The aim of our work is to obtain joint universality theorems for periodic Hurwitz zeta-functions. We prove two joint universality theorems for periodic Hurwitz zeta-function. In the first theorems, the set L is linearly independent over the field of national numbers, then the periodic Hurwitz zeta-functions are universality. In the second joint universality theorem, we consider the use then parameter alpha corresponds general periodic sequence. Then the set L is linearly independent over the field of national numbers and the rank hypothesis in this theorem is weaker then that in A. Laurinčikas (2008) work. Then the second periodic Hurwitz zeta-functions are universal too. / Magistro darbe yra nagrinėjamas Hurvico dzeta funkcijų rinkinio jungtinis universalumas. Yra įrodytos dvi jungtinės universalumo teoremos. Pirmoji teorema tvirtina, kad jei aibė L yra tiesiškai nepriklausoma virš racionaliųjų skaičių kūno, tai periodinės Hurvico dzeta funkcijos yra universalios. Ši teorema žymiai susilpnina sąlygas, kurioms esant, buvo gautas analogiškas rezultatas A. Javtoko ir A. Laurinčiko 2008 m. darbe. Antroje teoremoje yra nagrinėjamas atvejis, kai kiekvieną skaičių alpha atitinka periodinių sekų rinkinys. Kai sistema L yra tiesiškai nepriklausoma virš racionaliųjų skaičių kūno ir galioja vieno rango tipo sąlyga, silpnesnė negu A. Laurinčiko darbe (2008), tai periodinių Hurvico dzeta funkcijų rinkinys yra taip pat universalus.

Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function. / Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti.

Yra gautos Lercho dzeta funkcijos su algebriniu iracionaliuoju parametru ribinės teoremos silpno tikimybinių matų konvergavimo prasme. Yra įrodyta minėto tipo teorema kompleksinėje plokštumoje, jungtinė ribinė teorema Lercho dzeta funkcijų rinkiniui kompleksinėje plokštumoje ir teorema analizinių funkcijų erdvėje. Įrodytos teoremos charakterizuoja Lercho dzeta funkcijų asimptotinį elgesį ir gali būti taikomos šios funkcijos universalumui tirti. / Limit theorems in the sense of weak convergence of probability measures for the Lerch zeta-function with algebraic irrational parameter are obtained. A theorem of mentioned type on the complex plane, a joint limit theorem for a collection of Lerch zeta-functions on the complex plane as well as a limit theorem in the space of analytic functions are proved. The theorems obtained characterize the asymptotic behaviour of the Lerch zeta-function and can be applied in the investigation of the universality of that function.

Inverse problems occur in a wide variety of applications and are an active area of research in many disciplines. We consider inverse problems for a broad class of nonlinear systems of partial differential equations (PDEs). We develop collage-based approaches for solving inverse problems for nonlinear PDEs of elliptic, parabolic and hyperbolic type. The original collage method for solving inverse problems was developed in [29] with broad application, in particular to ordinary differential equations (ODEs). Using a consequence of Banach’s fixed point theorem, the collage theorem, one can bound the approximation error above by the so-called collage distance, which is more readily minimizable. By minimizing the collage distance the approximation error can be controlled. In the case of nonlinear PDEs we consider the weak formulation of the PDE and make use of the nonlinear Lax-Milgram representation theorem and Galerkin approximation theory in order to develop a similar upper-bound on the approximation error. Supporting background theory, including weak solution theory,is presented and example problems are solved for each type of PDE to showcase the methods in practice. Numerical techniques and considerations are discussed and results are presented. To demonstrate the practical applicability of this work, we study two real-world applications. First, we investigate a model for the migration of three fish species through floodplain waters. A development of the mathematical model is presented and a collage-based method is applied to this model to recover the diffusion parameters. Theoretical and numerical particulars are discussed and results are presented. Finally, we investigate a model for the “Gao beam”, a nonlinear beam model that incorporates the possibility of buckling. The mathematical model is developed and the weak formulation is discussed. An inverse problem that seeks the flexural rigidity of the beam is solved and results are presented. Finally, we discuss avenues of future research arising from this work. / Natural Sciences and Engineering Research Council of Canada, Department of Mathematics & Statistics

