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The action of any group on itself by conjugation and the corresponding conjugacy relation plays an important role in group theory. Generalizing the group theoretic notion of conjugacy to semigroups is one of the interesting problems, and semigroup theorists had produced substantial amount of research in this direction. The challenge to introduce a new notion of conjugacy in semigroups is to choose the suitable set of conjugating elements. A semigroup may contain a zero, and if zero lies in the conjugating set, then the relation reduces to the universal relation as can be seen in the notions ∼l, ∼p, and ∼o. To avoid this problem, various innovative notions of conjugacy in semigroups have been considered so far, and ∼n is one of these notions. ∼n is an equivalence relation in any semigroup, coincides with the usual group theoretic notion if the underlying semigroup is a group, and does not reduce to a universal relation even if S contains a zero. In this paper, we study ∼n notion of conjugacy in some classes of epigroups. Since epigroups are generalizations of groups, our results of this paper are innovative and generalize the existing results on other notions. After proving some fundamental results, we compare our results with existing ones and prove that they are worth contribution in the study of conjugacy in epigroups. [ABSTRACT FROM AUTHOR]

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 Z 2 = { 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When G = ∏ n Γ n for countable abelian Γ n , Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343], Ding and Gao showed that for such G, the orbit equivalence relation must in fact be potentially Π 6 0 , while conjecturing that the optimal bound could be Π 3 0 . We show that the optimal bound is D ( Π 5 0) by constructing an action of such a group G which is not potentially Π 5 0 , and show how to modify the analysis of [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343] to get this slightly better upper bound. It follows, using the results of Hjorth et al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets. [ABSTRACT FROM AUTHOR]

The theory of rough sets has been classified into several directions where the information granules are generated using tolerances, pre-orders and binary relations. Maximal Compatibility Block (MCB) plays an important role in case of tolerance relations. In the present work, the lattice structure of rough sets generated by MCBs is explored. The concepts are applied to establish equivalence between the isomorphism of graphs with isomorphism of rough lattices constructed using MCBs. [ABSTRACT FROM AUTHOR]

We show that for any Polish group G and any countable normal subgroup Γ ◃ G, the coset equivalence relation G/Γ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that the "stabilization" of any countable ergodic probability measure preserving (p.m.p.) equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions. [ABSTRACT FROM AUTHOR]

This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ' # x. F x = n if and only if ∃ n x   F x ', where # is the number-of operator and ∃ n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. [ABSTRACT FROM AUTHOR]

The basic methodology of rough set theory depends on an equivalence relation induced from the generated partition by the classification of objects. However, the requirements of the equivalence relation restrict the field of applications of this philosophy. To begin, we describe two kinds of closure operators that are based on right and left adhesion neighbourhoods by any binary relation. Furthermore, we illustrate that the suggested techniques are an extension of previous methods that are already available in the literature. As a result of these topological techniques, we propose extended rough sets as an extension of Pawlak's models. We offer a novel topological strategy for making a topological reduction of an information system for COVID-19 based on these techniques. We provide this medical application to highlight the importance of the offered methodologies in the decision-making process to discover the important component for coronavirus (COVID-19) infection. Furthermore, the findings obtained are congruent with those of the World Health Organization. Finally, we create an algorithm to implement the recommended ways in decision-making. [ABSTRACT FROM AUTHOR]

The authors propose a connection between mathematical and aesthetic reasoning for the case of color selection in oblique, open-work woven artifacts. The general approach is first modeled on a set of open-work squares. Rotational symmetries of the square provide a mathematica starting point for a set of four open-work mats woven with paper strips. A mathematical eye, guided by concepts such as symmetries and equivalence relations, helps determine aesthetic choices for the set, specifically with respect to color. This color selection algorithm is then applied to a more complicated woven artifact. [ABSTRACT FROM AUTHOR]

Let \mathscr {R} be an \mathrm {NIP} expansion of (\mathbb {R},<,+) by closed subsets of \mathbb {R}^n and continuous functions f : \mathbb {R}^m \to \mathbb {R}^n. Then \mathscr {R} is generically locally o-minimal. This follows from a more general theorem on \mathrm {NIP} expansions of locally compact groups, which itself follows from a result on quotients of definable sets in \aleph _1-saturated \mathrm {NIP} structures by equivalence relations which are both externally definable and \bigwedge-definable. We also show that \mathscr {R} is strongly dependent if and only if \mathscr {R} is either o-minimal or (\mathbb {R},<,+,\alpha \mathbb {Z})-minimal for some \alpha > 0. [ABSTRACT FROM AUTHOR]

