In this thesis we review some aspects of Yang-Baxter deformations of semi-symmetric space sigma models. We start by giving a short review of the sigma model description of superstrings and then we offer a self contained introduction to the Yang-Baxter deformation technique. We then show how to obtain an integrable deformation of the hybrid sigma model. Also, we show that the gravity dual of beta-deformed ABJM theory can be obtained as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to construct an Abelian r-matrix satisfying the classical Yang-Baxter equation. / Nesta tese revisamos alguns aspectos das deformações de Yang-Baxter de modelos sigma em espaços semi-simétricos. Damos uma breve revisão do modelo sigma de supercordas e, em seguida, oferecemos uma introdução ao método de deformação de Yang-Baxter. Em seguida, mostramos como obter uma deformação integrável do modelo sigma híbrido. Além disso, mostramos que o dual gravitacional da teoria ABJM beta-deformada pode ser obtida como uma deformação de Yang-Baxter. Isso é feito selecionando-se uma combinação conveniente de geradores de Cartan para construir uma matriz r Abeliana satisfazendo a equação clássica de Yang-Baxter.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-09-20T19:09:51Z No. of bitstreams: 2 Tese - Lucas Nunes Jorge - 2017.pdf: 5873752 bytes, checksum: 804e244800839331e5594855dcd427a8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-21T10:49:25Z (GMT) No. of bitstreams: 2 Tese - Lucas Nunes Jorge - 2017.pdf: 5873752 bytes, checksum: 804e244800839331e5594855dcd427a8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-21T10:49:25Z (GMT). No. of bitstreams: 2 Tese - Lucas Nunes Jorge - 2017.pdf: 5873752 bytes, checksum: 804e244800839331e5594855dcd427a8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-04 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work, we used a refined entropy sampling technique based on the Wang- Landau method and finite-size scaling techniques to study variations of the Baxter-Wu model, namely: spin-$1/2$, spin-$1$, spin-$1$ with the crystal field interaction and was done a three-dimensional proposal for the model. It was also verified characteristics in the order parameter to be adopted in the simulations. The universality class and the critical temperature were calculated for the spin-$1/2$ case, and the results founded were in good agreement with the exact ones found in the literature. We sought to determine the kind of the phase transition that the model suffers for the spin-$1$ case, being carried out a detailed study for continuous and discontinuous phase transitions. The Baxter-Wu model with crystal field, $D$, had its phase diagram constructed, as well as the determination of the point at which discontinuous transitions finalizes. The critical exponent, $\nu$, was evaluated for several values of the crystal field, where we verified is variation along the critical line, with the existence of a peak, corroborating the existence of a multicritic behavior of the model. We also observed the existence of an anomaly in the specific heat, associated to the Schottky defect. This anomaly appears more clearly for values of $D \geq 1.990$. In the study of the order parameter, we verified that in the simulations one should not, when considering lattice sizes multiple of three, use the order parameter as the total magnetization of the lattice, but to consider the magnetization by sub-lattices. When working with sizes of lattices that are not multiples of three, it is not a problem to adopt the order parameter as the total magnetization of the lattice. A three-dimensional proposal was also made for the Baxter-Wu model, and its phase transition was characterized. / Neste trabalho, utilizamos uma técnica de amostragem entrópica com refinamentos baseada no método de Wang-Landau e técnicas de escala de tamanho finito para estudar variações do modelo de Baxter-Wu, a saber: spin-$1/2$, spin-$1$, spin-$1$ na presença de anisotropia de campo cristalino, onde também foi feita uma proposta tridimensional para o modelo. Foram verificadas também características no parâmetro de ordem a ser adotado nas simulações. A classe de universalidade e a temperatura crítica foram calculadas para o caso spin-$1/2$, e os resultados encontrados tiveram boa concordância com os resultados exatos que constam na literatura. Buscou-se determinar o tipo de transição de fase que o modelo descreve para o caso spin-$1$, sendo feito um detalhado estudo para transições de fases contínua e descontínua. O modelo de Baxter-Wu com interação do campo cristalino, $D$, teve o seu diagrama de fases construído, bem como a determinação do ponto em que terminam as transições descontínuas. O expoente crítico, $\nu$, foi avaliado para diversos valores do campo cristalino, onde verificamos a sua variação ao longo da linha crítica, com o aparecimento de um pico, corroborando a existência de um comportamento multicrítico do modelo. Observamos também uma anomalia no calor específico, que pode ser associada ao defeito Schottky. Essa anomalia aparece de maneira mais significativa para valores de $D\geqslant1.990$. No estudo do parâmetro de ordem, verificamos que nas simulações não se deve, ao considerar tamanhos de redes múltiplos de três, utilizar o parâmetro de ordem como a magnetização total da rede, mas tomar a magnetização por sub-redes. Ao se trabalhar com tamanhos de redes que não sejam múltiplos de três, não há problema em adotá-lo como a magnetização total da rede. Foi feita também uma proposta tridimensional para o modelo de Baxter-Wu, e foi caracterizada a sua transição de fases.

