Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.  

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

In this paper, we study the convergence of the sequence defined by

where and is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

  

The local index formula in noncommutative geometry

In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.We dedicate this paper to Misha Gromov  

Chaotic Attractors with Discrete Planar Symmetries

Chaotic behavior is known to be compatible with symmetry and illustrations are constructed using functions equivariant with respect to the desired symmetries. Earlier investigations determined families of equivariant functions for a few of the discrete symmetry groups in the plane; those results are extended to all the discrete symmetry groups of the plane. This includes consideration of the all the frieze and two-dimensional crystallographic groups. © 1998 Elsevier Science Ltd. All rights reserved.  

A note on the three color problem

It is shown that a planar graph withouti-circuits, 4 i 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.The author's research was partially supported by the Office of Naval Research, Grant number N00014-92-J-1965.  

A strong convergence theorem for relatively nonexpansive mappings in a Banach space

In this paper, we prove a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Using this result, we also discuss the problem of strong convergence concerning nonexpansive mappings in a Hilbert space and maximal monotone operators in a Banach space.  

Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process  

Transportation cost for Gaussian and other product measures

Consider the canonical Gaussian measure _N on , a probability measure on ^N , absolutely continuous with respect to _N . We prove that the transportation cost of to _N , when the cost of transporting a unit of mass fromx toy is measured by x–y², is at most d. As a consequence we obtain a completely elementary proof of a very sharp form of the concentration of measure phenomenon in Gauss space. We then prove a result of the same nature when _N is replaced by the measure of density 2^–N exp (– _iN |x _i |). This yields a sharp form of concentration of measure in that space.Work partially supported by an NSF Grant.  

Martingale transforms and Hardy spaces

Summary Burkholder's martingale transforms are especially useful in studying predictable martingale Hardy spaces. Characterizations of such spaces via martingale transforms are provided. In particular, it is shown that for 0<p<, a martingale inh ^p , defined by the conditioned square function, is the martingale transform of a bmo₂ martingale with a multiplier sequence whose maximal function is inL ^p .