This thesis in analytic number theory consists of 3 parts and 13 individual papers. In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function. In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line. In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula. We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant. We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.

No description available.

No description available.

The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).

Os fundamentos teóricos desenvolvidos neste trabalho e que dão suporte aos métodos propostos garantem que as estimativas obtidas sejam sempre conservadoras (no sentido de que elas são sempre subconjuntos da região de estabilidade verdadeira) e, portanto, possuam elevado grau de confiança ao concluir sobre a estabilidade do sistema. Os métodos apresentados consistem em extensões dos métodos Closest UEP e CUEP, utilizados na análise de estabilidade transitória de sistemas elétricos de potência, para sistemas que admitem FEG. Embora os métodos Closest UEP e CUEP forneçam estimativas de forma rápida e precisa, sua aplicação está limitada à existência de uma Função Energia (FE) para o sistema, o que consiste em uma forte limitação. Muitos sistemas não admitem FE e, mesmo quando se pode provar a existência de uma FE, a impossibilidade de exibi-la impede a aplicação dos métodos citados. Outra contribuição deste trabalho consiste em um método computacional que permite a obtenção de uma FEG para sistemas polinomiais. O método apresentado também é aplicado a uma classe de problemas não polinomiais, provenientes da modelagem de sistemas elétricos de potência, mediante uma mudança não linear de variáveis que permite a construção de um sistema polinomial equivalente. Através dos métodos apresentados, visa-se disponibilizar métodos computacionais que permitam a obtenção de estimativas rápidas e precisas e que possam ser aplicados a uma ampla classe de sistemas: aqueles que admitem FEG. Com isso, almeja-se não somente contribuir para o desenvolvimento de métodos para análise de estabilidade de sistemas elétricos de potência mas, também, disponibilizá-los a outras áreas do conhecimento. / In this work, we develop computational methods to estimate stability regions and the relevant part of stability boundary of attracting sets of nonlinear dynamical systems. Such methods are based on Generalized Energy Function (GEF) theory and, therefore, can be applied to a larger class of problems than those based on Energy Functions (EF). The theoretical foundations developed in this work, which support the proposed methods, ensure that the estimates are always conservative (in the sense that they are subsets of the true stability region), providing high confidence level when asserting the stability of a system. The presented methods are extensions of the Closest UEP and the CUEP methods, used in the assessment of stability of electrical power systems, to the systems that admit GEF. Even though the Closest UEP and CUEP methods provide estimates in a fast and accurate way, they are only applicable to systems that admit EFs, which consists in a strong limitation for their usage. Many systems do not admit EF and, even if it is possible to prove the existence of an EF, the impossibility to exhibit it in the form of elementary mathematical functions prevents the application of such methods. Other contribution of this work is a computational method to obtain a GEF for polinomial systems. We also applied the presented method to a class of non polinomial systems arising from electrical power system models, after a nonlinear change of variables that provides an equivalent polinomial system. By means of the proposed methods, we aim to offer computational methods to allow fast and accurate stability region estimates which could be used in a broad class of dynamical systems: those that admit GEF. This way, we plan to contribute for the development of methods used in the assessment of stability of electrical power systems and make such tools available to systems from other areas of science.

Os fundamentos teóricos desenvolvidos neste trabalho e que dão suporte aos métodos propostos garantem que as estimativas obtidas sejam sempre conservadoras (no sentido de que elas são sempre subconjuntos da região de estabilidade verdadeira) e, portanto, possuam elevado grau de confiança ao concluir sobre a estabilidade do sistema. Os métodos apresentados consistem em extensões dos métodos Closest UEP e CUEP, utilizados na análise de estabilidade transitória de sistemas elétricos de potência, para sistemas que admitem FEG. Embora os métodos Closest UEP e CUEP forneçam estimativas de forma rápida e precisa, sua aplicação está limitada à existência de uma Função Energia (FE) para o sistema, o que consiste em uma forte limitação. Muitos sistemas não admitem FE e, mesmo quando se pode provar a existência de uma FE, a impossibilidade de exibi-la impede a aplicação dos métodos citados. Outra contribuição deste trabalho consiste em um método computacional que permite a obtenção de uma FEG para sistemas polinomiais. O método apresentado também é aplicado a uma classe de problemas não polinomiais, provenientes da modelagem de sistemas elétricos de potência, mediante uma mudança não linear de variáveis que permite a construção de um sistema polinomial equivalente. Através dos métodos apresentados, visa-se disponibilizar métodos computacionais que permitam a obtenção de estimativas rápidas e precisas e que possam ser aplicados a uma ampla classe de sistemas: aqueles que admitem FEG. Com isso, almeja-se não somente contribuir para o desenvolvimento de métodos para análise de estabilidade de sistemas elétricos de potência mas, também, disponibilizá-los a outras áreas do conhecimento. / In this work, we develop computational methods to estimate stability regions and the relevant part of stability boundary of attracting sets of nonlinear dynamical systems. Such methods are based on Generalized Energy Function (GEF) theory and, therefore, can be applied to a larger class of problems than those based on Energy Functions (EF). The theoretical foundations developed in this work, which support the proposed methods, ensure that the estimates are always conservative (in the sense that they are subsets of the true stability region), providing high confidence level when asserting the stability of a system. The presented methods are extensions of the Closest UEP and the CUEP methods, used in the assessment of stability of electrical power systems, to the systems that admit GEF. Even though the Closest UEP and CUEP methods provide estimates in a fast and accurate way, they are only applicable to systems that admit EFs, which consists in a strong limitation for their usage. Many systems do not admit EF and, even if it is possible to prove the existence of an EF, the impossibility to exhibit it in the form of elementary mathematical functions prevents the application of such methods. Other contribution of this work is a computational method to obtain a GEF for polinomial systems. We also applied the presented method to a class of non polinomial systems arising from electrical power system models, after a nonlinear change of variables that provides an equivalent polinomial system. By means of the proposed methods, we aim to offer computational methods to allow fast and accurate stability region estimates which could be used in a broad class of dynamical systems: those that admit GEF. This way, we plan to contribute for the development of methods used in the assessment of stability of electrical power systems and make such tools available to systems from other areas of science.

