Related papers: Modern Set
In this paper, we extend the classical arithmetic defined over the set of natural numbers N, to the set of all finite directed connected multigraphs having a pair of distinct distinguished vertices. Specifically, we introduce a model F on…
In a recent paper we have analyzed the role that a universal set of local symmetries plays in suppressing the superposition principle in classical mechanics via a path-integral formulation of classical mechanics itself. In this paper first…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…
A generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an…
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…
The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows…
Composite theories are the algebraic equivalent of distributive laws. In this paper, we delve into the details of this correspondence and concretely show how to construct a composite theory from a distributive law and vice versa. Using term…
We give a lightweight alternative construction of Jacobs's distributive law for multisets and distributions that does not involve any combinatorics. We first give a distributive law for lists and distributions, then apply a general theorem…
This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of `maximal formula', `segment' and `maximal segment' suitable to the system, and gives…
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
We show that the existing generalized separation statements including the conventional extremal principle and its extensions differ {in the ways norms on product spaces are defined}. We prove a general separation statement with arbitrary…
Bayesian networks provide a powerful tool for reasoning about probabilistic causation, used in many areas of science. They are, however, intrinsically classical. In particular, Bayesian networks naturally yield the Bell inequalities.…
We discuss various forms of the classical van der Corput's difference theorem and explore applications to and connections with the theory of uniform distribution, ergodic theory, topological dynamics and combinatorics.
This paper illustrates the richness of the concept of regular sets of time bounds and demonstrates its application to problems of computational complexity. There is a universe of bounds whose regular subsets allow to represent several time…
We provides some new equivalent forms of collection principle over some very weak set theories after reviewing the existing ones.
Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a…
We extend the classical notion of standardly stratified $k$-algebra (stated for finite dimensional $k$-algebras) to the more general class of rings, possibly without $1,$ with enough idempotents. We show that many of the fundamental…
We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the…