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Frisk, Anders. "On the structure of standardly stratified algebras /." Uppsala, 2004. http://www.math.uu.se/research/pub/Frisk5lic.pdf.
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Wiesnet, Franziskus Wolfgang Josef. "The computational content of abstract algebra and analysis." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/313875.
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In this dissertation we discuss several forms of proof interpretation based on examples in algebra and analysis. Our main goal is the construction of algorithms or functions out of proofs - constructive proofs and classical proofs. At the beginning of this dissertation the constructive handling of Zorn's lemma in proofs of algebra plays a main role. We consider maximal objects and their approximations in various algebraic structures. The approximation is inspired by Gödel's functional interpretation and methods from proof mining. Proof mining is a branch of mathematical logic which analyses classical existence statements to get explicit bounds. We describe a recursive algorithm which constructs a sequence of approximations until they become close enough. This algorithm will be applied to Krull's lemma and its generalisation: the universal Krull-Lindenbaum lemma. In some case studies and applications like the theorem of Gauß-Joyal and Kronecker's theorem we show explicit uses of this algorithm.
By considering Zariski's lemma we present how a manual construction of an algorithm from a constructive proof works. We formulate a constructive proof of Zariski's lemma, and use this proof as inspiration to formulate an algorithm and a computational interpretation of Zariski's lemma. Finally, we prove that the algorithm indeed fulfils the computational interpretation. As an outlook we sketch how this algorithm could be combined with our state-based algorithm from above which constructs approximations of maximal objects, to get an algorithm for Hilbert's Nullstellensatz. We then take a closer look at an example of an algorithm which is extracted out of a constructive proof by using a computer assistant. In our case we use the proof assistant Minlog. We use coinductively defined predicates to prove that the signed digit representation of real numbers is closed under limits of convergent sequences. From a formal proof in Minlog, the computer generates an algorithm, a soundness theorem and a proof of the soundness theorem corresponding to the convergence theorem out of our formalisation. The convergence theorem is applied to Heron's method to get an algorithm which computes the signed digit representation of the square root of a non-negative real number out of its signed digit representation. Here we use the programming language Haskell to display the computational content. As second application we show that the signed digit code is closed under multiplication and state the corresponding algorithm. This dissertation is concluded by an example of proof mining in analysis. We consider one convergence lemma with a classical proof, use techniques of proof mining to get a new lemma with a rate of convergence, and apply the new lemma to various fixed-point theorems for asymptotically weakly contractive maps and their variants. In the last step, beside extracting quantitative information - in our case rates of convergence - from proofs, we also discuss the usage of proof mining to generalise or combine theorems.
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Wiesnet, Franziskus Wolfgang Josef. "The computational content of abstract algebra and analysis." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/313875.
Full textAbstract:
In this dissertation we discuss several forms of proof interpretation based on examples in algebra and analysis. Our main goal is the construction of algorithms or functions out of proofs - constructive proofs and classical proofs.
At the beginning of this dissertation the constructive handling of Zorn's lemma in proofs of algebra plays a main role. We consider maximal objects and their approximations in various algebraic structures. The approximation is inspired by Gödel's functional interpretation and methods from proof mining. Proof mining is a branch of mathematical logic which analyses classical existence statements to get explicit bounds. We describe a recursive algorithm which constructs a sequence of approximations until they become close enough.
This algorithm will be applied to Krull's lemma and its generalisation: the universal Krull-Lindenbaum lemma. In some case studies and applications like the theorem of Gauß-Joyal and Kronecker's theorem we show explicit uses of this algorithm.
By considering Zariski's lemma we present how a manual construction of an algorithm from a constructive proof works. We formulate a constructive proof of Zariski's lemma, and use this proof as inspiration to formulate an algorithm and a computational interpretation of Zariski's lemma. Finally, we prove that the algorithm indeed fulfils the computational interpretation. As an outlook we sketch how this algorithm could be combined with our state-based algorithm from above which constructs approximations of maximal objects, to get an algorithm for Hilbert's Nullstellensatz.
We then take a closer look at an example of an algorithm which is extracted out of a constructive proof by using a computer assistant. In our case we use the proof assistant Minlog. We use coinductively defined predicates to prove that the signed digit representation of real numbers is closed under limits of convergent sequences. From a formal proof in Minlog, the computer generates an algorithm, a soundness theorem and a proof of the soundness theorem corresponding to the convergence theorem out of our formalisation.
The convergence theorem is applied to Heron's method to get an algorithm which computes the signed digit representation of the square root of a non-negative real number out of its signed digit representation. Here we use the programming language Haskell to display the computational content. As second application we show that the signed digit code is closed under multiplication and state the corresponding algorithm.
This dissertation is concluded by an example of proof mining in analysis. We consider one convergence lemma with a classical proof, use techniques of proof mining to get a new lemma with a rate of convergence, and apply the new lemma to various fixed-point theorems for asymptotically weakly contractive maps and their variants. In the last step, beside extracting quantitative information - in our case rates of convergence - from proofs, we also discuss the usage of proof mining to generalise or combine theorems.
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Juett, Jason Robert. "Some topics in abstract factorization." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/2534.
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Anderson and Frazier defined a generalization of factorization in integral domains called tau-factorization. If D is an integral domain and tau is a symmetric relation on the nonzero nonunits of D, then a tau-factorization of a nonzero nonunit a in D is an expression a = lambda a_1 ... a_n, where lambda is a unit in D, each a_i is a nonzero nonunit in D, and a_i tau a_j for i != j. If tau = D^# x D^#, where D^# denotes the nonzero nonunits of D, then the tau-factorizations are just the usual factorizations, and with other choices of tau we get interesting variants on standard factorization. For example, if we define a tau_d b if and only if (a, b) = D, then the tau_d-factorizations are the comaximal factorizations introduced by McAdam and Swan. Anderson and Frazier defined tau-factorization analogues of many different factorization concepts and properties, and proved a number of theorems either generalizing standard factorization results or the comaximal factorization results of McAdam and Swan. Some of these concepts include tau-UFD's, tau-atomic domains, the tau-ACCP property, tau-BFD's, tau-FFD's, and tau-HFD's. They showed the implications between these concepts and showed how each of the standard variations implied their tau-factorization counterparts (sometimes assuming certain natural constraints on tau). Later, Ortiz-Albino introduced a new concept called Gamma-factorization that generalized tau-factorization. We will summarize the known theory of tau-factorization and Gamma-factorization as well as introduce several new or improved results.
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Fernandes, Renato da Silva. "Combinatória: dos princípios fundamentais da contagem à álgebra abstrata." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-31012018-161438/.
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O objetivo deste trabalho é fazer um estudo amplo e sequencial sobre combinatória. Iniciase com os fundamentos da combinatória enumerativa, tais como permutações, combinações simples, combinações completas e os lemas de Kaplanski. Num segundo momento é apresentado uma abordagem aos problemas de contagem utilizando a teoria de conjuntos; são abordados o princípio da inclusão-exclusão, permutações caóticas e a contagem de funções. No terceiro momento é feito um aprofundamento do conceito de permutação sob a ótica da álgebra abstrata. É explorado o conceito de grupo de permutações e resultados importantes relacionados. Na sequência propõe-se uma relação de ordem completa e estrita para o grupo de permutações. Por fim, investiga-se dois problemas interessantes da combinatória: a determinação do número de caminhos numa malha quadriculada e a contagem de permutações que desconhecem padrões de comprimento três.The objective of this work is to make a broad and sequential study on combinatorics. It begins with the foundations of enumerative combinatorics, such as permutations, simple combinations, complete combinations, and Kaplanskis lemmas. In a second moment an approach is presented to the counting problems using set theory; the principle of inclusion-exclusion, chaotic permutations and the counting of functions are addressed. In the third moment a deepening of the concept of permutation is made from the perspective of abstract algebra. The concept of group of permutations and related important results is explored. A strict total order relation for the permutation group is proposed. Finally, we investigate two interesting combinatorial problems: the determination of the number of paths in a grid and the number of permutations that avoids patterns of length three.
