Related papers: Analysis and Probability on Infinite-Dimensional S…
Although some information-theoretic measures of uncertainty or granularity have been proposed in rough set theory, these measures are only dependent on the underlying partition and the cardinality of the universe, independent of the lower…
In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable…
This set of lecture notes constitutes the free textbook project I initiated towards the end of Summer 2015, while preparing for the Fall 2015 Analytical Methods in Physics course I taught to upper level undergraduates at the University of…
``Constants of Nature'' and cosmological parameters may in fact be variables related to some slowly-varying fields. In models of eternal inflation, such fields will take different values in different parts of the universe. Here I show how…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It…
After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the…
The principal goal of this paper is to pass all quantum probability formulas to the projective space associated to the complex Hilbert space of a given quantum system, providing a more complete geometrization of quantum theory. Quantum…
The concept of entropy, firstly introduced in information theory, rapidly became popular in many applied sciences via Shannon's formula to measure the degree of heterogeneity among observations. A rather recent research field aims at…
We present a general infinite volume limit construction of probability measures obeying the Glimm-Jaffe axioms of Euclidean quantum field theory in arbitrary space-time dimension. In particular, we obtain measures that may be interpreted as…
In Banach space theory, the ``local theory'' refers to the collection of finite dimensional methods and ideas which are used to study infinite dimensional spaces (see e.g. [P4,TJ]). It is natural to try to develop an analogous theory in the…
Notes for a Course on Probability and Statistics: L1: Elements of Probability; L2: Bayesian Inference; L3: Monte Carlo Methods
We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by…
In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class $\mathfrak M$ of normed spaces. The method produces, among other, the Gurari\u{\i} space $\mathcal G$ (the…
In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations…
We describe the interface between measure theoretic probability and causal inference by constructing causal models on probability spaces within the potential outcomes framework. We find that measure theory provides a precise and instructive…
These lectures notes give an introduction to the fast developing area of research dealing with perturbative descriptions of the gravitational instability in an expanding universe. I just sketch the outlines of some proofs, and many…
When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics…
Talk at the 25th Johns Hopkins Workshop "2001: A Relativistic Spacetime Odyssey", Firenze September 2001.
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March…