Sense Of Baire Category Research Articles

Quelques propriétés génériques en analyse

We consider some classical genericity results in the sense of Baire categories, and we study whether they can be extended to the settings supplied by prevalence and by HP-residual sets. To cite this article: A. Fraysse et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Properties of typical bounded closed convex sets in Hilbert space

For a nonempty separable convex subset X of a Hilbert space ℍ(Ω), it is typical (in the sense of Baire category) that a bounded closed convex set C ⊂ ℍ(Ω) defines an m‐valued metric antiprojection (farthest point mapping) at the points of a dense subset of X, whenever m is a positive integer such that m ≤ dimX + 1.

Baire category and nowhere differentiability for feasible real functions

AbstractA notion of resource‐bounded Baire category is developed for the class PC[0,1] of all polynomial‐time computable real‐valued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resource‐bounded Banach‐Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexity‐theoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Prevalence of backward stochastic differential equations with unique solution

We prove that in the sense of Baire category, almost all backward stochastic differential equations (BSDEs) with bounded and continuous coefficient have the properties of existence and uniqueness of solutions as well as the continuous dependence of solutions on the coefficient and the L2‐convergence of their associated successive approximations.

The Determination of a Convex Set from Its Angle Function

In this paper we discuss the following question: how can we decide whether a convex set is determined by its angle function or not? We give sufficient conditions for convex polygons and for regular convex sets which guarantee that the set is distinguishable. We also investigate the question: which sets are typical (in the sense of Baire category), those which are distinguishable or those which are not? We prove that the family of distinguishable sets is of second Baire category.

A note of stability of weakly efficient solution set for optimization with set‐valued maps

In this paper, we study the stability of weakly efficient solution sets for optimization problems with set‐valued maps. We introduce the concept of essential weakly efficient solutions and essential components of weakly efficient solution sets. We first show that most optimization problems with set‐valued maps (in the sense of Baire category) are stable. Secondly, we obtain some sufficient conditions for the existence of one essential weakly efficient solution or one essential component of the weakly efficient solution set .

Typical Properties of Correlation Dimension

Let (X, ρ) be a complete separable metric space and M be the set of all probability Borel measures on X. We show that if the space M is equipped with the weak topology, the set of measures having the upper (resp. lower) correlation dimension zero is residual. Moreover, the upper correlation dimension of a typical (in the sense of Baire category) measure is estimated by means of the local lower entropy and local upper entropy dimensions of X.

The number of nearest and farthest points to a compactum in Euclidean space

A typical (in the sense of Baire category) compactA inE, whereE is either the Euclidean spaceE 8,s≧2, or the separable Hilbert space ℍ, generates a dense subsetC n,m(A) of the underlying space, such that everyx∈C n,m(A) has exactlyn nearest andm farthest points fromA, whenevern andm are positive integers satisfyingn+m≦ dimE+2.

On the Box Dimension of Typical Measures

Let be a complete metric space and let be the space of all probability Borel measures on X. We give some estimations of the upper and lower box dimensions of the typical (in the sense of Baire category) measure in .

On the “prediction” problem

We prove that almost every (in the Baire category sense) weight w on a circle T satisfies the following property: any function from L 2(w, T) can be decomposed as a series ∑ n∈ Z + c(n) e int which converges in the norm. We discuss this result in the context of the classical Szegö–Kolmogorov “prediction” theorem. To cite this article: A. Olveskii, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 279–282.

A Variational Principle for Problems with Functional Constraints

In this paper we show that in several important classes of optimization problems, like mathematical programming with k-smooth data, quadratic programming in a Hilbert space, convex programming in a Banach space, semi-infinite programming, and optimal control of linear systems with quadratic cost, most of the problems (in the Baire category sense) are well-posed. This is derived from a general variational principle for problems with functional constraints.

On the typical structure of compact sets

Let $\Bbb E$ be a strictly convex separable Banach space of dimension at least 2. A compact set $K\subset \Bbb E$ has hispid structure if the nearest point mapping $p_K^{} : \Bbb E\to 2^K$ is not single valued on a dense subset of $\Bbb E$ . It is proved that if K is a union of disjoint compact sets K 1,K 2 and K has hispid structure, then K 1 and K 2 have hispid structure. Furthermore, it is shown that for typical (in the sense of Baire category) compact set K and arbitrary $x \in \Bbb E$ , the set of all r > 0 such that the intersection $K \cap B(x,r) \ne \emptyset $ and has no hispid structure is of Jordan measure zero.

Box and Packing Dimensions of Typical Compact Sets

Let (X,ρ) be a complete metric space and let dim A be the upper box dimension of the set \(\). We show that packing dimension of the typical (in the sense of Baire category) compact set is at least \(\).