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Awuah, Bernard Prince. "The effectiveness of the concrete / semi-concrete / abstract (CSA) appoach and drill- practice on grade 10 learners' ability to simplify addition and subtraction algebraic fractions." Thesis, University of Fort Hare, 2016. http://hdl.handle.net/10353/5105.
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This study was conducted in one of the education districts in the Eastern Cape Province of South Africa. The purpose was to analyse the effectiveness of the concrete/semi-concrete/abstract (CSA) approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The following two objectives were set. First, to identify the learners’ challenges in studying addition and subtraction of algebraic fractions in grade 10; and second to analyse the effectiveness of the CSA approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. Both threshold concepts and troublesome knowledge, Polya’s problem-solving techniques, CSA Approach theory and Drill-practice theory were all pertinent as a theoretical framework for the study. Positivism research paradigm was adopted for the study and it afforded the researcher opportunity to employ quantitative research approach. Based on the research question of this study, an experimental design was chosen as a suitable descriptive design. Purposive sampling method was used to select three schools which involved 135 grade 10 mathematics learners. Stratified random sampling method was thereafter employed to select 45 learners from each school for the study. The learners were grouped in each school as top, average and weak based on their performance in Algebra in term one. Pre-questionnaire and post-questionnaire were used to obtain data regarding challenges learners experience in simplifying addition and subtraction of algebraic fractions. Ethical clearance from the relevant school and university authorities were obtained. On the first two days, the researcher briefed the school authorities and learners and explained to them the purpose and details of the study. Day three was used to administer the pre-questionnaire test, thereafter, the next ten days were used to teach addition and subtraction of both numeric and algebraic fractions with same and different numerators and denominators. The next two days were used for revision and the last day was used to administer the postquestionnaire test out 25 marks. The respondent rate was 98.5%. The data collected were analysed by using SPSS version 16.10. Both descriptive and inferential statistics were used to analyse the data. The pre-questionnaire scores revealed that majority of the learners’ perceived fractions as two separate entities and as a result add or subtract numerator to numerator and denominator to denominator. It was also discovered that learners had a challenge in finding LCM of algebraic fractions. A t-Test for independent means was used to test the following hypotheses at 𝛼 = 0.05: 𝐇𝟎: The CSA approach and drill-practice intervention has no significant effect on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions; 𝐇𝟏: The CSA approach and drill-practice will significantly enhance Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The t-Test revealed a p-value of 0.139 which was statistically significant at 𝛼 = 0.05. Therefore, the researcher rejected the null hypothesis and concluded that the CSA approach and drill-practice have significantly enhanced the Grade 10 learners’ ability to simplify algebraic fractions.
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Rome, Zachary Robert. "Computational Abstract Algebra: Using Monomial Matrices to Represent Groups in GAP." Thesis, The University of Arizona, 2012. http://hdl.handle.net/10150/244772.
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A monomial matrix is a matrix with exactly one non-zero element in each row and column. We will utilize GAP to construct all (transitive) representations of a given group using monomial matrices. First, essential group theory definitions and theorems will be provided, as well as an in-depth look at table of marks and monomial matrices. After describing the necessary mathematics, we will explore the GAP programming needed to achieve this goal. Ultimately we want a table, where each row represents a subgroup of the given group and, within the row, the table will hold the linear characters fixed by the monomial matrices of that subgroup. We will furthermore explore how to represent monomial matrices computationally in different ways and how to create a data structure to represent them. Our final goal will require GAP functions for finding all homomorphisms from a subgroup to roots of unity, using these homomorphisms to create monomial matrix representations of the group, and iterating through the subgroups of the group (up to conjugacy) to find all (transitive) monomial matrix representations of the group.
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Ahlgren, Joyce Christine. "Ideals, varieties, and Groebner bases." CSUSB ScholarWorks, 2003. https://scholarworks.lib.csusb.edu/etd-project/2282.
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The topics explored in this project present and interesting picture of close connections between algebra and geometry. Given a specific system of polynomial equations we show how to construct a Groebner basis using Buchbergers Algorithm. Gröbner bases have very nice properties, e.g. they do give a unique remainder in the division algorithm. We use these bases to solve systems of polynomial quations in several variables and to determine whether a function lies in the ideal.
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SPINOSA, Leonardo. "Burnside and Mackey Theories for Abstract Groupoids." Doctoral thesis, Università degli studi di Ferrara, 2019. http://hdl.handle.net/11392/2488082.
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The objective of this thesis is the study of Burnside and Mackey theories for abstract groupoids, which are natural generalizations of groups.
In fact, a groupoid can be thought as a group with many objects.
In Chapter 1 we will start explaining the basic properties of groupoids and after that, we will prove some properties of the two monoidal structures on the category of groupoid-sets (generalizations of group-sets). Namely, the structure given by the coproduct, that is, the disjoint union, and the other one given by the fibre product over the set of objects of the groupoid, considered canonically as a groupoid-set.
In Chapter 2 we will develop a theory of conjugations for subgroupoids, showing the deep differences with the group case.
Moreover, we will use this theory to prove a version of the Burnside Theorem for groupoids, with the appropriate finiteness hypotheses.
In Chapter 3 we will prove a version of the famous Mackey formula for group-bisets in the case of groupoid-bisets, showing its efficacy with a specific example.
In Chapter 4 we will discuss the equivalence of two groupoids as categories and what this implies for the groupoid-biset categories involved.
In Chapter 5, following the classical path, we will develop a Burnside theory for groupoids, showing that the Burnside ring of a groupoid is isomorphic to the direct product of the Burnside rings of its isotropy group types. This puts in evidence that the classical methods only lead to the study of a part of the poset of subgroupoids, as the subgroupoids with several objects don't show up.
In Chapter 6 we will develop a categorification of the classical notion of groupoid-set, replacing it with an internal category in the category of groupoid-sets, called a categorified groupoid-set.
Subsequently, we will use this notion to construct a new Burnside theory for groupoids and we will finish showing that, even in this case, the categorified Burnside ring of a groupoid is isomorphic to the direct product of the categorified Burnside rings of its isotropy group types.
This makes manifest that the study of groupoids needs more sophisticated tools and techniques than the traditional ones.
In the appendixes we will explain how to pass from a rig (also called semiring), that is, a ring without additive inverses, to a ring, using a construction called Grothendieck functor.
This notion is crucial to both the Burnside theories we developed.
Moreover, in the last appendix we will collect some definitions about monoidal categories, to fix the employed terminology, and we will prove a known result on monoidal functors whose whole proof we have been unable to find elsewhere.L'obiettivo di questa tesi è lo studio delle teorie di Mackey e Burnside per i gruppoidi astratti, che sono generalizzazioni naturali dei gruppi. Infatti, un gruppoide può essere pensato come un gruppo con molti oggetti. Nel Capitolo 1 verranno innanzitutto illustrate le proprietà basilari dei gruppoidi; successivamente, verranno provate alcune proprietà delle due strutture monoidali sulla categoria dei groupoid-set (generalizzazioni dei group-set). Per la precisione, la struttura data dal coprodotto, cioè l'unione disgiunta, e la struttura data dal prodotto fibrato sull'insieme degli oggetti del gruppoide, considerato in modo canonico come un groupoid-set. Nel Capitolo 2 verrà sviluppata una teoria delle coniugazioni per i sottogruppoidi, mostrando le profonde differenze con il caso dei gruppi. Inoltre, questa teoria verrà utilizzata per provare una versione del teorema di Burnside per i gruppoidi, con le appropriate ipotesi di finitezza. Nel Capitolo 3 verrà provata una versione della famosa formula di Mackey per i group-biset nel caso dei groupoid-biset, mostrando la sua efficacia con uno specifico esempio. Nel Capitolo 4 verrà discussa l'equivalenza di due gruppoidi come categorie e cosa questo implichi per le categorie di groupoid-biset coinvolte. Nel Capitolo 5, seguendo la teoria classica, verrà sviluppata una teoria di Burnside per i gruppoidi, mostrando come l'anello di Burnside di un gruppoide sia isomorfo al prodotto diretto degli anelli di Burnside dei suoi gruppi di isotropia, uno per ogni componente connessa. Tutto ciò dimostra chiaramente come i metodi classici conducano allo studio solamente di una parte del poset dei sottogruppoidi, dato che i sottogruppoidi con più di un oggetto non compaiono. Nel Capitolo 6 verrà sviluppata una categorificazione della classica nozione di groupoid-set, sostituendolo con una categoria interna alla categoria dei groupoid-set, chiamata un groupoid-set categorificato. Successivamente, questa nozione verrà utilizzata per costruire una nuova teoria di Burnside per i gruppoidi e, infine, verrà dimostrato come, anche in questo caso, l'anello di Burnside categorificato di un gruppoide sia isomorfo al prodotto diretto degli anelli di Burnside categorificati dei suoi gruppi di isotropia, uno per ogni componente connessa. Tutto ciò dimostra come lo studio dei gruppoidi necessiti di tecniche e strumenti più sofisticati di quelli tradizionali. Nelle appendici verrà spiegato come passare da un rig (chiamato anche semianello), cioè un anello privo degli inversi additivi, a un anello, usando una costruzione chiamata funtore di Grothendieck. Questa nozione è cruciale per entrambe le teorie di Burnside sviluppate in questa tesi. Inoltre, nell'ultima appendice verranno riunite alcune definizioni sulle categorie monoidali, per fissare la terminologia utilizzata, e verrà provato un risultato conosciuto del quale, però, non si riesce a trovare altrove la dimostrazione.
