Subset Of Space Research Articles

On fuzzy random sets and their mathematical expectations

In this article, we construct an embedding of the space of fuzzy sets into a normed linear space. We then discuss several classes of random sets which take values either in the space of classical subsets or in the space of fuzzy subsets. The mathematical expectations of random sets are discussed. The embedding provided here can also be applied to study the calculus of set-valued functions and fuzzy functions.

On some classes of algebraic surfaces with an infinite set of subspaces of skew symmetry

Special classes of (m − 1)-dimensional algebraic surfaces F n in a space Emwith inifinite set ofsubspaces of skew symmetry (in particular, orthogonal) are studied. It is assumed that directions of symmetry, as a rule, are asymptotic for F n .

Some complete Σ¹₂ sets in harmonic analysis

We prove that several specific pointsets are complete Σ 2 1 \Sigma _2^1 (complete PCA). For example, the class of N 0 {N_0} -sets, which is a hereditary class of thin sets that occurs in harmonic analysis, is a pointset in the space of compact subsets of the unit circle; we prove that this pointset is complete Σ 2 1 \Sigma _2^1 . We also consider some other aspects of descriptive set theory, such as the nonexistence of Borel (and consistently with ZFC {\text {ZFC}} , the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Borel way (and consistently, no measurable way) to choose for each N 0 {N_0} -set, a trigonometric series witnessing that it is an N 0 {N_0} -set.

Chaotic iterations of fuzzy sets

Sufficient conditions are developed for the chaotic behaviour of iterative schemes involving continuous mappings from the metric space (En, D) of fuzzy subsets of Rn into itself. The application of these conditions is then illustrated by means of a simple example of a mapping from (E1,D) into itself.

The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity

We give geometric proofs of recent results of G. Beer [13]: the Young-Fenchel correspondence f → f ∗ f \to {f^*} is bicontinuous on the space of closed proper convex functions on a normed vector space X X endowed with the epidistance topology and the polarity operation is continuous on the space of closed convex subsets of X X with the bounded Hausdorff topology. Our methods are in the spirit of a famous result due to Walkup and Wets [31] about the isometric character of the polarity for closed convex cones. We also prove that the Hausdorff distance associated with the cosmic distance on the space of convex subsets of a normed vector space induces the bounded -Hausdorff topology. This shows a link between Beer’s results and the continuity results of [2].

Category of loop spaces of open subsets in euclidean space

has been studied in several recent papers (e.g. [l, 2, 5, 6, 91) in situations where the potential function V is singular, in the sense that it is defined and C’ only on an open subset U of Euclidean k-space Rk and not on the whole of Rk. If E = WGT2(R, Rk) is the Sobolev space of T-periodic functions from R to Rk and E(U) is the subset of E of functions which map into U, then the topology of E(U) plays a crucial role in these studies. In particular, the knowledge that the Lusternik-Schnirelmann category of E(U) is infinite, when used in conjunction with min-max methods, implies the existence of infinitely many critical points. Another subset E,,(U), which consists of those functions in E(U) which take the origin to a base point in U, plays a dual role. First of all, in problems concerning the existence of geodesics in U from one subset A of U to another B of U (see e.g. [3]), the category of E,(U) plays a role. Secondly, the general inequality cat E(U) 2 cat E,,(U) (see [6]), when U is connected and simply connected, can be applied to show that cat E(U) = cat E,,(U), thus providing a tool for computing cat E(U). The topological analogue of E(U) is the space of free loops AU = C’[S’, U] of continuous maps from the circle S’ to U and the analogue of E,(U) is the space of based loops RU which consists of maps in AU which take a base point in S’ to a base point in U. AU has the same homotopy type as E(U) and S2U has the same homotopy type as E,,(U). Thus, homotopy type invariants (e.g. category) of E(U) and E,(U) may be studied through AU and SZU, respectively. The following classical theorem of Serre [l l] is useful in this regard and is stated in terms of a general space X.

