Browsing by Subject "Hopf algebras"
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Item Deformations of Quantum Symmetric Algebras Extended by Groups(2012-07-16) Shakalli Tang, JeanetteThe study of deformations of an algebra has been a topic of interest for quite some time, since it allows us to not only produce new algebras but also better understand the original algebra. Given an algebra, finding all its deformations is, if at all possible, quite a challenging problem. For this reason, several specializations of this question have been proposed. For instance, some authors concentrate their efforts in the study of deformations of an algebra arising from an action of a Hopf algebra. The purpose of this dissertation is to discuss a general construction of a deformation of a smash product algebra coming from an action of a particular Hopf algebra. This Hopf algebra is generated by skew-primitive and group-like elements, and depends on a complex parameter. The smash product algebra is defined on the quantum symmetric algebra of a nite-dimensional vector space and a group. In particular, an application of this result has enabled us to find a deformation of such a smash product algebra which is, to the best of our knowledge, the first known example of a deformation in which the new relations in the deformed algebra involve elements of the original vector space. Finally, using Hochschild cohomology, we show that these deformations are nontrivial.Item Optimization of connection patterns in networks of oscillators(Texas Tech University, 2003-12) Navaratna, Menaka BandaraThe means by which pacemaker cells of the mammalian suprachiasmatic nucleus (SCN) are synchronized is unknown. In the absence of anatomical data on the interneuronal connections among SCN neurons, we have modeled the SCN network in terms of a number of possible connection topologies. We employ a mathematical model proposed by Achermann and Kunz (1999), to study the problem of interpreting synchronization in the SCN network from a dynamical systems viewpoint. We vary the proportion of local or nearest neighbor neuronal connections and global or long distance connections in the SCN, and compare time elapsed before synchronization is established. Time of resynchronization is the time elapsed before SCN neurons reestablish their phase-locked circadian response after complete randomization of initial phases. We consider two models where one, is a three-dimensional model with 8000 neurons connected as a torus and the second, an one-dimensional model consisting of 500 neurons (mean period=24 hr, S.D.=1 hr), each with Kronauer dynamics, with weak inhibitory coupling to each other, similar to the model described by Achermann and Kunz. Neurons are arranged in a ring in all directions, and connections are assumed to be symmetric with respect to cell locations. Mainly we studied two different dynamics: a three dimensional model of 8000 neurons with fixed connection patterns and one-dimensional model of 500 neurons with random symmetric connection patterns. The three dimensional models was simulated under four different light conditions: Absence of light, presence of light,10x10x10 core light and 16x16x16 core light. It was observed that the existence of few long distance connections make significant difference in the synchronization times. For instance, the synchronization times for pure locally connected and few long distance connected networks under the core light of 16x16x16 were 17 and 11 days, respectively. Results were similar for the other light conditions. In the second stage, we studied the synchronization and phase locking times of two types of randomly (symmetric) connected networks: Purely local and few long distance connections networks. It was clearly observed, with a light increase in long distance connections the phase locking time drops dramatically. For instance, if the number of long distance connections are increased from 2 to 10 (out of 30 connections), the phase locking time drops from 47 days to 6 days. In conjunction with our previous finding that completely interconnected global networks resynchronize much more quickly than the physiological oscillator in the SCN, these results suggest the possibility that the SCN topology is a "small world" network, i.e., a neuronal network with largely local interconnections plus a small number of long distance connections.Item Phase locking in the circadian rhythm(Texas Tech University, 2003-05) Wijeratne, Nilmini SaumyaExistence of an internal timing mechanism in mammals has been well established and known as the Circadian Rhythm which is generated in a bilateral structure contained in the hypothalamus called the Suprachaismatic Nucleus (SCN) which consists of 16,000 neurons. Individually, each neuron behaves like a clock, and the ensemble of neurons are capable of producing well-synchronized and phased-locked clock signals with precise time patterns. In this thesis, theory of Hopf Bifurcation in the presence of symmetries and the Center Manifold theory are used to explain the functionality and phase locking of the SCN. In addition, dynamical behavior of the system is simulated for different connection topologies and different bifurcation parameters. Also, system response in the presence of a second bifurcation parameter is also analyzed. Center Manifold theory describes a way to extract out a dynamical system representative of persistent dynamical effects. In order to compute the reduced order representative, the Taylor series expansion was incorporated into our work. Use of symmetries eliminates the necessity to calculate the Taylor series coefficients for all unstable states, thus reducing the calculations by a considerable amount.Item Tate Cohomology of Finite Dimensional Hopf Algebras(2014-06-19) Nguyen, Van CatLet A be a finite dimensional Hopf algebra over a field k. In this dissertation, we study the Tate cohomology ?* (A, k) and Tate-Hochschild cohomology (HH) ?* (A, A) of A, and their properties. We introduce cup products that make them become graded-commutative rings and establish the relationship between these rings. In particular, we show ?* (A, k) is an algebra direct summand of (HH) ?* (A, A) as a module over ?* (A, k). When A is a finite group algebra RG over a commutative ring R, we show that the Tate-Hochschild cohomology ring (HH) ?* (RG, RG) of RG is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of G. Moreover, our main result provides an explicit formula for the cup product in (HH) ?* (RG, RG) with respect to this decomposition. When A is symmetric, we show that there are finitely generated A-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring ?* (A, k) of A. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to A has finitely generated Tate cohomology, then so does every module in that component.