In this dissertation, we will give a generating set of the Kauffman bracket skein module over the field Q(A) of 0-framed surgery along the trefoil knot. This generating set is described as a certain subset of a known basis for the skein module over Z[A^±1] of the trefoil exterior.

Apples, porcupines, and the most obscure Bob Dylan song'is every topic a few clicks from Philosophy? Within Wikipedia, the surprising answer is yes: nearly all paths lead to Philosophy. Wikipedia is the largest, most meticulously indexed collection of human knowledge ever amassed. More than information about a topic, Wikipedia is a web of naturally emerging relationships. By following the first link in each article, we algorithmically construct a directed network of all 4.7 million articles: Wikipedia's First Link Network. Here we study the English edition of Wikipedia's First Link Network for insight into how the many inventions, places, people, objects, and events are related and organized. We traverse every path, measuring the accumulation of first links, path lengths, basins, cycles, and the influence each article exerts in shaping the network. We discover scale-free distributions describe path length, accumulation, and influence. Far from dispersed, first links disproportionately accumulate at a few articles'flowing from specific to general and culminating around fundamental notions such as Community, State, and Science. Philosophy shapes more paths than any other article by two orders of magnitude. Curiously, we also observe a gravitation towards topical articles such as Health Care and Fossil Fuel. These findings enrich our view of the connections and structure of Wikipedia's ever growing store of knowledge.

This paper examines print queue management for the graphics department of a financial services company. The current network configuration has proven to be sub-optimal. The IT department is currently undergoing testing of possible alternative network configurations. The objective is to improve performance by leveraging existing resources with new technology. In this paper, the effect of consolidating the queue into one primary queue manager is analyzed, along with prioritizing print jobs, and forecasting future printer needs. Analysis was performed using queuing theory concepts along with an analysis of both steady state and transient behavior using simulation modeling.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / For a scheme, let D be the sheaf of differential operators, assigning to any open subscheme it’s ring of differential operators. The study of D-modules advances their theory independently, but pervades many other areas of modern mathematics as well. Most notably, the theory provided a framework to solve Hilbert’s 21-st problem, and to develop the Riemann-Hilbert correspondence, and eventually led to the resolution of the Kazhdan-Lustig conjecture in representation theory. For an affine patch of the scheme having dimension n, the sheaf will assign the n-th Weyl algebra. In [1], Hayashi develops the quantized Weyl algebra, a deformation of this algebra, and in [2] Lunts and Rosenberg develop versions of β and quantum differential operators for a graded non-commutative algebra. Iyer and McCune compute in [3] the ring of these quantum differential operators of Lunts and Rosenberg over the polynomial algebra in n-variables, or, over affine n-space. In [4], Bischof examines how a reconciliation of the β deformation in [2] and a 2-cocycle deformation of the graded algebra influence the category of these quantum D-modules, and considers some localizations. One naturally wonders about the category of modules for these quantum differential operators on a non-commutative space; about it’s objects and it’s structure. With the aim of future study in non-commutative grassmannians and flag varieties, of U[subscript]q(sl[subscript]n), for example, we consider a non-commutative projective space glued together from a covering of 2-cocycle deformed polynomial rings, as proposed in [5] and [4]. We determine when there exists a deformed polynomial ring from which we can obtain this covering, and the category of quasi-coherent sheaves can be realized via the categorical Proj construction. With a guiding hand from Rosenberg’s [5] we develop a general ring structure for containing these quantum differential operators on polynomial algebras. Finally, towards the goal of defining holonomic quantum D-modules, we consider the GK-dimension of the corresponding associated graded algebra for the purpose of determining the dimension of what might be considered the singular support for a quantum D-module.

Doctor of Philosophy / Department of Mathematics / Todd Cochrane / Craig Spencer / See PDF file for full abstract.

We solve certain three-term recurrence relations for generating functions of map enumeration problems. These are combinatorial maps, an embedding of a graph into a surface in a particular way. The generating functions enumerate the maps according to an appropriate notion of a distance or height in the map. These problems were studied and the recurrence relations derived in [BDFG03] and [BM06].By viewing the three-term recurrence as giving a two-dimensional discrete dynamical system, these combinatorial problems are set in the context of discrete dynamical systems and integrable systems theory. The integrable nature of the system was made apparent by numerical study, and is confirmed by recognition that the recurrences are autonomous discrete Painleve-I equations. The autonomous discrete Painleve equations are known to be instances of the QRT Mapping, named for Quispel, Roberts, and Thompson [QRT88, QRT89], an integrable structure with explicitly-given invariant. Level sets of such invariants are in general elliptic curves, and thus orbits in the dynamical systems can be parametrized through elliptic functions. The solution to a recurrence relation for combinatorial generating functions is rigorously derived from the general elliptic parametrization of the dynamical system, as the combinatorial initial condition indicates that the combinatorial orbit actually lies on a stable manifold of a hyperbolic fixed point of the system. This special orbit thus lies on a separatrix of the system, which is given by a degeneration in the elliptic nature of the level sets of the invariant function. These solutions have a particularly nice algebraic form, which is seen to be a consequence of the degeneration of the elliptic parametrization. The framework and method are general, applicable to any combinatorial enumeration problem that arises with a similar QRT-type structure.

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<p>In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.</p><p>First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system, </p><p>and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons. </p><p>Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons. </p><p>Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively. </p><p>Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses </p><p>and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital </p><p>stability but no $L_2$ stability.</p> / Dissertation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1973. / Vita. / Bibliography: leaves 398-399. / Ph.D.