无约束最优化问题中一类新黄族及其性质

本文提出了一类新的用于解决无约束最优化问题的拟牛顿方法,并证明了这样的性质,在 精确线性搜索条件下,每一步该族所有方法所产生的迭代方向和迭代点列仅依赖于参数ρ.该方 法可视为拟牛顿方法中黄族的推广.     

Infrared Asymptotics of the Heat Kernel and Nonlocal Effective Action

We review recent results in the nonperturbative theory of the heat kernel and its late-time asymptotic properties responsible for the infrared behavior of the quantum effective action for massless theories. In particular, we derive a generalization of the Coleman-Weinberg potential for theories with an inhomogeneous background field. This generalization represents a new nonlocal, nonperturbative action accounting for the effects in a transition domain between the space-time interior and its infinity. In four dimensions, these effects delocalize the logarithmic Coleman-Weinberg potential, while in d > 4, they are dominated by a new powerlike, renormalization-independent nonlocal structure. We also consider the nonperturbative behavior of the heat kernel in a curved space-time with an asymptotically flat geometry. In particular, we analyze the conformal properties of the heat kernel for a conformally invariant scalar field and discuss the problem of segregating the local cosmological term from the nonlocal effective action.This paper was written at the request of the Editorial Board.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 328–356, June, 2005.     

Weakly Implicative (Fuzzy) Logics I: Basic Properties

This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass–the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for both classes, demonstrating their usefulness and importance.The work was supported by grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AVOZ10300504.     

Generalized Rigid Groups: Definitions,Basic Properties,and Problems

We find a natural generalization of the concept of rigid group. The generalized rigid groups are also called r-groups. The terms of the corresponding rigid series of every r-group can be characterized by both ?-formulas and ?-formulas. We find a recursive system of axioms for the class of r-groups of fixed solubility length. We define divisible r-groups and give an appropriate system of axioms. Several fundamental problems are stated.     

Multiscale Riesz Products and their Support Properties

This work defines and investigates the properties of multiscale Riesz product measures. These are product measures constructed on general locally compact Abelian groups in a process similar to that of the original example of Riesz. The multiscale element of the construction is the use of a general homomorphism of the group in place of the dilation factor. Furthermore, this construction allows for the use of generating functions that are piecewise constant, reminiscent of a wavelet approach, as well as trigonometric polynomials, which is a more classical Fourier approach. Results obtained include the characterization of the mutual absolute continuity or singularity of such Riesz products, in the spirit of Zygmund’s original dichotomy result. In addition, the differences regarding the support properties of measures based on each approach are analyzed and examples are constructed. Mathematics Subject Classifications (2000) Primary 43A05, 43A45; secondary 43A25, 43A46.John J. Benedetto: Supported in part by NSF-DMS Grant 0139759 (2002-2005) and the General Research Board of the University of Maryland.Ioannis Konstantinidis: Supported in part by DARPA Grant MDA 972011003.     

Riesz multipliers onL 1 (G)

This work was supported in part by the Italian Research Science Foundation (C.N.R.), Fondi Ministeriali 40%, 1986.     

高维Heisenberg群上热算子的分析

用留数定理及轨道积分方法,讨论了2n+1(n≥1)维Heisenberg群上热核及Green核的渐近性,并给出了更简明的渐近公式,彻底解决了Hueber,H等人遗留的问题.     

(L)2样本深度的基本性质

本文讨论由L^2深度修正得到的L^2深度相应的样本深度的性质，得到了样本深度的相合性和渐近正态性，并证明了它在任意紧集上的一致相合性．最后，基于上述性质简要讨论了样本深度等高的一些性质．     

样本深度的基本性质

本文讨论由L2深度修正得到的(L)2深度相应的样本深度的性质,得到了样本深度的相合性和渐近正态性,并证明了它在任意紧集上的一致相合性.最后,基于上述性质简要讨论了样本深度等高的一些性质.     

Unconstrained Methods for Generalized Nonlinear Complementarity and Variational Inequality Problems

In this paper, we construct unconstrained methods for the generalized nonlinear complementarity problem and variational inequalities. Properties of the correspondent unconstrained optimization problem are studied. We apply these methods to the subproblems in trust region method, and study their interrelationships. Numerical results are also presented.     

From Asymptotics to Spectral Measures: Determinate Versus Indeterminate Moment Problems

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna parametrization. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of the Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.     

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C _n (s₁, s_2,…, s _k ) C_2n(s₁, s_2,…, s _k , n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.     

Exact Algorithms for Large-Scale Unconstrained Two and Three Staged Cutting Problems

In this paper we propose two exact algorithms for solving both two-staged and three staged unconstrained (un)weighted cutting problems. The two-staged problem is solved by applying a dynamic programming procedure originally developed by Gilmore and Gomory [Gilmore and Gomory, Operations Research, vol. 13, pp. 94–119, 1965]. The three-staged problem is solved by using a top-down approach combined with a dynamic programming procedure. The performance of the exact algorithms are evaluated on some problem instances of the literature and other hard randomly-generated problem instances (a total of 53 problem instances). A parallel implementation is an important feature of the algorithm used for solving the three-staged version.     

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere ?? ^d in the Euclidean space ? ^d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r ^s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? ^d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? ^d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.     

On the stability of wavelet and Gabor frames (Riesz bases)

If the sequence of functions _{j, k} is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system _j,k which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of or is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarié and Meyer [17], then by Chui and Shi [9], and Long [16].     

The Minimality Properties of Chebyshev Polynomials and Their Lacunary Series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish best L _p , L and L ₁ approximation by partial sums of lacunary Chebyshev series of the form _i=0 a ⁱ _b ⁱ(x) where _n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided.