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Alghamdi, Mohamed A. M. A. "Some problems in algebraic topology : polynomial algebras over the Steenrod algebra." Thesis, University of Aberdeen, 1991. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=166808.
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We prove two theorems concerning the action of the Steenrod algebra in cohomology and homology. (i) Let A denote a finitely generated graded F<sub>p</sub> polynomial algebra over the Steenrod algebra whose generators have dimensions not divisible by p. The possible sets of dimensions of the generators for such A are known. It was conjectured that if we replaced the polynomial algebra A by a polynomial algebra truncated at some height greater than p over the Steenrod algebras, the sets of all possible dimensions would coincide with the former list. We show that the conjecture is false. For example F<sub>11</sub>[x<sub>6</sub>,x<sub>10</sub>]<sup>12</sup> truncated at height 12 supports an action of the Steenrod algebra but F<sub>11</sub>[x<sub>6</sub>,x<sub>10</sub>] does not. (ii) Let V be an elementary abelian 2-group of rank 3. The problem of determining a minimal set of generators for H*(BV,F<sub>2</sub>) over the Steenrod algebra was an unresolved problem for many years. (A solution was announced by Kameko in June 1990, but is not yet published.) A dual problem is to determine the subring M of the Pontrjagin ring H*(BV,F<sub>2</sub>). We determine this ring completely and in particular give a verification that the minimum number of generators needed in each dimension in cohomology is as announced by Kameko, but by using completely different techniques. Let v ε V - (0) and denote by a_5(v) ε H*(BV,F<sub>2</sub>) the image of the non-zero class in H<sub>2s-1</sub>(RP<sup>∞</sup>,F<sub>2</sub>) imeq F<sub>2</sub> under the homomorphism induced by the inclusion of F<sub>2 → V onto (0,v). We show that M is isomorphic to the ring generated by (a</sub>_s(v),s ≥ 1, v ε V - (0)) except in dimensions of the form 2^r+3 + 2^r+1 + 2^r - 3, r ≥ 0, where we need to adjoin our additional generator.
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Miscione, Steven. "Loop algebras and algebraic geometry." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=116115.
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This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.
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Bucicovschi, Orest. "Simple Lie algebras, algebraic prolongations and contact structures." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2008. http://wwwlib.umi.com/cr/ucsd/fullcit?p3307120.
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Thesis (Ph. D.)--University of California, San Diego, 2008.<br>Title from first page of PDF file (viewed July 1, 2008). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 82-85).
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Garrote, López Marina. "Algebraic and semi-algebraic phylogenetic reconstruction." Doctoral thesis, Universitat Politècnica de Catalunya, 2021. http://hdl.handle.net/10803/672316.
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Phylogenetics is the study of the evolutionary history and relationships among groups of biological entities (called taxa). The modeling of those evolutionary processes is done by phylogenetic trees whose nodes represent different taxa and whose branches correspond to the evolutionary processes between them. The leaves usually represent contemporary taxa and the root is their common ancestor. Nowadays, phylogenetic reconstruction aims to estimate the phylogenetic tree that best explains the evolutionary relationships of current taxa using solely information from their genome arranged in an alignment. We focus on the reconstruction of the topology of phylogenetic trees, which means reconstructing the shape of the tree considering labels at the leaves.To this end, one usually assumes that DNA sequences evolve according to a Markov process ruled by a prescribed model of nucleotide substitutions. These substitution models specify some transition matrices at the edges of the tree and a distribution of nucleotides at the root. Given a tree T and a substitution model, one can compute the distribution of nucleotide patterns at the leaves of T in terms of the model parameters. This joint distribution is represented by a vector whose entries can be expressed as polynomials on the model parameters and satisfy certain algebraic relationships. The study of these relationships and the geometry of the algebraic varieties defined by them (called phylogenetic varieties) have provided successful insight into the problem of phylogenetic reconstruction. However, from a biological perspective we are not interested in the whole variety, but only in the region of points that arise from stochastic parameters (the so-called phylogenetic stochastic region). The description of these regions leads to semi-algebraic constraints which play an important role since they characterize distributions with biological and probabilistic meaning. One of the main motivations for this thesis follows from the following question. Could the use of semi-algebraic tools improve the already existent algebraic tools for phylogenetic reconstruction?To answer this question, we compute the Euclidean distance of data points arising from an alignment of nucleotide to the phylogenetic varieties and their stochastic regions in a some scenarios of special interest in phylogenetics, such as trees with short external branches and/or subject to the long branch attraction phenomenon. In some cases, we compute these distances analytically and we can decide which tree has stochastic region closer to the data point. As a consequence, we can prove that, even if the data point was close to the phylogenetic variety of a given tree, it might be closer to the stochastic region of another tree. In particular, considering the stochastic phylogenetic region seems to be fundamental to cope with the phylogenetic reconstruction problem when dealing with the long branch attraction phenomenon.However, incorporating semi-algebraic tools into phylogenetic reconstruction methods can be extremely difficult and the procedure to do it is not evident at all. In this thesis, we present two phylogenetic reconstruction methods that combine algebraic and semi-algebraic conditions for the general Markov model. The first method we present is SAQ, which stands for Semi-Algebraic Quartet reconstruction method. Next, we introduce a more versatile method, ASAQ (for Algebraic and Semi-Algebraic Quartet reconstruction method}), which combines SAQ with the method Erik+2 (based on certain algebraic constraints). Both are phylogenetic reconstruction methods for DNA alignments on four taxa which have been proven to be statistically consistent.We test the suggested methods on simulated and real data to check their actual performance in several scenarios. Our simulation studies show that both methods SAQ and ASAQ are highly successful, even when applied to short alignments or data that violates their assumptions.