Search Results (33)

This paper introduces new structures ($\mathcal{DV}$, $\mathcal{FL}$) on a set $\mathrm{\Xi }$. They will be known as diving structures and floating structures. The pairs $(\mathrm{\Xi },\mathcal{DV})$, $(\mathrm{\Xi },\mathcal{FL})$ will be called diving spaces and floating spaces, respectively. These structures are not related to each other and are not related to any of the previous common structures such as topology, ideal, filter, grill or primal. We study some properties to characterize these new notions. Separation axioms and connectedness will be defined in these new spaces. When $\mathbb{R}$ is an equivalence relation on $\mathrm{\Xi }$, rough sets in diving spaces and rough sets in floating spaces will also be defined. Thus, $(\mathrm{\Xi },\mathbb{R},\mathcal{DV})$ and $(\mathrm{\Xi },\mathbb{R},\mathcal{FL})$ are called the diving approximation spaces and the floating approximation spaces, respectively. Full article

Partial information decompositions (PIDs) aim to categorize how a set of source variables provides information about a target variable redundantly, uniquely, or synergetically. The original proposal for such an analysis used a lattice-based approach and gained significant attention. However, finding a suitable underlying decomposition measure is still an open research question at an arbitrary number of discrete random variables. This work proposes a solution with a non-negative PID that satisfies an inclusion–exclusion relation for any f-information measure. The decomposition is constructed from a pointwise perspective of the target variable to take advantage of the equivalence between the Blackwell and zonogon order in this setting. Zonogons are the Neyman–Pearson region for an indicator variable of each target state, and f-information is the expected value of quantifying its boundary. We prove that the proposed decomposition satisfies the desired axioms and guarantees non-negative partial information results. Moreover, we demonstrate how the obtained decomposition can be transformed between different decomposition lattices and that it directly provides a non-negative decomposition of Rényi-information at a transformed inclusion–exclusion relation. Finally, we highlight that the decomposition behaves differently depending on the information measure used and how it can be used for tracing partial information flows through Markov chains. Full article

Based on equivalence relation R on X, equivalence class $\left[x\right]$ of a point and equivalence class $\left[A\right]$ of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space $(X,R)$ depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology ${\tau }_R$ on X is equivalent to the generated topologies associated with metric d, proximity $\delta$ and uniformity $\mathcal{U}$ on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology ${\tau }_R$ on X. Full article

It was established by Jensen in 1970 that there is a generic extension $\mathbf{L}\left[a\right]$ of the constructible universe $\mathbf{L}$ by a non-constructible real $a\notin \mathbf{L}$, minimal over $\mathbf{L}$, such that a is ${\Delta }_3^1$ in $\mathbf{L}\left[a\right]$. Our first main theorem generalizes Jensen’s result by constructing, for each $n\ge 2$, a generic extension $\mathbf{L}\left[a\right]$ by a non-constructible real $a\notin \mathbf{L}$, still minimal over $\mathbf{L}$, such that a is ${\Delta }_{n+1}^1$ in $\mathbf{L}\left[a\right]$ but all ${\mathit{\Sigma }}_n^1$ reals are constructible in $\mathbf{L}\left[a\right]$. Jensen’s forcing construction has found a number of applications in modern set theory. A problem was recently discussed as to whether Jensen’s construction can be reproduced entirely by means of second-order Peano arithmetic ${\mathbf{PA}}_2$, or, equivalently, ${\mathbf{ZFC}}^{-}$ (minus the power set axiom). The obstacle is that the proof of the key CCC property (whether by Jensen’s original argument or the later proof using the diamond technique) essentially involves countable elementary submodels of ${\mathbf{L}}_{{\omega }_2}$, which is way beyond ${\mathbf{ZFC}}^{-}$. We demonstrate how to circumvent this difficulty by means of killing only definable antichains in the course of a Jensen-like transfinite construction of the forcing notion, and then use this modification to define a model with a minimal ${\Delta }_{n+1}^1$ real as required as a class-forcing extension of a model of ${\mathbf{ZFC}}^{-}$ plus $\mathbf{V}=\mathbf{L}$. Full article

In this paper, we will explore alternative varieties of integer multiplication by modifying the product axiom of Dedekind–Peano arithmetic (PA). In addition to studying the elementary properties of the new models of arithmetic that arise, we will see that the truth or falseness of some classical conjectures will be equivalently in the new ones, even though these models have non-commutative and non-associative product operations. To pursue this goal, we will generalize the divisor and prime number concepts in the new models. Additionally, we will explore various general number properties and project them onto each of these new structures. This fact will enable us to demonstrate that indistinguishable properties on PA project different properties within a particular model. Finally, we will generalize the main idea and explain how each integer sequence gives rise to a unique arithmetic structure within the integers. Full article