The Canadian artist IAIN BAXTER&, known before 2005 as Iain Baxter, created an innovative Conceptual Art practice in the mid-60s that continues to make important contributions even today. He has maintained a strong collaborative element in his art, as witnessed by his role in the short-lived group IT (1965) and N.E. THING CO. (1966-1978)--an actual incorporated company consisting of BAXTER& and his first wife, Ingrid Baxter as chief officers--and by the addition of an ampersand to his legal name in 2005 to signify the open-ended quality of his work that relies on viewers‘ contributions to help determine its meaning. This dissertation introduces the term "Pop-inflected Conceptual Art" to describe how BAXTER& merges his use of information technologies, modern and ubiquitous materials, and pedestrian activities with a desire to question the received role and purpose of art through an epistemological approach. By presenting BAXTER&‘s key influences—Zen Buddhism, as described by D.T. Suzuki and Alan Watts, and the communications theory of Marshall McLuhan—this study describes five underlying principles that inform BAXTER&‘s work individually and in unison. These principles are: his preference for foregrounding the banal; creation of the "infoscape" that merges the natural world and the constant stream of information within North American culture; proclivity for experimenting with such unlikely media as plastics and telecommunication media; understanding of art‘s kinship with language; and usage of pseudonyms. This study describes the core terms of McLuhan‘s theory in order to analyze this thinker‘s significance for BAXTER&‘s work; moreover, it presents how the above five principles are evident throughout the three main divisions in BAXTER&‘s artistic career: before, during, and after his tenure with N.E. THING CO. Through an analysis of key examples of many of this artist‘s works, this study also determines affinities to both established Pop artists and his Conceptual Art peers, while distinguishing how his Zen and McLuhanesque hybrid approach, which includes a consistent reliance on humor to communicate definitively his ideas, sets him apart from these groups of artists and foregrounds his role as the precursor to such younger Vancouver photoconceptual artists as Ian Wallace, Jeff Wall, Rodney Graham, and Stan Douglas.

Pakeha identity has long been problematic, caught in a straddling stance between European co-ordinates of origin, and life in Aotearoa. This has been particularly evident over the last three decades: with the rise of tino rangatiratanga, Pakeha identity has undergone something of a crisis. Group identity, especially in such periods of crisis, requires �narratives� that re-imagine being and belonging. Poetry by Pakeha both displays the problem with identity at the cross cultural threshold between Maori and Pakeha, asking � what happens when Pakeha engage cross-culturally with te ao Maori, appropriating te reo Maori and drawing on Maoritanga, to re-vision and reconfigure identity? And how does such an approach shape the imagining of Pakeha identity?� I study, in parallel, the cross-cultural poetry of James K. Baxter and Glen Colquhoun in relation to these questions. Borrowing a conceptual metaphor from Colin McCahon, I examine these cross-cultural poems in detail as �load-bearing structures�. I pay particular attention to the way in which, in purpose, design and materials, they function to re-imagine Pakeha identity in reciprocal relationship with te ao Maori. Aware of the problems of culture-crossing, at the outset I establish a historical and interpretive framework for the poetry. Furthermore, I discuss the question of appropriation, arguing for an ethical distinction between appropriation and misappropriation based on a cross-cultural relationship of faithful and reciprocal engagement. I conclude that Baxter and Colquhoun are singular and radical in their reconfiguration of Pakeha identity. Baxter embraces te ao Maori in a direct challenge to Pakeha nationalism, prescribing the necessary corrective of the tuakana-teina dynamic to Pakeha identity and its relationship with te ao Maori. Writing after the Maori renaissance, Glenn Colquhoun irreverently opens up a further reconfiguration of cross-cultural relationship, pushing both Maori and Pakeha beyond a cultural dichotomy towards a mutually defining complementarity. Both place themselves on the cultural threshold of language, embracing the tensions of the cross-cultural scenario. The result is cross-cultural poetry, load-bearing structures that manifest the tension and ambivalence of the settler culture�s straddling identity, enacting what it is to be Pakeha.