<br>La filogenètica és l'estudi de la història evolutiva entre grups d'entitats biològiques (anomenades tàxons). Aquests processos evolutius estan modelitzats per arbres filogenètics els nodes dels quals representen diferents tàxons i les branques corresponen als processos evolutius entre ells. Les fulles normalment representen tàxons actuals i l'arrel és el seu avantpassat comú. Actualment, la reconstrucció filogenètica pretén estimar l'arbre filogenètic que millor explica les relacions evolutives de tàxons actuals utilitzant únicament informació del seu genoma organitzada en un alineament. En aquesta tesi ens centrem en la reconstrucció de la topologia dels arbres filogenètics, és a dir, reconstruir la forma de l'arbre tenint en compte els noms associats a les fulles. Amb aquesta finalitat, assumim que les seqüències d'ADN evolucionen segons un procés de Markov d'acord amb un model de substitució de nucleòtids. Aquests models de substitució assignem matrius de transició a les arestes d’un arbre i una distribució de nucleòtids a l'arrel. Donat un arbre i un model, es pot calcular la distribució de les possibles observacions de nucleòtids a les fulles en termes dels paràmetres del model. Aquesta distribució conjunta s’expressa en forma de vector, les entrades del qual es poden escriure com polinomis en funció dels paràmetres del model i satisfan certes relacions algebraiques. L'estudi d'aquestes relacions i de la geometria de les varietats algebraiques que defineixen (anomenades varietats filogenètiques) han servit per entendre millor el problema de la reconstrucció filogenètica. No obstant això, des d'una perspectiva biològica no estem interessats en tota la varietat, sinó només en la regió de punts que resulten de paràmetres estocàstics (l'anomenada regió estocàstica). La descripció d'aquestes regions condueix a restriccions semi-algebraiques que tenen un paper important ja que caracteritzen les distribucions amb significat biològic. Una de les principals motivacions d'aquesta tesi és la següent: Podria l'ús d'eines semi-algebraiques millorar les eines algebraiques ja existents per a la reconstrucció filogenètica? Per poder respondre, calculem la distància euclidiana entre punts de dades obtinguts a partir d’un alineament i varietats filogenètiques i les seves regions estocàstiques en escenaris d'especial interès en la filogenètica. En alguns casos, podem calcular aquestes distàncies de forma analítica i això ens permet demostrar que, fins i tot si un punt de dades fos proper a la varietat filogenètica d'un arbre donat, podria estar més a prop de la regió estocàstica d'un altre arbre. En particular, considerar la regió estocàstica sembla ser fonamental per fer front al problema de la reconstrucció filogenètica quan tractem amb del fenomen d'atracció de branques llargues. Tot i això, incorporar d'eines semi-algebraiques en els mètodes de reconstrucció filogenètica pot ser extremadament difícil i el procediment per fer-ho no és gens evident. En aquesta tesi, presentem dos mètodes de reconstrucció filogenètica que combinen condicions algebraiques i semi-algebraiques per al model general de Markov. El primer mètode que presentem és el SAQ, que rep el nom de Semi-Algebraic Quartet reconstruction method. A continuació, introduïm un mètode més versàtil, l'ASAQ (Algebraic and Semi-Algebraic Quartet reconstruction method), que combina el SAQ amb el mètode Erik+2 (basat en certes restriccions algebraiques). Tots dos són mètodes de reconstrucció filogenètica per a alineaments d'ADN per quatre tàxons i hem demostrat que tots dos són estadísticament consistents. Finalment, testem els mètodes proposats amb dades simulades i dades reals per comprovar el seu rendiment en diversos escenaris. Les nostres simulacions mostren que ambdós mètodes SAQ i ASAQ obtenen<br>Matemàtica aplicada
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Bowman, Christopher David. "Algebraic groups, diagram algebras, and their Schur-Weyl dualities." Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610216.
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Ronagh, Pooya. "The inertia operator and Hall algebra of algebraic stacks." Thesis, University of British Columbia, 2016. http://hdl.handle.net/2429/58120.
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We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line q = [A¹]. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks. Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting polynomials in q) and their corresponding eigenvalue decompositions are used to define a graded structure on the Grothendieck ring. We then define the structure of a Hall algebra on the space of stack functions. The commutative and non-commutative products of the Hall algebra respect the graded structure defined above. Moreover, the two multiplications coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. Finally, for any algebroid, an ε-element is defined and shown to be contained in the space of virtually indecomposables. This is a new approach to the theory of generalized Donaldson-Thomas invariants.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate
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Dias, Eduardo Manuel. "Algebraic covers." Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/80934/.
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The main goal of this thesis is the description of the section ring of a surface R(S,L) = O∞n=0 H0(S,nL) where L is an ample base point free divisor defining a covering map φL: S -> P2 such that φ*OS = OP2 O Ω1P2 O Ω1P2 O Op2(-3). This is an abelian surface with a polarization of type (1,3) which was studied before in [BL94, Cas99, Cas12]. Given a covering map φ: X -> Y, following the methods introduced by Miranda for general d covers, in chapter 3 we will define a cover homomorphism that will induce a commutative and associative multiplication in φ*OX. Chapter 4 focuses in the OP2-modules Hom (S2Ω1P2,Ω1P2) that will be used to define a commutative multiplication for our surface. Chapter 5 is about the associative condition. It is a computational method based on the paper [Rei90]. In the last chapter we use the ring R(S,L) to prove that the moduli space of abelian surfaces with a polarization of type (1,3) and canonical level structure is rational. We will also show how to use the same method to find models for covering maps such that φ*OS = OP2 O Ω1P2(-m1) O Ω1P2(-m2) O OP2(-m1-m2-3). The last section contains new problems whose goal is to construct and study algebraic varieties given by the vanishing of a high codimensional Gorenstein ideal.
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Milione, Piermarco. "Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/402209.
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The main purpose of this dissertation is to introduce Shimura curves from the non-Archimedean point of view, paying special attention to those aspects that can make this theory amenable for computations. Despite the fact that the theory of p-adic uniformization of Shimura curves goes back to the 1960s with the results of Cerednik and Drinfeld, only in the last years explicit examples related to these uniformizations have been computed.
The structure of this dissertation is as follows. In Chapter 1 we introduce Shimura curves starting from an indefinite quaternion algebra H over a totally real field F. This is done mostly following the fundamental paper of Shimura [Shi67]. We also give the definitions using the adelic approach of [Shi70b] and [Shi70c]. The point of view we adopt is the arithmetical one, since we try to make clear the link connecting Shimura curves to the arithmetic of quaternion algebras. In this sense, we give evidence of why Shimura curves have to be considered a geometric interpretation of most arithmetical phenomena in quaternion orders. Chapter 2 has the aim of introducing those non-Archimedean objects which appear later in the statements of the theorems of Cerednik and Drinfeld. In Chapter 3 we start the study of fundamental domains in Hp for the action of discrete and cocompact subgroups of PGL2(Qp) arising in the p-adic uniformization of Shimura curves. In Chapter 4 we associate to the p-adic uniformization of the Shimura curve X(DH;N) certain parameters in Hp(Cp) analogous to the complex multiplication parameters in H: we refer to them by p-imaginary multiplication paramters, since they are defined over the unramified quadratic extension of Qp. In the study of these parameters, we follow the p-adic analog of the line adopted in [AB04]. Specifically, we are able to recover these parameters as zeros of certain binary quadratic forms with p-adic coefficients.
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Sinn, Rainer [Verfasser]. "Algebraic Boundaries of Convex Semi-Algebraic Sets / Rainer Sinn." Konstanz : Bibliothek der Universität Konstanz, 2014. http://d-nb.info/1052418252/34.
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Sharif, H. "Algebraic functions, differentially algebraic power series and Hadamard operations." Thesis, University of Kent, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.235336.
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Wilson, David. "Advances in cylindrical algebraic decomposition." Thesis, University of Bath, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636529.
