Equivalence Relation Research Articles

A family of subfactors arising from a pair of complex Hadamard matrices

In this paper, we define a new equivalence relation ‘[Formula: see text]’ on the set of all Hadamard inequivalent complex Hadamard matrices of order [Formula: see text] and show that pairs [Formula: see text] of equivalent matrices [Formula: see text] produce an infinite family of potentially new subfactors of the hyperfinite type [Formula: see text] factor [Formula: see text]. All these subfactors are irreducible with the Jones index [Formula: see text], including all possibilities. We also show that this family contains infinitely many infinite-depth subfactors. As an application, we compute the Connes–Størmer relative entropy and the angle between the pair [Formula: see text] of spin model subfactors arising from the pair [Formula: see text] of equivalent matrices. On the other hand, pairs [Formula: see text] of inequivalent matrices [Formula: see text] lead to subalgebras of [Formula: see text] with infinite Pimsner–Popa index.

BOREL LINE GRAPHS

Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

Generating Bent Functions and Dynamic Filters: A Novel Equivalence-Based Approach

Boolean functions are fundamental building blocks in both discrete mathematics and computer science, with applications spanning from cryptography to coding theory. Bent functions, a subset of Boolean functions with maximal nonlinearity, are particularly valuable in cryptographic applications. This study introduces a novel equivalence relation among all Boolean functions and presents an algorithm to generate bent functions based on this relation. We systematically generated a collection of 10,000 bent functions over eight variables, all originating from the same equivalence class, and analyzed their structural complexity through rank determination. Our findings revealed the presence of at least five distinct affine classes of bent functions within this collection. By employing this construction, we devised an algorithm to generate a filter function capable of combining Boolean functions. This filter function can be dynamically adjusted based on a key, offering potential applications in symmetric cipher design, such as enhancing security or improving efficiency.

Supertropical Monoids III: Factorization and splitting covers

The category [Formula: see text] of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory [Formula: see text] of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.

Конструктивные алгоритмы автоматизированного решения позиционных задач

This paper presents the theoretical foundations for an approach to automated solutions of positional problems involving intersections of geometric objects in Euclidean space. A conceptual framework is developed based on concepts and definitions related to the equivalence relation known in set theory. This framework includes: an equivalence relation defined on the set of geometric objects in Euclidean space; the dimension and factorization of the object set, represented as a collection of equivalence classes; and the projected factor set of space. A set-theoretic basis for representing the projection operation is provided, enabling the development of constructive algorithms to solve positional problems involving intersections of spatial geometric objects. These algorithms have been implemented within the KOMPAS-3D CAD environment. The theoretical foundations and analysis of the algorithm functionality establish definitions for "geometric modeling" and "3D modeling." The proposed approach to automating positional problem solutions essentially constitutes a 3D model of these solutions through the creation of a virtual model. The paper includes examples of automated 3D solutions to intersection problems, descriptions of the algorithms used in their design, and examples of practical applications of these algorithms.

The Borel complexity of the space of left‐orderings, low‐dimensional topology, and dynamics

AbstractWe develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture.

Bipolar Complex Fuzzy Rough Sets and Their Applications in Multicriteria Decision Making

Bipolar complex fuzzy set (BCFS) is a more advanced and powerful phenomenon as it consists of two-dimensional data with positive and negative impacts of an element. It can solve the data consisting of the positive and negative impacts of an element which is a bipolar fuzzy set (BFS). It also covers the two-dimensional complex data which is a complex fuzzy set (CFS). Due to these attributes, BFS and CFS are less useful in comparison with BCFS to capture vagueness, complexity, and ambiguity in the data. Furthermore, lower and upper approximations based on equivalency relations constitute another significant phenomenon known as rough set (RS). This structure is also more powerful in dealing with real-life dilemmas. Rather than comparing the RS and BCFS, we combine both phenomena to handle the complexity more powerfully to deal with such types of phenomena that are not handled by other structures. So, by combining both phenomena, we introduce a novel structure known to be bipolar complex fuzzy rough set (BCFRS) in this manuscript. After that, we define some important operations, some significant properties related to this structure, and some aggregation operators (AOs) to solve decision-making (DM) problems related to cyber security. We address a practical application of cyber security (C-S) in computing for the protection of critical data to demonstrate the usefulness of the multi-attribute DM(MADM) approach. Based on the various criteria and attributes given by the experts, we find the best and better alternative to the C-S by applying the MADM approach. We get the A4 as the best and finest alternative by using bipolar complex fuzzy rough (BCFR) weighted arithmetic averaging (BCFRWAA), BCFR ordered weighted arithmetic averaging (BCFROWAA), and BCFR ordered weighted geometric averaging (BCFROWGA) operators. And, by using BCFR weighted geometric averaging (BCFRWGA), we get the A3 as the finest alternative. Lastly, to prove the superiority, validity, and generalization of our unique established theory, we give a detailed comparative study of our established work with several prevalent theories.

