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A brief overview of our current understanding of abundance and properties of dark energy and dark matter is presented. A more focused discussion of supersymmetric dark matter follows. Included is a frequentist approach to the supersymmetric…
For a real or complex one-dimensional map satisfying a weak hyperbolicity assumption, we study the existence and statistical properties of physical measures, with respect to geometric reference measures. We also study geometric properties…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
We provide a concise and accessible introduction to (geometric) string structures, highlighting their connection to loop spaces and outlining relationships with neighboring topics.
In this paper we discuss various philosophical aspects of the hyperstructure concept extending networks and higher categories. By this discussion we hope to pave the way for applications and further developments of the mathematical theory…
Family of replica matrices, related to general ultrametric spaces, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed.
The research field of spatial scientometrics is dedicated to measuring and analyzing science with spatial components (e.g., location, place, mapping). Because of the dynamic nature of this field, researchers from multidisciplinary domains…
Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework…
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…
Many tools and techniques measure local structure in materials in contexts ranging from biology to geology. We provide a survey of those tools and metrics that are especially useful for analyzing particulate soft matter. The metrics we…
Some geometric structures with associated Riemannian metrics have been considered in the book.
This short survey has been prepared in connection with the workshop on discrete metric spaces and their applications at Princeton, August, 2003, and tries to convey some of the ways that one might look at functions on metric spaces in…
Basic issues of the general model-building framework of the mechanics of complex bodies are discussed. Attention is focused on the representation of the material elements, the conditions for the existence of ground states in conservative…
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are…
This article discusses several matters related to Sobolev, Poincare, and isoperimetric inequalities in various settings.
Let (X,d) be a finite metric space. This paper first discusses the spectrum of the p-distance matrix of a finite metric space of p-negative type and then gives upper and lower bounds for the so called gap of a finite metric space of strict…
Here we briefly describe some topics along the lines of projective spaces and related geometric constructions connected to linear algebra, which provide fundamental examples in classical geometry and analysis.
We introduce the concept of protometric and present some properties of protometrics.
Some theoretical and experimental aspects regarding the direct dark matter field are mentioned. In particular some arguments, which play a relevant role in the evaluation of model dependent interpretations of experimental results and in…
Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry…