Journal articles: 'Equivalence classes (Set theory)'

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Liu, Dong Li. "The Equivalence Relationship of Matrix and the Corresponding Equivalence Classes." Applied Mechanics and Materials 651-653 (September 2014): 2211–15. http://dx.doi.org/10.4028/www.scientific.net/amm.651-653.2211.

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In order to further integrate the content of Linear Algebra, and deeply reveal the equivalence relationship of matrix, this paper discusses the three equivalence relationships on the set of matrices: matrix equivalence、matrix similarity and matrix contract, and gives the corresponding equivalence classes, which further enriches the theory of Linear Algebra.

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ILYUTKO, DENIS PETROVICH. "AN EQUIVALENCE BETWEEN THE SET OF GRAPH-KNOTS AND THE SET OF HOMOTOPY CLASSES OF LOOPED GRAPHS." Journal of Knot Theory and Its Ramifications 21, no. 01 (January 2012): 1250001. http://dx.doi.org/10.1142/s0218216511009649.

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In the present paper we construct an equivalence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related by this equivalence. This equivalence is given by a simple formula.

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Dakun, Zhang, Song Guozhi, and Huang Cui. "Free Triplet Conjecture and Equivalence Classes Derived Using Group Theory." Open Biotechnology Journal 9, no. 1 (October 27, 2015): 216–20. http://dx.doi.org/10.2174/1874070701509010216.

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All proteins are made up of 20 different amino acids which contain 4 kinds of nucleotides . Three consecutive nucleotides on the gene, called triplet codons, are used to code an amino acid, and 64 triplet codons comprise the genetic code table. Central dogma (DNA-RNA-protein) has been acknowledged, but the process and mechanism of mRNA passing through the nuclear membrane still require further investigation. For these two problems mentioned above, this paper proposed a conjecture of nucleotide free triplet and obtained 20 equivalence classes of mapping from free triplet vertex set to nucleotide set using group theory. Whether the four numbers 3, 4, 20 and 64 have relevance are taken into consideration here. Subsequently, the numbers 3, 4, 20 and 64 were connected together which was important for the analysis of triplet code and protein composition.

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Hjorth, Greg. "Some applications of coarse inner model theory." Journal of Symbolic Logic 62, no. 2 (June 1997): 337–65. http://dx.doi.org/10.2307/2275536.

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AbstractThe Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .

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Huh, JiSun, and Byungchan Kim. "The number of equivalence classes arising from partition involutions." International Journal of Number Theory 16, no. 05 (December 2, 2019): 925–39. http://dx.doi.org/10.1142/s1793042120500475.

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Involutions have played important roles in many research areas including the theory of partitions. In this paper, for various sets of partitions, we give relations between the number of equivalence classes in the set of partitions arising from an involution and the number of partitions satisfying a certain parity condition. We examine the number of equivalence classes arising from the conjugations on ordinary partitions, overpartitions, and partitions with distinct odd parts. We also consider other types of involutions on partitions into distinct parts, unimodal sequences with a unique marked peak, and partitions with distinct even parts.

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Mofrad, Asieh Abolpour, Anis Yazidi, Hugo L. Hammer, and Erik Arntzen. "Equivalence Projective Simulation as a Framework for Modeling Formation of Stimulus Equivalence Classes." Neural Computation 32, no. 5 (May 2020): 912–68. http://dx.doi.org/10.1162/neco_a_01274.

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Stimulus equivalence (SE) and projective simulation (PS) study complex behavior, the former in human subjects and the latter in artificial agents. We apply the PS learning framework for modeling the formation of equivalence classes. For this purpose, we first modify the PS model to accommodate imitating the emergence of equivalence relations. Later, we formulate the SE formation through the matching-to-sample (MTS) procedure. The proposed version of PS model, called the equivalence projective simulation (EPS) model, is able to act within a varying action set and derive new relations without receiving feedback from the environment. To the best of our knowledge, it is the first time that the field of equivalence theory in behavior analysis has been linked to an artificial agent in a machine learning context. This model has many advantages over existing neural network models. Briefly, our EPS model is not a black box model, but rather a model with the capability of easy interpretation and flexibility for further modifications. To validate the model, some experimental results performed by prominent behavior analysts are simulated. The results confirm that the EPS model is able to reliably simulate and replicate the same behavior as real experiments in various settings, including formation of equivalence relations in typical participants, nonformation of equivalence relations in language-disabled children, and nodal effect in a linear series with nodal distance five. Moreover, through a hypothetical experiment, we discuss the possibility of applying EPS in further equivalence theory research.

