M. Tech. Mathematical Technology. / Discusses the main problems in terms of how to derive mathematical models for a free, a forced and a damped spring pendulum and determining numerical solutions using a computer algebra system (CAS), because exact analytical solutions are not obvious. Hence this mini-dissertation mainly deals with how to derive mathematical models for the spring pendulum using the Euler-Lagrange equations both in the Cartesian and polar coordinate systems and finding solutions numerically. Derivation of the equations of motion are done for the free, forced and damped cases of the spring pendulum. The main objectives of this mini-dissertation are: firstly, to derive the equations of motion governing the oscillatory and rotational components of the spring pendulum for the free, the forced and damped cases of the spring pendulum ; secondly, to solve these equations numerically by writing the equations as initial value problems (IVP); and finally, to introduce a novel way of incorporating nonlinear damping into the Euler-Lagrange equations of motion as introduced by Joubert, Shatalov and Manzhirov (2013, [20]) for the spring pendulum and interpreting the numerical solutions using CAS-generated graphics.

Many mathematical models generally consist of either a continuous system like that of a system of differential equations, or a discrete system such as a discrete game theoretic model; however, there exist phenomena in which neither modeling approach alone is sufficient for capturing the behaviour of the intended real world system. This leads to the need to explore the use of combinations of such discrete and continuous processes, namely the use of mathematical modeling with what are known as hybrid dynamical systems. In what follows, we provide a blueprint for one approach to study several classes of equilibrium problems in non-equilibrium states through the direct use of hybrid dynamical systems. The motivation of our work stems from the fact that the real world is rarely, if ever, in a state of perfect equilibrium and that the behaviour of equilibrium problems in non-equilibrium states is just as complex and interesting (if not more so) than standard equilibrium solutions. Our approach consists of an association of classes of traffic equilibrium problems, noncooperative games, minimization problems, and complementarity problems to a class of hybrid dynamical system called projected dynamical systems. The purposed connection between equilibrium problems and projected dynamical system is made possible through mutual connections to the robust framework of variational inequalities. The results of our work include theoretical contributions such as showing how evolution solutions (non-equilibrium solutions) can be analyzed from a theoretical point of view and how they relate to equilibrium solutions; computational methods for tracking and visualizing evolution solutions and the development of numerical algorithms for simulation; and applications such as the effect of population vaccination decisions in the spread of infectious disease, dynamic traffic networks, dynamic vaccination games, and nonsmooth electrical circuits.

This thesis is devoted to a study of the effect of disorder on strongly correlated electrons. For non-interacting electrons, Anderson localization occurs if the amount of disorder is sufficient. For disorder-free systems, a Mott metal-insulator transition may occur if the electron-electron interactions are strong enough. The question we ask in this thesis is what happens when both disorder and interactions are present. We study the Anderson-Hubbard model, which is the simplest model to include both interactions and disorder, using a Gutzwiller variational wave function approach. We then study Anderson localization of electrons from the response of the Anderson-Hubbard Hamiltonian to an external magnetic field. An Aharonov-Bohm flux induces a persistent current in mesoscopic rings. Strong interactions result in two competing tendencies: they tend to suppress the current because of strong correlations, and they also screen the disorder potential and making the system more homogenous. We find that, for strongly interacting electrons, the localization length may be large, even though the current is suppressed by strong correlations. This unexpected result highlights how strongly correlated materials can be quiet di erent from weakly correlated ones. / Thesis (Ph.D, Physics, Engineering Physics and Astronomy) -- Queen's University, 2011-08-31 09:51:47.155

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research. Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution. Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method. In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula. Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.

Padalinimo paviršiai – greitas ir efektyvus glodžių paviršių konstravimas trimatėje erdvėje, naudojamas animacijoje. Nestacionarios padalinimo schemos, kitaip nei klasikinės (Catmull – Clark, Loop, Doo-Sabin), kiekviename padalinimo žingsnyje leidžia keisti padalinimo taisyklę, todėl galima išgauti žymiai platesnę paviršių aibę. Parašėme programinę įrangą, kuri pagal įvedamą tempimo parametrą γ konstruoja nestacionarias kaukes su trigonometrinėmis funkcijomis ir jas pritaiko paviršiams, turi galimybę pavaizduoti paviršiaus tinklelį 3D erdvėje ir gali jį eksportuoti į VRML formato failą. Nestacionarių padalinimo schemų ir sukimo paviršių pagalba, gavome plačią paviršių aibę į kurią įeina ir klasikinės CAD figūros. / Variational subdivision schemes as distinct from classical (Catmull – Clark, Loop, Doo-Sabin) in every subdivision step allows to change the subdivision rule. Therefore we are able to get wider set of smooth surfaces. We made a software that using the entered parameter γ constructs variational subdivision mask with trigonometric functions and applies it to surfaces. Software is able to show surface in 3D space with move, rotate and zoom abilities. Software also can export surfaces to VRML file format. Subdividing surfaces by our software we faced with border and extraordinary vertex problems. We suggested and realized two solving methods for both problems. Our software is able to control the surface's smoothness and sharpness in all subdivision steps. Using variational subdivision schemes and rotation methods, we can generate a wide set of surfaces witch also includes classical CAD figures (sphere, cylinder).

