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Relevant bibliographies by topics / Equivalence classes (Set theory)

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Academic literature on the topic 'Equivalence classes (Set theory)'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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Journal articles on the topic "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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Dissertations / Theses on the topic "Equivalence classes (Set theory)"

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Kashcheyeva, Olga S. "Monomialization of strongly prepared morphisms to surfaces /." free to MU campus, to others for purchase, 2003. http://wwwlib.umi.com/cr/mo/fullcit?p3091935.

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Roberts, Creta M. "Promoting generalization of coin value relations with young children via equivalence class formation." Virtual Press, 1999. http://liblink.bsu.edu/uhtbin/catkey/1137578.

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Sidman and Tailby (1982) established procedures to analyze the nature of stimulus to stimulus relations established by conditional discriminations. Their research describes specific behavioral tests to determine the establishment of properties that define the relations of equivalence. An equivalence relation requires the demonstration of three conditional relations: reflexivity, symmetry, and transitivity. The equivalence stimulus paradigm provides a method to account for novel responding. The research suggests that equivalence relations provide a more efficient and effective approach to the assessment, analysis, and instruction of skills. The present research examined the effectiveness of the formation of an equivalence class in teaching young children coin value relations. The second aspect of the study was to determine if there was a relationship between equivalence class formation and generalization of the skills established to other settings. Five children, 4- and 5-years old, were selected to participate in the study based on their lack of skills in the area of coin values and purchasing an item with dimes or quarters equaling fifth cents. The experimental task was presented on a Macintosh computer with HyperCard programming. The experimental stimuli consisted of pictures of dimes, quarters, and Hershey candy bars presented in match-to-sample procedures. Two conditional discriminations were taught (if A then B and if B then C.). The formation of an equivalence class was evaluated by if C then A. Generalization across settings was tested after the formation of an equivalence class by having the children purchase a Hershey candy bar with dimes at a play store. A multiple baseline experimentaldesign was used to demonstrate a functional relationship between the formation of an equivalence class and generalization of skills across settings. The present research provides supportive evidence that coin value relations can be taught to young children using equivalence procedures. The study also demonstrated generalization of novel, untaught stimuli across settings, after the formation of an equivalence class. A posttest on generalization across settings was conducted 3 months after the study. Long-term stability of equivalence relations was demonstrated by three of the subjects.
Department of Special Education

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Ndiweni, Odilo. "The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes." Thesis, University of Fort Hare, 2007. http://hdl.handle.net/10353/88.

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In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

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Metcalfe, Marta J. "Teaching phonics skills to young children via the formation of generalized equivalence classes." Virtual Press, 1999. http://liblink.bsu.edu/uhtbin/catkey/1137509.

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An equivalence class exists if the stimuli that comprise the class are related by the properties of reflexivity, symmetry, and transitivity. Through these properties, new behavior that has not been taught emerges. For example, when taught to match Set A stimuli to Set B stimuli and to match Set A stimuli to Set C stimuli, if equivalence classes have formed, subjects will (with no explicit instruction) match Set B stimuli to Set C and Set C stimuli to Set B stimuli. Although equivalence classes have been studied extensively, few studies have considered the application of this technology to educational concerns. The purpose of this study was (a) to determine if phonics skills could effectively and efficiently be taught to young children through the formation of equivalence classes and (b) to investigate the generality of those acquired skills. Using a conditional discrimination procedure, children were taught to match printed letters to dictated phonetic sounds and to match the initial sound of pictured items to dictated phonetic sounds. Test results indicated that equivalence classes had emerged and that generalization did occur. The children could match the initial sound of pictured items to printed letters and vice versa and could name letter sounds and initial sounds of pictured items. During generality testing, each child could identify the initial sound of several novel pictured items and could sound out the letters within the words. However, reading did not occur. Only 1 of 5 children could blend the sounds of letters into recognizable words. A significant difficulty encountered throughout the study was maintaining the children's motivation, possibly due to the children's inexperience in attending to academic tasks. This study did, however, demonstrate that the formation of equivalence classes is an effective and efficient method for teaching phonics and that the formation of generalized equivalence classes is effective in extending those taught relations to novel stimuli.
Department of Special Education

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Gomes, Lúcia Pereira dos Santos. "Estudo das relações." Mestrado Profissional em Matemática, 2013. http://ri.ufs.br:8080/xmlui/handle/123456789/6518.