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PONG, WAI YAN. "LENGTH SPECTRA OF NATURAL NUMBERS." International Journal of Number Theory 05, no. 06 (September 2009): 1089–102. http://dx.doi.org/10.1142/s1793042109002584.

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A natural number n can generally be written as a sum of m consecutive natural numbers for various values of m ≥ 1. The length spectrum of n is the set of these admissible m. Two numbers are spectral equivalent if they have the same length spectrum. We show how to compute the equivalence classes of this relation. Moreover, we show that these classes can only have either 1,2 or infinitely many elements.

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Zhao, Jie, Jia-ming Liang, Zhen-ning Dong, De-yu Tang, and Zhen Liu. "Accelerating information entropy-based feature selection using rough set theory with classified nested equivalence classes." Pattern Recognition 107 (November 2020): 107517. http://dx.doi.org/10.1016/j.patcog.2020.107517.

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Tradler, Thomas, Scott O. Wilson, and Mahmoud Zeinalian. "An elementary differential extension of odd K-theory." Journal of K-Theory 12, no. 2 (April 4, 2013): 331–61. http://dx.doi.org/10.1017/is013002018jkt218.

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AbstractThere is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

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DUATO, JOSÉ. "CHANNEL CLASSES: A NEW CONCEPT FOR DEADLOCK AVOIDANCE IN WORMHOLE NETWORKS." Parallel Processing Letters 02, no. 04 (December 1992): 347–54. http://dx.doi.org/10.1142/s0129626492000490.

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In previous papers, we have developed the theoretical background for the design of deadlock-free adaptive routing algorithms for store-and-forward and wormhole networks. Some definitions and theorems have been proposed, developing conditions to verify that an adaptive algorithm is deadlock-free, even when there are cyclic dependencies between channels. Also, two design methodologies have been proposed. In this paper, we propose a partial order between channels as well as an equivalence relation. This relation splits the set of channels into equivalence classes. Then, we extend our previous theory by considering equivalence classes (channel classes) instead of channels. This extension drastically simplifies the verification of deadlock freedom for adaptive routing algorithms with cyclic dependencies between channels. Finally, we present an example.

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Casacuberta, Carles, Javier J. Gutiérrez, and Jiří Rosický. "A generalization of Ohkawa’s theorem." Compositio Mathematica 150, no. 5 (April 3, 2014): 893–902. http://dx.doi.org/10.1112/s0010437x13007616.

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Enayat, Ali, Vladimir Kanovei, and Vassily Lyubetsky. "On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom." Mathematics 9, no. 14 (July 15, 2021): 1670. http://dx.doi.org/10.3390/math9141670.

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Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class.

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Conidis, Chris J. "Classifying model-theoretic properties." Journal of Symbolic Logic 73, no. 3 (September 2008): 885–905. http://dx.doi.org/10.2178/jsl/1230396753.

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AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

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CHEN, WEIMIN. "ON A NOTION OF MAPS BETWEEN ORBIFOLDS II: HOMOTOPY AND CW-COMPLEX." Communications in Contemporary Mathematics 08, no. 06 (December 2006): 763–821. http://dx.doi.org/10.1142/s0219199706002283.

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This is the second of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants — the homotopy groups, and (2) an analog of CW-complex theory. As a corollary of this machinery, the classical Whitehead theorem (which asserts that a weak homotopy equivalence is a homotopy equivalence) is extended to the orbifold category.

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MCCULLOUGH, DARRYL, and MARCUS WANDERLEY. "NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 481–509. http://dx.doi.org/10.1017/s0017089512000675.

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AbstractWe present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

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FOKINA, EKATERINA, BAKHADYR KHOUSSAINOV, PAVEL SEMUKHIN, and DANIEL TURETSKY. "LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS." Journal of Symbolic Logic 81, no. 2 (May 3, 2016): 463–82. http://dx.doi.org/10.1017/jsl.2015.11.

