Optimal 5-step nilpotent quadratic algebras

By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

Représentations et fusion des algèbres de Temperley-Lieb originale et diluée

Les algèbres de Temperley-Lieb originales, aussi dites régulières, apparaissent dans de nombreux modèles statistiques sur réseau en deux dimensions: les modèles d'Ising, de Potts, des dimères, celui de Fortuin-Kasteleyn, etc. L'espace d'Hilbert de l'hamiltonien quantique correspondant à chacun de ces modèles est un module pour cette algèbre et la théorie de ses représentations peut être utilisée afin de faciliter la décomposition de l'espace en blocs; la diagonalisation de l'hamiltonien s'en trouve alors grandement simplifiée. L'algèbre de Temperley-Lieb diluée joue un rôle similaire pour des modèles statistiques dilués, par exemple un modèle sur réseau où certains sites peuvent être vides; ses représentations peuvent alors être utilisées pour simplifier l'analyse du modèle comme pour le cas original. Or ceci requiert une connaissance des modules de cette algèbre et de leur structure; un premier article donne une liste complète des modules projectifs indécomposables de l'algèbre diluée et un second les utilise afin de construire une liste complète de tous les modules indécomposables des algèbres originale et diluée. La structure des modules est décrite en termes de facteurs de composition et par leurs groupes d'homomorphismes. Le produit de fusion sur l'algèbre de Temperley-Lieb originale permet de «multiplier» ensemble deux modules sur cette algèbre pour en obtenir un autre. Il a été montré que ce produit pouvait servir dans la diagonalisation d'hamiltoniens et, selon certaines conjectures, il pourrait également être utilisé pour étudier le comportement de modèles sur réseaux dans la limite continue. Un troisième article construit une généralisation du produit de fusion pour les algèbres diluées, puis présente une méthode pour le calculer. Le produit de fusion est alors calculé pour les classes de modules indécomposables les plus communes pour les deux familles, originale et diluée, ce qui vient ajouter à la liste incomplète des produits de fusion déjà calculés par d'autres chercheurs pour la famille originale. Finalement, il s'avère que les algèbres de Temperley-Lieb peuvent être associées à une catégorie monoïdale tressée, dont la structure est compatible avec le produit de fusion décrit ci-dessus. Le quatrième article calcule explicitement ce tressage, d'abord sur la catégorie des algèbres, puis sur la catégorie des modules sur ces algèbres. Il montre également comment ce tressage permet d'obtenir des solutions aux équations de Yang-Baxter, qui peuvent alors être utilisées afin de construire des modèles intégrables sur réseaux.

Knowledge Workers and Associative Activity: An Examination of Knowledge Work Systems and Settings

Thesis (Ph.D.)--University of Washington, 2016-08

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

Widening participation in higher education: the role of professional and social class identities and commitments

Since the neoliberal reforms to British education in the 1980s, education debates have been saturated with claims to the efficacy of the market as a mechanism for improving the content and delivery of state education. In recent decades with the expansion and ‘massification’ of higher education, widening participation (WP) has acquired an increasingly important role in redressing the under-representation of certain social groups in universities. Taken together, these trends neatly capture the twin goals of New Labour’s programme for education reform: economic competitiveness and social justice. But how do WP professionals negotiate competing demands of social equity and economic incentive? In this paper we explore how the hegemony of neoliberal discourse – of which the student as consumer is possibly the most pervasive – can be usefully disentangled from socially progressive, professional discourses exemplified through the speech and actions of WP practitioners and managers working in British higher education institutions.

Big bang bifurcations in von Bertalanffy’s dynamics with strong and weak Allee effects

The main purpose of this work was to study population dynamic discrete models in which the growth of the population is described by generalized von Bertalanffy's functions, with an adjustment or correction factor of polynomial type. The consideration of this correction factor is made with the aim to introduce the Allee effect. To the class of generalized von Bertalanffy's functions is identified and characterized subclasses of strong and weak Allee's functions and functions with no Allee effect. This classification is founded on the concepts of strong and weak Allee's effects to population growth rates associated. A complete description of the dynamic behavior is given, where we provide necessary conditions for the occurrence of unconditional and essential extinction types. The bifurcation structures of the parameter plane are analyzed regarding the evolution of the Allee limit with the aim to understand how the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is realized. To generalized von Bertalanffy's functions with strong and weak Allee effects is identified an Allee's effect region, to which is associated the concepts of chaotic semistability curve and Allee's bifurcation point. We verified that under some sufficient conditions, generalized von Bertalanffy's functions have a particular bifurcation structure: the big bang bifurcations of the so-called box-within-a-box type. To this family of maps, the Allee bifurcation points and the big bang bifurcation points are characterized by the symmetric of Allee's limit and by a null intrinsic growth rate. The present paper is also a significant contribution in the framework of the big bang bifurcation analysis for continuous 1D maps and unveil their relationship with the explosion birth and the extinction phenomena.

