Nowadays there is an ever increasing demand to obtain more accurate numericalsimulation results while at the same time using fewer computations. One area withsuch a demand is oil reservoir simulations, which builds upon Poisson's equation withvariable coefficients (PEWVC). This thesis focuses on applying and testing a high ordernumerical scheme to solve the PEWVC, namely Summation By Parts - SimultaneousApproximation Term (SBP-SAT). The thesis opens with proving that the method isconvergent at arbitrary high orders given sufficiently smooth coefficients. Theconvergence is furthermore verified in practice by test cases on the Poisson'sequation with smoothly variable permeability coefficients. To balance observed lowerboundary flux convergence, the SBP-SAT method was modified with additionalpenalty terms that were subsequently shown to work as expected. Finally theSBP-SAT method was tested on a semi-realistic model of an oil reservoir withdiscontinuous permeability. The correctness of the resulting pressure distributionvaried and it was shown that flux leakage was the probable cause. Hence theproposed SBP-SAT method performs, as expected, very well in continuous settingsbut typically allows undesirable leakage in discontinuous settings. There are possiblefixes, but these are outside the scope of this thesis.

Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.

Objetiva-se demonstrar que a prisão civil do depositário judicial infiel continua sendo possível no Brasil, mesmo em face da ratificação da Convenção Americana de Direitos Humanos de 1969. Busca-se estudar a natureza jurídica de cada espécie de depósito e, a partir desse referencial teórico, aferir-se que a hipótese do depositário judicial não possui índole contratual, mas de direito público, não se envolvendo com a dívida em execução, nem com nenhum outro elemento de direito privado, podendo o encargo recair sobre o próprio credor ou sobre terceiro. Nesse sentido, a prisão do depositário judicial infiel não seria por dívida, mas como decorrência do desacato (contempt) revelado pelo descumprimento dos encargos processuais de direito público por si assumidos perante o juiz da execução, não estando abrangida pela proscrição estabelecida por aquela norma internacional. Por outro lado, analisa-se o conflito de direitos fundamentais envolvendo a liberdade individual e as garantias de acesso à justiça e da efetividade da tutela jurisdicional, concluindo-se que os velhos critérios hermenêuticos de solução de antinomias são insuficientes ao exame da questão, devendo ser dada uma interpretação constitucional adequada em cada caso concreto com supedâneo no princípio da proporcionalidade. Analisa-se, também, hermenêutica e linguisticamente, a referência legislativa e cada um dos precedentes judiciais da súmula vinculante n. 25 do Supremo Tribunal Federal, verificando-se que o conflito foi apreciado jurisprudencialmente apenas sob o âmago do confronto entre os direitos de liberdade e de propriedade, turvando-se o olhar investigativo sobre a tensão que aquela decisão sumular, tal como redigida, enseja em relação às garantias de acesso à justiça e de efetividade da tutela jurisdicional. Desse modo, conclui-se que a súmula vinculante n. 25 precisa ser revisada pelo STF, e, até que isso aconteça, impõe-se que seja dada uma interpretação constitucional adequada à sua redação, a fim de se restringir sua destinação apenas aos depositários contratuais, não alcançando os depositários judiciais

Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

Wave phenomena appear in many fields of science such as acoustics, geophysics, and quantum mechanics. They can often be described by partial differential equations (PDEs). As PDEs typically are too difficult to solve by hand, the only option is to compute approximate solutions by implementing numerical methods on computers. Ideally, the numerical methods should produce accurate solutions at low computational cost. For wave propagation problems, high-order finite difference methods are known to be computationally cheap, but historically it has been difficult to construct stable methods. Thus, they have not been guaranteed to produce reasonable results. In this thesis we consider finite difference methods on summation-by-parts (SBP) form. To impose boundary and interface conditions we use the simultaneous approximation term (SAT) method. The SBP-SAT technique is designed such that the numerical solution mimics the energy estimates satisfied by the true solution. Hence, SBP-SAT schemes are energy-stable by construction and guaranteed to converge to the true solution of well-posed linear PDE. The SBP-SAT framework provides a means to derive high-order methods without jeopardizing stability. Thus, they overcome most of the drawbacks historically associated with finite difference methods. This thesis consists of three parts. The first part is devoted to improving existing SBP-SAT methods. In Papers I and II, we derive schemes with improved accuracy compared to standard schemes. In Paper III, we present an embedded boundary method that makes it easier to cope with complex geometries. The second part of the thesis shows how to apply the SBP-SAT method to wave propagation problems in acoustics (Paper IV) and quantum mechanics (Papers V and VI). The third part of the thesis, consisting of Paper VII, presents an efficient, fully explicit time-integration scheme well suited for locally refined meshes.

