L'AMPc té un paper clau com a missatger intracel.lular regulant la transmissió dels senyals extracel•lulars en diferents teixits i controlant múltiples processos cel lulars. Els nivells intracel.lulars d'AMPc es controlen mitjançant la seva síntesi, catalitzada per l'enzim adenilat ciclasa, i mitjançant la seva degradació a través de l'acció de les fosfodiesterases (PDE) de nucleòtids cíclics. Hi ha 11 famílies de PDEs. La PDE4 representa a una família de fosfodiesterases específiques d'AMPc formada per quatre gens paràlegs (PDE4-D), cadascun dels quals és capaç de generar múltiples variants d’splicing. La PDE4A, la PDE4B i la PDE4D es troben expressades en diferents tipus de cèl lules inflamatòries on tenen un important paper com reguladores dels processos inflamatoris. La inhibició selectiva, tant in vitro com in vivo, de PDEs ha demostrat tenir diferents efectes antiinflamatoris. En aquest treball es mostra la implicació de la isoforma PDE4B, i en concret la seva variant de splicing PDE4B2, durant el procés neuroinflamatori del model animal d’Esclerosis Múltiple, l’Encefalomielitis Autoimmune Experimental (d’EAE). Els resultats mostren un augment de l’expressió de l’ARNm de PDE4B2 a la medul•la espinal de ratolins EAE que correlaciona amb l’expressió d’alguns marcadors inflamatoris de forma dependent a la simptomatologia clínica dels animals. També s’observa que l’enzim PDE4B es trova localitzat principalment en cèl•lules presentadores d’antigen (APCs) com les cèl•lules dendrítques i els macròfags/micròglia. A més, els ratolins PDE4B-/- mostren una aparició temprana dels símptomes clínics en comparació amb els ratolins wildtype, amb alteracions en l’expressió de l’ARNm d’algunes citocines. L’alteració selectiva de la PDE4B2 en el model d’EAE en ratolí i la seva participació en el desenvolupament de la malaltia com s’ha observat en els animals PDE4B-/- presenta noves possibilitats sobre l’ús d’inhibidors selectius per les diverses isoformes (i variants d’splicing) tant per aplicacions terapèutiques com per investigar mecanismes d’inflamació en malalties neurodegeneratives. / Experimental autoimmune encephalomyelitis (EAE) is an animal model of multiple sclerosis that courses with neuroinflammation, axonal damage and demyelination. The model is characterized by T- and B-cell responses to myelin oligodendrocyte glycoprotein which produce a wide range of pro- and anti-inflammatory cytokines. The modulation of cAMP levels through pharmacological manipulation of phosphodiesterases (PDEs) provokes profound anti-inflammatory responses. In the EAE model, amelioration of the clinical signs and delayed onset is observed after PDE4 inhibition and the PDE4B gene has been related to the inflammatory immune response in mice. Here we analyzed post-immunization changes in the expression of mRNA coding for the PDE4B2 splice variant by semiquantitative real-time PCR and in situ hybridization. The results showed an upregulation of PDE4B2 mRNA in the spinal cord of EAE mice which correlates with FoxP3 and TGF-β mRNAs expression in a score-dependent manner. We also found that PDE4B enzyme is mainly localized in antigen-presenting cells (APCs) such as dendritic cells and microglia/macrophages. PDE4B-/- mice show an earlier onset of the disease compared to wildtype mice, with alterations in some cytokine mRNA expression. The results point to a protective role of the PDE4B enzyme and PDE4B2 splice variant in particular, during EAE pathogenesis by modulating cAMP levels in APCs and controlling the cytokine environment for T-cell differentiation.

A general methodology is described in which policyholder behaviour is decoupled from the pricing of a variable annuity based on the cost of hedging it, yielding two sequences of weakly coupled systems of partial differential equations (PDEs): the pricing and utility systems. The utility systems are used to generate policyholder withdrawal behaviour, which is in turn fed into the pricing systems as a means to determine the cost of hedging the contract. This approach allows us to incorporate the effects of utility-based pricing and factors such as taxation. As a case study, we consider the Guaranteed Lifelong Withdrawal and Death Benefits (GLWDB) contract. The pricing and utility systems for the GLWDB are derived under the assumption that the underlying asset follows a Markov regime-switching process. An implicit PDE method is used to solve both systems in tandem. We show that for a large class of utility functions, the two systems preserve homogeneity, allowing us to decrease the dimensionality of solutions. We also show that the associated control for the GLWDB is bang-bang, under which the work required to compute the optimal strategy is significantly reduced. We extend this result to provide the reader with sufficient conditions for a bang-bang control for a general variable annuity with a countable number of events (e.g. discontinuous withdrawals). Homogeneity and bang-bangness yield significant reductions in complexity and allow us to rapidly generate numerical solutions. Results are presented which demonstrate the sensitivity of the hedging expense to various parameters. The costly nature of the death benefit is documented. It is also shown that for a typical contract, the fee required to fund the cost of hedging calculated under the assumption that the policyholder withdraws at the contract rate is an appropriate approximation to the fee calculated assuming optimal consumption.

Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.

