Equivalence Relation Research Articles

On the Existence and Multiplicity of Classical and Weak Solutions of a Hamiltonian Integro-Differential System and Their Equivalence Relation

This paper is devoted to the study of existence and multiplicity of weak solutions to a Hamiltonian integro-differential system. The main tool used is the theory of min–max based on Mountain-Pass theorem. Hamiltonian integro-differential considered system is of Fredholm type and the imposed Dirichlet boundary conditions are occurred at the integral bounds. Furthermore, we demonstrate some cases in which the weak solutions are equivalent with classical solutions

Pan-integrals of set-valued functions based on fuzzy measures

We propose the set-valued pan-integral of set-valued function based on number-valued fuzzy measure and present some basic properties. By means of a preorder on the class of all nonempty set of R1, we show the monotonicity of set-valued pan-integral in the sense of the preorder. We introduce an equivalence relation based on the preorder and demonstrate the linearity of set-valued pan-integral in the sense of the equivalence relation. The relationships of the set-valued pan-integral and the set-valued Choquet integral are discussed. Chebyshev's inequality of set-valued pan-integrals is shown. An open problem concerning the linearity of the set-valued pan-integral is raised.

On the distribution of equivalence classes of random symmetric p‐adic matrices

AbstractWe consider random symmetric matrices with independent entries distributed according to the Haar measure on for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over has a non‐trivial zero.

In Memoriam: Peter J. Urcuioli (1952-2022).

In Memoriam: Peter J. Urcuioli (1952-2022).

On difunctions

The notion of a difunction was introduced by Jacques Riguet in 1948. Since then it has played a prominent role in database theory, type theory, program specification and process theory. The theory of difunctions is, however, less known in computing than it perhaps should be. The main purpose of the current paper is to give an account of difunction theory in relation algebra, with the aim of making the topic more mainstream.As is common with many important concepts, there are several different but equivalent characterisations of difunctionality, each with its own strength and practical significance. This paper compares different proofs of the equivalence of the characterisations.A well-known property is that a difunction is a set of completely disjoint rectangles. This property suggests the introduction of the (general) notion of the “core” of a relation; we use this notion to give a novel and, we believe, illuminating characterisation of difunctionality as a bijection between the classes of certain partial equivalence relations.

THE IMPLEMENTATION OF A ROUGH SET OF PROJECTIVE MODULE

In ring and module theory, one concept is the projective module. A module is said to be projective if it is a direct sum of independent modules. (U, R) is an approximation space with non-empty set and equivalence relation If X subset U, we can form upper approximation and lower approximation. X is rough set if upper Apr(X) is not equal to under Apr(X). The rough set theory applies to algebraic structures, including groups, rings, modules, and module homomorphisms. In this study, we will investigate the properties of the rough projective module.

Countable dense sets of dendrites

For each separable space X, we consider the family of all countable dense sets of X modulo an equivalence relation defined as follows: we say that M∼N if and only if there is a homeomorphism h:X→X such that h[M]=N. The countable dense homogeneity degree of X is defined as the cardinality of the set of the equivalence classes under the relation ∼. In case such degree equals 1, the space is said to be countable dense homogeneous. Countable dense homogeneity has become a classical concept and has been studied with interest in the last decades. In this paper we present a full characterization of the countable dense homogeneity degree for the class of dendrites.

ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS

AbstractGiven a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

On the Stabilization of Singular, Strictly Proper Transfer Function Matrices

This work focuses on the computation of stabilizing controllers for singular transfer function matrices. Stabilization of systems is a thoroughly studied subject, and numerous methodologies that enable the determination of stabilizing controllers have been developed over the years. In this work, we present a framework that unifies these methodologies. More explicitly, we utilize equivalence relations of rational matrices, allowing us to reduce the problem of stabilizing a singular transfer function to the problem of stabilizing a nonsingular transfer function matrix of smaller dimensions, thus connecting the stabilization of singular and nonsingular MIMO transfer functions. Furthermore, it is shown that the aforementioned nonsingular matrix can always be diagonal, in which case the problem of stabilizing it is reduced to stabilizing each of its diagonal elements, which are SISO transfer functions. Finally, it is proved that using this equivalence relation, similar results hold for the case of strong stabilizing controllers, that is, stabilizing controllers that are stable themselves.

Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality

The formalism of generalized probabilistic theories (GPTs) was originally developed as a way to characterize the landscape of conceivable physical theories. Thus, the GPT describing a given physical theory necessarily includes all physically possible processes. We here consider the question of how to provide a GPT-like characterization of a particular experimental setup within a given physical theory. We show that the resulting characterization is not generally a GPT in and of itself, rather, it is described by a more general mathematical object that we introduce and term an accessible GPT fragment. We then introduce an equivalence relation, termed cone equivalence, between accessible GPT fragments (and, as a special case, between standard GPTs). We give a number of examples of experimental scenarios that are best described using accessible GPT fragments, and where moreover, cone-equivalence arises naturally. We then prove that an accessible GPT fragment admits of a classical explanation if and only if every other fragment that is cone equivalent to it also admits of a classical explanation. Finally, we leverage this result to prove several fundamental results regarding the experimental requirements for witnessing the failure of generalized noncontextuality. In particular, we prove that neither incompatibility among measurements nor the assumption of freedom of choice is necessary for witnessing failures of generalized noncontextuality, and moreover, that such failures can be witnessed even when using arbitrarily inefficient detectors.

Equivalence Relations in Convex Optimization

Equivalence Relations in Convex Optimization

LogicENN: A Neural Based Knowledge Graphs Embedding Model with Logical Rules.

Knowledge graph embedding models have gained significant attention in AI research. The aim of knowledge graph embedding is to embed the graphs into a vector space in which the structure of the graph is preserved. Recent works have shown that the inclusion of background knowledge, such as logical rules, can improve the performance of embeddings in downstream machine learning tasks. However, so far, most existing models do not allow the inclusion of rules. We address the challenge of including rules and present a new neural based embedding model (LogicENN). We prove that LogicENN can learn every ground truth of encoded rules in a knowledge graph. To the best of our knowledge, this has not been proved so far for the neural based family of embedding models. Moreover, we derive formulae for the inclusion of various rules, including (anti-)symmetric, inverse, irreflexive and transitive, implication, composition, equivalence, and negation. Our formulation allows avoiding grounding for implication and equivalence relations. Our experiments show that LogicENN outperforms the existing models in link prediction.

Classifying word problems of finitely generated algebras via computable reducibility

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the [Formula: see text]-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of [Formula: see text]-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of [Formula: see text]-degrees which embeds rich structures and is large in several reasonable ways.

Classical Orthogonal Polynomials Revisited

This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel polynomials. It is presented a cyclical proof of some of the most relevant characterizations, particularly those due to Al-Salam and Chihara, Bochner, Hahn, Maroni, and McCarthy. Two apparently new characterizations are also added. Moreover, it is proved through an equivalence relation that, up to constant factors and affine changes of variables, the four families of polynomials named above are the only families of classical orthogonal polynomials.

National and Cultural Marking of Russian and Chinese Phraseological Units with Nominations of Alcoholic Beverages

