In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.

In this thesis, we consider semi-algebraic sets over a real closed field R defined by quadratic polynomials. Semi-algebraic sets of R^k are defined as the smallest family of sets in R^k that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the topological complexity of semi-algebraic sets over a real closed field R defined by quadratic polynomials, in terms of the parameters of the system of polynomials defining them, which improve the known results. We conclude the thesis with presenting two new algorithms along with their implementations.

Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)[x] where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)). We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.

Submitted by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-19T17:19:01Z No. of bitstreams: 1 arquivototal.pdf: 977282 bytes, checksum: 98d2394b44f8e76ed8a9986250386a2c (MD5) / Approved for entry into archive by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-19T17:19:18Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 977282 bytes, checksum: 98d2394b44f8e76ed8a9986250386a2c (MD5) / Made available in DSpace on 2015-05-19T17:19:18Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 977282 bytes, checksum: 98d2394b44f8e76ed8a9986250386a2c (MD5) Previous issue date: 2013-08-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this study, we evaluate if the congruence x2 a (mod m), where m is prime and (a;m) = 1, has or not solutions, highlighting the importance of Quadratic Residues and consequently the cooperation of the Legendre's Symbol, the Euler's Criterion and the Gauss' Lemma. Also, we demonstrate the Law of Quadratic Reciprocity generalizing situations for composite numbers, that is, the Jacobi's Symbol and its properties. We present some proposals of activities for the High School involving the subject matter and its possible applications, through an understandable language for students of this level. / Neste estudo, vamos avaliar se a congruência x2 a (mod m), onde m é primo e (a;m) = 1, apresenta ou não solução, destacando a importância dos Resíduos Quadráticos e, consequentemente da cooperação do Símbolo de Legendre, do Critério de Euler e do Lema de Gauss. Também, demonstraremos a Lei de Reciprocidade Quadrática generalizando situações para números compostos, ou seja, o Símbolo de Jacobi e suas propriedades. Apresentamos algumas propostas de atividades para o Ensino Médio envolvendo o assunto abordado e suas possíveis aplicações, através de uma linguagem compreensível aos alunos deste nível de ensino.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-25T12:30:08Z No. of bitstreams: 1 cristianemouradasilvabronsatocanella.pdf: 3616926 bytes, checksum: 56d40ead5f29f375d5ec1d6d47c6db80 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-25T16:34:02Z (GMT) No. of bitstreams: 1 cristianemouradasilvabronsatocanella.pdf: 3616926 bytes, checksum: 56d40ead5f29f375d5ec1d6d47c6db80 (MD5) / Made available in DSpace on 2016-07-25T16:34:02Z (GMT). No. of bitstreams: 1 cristianemouradasilvabronsatocanella.pdf: 3616926 bytes, checksum: 56d40ead5f29f375d5ec1d6d47c6db80 (MD5) Previous issue date: 2016-06-27 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho sobre funções quadráticas, destina-se a colaborar de forma contextualizada, com o estudo da função quadrática, desde sua parte histórica, passando pela equação quadrática, inequações quadráticas, sua representação gráfica, sua caracterização, contextualização através de problemas, resgatando a construção da parábola com régua e compasso e de forma atual, utilizando o GeoGebra. Mostramos também a utilização da propriedade refletora da parábola em diversas situações, bem como uma explicação geométrica para o fato. Espera-se que seja um instrumento colaborativo na construção do conhecimento e que seja principalmente, de fácil entendimento. / This work about quadratic functions is intended to collaborate in context with the study of the quadratic function from its historical part, passing by quadratic equation, quadratic inequalities, its graphic representation, its characterization, contextualization through problems, rescuing construction the parabola with ruler and compass, and current form using GeoGebra. Also we show the use of the reflective parabola property in several situations, as well as, a geometric explanation for the fact. It is expected to be a collaborative tool in the construction of knowledge and it is especially easy to understand.

The study of Stochastic Local Search (SLS) algorithms is becoming more pivotal these days, due to their vast number of applications in decision making. Prior to the implementation of algorithmic knowledge for decision making, many decisions were made based on manual calculation, on the fly, or even based on guts feeling. Nowadays, such an approach is more rarely seen, especially when the decisions that need to be made are high-risk, cost intensive, or time-consuming. The increasingly often used SLS algorithms are one of the options available to assist the decision making process these days.The work discussed in this thesis concerns the study of SLS algorithms for solving the Quadratic Assignment Problem (QAP), a prominent combinatorial optimization problem, which until today is very hard to solve. Our interest is to study the behavior and performance of SLS algorithms for solving QAP instances of different characteristics, such as size, sparsity, and structure. In this study, we have also proposed new variants of SLS algorithms, inspired by existing, well-performing SLS algorithms for solving the QAP. The new variants of SLS algorithms are then further extended for solving the bi-objective QAP (bQAP).One main focus in this study is to see how the performance of algorithms scales with instance size. We have considered instances that are much larger than the ones usually used in the studies of algorithms for solving the QAP. By understanding how the algorithms perform when the instance size changes, we might be able to solve other problems effectively by considering the similarity in their characteristics to the ones of the QAP, or by seeing common trends in the relative performance of the various available SLS methods. For single objective QAP instances we found that the structure and size of instances do have a significant impact on the performance of SLS algorithms. For example, comparisons between Tabu Search (TS) and Simulated Annealing (SA) on instances with randomly generated matrices show that the overall performance of TS is better than SA, irrespective the size of instances considered. The results on a class of structured instances however show that TS performs well on small-sized instances, while on the larger ones, SA shows better results. In another experiment, Hierarchical Iterated Local Search (HILS) has shown very good results compared to several Iterated Local Search (ILS) variants. This experiment was done on a class of structured instances of size from 100 to 500. An extensive experiment on a class of structured instances of size 30 to 300 using tuned parameter settings shows that population based algorithms perform very well on most of the instance classes considered. SA however, shows very good performance especially on large-sized instances with low sparsity level. For the bQAP, the correlation between the flow matrices does have a strong effect that determines the performance of algorithms for solving them. Hybrid Simulated Annealing (HSA) clearly outperforms Hybrid Iterative Improvement (HII). When compared to Multi Objective Ant Colony Optimization (MOACO) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), HSA shows very good performance, where HSA outperforms MOACO and SPEA2, especially on instances of large size, thus, offering a better scaling behavior. Based the results obtained in this study, it is possible to come up with a general idea on the suitability of SLS algorithms for solving instances with a certain characteristic. Given an unknown QAP instance, one can guess the most suitable algorithm for solving it depending on the type, size, and sparsity of the instance, while for a bQAP instance the most suitable algorithm can be guessed based on its size and correlation between the flow matrices. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic analogue of a well-known conjecture, namely the Vandiver's conjecture.