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Vernitski, Alexei. "Classes of abstract semigroups closed under the formation of subsemigroups and finitary direct products." Thesis, University of Essex, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284611.
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Zheng, Yi 1976. "Abstract data types and extended domain operations in a nested relational algebra." Thesis, McGill University, 2002. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=79213.
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This thesis documents the design and implementation of two enhancements to the Aldat database programming language: abstract data types (ADTs) and extensions to the domain algebra.Utilizing a nested relational model and an improved procedural abstraction facility, ADTs are declared as computations encapsulating states with their accessor/modifier methods. Objects of an ADT can be instantiated via a single join. As computation calls are embedded into updates and virtual domain actualization, objects are manipulated and accessed solely through the methods exported by the ADT.
The vertical domain algebra empowers Aldat with the capability to combine values along a domain using system defined operators. A mechanism has now been installed to run user defined computations as well. This, coupled with ADTs, opens up the opportunity for Aldat to handle applications such as GIS which require at once the capacity of a traditional DBMS and the computational power of a modern programming language.
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Fukawa-Connelly, Timothy Patrick. "A tale of two courses; teaching and learning in udergraduate abstract algebra." College Park, Md.: University of Maryland, 2007. http://hdl.handle.net/1903/7675.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2007.Thesis research directed by: Dept. of Curriculum and Instruction. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Wessel, Daniel. "Choice, extension, conservation. From transfinite to finite proof methods in abstract algebra." Doctoral thesis, Università degli studi di Trento, 2018. https://hdl.handle.net/11572/368524.
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Maximality principles such as the ones going back to Kuratowski and Zorn ensure the existence of higher type ideal objects the use of which is commonly held indispensable for mathematical practice. The modern turn towards computational methods, which can be witnessed to have a strong influence on contemporary foundational studies, encourages a reassessment within a constructive framework of the methodological intricacies that go along with invocations of maximality principles. The common thread that can be followed through the chapters of this thesis is explained by the attempt to put the widespread use of ideal objects under constructive scrutiny. It thus walks the tracks of a revised Hilbert’s programme which has inspired a reapproach to constructive algebra by finitary means, and for which Scott’s entailment relations have already shown to provide a vital and utmost versatile tool. In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory; the notion of Jacobson radical is brought from commutative rings to a general ideal theory for single-conclusion entailment relations; a flexible conservation criterion of Scott for multi-conclusion entailment relations is put into action; elementary and constructive variants for algebraic extension theorems such as Sikorski’s on the injectivity of complete atomic Boolean algebras are phrased and proved in terms of entailment relations; and a point-free version of Sikora’s theorem on spaces of orderings of groups is obtained by a revisitation with syntactical means of some of the classical criteria for groups to be orderable.
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Wessel, Daniel. "Choice, extension, conservation. From transfinite to finite proof methods in abstract algebra." Doctoral thesis, University of Trento, 2018. http://eprints-phd.biblio.unitn.it/2759/1/Wessel_thesis_final.pdf.
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Maximality principles such as the ones going back to Kuratowski and Zorn ensure the existence of higher type ideal objects the use of which is commonly held indispensable for mathematical practice. The modern turn towards computational methods, which can be witnessed to have a strong influence on contemporary foundational studies, encourages a reassessment within a constructive framework of the methodological intricacies that go along with invocations of maximality principles. The common thread that can be followed through the chapters of this thesis is explained by the attempt to put the widespread use of ideal objects under constructive scrutiny. It thus walks the tracks of a revised Hilbert’s programme which has inspired a reapproach to constructive algebra by finitary means, and for which Scott’s entailment relations have already shown to provide a vital and utmost versatile tool. In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory; the notion of Jacobson radical is brought from commutative rings to a general ideal theory for single-conclusion entailment relations; a flexible conservation criterion of Scott for multi-conclusion entailment relations is put into action; elementary and constructive variants for algebraic extension theorems such as Sikorski’s on the injectivity of complete atomic Boolean algebras are phrased and proved in terms of entailment relations; and a point-free version of Sikora’s theorem on spaces of orderings of groups is obtained by a revisitation with syntactical means of some of the classical criteria for groups to be orderable.
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Plaxco, David Bryant. "Relating Understanding of Inverse and Identity to Engagement in Proof in Abstract Algebra." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/56587.
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In this research, I set out to elucidate the relationships that might exist between students' conceptual understanding upon which they draw in their proof activity. I explore these relationships using data from individual interviews with three students from a junior-level Modern Algebra course. Each phase of analysis was iterative, consisting of iterative coding drawing on grounded theory methodology (Charmaz, 2000, 2006; Glaser and Strauss, 1967). In the first phase, I analyzed the participants' interview responses to model their conceptual understanding by drawing on the form/function framework (Saxe, et al., 1998). I then analyzed the participants proof activity using Aberdein's (2006a, 2006b) extension of Toulmin's (1969) model of argumentation. Finally, I analyzed across participants' proofs to analyze emerging patterns of relationships between the models of participants' understanding of identity and inverse and the participants' proof activity. These analyses contributed to the development of three emerging constructs: form shifts in service of sense-making, re-claiming, and lemma generation. These three constructs provide insight into how conceptual understanding relates to proof activity.Ph. D.
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Esteban, María. "Duality Theory and Abstract Algebraic Logic." Doctoral thesis, Universitat de Barcelona, 2013. http://hdl.handle.net/10803/125336.
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In this thesis we present the results of our research on duality theory for non-classical logics under the point of view of Abstract Algebraic Logic (AAL). Firstly, we propose an abstract Spectral-like duality and an abstract Priestley-style duality for every filter distributive finitary congruential logic with theorems. This proposal aims to unify the various dualities for concrete logics that we find in the literature, by showing the abstract template in which all of them fit. Secondly, the dual correspondence of some logical properties is examined. This serves to reveal the connection between our abstract dualities and the concrete dualities related wot concrete logics. We apply those results to get new dualities for suitable expansions of a well-known logic: the implicative fragment of intuitionistic logic. Finally, we develop a new strategy that can be modularly applied to simplify some of the dualities obtained.
The first part of the dissertation is devoted to introduce the preliminaries and the basic notation. In Chapter 1 we fix the mathematical concepts that we assume the reader is familiar with. Of particular interest is the section in which we introduce the basic concepts of AAL, such as "S-filter" or "S-algebra". The notion of "closure operator" plays a fundamental role in AAL, as well as in our dissertation. The notions of filter and ideal associated with a closure operator, and the separation lemmas between them are studied in detail in Chapter 2. Moreover, we briefly review the literature on duality theory for non classical logics in Chapter 3.
