Related papers: A new definition of random sets
We study from a statistical mechanics viewpoint some of the simplest mathematical objects, finite pure sets. Starting from the empty set, new generations are produced step by step, sets of the next generation being those whose elements are…
In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum…
The classical randomization criterion is an important result of statistical decision theory. Recently, a quantum analogue has been proposed, giving equivalent conditions for two sets of quantum states, ensuring existence of a quantum…
The concepts of similarity and distance are crucial in data mining. We consider the problem of defining the distance between two data sets by comparing summary statistics computed from the data sets. The initial definition of our distance…
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…
This work considers the notion of random tensors and reviews some fundamental concepts in statistics when applied to a tensor based data or signal. In several engineering fields such as Communications, Signal Processing, Machine learning,…
We introduce the concept of boundariness capturing the most efficient way of expressing a given element of a convex set as a probability mixture of its boundary elements. In other words, this number measures (without the need of any…
Thirty-three new definitions are presented, derived from neutrosophic set, neutrosophic probability, neutrosophic statistics, and neutrosophic logic. Each one is independent, short, with references and cross references like in a dictionary…
We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-L\"{o}f (ML) randomness, in the sense that…
We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion…
In wireless networks, the knowledge of nodal distances is essential for several areas such as system configuration, performance analysis and protocol design. In order to evaluate distance distributions in random networks, the underlying…
The probabilistic reachability problems of nondeterministic systems are studied. Based on the existing studies, the definition of probabilistic reachable sets is generalized by taking into account time-varying target set and obstacle. A…
The heat kernel on the symmetric space of positive definite Hermitian matrices is used to endow the spaces of Bergman metrics of degree k on a Riemann surface M with a family of probability measures depending on a choice of the background…
A new number system, the set of the non-Dedekindian numbers, is introduced and characterized axiomatically. It is then proved that any hypercontinous hyperreal number system is strictly included in the set of the Non-Dedekindian Numbers.…
In a series of papers Tsirelson constructed from measure types of random sets and generalised random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying…
In this research a new algebraic semantics of rough set theory including additional meta aspects is proposed. The semantics is based on enhancing the standard rough set theory with notions of 'relative ability of subsets of approximation…
We aim to link random fields and marked point processes and therefore introduce a new class of stochastic processes which are defined on a random set in R^d. Unlike for random fields, the mark covariance function of a marked random set is…
Mathematicians tend to use the phrase "arbitrarily close" to mean something along the lines of "every neighborhood of a point intersects a set". Taking the latter statement as a technical definition for arbitrarily close leads to an…
A new proposal for a Lorentz-invariant spontaneous localization theory is presented. It is based on the choice of a suitable set of macroscopic quantities to be stochastically induced to have definite values. Such macroscopic quantities…