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AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

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Mohd Noor, Alia Husna, Nor Haniza Sarmin, and Hamisan Rahmat. "Related graphs of the conjugacy classes of a 3-generator 5-group." Malaysian Journal of Fundamental and Applied Sciences 14 (October 25, 2018): 454–56. http://dx.doi.org/10.11113/mjfas.v14n0.1269.

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The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of which is of size two where and commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.

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DESCHRIJVER, GLAD, and ETIENNE E. KERRE. "CLASSES OF INTUITIONISTIC FUZZY T-NORMS SATISFYING THE RESIDUATION PRINCIPLE." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11, no. 06 (December 2003): 691–709. http://dx.doi.org/10.1142/s021848850300248x.

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Intuitionistic fuzzy sets constitute an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a non-membership degree. The only constraint on those two degrees is that the sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In the fuzzy case a t-norm satisfies the residuation principle if and only if it is left-continuous. Deschrijver, Cornelis and Kerre proved that for intuitionistic fuzzy t-norms the equivalence between the residuation principle and intuitionistic fuzzy left-continuity only holds for t-representable t-norms.1 In this paper we construct particular subclasses of intuitionistic fuzzy t-norms that satisfy the residuation principle but that are not t-representable and we show that a continuous intuitionistic fuzzy t-norm [Formula: see text] satisfying the residuation principle is t-representable if and only if [Formula: see text].

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DABKOWSKI, MIECZYSŁAW K., MAKIKO ISHIWATA, and JÓZEF H. PRZYTYCKI. "5-MOVE EQUIVALENCE CLASSES OF LINKS AND THEIR ALGEBRAIC INVARIANTS." Journal of Knot Theory and Its Ramifications 16, no. 10 (December 2007): 1413–49. http://dx.doi.org/10.1142/s0218216507005877.

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We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S3 along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. [Formula: see text]-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

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KAWAMURA, KATSUNORI. "UNIVERSAL ALGEBRA OF SECTORS." International Journal of Algebra and Computation 19, no. 03 (May 2009): 347–71. http://dx.doi.org/10.1142/s0218196709005172.

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We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C *-algebra [Formula: see text], a sector is a unitary equivalence class of unital *-endomorphisms of [Formula: see text]. We show that the set [Formula: see text] of all sectors of [Formula: see text] is a universal algebra with an N-ary sum which is not reduced to any binary sum when [Formula: see text] includes the Cuntz algebra [Formula: see text] as a C *-subalgebra with common unit for N ≥ 3. Next we explain that the set [Formula: see text] of all unitary equivalence classes of unital *-representations of [Formula: see text] is a right module of [Formula: see text]. An essential algebraic formulation of branching laws of representations is given by using submodules of [Formula: see text]. As an application, we show that the action of [Formula: see text] on [Formula: see text] distinguishes elements of [Formula: see text].

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Kharlampovich, Olga, and Alina Vdovina. "Low complexity algorithms in knot theory." International Journal of Algebra and Computation 29, no. 02 (March 2019): 245–62. http://dx.doi.org/10.1142/s0218196718500698.

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Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. This shows that (unless P[Formula: see text]NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational complexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace[Formula: see text]. Alternating knots with some additional combinatorial structure will be referred to as standard. As expected, almost all alternating knots of a given genus are standard. We show that the genus problem for these knots belongs to [Formula: see text] circuit complexity class. We also show, that the equivalence problem for such knots with [Formula: see text] crossings has time complexity [Formula: see text] and is in Logspace[Formula: see text] and [Formula: see text] complexity classes.

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Sherman, David. "On the dimension theory of von Neumann algebras." MATHEMATICA SCANDINAVICA 101, no. 1 (September 1, 2007): 123. http://dx.doi.org/10.7146/math.scand.a-15035.

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In this paper we study three aspects of $(\mathcal{P}(\mathcal{M})/{\sim})$, the set of Murray-von Neumann equivalence classes of projections in a von Neumann algebra $\mathcal M$. First we determine the topological structure that $(\mathcal{P}(\mathcal{M})/{\sim})$ inherits from the operator topologies on $\mathcal M$. Then we show that there is a version of the center-valued trace which extends the dimension function, even when $\mathcal M$ is not $\sigma$-finite. Finally we prove that $(\mathcal{P}(\mathcal{M})/{\sim})$ is a complete lattice, a fact which has an interesting reformulation in terms of representations.