In order to deal with coarse‐grained and multi‐grained calculation problems, as well as granularity transformation problems in information system, quotient space theory is introduced in rough set theory. The main idea of this research is to try to maintain the important properties of the original space into the quotient space. Aimed to preserve the micro properties and the macro properties, two pairs of approximation operators on the quotient space are defined. When it comes to the composite of quotient spaces, the idea of these operators shows greater advantages. Examples are cited to illustrate possible applications of these operators, and their matrix representations are also given to make the calculations easy. Finally, all approximation operators on the quotient space involved so far are compared and their relationships are shown through a diagram. [ABSTRACT FROM AUTHOR]

A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and 0-large generic extensions of the constructible universe L. More specifically, if a ∈ 2 ω is either Sacks generic or 0-large generic real over L, then it is true in L[a] that there is a lightface Π 2 1 equivalence relation Q on the Π 2 1 set U = 2 ω \ L with exactly two equivalence classes, and both those classes are non-OD sets. [ABSTRACT FROM AUTHOR]

We make some beginning observations about the category 피q of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations R and S is a mapping from the set of R-equivalence classes to that of S-equivalence classes, which is induced by a computable function. We also consider some full subcategories of 피q, such as the category E q Σ 1 0 of computably enumerable equivalence relations (called ceers), the category E q Π 1 0 of co-computably enumerable equivalence relations, and the category 피q(Dark*) whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in E q Σ 1 0 the epimorphisms coincide with the onto morphisms, but in E q Π 1 0 there are epimorphisms that are not onto. Moreover, 피q, E q Σ 1 0 , and 피q(Dark*) are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in E q Π 1 0 whose coequalizer in 피q is not an object of E q Π 1 0. [ABSTRACT FROM AUTHOR]

For a compact surface S = Sg,n with 3g + n≥4, we introduce a family of unitary representations of the mapping class group Mod(S) based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of Mod(S) on the Teichmüller space of S. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups. [ABSTRACT FROM AUTHOR]

We study cocycles of countable groups of Borel automorphisms of a standard Borel space (X,B) taking values in a locally compact second countable group G. We prove that for a hyperfinite group - the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the 2-odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G. We also provide a Borel version of Gottschalk-Hedlund theorem. [ABSTRACT FROM AUTHOR]

Let G be a finite group and k an algebraically closed field of characteristic p > 0. We define the notion of a Dade kG -module as a generalization of endo-permutation modules for p -groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade kG -modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group D (G) defined by Lassueur. We also consider the subgroup D Ω (G) of D (G) generated by relative syzygies Ω X , where X is a finite G -set. If C (G , p) denotes the group of superclass functions defined on the p -subgroups of G , there are natural generators ω X of C (G , p) , and we prove the existence of a well-defined group homomorphism Ψ G : C (G , p) → D Ω (G) that sends ω X to Ω X. The main theorem of the paper is the verification that the subgroup of C (G , p) consisting of the dimension functions of k -orientable real representations of G lies in the kernel of Ψ G. [ABSTRACT FROM AUTHOR]

Age-related audio-visual integration (AVI) has been investigated extensively; however, AVI ability is either enhanced or reduced with ageing, and this matter is still controversial because of the lack of systematic investigations. To remove possible variates, 26 older adults and 26 younger adults were recruited to conduct meaningless and semantic audio-visual discrimination tasks to assess the ageing effect of AVI systematically. The results for the mean response times showed a significantly faster response to the audio-visual (AV) target than that to the auditory (A) or visual (V) target and a significantly faster response to all targets by the younger adults than that by the older adults (A, V, and AV) in all conditions. In addition, a further comparison of the differences between the probability of audio-visual cumulative distributive functions (CDFs) and race model CDFs showed delayed AVI effects and a longer time window for AVI in older adults than that in younger adults in all conditions. The AVI effect was lower in older adults than that in younger adults during simple meaningless image discrimination (63.0 ms vs. 108.8 ms), but the findings were inverse during semantic image discrimination (310.3 ms vs. 127.2 ms). In addition, there was no significant difference between older and younger adults during semantic character discrimination (98.1 ms vs. 117.2 ms). These results suggested that AVI ability was impaired in older adults, but a compensatory mechanism was established for processing sematic audio-visual stimuli. [ABSTRACT FROM AUTHOR]

