Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

We prove a generalization of Debreu’s Open Gap Lemma. Given any subset of the real line, this lemma guarantees the existence of a strictly increasing real function such that all the gaps of the image of the subset are open. Now we extend it to the case of n subsets of the real line and study the existence of a strictly increasing real function such that all the gaps of the image of each set are open. This function does not exist in general so, we characterize the cases in which it exists. This generalization is not equivalent to Debreu’s lemma working on the union of the n subsets, neither on the intersection. Then we use it in order to obtain new results on continuous representations of biorders. We also generalize the concept of biorder.     

Continuity at the extreme points

We prove that a bounded convex lower semicontinuous function defined on a convex compact set K is continuous at a dense subset of extreme points. If there is a bounded strictly convex lower semicontinuous function on K, then the set of extreme points contains a dense completely metrizable subset.     

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its ε-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set $\mathcal{C}(X)$ of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded.     

Maximizing Strictly Convex Quadratic Functions with Bounded Perturbations

The problem of maximizing [(f)\tilde]=f+p\tilde{f}=f+p over some convex subset D of the n-dimensional Euclidean space is investigated, where f is a strictly convex quadratic function and p is assumed to be bounded by some s∈[0,+∞[. The location of global maximal solutions of [(f)\tilde]\tilde{f} on D is derived from the roughly generalized convexity of [(f)\tilde]\tilde{f}. The distance between global (or local) maximal solutions of [(f)\tilde]\tilde{f} on D and global (or local, respectively) maximal solutions of f on D is estimated. As consequence, the set of global (or local) maximal solutions of [(f)\tilde]\tilde{f} on D is upper (or lower, respectively) semicontinuous when the upper bound s tends to zero.     

A lattice-theoretic approach to arbitrary real functions on frames

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F?(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀-spaces.     

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.      

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.     

Separating sets by lower and upper semicontinuous functions with a closed graph

In this paper we characterize the pairs (A^?, A⁺) of disjoint subsets of perfectly normal topological space which can be separated by a lower and an upper semicontinuous function with a closed graph.     

Strict feasibility of pseudomonotone set-valued variational inequalities

In this article, we use degree theory developed in Kien et al. [B.T. Kien, M.-M. Wong, N.C. Wong, and J.C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res. 193 (2009), pp. 12–22.] to prove a result on the existence of solutions to set-valued variational inequality under a weak coercivity condition, provided that the set-valued mapping is upper semicontinuous with nonempty compact convex values. If the set-valued mapping is pseudomonotone in the sense of Karamardian and upper semicontinuous with nonempty compact convex values, it is shown that the set-valued variational inequality is strictly feasible if and only if its solution set is nonempty and bounded.     

On a Fixed Point Theorem of Ky Fan

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function ƒ:D→E, in several directions: (1) for noncompact sets D, when ƒ(D) precompact; (2) for compact D and upper semicontinuous multifunction ƒ and more generally, (3) for noncompact D and upper semicontinuous multifunction ƒ with ƒ(D) Hausdorff precompact. In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact. Received May 23, 2000, Accepted September 4, 2001     

A Nonstandard Delta Function in a Predicative Theory

In [1] Todorov has shown by means of axiomatic set theory that there exists a nonstandard function Δ: *?ⁿ → * ? such that for all continuous functions φ: ?ⁿ → ?, . Here *? and *? are the set of the nonstandard real numbers and the set of the nonstandard complex numbers, respectively, and *φ: *?ⁿ → *? is the nonstandard extension of φ In the present note we want to prove an analogous theorem by predicative means only.     

Mixed Semicontinuous Mappings and Their Applications to Differential Inclusions

A class of multivalued mappings, called by us mixed semicontinuous mappings, which are upper semicontinuous in some points and lower semicontinuous in remaining points is considered. A general selection theorem for mixed semicontinuous maps is proved. Then applications to differential inclusions with mixed semicontinuous right-hand sides are presented. Note that our applications generalize earlier existence results obtained both on the whole Euclidean space R ⁿ and on a compact proximate retract of R ⁿ .     

Mixed Optimal Stopping and Stochastic Control Problems with Semicontinuous Final Reward for Diffusion Processes

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.     

On Karamardian's theorem about lower semicontinuous strictly quasiconvex functions

Summary A well-known theorem ofKaramardian [1967] says: A lower semicontinuous and strictly quasiconvex real-valued function defined on a convex subset ofR ⁿ is quasiconvex. A proof is given for the following generalization: A connected quasiorder valued function, defined on a convex subset of a normed real linear space, which is lower semicontinuous and strictly midpointquasiconvex is actually both quasiconvex and strictly quasiconvex. A number of similar results are proved interrelating the concepts of midpoint-quasiconvexity, rational quasiconvexity, quasiconvexity and their strict counterparts.

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Zusammenfassung Ein bekannter Satz vonKaramardian [1967] lautet: Eine strikt quasikonvexe und nach unten halbstetige reellwertige Funktion auf einer konvexen Teilmenge desR ⁿ ist quasikonvex. Es wird folgende Verallgemeinerung bewiesen: Eine strikt mittelpunktsquasikonvexe und nach unten halbstetige konnexquasiordnungswertige Funktion auf einer konvexen Teilmenge eines normierten reellen linearen Raumes ist quasikonvex und strikt quasikonvex. Ferner wird eine Anzahl ähnlicher Aussagen bewiesen, die die Begriffe Mittelpunktsqausikonvexität, rationale Quasikonvexität und Quasikonvexität sowie deren strikte Entsprechungen miteinander in Beziehung setzen.

     

Qualitative Properties of the Solution Set for Time-Delayed Discontinuous Dynamics

This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness and connectedness of the solution and reachable sets in the lower semicontinuous case, regularity of the solution and reachable mappings with respect to parameters, and existence of solutions for a dynamic optimization problem.

     

Controlling Orthogonal Series with Irrational Rotations

We compute covering numbers associated to the set of partial sums of orthogonal series by means of irrational rotations. The device is a new metric inequality linking the increment's norm of partial sums to the one of ergodic averages of rotations acting on a suitable L ²-element of the torus. This allows to compute the number of balls covering the whole set of partial sums, by means of rotations.     

锥拟凸集值映射多目标优化的解集的连通性

在一般拓扑向量空间中,本文引进了一类锥进凸（锥严格拟凸）集值映射。并在目标映射是锥拟凸（锥严格拟凸）和上半连续的条件下,利用Minkowski泛函,证明了弱有效解（有效解）集是连通的。文[9]中的结论是本文所得结果大目标映射为单值和拓扑空间为格的情况下的特例。     

Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

LetH be a set of positive real numbers. We define the geometric graphG _H as follows: the vertex set isR ⁿ (or the unit circleS ¹) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets. In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.     

Fixed point results for multimaps in CAT(0) spaces

Common fixed point results for families of single-valued nonexpansive or quasi-nonexpansive mappings and multivalued upper semicontinuous, almost lower semicontinuous or nonexpansive mappings are proved either in CAT(0) spaces or R-trees. It is also shown that the fixed point set of quasi-nonexpansive self-mapping of a nonempty closed convex subset of a CAT(0) space is always nonempty closed and convex.     

Percolation clusters in hyperbolic tessellations

It is known that, for site percolation on the Cayley graph of a co-compact Fuchsian group of genus , infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set of limit points at of an infinite cluster is a perfect, nowhere dense set of Lebesgue measure 0. In this paper, a variational formula for the Hausdorff dimension is proved, and used to deduce that is a continuous, strictly increasing function of p that converges to 0 and 1 at the lower and upper boundaries, respectively, of the coexistence phase. Submitted: July 2000.