In the second part of the dissertation we present an abstract view of the duality theory for non-classical logics. In Chapter 4 we review previous works on this topic, in which our work relies, and we introduce the notions of "referential algebra", "irreducible and optimal S-filter" and "S-semilattice". This lead us to identify a set of necessary conditions that a logic should satisfy in order to develop a Spectra-like/Priestley-style duality for it. These conditions are: "filter distributivity","congruentiality", "finitarity" and "having theorems". Moreover, we carry out a brief digression in which we argue how those notions can also be used to develop an abstract theory of canonical extensions.
The core of the proposed theory consists of the definitions of dual objects and morphisms, for the category of S-algebras and homomorphisms, for any logic S that satisfies the mentioned properties. In Chapter 5 we define a Spectral-like duality and a Priestley-style duality for filter distributive finitary congruential logics with theorems, and we prove the respective duality theorems. Due to the abstraction of our approach, we obtain that the objects of both categories involved in the duality posses algebraic nature. However, through the analysis of the dual correspondence of several well-known logical properties, we can simplify the definitions of the dual categories, provided the logic under consideration satisfies such good logical properties. This analysis is interesting under the point of view of AAL, since our results can be regarded as bridge theorems between logical properties and properties of a Kripke-style semantics. And it is also interesting under the point of view of duality theory, since it confirms the strength of duality theory, that can be developed in a modular way beyond the distributive lattice setting. Moreover, our analysis shows the connection of the general theory proposed with the concrete results that we find in the literature, and lead us to explore the applications of such general theory to obtain new dualities.En esta tesis se presentan los resultados de nuestra investigación acerca de la teoría de la dualidad para lógicas no clásicas desde el punto de vista de la Lógica Algebráica Abstracta (LAA). Un estudio preliminar de las distintas nociones de filtros e ideales lógicos asociados a las álgebras de una lógica cualquiera, y los lemas de separación entre dichas nociones nos lleva a proponer una dualidad abstracta de tipo espectral, y otra de tipo Priestley, para cada lógica congruencial, filtro distributiva, finitaria y con teoremas. Esta propuesta pretende unificar las distintas dualidades de tipo espectral y de tipo Priestley para lógicas no clásicas que encontramos en la literatura, mostrando el esquema abstracto en el que todas ellas encajan e identificando. En segundo lugar es examinada la correspondencia dual de algunas propiedades lógicas, como la propiedad de la conjunción, la propiedad de la disyunción, el teorema de deducción, la propiedad del elemento inconsistente o la propiedad de introducción de la modalidad. Esto sirve, por una parte, para revelar la conexión que existe entre las dualidades abstractas propuestas y las dualidades concretas relacionadas con lógicas no clásicas que habían sido estudiadas previamente, y por otra parte, para obtener nuevas dualidades. Centrándonos en el fragmento implicativo de la lógica intuicionista y en sus expansiones que son filtro distributivas, congruenciales, finitarias y con teoremas, mostramos cómo las dualidades que habían sido estudiadas para algunas de esas lógicas se pueden obtener como casos particulares de la teoría general. Además obtenemos nuevas dualidades para varias de dichas expansiones, algunas de las cuales pueden ser simplificadas dado que las lógicas tienen buenas propiedades. Finalmente, desarrollamos una nueva estrategia que puede ser aplicada de forma modular para simplificar algunas de las dualidades obtenidas. En conclusión, en esta tesis se muestra que la Lógica Algebráica Abstracta provee un marco general teórico apropiado para desarrollar una teoría abstracta de la dualidad para lógicas no clásicas. Dicha teoría uniformiza los diferentes resultados de la literatura, y de ella se deducen nuevos resultados.
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Iori, Tomoyuki. "Symbolic-Numeric Approaches Based on Theories of Abstract Algebra to Control, Estimation, and Optimization." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263785.
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Sung, Edward William. "Using the concrete-representation-abstract instruction to teach algebra to students with learning disabilities." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3263.
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This project explored the Concrete to Representational to Abstract instruction (CRA instruction) as a strategy to teach abstract math concepts for secondary students with learning disabilities. Through the review of literature, multiple researchers suggested that students with learning disabilities need to be exposed to a variety of instructional strategies to develop problem solving skills in algebra concepts.
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Meacham, Desmond J. "Standards interoperability : application of contemporary software assurance standards to the evolution of legacy software /." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2006. http://library.nps.navy.mil/uhtbin/hyperion/06Mar%5FMeacham.pdf.
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Caprace, Pierre-Emmanuel. ""Abstract" homomorphisms of split Kac-Moody groups." Doctoral thesis, Universite Libre de Bruxelles, 2005. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210962.
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Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits.Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés.
En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$.
Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts.
Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis.
Doctorat en sciences, Spécialisation mathématiques
info:eu-repo/semantics/nonPublished
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Stiles, Megan E. "The Mathieu Groups." Youngstown State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1308861143.
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Cândido, Renato Markele Ferreira 1988. "Filtros de partículas aplicados a sistemas max plus." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259747.
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Orientador: Rafael Santos MendesDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
Made available in DSpace on 2018-08-24T01:12:10Z (GMT). No. of bitstreams: 1 Candido_RenatoMarkeleFerreira_M.pdf: 1921815 bytes, checksum: a5e82ec1bfadd836b1ba66fda5ce00ec (MD5) Previous issue date: 2013
Resumo: A principal contribuição desta dissertação é a proposta de algoritmos de filtragem por partículas em sistemas a eventos discretos nos quais predominam os problemas de sincronização. Esta classe de sistemas pode ser descrita por meio de equações lineares em uma álgebra não convencional usualmente conhecida como álgebra Max Plus. Os Filtros de Partículas são algoritmos Bayesianos sub-ótimos que realizam uma amostragem sequencial de Monte Carlo para construir uma aproximação discreta da densidade de probabilidade dos estados baseada em um conjunto de partículas com pesos associados. É apresentada uma revisão de sistemas a eventos discretos, de filtragem não linear e de filtros de partículas de um modo geral. Após apresentar esta base teórica, são propostos dois algoritmos de filtros de partículas aplicados a sistemas Max Plus. Em seguida algumas simulações foram apresentadas e os resultados apresentados mostraram a eficiência dos filtros desenvolvidos
Abstract: This thesis proposes, as its main contribution, particle filtering algorithms for discrete event systems in which synchronization phenomena are prevalent. This class of systems can be described by linear equation systems in a nonconventional algebra commonly known as Max Plus algebra. Particles Filters are suboptimal Bayesian algorithms that perform a sequential Monte Carlo sampling to construct a discrete approximation of the probability density of states based on a set of particles with associated weights. It is presented a review of discrete event systems, nonlinear filtering and particle filters. After presenting this theoretical background, two particle filtering algorithms applied to Max Plus systems are proposed. Finally some simulation results are presented, confirming the accuracy of the designed filters
Mestrado
Automação
Mestre em Engenharia Elétrica
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Adovasio, Ben. "A Character Theory Free Proof of Burnside's paqb Theorem." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1337975956.
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Wannenburg, Johann Joubert. "Varieties of De Morgan Monoids." Thesis, University of Pretoria, 2020. http://hdl.handle.net/2263/75178.
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De Morgan monoids are algebraic structures that model certain non-classical logics. The variety DMM of all De Morgan monoids models the relevance logic Rt (so-named because it blocks the derivation of true conclusions from irrelevant premises). The so-called subvarieties and subquasivarieties of DMM model the strengthenings of Rt by new logical axioms, or new inference rules, respectively. Meta-logical problems concerning these stronger systems amount to structural problems about (classes of) De Morgan monoids, and the methods of universal algebra can be exploited to solve them. Until now, this strategy was under-developed in the case of Rt and DMM.
The thesis contributes in several ways to the filling of this gap. First, a new structure theorem for irreducible De Morgan monoids is proved; it leads to representation theorems for the algebras in several interesting subvarieties of DMM. These in turn help us to analyse the lower part of the lattice of all subvarieties of DMM. This lattice has four atoms, i.e., DMM has just four minimal subvarieties. We describe in detail the second layer of this lattice, i.e., the covers of the four atoms. Within certain subvarieties of DMM, our description amounts to an explicit list of all the covers. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.