Let $\mathcal M=(M, be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin's work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the "geometric" simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes. [ABSTRACT FROM AUTHOR]

In this research, we introduce an approach to find a family of sixth order Sturm- Liouville problems having the same spectrum. Using Darboux Lemma and the fact that any second order Sturm-Liouville problem with the Dirichlet boundary conditions is equivalent to a sixth order Sturm-Liouville problem, the considered problems are formulated. [ABSTRACT FROM AUTHOR]

In modules there is a relation between supplemented and π-projective semimodules. This relation was introduced, explained and investigated by many authors. This research will firstly introduce a concept of "supplement subsemimodule" analogues to the case in modules: a subsemimodule Y of a semimodule W is said to be supplement of a subsemimodule X if it is minimal with the property X+Y=W. A subsemimodule Y is called a supplement subsemimodule if it is a supplement of some subsemimodule of W. Then, the concept of supplemented semimodule will be defined as follows: an S-semimodule W is said to be supplemented if every subsemimodule of W is a supplement. We also review other types of supplemented semimodules. Previously, the concept of π-projective semimodule was introduced. The main goal of the present study is to explain the relation between the two concepts, supplemented semimodule and π-projective semimodules, and prove these relations by many results. [ABSTRACT FROM AUTHOR]

Copyright of Tunnel Construction / Suidao Jianshe (Zhong-Yingwen Ban) is the property of Tunnel Construction Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this research and by using the concept of Nano- Ἷ-Semi-G-Closed Set (resp., Nano-Ἷ-Semi-g-open, Nano-Ἷ-Semi-g-S and Nano-Ἷ-Semi-g-ω-closed setset) was defined. Some properties and examples with illustrative table and an applied example were presented. [ABSTRACT FROM AUTHOR]

One of the significant aspects of fuzzy group theory is classification of the fuzzy subgroups of finite groups under a suitable equivalence relation. In this paper, we determine the number of distinct fuzzy subgroups of dicyclic groups in some particular cases by the new equivalence relation introduced by Tǎrnǎuceanu. In this case, the corresponding equivalence classes of fuzzy subgroups of a group G are closely connected to the automorphism group and the chains of subgroups of G. In fact, this new equivalence relation generalizes the natural equivalence relation defined on the lattice of fuzzy subgroups. [ABSTRACT FROM AUTHOR]

The accurate and efficient identification of effective reservoirs plays an important role in the real-time flood control operation of multireservoir systems. The redundancy of the initial criteria system affects the efficiency of identifying effective reservoirs. In this paper, a criteria reduction model based on the equivalence relation of rough set theory is proposed to solve this problem. Correspondingly, the decision rules are derived for identifying effective reservoirs. In addition, the reduction result of rough set theory is compared with the results of correlation analysis-principal component analysis (CA-PCA), vague set theory and back-propagation (BP) neural network methods. The proposed methodology is applied to the flood control system in the upper and middle reaches of the Huaihe River basin in China. The results show that the dimensionality of the initial criteria system is reduced and that the reduction result is reasonable. The proposed reduction model is an effective tool to reduce the initial criteria system for effective reservoir identification in real-time flood control operation. [ABSTRACT FROM AUTHOR]

Poincaré series of p-adic, definable equivalence relations have been studied in various cases since Igusa's and Denef's work related to counting solutions of polynomial equations modulo pn for prime pn. General semi-algebraic equivalence relations on local fields have been studied uniformly in p recently by Hrushovski, Martin and Rideau (2014). In this paper we generalize their rationality result to the analytic case, uniformly in p, we build further on their appendix given by Cluckers as well as on work by van den Dries (1992), on work by Cluckers, Lipshitz and Robinson (2006). In particular, the results hold for large positive characteristic local fields. We also introduce rational motivic constructible functions and their motivic integrals, as a tool to prove our main results. [ABSTRACT FROM AUTHOR]

