Phd Degree / Doktora
Permanent URI for this collectionhttps://hdl.handle.net/11147/2869
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Doctoral Thesis Randomization of certain operators in harmonic analysis(01. Izmir Institute of Technology, 2024) Sahillioğulları, Cihan; Temur, FarukBu tezde, stokastik süreçler aracılığıyla rastsallaştırılmış Hardy-Littlewood majorant problemi çalışılmıştır. Rastsallaştırma için durağan süreçler, rastgele yürüyüşler ve Poisson süreçleri kullanılmış ve bu süreçlerle pertürbe edilmiş deterministik kümeler için Hardy-Littlewood majorant özelliğinin neredeyse kesin olarak geçerli olduğu gösterilmiştir. Poisson süreçleri ile Green-Ruzsa kümesi de dahil olmak üzere çok büyük bir seyrek küme sınıfı pertürbe edilmiştir ve Hardy-Littlewood majorant özelliğinin ihmal edilebilir bir olasılıkla geçerliliğini sürdürdüğü gösterilmiştir. Ayrıca, frekansları daha büyük adım boyutuna sahip bir artimetik ilerleme oluşturan bir üstel toplamın L^2-normu ve L^4-normunun beklenen değerlerinin rastsallaştırmadan nasıl etkilendiği incelenmiştir. Dahası, Poisson süreçleriyle rastsallaştırılmış frekanslara sahip üstel toplamların L^n-normlarının, n ∈ 2N, beklenen değeri kestirilmiş ve bu normlar, ortalama anlamda, bölgeler üzerindeki tam sayı koordinatlı noktalar veya diyofant denklemlerinin çözümleri olarak yorumlanır.Doctoral Thesis Supersymmetric Coherent States and Superqubit Units of Quantum Information(01. Izmir Institute of Technology, 2024) Koçak Özvarol, Aygül; Pashaev, OktayBu tezde, hem fermiyonik hem de bozonik bileşenleri içeren, maksimum dolanık Bell tabanlı sÜper-eş uyumlu durumlar kümesini inceliyoruz. Aragone ve Zypmann tarafından tanıtılan süpersimetrik yok edici operatörü genişleterek, Bell iki-kübit kuantum durumlarıyla ilişkili dört farklı süpersimetrik eş uyumlu durum geliştiriyoruz. Bu Bell süper-kübit durumları, yer değiştirme operatörü kullanılarak inşa edilen Bell tabanlı süpersimetrik eş uyumlu durumların temelini oluşturur. Bu durumlar, süper-Bloch küresi üzerinde noktalar olarak temsil edilen ayrık bozonik eş uyumlu durumlarla birleştirildiğinde, ortaya çıkan yapıyı Bell tabanlı süper-eş uyumlu durumlar olarak adlandırılır.. Bozonik ve fermiyonik bileşenler arasındaki dolanıklık, süper-kübit referans durumu üzerinde etkili olan bir bozonik yer değiştirme operatörü aracılığıyla analiz edilir. Bu dolanık süper-eş uyumlu durumlar için belirsizlik ilişkileri concurrence $C$ ile ifade edilir. Belirsizlik ile concurrence arasındaki monoton ilişki, dolanıklığın belirsizlik ilişkileri üzerindeki etkisini göstermektedir. Daha sonra, konum ve momentum belirsizliklerinde kuadratür sıkışması gözlemliyoruz. Ayrıca, belirsizlik ilişkileri iki Fibonacci sayısının oranı ile karakterize edilen sonsuz bir süper-eş uyumlu durum dizisi tanımlıyoruz. Önceki sonuçları genelleştirmek amacıyla, tek bir süper-parçacık durumunun karmaşık bir parametre $\zeta$ ile tanımlandığı genel bir süper-kübit kuantum durumu tanıtıyoruz. Bu tanımlama, iki birim küre ile karakterize edilen PK-süper-kübit kuantum durumlarına yol açmaktadır. Bu durumlar, PK-süpersimetrik eş uyumlu durumlar olarak adlandırdığımız yapıların temelini oluşturur ve bu durumların dolanıklık özelliklerini inceliyoruz. Son olarak, pq-deforme süper-eş uyumlu durumları ve özel bir durum olarak q-deforme süper-eş uyumlu durumları ele alıyoruz.Doctoral Thesis Initial-Boundary Value Problem for the Higher-Order Nonlinear Schrödinger Equation on the Half-Line(01. Izmir Institute of Technology, 2024) Alkın, Aykut; Batal, Ahmet; Özsarı, TürkerWe establish local well-posedness in the sense of Hadamard for the higher-order nonlinear Schrödinger equation with a general power nonlinearity formulated on the halfline {x > 0}. We consider separately the two different scenarios of associated coefficients such that only one boundary condition is required, or exactly two boundary conditions are required. We assume a general nonhomogeneous boundary datum of Dirichlet type at x = 0 for the former case, and we add the Neumann type for the latter case. Our functional framework centers around fractional Sobolev spaces Hs x(R+) with respect to the spatial variable. We treat both high regularity (s > 1 2 ) and low regularity (s < 1 2 ) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initialboundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data.Doctoral Thesis Analysis of Damped and Viscoelastic Linear Wave Equations Exposed To External Neumann Manipulations(01. Izmir Institute of Technology, 2024) Susuzlu, İdem; Batal, Ahmet; Özsarı, TürkerIn this thesis, the main aim is to study the global existence and the stabilization of solutions for linear damped and viscoelastic wave equations evolving on a bounded medium in an arbitrarily large time interval subject to inhomogeneous Neumann manipulation on a part of the boundary. The analysis of these models reveals additional interesting features and challenges in comparison to their homogeneous counterparts, on which there are studies in the literature. This is due to the fact that, in the present context, the rate at which energy of solution is changed has a dependency on the boundary trace of temporal derivative. It is not clear how this quantity could be controlled in terms of given data according to Sobolev trace theory. Nevertheless, we achieve to establish global existence of solutions first using dynamic