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The purpose of this monograph is to make a detailed study about studying of relations. So, we will provide some definitions. Developing this Themes like binary relations, equivalence relations and order relations. At the same we will show ways to represent relations highlighting the representation in the form of graphs. And we will finish the job with Topological Ordering in completing some tasks of a project.
O objetivo desta monografia é fazer um estudo detalhado sobre relações. Para tanto, forneceremos algumas definições. Para o desenvolvimento deste foram abordados temas como relações binárias, relações de equivalência e relações de ordem. No desenvolvimento deste veremos formas de representar as relações dando destaque a representação na forma de Grafos. E finalizamos o trabalho com a Ordenação Topológica na conclusão de algumas tarefas de um projeto.

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Craft, Colin N. "Applications of a Model-Theoretic Approach to Borel Equivalence Relations." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1538768/.

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The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.

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Kieftenbeld, Vincent. "Three Topics in Descriptive Set Theory." Thesis, University of North Texas, 2010. https://digital.library.unt.edu/ark:/67531/metadc28441/.

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This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.

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Silguero, Russell V. "Do contingency-conflicting elements drop out of equivalence classes? Re-testing Sidman's (2000) theory." Thesis, University of North Texas, 2015. https://digital.library.unt.edu/ark:/67531/metadc848078/.

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Sidman's (2000) theory of stimulus equivalence states that all positive elements in a reinforcement contingency enter an equivalence class. The theory also states that if an element from an equivalence class conflicts with a programmed reinforcement contingency, the conflicting element will drop out of the equivalence class. Minster et al. (2006) found evidence suggesting that a conflicting element does not drop out of an equivalence class. In an effort to explain maintained accuracy on programmed reinforcement contingencies, the authors seem to suggest that participants will behave in accordance with a particular partitioning of the equivalence class which continues to include the conflicting element. This hypothesis seems to explain their data well, but their particular procedures are not a good test of the notion of "dropping out" due to the pre-establishment of equivalence classes before the conflicting member entered the class. The current experiment first developed unpartitioned equivalence classes and only later exposed participants to reinforcement contingencies that conflicted with pre-established equivalence classes. The results are consistent with the notion that a partition developed such that the conflicting element had dropped out of certain subclasses of the original equivalence class. The notion of a partitioning of an equivalence class seems to provide a fuller description of the phenomenon Sidman (1994, 2000) described as "dropping out" of an equivalence class.

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Montagnoli, Tathianna Amorim Souza. "Um procedimento para investigar aprendizagem discriminativa e formação de classes funcionais em cães (Canis familiaris)." Universidade Federal de São Carlos, 2012. https://repositorio.ufscar.br/handle/ufscar/6048.