Thereafter, we use these insights to identify strengthenings of Rt with certain desirable meta-logical features. In each case, we work with the algebraic counterpart of a meta-logical property. For example, we identify precisely the varieties of De Morgan monoids having the joint embedding property (any two nontrivial members both embed into some third member), and we establish convenient sufficient conditions for epimorphisms to be surjective in a subvariety of DMM. The joint embedding property means that the corresponding logic is determined by a single set of truth tables. Epimorphisms are related to 'implicit definitions'. (For instance, in a ring, the multiplicative inverse of an element is implicitly defined, because it is either uniquely determined or non-existent.) The logical meaning of epimorphism-surjectivity is, roughly speaking, that suitable implicit definitions can be made explicit in the corresponding logical syntax.Thesis (PhD)--University of Pretoria, 2020.
DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)
Mathematics and Applied Mathematics
PhD
Unrestricted
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Serbin, Kaitlyn Stephens. "Prospective Teachers' Knowledge of Secondary and Abstract Algebra and their Use of this Knowledge while Noticing Students' Mathematical Thinking." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104563.
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I examined the development of three Prospective Secondary Mathematics Teachers' (PSMTs) understandings of connections between concepts in Abstract Algebra and high school Algebra, as well as their use of this understanding while engaging in the teaching practice of noticing students' mathematical thinking. I drew on the theory, Knowledge of Nonlocal Mathematics for Teaching, which suggests that teachers' knowledge of advanced mathematics can become useful for teaching when it first helps reshape their understanding of the content they teach. I examined this reshaping process by investigating how PSMTs extended, deepened, unified, and strengthened their understanding of inverses, identities, and binary operations over time. I investigated how the PSMTs' engagement in a Mathematics for Secondary Teachers course, which covered connections between inverse functions and equation solving and the abstract algebraic structures of groups and rings, supported the reshaping of their understandings. I then explored how the PSMTs used their mathematical knowledge as they engaged in the teaching practice of noticing hypothetical students' mathematical thinking. I investigated the extent to which the PSMTs' noticing skills of attending, interpreting, and deciding how to respond to student thinking developed as their mathematical understandings were reshaped.
There were key similarities in how the PSMTs reshaped their knowledge of inverse, identity, and binary operation. The PSMTs all unified the additive identity, multiplicative identity, and identity function as instantiations of the same overarching identity concept. They each deepened their understanding of inverse functions. They all unified additive, multiplicative, and function inverses under the overarching inverse concept. They also strengthened connections between inverse functions, the identity function, and function composition. They all extended the contexts in which their understandings of inverses were situated to include trigonometric functions. These changes were observed across all the cases, but one change in understanding was not observed in each case: one PSMT deepened his understanding of the identity function, whereas the other two had not yet conceptualized the identity function as a function in its own right; rather, they perceived it as x, the output of the composition of inverse functions.
The PSMTs had opportunities to develop these understandings in their Mathematics for Secondary Teachers course, in which the instructor led the students to reason about the inverse and identity group axioms and reflect on the structure of additive, multiplicative, and compositional inverses and identities. The course also covered the use of inverses, identities, and binary operations used while performing cancellation in the context of equation solving.
The PSMTs' noticing skills improved as their mathematical knowledge was reshaped. The PSMTs' reshaped understandings supported them paying more attention to the properties and strategies evident in a hypothetical student's work and know which details were relevant to attend to. The PSMTs' reshaped understandings helped them more accurately interpret a hypothetical student's understanding of the properties, structures, and operations used in equation solving and problems about inverse functions. Their reshaped understandings also helped them give more accurate and appropriate suggestions for responding to a hypothetical student in ways that would build on and improve the student's understanding.Doctor of Philosophy
Once future mathematics teachers learn about how advanced mathematics content is related to high school algebra content, they can better understand the algebra content they may teach. The future teachers in this study took a Mathematics for Secondary Teachers course during their senior year of college. This course gave them opportunities to make connections between advanced mathematics and high school mathematics. After this course, they better understood the mathematical properties that people use while equation solving, and they improved their teaching practice of making sense of high school students' mathematical thinking about inverses and equation solving. Overall, making connections between the advanced mathematics content they learned during college and the algebra content related to inverses and equation solving that they teach in high school helped them improve their teaching practice.
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Johnson, Estrella Maria Salas. "Establishing Foundations for Investigating Inquiry-Oriented Teaching." PDXScholar, 2013. http://pdxscholar.library.pdx.edu/open_access_etds/1102.
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The Teaching Abstract Algebra for Understanding (TAAFU) project was centered on an innovative abstract algebra curriculum and was designed to accomplish three main objectives: to produce a set of multi-media support materials for instructors, to understand the challenges faced by mathematicians as they implemented this curriculum, and to study how this curriculum supports student learning of abstract algebra. Throughout the course of the project I took the lead investigating the teaching and learning in classrooms using the TAAFU curriculum. My dissertation is composed of three components of this research. First, I will report on a study that aimed to describe the experiences of mathematicians implementing the curriculum from their perspective. Second. I will describe a study that explores the mathematical work done by teachers as they respond to the mathematical activity of their students. Finally, I will discuss a theoretical paper in which I synthesize aspects of the instructional theory underlying the TAAFU curriculum in order to develop an analytic framework for analyzing student learning. This dissertation will serve as a foundation for my future research focused on the relationship between teachers' mathematical work and the learning of their students.
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Wohlgehagen, James L. (James Lee). "A Comparison of the Effectiveness of an Abstract and a Concrete Approach in Teaching Selected Algebraic Concepts to Ninth and Tenth Grade Students." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc331465/.
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One purpose of this study was to determine whether any differences in immediate achievement or retention existed between students using manipulatives and students not using manipulatives. Also addressed in this study is whether or not the use of manipulatives is more beneficial for girls than boys and whether the use of manipulatives is more beneficial for low-ability students than for high-ability students.
Students selected for this study were from a large suburban school district in Texas. The students were from eight intact classes, four of which were designated as the experimental group and the other four as the control group. The sample consisted of one hundred eighty-seven students.
All students were tested with a test developed by the researcher. This same test was administered as a pretest, posttest, and retention test. The following supplemental data were also gathered on the students: mathematics scores from the California Test of Basic Skills and scores from the mathematics section of the Texas Educational Assessment of Minimum Skills test.
Analysis of the data revealed no statistical difference in the mean scores of students instructed with or without manipulatives when the test was administered immediately after instruction. Nor was there any statistical difference in the mean scores when the test was administered two months after instruction. There was no statistical difference in the mean gain scores from the pretest to the posttest between boys and girls or between high- and low-achieving students. Nor was there any statistical difference between the mean gain scores from the pretest to the retention test between boys and girls or between high- and low-achieving students.
It is recommended that further studies be conducted to investigate achievement and retention of students using manipulatives at the secondary level. It is also recommended that variables other than achievement be studied to determine the effects of manipulatives on secondary students.
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Bojorquez, Betzabe. "Geometric Constructions from an Algebraic Perspective." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/237.
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Many topics that mathematicians study at times seem so unrelated such as Geometry and Abstract Algebra. These two branches of math would seem unrelated at first glance. I will try to bridge Geometry and Abstract Algebra just a bit with the following topics. We can be sure that after we construct our basic parallel and perpendicular lines, bisected angles, regular polygons, and other basic geometric figures, we are actually constructing what in geometry is simply stated and accepted, because it will be proven using abstract algebra. Also we will look at many classic problems in Geometry that are not possible with only straightedge and compass but need a marked ruler.
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Watt, Sarah Jean. "Teaching algebra-based concepts to students with learning disabilities: the effects of preteaching using a gradual instructional sequence." Diss., University of Iowa, 2013. https://ir.uiowa.edu/etd/2658.