As inverter-based distributed energy resources (DERs) such as photovoltaic (PV) and battery energy storage system (BESS) penetrate within the distribution system. New challenges regarding how to utilize these devices to improve power quality arises. Before, PV systems were required to disconnect from the grid during a large disturbance, but now smart inverters are required to have dynamically controlled functions that allows them to remain connected to the grid. Monitoring power flow at the point of common coupling is one of the many functions the controller should perform. Smart inverters can inject active power to pick up critical load or inject reactive power to regulate voltage within the electric grid. In this context, this thesis focuses on a high level and local control design that incorporates DERs. Different controllers are implemented to stabilize the microgrid in an Islanding and resiliency mode. The microgrid can be used as a resiliency source when the distribution is unavailable. An average model in the D-Q frame is calculated to analyze the inherent dynamics of the current controller for the point of common coupling (PCC). The space vector approach is applied to design the voltage and frequency controller. Secondly, using inverters for Volt/VAR control (VVC) can provide a faster response for voltage regulation than traditional voltage regulation devices. Another objective of this research is to demonstrate how smart inverters and capacitor banks in the system can be used to eliminate the voltage deviation. A mixed-integer quadratic problem (MIQP) is formulated to determine the amount of reactive power that should be injected or absorbed at the appropriate nodes by inverter. The Big M method is used to address the nonconvex problem. This contribution can be used by distribution operators to minimize the voltage deviation in the system. / Master of Science / Reliable power supply from the electric grid is an essential part of modern life. This critical infrastructure can be vulnerable to cascading failures or natural disasters. A solution to improve power systems resilience can be through microgrids. A microgrid is a small network of interconnected loads and distributed energy resources (DERs) such as microturbines, wind power, solar power, or traditional internal combustion engines. A microgrid can operate being connected or disconnected from the grid. This research emphases on the potentially use of a Microgrid as a resiliency source during grid restoration to pick up critical load. In this research, controllers are designed to pick up critical loads (i.e hospitals, street lights and military bases) from the distribution system in case the electric grid is unavailable. This case study includes the design of a Microgrid and it is being tested for its feasibility in an actual integration with the electric grid. Once the grid is restored the synchronization between the microgrid and electric must be conducted. Synchronization is a crucial task. An abnormal synchronization can cause a disturbance in the system, damage equipment, and overall lead to additional system outages. This thesis develops various controllers to conduct proper synchronization. Interconnecting inverter-based distributed energy resources (DERs) such as photovoltaic and battery storage within the distribution system can use the electronic devices to improve power quality. This research focuses on using these devices to improve the voltage profile within the distribution system and the frequency within the Microgrid.

Square Form Factoring is an <i>O</i>(<i>N</i>^1/4) factoring algorithm developed by D. Shanks using certain properties of quadratic forms. Central to the original algorithm is an iterative search for a square form. We propose a new subexponential-time algorithm called SQUFOF2, based on ideas of D. Shanks and R. de Vogelaire, which replaces the iterative search with a sieve, similar to the Quadratic Sieve.

Commercial fresh vegetable production is one of the most rewarding and risky farming activities. The price and yield variations throughout the production year, the special characteristics of fresh vegetable produce (i.e. perishability), and the changing consumer demands are some of the factors contributing to the increased uncertainty faced by vegetable producers. This dissertation combined mathematical programming and econometric techniques to: 1) investigate the optimal production and marketing practices under different price distribution information scenarios, risk aversion levels and marketing outlets and 2) examine growers’ preferences as well the effect of risk aversion levels and growers’ risk perception on the choice of marketing contracts. Specifically, the following three modeling approaches were adopted in order to achieve the dissertation objectives: 1) quadratic programming under a mean-variance framework, 2) discrete choice experiments and 3) a combination of quadratic and integer programming embodied in a meanvariance framework. The findings indicate that optimal production practices and the resulting net returns are substantially influenced not only by the choice of marketing channel but also by growers’ risk aversion levels as well as price knowledge. Furthermore, regarding the choice of marketing contracts, the results highlight the existence of heterogeneity in preferences and illustrate the importance of certification cost, in line with the previous literature. Lastly, the findings indicate that risk aversion and risk preferences do not play a significant role in the choice of contractual agreements by farmers.