Många sjöar är idag påverkade av mänsklig aktivitet, bland annat är 1 % av Sveriges sjöar eutrofierade, däribland Ekoln. Ekoln är en stor sjö i Uppsala län söder om Uppsala som länge varit eutrofierad. Fyrisån och Örsundaån är de största inflödena till Ekoln och Sävjaån är ett betydande biflöde till Fyrisån. Kungsängsverket är Uppsalas reningsverk och har sitt utlopp i Fyrisån och som ett led i att minska eutrofieringen byggdes fosforreningen ut 1972. Fosforhalterna i Ekoln sjönk och algblomningarna blev färre. Kvävereningen byggdes ut 1999 och i detta arbete utvärderas detta reningssteg.   Hypotesen är att när ammonium minskar i bottenvattnet kommer syrgasförhållandena att förbättras på grund av minskad ammoniumnedbrytning som förbrukar syrgas. Med ökad halt syrgas är det känt sedan tidigare studier att den interna fosforbelastningen och algblomningen minskar.   För att undersöka om det skett en signifikant minskning av näringsämnen i intransporten till Ekoln och av halterna i Ekoln utfördes statistiska test av ämnena för perioden före och perioden efter införandet av kväverening. Intransporterna modellerades även i StormTac.   Resultatet visar att det skett en minskning av kväve- och ammoniumtransport och en ökning av fosfor-, fosfat- och TOC-transport. I Ekoln har en minskning påvisats i ammonium-, totalfosforhalt och en ökning i fosfat- och TOC-halt. Ammoniumhalten har minskat i bottenvattnet så den minskade intransporten av ammonium gett en effekt. Fosfor- och fosfathalten har inte förändrats signifikant i bottenvattnet, men det har skett en viss minskning av fosfor och en viss ökning av fosfat i Ekoln perioden 1990-1998 till 2000-2010. Även syrgashalten har ökat i bottenvattnet.   Att det inte skett signifikanta förändringar i fosfor, fosfat och syrgas tyder på att intransporten till vattenmassan inte förändrats. Ökade TOC-halter leder till ökad nedbrytning där syrgas förbrukas i bottenvattnet. Det är möjligt att situationen sett ännu värre ut i Ekoln om inte kväveutsläppen minskat.

This thesis explores the structure of games, including both the internal structure of various games and also the structure of classes of games as partially ordered sets. Internal structure is explored through consideration of juxtapositions of game positions and how the underlying games interact. We look at ordinal sums and introduce side-sums as a means of understanding this interaction, giving a full solution to a Toppling Dominoes variant through its application. Loopy games in which only one player is allowed a pass move, referred to as Oslo games, are introduced and their game structure explored. The poset of Oslo games is shown to form a distributive lattice. The Oslo forms of Wythoff’s game, Grundy’s game and octal .007 are introduced and full solutions given. Finally, the poset of option-closed games is given up to day 3 and all are shown to form a planar lattice. The option-closed game of Cricket Pitch is also fully analyzed.

