Contractibility of Hyperspaces C_k(X)

This paper proves the following results: let X be a continuum, let k, m ∈ N, and let B ∈ C m (X), consider the continuous surjection f k : C k (X) → C k (X). We define the mapping B : C k (X) → C k+m (X): by B (A) = f k (A) B. Then following assertions are equivalent: (1) The hyperspace C k (X) is g-contractible; (2) For each m ∈ N and for each B ∈ C m (X) the mapping B is a W -deformation in C k+m (X); (3) For each m ∈ N there exists B ∈ C m (X) such that the mapping B is a W -deformation in C k+m (X); (4) There exists m ∈ N such that for each B ∈ C m (X) the mapping B is a W -deformation in C k+m (X); (5) There exist m ∈ N and B ∈ C m (X) such that the mapping B is a W -deformation in C k+m (X).

On the Behaviors of Hyperspace Topologies for Subspaces

For a topological space X we denote by CL(X) the collection of all nonempty closed subsets of X. Suppose we have a map T which assigns in some coherent way to every topological space X some topology T(X) on CL(X). In this paper we study continuity and inverse continuity of the map iA,X :(CL(A),T{A)) → (CL(X),T(X)) defined by iA,x(F) = F whenever F ∈CL(A), for various assignment T; in particular, for locally finite topology, upper Kuratowski topology, and Attouch-Wets topology, etc.

Agent-Oriented Architecture for Ubiquitous Computing in Smart Hyperspace

Agent-oriented approach is increasingly showing its magic power in a diversity of fields, specifically, ubiquitous computing and smart environment. Meanwhile, it is considered the next creative issue is to interconnect and integrate isolated smart spaces in real world together into a higher level space known as a hyperspace. In this paper, an agent-oriented architecture, which involves the techniques of mobile agents, middleware, and embedded artificial intelligence, is proposed. Detailed implementations describe our efforts on the design of terminal device, user interface, agents, and AI

Making Holes in the Hyperspace of Subcontinua of Some Continua

Let be a metric continuum. Let , is said to make a hole in , if is not unico-herent. In this paper, we characterize elements such that makes a hole in , where is either a smooth fan or an Elsa continuum.

The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification

Let L be a continuous semilattice. We use USC(X, L) to denote the family of all lower closed sets including X × {0} in the product space X × AL and ↓1 C(X,L) the one of the regions below of all continuous maps from X to AL. USC(X, L) with the Vietoris topology is a topological space and ↓C(X, L) is its subspace. It will be proved that, if X is an infinite locally connected compactum and AL is an AR, then USC(X, L) is homeomorphic to [-1,1]ω. Furthermore, if L is the product of countably many intervals, then ↓ C(X, L) is homotopy dense in USC(X,L), that is, there exists a homotopy h : USC(X,L) × [0,1] →USC(X,L) such that h0 = idUSC(X,L) and ht(USC(X,L)) C↓C(X,L) for any t > 0. But ↓C(X, L) is not completely metrizable.

The Hyperspace of the Regions below Continuous Maps with the Fell Topology

For a Tychonoff space X, we use ↓USCF（X） and↓CF（X） to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0, 1] with the subspace topologies of the hyperspace Cldf（X × I） consisting of all non-empty closed sets in X × I endowed with the Fell topology. In this paper, we shall show that there exists a homeomorphism h： ↓USCF（X） → Q = [-1, 1]^∞ such that h（↓ CF（X）） ： co ： {（Xn） E Q ｜ limn→ ∞ xn = O} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.

Theory of Selection Operators on Hyperspaces and Multivalued Stochastic Processes

In this paper, a new concept of selection operators on hyperspaces (subsets spaces) is introduced, and the existence theorems for several kinds of selection operators are proved. Using the methods of selection operators, we give a selection characterization of identically distributed multivalued random variables and completely solve the vector-valued selection problem for sequences of multivalued random variables converging in distribution. The regular selections and Markov selections for multivalued stochastic processes are studied, and a discretization theorem for multivalued Markov processes is established. A theorem on the asymptotic martingale selections for compact and convex multivalued asymptotic martingale is proved.