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Reziumė Disertacijoje nagrinėjamos atsitiktinių keitinių problemos yra priskirtinos tikimybinei kombinatorikai. Gauti rezultatai aprašo visiškai adityviųjų funkcijų, apibrėžtų simetrinėje grupėje, reikšmių asimptotinius skirstinius Evenso tikimybinio mato atžvilgiu, kai grupės eilė neaprėžtai didėja. Išvestos adityviųjų funkcijų laipsninių ir faktorialinių momentų formulės. Funkcijų, išreiškiančių atsitiktinio keitinio ciklų su bet kokiais apribojimais skaičius, atveju rastos būtinos ir pakankamos ribinių tikimybinių dėsnių egzistavimo sąlygos. Išsamiai išnagrinėtas konvergavimas į Puasono, quasi-Puasono, Bernulio, binominio ir kitus skirstinius, sukoncentruotus sveikųjų neneigiamų skaičių aibėje. Rezultatai apibendrinti sveikareikšmių visiškai adityviųjų funkcijų klasėje. Darbe įrodytas bendras silpnasis didžiųjų skaičių dėsnis, rastos būtinos ir pakankamos adityviųjų funkcijų sekų pasiskirstymo funkcijų konvergavimo į išsigimusį nuliniame taške dėsnį egzistavimo sąlygos. Sprendžiamos problemos yra susietos su tikimybiniais vektorių, turinčių sveikąsias neneigiamas koordinates, uždaviniais. Adicinėje tokių vektorių pusgrupėje išnagrinėti multiplikatyviųjų funkcijų vidurkiai tikimybinio mato, vadinamo Evanso atrankos formule, atžvilgiu. Gauti tikslūs viršutinieji ir apatinieji įverčiai. Iš jų išplaukia svarbios atsitiktinių keitinių tikimybių savybės. Disertacijoje plėtojami faktorialinių momentų ir kiti kombinatoriniai bei tikimybiniai metodai. / In the thesis the examining problems of random permutations are attributed to the probabilistic combinatorics. Obtained results describe asymptotical distributions of completely additive functions values defined on a symmetric group with respect to Ewens probability measure, if the group order unbounded increases. Power and factorial moments formulae of additive functions are derived. There are established necessary and sufficient conditions under which the distributions of a number of cycles with restricted lengths obey the limit probability laws. The convergence to the Poisson, quasi-Poisson, Bernoulli, binomial and other distributions, defined on the positive whole - number set are exhaustively investigated. The results are generalized on the class of whole - number completely additive functions. The general weak law of large numbers is proved in the thesis, necessary and sufficient existence conditions, under which the distributions of the sequences of additive functions converge to the degenerate at the point zero limit law are established. Examining problems are related to the probability tasks of the vectors, which have whole - numbered nonnegative coordinates. The mean values of multiplicative functions defined on those vectors’ additive semigroup with respect to the Ewens measure, called Ewens Sampling Formula, and investigated. Lower and upper sharp estimates are obtained. From the latter results follow important probabilities’ properties of random... [to full text]

Bismuth ferrite (BFO) is a multiferroic material with cross-correlation between magnetic and electric orders. With no applied external fields the spin structure of BFO is anitferromagnetic and cycloidal. This ordering prevents the detection of the weak ferromagnetism known to exist in the material. The application of magnetic and electric fields of suitable strength and direction is capable of compelling the Fe3+ spins to align in a homogeneous, antiferromagnetic fashion. This report details how numerical methods were used to simulate the spin alignment of a BFO system under different fields. The results were compiled into electric field-magnetic field phase diagrams of BFO to show the divide between cycloidal and homogeneous systems. / Graduate / 0607 / 0611 / marca@uvic.ca

In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems. The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case. The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach. Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks. Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability. Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

Cette thèse traite des équations différentielles stochastiques rétrogrades réfléchies du second ordre dans une filtration générale . Nous avons traité tout d'abord la réflexion à une barrière inférieure puis nous avons étendu le résultat dans le cas d'une barrière supérieure. Notre contribution consiste à démontrer l'existence et l'unicité de la solution de ces équations dans le cadre d'une filtration générale sous des hypothèses faibles. Nous remplaçons la régularité uniforme par la régularité de type Borel. Le principe de programmation dynamique pour le problème de contrôle stochastique robuste est donc démontré sous les hypothèses faibles c'est à dire sans régularité sur le générateur, la condition terminal et la barrière. Dans le cadre des Équations Différentielles Stochastiques Rétrogrades (EDSRs ) standard, les problèmes de réflexions à barrières inférieures et supérieures sont symétriques. Par contre dans le cadre des EDSRs de second ordre, cette symétrie n'est plus valable à cause des la non linéarité de l'espérance sous laquelle est définie notre problème de contrôle stochastique robuste non dominé. Ensuite nous un schéma d'approximation numérique d'une classe d'EDSR de second ordre réfléchies. En particulier nous montrons la convergence de schéma et nous testons numériquement les résultats obtenus. / This thesis deals with the second-order reflected backward stochastic differential equations (2RBSDEs) in general filtration. In the first part , we consider the reflection with a lower obstacle and then extended the result in the case of an upper obstacle . Our main contribution consists in demonstrating the existence and the uniqueness of the solution of these equations defined in the general filtration under weak assumptions. We replace the uniform regularity by the Borel regularity(through analytic measurability). The dynamic programming principle for the robust stochastic control problem is thus demonstrated under weak assumptions, that is to say without regularity on the generator, the terminal condition and the obstacle. In the standard Backward Stochastic Differential Equations (BSDEs) framework, there is a symmetry between lower and upper obstacles reflection problem. On the contrary, in the context of second order BSDEs, this symmetry is no longer satisfy because of the nonlinearity of the expectation under which our robust stochastic non-dominated stochastic control problem is defined. In the second part , we get a numerical approximation scheme of a class of second-order reflected BSDEs. In particular we show the convergence of our scheme and we test numerically the results.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L^p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H^r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H^r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.