Existence of continuous selections on spaces of subsets in o?an-type topologies

Existence of continuous selections on spaces of subsets in o?an-type topologies

A note on compact sets in spaces of subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

Choice, hull, continuity and fidelity

Continuity and fidelity (i.e., ‘path-independence’) conditions are studied for choices (picking subsets), hulls (picking supersets) and compositions of these. Examples of hulls are topological closure and convex hull, both of which are faithful. Using a continuity theorem of Sertel, a sufficient condition is given for closed convex hull, d, to be both continuous and faithful on the space of compact subsets of a locally convex topological vector space. A sufficient condition is also given for the joint continuity and fidelity of the composition, sd, of a choice, s, and d. In contrast with the Kalai and Megiddo theorem that singleton-valued maps of this form cannot be faithful and at the same time continuous on the space of finite subsets of E n , the conjunction of (upper semi-)continuity and fidelity is shown to be commonplace for choices or maps of the above form (not constrained to be singleton-valued).

Almost sure limit sets of random samples in ℝd

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

Almost sure limit sets of random samples in ℝd

If {Xj , } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

A compactness result for perfect matchings

We show that an infinite set system ( V, E ), for which E is a subspace of the vector space of all finite subsets of V, has a perfect matching if and only if every finite subset of V is covered by a matching of ( V, E ).

Curvature measures and random sets II

In choosing models of stochastic geometry three general problems play a role which are closely connected with each other: In the present paper second order local geometric properties of random subsets of R d are of interest. These properties are described by signed curvature measures in a measure geometric context. The theory of point processes on general spaces (here on the space of subsets with positive reach) provides an appropriate framework for solving construction and measurability problems. Mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties. Ergodic interpretations of the curvature densities are also given. The appendix provides auxiliary results for random signed Radon measures in locally compact separable Hausdorff spaces.

Monotone reducibility and the family of infinite sets

AbstractLet A and B be subsets of the space 2N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2N into 2N such that A = Φ−1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, a ⊂ b implies Φ(a) ⊂ Φ(b). The set A is said to be monotone if a ∈ A and a ⊂ b imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The sets are all reducible to the ( but not ) sets, which are in turn all reducible to the strictly sets, which are all in turn reducible to the strictly sets. In addition, the nontrivial sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly monotone sets which have different monotone degrees. We show that every monotone set is actually positive. We also consider reducibility for subsets of the space of compact subsets of 2N. This leads to the result that the finitely iterated Cantor-Bendixson derivative Dn is a Borel map of class exactly 2n, which answers a question of Kuratowski.

A nonlocally connected continuum $X$ such that $C(X)$ is a retract of $2\sp{X}$

Let $X$ be a metric continuum and let ${2^X}\left ( {C\left ( X \right )} \right )$ denote the hyperspace of closed subsets (subcontinua) of $X$. An example is given of a nonlocally connected continuum $X$ such that $C\left ( X \right )$ is a retract of ${2^X}$.

Normality of the space of closed subsets in Ochan-type topologies. II

Normality of the space of closed subsets in Ochan-type topologies. II

Theory of fuzzy systems

In this paper, a theory of dynamical fuzzy systems is proposed. It is shown that a fuzzy system can be represented by a flow, describing the evolution of this fuzzy system in the space of fuzzy subsets of the state-space. The topological properties of the flow are then established.

Isometries of spaces of compact or compact convex subsets of metric manifolds

The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.

Compact supported endographs and fuzzy sets

A metric is defined on a space of functions from a locally compact metric space X into the unit interval I in terms of the Hausdorff metric distance between their compact supported endographs in X × I. Convergence in this metric is shown to be equivalent to the conjunction of the Hausdorff metric convergence of supports in X and two conditions involving numerical values of the functions. The space of nonempty compact subsets of X with the Hausdorff metric is imbedded in the above function space by the characteristic function on subsets of X. Applications of these results to fuzzy subsets of X and fuzzy dynamical systems on X are indicated.