Internal sets and internal functions in Colombeau theory

This paper is devoted to the study of generalized functions as pointwise functions (so-called internal functions) on certain sets of generalized points (so-called internal sets). We treat the case of the Colombeau algebras of generalized functions, for which these notions have turned out to constitute a fundamental technical tool. We provide general foundations for the notion of internal functions and internal sets and prove a saturation principle. Various applications to Colombeau algebras are given.     

The existence of generalized mix functions

To enrich the message space of a cipher and guarantee security, Ristenpart and Rogaway defined mix functions on two sets of equal size. To mix inputs from two sets of different sizes, Stinson generalized the definition of mix functions (called generalized mix functions), and established an existence result for generalized mix functions with 10 undetermined pairs of input sizes. In this paper, we complete the solution to the existence problem for generalized mix functions.      

广义凸函数的特征性质

提出广义凸集、广义凸函数、中间点广义凸函数、端点广义凸函数四个定义,通过定义条件P1,研究条件P1所蕴含的等式关系,进而得到一个基础性定理一稠密性定理和一个相对条件较弱的推论,最后将结果应用于若干不同类型的广义凸函数类,尤其是s-凸函数、几何凸函数、rp-凸函数,得到它们所共有的一个特征性质,即满足稠密性定理．     

Fenchel Problem of Level Sets

The Fenchel problem of level sets in the smooth case is solved by deducing a new and nice geometric necessary and sufficient condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides a complete geometric characterization of a new subclass of pseudoconvex functions originating from analytical mechanics and a new view on convexlike and generalized convexlike mappings in image analysis. This paper is dedicated to the memory of Guido Stampacchia. The author thanks K. Balla for assistance in solving the system of differential equations of Example 2.1 and for helpful remarks. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy. An erratum to this article is available at .     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.     

广义差集与几乎完美序列

提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的.     

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.     

极大值复合函数Clarke广义梯度计算的一个新方法

对于一类非光滑函数－－光滑函数的有限次极大值复合函数,给出了计算它在一点处Ｃｌａｒｋｅ广义梯度呈一个元素的新方法,与以往方法比较,本文的方法不需判别线性不等式组的相容性,因而易于实现。     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

Generalized Flows and Singular ODEs on Differentiable Manifolds

Based on the concept of manifold-valued generalized functions, we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right-hand sides in a global setting. After establishing several existence and uniqueness results for solutions of such equations and flows of singular vector fields, we compare the solution concept employed here with the purely distributional setting. Finally, we derive criteria securing that a sequence of smooth flows corresponding to the regularization of a given singular vector field converges to a measurable limiting flow.     

On the optimal control method in quaternionic analysis

Gaveau?s optimal control method for real and complex Monge–Ampere operators is generalized to that for quaternionic Monge–Ampere operator. It is also applied to investigate quaternionic regular functions: the characterization of the Silov boundary of a smooth quaternionic pseudoconvex domain.     

Derivatives of generalized farthest functions and existence of generalized farthest points

The relationship between directional derivatives of generalized farthest functions and the existence of generalized farthest points in Banach spaces is investigated. It is proved that the generalized farthest function generated by a bounded closed set having a one-sided directional derivative equal to 1 or −1 implies the existence of generalized farthest points. New characterization theorems of (compact) locally uniformly convex sets are given.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Approximate solution of a mixed problem for a parabolic equation by means of a special basis of functions

Using the eigenfunctions of two Sturm-Liouville problems (with the same operator of the most general form but two different sets of boundary conditions), we propose a method for construction of specific basis functions such that the corresponding expansions of smooth and piecewise smooth functions lead to fast converging series. The last circumstance can be successfully employed for approximate solution of mixed problems for a parabolic equation when the sought function is approximated in the spatial variables by few basis functions. First, the case of one spatial coordinate is elaborated and the two-dimensional case is briefly discussed in Section 5. The method aims primarily at the case when the sought function is piecewise smooth in the spatial variable and the implementation of the method bases on the concept of generalized solution.     

THIRD-ORDER DIFFERENTIAL SUBORDINATION RESULTS FOR ANALYTIC FUNCTIONS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z(Bcκ+1f(z))′= κBcκf(z)-(κ- 1)Bcκ+1f(z),where b, c, p ∈ C and κ = p +(b + 1)/2 ∈ C \ Z-0(Z-0= {0,-1,-2, ··· }). The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.     

Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007. L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.