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Teaching algebra-based concepts to students with learning disabilities: The effects of preteaching using a gradual instructional sequence
by Sarah Jean Watt
An Abstract of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Teaching and Learning Special Education) in the Graduate College of The University of Iowa May 2013
Thesis Supervisor: Associate Professor William J. Therrien
Research to identify validated instructional approaches to teach math to students with LD and those at-risk for failure in both core and supplemental instructional settings is necessary to assist teachers in closing the achievement gaps that exist across the country. The concrete-to-representational-to-abstract instructional sequence (CRA) has been identified through the literature as a promising approach to teaching students with and without math difficulties (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Cass, Cates, Smith, & Jackson (e.g. CSA), 2003; Flores, 2010). The CRA sequence transitions students from the use of concrete manipulatives to abstract symbols through the use of explicit instruction to increase computational and conceptual understanding.
The main purpose of this study was to assess the effects of preteaching essential pre-algebra skills on the overall algebra achievement scores for students with disabilities and those at-risk for failure in math. Specifically the study examined the following research questions: (1) What are the effects of preteaching math units using the CRA instructional sequence on the algebra achievement of students with LD and those at risk for math failure? (2) What are the effects of preteaching math units using the CRA instructional sequence on the transfer of algebra-based skills of students with LD and those at risk for math failure to the general education setting? (3) What are the effects of preteaching math units using the CRA instructional sequence on the maintenance of algebra-based skills for students with LD and those at risk for math failure?
Summary of Study Design and Findings
Thirty-two students enrolled in one of four 6th grade classrooms across two elementary schools participated in this study. Sixth grade students who currently receive tier 2 or tier 3 supplemental and intensive instruction in math; and those identified as having a math learning disability will be participants. A treatment and control, pre/post experimental design was used to examine the effect of the intervention on students' math achievement. The intervention was replicated across two math units related to teaching algebra-based concepts: Solving Equations and Fractions. The treatment condition consisted of a combination of preteaching and the use of the CRA instructional sequence. Prior to each unit, Solving Equations and Fractions, researchers pretaught students 3 essential prerequisite skills necessary for success in the upcoming unit, at the concrete, representational, and abstract levels of learning. Each preteaching session lasted for ten days, 30 minutes each day. Immediate, delayed, and follow-up measures were used to support the examination of the research questions and hypotheses.
Overall findings indicate that the combination of preteaching using the CRA gradual sequence is effective at improving the overall algebra performance for students with disabilities.
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Crawford, Simon Philip. "Singularities of noncommutative surfaces." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31543.
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The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.
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Vaughan, Stephen N. (Stephen Nick). "Some Properties of Noetherian Rings." Thesis, North Texas State University, 1986. https://digital.library.unt.edu/ark:/67531/metadc501109/.
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This paper is an investigation of several basic properties of noetherian rings. Chapter I gives a brief introduction, statements of definitions, and statements of theorems without proof. Some of the main results in the study of noetherian rings are proved in Chapter II. These results include proofs of the equivalence of the maximal condition, the ascending chain condition, and that every ideal is finitely generated. Some other results are that if a ring R is noetherian, then R[x] is noetherian, and that if every prime ideal of a ring R is finitely generated, then R is noetherian.
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Ferreira, Nilton Cezar. "Uma proposta de ensino de álgebra abstrata moderna, com a utilização da metodologia de ensino-aprendizagem-avaliação de matemática através da resolução de problemas, e suas contribuições para a formação inicial de professores de matemática /." Rio Claro, 2017. http://hdl.handle.net/11449/149213.
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Orientador: Lourdes de la Rosa OnuchicBanca: Rosa Lúcia Sverzut Baroni
Banca: Henrique Lazari
Banca: Glen César Lemos
Banca: José Pedro Machado Ribeiro
Resumo: Este trabalho teve como principal objetivo investigar as contribuições que a Álgebra Abstrata Moderna (onde se trabalham as teorias de Grupos, Anéis e Corpos, dentre outras), ministrada como uma disciplina em cursos de Licenciatura em Matemática no Brasil, poderia dar à Formação Inicial de Professores de Matemática. Esta pesquisa teve caráter qualitativo e foi apoiada no Modelo Metodológico de Romberg-Onuchic. Visando alcançar esse objetivo, propusemos uma pesquisa de campo, desenvolvida em 2015, com uma turma do quinto período de Licenciatura em Matemática do Instituto Federal de Goiás (IFG). Para isso, elaboramos e implementamos um projeto de ensino com o propósito de levar os alunos dessa turma a construírem um conhecimento satisfatório de Álgebra Abstrata Moderna e mostrar a relação de seus conteúdos com os da Educação Básica. Para a construção desse conhecimento, fizemos uso da Metodologia de Ensino-Aprendizagem-Avaliação de Matemática através da Resolução de Problemas, figurada no campo da Educação Matemática e consolidada por diversas pesquisas como eficiente no processo de ensino, aprendizagem e avaliação de Matemática em diversos níveis - Fundamental, Médio e Superior. A correlação entre os conteúdos de Álgebra Abstrata Moderna e os da Educação Básica se deu através da proposição, aos estudantes da referida turma, de atividades extraclasse, que, sempre, em um momento posterior, eram discutidas, em sala de aula, por todos os integrantes desse processo: alunos, pesquis... (Resumo completo, clicar acesso eletrônico abaixo)
Abstract: The main purpose of the present work was to investigate the contributions that Modern Abstract Algebra (which the theories of Groups, Rings and Fields, among others, are worked on), as a discipline in Degree courses in Mathematics in Brazil, might give to initial Teacher Education in Mathematics. The present research has a qualitative approach and it was grounded on the Methodological Model of Romberg-Onuchic. In order to achieve that goal, we proposed a field research, developed in 2015, involving a class of fifth semester students of Degree in Mathematics at Instituto Federal de Goiás (IFG). To that end, we elaborated and implemented a teaching project with the purpose of enabling that group of students to build satisfactory knowledge on Modern Abstract Algebra and showing the relationship of its contents to the ones of Elementary Education. In order to build such knowledge, we used the Methodology of Teaching-Learning-Evaluation in Mathematics through Problem Solving, found in the field of Mathematics Education and consolidated by several researches as effective in the process of Mathematics teaching, learning and evaluation in several levels - Elementary, Middle and Higher Education. The correlation between the contents of Modern Abstract Algebra and the ones of Elementary Education came about through the proposition to that group of students of extracurricular activities which were always discussed further in classroom by all people involved in that process: students, researcher and teacher. There were also two meetings with the only purpose of working, discussing and analysing this association - Modern Abstract Algebra and Elementary Education. The evidence-gathering was made through the researcher's observation during the project application, the materials produced by the students, the media (audio and video recordings of the meetings) and a diagnostic ... (Complete abstract electronic access below)
Doutor
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Chadman, Corey S. "Functional Limits in Topology." Youngstown State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1371035042.
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Aslanyan, Vahagn. "Ax-Schanuel type inequalities in differentially closed fields." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:bced8c2d-22df-4a21-9a1f-5e4204b6c85d.
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In this thesis we study Ax-Schanuel type inequalities for abstract differential equations. A motivating example is the exponential differential equation. The Ax-Schanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamation-with-predimension technique to refute Zilber's Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by Ax-Schanuel. In this way one constructs a limit structure whose theory turns out to be precisely the first-order theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber's work on pseudo-exponentiation). One says in this case that the inequality is adequate. Thus, by an Ax-Schanuel type inequality we mean a predimension inequality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the Ax-Schanuel theorem. Further, the question turns out to be closely related to the problem of recovering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some criteria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a non-trivial adequate predimension inequality. Another example of a predimension inequality is the analogue of Ax-Schanuel for the differential equation of the modular j-function due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the first-order theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triviality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set.
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Gonda, Jessica Lynn. "Subgroups of Finite Wreath Product Groups for p=3." University of Akron / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=akron1460027790.
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Margraff, Aaron Thaddeus. "An Exposition on Group Characters." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1397492784.
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Thomas, Teri M. "A Generalization of Sylow’s Theorem." Youngstown State University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1256911896.
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Molina, Aristizabal Sergio D. "Semi-Regular Sequences over F2." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1445342810.