Le sujet de cette thèse est l'établissement d'une nouvelle solution de l'équation de Yang-Baxter. Cette équation est présente dans de très nombreux domaines de la physique théorique (systèmes intégrables, mécanique statistique, QISM,...) ou des mathématiques (théorie des nœuds, groupes quantiques,...), mais l'étude de ses solutions est difficile (équations non-linéaires, variables non-commutatives, etc.). Une solution de l'équation de Yang-Baxter est aussi appelée tressage.<br /><br />Dans une première partie, nous présentons des résultats généraux sur le groupe des tresses et son algèbre de groupe. Nous nous intéressons ensuite aux analogues tressés que l'on peut considérer comme des analogues non-commutatifs de q-analogues. Nous présentons entre autres des analogues pour les coefficients binomiaux, les symboles de Pochhammer et les nombres de Fuß-Catalan, ainsi que pour le développent binomial et la convolution de Vandermonde. Ces deux premiers chapitres contiennent des résultats plus ou moins standards et forment l'assise des résultats qui suivent. La définition des nombres de Fuß-Catalan est toutefois originale.<br /><br />Dans une seconde partie, nous abordons les tressages d'espaces de tenseurs. Nous commençons par présenter les équations qui doivent être satisfaites par un tel objet et nous donnons une solution dont nous montrons l'unicité. Dans un dernier chapitre, nous plaçons ce tressage dans un contexte plus général et nous présentons les tressages dits « zébrés » qui prennent en compte une éventuelle cyclicité dans l'ordre des tenseurs sur lesquels ils se projettent. Le contenu de ces deux derniers chapitres est original. Nous fournissons ainsi une nouvelle solution de l'équation de Yang-Baxter et explorons ses propriétés.

Ce travail est au confluent de la théorie des algèbres d'opérateurs<br />et des probabilités non-commutatives. Nous étudions les propriétés<br />des algèbres de von Neumann, $\Gamma_{q}(H_{\R})$, engendrées par<br />des variables Gaussiennes non-commutatives et $q$-déformées. Ces<br />variables $q$-Gaussiennes sont des opérateurs agissant sur l'espace<br />de Fock $q$-déformé, où sont réalisées les relations de<br />$q$-commutations canoniques.<br /><br />Dans la première partie de ce mémoire, nous établissons des<br />inégalités à coefficients opérateurs de type Khintchine-$L^{\infty}$<br />pour les produits de Wick des algèbres $q$-Gaussiennes. Ces<br />inégalités étendent d'un côté les inégalités scalaires dues à<br />Haagerup dans le cas libre et d'un autre côté les inégalités à<br />coefficients opérateurs, pour les $q$-Gaussiens, dues à Bo\.zejko et<br />Speicher. A l'aide de ces inégalités nous en déduisons que les<br />algèbres $\Gamma_q(H_{\R})$ sont non injectives dès que<br />$\dim_{\R}(H_{\R})\ge 2$.<br /><br />La deuxième partie est dédiée à la construction d'un modèle<br />asymptotique matriciel pour les variables $q$-Gaussiennes.<br />L'existence d'un tel modèle nous permet de prouver que les algèbres<br />$\Gamma_{q}(H_{\R})$ sont QWEP.<br /><br />Chemin faisant, nous traitons également le cas $C^*-$algébrique et<br />étudions diverses généralisations des résultats précédents pour les<br />déformations par opérateur de Yang-Baxter et pour les déformations<br />$q$-Gaussiennes de type $I\!I\!I$.

Cette thèse s'inscrit dans le cadre général de la théorie des systèmes intégrables avec bords et le développement des structures algébriques associées. D'une part, nous nous attaquons au problème de la diagonalisation de l'hamiltonien du modèle XXZ avec bords non diagonaux. Nous exhibons les deux ensembles d'états propres et valeurs propres du modèle si les paramètres de bords satisfont deux conditions. D'autre part, nous introduisons un modèle de physique statistique que nous appelons le modèle face avec un bord réfléchissant. Nous calculons exactement sa fonction de partition et nous montrons que cette dernière se représente simplement sous la forme d'un unique déterminant matriciel. Nous montrons que ces deux problèmes sont reliés par la transformation vertex-face et exhibent une structure algébrique commune, l'algèbre de réflexion dynamique. Nous nous intéressons aux aspects mathématiques de cette algèbre dans le cas elliptique général, et nous introduisons deux classes de ces représentations, la représentation de co-module d'évaluation et sa duale. Nous pensons que cette algèbre est la structure clef pour l'analyse des modèles faces avec bords. En particulier, nous montrons à l'aide de twists de Drinfel'd que leur fonction de partition se représente simplement dans le cas général.

Thesis (D. Min.)--Trinity International University, 2001. / Abstract. Includes bibliographical references (leaves 238-254).