A Study of Bases, Continuity and Homeomorphism in Hausdorff Topology

This paper surveys the essential concepts of bases, continuity, and homeomorphism within the context of Hausdorff topology. We examine the characterization of Hausdorff spaces through various types of bases, including minimal generative bases, which serve to illuminate the intrinsic properties of these spaces. The interplay between bases and morphisms is explored, emphasizing how morphisms facilitate continuity of mappings between Hausdorff spaces and their quotient images. We investigate the implications of these relationships for determining homeomorphisms, establishing conditions under which two Hausdorff spaces can be deemed equivalent. Key results include the construction of quotient images using equivalence relations, as well as criteria for continuity and homeomorphism that are uniquely suited to Hausdorff spaces. The findings on bases also demonstrate that a topological space X is Hausdorff if and only if there exists a minimal generative base BM for X such that for any Subfamily B′M of BM that is also a base for X, every element of BM can be expressed as a union of elements from B′M . This result provides an important tool for studying the properties of Hausdorff spaces and their relationships with other topological spaces.

A rational function quantum equivalence relation on the set of all prime numbers

In this paper, we prove new results which make it possible to vastly simplify solution of the problem of Nathanson concerning the enlargement of support bases and thus of the corresponding semigroup supports of rational function solutions to the quantum functional equations arising from multiplication of quantum integers. In particular, we establish a certain quantum equivalence relation on the set of all prime numbers which encodes the symmetry of these solutions and thus solve a problem which simplify in crucial ways the criteria for the problem posed by Nathanson concerning extensions of the support bases of these solutions.

Fuzzy Topological Spaces Generated by Fuzzy Digraphs

In this paper, we defined various fuzzy topological spaces on fuzzy digraph and study interrelation between them. Using adjacency relation on fuzzy vertices and fuzzy edges of fuzzy digraph we defined, namely two types of fuzzy topologies, left(right) fuzzy vertex topology and left(right) fuzzy edge topology on fuzzy digraph, respectively. We have obtained results related to fuzzy topology on fuzzy subdigraph [Formula: see text] of fuzzy digraph [Formula: see text]. Separation axioms [Formula: see text], [Formula: see text] and [Formula: see text] for these fuzzy topological spaces are studied. Some characterization results for fuzzy topological spaces related to isomorphic fuzzy digraphs are given. Further, we have shown that two fuzzy digraphs are isomorphic if and only if their corresponding fuzzy topologies are homeomorphic, this relation on set of fuzzy digraphs forms an equivalence relation. We have also defined and studied fuzzy bitopological spaces on fuzzy digraphs.

Intrinsically Hölder Sections in Metric Spaces

We introduce the notion of intrinsically Holder graphs in metric spaces that generalized the one of ¨ intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to Holder maps but ¨ is profoundly different from them. We prove some relevant results as the Ascoli-Arzela compactness Theorem, ` Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over R or C, a convex set and an equivalence relation for intrinsically Holder graphs. These last three properties are new also in the Lipschitz case.

On some building blocks of hypergraphs

In this study, we explore the interrelation between hypergraph symmetries represented by equivalence relations on the vertex set and the spectra of operators associated with the hypergraph. We introduce the idea of equivalence relation compatible operators related to hypergraphs. Some eigenvalues and the corresponding eigenvectors can be computed directly from the equivalence classes of the equivalence relation. The other eigenvalues can be computed from a quotient operator obtained by identifying each equivalence class as an element. The equivalence classes of specific equivalence relations on the vertex set determine the structures of the hypergraph. We collectively classify them as building blocks of hypergraphs. For instance, an equivalence relation R s on the vertex set of a hypergraph is such that two vertices are R s -related if they belong to the same set of hyperedges. The R s -equivalence classes are named as units. Using units, we explore another symmetric substructure of hypergraphs called twin units. We show that these building blocks leave certain traces in the spectrum and the corresponding eigenspaces of the R s -compatible operators associated with the hypergraph. We also show that, conversely, some specific footprints in the spectrum and the corresponding eigenvectors retrace the presence of some of these building blocks in the hypergraph. Besides the spectra of R s -compatible operators, building blocks are also interrelated with hypergraph colouring, distances in hypergraphs, hypergraph automorphisms, and random walks on hypergraphs.