The Hilbert-P\'olya conjecture asserts that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ correspond (in a certain canonical way) to the eigenvalues of some positive operator. R. Meyer constructed a differential operator $D_-$ acting on a function space $\H$ and showed that the eigenvalues of the adjoint of $D_-$ are exactly the nontrivial zeros of $\zeta(s)$ with multiplicity correspondence. We follow Meyer's construction with a slight modification. Specifically, we define two function spaces $\H_\cap$ and $\H_-$ on $(0,\infty)$ and characterize them via the Mellin transform. This allows us to show that $Z\H_\cap\subseteq\H_-$ where $Zf(x)=\sum_{n=1}^\infty f(nx)$. Also, the differential operator $D$ given by $Df(x)=-xf'(x)$ induces an operator $D_-$ on the quotient space $\H=\H_-/Z\H_\cap$. We show that the eigenvalues of $D_-$ on $\H$ are exactly the nontrivial zeros of $\zeta(s)$. Moreover, the geometric multiplicity of each eigenvalue is one and the algebraic multiplicity of each eigenvalue is its vanishing order as a nontrivial zero of $\zeta(s)$. We generalize our construction on the Riemann zeta function to some $L$-functions, including the Dirichlet $L$-functions and $L$-functions associated with newforms in $\mathcal S_k(\Gamma_0(M))$ with $M\ge1$ and $k$ being a positive even integer. We give spectral interpretations for these $L$-functions in a similar fashion.

<p>Disertacija se bavi sumiranjem redova sa specijalnim funkcijama. Ovi redovi se posredstvom trigonometrijskih redova svode na redove sa Riemannovom zeta funkci&shy;jom i srodnim funkcijama. U određenim slučajevima sumacione formule se mogu dovesti na takozvani zatvoreni oblik, &scaron;to znači da se beskonačni redovi predstavljaju konačnim sumama. Predloženi metodi sumacije omogućavaju ubrzanje konvergencije, a mogu se primeniti i kod nekih graničnih problema matematičke fizike. Sumacione formule uključuju kao specijalne slučajeve neke formule poznate iz literature, ali i nove sume, s obzirom da su op&scaron;teg karaktera. Pomoću ovih formula sumirani su i redovi sa integralima trigonometrijskih i specijalnih funkcija.</p> / <p>This dissertation deals with the summation of series over special functions. Through<br />trigonometric series these series are reduced to series in terms of Riemann zeta and<br />related functions. They can be brought in closed form in some cases, i.e. infinite<br />series are expressed as finite sums. Closed form formulas make it possible to accele&shy;<br />rate the convergence of some series, and have many applications in various scientific<br />fields as well. For example, closed form solutions of the boundary value problem in<br />mathematical physics can be obtained. Summation formulas include particular cases<br />known from the literature, but because of their general character one can come to<br />new sums. By means of these formul&aacute;is the sums of series over integrals containing<br />trigonometric or special functions have been found.</p>

The main aim of this thesis was to evaluate the performance of the contrast sensitivity clock (CSC), a new screening device for measuring contrast sensitivity (CS) and glare. This device allows CS without glare, with glare and disability glare scores to be recorded. After initial data collection the design of the CSC was slightly amended improving the performance of the device. The amended design of the CSC was shown to be a valid, discriminative and repeatable measure for purpose. The CSC is also a quick test to perform and is relatively cheap to produce. If all these factors are considered it shows potential to become the test of choice for the assessment of visual glare. A head-mounted display system was also evaluated in terms of the glare effects it may cause. The monocular display screen of the device significantly reduced the CS of the eye directly exposed but also had an effect on binocular performance, reducing amounts of binocular summation. Electronic devices, including head-mounted displays and satellite navigation systems can seriously affect CS at low luminance levels, similar to those found when driving at night.

