the number of link components.

     

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C², or even C¹, or even Fréchet-differentiable.     

A mean value theorem in Asplund spaces

In this paper a nonsmooth mean value theorem in Asplund spaces, under invexity, is provided.     

Beurling's theorem for Riemannian symmetric spaces II

We prove two versions of Beurling's theorem for Riemannian symmetric spaces of arbitrary rank. One of them uses the group Fourier transform and the other uses the Helgason Fourier transform. This is the master theorem in the quantitative uncertainty principle.

     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

A transplantation theorem for Jacobi series in weighted Hardy spaces

A transplantation theorem for Jacobi series in weighted Hardy spaces is proved.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

An intersection theorem in L-convex spaces with applications

A new intersection theorem is obtained in L-convex spaces without linear structure. As its applications, a fixed point theorem, a maximal element theorem, a coincidence theorem, some new minimax inequalities and a saddle point theorem are given in L-convex spaces. Our results generalize many known theorems in the literature.     

The Kreps-Yan theorem for Banach ideal spaces

Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions C∩X ₊ = {0} and C??X ₊ guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Analysis of Low Hamming Weight Products

Hoffstein and Silverman suggested a use of Low Hamming Weight Product (LHWP) to compute a random power in a group or a multiple of an element in a ring. It reduces the computation of powers in a group with fast endomorphisms such as the Galois field F_2ⁿ and Koblitz elliptic curves. In this paper, we introduce a reduced representation of LHWP and apply them to attack the relevant cryptosystems.     

Weighted Hamming metric structures

It is known that the binary generalized Goppa codes are perfect codes for the weighted Hamming metrics. In this paper, we present the existence of a weighted Hamming metric that admits a binary Hamming code (resp. an extended binary Hamming code) to be perfect code. For a special weighted Hamming metric, we also give some structures of a 2-perfect code, show how to construct a 2-perfect linear code and obtain the weight distribution of a 2-perfect code from the partial information of the code.     

An off-diagonal Marcinkiewicz interpolation theorem on Lorentz spaces

Let (X, μ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L ^p,q (X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ∈ [1,∞) and q ∈ [1,∞), was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].     

哈明距离下的网络逆问题研究综述

逆优化问题研究的是如何改变原问题中的权参数,使得某些给定的解是问题在新的权参数下的最优解,且使总的改造费用尽可能少.作为逆优化问题中相对较新的一个分支,哈明距离下的网络逆问题具有较大的理论研究及实际应用价值.此文首先介绍了逆优化问题和哈明距离下的网络逆问题以及它们的应用,然后详细介绍了哈明距离下的网络逆问题的研究动态及使用的研究方法.最后给出了该领域中的一些值得研究的问题.     

Constrained Inverse Minimum Spanning Tree Problems under the Bottleneck-Type Hamming Distance

In this paper, we consider the inverse minimum spanning tree problem under the bottleneck-type Hamming distance, where the weights of edges can be modified only within given intervals. We further consider the constrained case in which the total modification cost cannot exceed a given upper bound. It is shown that these inverse problems can be transformed into a minimum node cover problem on a bipartite graph, and we give a strongly polynomial time algorithm to solve this type of node cover problems. This work is supported by The National Natural Science Foundation of China (60021201), The Hong Kong Research Grant Council under the grant CERG 9040883 (CITYU 103003), and the Doctoral Foundation of Hohai University (2005-02).