Efficient algorithms to solve scheduling problems with a variety of optimization criteria

La programmation par contraintes est une technique puissante pour résoudre, entre autres, des problèmes d’ordonnancement de grande envergure. L’ordonnancement vise à allouer dans le temps des tâches à des ressources. Lors de son exécution, une tâche consomme une ressource à un taux constant. Généralement, on cherche à optimiser une fonction objectif telle la durée totale d’un ordonnancement. Résoudre un problème d’ordonnancement signifie trouver quand chaque tâche doit débuter et quelle ressource doit l’exécuter. La plupart des problèmes d’ordonnancement sont NP-Difficiles. Conséquemment, il n’existe aucun algorithme connu capable de les résoudre en temps polynomial. Cependant, il existe des spécialisations aux problèmes d’ordonnancement qui ne sont pas NP-Complet. Ces problèmes peuvent être résolus en temps polynomial en utilisant des algorithmes qui leur sont propres. Notre objectif est d’explorer ces algorithmes d’ordonnancement dans plusieurs contextes variés. Les techniques de filtrage ont beaucoup évolué dans les dernières années en ordonnancement basé sur les contraintes. La proéminence des algorithmes de filtrage repose sur leur habilité à réduire l’arbre de recherche en excluant les valeurs des domaines qui ne participent pas à des solutions au problème. Nous proposons des améliorations et présentons des algorithmes de filtrage plus efficaces pour résoudre des problèmes classiques d’ordonnancement. De plus, nous présentons des adaptations de techniques de filtrage pour le cas où les tâches peuvent être retardées. Nous considérons aussi différentes propriétés de problèmes industriels et résolvons plus efficacement des problèmes où le critère d’optimisation n’est pas nécessairement le moment où la dernière tâche se termine. Par exemple, nous présentons des algorithmes à temps polynomial pour le cas où la quantité de ressources fluctue dans le temps, ou quand le coût d’exécuter une tâche au temps t dépend de t.

Endomorphisms of Banach algebras of infinitely differentiable functions

In this note we study the endomorphisms of certain Banach algebras of infinitely differentiable functions on compact plane sets, associated with weight sequences M. These algebras were originally studied by Dales, Davie and McClure. In a previous paper this problem was solved in the case of the unit interval for many weights M. Here we investigate the extent to which the methods used previously apply to general compact plane sets, and introduce some new methods. In particular, we obtain many results for the case of the closed unit disc. This research was supported by EPSRC grant GR/M31132

Dancing the Archive: Rhythms of Change in Post-Volcano Identities on Montserrat, West Indies

In this dissertation, I demonstrate how improvisations within the structures of performance during Montserrat’s annual festivals produce “rhythms of change” that contribute to the formation of cultural identities. Montserrat is a small island of 39.5 square miles in the Caribbean’s Leeward Islands, and a volcanic disaster in the 1990s led to the loss of villages, homes, and material possessions. The crisis resulted in mass displacement and emigration, and today’s remaining population of 5,000 is now in a stage of post-volcano redevelopment. The reliability of written archives for establishing cultural knowledge is tenuous, and the community is faced with re-energizing cherished cultural traditions. This ethnographic research traces my embodied search for Montserrat’s history through an archive that is itself intangible and performative. Festivals produce some of the island’s most visible and culturally political events, and music and dance performances prompt on- and off-stage discussions about the island’s multifaceted heritage. The festival cycle provides the structure for ongoing renegotiations of what it means to be “Montserratian.” I focus especially on the island’s often-discussed and debated “triangular” heritage of Irishness, Africanness, and Montserratianness as it is performed during the festivals. Through my meanderings along the winding hilly roads of Montserrat, I explored reconfigurations of cultural memory through the island’s masquerade dance tradition and other festival celebrations. In this work, I introduce a “Cast of Characters,” each of whose scholarly, artistic, and public service work on Montserrat contributes to the shape and transformation of the island’s post-volcano cultural identities today. This dissertation is about the kinesthetic transmission of shared (and sometimes unshared) cultural knowledge, the substance of which echoes in the rhythms of Montserrat’s music and dance practices today.

Life Stories, Cultural Metissage, and Personal Identities

This article encompasses an underlying notion of personal identities and processes of interaction, which distinguish essentialist identity from relational identity in contexts involving subjects, fields of possibilities, and cultural metamorphosis. It addresses the idea of the individual and her/his transformations: “I am who I want to be if I can be that person.” Any one of us could hypothetically have been someone else. The question of the reconstruction of individual identities is a vital aspect in the relationship between objective social conditions and what each person subjectively does with them, in terms of auto-construction. The complexity of this question reflects the idea of a cultural kaleidoscope, in which similar social conditions experienced by different individuals can produce differentiated identities. The title and structure of this text also seek to encompass the idea that in a personal life story, the subject lives between various spheres and sociocultural contexts, with a composite, mestizo, and superimposed or displaced identity, in each context. This occurs as the result of a cultural metamorphosis, which is constructed both by the individual as well as by heterogeneous influences between the context of the starting and finishing points at a given moment. This complex process of cultural metamorphosis—the fruit of interweaving subjective and objective forces—reveals a new dimension: the truly composite nature of personal identities.

