Related papers: Na\"ive Infinitesimal Analysis: Its Construction a…
The vertices of a finite state system are usually a subset of the natural numbers. Most algorithms relative to these systems only use this fact to select vertices. For infinite state systems, however, the situation is different: in…
From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities…
We describe a new, generally applicable strategy for the systematic construction of basis invariants (BIs). Our method allows one to count the number of mutually independent BIs and gives controlled access to the interrelations (syzygies)…
When dealing with certain kind of complex phenomena the theoretician may face some difficulties -- typically a failure to have access to information for properly characterize the system -- for applying the full power of the standard…
Using methods developed by Robinson, we find a complete theory suitable for a first order description of infintesimal neighborhoods. We use this to construct a specialisation having universal properties and to find a recursively enumerable…
This paper aims to present objective methods for constructing new fuzzy sets from known fuzzy or classical sets, defined over the elements of a finite universe's superstructure. The paper proposes rules for assigning membership functions to…
It has been observed that representations learned by distinct neural networks conceal structural similarities when the models are trained under similar inductive biases. From a geometric perspective, identifying the classes of…
We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size…
We propose a new model of computation based on nonstandard analysis. Intuitively, the role of "algorithm" is played by a new notion of finite procedure, called Omega-invariance and inspired by physics, from nonstandard analysis. Moreover,…
We provide a systematic, thorough treatment of the foundations of probability theory and stochastic processes along the lines of E. Bishop's constructive analysis. Every existence result presented shall be a construction; and the input…
We introduce the notion of "functional extension" of a set X, by means of two natural algebraic properties of the operator * on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple…
This paper introduces a space of variable lotteries and proves a constructive version of the expected utility theorem. The word ``constructive'' is used here in two senses. First, as in constructive mathematics, the logic underlying proofs…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…
Mixture models are one of the most widely used statistical tools when dealing with data from heterogeneous populations. This paper considers the long-standing debate over finite mixture and infinite mixtures and brings the two modelling…
A conditional knowledge base R is a set of conditionals of the form "If A, the usually B". Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure…
The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, whichinvolve norms not only of the initial data, but also of nonlinear functions of the initial…
In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential…
As David Berlinski writes (1997), the existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself. Here we analyze the essence of mathematics making the main emphasis…
In literature, many important combinatorial properties of subsets of N have been studied both with nonstandard techniques and from the point of view of N. In this thesis we mix these two different approaches in a technique that, at the same…