In this dissertation, we study two areas of interest in knot theory: Random knots in the unit cube, and the Yang-Baxter solutions constructed from Frobenius algebras. The study of random knots can be thought of as a model of DNA strings situated in confinement. A random knot with n vertices is a polygonal loop formed by selecting n distinct points in the unit cube, for a positive integer n, and connecting these points by straight line segments successively, such that the last point selected is joined with the first one. We present a step by step description of our algorithm and Maple codes for generating random knots in the unit cube, with a given vertex number n. To detect non-trivial knots, we use a knot invariant called the determinant. We present an algorithm and its Maple code for computing the determinant for random knots. For each vertex number n, we generate large number of random knots and form data sets of values of the determinant. Then we analyze our data sets in various ways. For instance, for each vertex number n, we form data sets of the number of p-colorable random knots by finding the set of prime divisors of each determinant output. We define the stick number for p-colorability to be the minimum number of line segments required to form a p-colorable knot. We use our data sets to find upper bounds for stick numbers for p-colorability, for primes p _ 191. We also find distributions of p-colorable knots and small determinant values. The second topic on random knots is the linking number of random links. A random link is a collection of disjoint random knots produced simultaneously. We present descriptions of our algorithm and its Maple code for constructing random links of two components, and calculating their linking numbers in detail. By running the code for 1000 times, for the vertex number n less than or equal to 30, we obtain data sets of linking numbers for two-component random links such that each component is a random knot with n vertices. Then we find the distribution of linking numbers and calculate upper bounds for the stick number for the linking numbers ` _ 15. The second area we investigate is applications of Fobenius algebras to knot theory. Chain complexes and Yang-Baxter solutions (R-matrices) are constructed by the skein theoretic approach using Frobenius algebras, and deformed R-matrices are constructed by using 2-cocyles. We compute cohomology groups, Yang-Baxter solutions and their cocycle deformations for group algebras, polynomial algebras and complex numbers. We construct knot and link invariants using these R-matrices from Frobenius algebras via Turaev’s criteria. Then a series of skein relations of the invariant are introduced for oriented knot or link diagrams. We also present calculations of the Frobenius skein invariant for various knots and links.

While Richard Baxter (1615-91) is well known and rightly held in high esteem for his practical divinity and his evangelistic zeal, he has hitherto been conspicuously neglected as a theologian. In particular there have been no major studies of him with respect to the renewed paradigm of Protestant Scholasticism and none at all of his Methodus Theologiae (1681), which represents the fruit of a lifetime of theological reflection and study and which is arguably, in both scope and vision, one of the last great Summas of English scholastic divinity. This thesis focuses on the Methodus and on Baxter‟s theological method, which he took, though imperfect, to be the closest to the true Scripture method of theology that anyone had yet come. Baxter believed that every level of (active) created reality reflected the impress of God‟s Triune being in metaphysical composition, structure and activity. This he described, following the Italian metaphysicist Tommaso Campanella, in terms of the divine primalities or principles of Power, Wisdom and Love. In the Methodus these insights are systematised into a kind of Trinitarian logic. Baxter held that human reason should be sanctified in order to conform to the Trinitarian structure of created reality, and therefore espoused a method of trichotomising organised according to these same divine principles, derivative of both Ramist and Lullist method. This thesis argues that the whole of Baxter‟s mature thought is structured in a Trinitarian fashion according to his own „hallowed logic‟ and that two themes, often interlinked, are the key to interpreting his thought: the metaphysics of the divine principles and the Christian‟s baptismal covenant with the Triune God. Furthermore it examines Baxter‟s analogical ascent from the general vestigia Trinitatis present in the whole created order through the special vestigium of man‟s soul fashioned in the image of God and finally to the Trinity itself. This detailed exposition provides the basis, in the concluding chapter, for an examination of the whole of the Methodus and a demonstration that this represents a methodological unfolding of the covenant between the believer and God the Father, Son and Holy Spirit in the threefold Kingdom of Nature, Grace and Glory. In this way the Methodus may be seen as having taken its inspiration from the Theo-Politica (1659) of Baxter‟s friend George Lawson. Finally this thesis concludes that Baxter‟s thought has pronounced Scotist and Nominalist accents. His Scotism in particular runs deep and has strong ties with his Trinitarian thought, which is especially significant in light of the recent increasingly vocal discussions of the Scotist character of Protestant Scholasticism. Overall therefore it is suggested that Baxter is a neglected figure who deserves to be rediscovered and whose mature theology represents a fascinating reconstrual of biblical ideas according to a Trinitarian and scholastic paradigm.