Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω() for the skew-product maps of the unit square , (x,y)→(f(x),g(x,y)), with base f having closed set of periodic points. For every and every point (x,y) with x periodic of period p_x by f and y not chain recurrent of F^p_x|_Ix, where , we prove that (x,y)Ω(F). On the other hand we construct a map with an isolated fixed point x₀ of f and y₀Ω(F|_Ix0) such that (x₀,y₀)Ω(F₀).     

Density of the set of positive proper minimal points in the set of minimal points

One of the important problems of vector optimization concerns the density of the set of positive proper minimal points in the set of minimal points. We use the concepts of dentable point and approximating cones to derive sufficient conditions guaranteeing that the set of minimal points is contained in the closure of the set of positive proper minimal points. The result can be applied to obtain a density result for the unit ball in 1 ^p , 1<p<+, which does not follow from any other well-known density theorem.The author would like to thank Professor W. T. Fu for helpful comments. Moreover, the author is grateful to Professor H. P. Benson and the referees for valuable remarks and suggestions concerning a previous draft of this paper.     

Peak set without peak points

We give an example of a natural Banach function algebra on the unit disc such that a smaller disc is a peak set for the algebra, but it does not contain any peak point.

     

Families of special Weierstrass points

The purpose of this Note is to show that loci of (special) Weierstrass points on the fibers of a family $\pi :\mathfrak{X}\to S$ of smooth curves of genus $g?2$ can be studied by simply pulling back the Schubert calculus naturally living on a suitable Grassmann bundle over $\mathfrak{X}$. Using such an idea we prove new results regarding the decomposition in $A_1(\mathfrak{X})$ of the class of the locus of Weierstrass points having weight at least 3 as the sum of classes of Weierstrass points having “bounded from below” gaps sequences. To cite this article: L. Gatto, P. Salehyan, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

The extreme points of the set of decreasing failure rate distributions

Summary Since the class of extended decreasing failure rate (EDFR) life distributions (i.e., distributions with support in [0, ]) is compact and convex, it follows from Choquet's Theorem that every EDFR life distribution can be represented as a mixture of extreme points of the EDFR class. We identify the extreme points of this class and of the standard class of decresing failure rate (DFR) life distributions. Further, we show that even though the convex class of DFR life distributions is not compact, every DFR life distribution can be represented as a mixture of extreme points of the DFR class.Research sponsored by the Air Force Office of Scientific Research, AFSC, USAF, under Grant AFOSR 78-3678.Research sponsored by the National Science Foundation MCS-7904698.     

Distinct distances between a collinear set and an arbitrary set of points

We consider the number of distinct distances between two finite sets of points in ${\mathbb{R}}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O\left(1\right)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then

$\Omega \left(min\left\{n^{2∕3}{m}^{2∕3},\frac{n^{10∕11}{m}^{4∕11}}{{log}^{2∕11}m},n^2,m^2\right\}\right).$

Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.     

The essential components of the set of equilibrium points for set-valued maps

In this article, we discuss the stability of equilibrium points for set-valued maps. We introduce the notion of essential components of the set of equilibrium points for set-valued maps and prove an existence theorem of essential components. As applications, we deduce the existence of essential components of the set of coincidence points and show that every Marchaud dynamical economy (cf. [J.P. Aubin, Dynamic Economic Theory, Springer-Verlag, 1997]) possesses at least one essential component of the set of its viable equilibrium points.     

The topological space of all extreme points of a compact convex set

G. Choquet and R. Haydon have proved that every Polish (i.e. completely metrizable and separable) spaceX is homeomorphic to the space of all extreme points of a metrizable compact convex setK which can be chosen to be even a Choquet simplex. In our paper we generalize partially this result to the case of a complete metric spaceX of zero covering dimension, without requiring any separability property forX.     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

The strip of minimum width covering a centrally symmetric set of points

In this paper the following is proved: let be a centrally symmetric set of points, such that the distance between any pair of points is at least 1 and every three of them can be covered by a strip of width 1. Then there is a strip of width √2 covering . Supported by CONACYT, SNI 38848     

Extreme points of the set of Banach limits

Let ${\mathbb{N}}$ be the set of all natural numbers and ${\ell_\infty=\ell_\infty (\mathbb{N})}$ be the Banach space of all bounded sequences x = (x ₁, x ₂ . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let ${\ell_\infty^*}$ be its Banach dual. Let ${\mathfrak{B} \subset \ell_\infty^*}$ be the set of all normalised positive translation invariant functionals (Banach limits) on ? _∞ and let ${ext(\mathfrak{B})}$ be the set of all extreme points of ${\mathfrak{B}}$ . We prove that an arbitrary sequence (B _j ) _{j ≥ 1}, of distinct points from the set ${ext(\mathfrak{B})}$ is 1-equivalent to the unit vector basis of the space ? ₁ of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of ${ext(\mathfrak{B})}$ does not contain a Cesáro-invariant Banach limit.     

On the choice of special Pareto points

Abstract

We propose two strategies for choosing Pareto solutions of constrained multiobjective optimization problems. The first one, for general problems, furnishes balanced optima, i.e. feasible points that, in some sense, have the closest image to the vector whose coordinates are the objective components infima. It consists of solving a single scalar-valued problem, whose objective requires the use of a monotonic function which can be chosen within a large class of functions. The second one, for practical problems for which there is a preference among the objective’s components to be minimized, gives us points that satisfy this order criterion. The procedure requires the sequential minimization of all these functions. We also study other special Pareto solutions, the sub-balanced points, which are a generalization of the balanced optima.