As a generalization of fuzzy sets, the notion of intuitionistic fuzzy sets was introduced by Atanassov [6], and applications of intuitionistic fuzzy concepts have already been done by Atanassov and many others in algebra, topological space, knowledge engineering, natural language, and neural network etc. The concept of hyperstructure first was introduced by Marty [33]. Vougiouklis [43], in the fourth AHA congress (1990), introduced the notion of Hv-structures. Recently, well known authors such as Davvaz, Dudek, Jun, Zhan, Cristea etc. have studied and discussed the intuitionistic fuzzification of different kinds of hyperstructures. The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we deal with Hv- Hv- Γsemigroups which is a generalization of Hv-Γsemigroups and Hv-Γsemigroups. Using Atanassov idea, we apply the concept of intuitionistic fuzzy sets to Hv- Hv-Γsemigroups initiating this kind of study. We introduce the notion of an intuitionistic fuzzy Hv-ideal of an Hv- Hv-semigroup and different properties and characterizations of them are investigated and obtained extending some results obtained in Hv-rings. Also some natural equivalence relations on the set of all intuitionistic fuzzy Hv-ideals of an Hv- Hv-semigroup are investigated. [ABSTRACT FROM AUTHOR]

For a finite group G, we define an equivalence relation ∼ on G as follows: where xG is the conjugacy class of x in G. The set of sizes of equivalence classes with respect to this relation is called the same-size conjugate set of G and denote it by U(G). In this paper, we consider the influence of U(G) on the structure of a group G. In fact, we conjecture that all finite non-abelian simple groups G are characterizable by U(G) and we confirm this conjecture for the projective special linear groups PSL3(3) and PSL2(q), where. [ABSTRACT FROM AUTHOR]

Ion permeation through transmembrane channels has traditionally been modeled using two different approaches. In one approach, the translocation of the permeant ion through the channel pore is modeled as continuous diffusion and the rate of ion transport is obtained from solving the steady-state diffusion equation. In the other approach, the translocation of the permeant ion through the pore is modeled as hopping along a discrete set of internal binding sites and the rate of ion transport is obtained from solving a set of steady-state rate equations. In a recent work [Zhou, J. Phys. Chem. Lett. 1, 1973 (2010)], the rate constants for binding to an internal site were further calculated by modeling binding as diffusion-influenced reactions. That work provided the foundation for bridging the two approaches. Here we show that, by representing a binding site as an energy well, the two approaches indeed give the same result for the rate of ion transport. [ABSTRACT FROM AUTHOR]

Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count, count-distinct, min, max, and sum. Essentially, this class contains unnested SQL queries with the above aggregate operators, with a where clause consisting of a conjunction of comparisons, and without a having clause. The comparisons are either interpreted over a domain with a dense order (like the rationals) or with a discrete order (like the integers). Characterizations of equivalence differ for the two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable in polynomial space. [ABSTRACT FROM AUTHOR]

We solve an open question of Milner [1984].We define a set of so-called well-behaved finite automata that, modulo bisimulation equivalence, corresponds exactly to the set of regular expressions, and we show how to determine whether a given finite automaton is in this set. As an application, we consider the star height problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class. [ABSTRACT FROM AUTHOR]

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups. [ABSTRACT FROM AUTHOR]

For a semiring ℛ \mathcal{R} , the relations of shift equivalence over ℛ \mathcal{R} ( SE- ⁢ ℛ \textup{SE-}\mathcal{R}) and strong shift equivalence over ℛ \mathcal{R} ( SSE- ⁢ ℛ \textup{SSE-}\mathcal{R}) are natural equivalence relations on square matrices over ℛ \mathcal{R} , important for symbolic dynamics. When ℛ \mathcal{R} is a ring, we prove that the refinement of SE- ⁢ ℛ \textup{SE-}\mathcal{R} by SSE- ⁢ ℛ \textup{SSE-}\mathcal{R} , in the SE- ⁢ ℛ \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient N ⁢ K 1 ⁢ (ℛ) / E ⁢ (A , ℛ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group N ⁢ K 1 ⁢ (ℛ) NK_{1}(\mathcal{R}). Here, E ⁢ (A , ℛ) E(A,\mathcal{R}) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over ℛ \mathcal{R} that the refinement of its SE- ⁢ ℛ \textup{SE-}\mathcal{R} class into SSE- ⁢ ℛ \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the GL ⁢ (ℛ ⁢ [ t ]) \mathrm{GL}(\mathcal{R}[t]) equivalence class of I - t ⁢ A I-tA into El ⁢ (ℛ ⁢ [ t ]) \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with N ⁢ K 1 ⁢ (ℛ) / E ⁢ (A , ℛ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}). For a general ring ℛ \mathcal{R} and A invertible, the proof that E ⁢ (A , ℛ) E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For ℛ \mathcal{R} commutative, we show ∪ A E ⁢ (A , ℛ) = N ⁢ S ⁢ K 1 ⁢ (ℛ) \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev's presentation of K 1 K_{1} of an exact category. [ABSTRACT FROM AUTHOR]