Characterizing neural tissues with mass spectrometry imaging via enhanced computational approaches

Neuropeptides affect the activity of the myriad of neuronal circuits in the brain. They are under tight spatial and chemical control and the dynamics of their release and catabolism directly modify neuronal network activity. Understanding neuropeptide functioning requires approaches to determine their chemical and spatial heterogeneity within neural tissue, but most imaging techniques do not provide the complete information desired. To provide chemical information, most imaging techniques used to study the nervous system require preselection and labeling of the peptides of interest; however, mass spectrometry imaging (MSI) detects analytes across a broad mass range without the need to target a specific analyte. When used with matrix-assisted laser desorption/ionization (MALDI), MSI detects analytes in the mass range of neuropeptides. MALDI MSI simultaneously provides spatial and chemical information resulting in images that plot the spatial distributions of neuropeptides over the surface of a thin slice of neural tissue. Here a variety of approaches for neuropeptide characterization are developed. Specifically, several computational approaches are combined with MALDI MSI to create improved approaches that provide spatial distributions and neuropeptide characterizations. After successfully validating these MALDI MSI protocols, the methods are applied to characterize both known and unidentified neuropeptides from neural tissues. The methods are further adapted from tissue analysis to be able to perform tandem MS (MS/MS) imaging on neuronal cultures to enable the study of network formation. In addition, MALDI MSI has been carried out over the timecourse of nervous system regeneration in planarian flatworms resulting in the discovery of two novel neuropeptides that may be involved in planarian regeneration. In addition, several bioinformatic tools are developed to predict final neuropeptide structures and associated masses that can be compared to experimental MSI data in order to make assignments of neuropeptide identities. The integration of computational approaches into the experimental design of MALDI MSI has allowed improved instrument automation and enhanced data acquisition and analysis. These tools also make the methods versatile and adaptable to new sample types.

Truncation identities for the small polaron fusion hierarchy

We study a one-dimensional lattice model of interacting spinless fermions. This model is integrable for both periodic and open boundary conditions; the latter case includes the presence of Grassmann valued non-diagonal boundary fields breaking the bulk U(1) symmetry of the model. Starting from the embedding of this model into a graded Yang-Baxter algebra, an infinite hierarchy of commuting transfer matrices is constructed by means of a fusion procedure. For certain values of the coupling constant related to anisotropies of the underlying vertex model taken at roots of unity, this hierarchy is shown to truncate giving a finite set of functional equations for the spectrum of the transfer matrices. For generic coupling constants, the spectral problem is formulated in terms of a functional (or TQ-)equation which can be solved by Bethe ansatz methods for periodic and diagonal open boundary conditions. Possible approaches for the solution of the model with generic non-diagonal boundary fields are discussed.

Securing Disunion: young people’s nationalism, identities and (in)securities in the campaign for an independent Scotland.

This paper explores ethnic and religious minority youth perspectives of security and nationalism in Scotland during the independence campaign in 2014.  We discuss how young people co-construct narratives of Scottish nationalism alongside minority ethnic and faith identities in order to feel secure. By critically combining literatures from feminist geopolitics, international relations (IR) and children’s emotional geographies, we employ the concept of ‘ontological security’. The paper departs from state-centric approaches to security to explore the relational entanglements between geopolitical discourses and the ontological security of young people living through a moment of political change. We examine how everyday encounters with difference can reflect broader geopolitical narratives of security and insecurity, which subsequently trouble notions of ‘multicultural nationalism’ in Scotland and demonstrate ways that youth ‘securitize the self’ (Kinnvall, 2004). The paper responds to calls for empirical analyses of youth perspectives on nationalism and security (Benwell, 2016) and on the nexus between security and emotional subjectivity in critical geopolitics (Pain, 2009; Shaw et al., 2014). Funded by the Arts and Humanities Research Council (AHRC), this paper draws on focus group and interview data from 382 ethnic and religious minority young people in Scotland collected over the 12-month period of the campaign. Keywords: nationalism, young people, race and ethnicity, ontological security, everyday geopolitics

Intersections of Gay and Bisexual Identity with Fatness

This thesis examines the intersections of gay and bisexual identity with body size, or fatness. Gay and bisexual identity and fatness are marginalized social identities that seem to be incompatible (Bond, 2013). While a sense of collective identity with the gay and bisexual community has been shown to be a protective factor against internalized homonegativity in gay and bisexual men (Halpin & Allen, 2004), the degree to which this protective factor persists for fat people in an anti-fat environment like the gay and bisexual community (Wrench & Knapp, 2008) has not been explored. This intersection of identities and anti-fat culture seemed to suggest there might be a relationship between fatness and internalized homophobia. Fatness did not moderate the relationship between sense of belonging to the gay and bisexual community and internalized homonegativity, but a significant positive relationship was found between belongingness to the gay and bisexual community and body shame.