Equivalence Relation Research Articles

Co-spectral radius for countable equivalence relations

Abstract We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

Chow–Lefschetz motives

We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.

Characterization of neighborhood operators based on neighborhood relationships

Abstract Neighborhood relationships play a pivotal role in rough set theory, addressing the limitations of equivalence relations. This article focuses on defining upper and lower approximation operators using neighborhood relationships and exploring their properties in terms of serialization, inverse serialization, reflexivity, symmetry, transitivity, and Euclidean relations. Furthermore, a necessary and sufficient condition for the upper approximation operator to function as a topological closure operator is derived. Overall, this research sheds light on the significance of neighborhood relationships and their implications within rough set theory.

More fermionic supersymmetric Wilson loops in four dimensions

We construct supersymmetric fermionic Wilson loops along general curves in four-dimensional N=4 super Yang-Mills theory and along general planar curves in N=2 superconformal SU(N)×SU(N) quiver theory. These loops are generalizations of the Zarembo loops and are cohomologically equivalent to them. In N=4 super Yang-Mills theory, we compute their expectation values and verify the cohomological equivalence relation up to the order g4 in perturbation theory.

Prime 2-structures

The 2-structures generalize arc-coloured digraphs. They are obtained from a vertex set by considering an equivalence relation between the ordered pairs of distinct vertices. The 2-structures provide a suitable framework to introduce the modular decomposition. A vertex subset of a 2-structure is a module whenever each vertex outside is linked in an equivalent way to all the vertices inside. A 2-structure is prime if it does not admit proper modules with at least 2 elements. Our purpose is to study the possible cardinalities of the prime 2-substructures in a finite or infinite prime 2-structure.

New counterexamples to the birational Torelli theorem for Calabi–Yau manifolds

We produce counterexamples to the birational Torelli theorem for Calabi–Yau manifolds in arbitrarily high dimension: this is done by exhibiting a series of non-birational pairs of Calabi–Yau ( n 2 − 1 ) (n^2-1) -folds which, for n ≥ 2 n \geq 2 even, admit an isometry between their middle cohomologies. These varieties also satisfy an L \mathbb {L} -equivalence relation in the Grothendieck ring of varieties, i.e. the difference of their classes annihilates a power of the class of the affine line. We state this last property for a broader class of Calabi–Yau pairs, namely all those which are realized as pushforwards of a general ( 1 , 1 ) (1,1) -section on a homogeneous roof in the sense of Kanemitsu, along its two extremal contractions.

Locally finite and finitely approximated unoids over computably separable equivalences

We prove that every coinfinite set is a characteristic transversal of a suitably computably separable equivalence relation, over which only locally finite, locally finite separable and finitely approximable unary algebras are represented. Similar properties for uniformly computable separable equivalences are considered.

The Enumeration of (⊙,∨)-Multiderivations on a Finite MV-Chain

In this paper, (⊙,∨)-multiderivations on an MV-algebra A are introduced, the relations between (⊙,∨)-multiderivations and (⊙,∨)-derivations are discussed. The set MD(A) of (⊙,∨)-multiderivations on A can be equipped with a preorder, and (MD(A)/∼,≼) can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain Ln, (MD(Ln)/∼,≼) becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of MD(Ln).

A New Approach to Understanding Quantum Mechanics: Illustrated Using a Pedagogical Model over ℤ2

The new approach to quantum mechanics (QM) is that the mathematics of QM is the linearization of the mathematics of partitions (or equivalence relations) on a set. This paper develops those ideas using vector spaces over the field Z2={0.1} as a pedagogical or toy model of (finite-dimensional, non-relativistic) QM. The 0,1-vectors are interpreted as sets, so the model is “quantum mechanics over sets” or QM/Sets. The key notions of partitions on a set are the logical-level notions to model distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. Those pairs of concepts are the key to understanding the non-classical ‘weirdness’ of QM. The key non-classical notion in QM is the notion of superposition, i.e., the notion of a state that is indefinite between two or more definite- or eigen-states. As Richard Feynman emphasized, all the weirdness of QM is illustrated in the double-slit experiment, so the QM/Sets version of that experiment is used to make the key points.

Relational generalized iterated function systems

In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if we consider the total equivalence relation, we recover the classical generalized iterated function systems. Our main result states that each compact subset of the underlying metric space generates, via a sequence of iterates, a fixed point of the associated fractal operator, called an attractor of the system. We also establish a structure result for the attractors and a theorem concerning the continuous dependence of the attractor on the associated compact set. Ultimately, we provide some examples which illustrate our main results.

Universal Mass-Energy Equivalence Relation is Expressed by Mechanical Energy of a Falling Body on Earth at Constant Acceleration Due to Gravity

Universal Mass-Energy Equivalence Relation is Expressed by Mechanical Energy of a Falling Body on Earth at Constant Acceleration Due to Gravity

Generalized exponentially bounded integrated semigroups

The subject of this paper is the analysis of sequences of infinitesimal generators and exponentially bounded integrated semigroups which are related to Cauchy problems(1)∂∂tu(t,x)−a(D)u(t,x)=f(t,x),u(0,x)=u0(x),t≥0,x∈Rd, with distributional initial data u0 and distributional right hand sides f through sequences of equations with regularized u0 and f, and a(D) approximated by suitable sequences of (pseudo)differential operators an(D). Mainly, the paper deals with the comparison of sequences of infinitesimal generators and the determination of corresponding sequences of integrated semigroups. For this purpose, we introduce association, the relation of equivalence for infinitesimal generators on one side and the corresponding relations of equivalence of integrated semigroups on another side. The order of involved assumptions on generators essentially characterizes the mutual dependence of sequences of infinitesimal generators and the corresponding sequences of integrated semigroups. Our motivation is presented in Section 3 where we show how Cauchy problem (1) with certain type of singularities is transferred to a sequence of Cauchy problems with classical solutions presented by one time integrated semigroups, and then discuss the solution to (1) in a weak sense (in the sense of distributions) or a very weak sense (when only a sequence of solutions exists without a weak limit). The effects of perturbations of infinitesimal generators to integrated semigroups are the core of the paper.