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Made available in DSpace on 2016-06-02T20:30:56Z (GMT). No. of bitstreams: 1 5453.pdf: 1352961 bytes, checksum: 6c7a8392ec5e53e805f81c028682190a (MD5) Previous issue date: 2012-07-30
Universidade Federal de Minas Gerais
The investigation of symbolic or pre-symbolic behavior in dogs requires the establishment of arbitrary relations among stimuli. A way to teach such relations is requiring a common response in the presence of each one of them, which can result in a class of functionally equivalent stimuli. This study aimed to investigate the formation of functional classes with four dogs. For this purpose it was used an automated device for emission and recording of operant responses and for the presentation of visual stimuli. The operant response was to nose poke a stimuli which were presented in a touch-screen monitor. Three sets of two stimuli (A1/A2; B1/B2; C1/C2) were used for three dogs and two sets of two stimuli (C1/C2; E1/E2) were used for the other dog. In each set, one stimulus was related to reinforcement (S+) and the other to extinction (S-), regarding tasks of simultaneous simple discrimination. Five experimental phases were programmed for three dogs: I) Training and reversals of pair A stimuli; II) Training and reversals of pair B; III) Training and reversions of AB being presented in the same session with functional class formation probes; IV) Training and reversals of pair C; and V) Training and reversals of ABC with functional class formation probes. On the third and fifth phases stimuli of the same set (A1 and B1 or A1, B1 and C1, respectively) were established as S+, and the stimuli of the other set (A2 and B2 or A2, B2 and C2) as S-. After baseline was acquired and reached stability the functions of the stimuli were reversed repeatedly (S+ began to function as S- and vice versa), and it was appraised if the animals reverted the functions of the other stimuli given the reversal of the first pair in one of the sets, and that before direct exposure to new contingencies which would indicate functional classes formation (S+ class and S- class). For the fourth dog four experimental phases were programmed: I) Training of pair E stimuli (without reversal); II) Training of pair C stimuli; III) Training of CE being presented in the same session; and IV) Three reversions of CE being presented in the same session. The results show that the procedure was able to establish a complex and flexible discriminative repertoire in dogs, but insufficient to demonstrate relational responding in the only animal to be exposed to the functional class probes. Nevertheless, considerations were made about the positive aspects of the proposed procedure and learning set formation.
A investigação de comportamentos simbólicos ou pré-simbólicos em cães requer o estabelecimento de relações arbitrárias entre estímulos. Uma alternativa para ensinar tais relações é a exigência de uma resposta comum na presença de cada um deles, o que pode resultar em uma classe de estímulos funcionalmente equivalentes. O presente estudo teve por objetivo investigar a formação de classes funcionais em quatro cães. Para esta finalidade foi utilizado um equipamento automático para a emissão e registro das respostas operantes e apresentação dos estímulos visuais. A resposta operante era tocar com o focinho sobre estímulos projetados em um monitor equipado com tela sensível ao toque. Para três cães foram empregados três conjuntos de dois estímulos (A1/A2; B1/B2; C1/C2) e para o outro cão dois conjuntos de dois estímulos (C1/C2; E1/E2). Em cada conjunto um dos estímulos do par era relacionado com reforço (S+) e o outro era relacionado com extinção (S-), em tarefas de discriminação simples simultânea. Para os três primeiros cães foram programadas cinco fases experimentais: I) Treino e reversões com estímulos do par A; II) Treino e reversões com o par B; III) Treino e reversões de AB, apresentados em uma mesma sessão, com sondas de formação de classes funcionais; IV) Treino e reversões do par C e V) Treino e reversões de ABC, com apresentação de sondas de formação de classes funcionais. Na terceira e quinta fases os estímulos de um mesmo conjunto (A1 e B1 ou A1, B1 e C1, respectivamente) eram estabelecidos como S+ e os estímulos do outro conjunto (A2 e B2 ou A2, B2 e C2) como S-. Após aquisição e estabilidade da linha de base, as funções dos estímulos eram revertidas (os S+ passavam a funcionar como S- e vice-versa) repetidas vezes e era avaliado se, a partir da reversão do primeiro par de estímulos de um dos conjuntos, os animais revertiam as funções dos outros estímulos, antes da exposição direta às novas contingências, evidenciando formação de classes funcionais (a classe dos S+ e a classe dos S-). Para o quarto animal foram programadas quatro fases experimentais: I) Treino do par E (sem reversões) II) Treino do par C III) Treino dos pares C e E apresentados em uma mesma sessão e IV) Três reversão sucessivas dos pares C e E apresentados na mesma sessão. Os resultados obtidos mostram que o procedimento adotado foi capaz de estabelecer um repertório discriminativo complexo e flexível em cães, porém insuficiente para demonstrar responder relacional no único animal a passar pelas sondas de formação de classe funcional. Apesar disso, foram feitas considerações a respeito de aspectos positivos sobre o procedimento proposto e a formação de learning set.

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Holshouser, Jared Kenneth. "Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy." Thesis, University of North Texas, 2017. https://digital.library.unt.edu/ark:/67531/metadc984121/.

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In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

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Books on the topic "Equivalence classes (Set theory)"

1

Borel equivalence relations: Structure and classification. Providence, R.I: American Mathematical Society, 2008.

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Hjorth, Greg. Classification and orbit equivalence relations. Providence, RI: American Mathematical Society, 2000.

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G, Sushkov B., ed. Preobrazovanie logicheskikh funkt͡siĭ na klasse ėkvivalentnykh dvoichnykh derevʹev. Moskva: Vychislitelʹnyĭ t͡sentr AN SSSR, 1986.

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Kechris, A. S. Topics in orbit equivalence. Berlin: Springer, 2004.