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Iqbal, Saood, Abdul Shahid, Muhammad Roman, Zahid Khan, Shaha Al-Otaibi, and Lisu Yu. "TKFIM: Top-K frequent itemset mining technique based on equivalence classes." PeerJ Computer Science 7 (March 8, 2021): e385. http://dx.doi.org/10.7717/peerj-cs.385.

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Frequently used items mining is a significant subject of data mining studies. In the last ten years, due to innovative development, the quantity of data has grown exponentially. For frequent Itemset (FIs) mining applications, it imposes new challenges. Misconceived information may be found in recent algorithms, including both threshold and size based algorithms. Threshold value plays a central role in generating frequent itemsets from the given dataset. Selecting a support threshold value is very complicated for those unaware of the dataset’s characteristics. The performance of algorithms for finding FIs without the support threshold is, however, deficient due to heavy computation. Therefore, we have proposed a method to discover FIs without the support threshold, called Top-k frequent itemsets mining (TKFIM). It uses class equivalence and set-theory concepts for mining FIs. The proposed procedure does not miss any FIs; thus, accurate frequent patterns are mined. Furthermore, the results are compared with state-of-the-art techniques such as Top-k miner and Build Once and Mine Once (BOMO). It is found that the proposed TKFIM has outperformed the results of these approaches in terms of execution and performance, achieving 92.70, 35.87, 28.53, and 81.27 percent gain on Top-k miner using Chess, Mushroom, and Connect and T1014D100K datasets, respectively. Similarly, it has achieved a performance gain of 97.14, 100, 78.10, 99.70 percent on BOMO using Chess, Mushroom, Connect, and T1014D100K datasets, respectively. Therefore, it is argued that the proposed procedure may be adopted on a large dataset for better performance.

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Calvert, Wesley, and Julia F. Knight. "Classification from a Computable Viewpoint." Bulletin of Symbolic Logic 12, no. 2 (June 2006): 191–218. http://dx.doi.org/10.2178/bsl/1146620059.

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Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

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Lins, Sóstenes L., and Diogo B. Henriques. "Closed, oriented, connected 3-manifolds are subtle equivalence classes of plane graphs." Journal of Knot Theory and Its Ramifications 27, no. 14 (December 2018): 1850077. http://dx.doi.org/10.1142/s0218216518500773.

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A blink is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, it is shown that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a finite sequence of local moves, where each one appears in a concrete list of 64 moves: they are organized in 8 types, each being essentially the same move on 8 simply related configurations. The size of the list can be substantially decreased at the cost of loosing symmetry, just by keeping a very simple move type, the ribbon moves denoted [Formula: see text] (which are in principle redundant). The inclusion of [Formula: see text] implies that all the moves corresponding to plane duality (the starred moves), except for [Formula: see text] and [Formula: see text], are redundant and the coin calculus is reduced to 36 moves on 36 coins. A residual fraction link or a flink is a new object which generalizes blackboard-framed link. It plays an important role in this work. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs.

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Bertrand, Daniel. "Extensions panachées autoduales." Journal of K-Theory 11, no. 2 (March 6, 2013): 393–411. http://dx.doi.org/10.1017/is013001030jkt213.

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AbstractWe study self-duality of Grothendieck's blended extensions in the context of a tannakian category. The set of equivalence classes of symmetric, resp. antisymmetric, blended extensions is naturally endowed with a torsor structure, which enables us to compute the unipotent radical of the associated monodromy groups in various situations.

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Friedman, Greg, and Efton Park. "Unitary equivalence of normal matrices over topological spaces." Journal of Topology and Analysis 08, no. 02 (March 15, 2016): 313–48. http://dx.doi.org/10.1142/s1793525316500126.

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Let [Formula: see text] and [Formula: see text] be normal matrices with coefficients that are continuous complex-valued functions on a topological space [Formula: see text] that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point of [Formula: see text]. We use obstruction theory to establish a necessary and sufficient condition for [Formula: see text] and [Formula: see text] to be unitarily equivalent. We also determine bounds on the number of possible unitary equivalence classes in terms of cohomological invariants of [Formula: see text].

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Schenkel, Alexander. "Module parallel transports in fuzzy gauge theory." International Journal of Geometric Methods in Modern Physics 11, no. 03 (March 2014): 1450021. http://dx.doi.org/10.1142/s0219887814500212.