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Bounds, Jordan. "On the quasi-isometric rigidity of a class of right-angled Coxeter groups." Bowling Green State University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1561561078356503.
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Melhuish, Kathleen Mary. "The Design and Validation of a Group Theory Concept Inventory." PDXScholar, 2015. https://pdxscholar.library.pdx.edu/open_access_etds/2490.
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Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds. Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students' understanding of group theory concepts. Through careful creation and extensive field-testing, the GCI has been shown to be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale.
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Kioulos, Charalambos. "From Flag Manifolds to Severi-Brauer Varieties: Intersection Theory, Algebraic Cycles and Motives." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40716.
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The study of algebraic varieties originates from the study of smooth manifolds. One
of the focal points is the theory of differential forms and de Rham cohomology. It’s
algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing
and taking the pseudo-abelian envelope of the category of smooth projective varieties,
one obtains the category of pure motives.
In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer
varieties. This has been a subject of intensive investigation for the past twenty years,
with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin,
[Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2];
Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen];
Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the
thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in
the paper [Cal] by providing new examples of motivic decompositions of generalized
Severi-Brauer varieties.
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Florida, Julie. "Interventions in Solving Equations for Students with Mathematics Learning Disabilities : A Systematic Literature Review." Thesis, Högskolan för lärande och kommunikation, Högskolan i Jönköping, HLK, CHILD, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:hj:diva-30853.
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Approximately 5 to 14% of school age children are affected by mathematics learning disabilities. With the implementation of inclusion, many of these children are now being educated in the regular education class- room setting and may require additional support to be successful in algebra. Therefore, teachers need to know what interventions are available to them to facilitate the algebraic learning of students with mathemat- ics learning disabilities. This systematic literature review aims to identify, and critically analyze, interventions that could be used when teaching algebra to these students. The five included articles focused on interven- tions that can be used in algebra, specifically when solving equations. In the analysis of the five studies two types of interventions emerged: the concrete-representational-abstract model and graphic organizers. The concrete-representational-abstract model seems to show it can be used successfully in a variety of scenarios involving solving equations. The use of graphic organizers also seems to be helpful when teaching higher- level algebra content that may be difficult to represent concretely. This review discovered many practical implications for teachers. Namely, that the concrete-representational-abstract model of intervention is easy to implement, effective over short periods of time and appears to positively influence the achievement of all students in an inclusive classroom setting. The graphic organizer showed similar results in that it is easy to implement and appears to improve all students’ learning. This review provided a good starting point for teachers to identify interventions that could be useful in algebra; however, more research still needs to be done. Future research is suggested in inclusive classroom settings where the general education teacher is the instructor and also on higher-level algebra concepts.
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Silva, Elisabete Santana de ávila e. "Um código co-dígito verificador baseado em D5 : uma aplicação dos grupos de simetria." Mestrado Profissional em Matemática, 2013. https://ri.ufs.br/handle/riufs/6496.
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This present work to describe the code based on D5 as part of the application of Abstract Algebra, through Symmetry Groups, as well as its advantages over other codes in the case of detection of typos. To this end, we provide some definitions and theorems of the theory of groups useful for understanding this work. Study groups Permutation Groups and Symmetry, issues of great relevance to the study of dihedral groups, being these, particularly if those groups and the basis for the development of the code described herein.Este trabalho tem como objetivo descrever o Código baseado em D5 como aplicação de parte da Álgebra Abstrata, através dos Grupos de Simetria, bem como suas vantagens em relação a outros códigos, em se tratando da detecção de erros de digitação. Para tanto, fornecemos algumas definições e teoremas da teoria dos Grupos úteis à compreensão deste trabalho. Estudamos os Grupos de Permutação e os Grupos de Simetria, assuntos de grande relevância para o estudo dos Grupos Diedrais, por serem, estes, caso particular dos grupos citados e base para o desenvolvimento do código aqui descrito.
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Ferreira, Nilton Cezar [UNESP]. "Uma proposta de ensino de álgebra abstrata moderna, com a utilização da metodologia de ensino-aprendizagem-avaliação de matemática através da resolução de problemas, e suas contribuições para a formação inicial de professores de matemática." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/149213.
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Este trabalho teve como principal objetivo investigar as contribuições que a Álgebra Abstrata Moderna (onde se trabalham as teorias de Grupos, Anéis e Corpos, dentre outras), ministrada como uma disciplina em cursos de Licenciatura em Matemática no Brasil, poderia dar à Formação Inicial de Professores de Matemática. Esta pesquisa teve caráter qualitativo e foi apoiada no Modelo Metodológico de Romberg-Onuchic. Visando alcançar esse objetivo, propusemos uma pesquisa de campo, desenvolvida em 2015, com uma turma do quinto período de Licenciatura em Matemática do Instituto Federal de Goiás (IFG). Para isso, elaboramos e implementamos um projeto de ensino com o propósito de levar os alunos dessa turma a construírem um conhecimento satisfatório de Álgebra Abstrata Moderna e mostrar a relação de seus conteúdos com os da Educação Básica. Para a construção desse conhecimento, fizemos uso da Metodologia de Ensino-Aprendizagem-Avaliação de Matemática através da Resolução de Problemas, figurada no campo da Educação Matemática e consolidada por diversas pesquisas como eficiente no processo de ensino, aprendizagem e avaliação de Matemática em diversos níveis – Fundamental, Médio e Superior. A correlação entre os conteúdos de Álgebra Abstrata Moderna e os da Educação Básica se deu através da proposição, aos estudantes da referida turma, de atividades extraclasse, que, sempre, em um momento posterior, eram discutidas, em sala de aula, por todos os integrantes desse processo: alunos, pesquisador e professor da disciplina. Contou, ainda, com dois encontros exclusivos para se trabalhar, discutir e analisar essa associação – Álgebra Abstrata Moderna e Educação Básica. A coleta de evidências foi feita através da observação do pesquisador durante a aplicação do projeto; materiais produzidos pelos alunos; mídias (gravações em áudio e vídeo dos encontros realizados); e uma avaliação diagnóstica que teve como foco: Formação de Professores, Álgebra e Resolução de Problemas. Os resultados confirmaram que a Álgebra Abstrata, se trabalhada de forma adequada, poderá trazer contribuições significativas à formação de professores de Matemática.
The main purpose of the present work was to investigate the contributions that Modern Abstract Algebra (which the theories of Groups, Rings and Fields, among others, are worked on), as a discipline in Degree courses in Mathematics in Brazil, might give to initial Teacher Education in Mathematics. The present research has a qualitative approach and it was grounded on the Methodological Model of Romberg-Onuchic. In order to achieve that goal, we proposed a field research, developed in 2015, involving a class of fifth semester students of Degree in Mathematics at Instituto Federal de Goiás (IFG). To that end, we elaborated and implemented a teaching project with the purpose of enabling that group of students to build satisfactory knowledge on Modern Abstract Algebra and showing the relationship of its contents to the ones of Elementary Education. In order to build such knowledge, we used the Methodology of Teaching-Learning-Evaluation in Mathematics through Problem Solving, found in the field of Mathematics Education and consolidated by several researches as effective in the process of Mathematics teaching, learning and evaluation in several levels – Elementary, Middle and Higher Education. The correlation between the contents of Modern Abstract Algebra and the ones of Elementary Education came about through the proposition to that group of students of extracurricular activities which were always discussed further in classroom by all people involved in that process: students, researcher and teacher. There were also two meetings with the only purpose of working, discussing and analysing this association – Modern Abstract Algebra and Elementary Education. The evidence-gathering was made through the researcher’s observation during the project application, the materials produced by the students, the media (audio and video recordings of the meetings) and a diagnostic evaluation focused on Teacher Education, Algebra and Problem Solving. The results confirmed that Abstract Algebra, if properly worked on, might bring significant contributions to Teacher Education in Mathematics.
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Yannotta, Mark Alan. "Conventionalizing and Axiomatizing in a Community College Mathematics Bridge Course." PDXScholar, 2016. http://pdxscholar.library.pdx.edu/open_access_etds/3105.
