QSpace Collection:http://hdl.handle.net/1974/7582016-08-23T21:13:28Z2016-08-23T21:13:28ZGeometry 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 13:21:39.4382016-04-25T04:00:00ZOn the role of regularity in mathematical control theoryJafarpour, SABERhttp://hdl.handle.net/1974/141992016-04-09T20:19:40Z2016-04-08T04:00:00ZTitle: On the role of regularity in mathematical control theory
Authors: Jafarpour, SABER
Abstract: In this thesis, we develop a coherent framework for
studying time-varying vector fields of different regularity classes and their flows. This setting has the benefit of unifying all classes of regularity. In particular, it includes the real analytic regularity and provides us with tools and techniques for studying holomorphic extensions of real analytic vector fields. We show that under suitable integrability conditions, a time-varying real analytic vector field on a manifold
can be extended to a time-varying holomorphic vector field on a neighbourhood of that manifold. Moreover, in this setting, the nonlinear differential equation governing the flow of a
time-varying vector field can be considered as a linear
differential equation on an infinite dimensional locally convex vector space. We show that, in the real analytic case, the integrability of the time-varying vector field ensures convergence of the sequence of Picard iterations for this linear differential equation, giving us a series representation for the flow of a time-varying real analytic vector field.
Using the framework we develop in this thesis, we study a parametization-independent model in
control theory called tautological control system. In the
tautological control system setting, instead of defining a
control system as a parametrized family of vector fields on a
manifold, it is considered as a subpresheaf of the sheaf of vector fields on that manifold. This removes the explicit dependence of the systems on the control parameter and gives us a suitable framework for studying regularity
of control systems. We also study the relationship between tautological control systems and classical control systems. Moreover, we introduce a suitable notion of
trajectory for tautological control systems.
Finally, we generalize the orbit theorem of Sussmann and Stefan to the tautological framework. In particular, we show that orbits of a tautological control system are immersed submanifolds of the state manifold. It turns out that the presheaf structure on the family of vector fields of a system plays an important role in characterizing the tangent space to the orbits of the system. In particular, we prove that, for globally defined real analytic tautological control systems, every tangent space to the orbits of the system is generated by the Lie brackets of the vector fields of the system.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-04-08 11:57:25.9872016-04-08T04:00:00Z