Abstract This paper provides new equivalence theorems for regular chains and regular differential chains, which are generalizations of Ritt's characteristic sets. These theorems focus on regularity properties of elements of residue class rings defined by these chains, which are revealed by resultant computations. New corollaries to these theorems have quite simple formulations. [ABSTRACT FROM AUTHOR]

Highlights • A simple but effective coherent dropout is proposed to improve the ability of preventing overfitting in VQA model. • A siamese dropout mechanism is proposed to explicitly decrease the output variance of VQA model during training. • We further develop a deeper and wider encoding module called Multi-path Stacked Residual RNNs to enhance the representation ability of question encoder. • The proposed methods can bring clear improvements to the state-of-the-art VQA models on VQA-v1 and VQA-v2 datasets. Abstract Using dropout in Visual Question Answering (VQA) is a common practice to prevent overfitting. However, the current way to use dropout in multi-path networks may cause two problems: the co-adaptations of neurons and the explosion of output variance. In this paper, we propose coherent dropout and siamese dropout mechanism to solve the two problems, respectively. Specifically, in coherent dropout, the relevant dropout layers in multiple paths are forced to work coherently to maximize the ability of preventing neuron co-adaptations. We show that the coherent dropout is simple in implementation but very effective to overcome overfitting. As for the explosion of output variance, we develop a siamese dropout mechanism to explicitly minimize the difference between the two output vectors produced from the same input data during training phase. Such mechanism can reduce the gap between training and inference phases and make the VQA model more robust. With the help of the two techniques, we further design an enhanced question encoder called Multi-path Stacked Residual RNNs which is deeper and wider and more powerful than current shallow question encoder. Extensive experiments are conducted to verify the effectiveness of coherent dropout, siamese dropout and the enhanced question encoder. And the results show that our methods can bring clear improvements to the state-of-the-art VQA models on VQA-v1 and VQA-v2 datasets. [ABSTRACT FROM AUTHOR]

We say that the submodules A, B of an R-module M are μ-equivalent, AμB if and only if A + B/A<<μ M/A and A + B/B <<μ M/B . We show that μ relation is an equivalent relation and has good behavior with respect to addition of submodules, homorphisms, and direct sums, we apply these results to introduced the class of H- μ-supplemented modules. We say that a module M is H-μ-supplemented module if for every submodule A of M, there is a direct summand D of M such that AμD. Various properties of these modules are given. [ABSTRACT FROM AUTHOR]

We develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples AK ⁡ (n) {\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK ⁡ (3) {\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to AK ⁡ (3) {\operatorname{AK}(3)} with relations of lengths up to 20 (previous record was 17). [ABSTRACT FROM AUTHOR]

An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3‐D large‐scale gravity inversion problems. The 3‐D Gauss‐Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately 2 orders of magnitude, and the memory requirement is reduced by N′λ times compared with the traditional GLQ method, where N′λ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3‐D large‐scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hr and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3‐D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. Three‐dimensional density distributions of the Imbrium and Serenitatis basins are obtained, and high‐density bodies are found at the depths 10–60 km, likely indicating a significant uplift of the high‐density mantle beneath the two mascon basins. Key Points: An efficient forward modeling algorithm with kernel matrix equivalence is developedComputation time and memory requirement of the algorithm are significantly reducedThree‐dimensional density distributions of the Imbrium and Serenitatis basins are obtained [ABSTRACT FROM AUTHOR]

Abstract T -generable indistinguishability operators are operators E that can be expressed in the form E = T (E μ 1 , E μ 2 ,... , E μ m ) , where T is a t -norm and E μ is the fuzzy relation generated by the fuzzy subset μ. In this paper we analyse their relation with powers with respect to the t -norm T and with quasi-arithmetic means. For non-strict continuous Archimedean t-norms they are completely characterised as generable by crisp equivalence relations. These fuzzy relations are used to define a method, called JADE , useful for feature selection and classification tasks. JADE is based on minimising the distance between two indistinguishability measures: the one given by weighting the attribute-values describing the domain objects and the other one given by the correct classification taken as an equivalence relation. The preliminary experiments we carried out with JADE are promising concerning the accuracy in solving classification tasks. We also report some issues of the method that could be improved in the future. [ABSTRACT FROM AUTHOR]