Padalinimo paviršiai – greitas ir efektyvus glodžių paviršių konstravimas trimatėje erdvėje, naudojamas animacijoje. Nestacionarios padalinimo schemos, kitaip nei klasikinės (Catmull – Clark, Loop, Doo-Sabin), kiekviename padalinimo žingsnyje leidžia keisti padalinimo taisyklę, todėl galima išgauti žymiai platesnę paviršių aibę. Parašėme programinę įrangą, kuri pagal įvedamą tempimo parametrą γ konstruoja nestacionarias kaukes su trigonometrinėmis funkcijomis ir jas pritaiko paviršiams, turi galimybę pavaizduoti paviršiaus tinklelį 3D erdvėje ir gali jį eksportuoti į VRML formato failą. Nestacionarių padalinimo schemų ir sukimo paviršių pagalba, gavome plačią paviršių aibę į kurią įeina ir klasikinės CAD figūros. / Variational subdivision schemes as distinct from classical (Catmull – Clark, Loop, Doo-Sabin) in every subdivision step allows to change the subdivision rule. Therefore we are able to get wider set of smooth surfaces. We made a software that using the entered parameter γ constructs variational subdivision mask with trigonometric functions and applies it to surfaces. Software is able to show surface in 3D space with move, rotate and zoom abilities. Software also can export surfaces to VRML file format. Subdividing surfaces by our software we faced with border and extraordinary vertex problems. We suggested and realized two solving methods for both problems. Our software is able to control the surface's smoothness and sharpness in all subdivision steps. Using variational subdivision schemes and rotation methods, we can generate a wide set of surfaces witch also includes classical CAD figures (sphere, cylinder).

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.

Interest in the variation of fundamental constants has recently been stimulated by claims that the fine structure constant, α, was smaller in the past. Physicists are investigating whether α is currently varying using a number of methods including atomic clock experiments and quasar absorption spectra. To date atomic clock experiments have not reached the same level of precision as the quasar results but the precision to which transition frequencies are being measured is increasing dramatically and very soon atomic clock experiments based on Earth will be able to rival or surpass the quasar results. In order to relate the change in transition frequencies to a variation of α accurate calculations of relativistic effects in atoms and their dependence upon α are needed. Other effects, such as the small shift of transition frequencies due to blackbody radiation also need to be accounted for. In this thesis we perform accurate calculations of the dependence of transition frequencies in two-valence-electron atoms and ions on a variation of α. The relativistic Hartree-Fock method is used with many-body perturbation theory and configuration interaction methods to calculate transition frequencies. We also consider transitions with an enhanced sensitivity to α variation. In particular, narrow lines that correspond to atomic transitions between close lying, long-lived atomic states of different configurations. The small transition frequency, coupled with differences in the electron structure ensures a strong enhancement of the relative frequency change compared to a possible change in α . We also show that using the modified form of the Dirac Hamiltonian, as suggested by Bekenstein, does not affect the analysis of the quasar data pertaining to a measurement of α variation, nor does it affect atomic clock experiments. Finally we have performed calculations of the size of the frequency shift induced by a static electric field on the clock transition frequencies of the hyperfine splitting in Y b+, Rb, Cs, Ba+, and Hg+. The calculations are used to find the frequency shifts due to blackbody radiation which are needed for accurate frequency measurements and improvements of the limits on variation of α. Our result for Cs [??v/=E2 = -2:26(2) x 10-10Hz/(V/m)2] is in good agreement with early measurements and ab initio calculations. We present arguments against recent claims that the actual value might be smaller. The difference (~ 10%) is due to the continuum spectrum in the sum over intermediate states.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 101-103).