extension method to homogenize boundary conditions. Next, we construct the weak solutions of the homogenized models. For the damped wave equation, we rely on the semigroup approach while for the viscoelastic model we use Faedo-Galerkin method. The global unique solutions of the original models are obtained through a reunification argument. Then, we also prove uniform stabilization of solutions with decay rates characterized by the decay behavior of Neumann input using the multiplier (energy) technique. The latter requires a subtle analysis of boundary integrals in energy estimates involving unknown trace terms. We also develop numerical solutions of the models. For the damped wave equation, we rely on the explicit method while for the viscoelastic model we use the Crank-Nicolson method. We support our theoretical result with numerical simulations satisfying given assumptions. We supplement these with further numerical simulations in which data do not necessarily satisfy the given assumptions for decay. The latter offers, at the numerical level, essential physical insights into how energy might change in the presence of, for instance, improper boundary data.Doctoral Thesis Stability Analysis of Nonlinear Dynamical Systems With Lévy Typeperturbations(01. Izmir Institute of Technology, 2023) Tamcı, Ege; Batal, Ahmet; Savacı, Ferit AcarIn order to model the noise in power networks, generally, normal distribution is used. However, normal distribution is not convenient in modelling noise which has sudden peaks. Instead, combination of a continuous process and a jump processes is much more suitable. With this idea in mind, in this thesis, the stability and control of two equations used in modeling power grids is analyzed, under the assumption that they are exposed to Lévy process noise which includes jumps. These equations are the swing equation and the Kuramoto Model. The swing equation is used to model the single machine infinite bus system (SMIBS). Kuramoto Model is used when a large number of generators are considered as a network of coupled oscillators with their own natural frequencies. In our stability control study in the SMIBS, the noise in the system has sudden and finite changes is assumed and therefore should be modelled with a modified tempered α-stable process obtained by adding a finite jump condition on the tempered α-stable process when α < 1. The control functions depending on the mechanical power input and damping parameters are designed in order to make the system stable in probability and exponential stable at its equilibrium point. These theoretical results are supported by numerical studies. For Kuromato model, assuming that the power network consists of two layers, namely oscillator, and control layers and that is affected with a general Lévy process which has finite jumps, functions which provide the stability of phase and frequencies are obtained, depending on oscillator and coupling strengths. In the light of the numerical studies, the control of frequency and phase synchronization up to a certain noise intensity level can be evaluated, but it is not possible beyond that level is concluded.Doctoral Thesis Boundary Feedback Stabilization of Some Evolutionary Partial Differential Equations(01. Izmir Institute of Technology, 2022) Yılmaz, Kemal Cem; Batal, Ahmet; Özsarı, Türker; Özsarı, Türker; Batal, AhmetThe purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.Doctoral Thesis Exactly Solvable Burgers Type Equations With Variable Coefficients and Moving Boundary Conditions(01. Izmir Institute of Technology, 2022) Bozacı, Aylin; Atılgan Büyükaşık, ŞirinIn this thesis, firstly, a generalized diffusion type equation is considered. A family of analytical solutions to an initial value problem on the whole line for this equation is obtained in terms of solutions to the characteristic ordinary differential equation and the standard heat model by using Wei-Norman Lie algebraic approach for finding the evolution operator of the associated diffusion type equation. Then, initial-boundary value problems on half-line and an initial-boundary value problem with moving boundary for this equation are studied. It is shown that if the boundary propagates according to an associated classical equation of motion determined by the time-dependent parameters, then the analytical solution is obtained in terms of the heat problem on the half-line. For this, a non-linear Riccati type dynamical system, that simultaneously determines the solution of the diffusion type problem and the moving boundary is solved by a linearization procedure. The mean position of the solutions, the influence of the moving boundaries and the variable parameters are examined by constructing exactly solvable models. Then, an initial value problem for a generalized Burgers type equation on whole real line is discussed. By using Cole-Hopf linearization and solution of the corresponding generalized linear diffusion type equation, a family of analytical solution is obtained in terms of solutions to the characteristic equation and the standard heat or Burgers model. Exactly solvable models are constructed and the influence of the variable coefficients are examined. Later, an initial-boundary value problem for the generalized Burgers type equation with Dirichlet boundary condition defined on the half-line is studied. Finally, an initial-boundary value