Abstract For any pattern α of length at most two, we enumerate equivalence classes of Łukasiewicz paths of length n ≥ 0 where two paths are equivalent whenever the occurrence positions of α are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Łukasiewicz paths. [ABSTRACT FROM AUTHOR]

Abstract Let R ⊆ Q be rings such that the inclusion φ : R → Q is an epimorphism in the category of rings and Q is flat both as a right and as a left R -module. Assume that F. dim (Q Q ) = 0 , that is, that every left Q -module has projective dimension 0 or ∞. We prove that the following conditions are equivalent: (i) Flat left R -modules are strongly flat. (ii) Matlis-cotorsion left R -modules are Enochs-cotorsion. (i i i) h -divisible left R -modules are weak-injective. (iv) Homomorphic images of weak-injective left R -modules are weak-injective. (v) Homomorphic images of injective left R -modules are weak-injective. (vi) Left R -modules of weak dimension ≤1 are of projective dimension ≤1. (vii) The cotorsion pairs (P 1 , H D) and (F 1 , W I) coincide. Moreover, if R is a right and left Ore ring, S is its set of regular elements and Q = R [ S − 1 ] = [ S − 1 ] R is the classical ring of quotients of R , then the seven conditions above are also equivalent to: (v i i i) Divisible left R -modules are weak-injective. This extends a result by Fuchs and Salce (2018) [10] for modules over a commutative ring R. [ABSTRACT FROM AUTHOR]

The rough set theory is a successful tool to study the vagueness in data, while the fuzzy bipolar soft sets have ability to handle the uncertainty, as well as bipolarity of the information in many situations. We connect the Pawlak's rough sets with the fuzzy bipolar soft sets and introduce the concept of rough fuzzy bipolar soft sets. We also examine some structural properties of rough fuzzy bipolar soft sets and study the effects of the equivalence relation in Pawlak approximation space on the roughness of the fuzzy bipolar soft sets. We also discuss some similarity relations among the fuzzy bipolar soft sets, based on their roughness. At the end, an application of the rough fuzzy bipolar soft sets in a decision-making problem is discussed and an algorithm for this application is proposed, supported by an example. [ABSTRACT FROM AUTHOR]

An AG-groupoid S satisfying the identity x(yz) = z(xy) for all x, y, z ∈ S is called a CA-AG-groupoid. In this article the notions of equivalence relation and congruence is extended to CA-AG-groupoids and various congruences on CA-AG-groupoid and inverse CA-AG-groupoid are defined and investigated. Furthermore, it is shown that a suitably defined relation ρ on inverse CA-AG-groupoid S is a maximal idempotentseparating congruence, that S/ρ is fundamental and that the semilattice of idempotents of S is isomorphic to the semilattice of idempotents on S/ρ. [ABSTRACT FROM AUTHOR]

We generalize the well-known correspondence between partitions and equivalence relations on a set to the case of graphs and hypergraphs. This is motivated by the role that partitions and equivalence relations play in Rough Set Theory and the results provide some of the foundations needed to develop a theory of rough graphs. We use one notion of a partition of a hypergraph, which we call a graphical partition, and we show how these structures correspond to relations on a hypergraph having additional properties. In the case of a hypergraph with only nodes and no edges these properties are exactly the usual reflexivity, symmetry and transitivity properties required for equivalence relations on a set. We present definitions for upper and lower approximations of a subgraph with respect to a graphical partition. These generalize the well-known approximations in Rough Set Theory. We establish fundamental properties of our generalized approximations and provide examples of these constructions on some graphs. [ABSTRACT FROM AUTHOR]

Networks and graphs are highly relevant in modeling real-life communities and their interactions. In order to gain insight in their structure, different roles are attributed to vertices, effectively clustering them in equivalence classes. A new formal definition of regular equivalence is presented in this paper, and the relation with other equivalence types is investigated and mathematically proven. An efficient algorithm is designed, able to detect all regularly equivalent roles in large-scale complex networks. We apply it to both Barabási–Albert random networks, as well as real-life social networks, which leads to interesting insights. [ABSTRACT FROM AUTHOR]