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Since their conception by Collins in 1975, Cylindrical Algebraic Decompositions (CADs) have been used to analyse the real algebraic geometry of systems of polynomials. Applications for CAD technology range from quantifier elimination to robot motion planning. Although of great use in practice, the CAD algorithm was shown to have doubly exponential complexity with respect to the number of variables for the problem, which limits its use for large examples. Due to the high complexity of CAD, much work has been done to improve its performance. In this thesis new advances will be discussed that improve the practical efficiency of CAD for a variety of problems, with a new complexity result for one set of algorithms. A new invariance condition, truth table invariance (TTICAD), and two algorithms to construct TTICADs are given and shown to be highly efficient. The idea of restricting the output of CADs, allowing for greater efficiency, is formalised as sub-decompositions and two particular ideas are investigated in depth. Efficient selection of various formulation choices for a CAD problem are discussed, with a collection of heuristics investigated and machine learning applied to assist in choosing an optimal heuristic. The mathematical expression of a problem is shown to be of great importance, with preconditioning and reformulation investigated. Finally, these advances are collected together in a general framework for applying CAD in an efficient manner to a given problem. It is shown that their combination is not cumulative and care must be taken. To this end, a prototype software CADassistant is described to help users take advantage of the advances without knowledge of the underlying theory. The effects of the various advances are demonstrated through a guiding example originally considered by Solotareff, which describes the approximation of a cubic polynomial by a linear function. Naïvely applying CAD to the problem takes 916.1 seconds of construction (from which a solution can easily be derived), which is reduced to 20.1 seconds by combining various advances from this thesis.
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Shammu, Nizar Miekha. "Algebraic and categorical structure of categories of crossed modules of algebras." Thesis, Bangor University, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.304282.
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Wagner, Fabian. "Exactly solvable models, Yang-Baxter algebras and the algebraic Bethe Ansatz." Thesis, University of Cambridge, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621030.
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Ito, Kazuhiro. "Algebraic cycles and cohomology with torsion coefficients of algebraic varieties." Kyoto University, 2021. http://hdl.handle.net/2433/261597.
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Cottrell, Thomas. "Comparing algebraic and non-algebraic foundations of n-category theory." Thesis, University of Sheffield, 2014. http://etheses.whiterose.ac.uk/5324/.
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Many definitions of weak n-category have been proposed. It has been widely observed that each of these definitions is of one of two types: algebraic definitions, in which composites and coherence cells are explicitly specified, and non-algebraic definitions, in which a coherent choice of composites and constraint cells is merely required to exist. Relatively few comparisons have been made between definitions, and most of those that have concern the relationship between definitions of just one type. The aim of this thesis is to establish more comparisons, including a comparison between an algebraic definition and a non-algebraic definition. The thesis is divided into two parts. Part 1 concerns the relationships between three algebraic definitions of weak n-category: those of Penon and Batanin, and Leinster's variant of Batanin's definition. A correspondence between the structures used to define composition and coherence in the definitions of Batanin and Leinster has long been suspected, and we make this precise for the first time. We use this correspondence to prove several coherence theorems that apply to all three definitions, and also to take the first steps towards describing the relationship between the weak n-categories of Batanin and Leinster. In Part 2 we take the first step towards a comparison between Penon's definition of weak n-category and a non-algebraic definition, Simpson's variant of Tamsamani's definition, in the form of a nerve construction. As a prototype for this nerve construction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction is a Tamsamani--Simpson weak 2-category. We then define our nerve functor for Penon weak n-categories. We prove that the nerve of a Penon weak 2-category is a Tamsamani--Simpson weak 2-category, and conjecture that this result holds for higher n.
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Cardó, Carles 1975. "Algebraic dependency grammar." Doctoral thesis, Universitat Politècnica de Catalunya, 2018. http://hdl.handle.net/10803/463326.
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We propose a mathematical formalism called Algebraic Dependency Grammar with applications to formal linguistics and to formal language theory. Regarding formal linguistics we aim to address the problem of grammaticality with special attention to cross-linguistic cases. In the field of formal language theory this formalism provides a new perspective allowing an algebraic classification of languages. Notably our approach suggests the existence of so-called anti-classes of languages associated to certain classes of languages.
Our notion of a dependency grammar is as of a definition of a set of well-constructed dependency trees (we call this algebraic governance) and a relation which associates word-orders to dependency trees (we call this algebraic linearization).
In relation to algebraic governance, we define a manifold which is a set of dependency trees satisfying an agreement condition throughout a pattern, which is the algebraic form of a collection of syntactic addresses over the dependency tree. A boolean condition on the words formalizes the notion of agreement.
In relation to algebraic linearization, first we observe that the notion of projectivity is quintessentially that certain substructures of a dependency tree always form an interval in its linearization. So we have to establish well what is a substructure; we see again that patterns proportion the key, generalizing the notion of projectivity with recursive linearization procedures.
Combining the above modules we have the formalism: an algebraic dependency grammar is a manifold together with a linearization.
Notice that patterns sustain both manifolds and linearizations. We study their interrelation in terms of a new algebraic classification of classes of languages.
We highlight the main contributions of the thesis. Regarding mathematical linguistics, algebraic dependency grammar considers trees and word-order different modules in the architecture, which allows description of languages with varied word-order. Ellipses are permitted; this issue is usually avoided because it makes some formalisms non-decidable.
We differentiate linguistic phenomena structurally by their algebraic description. Algebraic dependency grammar permits observance of affinity between linguistic constructions which seem superficially different.
Regarding formal language theory, a new system for understanding a very large family of languages is presented which permits observation of languages in broader contexts. We identify a new class named anti-context-free languages containing constructions structurally symmetric to context-free languages. Informally we could say that context-free languages are well-parenthesized, while anti-context-free languages are cross-serial-parenthesized. For example copy languages and respectively languages are anti-context-free.<br>Es proposa un formalisme matemàtic anomenat Gramàtica de Dependències Algebraica amb aplicacions a la lingüística formal i a la teoria de llenguatges formals. Pel que fa a la lingüística formal es pretén abordar el problema de la gramaticalitat, amb un èmfasi especial en la transversalitat, això és, que el formalisme sigui apte per a un bon nombre de llengües. En el camp dels llenguatges formals aquest formalisme proporciona una nova perspectiva que permet una classificació algebraica dels llenguatges. Aquest enfocament suggereix a més a més l'existència de les aquí anomenades anti-classes de llenguatges associades a certes classes de llenguatges. La nostra idea d'una gramàtica de dependències és en un conjunt de sintagmes ben construïts (d'això en diem recció algebraica) i una relació que associa ordres de paraules als sintagmes d'aquest conjunt (d'això en diem linearització algebraica). Pel que fa a la recció algebraica, introduïm el concepte de varietat sintàctica com el conjunt de sintagmes que satisfan una concordança sobre un determinat patró. Un patró és un conjunt d'adreces sintàctiques descrit algebraicament. La concordança es formalitza a través d'una condició booleana sobre el vocabulari. En relació amb linearització algebraica, en primer lloc, observem que l'essencial de la noció clàssica de projectivitat rau en el fet que certes subestructures d'un arbre de dependències formen sempre un interval en la seva linearització. Així doncs, primer hem d'establir bé que vol dir subestructura. Un cop més veiem que els patrons en proporcionen la clau, tot generalitzant la noció de projectivitat a través d'un procediment recursiu de linearització. Tot unint els dos mòduls anteriors ja tenim el nostre formalisme a punt: una gramàtica de dependències algebraica és una varietat sintàctica juntament amb una linearització. Notem que els patrons són a la base de tots dos mòduls: varietats i linearitzacions, així que resulta del tot natural estudiar-ne la interrelació en termes d'un nou sistema de classificació algebraica de classes de llenguatges. Destaquem les principals contribucions d'aquesta tesi. Pel que fa a la matemàtica lingüística, la gramàtica de dependències algebraica considera els arbres i l'ordre de les paraules diferents mòduls dins l'arquitectura la qual cosa permet de descriure llenguatges amb una gran varietat d'ordre. L'ús d'el·lipsis és permès; aquesta qüestió és normalment evitada en altres formalismes per tal com la possibilitat d'el·lipsis fa que els models es tornin no decidibles. El nostre model també ens permet classificar estructuralment fenòmens lingüístics segons la seva descripció algebraica, així com de copsar afinitats entre construccions que semblen superficialment diferents. Pel que fa a la teoria dels llenguatges formals, presentem un nou sistema de classificació que ens permet d'entendre els llenguatges en un context més ampli. Identifiquem una nova classe que anomenem llenguatges anti-lliures-de-context que conté construccions estructuralment simètriques als llenguatges lliures de context. Informalment podríem dir que els llenguatges lliures de context estan ben parentetitzats, mentre que els anti-lliures-de-context estan parentetitzats segons dependències creuades en sèrie. En són mostres d'aquesta classe els llenguatges còpia i els llenguatges respectivament.
