QSpace Community: Queen's UniversityInformationQueen's UniversityInformationhttp://hdl.handle.net/1974/62016-07-27T09:36:04Z2016-07-27T09:36:04ZA Review on Repeated Games and Reputations with Incomplete InformationVerlezza, Michaelhttp://hdl.handle.net/1974/146632016-07-21T05:11:09Z2016-07-20T04:00:00ZTitle: A Review on Repeated Games and Reputations with Incomplete Information
Authors: Verlezza, Michael
Abstract: In this project we review the effects of reputation within the context of game theory.
This is done through a study of two key papers. First, we examine a paper from
Fudenberg and Levine: Reputation and Equilibrium Selection in Games with a Patient
Player (1989). We add to this a review Gossnerâ€™s Simple Bounds on the Value of a
Reputation (2011). We look specifically at scenarios in which a long-run player faces
a series of short-run opponents, and how the former may develop a reputation. In
turn, we show how reputation leads directly to both lower and upper bounds on
the long-run playerâ€™s payoffs.2016-07-20T04:00:00ZGeometry of Dirac OperatorsBeheshti Vadeqan, Babakhttp://hdl.handle.net/1974/146332016-07-06T05:10:44Z2016-07-05T04:00:00ZTitle: Geometry of Dirac Operators
Authors: Beheshti Vadeqan, Babak
Abstract: Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$.
Then
\[
\textsc{Atiyah Singer: } \quad
\text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V})
\]
where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$.
We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$.
The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel.
The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the
kernel we will see that in the computation of the index only one term matters.
The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-07-04 20:27:20.3862016-07-05T04:00:00ZComputationally Intensive Methods for Spectrum EstimationPohlkamp-Hartt, JOSHUAhttp://hdl.handle.net/1974/142912016-05-01T12:24:51Z2016-04-27T04:00:00ZTitle: Computationally Intensive Methods for Spectrum Estimation
Authors: Pohlkamp-Hartt, JOSHUA
Abstract: Spectrum estimation is an essential technique for analyzing time series data. A leading method in the field of spectrum estimation is the multitaper method. The multitaper method has been applied to many scientific fields and has led to the development of new methods for detection signals and modeling periodic data. Within these methods there are open problems concerning parameter selection, signal detection rates, and signal estimation. The focus of this thesis is to address these problems by using techniques from statistical learning theory. This thesis presents three theoretical contributions for improving methods related to the multitaper spectrum estimation method: (1) two hypothesis testing procedures for evaluating the choice of time-bandwidth, NW, and number of tapers, K, parameters for the multitaper method, (2) a bootstrapping procedure for improving the signal detection rates for the F-test for line components, and (3) cross-validation, boosting, and bootstrapping methods for improving the performance of the inverse Fourier transform periodic data estimation method resulting from the F-test. We additionally present two applied contributions: (1) a new atrial signal extraction method for electrocardiogram data, and (2) four new methods for analyzing, modeling, and reporting on hockey game play at the Major Junior level.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-04-27 14:03:57.7572016-04-27T04:00:00ZHigher rank sieves and applicationsVatwani, Akshaahttp://hdl.handle.net/1974/142762016-04-25T18:59:26Z2016-04-25T04:00:00ZTitle: Higher rank sieves and applications
Authors: Vatwani, Akshaa
Abstract: This thesis focuses on some of the key sieve theoretic ideas behind recent progress on bounded gaps between the primes. One such idea is the notion of higher rank sieve weights, first proposed by Atle Selberg and applied successfully to the context of prime k-tuples by J. Maynard and T. Tao. We develop an axiomatic formulation for a general higher rank sieve, in the spirit of Selberg's own treatment of his classical sieve. We apply this theory to an assortment of problems such as almost prime k-tuples and prime k-tuples in imaginary quadratic fields with class number 1.
Another novel idea that was brought to the forefront by the path-breaking work of Yitang Zhang is that of obtaining new equidistribution results for the primes by making the moduli "smooth" or free of large prime factors. We develop a general method to incorporate the technique of smoothing the moduli into the higher rank sieve and apply this to prime k-tuples.
In a different vein, the last chapter of the thesis expands upon the well-known parity principle in sieve theory. We show that sufficient "randomness" in the sign of the M\"obius function, combined with another conjecture about the equidistribution of the primes in arithmetic progressions, can be used to break the parity barrier and yield infinitely many twin primes.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-04-23 13:21:39.4382016-04-25T04:00:00Z