关于E-凸函数及E-凸规划几个错误结论的修正

本文研究Youness在1999年建立的有关E凸函数和E规划的结论.利用E凸函数和E凸规划的基本性质和优化分析技术,获得了有关E凸函数E凸规划的几个错误结论的修正..     

Indefinite integrals of quotients of special functions

A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica.     

Weak convergence of probability measures in spaces of smooth functions

A large number of results are available about the weak convergence of probability measures in spaces of continuous functions and spaces of cadlag functions. This paper presents similar results for spaces of smooth functions.     

Smoothness of scale functions for spectrally negative L��vy processes

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than It? calculus. The reason for the latter is that standard It? calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.     

Removable singularities for analytic or subharmonic functions

In this paper, results on removable singularities for analytic functions, harmonic functions and subharmonic functions by Besicovitch, Carleson, and Shapiro are extended. In each theorem, we need not assume thatf has the global property at any point, so we are able to allow dense sets of singularities. We do not state our results in terms of exceptional sets, but each one leads to a series of results implying that certain sets are removable for appropriate classes of functions.     

On the moments and distribution of discrete Choquet integrals from continuous distributions

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [J.-L. Marichal, I. Kojadinovic, Distribution functions of linear combinations of lattice polynomials from the uniform distribution, Statistics & Probability Letters 78 (2008) 985–991] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.     

Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on L² traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions.     

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.     

Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions

Rates of convergence of the Komlos-Major-Tusnady type in the invariance principle for empirical processes indexed by functions are obtained. The conditions are given in terms of the empirical entropy and the accuracy of the Haar type approximation for the corresponding classes of functions. The recent results of Massart⁽²⁵⁾ as well as some new results for empirical characteristic functions are obtained as a corollary.     

The permutation enumeration of wreath products Ck4Sn of cyclic and symmetric groups

Brenti introduced a homomorphism from the symmetric functions to polynomials in one variable with rational coefficients. His map is defined on the elementary symmetric functions. When it is applied to the homogeneous and power symmetric functions the results are generating functions for descents and excedences of permutations in the symmetric group, respectively. Beck and Remmel gave proofs of Brenti's results based on combinatorial definitions of the transition matrices between the bases. In addition, they gave an analog for the hyperoctahedral group. We extend the ideas of these proofs to obtain similar results for wreath products of an arbitrary cyclic group with the symmetric group.     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

On the uniqueness of Cournot equilibrium in case of concave integrated price flexibility

We consider a class of homogeneous Cournot oligopolies with concave integrated price flexibility and convex cost functions. We provide new results about the semi-uniqueness and uniqueness of (Cournot) equilibria for the oligopolies that satisfy these conditions. The condition of concave integrated price flexibility is implied by (but does not imply) the log-concavity of a continuous decreasing price function. So, our results generalize previous results for decreasing log-concave price functions and convex cost functions. We also discuss the particular type of quasi-concavity that characterizes the conditional revenue and profit functions of the firms in these oligopolies and we point out an error of the literature on the equilibrium uniqueness in oligopolies with log-concave price functions. Finally, we explain how the condition of concave integrated price flexibility relates to other conditions on the price and aggregate revenue functions usually considered in the literature, e.g., their concavity.     

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.     

Removable singularities for analytic or subharmonic functions

In this paper, results on removable singularities for analytic functions, harmonic functions and subharmonic functions by Besicovitch, Carleson, and Shapiro are extended. In each theorem, we need not assume thatf has the global property at any point, so we are able to allow dense sets of singularities. We do not state our results in terms of exceptional sets, but each one leads to a series of results implying that certain sets are removable for appropriate classes of functions. Partially supported by an NSF-Grant and an XL-Grant at Purdue respectively.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

Optimality and duality with generalized convexity

Hanson and Mond have given sets of necessary and sufficient conditions for optimality and duality in constrained optimization by introducing classes of generalized convex functions, called type I and type II functions. Recently, Bector defined univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, optimality and duality results for several mathematical programs are obtained combining the concepts of type I and univex functions. Examples of functions satisfying these conditions are given.     

SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS

In this paper, theq-analogue of the Stirling formula for theq-gamma function(Moak formula) is exploited to prove the complete monotonicity properties of some functions involving theq-gamma and theq-polygamma functions for all real numberq 0. The monotonicity of these functions is used to establish sharp inequalities for theq-gamma and theq-polygamma functions and theq-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].     

An approximate subgradient algorithm for unconstrained nonsmooth,nonconvex optimization

In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method.      

On the extremality of maximal dual feasible functions

Dual feasible functions have been used to compute bounds and valid inequalities for combinatorial optimization problems. Here, we analyze the properties of some of the best functions proposed so far. Additionally, we provide new results for composed functions. These results will allow improving the computation of bounds and valid inequalities.     

与条形区域有关的双向单叶解析函数子类的系数估计

本文引入了条形区域内由Salagean算子刻画的解析函数新子类, 讨论了该函数子类的系数估计、Fekete-Szego不等式以及与该函数子类有关的双向单叶函数的系数估计, 所得结果推广了前人的一些工作.