problem for the generalized Burgers type equations with Dirichlet boundary condition imposed at a moving boundary is considered. The analytical solution is obtained in terms of solution to characteristic equation and the standard heat or Burgers model, if the moving boundary propagates according to an associated classical equation of motion. In order to show certain aspects of the general results, some exactly solvable models are introduced and solutions corresponding to different types of initial and homogeneous/inhomogeneous boundary conditions are discussed by examining the influence of the moving boundaries.Doctoral Thesis Direct and Interior Inverse Generalized Impedance Problems for the Modified Helmholtz Equation(01. Izmir Institute of Technology, 2022) Özdemir, Gazi; Ivanyshyn Yaman, Olha; Yılmaz, OğuzOur research is motivated by the classical inverse scattering problem to reconstruct impedance functions. This problem is ill-posed and nonlinear. This problem can be solved by Newton-type iterative and regularization methods. In the first part, we suggest numerical methods for resolving the generalized impedance boundary value problem for the modified Helmholtz equation. We follow some strategies to solve it. The strategies of the first method are founded on the idea that the problem can be reduced to the boundary integral equation with a hyper-singular kernel. While the strategy of the second approach makes use of the concept of numerical differentiation, the first approach treats the hyper singular integral operator by splitting off the singularity. We also show the convergence of the first method in the Sobolev sense and the solvability of the boundary integral equation. We give numerical examples which show exponential convergence for analytical data. In the second part of this work, we take into account the inverse scattering problem of reconstructing the cavity’s surface impedance from sources and measurements positioned on a curve within it. For the approximate solution of an ill-posed and nonlinear problem, we propose a direct and hybrid method which is a Newton-type method based on a boundary integral equation approach for the boundary value problem for the modified Helmholtz equation. As a consequence of this, the numerical algorithm combines the benefits of direct and iterative schemes and has the same level of accuracy as a Newton-type method while not requiring an initial guess. The results are confirmed by numerical examples which show that the numerical method is feasible and effective.Doctoral Thesis Exactly Solvable Quantum Parametric Oscillators in Higher Dimensions(Izmir Institute of Technology, 2022) Çayiç, Zehra; Atılgan Büyükaşık, ŞirinThe purpose of this thesis is to study the dynamics of the generalized quantum parametric oscillators in one and higher dimensions and present exactly solvable models. First, time-evolution of the nonclassical states for a one-dimensional quantum parametric oscillator corresponding to the most general quadratic Hamiltonian is found explicitly, and the squeezing properties of the wave packets are analyzed. Then, initial boundary value problems for the generalized quantum parametric oscillator with Dirichlet and Robin boundary conditions imposed at a moving boundary are introduced. Solutions corresponding to different types of initial data and homogeneous boundary conditions are found to examine the influence of the moving boundaries. Besides, an N-dimensional generalized quantum harmonic oscillator with time-dependent parameters is considered and its solution is obtained by using the evolution operator method. Exactly solvable quantum models are introduced and for each model, the squeezing and displacement properties of the time-evolved coherent states are studied. Finally, time-dependent Schrödinger equation describing a generalized two-dimensional quantum coupled parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. The propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, a Cauchy-Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields is introduced. Based on the explicit results, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.Doctoral Thesis Krull-Schmidt Properties Over Non-Noetherian Rings(Izmir Institute of Technology, 2022) Gürbüz, Ezgi; Ay Saylam, BaşakLet R be a commutative ring and C a class of indecomposable R-modules. The Krull-Schmidt property holds for C if, whenever G1 ⊕ ·· · ⊕ Gn H1 ⊕ ·· · ⊕ Hm for Gi, Hj ∈ C, then n = m and, after reindexing, Gi Hi for all i ≤ n. The main purpose of this thesis is to investigate Krull-Schmidt properties of certain classes of modules over Non-Noetherian rings. Particularly weakly Matlis domains, strong Mori domains and Marot rings, all of which are among the class of Non-Noetherian rings, are studied. wweak isomorphism types are defined and the conditions when they coincide for torsionless modules over weakly Matlis domains are discussed. With the help of this comparison, the Krull-Schmidt property of w-ideals of a strong Mori domain is characterized. Also, the same property for overrings of a strong Mori domain is examined. Some useful results for a Marot ring with ascending condition on its regular ideals are obtained. Krull-Schmidt property on regular ideals of such a ring is studied and a characterization is given. Furthermore, the same property is discussed for overrings of a Marot ring.