Full textAbstract:
This dissertation consists of three related papers. The first paper, Rethinking mathematics bridge courses--An inquiry model for community colleges, introduces the activities of conventionalizing and axiomatizing from a practitioner perspective. In the paper, I offer a curricular model that includes both inquiry and traditional instruction for two-year college students interested in mathematics. In particular, I provide both examples and rationales of tasks from the research-based Teaching Abstract Algebra for Understanding (TAAFU) curriculum, which anchors the inquiry-oriented version of the mathematics bridge course.
The second paper, the role of past experience in creating a shared representation system for a mathematical operation: A case of conventionalizing, adds to the existing literature on mathematizing (Freudenthal, 1973) by "zooming in" on the early stages of the classroom enactment of an inquiry-oriented curriculum for reinventing the concept of group (Larsen, 2013). In three case study episodes, I shed light onto "How might conventionalizing unfold in a mathematics classroom?" and offer an initial framework that relates students' establishment of conventions in light of their past mathematical experiences.
The third paper, Collective axiomatizing as a classroom activity, is a detailed case study (Yin, 2009) that examines how students collectively engage in axiomatizing.
In the paper, I offer a revision to De Villiers's (1986) model of descriptive axiomatizing. The results of this study emphasize the additions of pre-axiomatic activity and axiomatic creation to the model and elaborate the processes of axiomatic formulation and analysis within the classroom community.
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Franco, Hernando José Rocha. "Os diversos conflitos observados em alunos de licenciatura num curso de álgebra: identificação e análise." Universidade Federal de Juiz de Fora (UFJF), 2011. https://repositorio.ufjf.br/jspui/handle/ufjf/3250.
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Neste trabalho, investigam-se os conflitos de aprendizagem que emergem quando estudantes de Licenciatura em Matemática estão diante de um primeiro curso de Álgebra Abstrata. Ao longo de um semestre, acompanhamos doze alunos, licenciandos em Matemática, durante as aulas da disciplina Álgebra I, cuja ementa contempla os conceitos de anéis, ideais, corpos e polinômios. O estudo fundamentou-se nos processos constituintes do Pensamento Matemático Avançado, na teoria da imagem e definição conceituais e nos níveis de sofisticação do pensamento matemático – procedimento, processo e proceito. Outros subsídios teóricos vieram com o levantamento de aspectos históricos da Álgebra como Ciência e como disciplina curricular da Educação Matemática. O contato direto com a turma durante as aulas, a aplicação de questionários e a observação das avaliações possibilitaram a coleta dos dados da pesquisa. Identificadas as dificuldades de aprendizagem, buscamos discuti-las à luz das interações entre a definição formal do objeto matemático e as imagens conceituais que os alunos formaram desse objeto. Ao final, apresentamos uma categorização dos conflitos analisados com base nas compreensões do fenômeno estudado.
In this work, the learning conflicts are investigated that emerge when students of degree in Mathematics are ahead of a first course of Abstract Algebra. Throughout a semester we follow twelve pupils, undergraduates in Mathematics, during the lessons of disciplines Algebra I, whose summary contemplates the ring concepts, ideals, fields and polynomials. The study it was based on the constituent processes of the Advanced Mathematical Thinking, on the theory of the conceptual image and definition and on the levels of sophistication of the mathematical thinking - procedure, process and procept. Other theoretical subsidies had come with the survey of historical aspects of Algebra as Science and as discipline curricular of the Mathematical Education. The direct contact with the group during the lessons, the application of questionnaires and the comment of the evaluations makes possible the collection of the data of the research. Identified the learning difficulties, we search discutiz them it the light of the interactions between the formal definition of the mathematical object and the conceptual images that the pupils had formed of this object. To the end, we present a categorization of the analyzed conflicts on the basis of the understandings of the studied phenomenon.
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Frisk, Anders. "On Stratified Algebras and Lie Superalgebras." Doctoral thesis, Uppsala : Department of Mathematics, Uppsala university, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7781.
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Stewart, K. J. "Concrete and abstract models of computation over metric algebras." Thesis, Swansea University, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.639113.
Full textAbstract:
In this thesis we study the computability theory of partial functions defined over metric algebras. We integrate aspects of two broad approaches to generalising the theory of computability on the natural numbers to uncountable algebraic structures - abstract models that use generalised models of computation to compute over structures which are considered independently of any representation or implementation, and concrete models for which connections with computability on the naturals and physical realisability are the main focus. The abstract models with which we work are those of the imperative style 'while' programming language and generalisations of Kleene's μ-recursive function schemes from the naturals to abstract algebras. We study computation by such models over effective metric partial algebras. The effectivity of such algebras is based on the choice of a computable dense metric sub-space. The notion of a computable function over these types of spaces is well established. We review some simple connections with other models. We prove the closure of these functions under abstract 'while' language computation and approximation. We prove analogous results for recursive function schemes. We consider some of these ideas in several applications connected with the real numbers, matrix algebra, deterministic parallel models of computation and Hilbert and Banach spaces. We analyse the connection between sets with computable projection functions and computable distance functions in Hilbert space.
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Wilder, A. J. "Algebraic tables : abstract computability and system documentation." Thesis, Swansea University, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636599.
Full textAbstract:
This thesis builds on the work of D. Parnas and other collaborators on the Naval Research Laboratory's pilot Software Cost Reduction Scheme for the A-7E aircraft. This thesis incorporates the tabular approach pioneered by this project into an algebraic environment to benefit the writers of algebraic specifications. Using generic techniques from research from the Software Engineering Research Group at McMaster this thesis defines six classes of function tables which may be used to define algebraic operations. Four of the six classes of function tables are: simple (finite non-recursive), nested, infinite and recursive. The remaining two are constructed by combining nested infinite and nested recursive function tables. Using Effective Definition Schemes (eds) of Friedman as a model of computation, we define the semantics of the classes of infinite function tables (simple or nested). For the class of finite function tables we restrict eds to finite eds. For the class of recursive function tables we extend eds to recursive eds. For all three models of computation we compare their computability with While and Straight Line high level programs. In addition, for the recursive eds we construct both their denotional and operational semantics and prove, in detail, their equivalence. The thesis concludes by applying the defined function tables to specifying embedded-systems, or interactive deterministic systems, which are not necessarily safety-critical. The hope is that these techniques can be used to engineer software to higher standards at the design stage of a project to reduce expensive maintenance costs. To illustrate the feasibility of this aim, we describe our experiences (with the supporting company Digita International) at applying these algebraic tables to documenting a commercial software feature.
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Greenfield, David. "Semigroup representations : an abstract approach." Thesis, University of Oxford, 1994. http://ora.ox.ac.uk/objects/uuid:ebe5ecaf-e400-41de-bcd2-e168475ac76e.
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Chapter One After the definitions and basic results required for the rest of the thesis, a notion of spectrum for semigroup representations is introduced and some relevant examples given. Chapter Two Any semigroup representation by isometries on a Banach space may be dilated to a group representation on a larger Banach space. A new proof of this result is presented here, and a connection is shown to exist between the dilation and the trajectories of the dual representation. The problem of dilating various types of spaces, including partially ordered spaces, C*-algebras, and reflexive spaces, is discussed, and new dilation theorems are given for dual Banach spaces and von Neumann algebras. Chapter Three In this chapter the spectrum of a representation is examined more closely with the aid of methods from Banach algebra theory. In the case where the representation is by isometries it is shown that the spectrum is non-empty, that it is compact if and only if the representation is norm-continuous, and that any isolated point in the unitary spectrum is an eigenvalue. Chapter Four An analytic characterisation is given of the spectral conditions that imply a representation by isometries is invertible. For representations of Z+n this con- dition is shown to be equivalent to polynomial convexity. Some topological conditions on the spectrum are also shown to imply invertibility. Chapter Five The ideas of the previous chapters are applied to problems of asymptotic behaviour. Asymptotic stability is described in terms of the behaviour of the dual of a representation. Finally, the case when the unitary spectrum is countable is discussed in detail.