QSpace Collection:
http://hdl.handle.net/1974/758
2015-11-26T08:48:44ZArithmetic and Intermediate Jacobians of Calabi-Yau threefolds
http://hdl.handle.net/1974/13588
Title: Arithmetic and Intermediate Jacobians of Calabi-Yau threefolds
Authors: Molnar, Alexander
Abstract: This thesis is centered around particular Calabi-Yau threefolds. Borcea \cite{Borcea} and Voisin \cite{Voisin} construct Calabi-Yau threefolds using elliptic curves and K3 surfaces with non-symplectic involutions. This family has an incredible property, that a general member has a mirror pair within this family. We start by investigating if this construction works only for Calabi-Yau threefolds with non-symplectic involutions or with non-symplectic automorphisms of higher order as well. Thereafter, we generalize this construction to Calabi-Yau fourfolds.
After this, we focus on the underlying construction that lead Borcea to the families above, using a product of three elliptic curves with non-symplectic involutions. These threefolds do not come in families, so we cannot ask about mirror symmetry, but if we have models defined over $\Qbb$, we may ask arithmetic questions. Many arithmetic properties of the Calabi-Yau threefolds can be studied via the underlying elliptic curves. In particular, we are able to show (re-establish in the rigid case) that the Calabi-Yau threefolds are all modular by computing their $L$-functions. Then, guided by a conjecture of Yui, we investigate their (Griffiths) intermediate Jacobians and a relationship between their respective $L$-functions.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-09-10 15:28:19.7432015-09-10T04:00:00ZDistributed Online Optimization on time-varying networks
http://hdl.handle.net/1974/13551
Title: Distributed Online Optimization on time-varying networks
Authors: Akbari Varnousfaderani, Mohammad Jr
Abstract: This thesis introduces two classes of discrete-time distributed online optimization algorithms, with a group of agents which communicate over a network. At each time, a private convex objective function is revealed to each agent. In the next time step, each agent updates its state using its own objective function and the information gathered from its immediate in-neighbours at that time. We design algorithms distributed over the network topologies, which guarantee that the individual regret, the diﬀerence between the network cost incurred by the agent’s states estimation and the cost incurred by the best ﬁxed choice, grows only sublinearly. One algorithm is based on gradient-ﬂow, which provably works for a sequence of time-varying uniformly strongly connected graphs. The other one is based on Alternating Direction Method of Multipliers, which works on ﬁxed undirected graphs and gives an explicit
regret bound in terms of the size of the network. We implement the proposed algorithms on a sensor network and the results demonstrate the good performance for both algorithms.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2015-08-31 09:59:41.2612015-09-01T04:00:00ZCurves of low genus on surfaces and applications to Diophantine problems
http://hdl.handle.net/1974/13545
Title: Curves of low genus on surfaces and applications to Diophantine problems
Authors: Garcia, Natalia
Abstract: We describe in detail a technique due to Vojta for finding the explicit set of curves of low genus on certain algebraic surfaces of general type, and refine some of its aspects. We then provide applications of this method to three Diophantine problems.
We prove under the Bombieri-Lang Conjecture that there are finitely many non-trivial sequences of integers of length 11 whose squares have constant second differences, and we prove unconditionally the analogous result for function fields of characteristic zero.
We prove under the Bombieri-Lang Conjecture that there are finitely many integer sequences of length 8 whose k-th powers have second differences equal to 2, we give an unconditional result for function fields of characteristic zero. Moreover, this gives new examples of surfaces having no curves of genus 0 or 1.
The third application is related to the surface parametrizing perfect cuboids. We give some new properties about their curves of genus 0 or 1 and we give new bounds for the degree of curves in this surface, in terms of their genus.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-08-28 10:03:04.0562015-08-31T04:00:00ZOptimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
http://hdl.handle.net/1974/13457
Title: Optimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
Authors: Wood, RICHARD
Abstract: Optimal zero-delay coding of a finite state Markov source through quantization is considered. Building on previous literature, the existence and structure of optimal policies are studied using a stochastic control problem formulation. In the literature, the optimality of deterministic Markov coding policies (or Walrand-Varaiya type policies) for infinite horizon problems has been established. This work expands on this result for systems with finite source alphabets, proving the optimality of de- terministic and stationary Markov coding policies for the infinite horizon setup. In addition, the ε-optimality of finite memory quantizers is established and the depen- dence between the memory length and ε is quantified. An algorithm to find the optimal policy for the finite time horizon problem is presented. Numerical results produced using this algorithm are shown.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2015-07-27 15:57:18.6672015-07-28T04:00:00Z