ON THE STRUCTURE OF PUNCTUAL LINEAR ORDERS ISOMORPHIC TO THE LEAST LIMIT ORDINAL

The algorithmic complexity of presentations for various structures receives significant attention in modern literature. The main tool for determining such complexities is reducibility. It is a mapping that preserves relations of signature (for example, equivalence relation, orders, and so on). This work is dedicated to the study of punctual representations of the least limit ordinal with respect to primitive recursive reducibility. We denote this structure as PR(ω). In particular, the paper examines the properties of structures Ω, consisting of computable copies of the least limit ordinal with respect to computable reducibility, and Peq, consisting of punctual equivalence relations with respect to primitive recursive reducibility. We say that the linear order L is reducible to the linear order R, if there exists a total function ρ such that (χ, γ) Є L if and only if (ρ(χ), ρ(γ)) Є R. Reducibility is called computable (primitive recursive) if the function that performs the reducibility is computable (primitive recursive). It is shown that the degree of ω is not the least degree in PR(ω), as it was in Ω. The structure PR(ω) does not contain maximal degrees, and this structure is not dense. Also, an example of an incomparable pair that has the least upper bound is given.

FoX: Formation-Aware Exploration in Multi-Agent Reinforcement Learning

Recently, deep multi-agent reinforcement learning (MARL) has gained significant popularity due to its success in various cooperative multi-agent tasks. However, exploration still remains a challenging problem in MARL due to the partial observability of the agents and the exploration space that can grow exponentially as the number of agents increases. Firstly, in order to address the scalability issue of the exploration space, we define a formation-based equivalence relation on the exploration space and aim to reduce the search space by exploring only meaningful states in different formations. Then, we propose a novel formation-aware exploration (FoX) framework that encourages partially observable agents to visit the states in diverse formations by guiding them to be well aware of their current formation solely based on their own observations. Numerical results show that the proposed FoX framework significantly outperforms the state-of-the-art MARL algorithms on Google Research Football (GRF) and sparse Starcraft II multi-agent challenge (SMAC) tasks.

Parabolic Tamari Lattices in Linear Type B

We study parabolic aligned elements associated with the type-$B$ Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (Mühle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-$B$ case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-$B$ Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-$B$ analogue of the parabolic Tamari lattice introduced for type $A$ in (Mühle and Williams, 2019).

On Proximity Spaces Constructed on Rough Sets

Based on equivalence relation R on X, equivalence class [x] of a point and equivalence class [A] 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 τR on X is equivalent to the generated topologies associated with metric d, proximity δ and uniformity 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 τR on X.

Approximate homomorphisms and sofic approximations of orbit equivalence relations

Abstract We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

Equivalent Condition of the Measure Shadowing Property on Metric Spaces

The concept referred to as the measure shadowing property for a dynamical system on compact metric space has recently been introduced, acting as an extension of the classical shadowing property by using the property of the Borel measures on the phase space. In this paper, we extend the concept of the measure shadowing property of continuous flows from compact metric spaces to the general metric spaces and demonstrate the equivalence relation between the measure shadowing property and the shadowing property for flows on metric spaces via the shadowable points.

Covering-based $$(\alpha , \beta )$$-multi-granulation bipolar fuzzy rough set model under bipolar fuzzy preference relation with decision-making applications

AbstractThe rough set (RS) and multi-granulation rough set (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this paper, with the help of bipolar fuzzy preference relation (BFPR) and bipolar fuzzy preference $$\delta $$ δ -covering (BFP$$\delta $$ δ C), we put forward the idea of BFP$$\delta $$ δ C based optimistic multi-granulation bipolar fuzzy rough set (BFP$$\delta $$ δ C-OMG-BFRS) model and BFP$$\delta $$ δ C based pessimistic multi-granulation bipolar fuzzy rough set (BFP$$\delta $$ δ C-PMG-BFRS) model. We examine several significant structural properties of BFP$$\delta $$ δ C-OMG-BFRS and BFP$$\delta $$ δ C-PMG-BFRS models in detail. Moreover, we discuss the relationship between BFP$$\delta $$ δ C-OMG-BFRS and BFP$$\delta $$ δ C-PMG-BFRS models. Eventually, we apply the BFP$$\delta $$ δ C-OMG-BFRS model for solving multi-criteria decision-making (MCDM). Furthermore, we demonstrate the effectiveness and feasibility of our designed approach by solving a numerical example. We further conduct a detailed comparison with certain existing methods. Last but not least, theoretical studies and practical examples reveals that our suggested approach dramatically enriches the MGRS theory and offers a novel strategy for knowledge discovery, which is practical in real-world circumstances.

An extension of several properties for fuzzy t-norm and vague t-norm

Rosenfeld defined a fuzzy subgroup of a given group as a fuzzy subset with two special conditions and Mustafa Demirci proposed the idea of fuzzifying the operations on a group through a fuzzy equality and a fuzzy equivalence relation. This paper mainly focuses on fuzzy subsets and vague sets of monoids with several extended algebraic properties. Firstly, we generalize some algebraic properties of t-norms to fuzzy t-norms, this allows for a broader analysis and classification of fuzzy t-norms, enabling their wider application. Furthermore, we explore specific research on the properties of vague t-norms. Finally, selected conclusions about fuzzy t-norms are extended to bounded lattices.