%e report the f~rst experimental verification of Berry's topological phase. The key element in the experiment was a single-mode, helica11y wound optica1 fiber, inside which a photon of a given helicity could be adiabatically transported around a closed path in momentum space. The experiment confirmed at the classical level that the angle of rotation of linearly polarized light in this fiber giv...

In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...

We develop a theory for sensitivity with respect to parameters in a convex subset of a topological vector space of dynamical systems in a Banach space. Specific motivating examples for probability measure dependent differential, partial differential and delay differential equations are given. Schemes that approximate the measures in the Prohorov sense are illustrated with numerical simulations ...

The present paper is a continuation of the work, carried out in [4] and [5] of investigating the relationship between a Gaussian process and its reproducing kernel Hilbert subspace. Our main result gives a characterization of the topological support of a Gaussian measure defined on a linear topological space of functions on an arbitrary set. As special cases we consider Gaussian processes on Ba...

Let K be a compact Hausdorff space and let )~ be a probability measure on K. We denote by Lo(K,2) the space of all Borel functions f:K--*N with the topology of convergence in measure. Lo(K, 2) is an F-space (complete metric topological vector space) if, as usual, we identify functions equal almost everywhere. Spaces of the type Lo(K,2 ) are probably the most studied examples of non-locally conv...

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumption...

It is shown that in an infinite-dimensional dually separated second category topological vector space X there does not exist a probability measure μ for which the kernel coincides with X. Moreover, we show that in “good” cases the kernel has the full measure if and only if it is finitedimensional. Also, the problem posed by S. Chevet [5, p. 69] is solved by proving that the annihilator of the k...

The conventional definition of a topological metric over a space specifies properties that must be obeyed by any measure of “how separated” two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measu...

in this paper we give conditions for a tuple of commutative bounded linear operators which holds in the property of the hypercyclicity criterion. we characterize topological transitivity and semi-hereiditarily of a dynamical system given  by an n-tuple of operators acting on a separable infinite dimensional banach space .

In many decisions under uncertainty, technological constraints restrict both the acts which an agent can perform and the events which she can observe. To address such situations, we assume that the set of possible states of the world and the set of possible outcomes each have a topological structure. The only feasible acts are continuous functions from states to outcomes, and the only observabl...