In this article, we describe from linguocultural point of view phraseological units of the Russian and Chinese languages with the nominations of alcoholic beverages, which form a separate group in the thematic field “Food”; we are taking into account the priority of the anthropocentric paradigm in modern linguistics. We also study the degree of their phraseoactivity, carry out a comparative analysis of set expressions in order to identify the universal and unique in the semantics and figurativeness of these units, and consider the main types of interlingual correlation (coincidences and discrepancies) of phraseological units. The material of the study was represented by 107 Russian and 95 Chinese phraseological units with the nomination/nominations of alcoholic beverages selected in accordance with a wide understanding of phraseology by continuous sample from one- and bilingual phraseological, linguistic, explanatory and etymological dictionaries of the Chinese and Russian languages. The analysis of the studied material showed that a distinctive feature of Russian phraseologisms is a variety of nominations of alcohol (вино ‘wine’, пиво ‘beer’, мед ‘mead’, водка ‘vodka’, брага ‘mash’, хмельное ‘hoppy’), the presence of diminutive variants of the names, as well as the use of various seeds in the composition one expression. The most phraseoactive in Russian sustainable expressions is the вино (‘wine’) component (about 55 %) in two meanings: ‘grape wine’ and ‘vodka’. In the Chinese phraseology, the general designation of alcohol 酒 [jiǔ] (‘alcohol’; ‘alcoholic drinks’;‘vodka’) constitutes 95 %, all other nominations include this morpheme and are presented by isolated phenomena. The anthropocentric nature of phraseological units is reflected in the description of the properties of a human nature, personal qualities, emotional and physical state, the characteristics of social behavior and interpersonal relations. In relations of partial equivalence, about 10 % of the studied phraseologisms are located, 90 % are non-equivalent units with national and cultural originality, which is manifested in structural, grammatical, lexical, stylistic, semantic properties, as well as in figurative component.

Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal

We apply our previous work on the relation between groupoid homology and K -theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.

Replies

Replies

Similarity measure-based three-way decisions in Pythagorean fuzzy information systems and its application in FANETs

The existing researchers generalize the decision-theoretic rough sets (DTRSs) model from the viewpoint of the cost function, whether the information system is complete, and so on. Few of them consider multiple different strategies to rank the expected losses. Furthermore, under the circumstance of Pythagorean fuzzy, we can’t directly define the partition of the objects set by employing equivalence relation, there is a need for constructing the general binary relation. Aiming at these problems, in present paper, we propose the similarity measure-based three-way decisions (3WD) in Pythagorean fuzzy information systems, both the binary relation and the similarity neighborhood are induced by similarity measure between objects. Each object has its own losses, different strategies are designed to rank the expected losses. Further, the similarity measure-based DTRSs dealing with crisp concept and the similarity measure-based Pythagorean fuzzy DTRSs dealing with Pythagorean fuzzy concept are developed to establish the three regions of similarity measure-based 3WD. Finally, the proposed models are used to make decisions for classifying the network nodes of flying ad-hoc networks (FANETs) into normal nodes also called safe nodes, suspicious nodes, and malicious nodes also called unsafe nodes under the evaluation of Pythagorean fuzzy information.

Metric hypergraphs and metric-line equivalences

In a metric space with distance function dist, we say that v is between u and w if u,v,w are pairwise distinct and dist(u,w)=dist(u,v)+dist(v,w). All triples {u,v,w} such that v is between u and w constitute a hypergraph induced by this metric space. In answer to a question posed in [3], we construct an infinite family of 3-uniform hypergraphs induced by no metric space.The line determined by a pair of distinct points x and y in a metric space is the set consisting of x, y, and all points z such that one of x,y,z is between the remaining two. Two pairs of distinct points are said to be equivalent if they determine the same line. The resulting equivalence relation is a metric-line equivalence. We construct an infinite family of genuinely distinct obstacles that prevent an abstract equivalence relation from being a metric-line equivalence.

ON THE NUMBER OF FUZZY SUBGROUPS AND FUZZY NORMAL SUBGROUPS OF S2, S3 AND A4

Counting fuzzy subgroups of a finite group is a fundamental problem of fuzzy group theory. Many researchers have made significant contributions to the rapid growth of this topic in recent years. The number of fuzzy subgroups of any group is infinite without the aid of equivalence relation. Some authors have used the equivalence relation of fuzzy sets to study the equivalence of fuzzy subgroups ([5], [6], [16]). The problem of counting the number of distinct fuzzy subgroups of a finite group is relative to the choice of the equivalence relation. The number of fuzzy subgroups of a particular group varies from one equivalence relation to the other. The equivalence relation applied in our computation can be seen in the existing literature. Sulaiman and Abd Ghafur [10] define an equivalence relation for counting fuzzy subgroups of group G. We have used this relation to find fuzzy subgroups and fuzzy normal subgroups of S2, S3 and A4. Lattice subgroup diagrams were used in our computation.