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Hartsell, Melanie Lynne. "Algebraic Number Fields." Thesis, University of North Texas, 1991. https://digital.library.unt.edu/ark:/67531/metadc501201/.
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This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
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Nikazad, Touraj. "Algebraic Reconstruction Methods." Doctoral thesis, Linköping : Linköpings universitet, Department of Mathematics Scientific Computing, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11670.
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Koudenburg, Seerp Roald. "Algebraic weighted colimits." Thesis, University of Sheffield, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.616958.
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Stephens, S. "Algebraic stream processing." Thesis, Swansea University, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.639104.
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We identify and analyse the typically higher-order approaches to stream processing in the literature. From this analysis we motivate an alternative approach to the specification of SPSs as STs based on an essentially first-order equational representation. This technique is called <I>Cartesian form specification.</I> More specifically, while STs are properly second-order objects we show that using Cartesian forms, the second-order <I>models</I> needed to formalise STs are so weak that we may use and develop well-understood first-order methods from computability theory and mathematical logic to reason about their properties. Indeed, we show that by specifying STs equationally in Cartesian form as <I>primitive recursive functions </I>we have the basis of a new, general purpose and mathematically sound <I>theory</I> of stream processing that emphasises the formal specification and formal verification of STs. The main topics that we address in the development of this theory are as follows. We present a theoretically well-founded general purpose stream processing language <I>ASTRAL</I> (<I>A</I>lgebraic <I>S</I>tream <I>TRA</I>nsformer <I>L</I>anguage) that supports the use of modular specification techniques for full second-order STs. We show how ASTRAL specifications can be given a Cartesian form semantics using the language <I>PREQ</I> that is an equational characterisation of the primitive recursive functions. In more detail, we show that by compiling ASTRAL specifications into an equivalent Cartesian form in PREQ we can use first-order equational logic with induction as a logical calculus to reason about STs. In particular, using this calculus we identify a syntactic class of correctness statements for which the verification of ASTRAL programmes is decidable relative to this calculus. We define an effective algorithm based on term re-writing techniques to implement this calculus and hence to automatically verify a very broad class of STs including conventional hardware devices. Finally, we analyse the properties of this abstract algorithm as a proof assistant and discuss various techniques that have been adopted to develop software tools based on this algorithm.
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Gibson, John Keith. "Algebraic coded cryptosystems." Thesis, Royal Holloway, University of London, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.321816.
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Steers, Luke. "Algebraic finite domination." Thesis, Queen's University Belfast, 2017. https://pure.qub.ac.uk/portal/en/theses/algebraic-finite-domination(f1e6eccf-ecc3-4601-9fab-6d4aa25bfd57).html.
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Eghosa, Edeghagba Elijah. "Ω-Algebraic Structures". Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2017. https://www.cris.uns.ac.rs/record.jsf?recordId=104206&source=NDLTD&language=en.
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The research work carried out in this thesis is aimed at fuzzifying algebraic and relational structures in the framework of Ω-sets, where Ω is a complete lattice.Therefore we attempt to synthesis universal algebra and fuzzy set theory. Our investigations of Ω-algebraic structures are based on Ω-valued equality, satisability of identities and cut techniques. We introduce Ω-algebras, Ω-valued congruences, corresponding quotient Ω-valued-algebras and Ω-valued homomorphisms and we investigate connections among these notions. We prove that there is an Ω-valued homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernelof an Ω-valued homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-valued homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an Ω-valued congruence determines a closure system of classical congruences on cut subalgebras. In addition, identities are preserved under Ω-valued homomorphisms. Therefore in the framework of Ω-sets we were able to introduce Ω-attice both as an ordered and algebraic structures. By this Ω-poset is defined as an Ω-set equipped with Ω-valued order which is antisymmetric with respect to the corresponding Ω-valued equality. Thus defining the notion of pseudo-infimum and pseudo-supremum we obtained the definition of Ω-lattice as an ordered structure. It is also defined that the an Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality fulfilling some particular lattice Ω-theoretical formulas. Thus using axiom of choice we proved that the two approaches are equivalent. Then we also introduced the notion of complete Ω-lattice based on Ω-lattice. It was defined as a generalization of the classical complete lattice.We proved results that characterizes Ω-structures and many other interesting results.Also the connection between Ω-algebra and the notion of weak congruences is presented.We conclude with what we feel are most interesting areas for future work.<br>Tema ovog rada je fazifikovanje algebarskih i relacijskih struktura u okviru omega- skupova, gdeje Ω kompletna mreza. U radu se bavimo sintezom oblasti univerzalne algebre i teorije rasplinutih (fazi) skupova. Naša istraživanja omega-algebarskih struktura bazirana su na omega-vrednosnoj jednakosti,zadovoljivosti identiteta i tehnici rada sa nivoima. U radu uvodimo omega-algebre,omega-vrednosne kongruencije, odgovarajuće omega-strukture, i omega-vrednosne homomorfizme i istražujemo veze izmedju ovih pojmova. Dokazujemo da postoji Ω -vrednosni homomorfizam iz Ω -algebre na odgovarajuću količničku Ω -algebru. Jezgro Ω -vrednosnog homomorfizma je Ω- vrednosna kongruencija. U vezi sa nivoima struktura, dokazujemo da Ω -vrednosni homomorfizam odredjuje klasične homomorfizme na odgovarajućim količničkim strukturama preko nivoa podalgebri. Osim toga, Ω-vrednosna kongruencija odredjuje sistem zatvaranja klasične kongruencije na nivo podalgebrama. Dalje, identiteti su očuvani u Ω- vrednosnim homomorfnim slikama.U nastavku smo u okviru Ω-skupova uveli Ω-mreže kao uredjene skupove i kao algebre i dokazali ekvivalenciju ovih pojmova. Ω-poset je definisan kao Ω -relacija koja je antisimetrična i tranzitivna u odnosu na odgovarajuću Ω-vrednosnu jednakost. Definisani su pojmovi pseudo-infimuma i pseudo-supremuma i tako smo dobili definiciju Ω-mreže kao uredjene strukture. Takodje je definisana Ω-mreža kao algebra, u ovim kontekstu nosač te strukture je bi-grupoid koji je saglasan sa Ω-vrednosnom jednakošću i ispunjava neke mrežno-teorijske formule. Koristeći aksiom izbora dokazali smo da su dva pristupa ekvivalentna. Dalje smo uveli i pojam potpune Ω-mreže kao uopštenje klasične potpune mreže. Dokazali smo još neke rezultate koji karakterišu Ω-strukture.Data je i veza izmedju Ω-algebre i pojma slabih kongruencija.Na kraju je dat prikaz pravaca daljih istrazivanja.