In axiomatic quantum field theory, the postulate of the uniqueness of the vacuum (a pure vacuum state) is independent from the other axioms and equivalent to the cluster decomposition property. The latter, however, implies a Coulomb or Yukawa attenuation of the interactions at growing distances and hence cannot accommodate the confining properties of the strong interaction. Thesolution of the Yang–Mills quantum theory given previously uses an auxiliary field to incorporate Gauss’s law and demonstrates the existence of two separate vacua, the perturbative and the confining vacuum, therefore resulting in a mixed vacuum state, deriving confinement, as well as the related, expected properties of the strong interaction. The existence of multiple vacua is, in fact, expected by the axiomatic, algebraic quantum field theory, via the decomposition of the vacuum state to eigenspaces of the auxiliary field. The general vacuum state is a mixed quantum state, and the cluster decomposition property does not hold. Because of the energy density difference between the two vacua, the physics of the strong interactions does not admit a Lagrangian description. I clarify the above remarks related to the previous solution of the Yang–Mills interaction and conclude with some discussion a criticism of a related mathematical problem and some tentative comments regarding the spin-2 case. Full article

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression $d(x,y)=t$. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value $d(x,y)$ and the other is a number within the range $[0,+\infty )$. Full article

The notions of continuity and separation axioms have significance in topological spaces. As a result, there has been a substantial amount of research on continuity and separation axioms, leading to the creation of several modifications of these axioms. In this paper, the concepts of soft slight $\omega$-continuity, soft ultra-Hausdorff, soft ultra-regular, and soft ultra-normal are initiated and investigated. Their characterizations and main features are determined. Also, the links between them and some other relevant concepts are obtained with the help of examples. Moreover, the equivalency between these notions and other related concepts is given under some necessary conditions. In addition, the inverse image of the introduced types of soft separation axioms under soft slight continuity and soft slight $\omega$-continuity is studied, and their reciprocal relationships with respect to their parametric topological spaces are investigated. Full article

The partial information decomposition (PID) framework is concerned with decomposing the information that a set of random variables has with respect to a target variable into three types of components: redundant, synergistic, and unique. Classical information theory alone does not provide a unique way to decompose information in this manner, and additional assumptions have to be made. Recently, Kolchinsky proposed a new general axiomatic approach to obtain measures of redundant information based on choosing an order relation between information sources (equivalently, order between communication channels). In this paper, we exploit this approach to introduce three new measures of redundant information (and the resulting decompositions) based on well-known preorders between channels, contributing to the enrichment of the PID landscape. We relate the new decompositions to existing ones, study several of their properties, and provide examples illustrating their novelty. As a side result, we prove that any preorder that satisfies Kolchinsky’s axioms yields a decomposition that meets the axioms originally introduced by Williams and Beer when they first proposed PID. Full article

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning. Full article

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, $\mathbf{R}\times ({\mathbf{R}}^3\setminus \left\{(0,0,0)\right\})$ and $S^1\times ({\mathbf{R}}^3\setminus \left\{(0,0,0)\right\})$. The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on $\mathbf{R}\times ({\mathbf{R}}^3\setminus \left\{(0,0,0)\right\})$ and a winding mapping of the real line $\mathbf{R}$ onto the circle $S^1$. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to $S^1\times ({\mathbf{R}}^3\setminus \left\{(0,0,0)\right\})$ by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom. Full article

Based on the axioms of quantum theory, we identify a class of topological singularities that encode a fundamental difference between classic and quantum probability, and explain quantum theory’s puzzles and phenomena in simple mathematical terms so they are no longer ‘quantum paradoxes’. The singularities provide also new experimental insights and predictions that are presented in this article and establish a surprising new connection between the physical and social sciences. The key is the topology of spaces of quantum events and of the frameworks postulated by these axioms. These are quite different from their counterparts in classic probability and explain mathematically the interference between quantum experiments and the existence of several frameworks or ‘violation of unicity’ that characterizes quantum physics. They also explain entanglement, the Heisenberg uncertainty principle, order dependence of observations, the conjunction fallacy and geometric phenomena such as Pancharatnam–Berry phases. Somewhat surprisingly, we find that the same topological singularities explain the impossibility of selecting a social preference among different individual preferences: which is Arrow’s social choice paradox: the foundations of social choice and of quantum theory are therefore mathematically equivalent. We identify necessary and sufficient conditions on how to restrict experiments to avoid these singularities and recover unicity, avoiding possible interference between experiments and also quantum paradoxes; the same topological restriction is shown to provide a resolution to the social choice impossibility theorem of Chichilnisky. Full article

As daily problems involve a great deal of data and ambiguity, it has become vital to build new mathematical ways to cope with them, and soft set theory is the greatest tool for doing so. As a result, we study methods of generating soft topologies through several soft set operators. A soft topology is known to be determined by the system of special soft sets, which are called soft open (dually soft closed) sets. The relationship between specific types of soft topologies and their classical topologies (known as parametric topologies) is linked to the idea of symmetry. Under this symmetry, we can study the behaviors and properties of classical topological concepts via soft settings and vice versa. In this paper, we show that soft topological spaces can be characterized by soft closure, soft interior, soft boundary, soft exterior, soft derived set, or co-derived set operators. All of the soft topologies that result from such operators are equivalent, as well as being identical to their classical counterparts under enriched (extended) conditions. Moreover, some of the soft topologies are the systems of all fixed points of specific soft operators. Multiple examples are presented to show the implementation of these operators. Some of the examples show that, by removing any axiom, we will miss the uniqueness of the resulting soft topology. Full article

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration. Full article

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic. Full article