Fundação de Amparo a Pesquisa no Estado do Rio Grande do Sul / The aim of this study was to evaluate the growth and development of watermelon hybrids in relation to air temperature by obtaining the thermal time for vegetative and reproductive subperiods for the hybrid cultivar Manchester and Top Gun, both early cycle were used. The experiment was conducted in an area belonging to the Federal University of Santa Maria/Campus Frederico Westphalen-RS, from September to December 2014. During the execution of the experiment, evaluations were performed every two days for the following characteristics: emission nodes/leaves, development of primary and secondary vines, and issuance of staminate flowers and pistillate. The spacing used was 1.5 m between plants and 3.0 m between rows of the crop. The values obtained for the evaluated criteria differ significantly where the hybrid Manchester demonstrated superiority over the hybrid Top Gun which presented average values of plastichrone with 16.6 °C day-1 node, the amount of nodes on the main vine was 45.8 nodes, the thermal sum for the issuance of secondary vines was 18.1 °C growing degree days-1 vine, and the final number of secondary vines and vines of 26.6, and a total number of growing degree days for staminate flower emission of 9.6 °C day-1 flower. The hybrid Top Gun was superior to the hybrid Manchester for the final number of primary vines, emitting on average 14.6 vines per plant and an accumulated growing degree day for pistillate flower emission of 51.9 °C day-1 flower. In the variables of thermal time on the main stem, primary stem, and growing degree day, the thermal time for primary and secondary vines, final number of staminate flowers and pistillate hybrids showed no statistical difference in mean values. / O objetivo deste estudo foi avaliar a velocidade de emissão de órgãos vegetativos e reprodutivos para dois híbridos de melancieira em relação a temperatura do ar, com a obtenção da soma térmica acumulada. Foram utilizados os híbridos Manchester e Top Gun, ambos de ciclo precoce. O experimento foi conduzido em área pertencente à Universidade Federal de Santa Maria/Campus Frederico Westphalen RS, no período de setembro a dezembro de 2014. Durante a execução do experimento, foram realizadas avaliações a cada dois dias sobre as seguintes características: emissão de nós, emissão de ramificações primárias, secundárias, emissão de flores estaminadas e pistiladas. As variáveis foram estimadas pelo inverso do coeficiente angular da regressão linear do órgão visível com a soma térmica diária acumulada a partir do transplante para o campo. Os resultados obtidos para os critérios avaliados diferiram significativamente, sendo que o híbrido Manchester demonstrou superioridade em relação ao híbrido Top Gun apresentando valores médios de plastocrono com 16,6 ºC dia nó-1, número final de nó na haste principal de 45,8 nós, soma térmica para emissão de ramificação secundária de 18,1 ºC dia-1 ramificação, número final de ramificações secundárias de 26,6 ramificações e soma térmica para emissão de flor estaminada de 9,6 ºC dia-1 flor. O híbrido Top Gun foi superior ao híbrido Manchester para as variáveis número final de ramificações primárias emitindo em média 14,6 ramificações por planta e soma térmica acumuladas para emissão de flor pistilada de 51,9 ºC dia flor-1. Nas variáveis soma térmica acumulada na haste principal, soma térmica para ramificação primária, soma térmica acumulada para ramificações primárias e secundárias, número final de flores estaminadas e pistiladas os híbridos não apresentaram diferenciação estatística dos valores médios.

In this work, we study two problems: first is one of the central problem in learning theory of learning sparse parities and the other k-Vector-SUM is an extension of the not oriousk-SUM problem. We first consider the problem of learning k-parities in the on-line mistake-bound model: given a hidden vector ∈ {0,1}nwith|x|=kand a sequence of “questions” a ,a ,12··· ∈{0,1}n, where the algorithm must reply to each question with〈a ,xi〉(mod 2), what is the best trade off between the number of mistakes made by the algorithm and its time complexity? We improve the previous best result of Buhrman et. al. By an exp (k) factor in the timecomplexity. Next, we consider the problem of learning k-parities in the presence of classification noise of rate η∈(0,12). A polynomial time algorithm for this problem (whenη >0 andk=ω(1))is a longstanding challenge in learning theory. Grigorescu et al. Showed an algorithm running in time(no/2)1+4η2+o(1). Note that this algorithm inherently requires time(nk/2)even when the noise rateη is polynomially small. We observe that for sufficiently small noise rate, it ispossible to break the(nk/2)barrier. In particular, if for some function f(n) =ω(logn) andα∈[12,1),k=n/f(n) andη=o(f(n)−α/logn), then there is an algorithm for the problem with running time poly(n)·( )nk1−α·e−k/4.01.Moving on to the k-Vector-SUM problem, where given n vectors〈v ,v ,...,v12n〉over the vector space Fdq, a target vector tand an integer k>1, determine whether there exists k vectors whose sum list, where sum is over the field Fq. We show a parameterized reduction fromk-Clique problem to k-Vector-SUM problem, thus showing the hardness of k-Vector-SUM. In parameterized complexity theory, our reduction shows that the k-Vector-SUM problem is hard for the class W[1], although, Downey and Fellows have shown the W[1]-hardness result for k-Vector-SUM using other techniques. In our next attempt, we try to show connections between k-Vector-SUM and k-LPN. First we prove that, a variant of k-Vector-SUM problem, called k-Noisy-SUM is at least as hard as the k-LPN problem. This implies that any improvements ink-Noisy-SUM would result into improved algorithms fork-LPN. In our next result, we show a reverse reduction from k-Vector-SUM to k-LPN with high noise rate. Providing lower bounds fork-LPN problem is an open challenge and many algorithms in cryptography have been developed assuming its1 2hardness. Our reduction shows that k-LPN with noise rate η=12−12·nk·2−n(k−1k)cannot be solved in time no(k)assuming Exponential Time Hypothesis and, thus providing lower bound for a weaker version of k-LPN