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In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

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Xu, Weihua, Xiaoyan Zhang, and Wenxiu Zhang. "Two New Types of Multiple Granulation Rough Set." ISRN Applied Mathematics 2013 (February 27, 2013): 1–16. http://dx.doi.org/10.1155/2013/791356.

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In the paper proposed two new types of the multiple granulation rough set models, where a target concept is approximated from two different kinds of views by using the equivalence classes induced by multiple granulations. A number of important properties of the two types of MGRS are investigated. From the properties, it can be found that Pawlak’s and Qian’s rough set models are special instances of those of our MGRS. Moreover, several important measures are presented in two types of MGRS, such as rough measure and quality of approximation. Furthermore, the relationship and difference are discussed carefully among Pawlak’s rough set, Qian’s MGRS, and two new types of MGRS. In order to illustrate our multiple granulations rough set model, some examples are considered, which are helpful for applying this theory in practical issues. One can get that the paper is meaningful both in the theory and in application for the issue of knowledge reduction in complex information systems.

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Dadarlat, Marius, and Wilhelm Winter. "On the $KK$-theory of strongly self-absorbing $C^{*}$-algebras." MATHEMATICA SCANDINAVICA 104, no. 1 (March 1, 2009): 95. http://dx.doi.org/10.7146/math.scand.a-15086.

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Let $\mathcal D$ and $A$ be unital and separable $C^{*}$-algebras; let $\mathcal D$ be strongly self-absorbing. It is known that any two unital ${}^*$-homomorphisms from $\mathcal D$ to $A \otimes \mathcal D$ are approximately unitarily equivalent. We show that, if $\mathcal D$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\mathcal D$ is asymptotically inner. Moreover, the space of automorphisms of $\mathcal D$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,(\mathrm{Aut}(\mathcal D)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\mathcal D$. As an application, we give a description of the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\mathcal D, A\otimes \mathcal D)$ is isomorphic to $K_0(A\otimes \mathcal D)$.

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31

Casals, Roger, and Oldřich Spáčil. "Chern–Weil theory and the group of strict contactomorphisms." Journal of Topology and Analysis 08, no. 01 (February 23, 2016): 59–87. http://dx.doi.org/10.1142/s1793525316500035.

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In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group [Formula: see text] is homotopically essential in the group of contactomorphisms of the standard contact sphere [Formula: see text] and present a proof of the homotopy equivalence [Formula: see text]. In the second part of the paper we focus on the group of strict contactomorphisms — using the framework of Chern–Weil theory we introduce and study contact characteristic classes analogous to the Reznikov Hamiltonian classes in symplectic topology. We carry out several explicit calculations illustrating the nontriviality of these contact characteristic classes.

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Kirschmer, Markus, and Gabriele Nebe. "Quaternary quadratic lattices over number fields." International Journal of Number Theory 15, no. 02 (March 2019): 309–25. http://dx.doi.org/10.1142/s1793042119500131.

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We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space [Formula: see text] with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of [Formula: see text]. This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in [Formula: see text].

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IMAI, HIROSHI, TOMONARI MASADA, FUMIHIKO TAKEUCHI, and KEIKO IMAI. "ENUMERATING TRIANGULATIONS IN GENERAL DIMENSIONS." International Journal of Computational Geometry & Applications 12, no. 06 (December 2002): 455–80. http://dx.doi.org/10.1142/s0218195902000980.

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We propose algorithms to enumerate (1) regular triangulations, (2) spanning regular triangulations, (3) equivalence classes of regular triangulations with respect to symmetry, and (4) all triangulations. All of the algorithms are for arbitrary points in general dimension. They work in output-size sensitive time with memory only of several times the size of a triangulation. For the enumeration of regular triangulations, we use the fact by Gel'fand, Zelevinskii and Kapranov that regular triangulations correspond to the vertices of the secondary polytope. We use reverse search technique by Avis and Fukuda, its extension for enumerating equivalence classes of objects, and a reformulation of a maximal independent set enumeration algorithm. The last approach can be extended for enumeration of dissections.

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Jan, Naeem, Saif Ur Rehman, Abdul Nasir, Hassen Aydi, and Sami Ullah Khan. "Analysis of Economic Relationship Using the Concept of Complex Pythagorean Fuzzy Information." Security and Communication Networks 2021 (June 8, 2021): 1–12. http://dx.doi.org/10.1155/2021/4513992.