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Castrén, Marcus. Recrel: A similarity measure for set-classes. Helsinki: Sibelius Academy, 1994.

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Hjorth, Greg. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Providence, RI: American Mathematical Society, 2005.

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K, Lewis David. Parts of classes. Oxford, UK: B. Blackwell, 1991.

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Parts of classes. Oxford: Basil Blackwell, 1990.

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Fulman, Igor. Crossed products of von Neumann algebras by equivalence relations and their subalgebras. Providence, R.I: American Mathematical Society, 1997.

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Weekly Reader Early Learning Library (Firm), ed. I know same and different. Milwaukee: Weekly Reader Early Learning Library, 2006.

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Book chapters on the topic "Equivalence classes (Set theory)"

1

Adamson, Iain T. "Equivalence Relations." In A Set Theory Workbook, 35–39. Boston, MA: Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-0-8176-8138-8_6.

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Hjorth, Greg. "Borel Equivalence Relations." In Handbook of Set Theory, 297–332. Dordrecht: Springer Netherlands, 2009. http://dx.doi.org/10.1007/978-1-4020-5764-9_5.

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Terefenko, Dariusz. "Set Classes in Jazz." In Jazz Theory, 336–55. Second edition. | New York ; London : Routledge, 2017.: Routledge, 2017. http://dx.doi.org/10.4324/9781315305394-25.

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Thomas, Simon. "Some applications of superrigidity to Borel equivalence relations." In Set Theory, 129–34. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/dimacs/058/10.

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Terefenko, Dariusz. "Set Classes in Jazz." In Jazz Theory Workbook, 88–94. New York ; London : Routledge, 2019.: Routledge, 2019. http://dx.doi.org/10.4324/9780429445477-25.

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Kanovei, Vladimir. "Some theorems of descriptive set theory." In Borel Equivalence Relations, 19–39. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/03.

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Kuga, Michio. "The Second Week: Equivalence Classes." In Galois’ Dream: Group Theory and Differential Equations, 19–22. Boston, MA: Birkhäuser Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0329-2_3.

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Fokina, Ekaterina B., and Sy-David Friedman. "Equivalence Relations on Classes of Computable Structures." In Mathematical Theory and Computational Practice, 198–207. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-03073-4_21.

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Nyikos, Peter J. "Classes of compact sequential spaces." In Set Theory and its Applications, 135–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/bfb0097337.

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Horadam, Kathy J., and Mercé Villanueva. "Relationships Between CCZ and EA Equivalence Classes and Corresponding Code Invariants." In Sequences and Their Applications - SETA 2014, 3–17. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-12325-7_1.

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Conference papers on the topic "Equivalence classes (Set theory)"

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Castillo Herrera, Ester, Luis Jimenez Linares, Luis Rodriguez Benitez, Juan Giralt Muina, and Juan Moreno Garcia. "Reduction of a Set of Fuzzy Classifiers by Equivalence Classes." In 2015 10th International Conference on Intelligent Systems and Knowledge Engineering (ISKE). IEEE, 2015. http://dx.doi.org/10.1109/iske.2015.41.

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Le Bars, Jean-Marie, and Alfredo Viola. "Equivalence classes of boolean functions for first-order correlation." In 2007 IEEE International Symposium on Information Theory. IEEE, 2007. http://dx.doi.org/10.1109/isit.2007.4557223.

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Sprintson, Alex. "Reductions techniques for establishing equivalence between different classes of network and index coding problems." In 2014 IEEE Information Theory Workshop (ITW). IEEE, 2014. http://dx.doi.org/10.1109/itw.2014.6970795.

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Voronin, Vladimir. "EQUIVALENCE OF DEFECTS IN TECHNICAL DIAGNOSTICS." In CAD/EDA/SIMULATION IN MODERN ELECTRONICS 2019. Bryansk State Technical University, 2019. http://dx.doi.org/10.30987/conferencearticle_5e0282101fc825.85024100.

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Equivalence relations are analyzed on the set of possible defects. Different classes of relative equivalence of defects are distinguished. The concept of sensitivity of diagnostic indicators to the elements of the set of possible defects is introduced.