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Lurie, Jacob 1977. "Derived algebraic geometry." Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/30144.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.<br>Includes bibliographical references (p. 191-193).<br>The purpose of this document is to establish the foundations for a theory of derived algebraic geometry based upon simplicial commutative rings. We define derived versions of schemes, algebraic spaces, and algebraic stacks. Our main result is a derived analogue of Artin's representability theorem, which provides a precise criteria for the representability of a moduli functor by geometric objects of these types.<br>by Jacob Lurie.<br>Ph.D.
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Schmidt, Renate Anneliese. "Algebraic terminological representation." Master's thesis, University of Cape Town, 1991. http://hdl.handle.net/11427/22147.
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This thesis investigates terminological representation languages, as used in KL-ONE-type knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntactic types, called concepts and roles, which are usually interpreted model-theoretically as sets and relations, respectively. I propose an algebraic rather than a model-theoretic approach. I show that terminological representations can be naturally accommodated in equational algebras of sets interacting with relations, and I use equational logic as a vehicle for reasoning about concepts interacting with roles.
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Garibaldi, Skip. "Trialitarian algebraic groups /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1998. http://wwwlib.umi.com/cr/ucsd/fullcit?p9906492.
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Andres, Wolf Daniel [Verfasser]. "Noncommutative computer algebra with applications in algebraic analysis / Wolf Daniel Andres." Aachen : Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2014. http://d-nb.info/1049821475/34.
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Townsend, Brian E. "Examining secondary students algebraic reasoning flexibility and strategy use /." Diss., Columbia, Mo. : University of Missouri-Columbia, 2005. http://hdl.handle.net/10355/4131.
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Thesis (Ph. D.)--University of Missouri-Columbia, 2005.<br>The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (November 14, 2006) Vita. Includes bibliographical references.
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Nickerson, Susan Denise. "Supporting students' understanding of algebra : symbolizing in a technology-enhanced classroom /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC IP addresses, 2001. http://wwwlib.umi.com/cr/ucsd/fullcit?p3022703.
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at, michor@esi ac. "The Generalized Cayley Map from an Algebraic Group to its Lie Algebra." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1066.ps.
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Pierce, Robyn U. "An exploration of algebraic insight and effective use of computer algebra systems /." Connect to thesis, 2001. http://eprints.unimelb.edu.au/archive/00000739.
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Kedzierska, Anna Magdalena. "Algebraic tools in phylogenomics." Doctoral thesis, Universitat Politècnica de Catalunya, 2012. http://hdl.handle.net/10803/81566.
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En aquesta tesi interdisciplinar desenvolupem eines algebraiques per a problemes en filogenètica i genòmica.
Per estudiar l'evolució molecular de les espècies sovint s'usen models evolutius estocàstics. L'evolució es representa en un arbre (anomenat filogenètic) on les espècies actuals corresponen a fulles de l'arbre i els nodes interiors corresponen a ancestres comuns a elles. La longitud d'una branca de l'arbre representa la quantitat de mutacions que han ocorregut entre les dues espècies adjacents a la branca. Llavors l'evolució de seqüències d'ADN en aquestes espècies es modelitza amb un procés Markov ocult al llarg de l'arbre. Si el procés de Markov se suposa a temps continu, normalment s'assumeix que també és homogeni i, en tal cas, els paràmetres del model són les entrades d'una raó de mutació instantània i les longituds de les branques. Si el procés de Markov és a temps discret, llavors els paràmetres del model són les probabilitats condicionades de substitució de nucleòtids al llarg de l'arbre i no hi ha cap hipòtesi d'homogeneïtat. Aquests últims són els tipus de models que considerem en aquesta tesi i són, per tant, més generals que els de temps continu.
Des d'aquesta perspectiva s'estudien els problemes més bàsics de la filogenètica: donat un conjunt de seqüències d'ADN, com decidim quin és el model evolutiu més adequat? com inferim de forma eficient els paràmetres del model? I fins i tot, tal i com també hem provat en aquesta tesi, és possible que les espècies no hagin evolucionat seguint un sol arbre sinó una mescla d'arbres i llavors cal abordar aquestes preguntes en aquest cas més general. Per a models evolutius a temps continu i homogenis, s'ha proposat solucions diverses a aquestes preguntes al llarg de les últimes dècades. En aquesta tesi resolem aquests dos problemes per a models evolutius a temps discret usant tècniques algebraiques provinents d'àlgebra lineal, teoria de grups, geometria algebraica i estadística algebraica. A més a més, la nostra solució per al primer problema és vàlida també per a mescles filogenètiques.
Hem fet tests dels mètodes proposats en aquesta tesi sobre dades simulades i dades reals del projectes ENCODE (Encyclopedia Of DNA Elements). Per tal de provar els nostres mètodes hem donat algoritmes per a generar seqüències evolucionant sota un model a temps discret amb un nombre esperat de mutacions prefixat. I així mateix, hem demostrat que aquests algorismes generen totes les seqüències possibles (per la majoria de models). Els tests sobre dades simulades mostren que els mètodes proposats són molt acurats i els resultats sobre dades reals permeten corroborar hipòtesis prèviament formulades. Tots els mètodes proposats en aquesta tesi han estat implementats per a un nombre arbitrari d'espècies i estan disponibles públicament.<br>In this thesis we develop interdisciplinary algebraic tools for genomic and phylogenetic problems.
To study the molecular evolution of species one often uses stochastic evolutionary models. The evolution is represented in a tree (called phylogenetic tree) whose leaves represent current species and whose internal nodes correspond to their common ancestors. The length of a branch of the tree represents the number of mutations that have occurred between the two species adjacent to the branch. Then ,the evolution of DNA sequences in these species is modeled with a hidden Markov process along the tree. If the Markov process is assumed to be continuous in time, it is usually assumed homogeneous as well and, if so, the model parameters are the instantaneous rate of mutation and the lengths of the branches. If the Markov process is discrete in time, then the model parameters are the conditional probabilities of nucleotide substitution along the tree and there is no assumption of homogeneity. The latter are the types of models we consider in this thesis and are therefore more general than the homogeneous continuous ones.