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Fuzzy set theory and fuzzy logics are the powerful mathematical tools to model the imprecision and vagueness. In this research, the novel concept of complex Pythagorean fuzzy relation (CPFR) is introduced. Furthermore, the types of CPFRs are explained with appropriate examples such as CPF composite relation, CPF equivalence relation, CPF order relation, and CPF equivalence classes. Moreover, numerous results and interesting properties of CPFRs are discussed in detail. Furthermore, the impacts of economic parameters over each other are studied through the proposed concepts of CPFRs. In addition, the application also discusses the effects of economic parameters of one country over the other countries’ economic parameters.

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ZHANG, SHUANG. "K-THEORY AND HOMOTOPY OF CERTAIN GROUPS AND INFINITE GRASSMANN SPACES ASSOCIATED WITH C*-ALGEBRAS." International Journal of Mathematics 05, no. 03 (June 1994): 425–45. http://dx.doi.org/10.1142/s0129167x94000243.

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We determine, in terms of [Formula: see text] and [Formula: see text], the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*-[Formula: see text]-module. These results provide various interpretations of [Formula: see text].

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Funakoshi, Yukari, Megumi Hashizume, Noboru Ito, Tsuyoshi Kobayashi, and Hiroko Murai. "A distance on the equivalence classes of spherical curves generated by deformations of type RI." Journal of Knot Theory and Its Ramifications 27, no. 12 (October 2018): 1850066. http://dx.doi.org/10.1142/s0218216518500669.

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In this paper, we introduce a distance [Formula: see text] on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text] satisfy a certain technical condition, then [Formula: see text] is obtained from [Formula: see text] by a single weak RIII only. In Theorem 3, we show that if [Formula: see text] and [Formula: see text] satisfy other conditions, then [Formula: see text] is ambient isotopic to a spherical curve that is obtained from [Formula: see text] by a sequence of a particular local deformations, which realizes [Formula: see text].

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37

Hashemi, Ebrahim, Mona Abdi, and Abdollah Alhevaz. "On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings." Journal of Algebra and Its Applications 18, no. 07 (July 2019): 1950126. http://dx.doi.org/10.1142/s0219498819501263.

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Let [Formula: see text] be an associative ring with nonzero identity. The zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonzero zero-divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.

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Abd El-Monsef, M. M. E., and N. M. Kilany. "Decision Analysis via Granulation Based on General Binary Relation." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–13. http://dx.doi.org/10.1155/2007/12714.

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Decision theory considers how best to make decisions in the light of uncertainty about data. There are several methodologies that may be used to determine the best decision. In rough set theory, the classification of objects according to approximation operators can be fitted into the Bayesian decision-theoretic model, with respect to three regions (positive, negative, and boundary region). Granulation using equivalence classes is a restriction that limits the decision makers. In this paper, we introduce a generalization and modification of decision-theoretic rough set model by using granular computing on general binary relations. We obtain two new types of approximation that enable us to classify the objects into five regions instead of three regions. The classification of decision region into five areas will enlarge the range of choice for decision makers.

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Pillay, Anand. "Some remarks on definable equivalence relations in O-minimal structures." Journal of Symbolic Logic 51, no. 3 (September 1986): 709–14. http://dx.doi.org/10.2307/2274024.

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Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n-tuples in M, to study definable (in M) equivalence relations on Mn. In particular, we show that if E is an A-definable equivalence relation on Mn (A ⊂ M) then E has only finitely many classes with nonempty interior in Mn, each such class being moreover also A-definable. As a consequence, we are able to give some conditions under which an O-minimal theory T eliminates imaginaries (in the sense of Poizat [P]).If L is a first order language and M an L-structure, then by a definable set in M, we mean something of the form X ⊂ Mn, n ≥ 1, where X = {(a1…,an) ∈ Mn: M ⊨ϕ(ā)} for some formula ∈ L(M). (Here L(M) means L together with names for the elements of M.) If the parameters from come from a subset A of M, we say that X is A-definable.M is said to be O-minimal if M = (M, <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals (a, b) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th(M) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M.

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Zhang, Qing-Hua, Long-Yang Yao, Guan-Sheng Zhang, and Yu-Ke Xin. "The Incremental Knowledge Acquisition Based on Hash Algorithm." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24, no. 03 (June 2016): 347–66. http://dx.doi.org/10.1142/s0218488516500173.