Metric, Fractal Dimensional and Baire Results on the Distribution of Subsequences

Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M(x) of limit measures of x. It is defined as the set of accumulation points of the sequence of the discrete measures induced by x. Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = (xn)n∈ℕ ∈ Xℕ and every a I there corresponds a subsequence denoted by ax. We investigate the set M(ax) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.

A note on a question of R. Pol concerning light maps

Let f :X→Y be an onto map between compact spaces such that all point-inverses of f are zero-dimensional. Let A be the set of all functions u :X→I=[0,1] such that u[f ←(y)] is zero-dimensional for all y∈Y. Do almost all maps u :X→I , in the sense of Baire category, belong to A? Toruńczyk proved that the answer is yes if Y is countable-dimensional. We extend this result to the case when Y has property C.

On typical Markov operators acting on Borel measures

It is proved that, in the sense of Baire category, almost every Markov operator acting on Borel measures is asymptotically stable and the Hausdorff dimension of its invariant measure is equal to zero.

HÖLDER SPECTRUM OF TYPICAL MONOTONE CONTINUOUS FUNCTIONS

We compute the Holder spectrum and the moment sums of the typ- ical (in the sense of Baire category) monotone continuous function, f. We show that these functions are of multifractal nature, for example equals the Hausdor dimension of those points where such a function, f is of exact Holder class 2 (0; 1): We also prove that f \grows only at those points where it has bad Holder properties, that is, at those points where it is of Holder class 0: The upper moment sum f (q) =1 ifq < 0; f (q) = 1 q if 0 q 1; and f (q) = 0 if 1 < q: The lower moment sum f (q) = 1 q if q < 0; f (q) = 0 if 0 q 1; and f (q) = 1 q if 1 < q:

On the completeness of multifractal spectra

We study the behavior of multifractal spectra on the boundary of their domains of definition. In particular, we show that the dimension of the set of points having minimal (or maximal) pointwise dimension is not necessarily zero. However, this situation may be neglected in the sense of Baire category for Gibbs measures on expanding conformal repellers.

Essential equilibria of n-person noncooperative games

In this paper, we prove that most of the games (in the sense of Baire Category) are essential or weakly essential for the following three models: (A) the games parametrized by payoffs; (B) games parametrized by payoffs and strategy spaces; and (C) games parametrized by payoffs and feasible strategy correspondences.

Cardinality of the metric projection on typical compact sets in Hilbert spaces

The metric projection mapping πX plays an important role in nonlinear approximation theory. Usually X is a closed subset of a Banach space [ ] and, for each e∈[ ], πX(e) is the set, perhaps empty, of all points in X which are nearest to e. From a classical theorem due to Stečkin [7] it is known that, when [ ] is uniformly convex, the metric projection πX(e) is single valued at each typical point e of [ ] (in the sense of the Baire categories), i.e. at each point e of a residual subset of [ ]. More recently Zamfirescu [8] has proven that, if X is a typical compact set in ℝn (in the sense of Baire categories) and n[ges ]2, then the metric projection πX(e) has cardinality at least 2 at each point e of a dense subset of ℝn. This result has been extended in several directions by Zhivkov [9, 10], who has also considered the case of the metric antiprojection mapping νX (which associates with each e∈[ ] the set νX(e), perhaps empty, of all ∈X which are farthest from e). For this mapping De Blasi [2] has shown that, if [ ] is a real separable Hilbert space with dim[ ]=+∞ and n is an arbitrary natural number not less than 2, then, for a typical compact convex set X⊂[ ], the metric antiprojection νX(e) has cardinality at least n at each point e of a dense subset of [ ]. A systematic discussion of the properties of the maps πX and νX, and additional bibliography, can be found in Singer [5, 6] and Dontchev and Zolezzi [3].In the present paper we consider some further properties of the metric projection mapping πX, with X a compact set in a real separable Hilbert space [ ]. If dim[ ]=n and 2[ges ]n&lt;+∞, it is proven that for a typical compact set X⊂[ ], the metric projection πX(e) has cardinality exactly n+1 at each point e of a dense subset of [ ], while the set of those points e∈[ ] where πX(e) has cardinality at least n+2 is empty. Furthermore it is shown that, if dim[ ]=+∞, then for a typical compact set X⊂[ ] the metric projection πX(e) has cardinality at least n (for arbitrary n[ges ]2) at each point e of a dense subset of [ ]. Incidentally we obtain a characterization of the dimension of the space [ ] by means of a typical property holding in the space of the compact subsets of [ ].