Asian options are exotic financial derivative products which price must be calculated by numerical evaluation. In this thesis, we study certain ways of solving partial differential equations, which are associated with these derivatives. Since standard numerical techniques for Asian options are often incorrect and impractical, we discuss their variations, which are efficiently applicable for handling frequent numerical instabilities reflected in form of oscillatory solutions. We will show that this crucial problem can be treated and eliminated by adopting flux limiting techniques, which are total variation dimishing.

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.

A Pulse Detonation Engine (PDE) is a propulsion device that takes advantage of the pressure rise inherent to the efficient burning of fuel-air mixtures via detonations. Detonation initiation is a critical process that occurs in the cycle of a PDE. A practical method of detonation initiation is Deflagration-to-Detonation Transition (DDT), which describes the transition of a subsonic deflagration, created using low initiation energies, to a supersonic detonation. This thesis presents the effects of obstacle spacing, blockage ratio, DDT section length, and airflow on DDT behavior in hydrogen-air and ethylene-air mixtures for a repeating PDE. These experiments were performed on a 2 diameter, 40 long, continuous-flow PDE located at the General Electric Global Research Center in Niskayuna, New York. A fundamental study of experiments performed on a modular orifice plate DDT geometry revealed that all three factors tested (obstacle blockage ratio, length of DDT section, and spacing between obstacles) have a statistically significant effect on flame acceleration. All of the interactions between the factors, except for the interaction of the blockage ratio with the spacing between obstacles, were also significant. To better capture the non-linearity of the DDT process, further studies were performed using a clear detonation chamber and a high-speed digital camera to track the flame chemiluminescence as it progressed through the PDE. Results show that the presence of excess obstacles, past what is minimally required to transition the flame to detonation, hinders the length and time to transition to detonation. Other key findings show that increasing the mass flow-rate of air through the PDE significantly reduces the run-up time of DDT, while having minimal effect on run-up distance. These experimental results provided validation runs for computational studies. In some cases as little as 20% difference was seen. The minimum DDT length for 0.15 lb/s hydrogen-air studies was 8 L/D from the spark location, while for ethylene it was 16 L/D. It was also observed that increasing the airflow rate through the tube from 0.1 to 0.3 lbs/sec decreased the time required for DDT by 26%, from 3.9 ms to 2.9 ms.

Mathcad gilt in den unterschiedlichsten Bereichen, z.B. in den Ingenieurwissenschaften, der Mathematik, der Physik, der Biologie oder sogar der Qualitätssicherung als hervorragendes Werkzeug zur übersichtlichen Darstellung komplexer Berechnungen. Sollten die enthaltenen Funktionalitäten nicht mehr ausreichen, besteht die Möglichkeit, Mathcad mit Hilfe von User-DLLs zu erweitern. Diese Erweiterung kann perfekt als Schnittstelle zwischen Mathcad und anderen Softwarepaketen genutzt werden. Die von der inuTech GmbH entwickelte Klassenbibliothek Diffpack zur Simulation und numerischen Lösung von Differentialgleichungen aus den verschiedensten Bereichen eignet sich hervorragend, erforderliche Funktionalitäten für Mathcad zu implementieren. Mathcad kann somit zur Parametrisierung, für Berechnungen und zur Darstellung der Ergebnisse verwendet werden, während Diffpack die Lösung der partiellen Differentialgleichung, z.B. mittels FEM, übernimmt.

This research seeks to expand the role of variational and adjoint processing methods into segments of the sonar, radar, and nondestructive testing communities where they have not yet been widely introduced. First, synthetic aperture reconstruction is expressed in terms of the adjoint operator. Many, if not all, practical imaging modalities can be traced back to this general result, as the adjoint is the foundation for backprojection-type algorithms. Next, active surfaces are developed in the context of the Helmholtz equation for the cases of opaque scatterers (i.e., with no interior field) embedded in free space, and penetrable scatterers embedded in a volume which may be bounded. The latter are demonstrated numerically using closed-form solutions based on spherical harmonics. The former case was chosen as the basis for a laboratory experiment using Lamb waves in an aluminum plate. Lamb wave propagation in plates is accurately described by the Helmholtz equation, where the field quantity is the displacement potential. However, the boundary conditions associated with the displacement potential formulation of Lamb waves are incompatible with the shape gradient derived for the Helmholtz equation, except for very long or very short wavelengths. Lastly, optical flow is used to solve a new and unique problem in the field of synthetic aperture sonar. Areas of acoustic focusing and dilution attributable to refraction can sometimes resemble the natural bathymetry of the ocean floor. The difference is often visually indistinguishable, so it is desirable to have a means of detecting these transient refractive effects without having to repeat the survey. Optical flow proved to be effective for this purpose, and it is shown that the parameters used to control the algorithm can be linked to known properties of the data collection and scattering physics.

In this thesis, first the tanh method, a method for obtaining exact traveling wave solutions to nonlinear differential equations, is introduced and described. Then the method is applied to two classes of Nonlinear Partial Differential Equations. The first one is a system of two (1 + 1)-dimensional nonlinear Korteweg-de Vries (KdV) type equations. The second one is a (3 + 1)-dimensional nonlinear wave equation. At the end, a few graphic representations of the obtained solitary wave solutions are provided, in correspondence to different values of the parameters used in the equations.