From this perspective we study the basic problems of phylogenetics: Given a set of DNA sequences, what is the evolutionary model that best fits the data? how can we efficiently infer the model parameters? Also, as we also checked in this thesis, it is possible that species have not evolved along a single tree but a mixture of trees so that we need to address these questions in this more general case. For continuous-time, homogeneous, evolutionary models, several solutions to these questions have been proposed during the last decades. In this thesis we solve these two problems for discrete-time evolutionary models, using algebraic techniques from linear algebra, group theory, algebraic geometry and algebraic statistics. In addition, our solution to the first problem is also valid for phylogenetic mixtures.
We have made tests of the methods proposed in this thesis on simulated and real data from ENCODE Project (Encyclopedia Of DNA Elements). To test our methods, we also provide algorithms to generate sequences evolving under discrete-time models with a given expected number of mutations. Even more, we have proved that these algorithms generate all possible sequences (for most models). Tests on simulated data show that the methods are very accurate and our results on real data confirm hypotheses previously formulated. All the methods in this thesis have been implemented for an arbitrary number of species and are publicly available.
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Alexander, Nicholas Charles. "Algebraic Tori in Cryptography." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/1154.
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Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP<sup>+</sup>05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.
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Redelmeier, Daniel. "Hyperpfaffians in Algebraic Combinatorics." Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/1055.
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The pfaffian is a classical tool which can be regarded as a generalization of the determinant. The hyperpfaffian, which was introduced by Barvinok, generalizes the pfaffian to higher dimension. This was further developed by Luque, Thibon and Abdesselam. There are several non-equivalent definitions for the hyperpfaffian, which are discussed in the introduction of this thesis. Following this we examine the extension of the Matrix-Tree theorem to the Hyperpfaffian-Cactus theorem by Abdesselam, proving it without the use of the Grassman-Berezin Calculus and with the new terminology of the non-uniform hyperpfaffian. Next we look at the extension of pfaffian orientations for counting matchings on graphs to hyperpfaffian orientations for counting matchings on hypergraphs. Finally pfaffian rings and ideal s are extended to hyperpfaffian rings and ideals, but we show that under reason able assumptions the algebra with straightening law structure of these rings cannot be extended.
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de, Almeida Otterson James Joaquim. "Curves in algebraic surfaces." Thesis, Imperial College London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.525234.
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Mahmoud, Ola. "Second-order algebraic theories." Thesis, University of Cambridge, 2011. https://www.repository.cam.ac.uk/handle/1810/241035.
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Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This workcompletes the foundations of the subject from the viewpoint of categorical algebra. Specifically, this thesis introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semanticlevel, that of second-order algebras and second-order functorial models. The development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
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Loeffler, David. "Overconvergent algebraic automorphic forms." Thesis, Imperial College London, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.487962.
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I present a general theory of overconvergent p-adic automorphic forms for reductive algebraic groups whose real points are compact (or, more generally, whose arithmetic subgroups are finite, as in the work of Gross). Let G be such a group, p a prime and P a parabolic subgroup of G defined over Qp. Given a fixed representation V of the Levi factor of P, I construct a family of p-adic locally analytic representations of the parahoric subgroup associated to P by induction from twists of V by characters. By considering functions from G(Af ) to these representations satisfying suitable equivariance conditions, one obtains a p-adic Banach module over a certain weight space, with a Heeke action, whose fibre over an integer weight naturally contains the Heeke module of classical automorphic forms of that weight. Using Buzzard's eigenvariety machine, this gives a construction of an eigenvariety parametrising finite slope overconvergent eigenforms. I also prove an analogue in this situation of Coleman's theorem that forms of small slope are classical, implying that this eigenvariety contains a dense set of points corresponding to classical eigenforms. A convenient property of the spaces constructed by these methods is that the definition is sufficiently concrete to allow computer calculations. This is also true for classical automorphic forms on groups satisfying the above condition, as has been noted by various authors. I give a detailed description of an algorithm for calculating classical automorphic forms in the case where G is a definite unitary group, and discuss how the results may be interpreted in terms of Galois representations. In future work I intend to extend this to calculate overconvergent automorphic forms on such groups. In addition, the thesis includes a chapter devoted to the original, motivating example where the theory overconvergent automorphic forms was developed, the case of modular forms for congruence subgroups of GL2 . The main result of this section is the proof of an instance of a conjecture due to Gouvea and Mazur, which is that the overconvergent Heeke eigenforms should form a basis for the entire space of overconvergent forms.
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Levy, D. "Algebraic properties of anomalies." Thesis, Imperial College London, 1987. http://hdl.handle.net/10044/1/47071.
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Balchin, Scott Lewis. "Augmented homotopical algebraic geometry." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/40623.
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In this thesis we are interested in extending the theory of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. We introduce the concept of augmentation categories, which are a class of generalised Reedy categories. An augmentation category is a category which has enough structure that we can mirror the simplicial constructions which make up the theory of homotopical algebraic geometry. In particular, we construct a Quillen model structure on their presheaf categories, and introduce the concept of augmented hypercovers to define a local model structure on augmented presheaves. As an application, we show that a crossed simplicial group is an example of an augmentation category. The resulting augmented geometric theory can be thought of as being equivariant. Using this, we define equivariant cohomology theories as special mapping spaces in the category of equivariant stacks. We also define the SO(2)-equivariant derived stack of local systems by using a twisted nerve construction. Moreover, we prove that the category of planar rooted trees appearing in the theory of dendroidal sets is also an augmentation category. The augmented geometry over this setting should be thought of as being stable in the spectral sense of the word. Finally, we show that we can combine the two main examples presented using a categorical amalgamation construction.
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Luo, Huazhang 1971. "Stability of algebraic manifolds." Thesis, Massachusetts Institute of Technology, 1998. http://hdl.handle.net/1721.1/47463.
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Sustretov, Dmitry. "Non-algebraic Zariski geometries." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412.
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The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H<sup>1</sup>(k(x), μ<sub>n</sub>). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.
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Vicinansa, Guilherme Scabin. "Algebraic estimators with applications." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-21092018-150106/.
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In this work we address the problem of friction compensation in a pneumatic control valve. It is proposed a nonlinear control law that uses algebraic estimators in its structure, in order to adapt the controller to the aging of the valve. For that purpose we estimate parameters related to the valve\'s Karnopp model, necessary to friction compensation, online. The estimators and the controller are validated through simulations.<br>Nessa pesquisa, estudamos o problema de compensação de atrito em válvulas pneumáticas. É proposta uma lei de controle não linear que tem estimadores algébricos em sua estrutura, para adaptar o controlador ao envelhecimento da válvula. Para isso, estimam-se os valores de parâmetros relacionados ao modelo de Karnopp da válvula, necessários à compensação do atrito, de maneira online. Os estimadores e o controlador são validados através de simulações.
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Ashford, Matthew. "Graphs of algebraic objects." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:a666c588-7653-4e1c-a60d-5841f1019a1e.
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Al-Zamil, Qusay Soad. "Algebraic topology of PDES." Thesis, University of Manchester, 2012. https://www.research.manchester.ac.uk/portal/en/theses/algebraic-topology-of-pdes(6e25e379-5e32-4db8-abd1-e0a892cecea6).html.