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In this paper, a new incremental knowledge acquisition method is proposed based on rough set theory, decision tree and granular computing. In order to effectively process dynamic data, describing the data by rough set theory, computing equivalence classes and calculating positive region with hash algorithm are analyzed respectively at first. Then, attribute reduction, value reduction and the extraction of rule set by hash algorithm are completed efficiently. Finally, for each new additional data, the incremental knowledge acquisition method is proposed and used to update the original rules. Both algorithm analysis and experiments show that for processing the dynamic information systems, compared with the traditional algorithms and the incremental knowledge acquisition algorithms based on granular computing, the time complexity of the proposed algorithm is lower due to the efficiency of hash algorithm and also this algorithm is more effective when it is used to deal with the huge data sets.

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Harnik, Victor, and Michael Makkai. "Lambek's categorical proof theory and Läuchli's abstract realizability." Journal of Symbolic Logic 57, no. 1 (March 1992): 200–230. http://dx.doi.org/10.2307/2275186.

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In this paper we give an introduction to categorical proof theory, and reinterpret, with improvements, Läuchli's work on abstract realizability restricted to propositional logic (but see [M1] for predicate logic). Partly to make some points of a foundational nature, we have included a substantial amount of background material. As a result, the paper is (we hope) readable with a knowledge of just the rudiments of category theory, the notions of category, functor, natural transformation, and the like. We start with an extended introduction giving the background, and stating what we do with a minimum of technicalities.In three publications [L1, 2, 3] published in the years 1968, 1969 and 1972, J. Lambek gave a categorical formulation of the notion of formal proof in deductive systems in certain propositional calculi. The theory is also described in the recent book [LS]. See also [Sz].The basic motivation behind Lambek's theory was to place proof theory in the framework of modern abstract mathematics. The spirit of the latter, at least for the purposes of the present discussion, is to organize mathematical objects into mathematical structures. The specific kind of structure we will be concerned with is category.In Lambek's theory, one starts with an arbitrary theory in any one of several propositional calculi. One has the (formal) proofs (deductions) in the given theory of entailments A ⇒ B, with A and B arbitrary formulas. One introduces an equivalence relation on proofs under which, in particular, equivalent proofs are proofs of the same entailment; equivalence of proofs is intended to capture the idea of the proofs being only inessentially different. One forms a category whose objects are the formulas of the underlying language of the theory, and whose arrows from A to B, with the latter arbitrary formulas, are the equivalence classes of formal proofs of A ⇒ B.

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BEATTIE, M., and C. WEATHERBY. "PYTHAGOREAN TRIPLES AND UNITS IN INTEGRAL GROUP RINGS." Journal of Algebra and Its Applications 04, no. 04 (August 2005): 355–67. http://dx.doi.org/10.1142/s021949880500123x.

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In this note, we describe a simple method for finding units of group rings of the form Z[G] = Z[H]#Z[C2] for H an abelian group, and apply this to the case G = D4, the dihedral group of order 8. Here, units may be described as integer points on hyperboloids, and, defining units u, v to be equivalent if they differ by an inner automorphism, we see that this equivalence relation partitions each hyperboloid into finitely many classes.

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Hjorth, Greg, and Alexander S. Kechris. "New Dichotomies for Borel Equivalence Relations." Bulletin of Symbolic Logic 3, no. 3 (September 1997): 329–46. http://dx.doi.org/10.2307/421148.

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We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).Intuitively, E ≤BF might be interpreted in any one of the following ways:(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).(ii) One can classify E by using as invariants F-equivalence classes.(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.

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Cheng, Linhai, Yu Zhang, Yingying He, and Yuejin Lv. "Rough set models of interval rough number information system." Journal of Intelligent & Fuzzy Systems 40, no. 1 (January 4, 2021): 1655–66. http://dx.doi.org/10.3233/jifs-191096.