ITERATION OF FIXED POINTS ON HYPERSPACES

LetXbe a compact,convex subset of a Banach spaceEandCC(X)be the collection of all non empty compact,coonvex subset ofXequipped with the Hausdorff metrich.Supposeκis a compact,convex subset ofCC(X)and T:(κ,h)(κ,h)is a nonexpansive mapping.Then for anyA 0∈κ,the sequence{A n}defined byA n+1=(A n+TA n)/2converges to a fixed point ofT.The special case thatκconsists of singletons only yields results previously obtained by H.Schaefer,M.Edelstein and S.Ishikawa respectively.

Continuous selections for pseudo-lower semicontinuous set-valued mappings on convex structure spaces and hyperspace contractibility

SINCE 1956, Michael’s continuous selection theory has been applied to functional analysis,topology, approximation theory and other mathematical fields. In this letter, the concept ofthe pseudo-lower semicontinuity is introduced, and a convex structure of metric space is de-fined. A continuous selection theorem for pseudo-lower semicontinuity is given. This

SEMI-OPENNESSAND ALMOST-OPENNESS OF INDUCED MAPPINGS

Given a mapping f between continua.Let 2f and C(f) mean the induced mappings between hyperspaces.Relations are studied under the conditions:f is semi-open(almost open,respectively),2f is semi-open(almost open,respectively) and C(f) is semi-open(almost open,respectively).

The New Method for CNC Trajectory Machining Motion Planning Based on Reference Time

In order to make motion planning fitting practice,many characteristic of CNC trajectory motion are discussed, such as the geometric function,the motion and the time.It is found that the relation between orbit function and motional parame- ter,so the differential equation about the trajectory motion be set-up by the goal of trajectory motion.The actual motion process is defined as reference time to link planning and practice.Present a new movement planning method based on self-defining time.At rest state,the differential simultaneous equation can be calculated according geometric characteristic analysis,it can be get that simple function consisted of coordinate and reference time variants.At motive state,dynamic parameter can be worked out accord- ing practical value of reference time,It is proved by experiment and simulation that it is a good way to control geometry and motion comprehensively,to reduce computation times and to increase the ability of environmental adaptation for path

The Schrödinger Equation of the Hilbert Hyper Space

Quantum measurement requires an observer to prepare a specific macroscopic measuring device from various options. In previous papers we redefined this observer role through a new concept: the observer determination, that is, the observer’s unique selection between the various measurement-devices. Unlike the measurement itself that is rationalized as dictated by nature, we presented the observer determination as a selection that cannot be disputed since it can neither be measured nor proven to be true or false. In general, we suggest that every action or decision made by the observer is eventually an output of some measurement. The apparently contradiction between the observer free determination and the deterministic measurement output was solved by extending the Hilbert space into a Hyper Hilbert space that is a space with hierarchy. In that frame the so called free selection of the observer determination in a certain level turns out to be a deterministic measurement output in the next higher level of the hierarchy. An important role of the conventional Hilbert space is played by the Schr?dinger equation. It determines a basis of stationary states. In this paper we define the Schr?dinger equation that corresponds with the various levels and we show that each level can be characterized by a unique time scale. We also show how various levels can be synchronized. We believe that this hyperspace level represents a certain level in the physics of consciousness and therefore a level unique time scale can contribute to the time perception of the mind.

The Multi-sensitivity and Topological Sequence Entropy of Dynamical System with Group Action

In this paper,we study the relationship between the multi-sensitivity and the topological maximal sequence entropy of dynamical systems for general group action.Furthermore,we also discuss the consistency of multi-sensitivity of a dynamical system(G■X)and its hyperspace dynamical system G■K(X).Moreover,we research the relationship between the multi-sensitivity of two dynamical systems and the multi-sensitivity of their product space dynamical system.Finally,we prove that if the topological sequence entropy of G■X vanishes,then so does that of its induced system G■M(X);if the topological sequence entropy of G■X is positive,then that of its induced system G■M(X)is infinity.