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We consider a compact, oriented,smooth Riemannian manifold $M$ (with or without boundary) and wesuppose $G$ is a torus acting by isometries on $M$. Given $X$ in theLie algebra of $G$ and corresponding vector field $X_M$ on $M$, onedefines Witten's inhomogeneous coboundary operator $\d_{X_M} =\d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariantforms on $M$) and its adjoint $\delta_{X_M}$. First, Witten [35] showed that the resulting cohomology classeshave $X_M$-harmonic representatives (forms in the null space of$\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomologygroups are isomorphic to the ordinary de Rham cohomology groups ofthe set $N(X_M)$ of zeros of $X_M$. The first principal purpose isto extend Witten's results to manifolds with boundary. Inparticular, we define relative (to the boundary) and absoluteversions of the $X_M$-cohomology and show the classes haverepresentative $X_M$-harmonic fields with appropriate boundaryconditions. To do this we present the relevant version of theHodge-Morrey-Friedrichs decomposition theorem for invariant forms interms of the operators $\d_{X_M}$ and $\delta_{X_M}$; the proofinvolves showing that certain boundary value problems are elliptic.We also elucidate the connection between the $X_M$-cohomology groupsand the relative and absolute equivariant cohomology, followingwork of Atiyah and Bott. This connection is then exploited to showthat every harmonic field with appropriate boundary conditions on$N(X_M)$ has a unique corresponding an $X_M$-harmonic field on $M$to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of $X_M$-cohomology and then we definethe \emph{$X_M$-Poincar\' duality angles} between the interiorsubspaces of $X_M$-harmonic fields on $M$ with appropriate boundaryconditions.Second, In 2008, Belishev and Sharafutdinov [9] showed thatthe Dirichlet-to-Neumann (DN) operator $\Lambda$ inscribes into thelist of objects of algebraic topology by proving that the de Rhamcohomology groups are determined by $\Lambda$.In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?.Based on the results in the first part above, we define an operator$\Lambda_{X_M}$ on invariant forms on the boundary $\partial M$which we call the $X_M$-DN map and using this we recover the longexact $X_M$-cohomology sequence of the topological pair $(M,\partialM)$ from an isomorphism with the long exact sequence formed from thegeneralized boundary data. Consequently, This shows that for aZariski-open subset of the Lie algebra, $\Lambda_{X_M}$ determinesthe free part of the relative and absolute equivariant cohomologygroups of $M$. In addition, we partially determine the mixed cup product of$X_M$-cohomology groups from $\Lambda_{X_M}$. This shows that $\Lambda_{X_M}$ encodes more information about theequivariant algebraic topology of $M$ than does the operator$\Lambda$ on $\partial M$. Finally, we elucidate the connectionbetween Belishev-Sharafutdinov's boundary data on $\partial N(X_M)$and ours on $\partial M$.Third, based on the first part above, we present the(even/odd) $X_M$-harmonic cohomology which is the cohomology ofcertain subcomplex of the complex $(\Omega^{*}_G,\d_{X_M})$ and weprove that it is isomorphic to the total absolute and relative$X_M$-cohomology groups.
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Trenn, Stephan. "Distributional differential algebraic equations." Ilmenau Univ.-Verl, 2009. http://d-nb.info/99693197X/04.
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Bonfanti, M. A. "ALGEBRAIC SURFACES WITH AUTOMORPHISMS." Doctoral thesis, Università degli Studi di Milano, 2015. http://hdl.handle.net/2434/345557.
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In my thesis I worked on two different projects, both related with projective surfaces with automorphisms. In the first one I studied Abelian surfaces with an automorphism and quaternionic multiplication: this work has already been accepted for publication in the Canadian Journal of Mathematics. In the second project I treat surfaces isogenous to a product of curves and their cohomology.
Abelian Surfaces with an Automorphism
The Abelian surfaces, with a polarization of a fixed type, whose endomorphism ring is an order in a quaternion algebra are parametrized by a curve, called Shimura curve, in the moduli space of polarized Abelian surfaces. There have been several attempts to find concrete examples of such Shimura curves and of the Abelian surfaces over this curve. In [HM95] Hashimoto and Murabayashi find Shimura curves as the intersection, in the moduli space of principally polarized Abelian surfaces, of Humbert surfaces. Such Humbert surfaces are now known “explicitly” in many other cases and this might allow one to find explicit models of other Shimura curves. Other approaches are taken in [Elk08] and [PS11].
We consider the rather special case where one of the Abelian surfaces in the family is the selfproduct of an elliptic curve. We assume this elliptic curve to have an automorphism of order three or four. For a fixed product polarization of type (1, d), we denote by Hj,d the set of the deformations of the selfproduct with the automorphism of order j. We prove the following theorem:
Theorem. Let j ∈ {3,4}, d ∈ Z, d > 0 and let τ ∈ Hj,d, so that the Abelian surface Aτ,d has an automorphism φj of order j. Then the endomorphism algebra of Aτ,d also contains an element ψj with ψj^2 = d. Moreover, for a general τ ∈ Hj,d one has
End(Aτ,d)Q =(−j,d)/Q
where (a, b)/Q := Q1 ⊕ Qi ⊕ Qj ⊕ Qk is the quaternion algebra with i^2 = a, j^2 =b and ij=−ji.
It is easy compute for which d the quaternion algebra (−j,d)/Q is a skew field: for these d the general Abelian surface in the family Hj,d is simple. In particular this provides examples of simple Abelian surfaces with an automorphism of order three and four. This construction, together with well-known results about automorphisms of Abelian surfaces (see [BL04, Chapter 13]), leads to:
Theorem. Let A be a simple Abelian surface and φ ∈ Aut(A) a non-trivial automorphism of finite order. Then ord(φ) ∈ {3, 4, 5, 6, 10}.
We focus in particular on the family H3,2 of Abelian surfaces with an automorphism of order three and a polarization of type (1,2). In [Bar87] Barth provides a description of a moduli space M2,4, embedded in P5, of (2, 4)-polarized Abelian surfaces with a level structure. Since the polarized Abelian surfaces we consider have an automorphism of order three, the corresponding points in M2,4 are fixed by an automorphism of order three of P^5. This allows us to explicitly identify the Shimura curve in M2,4 that parametrizes the Abelian surfaces with quaternionic multiplication by the
maximal order O6 in the quaternion algebra with discriminant 6. It is embedded as a line in M2,4 ⊂ P5 and the symmetric group S4 acts on this line by changing the level structures. According to Rotger [Rot04], an Abelian surface with endomorphism ring O6 is the Jacobian of a unique genus two curve. We show explicitly how to find such genus two curves, or rather their images in the Kummer surface embedded in P5 with a (2,4)-polarization.
These curves were already been considered by Hashimoto and Murabayashi in [HM95]: we give the explicit relation between their description and ours. Moreover we find a Humbert surface in M2,4 that parametrizes Abelian surfaces with Z( 2) in the endomorphism ring.
