This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vector valued Gaussian measures with the Radon-Nikodym property. Intended crystallographic beneets in...

The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing l(N). Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing l(N) and not being itself a W.C.G. space. INTRODUCTION In the present paper we deal wit...

The following paper develops the basics behind stochastic calculus, which extends the theory of integration to stochastic (random) processes. The paper starts off with developing Brownian motion and its properties, which are used to develop the theory behind Itô integration. Several forms of the Itô integral are presented. A brief overview of the Radon-Nikodym and Girsanov Theorems are presente...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

This paper considers delta estimators of the Radon-Nikodym derivative of a probability function with respect to a a-finite measure. We provide sufficient conditions for universal consistency, which are checked for some wide classes of non-parametric estimators. Let P be a probability measure in the Borel space (]l~d,~d), absolutely continuous with respect to the a-finite measure # and f = dP/d#...

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of ...

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that l-valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Ni...

In this paper, I provide the basis for a measureand integral-theoretic formulation of possibility theory. It is shown that, using a general definition of possibility measures, and a generalization of Sugeno’s fuzzy integral – the seminormed fuzzy integral, or possibility integral –, a unified and consistent account can be given of many of the possibilistic results extant in the literature. The ...

As particular case of the Radon-Nikodym theorem we show that any count distribution is a weighted Poisson distribution (WPD) and, more generally, a weighted version of any other count distribution with suitable support. Several properties and consequences of this result are then given. With a condition on their supports, we deduce that two count distributions are connected by the weightening op...

In this note we described the robustness properties of optimal risksensitive controllers for quantum systems. We consider a quantum generalization of risk-sensitive criteria using the framework of (James, 2004). The robustness properties are derived by evaluating certain Radon-Nikodym derivatives of the quantum models and of the cost criteria. In addition to induced perturbations in the quantum...