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Oguni, Shin-ichi. "Dilatational Equivalence Classes and Novikov-Shubin Type Capacities of Groups, and Random Walks." In Proceedings of the Noncommutative Geometry and Physics 2008, on K-Theory and D-Branes & Proceedings of the RIMS Thematic Year 2010 on Perspectives in Deformation Quantization and Noncommutative Geometry. WORLD SCIENTIFIC, 2013. http://dx.doi.org/10.1142/9789814425018_0016.

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Lin, He, Mengyao Nie, and Lingyue Li. "The extension of the rough set theory based on parallel equivalence operator in pansystems." In 2014 5th International Conference on Computing, Communication and Networking Technologies (ICCCNT). IEEE, 2014. http://dx.doi.org/10.1109/icccnt.2014.6963103.

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Wang, Fengxia, and Anil K. Bajaj. "On the Equivalence of Normal Form Theory and Multiple Time Scale Method." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35603.

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Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

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Kim, Kyoung-Yun, and Keunho Choi. "Rough Set-Based Design Rule Selection for Collaborative Assembly Design." In ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49191.

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While the modern product development requires more knowledge-intensive and collaborative environment, the capture, retrieval, accessibility, and reusability of that design knowledge are increasing critical. In this paper, a rough set theory generates demanded rules and selects the appropriate minimal rules among the demanded design rules associated to the assembly design knowledge. The design rules are infrequently captured and often ignored due to its complexity. Rough set theory synthesizes approximation of concepts, analyzes data by discovering patterns, and classifies into certain decision classes. Such patterns can be extracted from data by means of methods based on Boolean reasoning and discernibility. This paper shows the feasibility of rough-set based rule selection considering complex design data objects in order to obtain efficient assembly design decision.

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Scanlan, Kirk M., Michael T. Hendry, and C. Derek Martin. "Evaluating the Equivalency Between Track Quality Indices and the Minimum Track Geometry Threshold Exceedances Along a Canadian Freight Railway." In 2016 Joint Rail Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/jrc2016-5748.

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Railway regulators require that track geometry measurements meet a specific set of minimum safety thresholds. A proper interpretation of track geometry survey data is fundamental for the detection of track exceeding these thresholds and in need of corrective maintenance. Irregular track geometry independent of the minimum safety thresholds can also be used as evidence of degradation in the railway foundation. Therefore, multiple evaluation methods must be applied to the track geometry survey data when assessing foundation degradation. In this study, we compare multiple track geometry evaluation methods in order to assess if they equally identify sections of irregular track geometry along a 335 kilometer section of a Canadian freight railway. The track geometry evaluation methods investigated are the Transport Canada Class 5 minimum safety threshold exceedances and three literature-suggested track quality indices; the Overall Track Geometry Index, the Polish J Index and the Swedish Q Index. Furthermore, this study also investigates the ability of the track quality indices to provide additional insight into track geometry variability in sections without a minimum safety threshold exceedance. The track under investigation is not a Class 5, however, Class 5 minimum safety thresholds were used to produce enough threshold exceedances to allow for the comparison to the track quality indices. The results of the analysis reveal that while the large-scale variability in the three track quality indices is similar, the individual equivalency with the occurrence of Class 5 threshold exceedances is highly variable. Furthermore, only the Overall Track Geometry Index demonstrates the potential to provide consistent additional track geometry variability information.

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Lerbet, Jean. "Stability of Singularities of a Kinematical Chain." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84126.

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Since few years, the geometry of singularities of kinematical chains is better known. Using both Lie group theory and concepts of differential or analytic geometry, we already classified these singularities according to the nature of the function f associated to the chain (f is a product of exponential mappings of a Lie group). In the most general case, the tangent cone at a singularity has been explicitely given. Here, a different (and more difficult) aspect of the problem is studied. The concrete realisation of a kinematical chain is never perfect. That means that the vectors of the Lie agebra defining the function f are not exactly those of the chain: they are deformed. What happens for the singularities in this case? Are they remaining or do they disappear during the deformation? First, the mechanical problem is analysed as this of the stability of the fuction f and mathematical tools concerning stable mappings are given. Stability of a mapping means that the orbit of f under the action of diffeomorphisms in the source and in the target is an open set and its infinitesimal equivalent formulation is noted the inf-stability. Then we prove that the set of singularities of first class itself is a sbmanifold and we analyse a condition of normal crossing of the restriction of f to its manifold of singularities. Applying a result of the general theory, the stability of f is analysed.

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