Second Order Nonlinear Evolution Inclusions Ⅱ： Structure of the Solution Set

We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.     

On Normed Lattices and Their Banach Completions

It is known that the Banach completion Y = bX of a normed lattice X need not preserve the properties to be Dedekind complete or σ-Dedekind complete. In this paper it is proved that the countable interpolation property and the property to be sequentially order complete are preserved under the Banach completion. To prove this results we found some sufficient conditions (which are close to necessary ones) on X which secure for Y to have the countable interpolation property and (respectively) to be sequentially order complete. These conditions are obtained with the help of the newly developed techniques based on representations of normed lattices. It is well known that order continuity, and σ-order continuity of a norm are preserved under the Banach completion. Here necessary and sufficient conditions on X to secure these properties in Y are discussed. Mathematics Subject Classification 2000: 46B42, 46E15     

More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

For a sample of iid observations {(X_i, Y_i)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y_[]=(Y_[1], Y_[2], …, Y_[n]) and the stochastic order of subsets of Y_[] are studied. If (X, Y) is totally positive dependent of order 2, Y_[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, h_YX(yx), is decreasing in x for every y, Y_[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order.     

A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgiev's parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D(∂h) is pathwise connected and R(∂h) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R(∂h) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.     

Some subgroups of semilinearly ordered groups

Let G be a semilinearly ordered group with a positive cone P. Denote byn(G) the greatest convex directed normal subgroup of G, byo(G) the greatest convex right-ordered subgroup of G, and byr(G) a set of all elements x of G such that x and x⁻¹ are comparable with any element of P^± (the collection of all group elements comparable with an identity element). Previously. it was proved thatr(G) is a convex right-ordered subgroup of G. andn(G) ⊆r(G) ⊆o(G). Here, we establish a new property ofr(G). and show that the inequalities in the given system of inclusions are, generally, strict. Supported by RFFR grant No. 99-01-00156. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 465–479, July–August, 2000.     

Increasing convex ordering of queue length in bulk queues

This paper considers single-server bulk queues M^(X)/G^(Y)/1 and G^(X)/M^(Y)/1. In the former queue, service times and service capacity are dependent, while in the latter queue, inter-arriving times and arriving group size are dependent. We show that stronger dependence between those leads to shorter queue lengths in the increasing convex ordering sense.     

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

Polynomial normal densities generated by Hermite polynomials

It is known that for a normalN(0, 1) random variable (r.v.) Y₀ the expectation of the Hermite polynomial H_n in Y₀ is equal to zero, i.e.,E[H_n(Y₀)]=0, n≥1. We give examples of other distributions satisfying this condition as well as some characterizations of these distributions. We show that for some subsets of Hermite polynomials the orthogonality measure is not unique. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n–k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.     

Characterizations of counter-monotonicity and upper comonotonicity by (tail) convex order

In this paper, we characterize counter-monotonic and upper comonotonic random vectors by the optimality of the sum of their components in the senses of the convex order and tail convex order respectively. In the first part, we extend the characterization of comonotonicity by  Cheung (2010) and show that the sum of two random variables is minimal with respect to the convex order if and only if they are counter-monotonic. Three simple and illuminating proofs are provided. In the second part, we investigate upper comonotonicity by means of the tail convex order. By establishing some useful properties of this relatively new stochastic order, we prove that an upper comonotonic random vector must give rise to the maximal tail convex sum, thereby completing the gap in  Nam et al. (2011)’s characterization. The relationship between the tail convex order and risk measures along with conditions under which the additivity of risk measures is sufficient for upper comonotonicity is also explored.     

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.     

Oscillation Theorems of the Second Order Linear Matrix Differential System with Damping

By using the Riccati technique and the technique, new oscillation criteria are obtained for the second order matrix differential system(P(t)Y′(t))′ r(t)P(t)Y′(t) Q(t)Y(t) = 0, t≥t0.These results in the present paper generalize and improve many known conclusions. Furthermore, some results are different from the most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. In particular, our results complement a number of existing results and handle the ease that is not covered by the known criteria.     

New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set P(Y)\mathcal{P}(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \preccurlyeq\preccurlyeq: minimal elements, semicompactness, completeness, domination property of a subset of Q\mathcal{Q}, and semicontinuity of a set-valued map with values in Q\mathcal{Q} in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) from which one can easily derive similar results for the case, when F takes values on P(Y)\mathcal{P}(Y) equipped with various order relations.     

Strong Subdifferentiability of Convex Functionals and Proximinality

Using strong subdifferentiability of convex functionals, we give a new sufficient condition for proximinality of closed subspaces of finite codimension in a Banach space. We apply this result to the Banach space K(l₂) of compact operators on l₂ and we show that a finite codimensional subspace Y of K(l₂) is strongly proximinal if and only if every linear form which vanishes on Y attains its norm.     

The strong conical hull intersection property for convex programming

The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space, is a continuous convex function and g:X→Y is a continuous S-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP. The author is grateful to the referees for their constructive comments and valuable suggestions which have contributed to the final preparation of the paper.     

Biconvex sets and optimization with biconvex functions: a survey and extensions

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG).     

Some Remarks on Relative Chebyshev Centers

In this work we study a relative Chebyshev center ofKwith respect toY, whereKis a closed bounded convex subset of a Hilbert spaceX, andYis a closed convex subset ofX. Some results of Amir and Mach [J. Approx. Theory40, (1984), 364–374] are extended.     

Singular Set of a Convex Potential in Two Dimensions

Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C ¹ curves. Or we can say that the singular set is a C ¹ manifold.     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.