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Classical rough set theory (RST) is based on equivalence relations, and does not have an effective mechanism when the attribute value of the objects is uncertain information. However, the information in actual problems is often uncertain, and an accurate or too vague description of the information can no longer fully meet the actual needs. Interval rough number (IRN) can reflect a certain degree of certainty in the uncertainty of the data when describing the uncertainty of the data, and can enable decision makers to make decisions more in line with actual needs according to their risk preferences. However, the current research on rough set models (RSMs) whose attribute values are interval rough numbers is still very scarce, and they cannot analyze the interval rough number information system (IRNIS) from the perspective of similar relation. therefore, three new interval rough number rough set models (IRNRSMs) based on similar relation are proposed in this paper. Firstly, aiming at the limitations of the existing interval similarity degree (ISD), new interval similarity degree and interval rough number similarity degree (IRNSD) are proposed, and their properties are discussed. Secondly, in the IRNIS, based on the newly proposed IRNSD, three IRNRSMs based on similar class, β-maximal consistent class and β-equivalent class are proposed, and their properties are discussed. And then, the relationships between these three IRNRSMs and those between their corresponding approximation accuracies are researched. Finally, it can be found that the IRNRSM based on the β-equivalent classes has the highest approximation accuracy. Proposing new IRNRSMs based on similar relation is a meaningful contribution to extending the application range of RST.

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Yamashita, Shigeru, and Igor L. Markov. "Fast equivalence -- checking for quantum circuits." Quantum Information and Computation 10, no. 9&10 (September 2010): 721–34. http://dx.doi.org/10.26421/qic10.9-10-1.

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We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage existing techniques for simplification of quantum circuits. For reversible circuits which arise as runtime bottlenecks of key quantum algorithms, we develop several verification techniques and empirically compare them. We also combine existing quantum verification tools with the use of SAT-solvers. Experiments with circuits for Shor's number-factoring algorithm, containing thousands of gates, show improvements in efficiency by four orders of magnitude.

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LIN, HUAXIN. "C*-ALGEBRAS OF TRIVIAL K-THEORY AND SEMILATTICES." International Journal of Mathematics 10, no. 01 (February 1999): 93–128. http://dx.doi.org/10.1142/s0129167x99000057.

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We give a class of nuclear C*-algebras which contains [Formula: see text] and is closed under stable isomorphism, ideals, quotients, hereditary subalgebras, tensor products, direct sums, direct limits as well as extensions. We show that this class of C*-algebras is classified by their equivalence classes of projections and there is a one to one correspondence between (unital) C*-algebras in the class and countable distributive semilattices (with largest elements). One of the main results is that essential extensions of a C*-algebras which is a direct limit of finite direct sums of corners of [Formula: see text] by the same type of C*-algebras are still direct limits of finite direct sums of corners of [Formula: see text].

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ARMSTRONG, JOHN. "FUNCTORS EXTENDING THE KAUFFMAN BRACKET." Journal of Knot Theory and Its Ramifications 18, no. 07 (July 2009): 985–98. http://dx.doi.org/10.1142/s0218216509007282.

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In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category [Formula: see text] of framed tangles. We show that many such functors exist, and that they correspond up to equivalence to bilinear forms on free, finitely-generated modules over commutative rings R.

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48

Ara, Pere. "On the symmetric algebra of quotients of a C*-algebra." Glasgow Mathematical Journal 32, no. 3 (September 1990): 377–79. http://dx.doi.org/10.1017/s0017089500009460.

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Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

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Phoa, Wesley. "Building domains from graph models." Mathematical Structures in Computer Science 2, no. 3 (September 1992): 277–99. http://dx.doi.org/10.1017/s0960129500001481.

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In this paper we study partial equivalence relations (PERs) over graph models of the λcalculus. We define categories of PERs that behave like predomains, and like domains. These categories are small and complete; so we can solve domain equations and construct polymorphic types inside them. Upper, lower and convex powerdomain constructions are also available, as well as interpretations of subtyping and bounded quantification. Rather than performing explicit calculations with PERs, we work inside the appropriate realizability topos: this is a model of constructive set theory in which PERs, can be regarded simply as special kinds of sets. In this framework, most of the definitions and proofs become quite smple and attractives. They illustrative some general technicques in ‘synthetic domain theory’ that rely heavily on category theory; using these methods, we can obtain quite powerful results about classes of PERs, even when we know very little about their internal structure.

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Manturov, Vassily Olegovich. "Reidemeister moves and groups." Journal of Knot Theory and Its Ramifications 24, no. 10 (September 2015): 1540006. http://dx.doi.org/10.1142/s0218216515400064.

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Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

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Journal articles on the topic 'Equivalence classes (Set theory)'

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