The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

Finite-Timeless Local Observability for Linear Control Systems on Lie Groups

In this work, we introduce finite-timeless observability for linear control systems on Lie groups. Observability problem is to distinguish the elements of the state space by looking the images of the solutions passing through those elements in the output space. For this aim, we define almost indistinguishability and show that it is an equivalence relation. We consider linear control systems on Lie groups, where the output function is a Lie group homomorphism and study properties of the image of almost indistinguishable points from the neutral element. Thus, we relate local observability of this kind of systems to its finite-timeless local observability.

Wanted dead or alive: epistemic logic for impure simplicial complexes

Abstract We propose a logic of knowledge for impure simplicial complexes. Impure simplicial complexes represent synchronous distributed systems under uncertainty over which processes are still active (are alive) and which processes have failed or crashed (are dead). Our work generalizes the logic of knowledge for pure simplicial complexes, where all processes are alive, by Goubault et al. In our semantics, given a designated face in a complex, a formula can only be true or false there if it is defined. The following are undefined: dead processes cannot know or be ignorant of any proposition, and live processes cannot know or be ignorant of factual propositions involving processes they know to be dead. The semantics are therefore three-valued, with undefined as the third value. We propose an axiomatization that is a version of the modal logic S5. We also show that impure simplicial complexes correspond to certain Kripke models where agents’ accessibility relations are equivalence relations on a subset of the domain only.

Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods

In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation ut−Δu=λu+f(x,u)+g(x,t) associated with the Dirichlet boundary condition, where f(x,u) satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.

Homotopy classification of knotted defects in ordered media

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere S 3 by the quaternion group Q . As our mathematical model, we consider continuous maps from complements of spatial graphs to the space S 3 / Q modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of Q . Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of Q ; once we pass to planar diagrams, these labels can be refined to elements of Q associated with each arc. The same classification scheme applies not only in the case of Q but also to arbitrary groups.

Discerning equivalence relations from displayed but not previously learned stimulus pairs

A method was devised to determine whether participants could discern stimulus equivalence (SE) relations between stimuli without prior training, but simply via visual inspection of an array of premise pairs. On each of a series of slides nine circles containing different colours were paired together in boxes in different ways on each slide to embody three 3-member equivalence classes. Below this “study” array of boxes were two “choice” boxes, with two colours derived from the same equivalence class in one, and colours from two different classes in the other. Participants were instructed to choose the box in which the colours were seemingly combined “according to the pairs above”. Then on a final “sort” slide they were asked, in relation to its array, to allocate the nine colours into groups of their own choosing. Twenty-two undergraduate students, some naïve and some with some prior experience of standard equivalence experiments, participated in this experiment. Participants 1–11, in addition to trials in which the within-class choice was a symmetric or transitive pair, were given extra “baseline” trials with a simple copy of one of the array pairs in the within-class choice box. Such “baseline” trials were omitted for the remaining participants. On the choice trials about half of the participants systematically selected the within-class alternatives. There was some indication that prior experience made this more likely, whereas inclusion of “baseline” trials seemingly had a negative effect. Participants who systematically selected the within-class alternatives mostly also sorted the colours into the appropriate three equivalence groups. Some methodological refinements and extensions of this “plain sight” procedure are proposed, and the analytical potential of comparing performance on these with that on standard MTS procedures is discussed.

Orbit pseudometrics and a universality property of the Gromov-Hausdorff distance

We consider the notion of Borel reducibility between pseudometrics on standard Borel spaces introduced and studied recently by Cúth, Doucha and Kurka, as well as the notion of an orbit pseudometric, a continuous version of the notion of an orbit equivalence relation. It is well known that the relation of isometry of Polish metric spaces is bireducible with a universal orbit equivalence relation. We prove a version of this result for pseudometrics, showing that the Gromov-Hausdorff distance of Polish metric spaces is bireducible with a universal element in a certain class of orbit pseudometrics.

The excedance quotient of the Bruhat order, quasisymmetric varieties, and Temperley–Lieb algebras

AbstractLet be the ring of polynomials in variables and consider the ideal generated by quasisymmetric polynomials without constant term. It was shown by J. C. Aval, F. Bergeron, and N. Bergeron that the th Catalan number. In the present work, we explain this phenomenon by defining a set of permutations with the following properties: first, is a basis of the Temperley–Lieb algebra , and second, when considering as a collection of points in , the top‐degree homogeneous component of the vanishing ideal is . Our construction has a few byproducts that are independently noteworthy. We define an equivalence relation on the symmetric group using weak excedances and show that its equivalence classes are naturally indexed by noncrossing partitions. Each equivalence class is an interval in the Bruhat order between an element of and a 321‐avoiding permutation. Furthermore, the Bruhat order induces a well‐defined order on . Finally, we show that any section of the quotient gives an (often novel) basis for .