Schaefer’s fixed point theorem is used to study the controllability in an infinite delay system x′(t)=G(t,xt) + (Bu)(t). A compact map or homotopy is constructed enabling us to show that if there is an a priori bound on all possible solutions of the companion control system x′(t) = λ[G(t,xt) + (Bu)(t)], 0 < λ < 1, then there exists a solution for λ = 1. The a priori bound is established by mean...

In this paper, we study a class of Banach spaces, called φspaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.

We shall be concerned with Lα → Lqβ mapping properties of operators in I(X,Y ; C′) (here Lqβ denotes the L q Sobolev space). These are well known in case that C is locally the graph of a canonical transformation; this means that the projections πL : C → T X, πR : C → T Y are locally diffeomorphisms. In particular dX = dY := d. Then F ∈ I(X,Y, C′) maps Lα,comp(Y ) into Lβ,loc(X) if β ≤ α−μ. This...

In this paper, we study a class of Banach spaces, called φ-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.

Standard cosmology is based on the assumption that the universe is spatially homogeneous. However the consensus on a homogeneous matter structure, even on very large scales, has never been complete. The advantage of correlation dimension calculations is that they enable one to obtain the transition scale to a homogeneous distribution, whereas other methods, such as those using the two-point cor...

We identify a deformation of the N = 2 supersym-metric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkähler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sig...

A linear and continuous operator between Banach spaces is said to be absolutely summing if it maps unconditionally convergent series into absolutely convergent series. Moreover, it improves properties of stochastic processes. Indeed, N.Ghoussoub in [7] proved that an operator is absolutely summing if and only if it maps amarts (asymptotic martingales) into uniform amarts. In this paper we go a ...

This paper shows how the invariance of the arc-sine distribution on (0, 1) under a family of rational maps is related on the one hand to various integral identities with probabilistic interpretations involving random variables derived from Brownian motion with arc-sine, Gaussian, Cauchy and other distributions, and on the other hand to results in the analytic theory of iterated rational maps.

We present a string theory that reproduces the large-N expansion of two dimensional YangMills gauge theory on arbitrary surfaces. First, a new class of topological sigma models is introduced, with path integrals localized to the moduli space of harmonic maps. The Lagrangian of these harmonic topological sigma models is of fourth order in worldsheet derivatives. Then we gauge worldsheet diffeomo...