Sompong Dhompongsa

Fixed Points for Multivalued Mappings and the Metric Completeness

We consider the equivalence of the existence of fixed points of single-valued mappings and multivalued mappings for some classes of mappings by proving some equivalence theorems for the completeness of metric spaces.

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that J(X)<(1+5)/2, improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.

Fixed Point Theorems by Ways of Ultra-Asymptotic Centers

We use an approach on ultra-asymptotic centers to obtain fixed point theorems for two classes of nonself multivalued mappings. The results extend and improve several known ones.

The Fixed Point Property of Unital Abelian Banach Algebras

We give a general condition for infinite dimensional unital Abelian Banach algebras to fail the fixed point property. Examples of those algebras are given including the algebras of continuous functions on compact sets.

A measure of multivariate mutual complete dependence

Abstract The authors propose a multivariate version of Siburg and Stoimenov’s measure of mutual complete dependence. This multivariate version is, however, not the distance between a copula and the product copula C I under the modified Sobolev norm since the set of mutual complete dependence copulas does not lie on the sphere centered at C I . To overcome this difficulty, the authors choose another center and define measures of complete dependence based on the modified Sobolev norm and this center. The measure of multivariate mutual complete dependence is then defined as the summation of the (normalized) measures of complete dependence.

On topological properties of the Choquet weak convergence of capacity functionals of random sets

In view of the recent interests in random sets in information technology, such as models for imprecise data in intelligent systems, morphological analysis in image processing, we present, in this paper, some contributions to the foundation of random set theory, namely, a complete study of topological properties of capacity functionals of random sets, generalizing weak convergence of probability measures. These results are useful for investigating the concept of Choquet weak convergence of capacity functionals leading to tractable criteria for convergence in distribution of random sets. The weak topology is defined on the space of all capacity functionals on R^d. We show that this topological space is separable and metrizable.

Repeated-root constacyclic codes of prime power length over F p m u { u a } and their duals

The units of the chain ring R a = F p m u { u a } = F p m + u F p m + ? + u a - 1 F p m are partitioned into a distinct types. It is shown that for any unit ? of Type k , a unit λ of Type k ? can be constructed, such that the class of λ -constacyclic of length p s of Type k ? codes is one-to-one correspondent to the class of ? -constacyclic codes of the same length of Type k via a ring isomorphism. The units of R a of the form ? = ? 0 + u ? 1 + ? + u a - 1 ? a - 1 , where ? 0 , ? 1 , ? , ? a - 1 ? F p m , ? 0 ? 0 , ? 1 ? 0 , are considered in detail. The structure, duals, Hamming and homogeneous distances of ? -constacyclic codes of length p s over R a are established. It is shown that self-dual ? -constacyclic codes of length p s over R a exist if and only if a is even, and in such case, it is unique. Among other results, we discuss some conditions when a code is both α - and β -constacyclic over R a for different units α , β .

Rodé's theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces

We extend Rodés theorem on common fixed points of semigroups of nonexpansive mappings in Hilbert spaces to the CAT(0) space setting.2000 Mathematics Subject Classification: 47H09; 47H10.

Diametrically Contractive Multivalued Mappings

Diametrically contractive mappings on a complete metric space are introduced by V. I. Istratescu. We extend and generalize this idea to multivalued mappings. An easy example shows that our fixed point theorem is more applicable than a former one obtained by H. K. Xu. A convergence theorem of Picard iteratives is also provided for multivalued mappings on hyperconvex spaces, thereby extending a Proinovs result.

Measures of the functional dependence of random vectors

In this work, we define a set of properties that any measure of functional dependence that exists between random vectors should possess. We also construct measures of functional dependence and show that they satisfy the properties mentioned above. Relationships between these measures and previously defined measures of functional dependence between random variables are discussed. The measure of functional dependence for random vectors is proposed.Calculated examples related to this measure are provided.Some basic properties of this measure are proved and discussed.