Cohomology of surfaces isogenous to a product
Surfaces isogenous to a product of curves provide examples of surfaces of general type with many different geometrical invariants. They have been introduced by Catanese in [Cat00]:
Definition. A smooth surface S is said to be isogenous to a product (of curves) if it is isomorphic to a quotient (C×D)/G where C and D are curves of genus at least one and G is a finite group acting freely on C × D.
We say that a surface isogenous to a product is of mixed type if there exists a element of G interchanging the two curves; otherwise, if G acts diagonally on the product, we say that the surface is of unmixed type. A surface isogenous to a product is of general type if the genus of both curves, C and D, is greater or equal to 2: in this case we say that the surface is isogenous to a higher product.
The cohomology groups of a surface S = (C × D)/G isogenous to a product of unmixed type are determined by the action of the group G on the cohomology groups of the curves. Moreover the action of an automorphism group G on a smooth curve C forces a decomposition of the first cohomology group, as described in [BL04, section 13.6] and in [Roj07]:
Proposition (Group algebra decomposition). Let G be a finite group acting on a curve C. Let W1, ..., Wr denote the irreducible rational representations of G and let ni := dimDi (Wi), with Di := EndG(Wi), for i = 1, ..., r. Then there are rational Hodge substructures B1, ..., Br such that
H1(C, Q) ≃ n1B1+...+nrBr.
From this a decomposition of the cohomology groups of the surface S follows directly. We apply these results to surfaces isogenous to a higher product of unmixed type with χ(OS) = 2 and q(S) = 0: they have been studied and classified by Gleissner in [Gle13]. For these surfaces the Hodge diamond is fixed and in particular the Hodge numbers of the second coho- mology groups are the same as those of an Abelian surface.
From Gleissner’s classification we obtain a complete list of the 21 possible groups G. We proved that the second cohomology group of these surfaces can be described explicitly as follows:
Theorem. Let S be a surface isogenous to a higher product of unmixed type with χ(OS) = 2, q(S) = 0. Then there exist two elliptic curves E1 and E2 such that H2(S, Q) ∼= H2(E1 × E2, Q) as rational Hodge structures.
In general it is not possible to construct these elliptic curves “geometrically” using the action of G. More precisely there are no intermediate coverings πF : C → C/F and πH : D → D/H, F, H ≤ G with C/F = E1 and D/H = E2: we can only prove that such elliptic curves must exist. The proof of the theorem is standard for all but four groups: in these cases we study one by one the corresponding surfaces in order to construct the elliptic curves. As a further application we use this approach to study some surfaces isogenous to a higher product with pg = q = 2, in particular those are of Albanese general type.
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Preslicka, Anthony J. "The Topology and Algebraic Functions on Affine Algebraic Sets Over an Arbitrary Field." Digital Archive @ GSU, 2012. http://digitalarchive.gsu.edu/math_theses/121.
Full textAbstract:
This thesis presents the theory of affine algebraic sets defined over an arbitrary field K. We define basic concepts such as the Zariski topology, coordinate ring of functions, regular functions, and dimension. We are interested in the relationship between the geometry of an affine algebraic set over a field K and its geometry induced by the algebraic closure of K. Various versions of Hilbert-Nullstellensatz are presented, introducing a new variant over finite fields. Examples are provided throughout the paper and a question on the dimension of irreducible affine algebraic sets is formulated.
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Deshpande, D. V. "Topological methods in algebraic geometry : cohomology rings, algebraic cobordism and higher Chow groups." Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.598515.
Full textAbstract:
This thesis is divided into three chapters. The first chapter is about the cohomology ring of the space of rotational functions. In the second chapter, we define algebraic cobordism of classifying spaces, Ω*(<i>BG</i>) and <i>G</i>-equivariant algebraic cobordism Ω*<i><sub>G</sub></i>(-) for a linear algebraic group <i>G. </i>We prove some properties of the coniveau filtration on algebraic cobordism, denoted <i>F<sup>j</sup></i>(Ω*(-)); which are required for the definition to work. We show that <i>G</i>-equivariant cobordism satisfies the localization exact sequence. We compute Ω*(<i>BG</i>) for algebraic groups over the complex numbers corresponding to classical Lie groups <i>GL</i>(<i>n</i>), <i>SL</i>(<i>n</i>), <i>Sp</i>(<i>n</i>), <i>O</i>(<i>n</i>) and <i>SO</i>(2<i>n</i> + 1). We also compute Ω*(<i>BG</i>) when <i>G</i> is a finite abelian group. A finite non-abelian group for which we compute Ω*(<i>BG</i>) is the quaternion group of order 8. In all the above cases we check that Ω*(<i>BG</i>) is isomorphic to <i>MU</i>*(<i>BG</i>). The third chapter is work-in-progress on Steenrod operations on higher Chow groups. Voevodsky has defined motivic Steenrod operations on motivic cohomology and used them in his proof of the Minor Conjecture.
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Rüschoff, Christian [Verfasser], and Otmar [Akademischer Betreuer] Venjakob. "Relative algebraic K-theory and algebraic cyclic homology / Christian Rüschoff ; Betreuer: Otmar Venjakob." Heidelberg : Universitätsbibliothek Heidelberg, 2016. http://d-nb.info/1180737873/34.
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Junkins, Caroline. "The Grothendieck Gamma Filtration, the Tits Algebras, and the J-invariant of a Linear Algebraic Group." Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31331.
Full textAbstract:
Consider a semisimple linear algebraic group G over an arbitrary field F, and a projective homogeneous G-variety X. The geometry of such varieties has been a consistently active subject of research in algebraic geometry for decades, with significant contributions made by Grothendieck, Demazure, Tits, Panin, and Merkurjev, among others. An effective tool for the classification of these varieties is the notion of a cohomological (or alternatively, a motivic) invariant. Two such invariants are the set of Tits algebras of G defined by J. Tits, and the J-invariant of G defined by Petrov, Semenov, and Zainoulline. Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these invariants, which they developed through use of the second Chern class map.
The first goal of the present thesis is to extend this connection through the use of higher Chern class maps. Our main technical tool is the Steinberg basis, which provides explicit generators for the γ-filtration on the Grothendieck group K_0(X) in terms of characteristic classes of line bundles over X. As an application, we establish a connection between the J-invariant and the Tits algebras of a group G of inner type E6.
The second goal of this thesis is to relate the indices of the Tits algebras of G to nontrivial torsion elements in the γ-filtration on K_0(X). While the Steinberg basis provides an explicit set of generators of the γ-filtration, the relations are not easily computed. A tool introduced by Zainoulline called the twisted γ-filtration acts as a surjective image of the γ-filtration, with explicit sets of both generators and relations. We use this tool to construct torsion elements in the degree 2 component of the γ-filtration for groups of inner type D2n. Such a group corresponds to an algebra A endowed with an orthogonal involution having trivial discriminant. In the trialitarian case (i.e. type D4), we construct a specific element in the γ-filtration which detects splitting of the associated Tits algebras. We then relate the non-triviality of this element to other properties of the trialitarian triple such as decomposability and hyperbolicity.