<p>In order to understand how the brain functions, under normal as well as pathological conditions, it is important to study the mechanisms underlying information integration. Depending on the nature of an input arriving at a synapse, different strategies may be used by the neuron to integrate and respond to the input. Naturally, if a short train of high-frequency synaptic input arrives, it may be beneficial for the neuron to be equipped with a fast mechanism that is highly sensitive to inputs on a short time scale. If, on the contrary, inputs arriving with low frequency are to be processed, it may be necessary for the neuron to possess slow mechanisms of integration. For example, in certain working memory tasks (e. g. delay-match-to-sample), sensory inputs may arrive separated by silent intervals in the range of seconds, and the subject should respond if the current input is identical to the preceeding input. It has been suggested that single neurons, due to intrinsic mechanisms outlasting the duration of input, may be able to perform such calculations. In this work, I have studied a mechanism thought to be particularly important in supporting the integration of low-frequency synaptic inputs. It is mediated by a cascade of events that starts with activation of group I metabotropic glutamate receptors (mGlu1/5), and ends with a membrane depolarization caused by a current that is mediated by canonical transient receptor potential (TRPC) ion channels. This current, denoted I_TRPC, is the focus of this thesis.</p><p>A specific objective of this thesis is to study the role of I_TRPC in the integration of synaptic inputs arriving at a low frequency, < 10 Hz. Our hypothesis is that, in contrast to the well-studied, rapidly decaying AMPA and NMDA currents, I_TRPC is well-suited for supporting temporal summation of such synaptic input. The reason for choosing this range of frequencies is that neurons often communicate with signals (spikes) around 8 Hz, as shown by single-unit recordings in behaving animals. This is true for several regions of the brain, including the entorhinal cortex (EC) which is known to play a key role in producing working memory function and enabling long-term memory formation in the hippocampus.</p><p>Although there is strong evidence suggesting that I_TRPC is important for neuronal communication, I have not encountered a systematic study of how this current contributes to synaptic integration. Since it is difficult to directly measure the electrical activity in dendritic branches using experimental techniques, I use computational modeling for this purpose. I implemented the components necessary for studying I_TRPC, including a detailed model of extrasynaptic glutamate concentration, mGlu1/5 dynamics and the TRPC channel itself. I tuned the model to replicate electrophysiological in vitro data from pyramidal neurons of the rodent EC, provided by our experimental collaborator. Since we were interested in the role of I_TRPC in temporal summation, a specific aim was to study how its decay time constant (τ_decay) is affected by synaptic stimulus parameters.</p><p>The hypothesis described above is supported by our simulation results, as we show that synaptic inputs arriving at frequencies as low as 3 - 4 Hz can be effectively summed. We also show that τ_decay increases with increasing stimulus duration and frequency, and that it is linearly dependent on the maximal glutamate concentration. Under some circumstances it was problematic to directly measure τ_decay, and we then used a pair-pulse paradigm to get an indirect estimate of τ_decay.</p><p>I am not aware of any computational model work taking into account the synaptically evoked I_TRPC current, prior to the current study, and believe that it is the first of its kind. We suggest that I_TRPC is important for slow synaptic integration, not only in the EC, but in several cortical and subcortical regions that contain mGlu1/5 and TRPC subunits, such as the prefrontal cortex. I will argue that this is further supported by studies using pharmacological blockers as well as studies on genetically modified animals.</p> / QC 20101005

Advances in numerical optimization have raised the possibility that efficient and novel aircraft configurations may be ``discovered'' by an algorithm. To begin exploring this possibility, a fast and robust set of tools for aerodynamic shape optimization is developed. Parameterization and mesh-movement are integrated to accommodate large changes in the geometry. This integrated approach uses a coarse B-spline control grid to represent the geometry and move the computational mesh; consequently, the mesh-movement algorithm is two to three orders faster than a node-based linear elasticity approach, without compromising mesh quality. Aerodynamic analysis is performed using a flow solver for the Euler equations. The governing equations are discretized using summation-by-parts finite-difference operators and simultaneous approximation terms, which permit nonsmooth mesh continuity at block interfaces. The discretization results in a set of nonlinear algebraic equations, which are solved using an efficient parallel Newton-Krylov-Schur strategy. A gradient-based optimization algorithm is adopted. The gradient is evaluated using adjoint variables for the flow and mesh equations in a sequential approach. The flow adjoint equations are solved using a novel variant of the Krylov solver GCROT. This variant of GCROT is flexible to take advantage of non-stationary preconditioners and is shown to outperform restarted flexible GMRES. The aerodynamic optimizer is applied to several studies of induced-drag minimization. An elliptical lift distribution is recovered by varying spanwise twist, thereby validating the algorithm. Planform optimization based on the Euler equations produces a nonelliptical lift distribution, in contrast with the predictions of lifting-line theory. A study of spanwise vertical shape optimization confirms that a winglet-up configuration is more efficient than a winglet-down configuration. A split-tip geometry is used to explore nonlinear wake-wing interactions: the optimized split-tip demonstrates a significant reduction in induced drag relative to a single-tip wing. Finally, the optimal spanwise loading for a box-wing configuration is investigated.