Phd Degree / Doktora
Permanent URI for this collectionhttps://hdl.handle.net/11147/2869
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Doctoral Thesis Exactly Solvable Quantum Parametric Oscillators in Higher Dimensions(Izmir Institute of Technology, 2022) Atılgan Büyükaşık, Şirin; Atılgan Büyükaşık, Şirin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe 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) Ay Saylam, Başak; Ay Saylam, Başak; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyLet 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.Doctoral Thesis When Certain Relative Projectivity and Injectivity Conditions Imply the Global Projectivity and Injectivity(Izmir Institute of Technology, 2022) Büyükaşık, Engin; Büyükaşık, Engin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyA right R-module M is called R-projective provided that it is projective relative to the right R-module RR. One of the parts of this thesis deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that RR is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Σ-CS and flat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(RR). As a second goal, we deal with simple-injective modules. For a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is quasi-Frobenius if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is quasi-Frobenius and B is hereditary. An abelian group is simpleinjective if and only if its torsion part is injective.Doctoral Thesis Reidemeister torsion of closed л-manifolds(Izmir Institute of Technology, 2021) Dirican Erdal, Esma; Erman, Fatih; Sözen, Yaşar; Erman, Fatih; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyLet M be a closed orientable 2n-dimensional л-manifold such that n , 2 and M is either (n-2)-connected or (n-1)-connected. Such a manifold M can be decomposed as a connected sum of certain simpler manifolds. In this thesis, by using such connected sum decompositions, we develop multiplicative gluing formulas that express the Reidemeister torsion of M with untwisted R-coefficients in terms of Reidemeister torsions of its building blocks in the decomposition. Then we apply these results to handlebodies, compact orientable smooth (2n+1)-dimensional manifolds whose boundary is a (n-2)-connected 2n-dimensional closed л-manifold, and product manifolds.Doctoral Thesis Analysis and Application of Linearization Technique for Nonlinear Problems(Izmir Institute of Technology, 2020) İmamoğlu Karabaş, Neslişah; Tanoğlu, Gamze; İmamoğlu Karabaş, Neslişah; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe purpose of this thesis is to investigate the implementation of linearization technique combining with the multiquadric radial basis function method to nonlinear problems which appears in engineering and physics. Presented linearization technique is formed by the Frechet derivatives and Newton Raphson method. This technique is applied to Burgers' equation, Coupled Burgers' equation and 2-D cubic nonlinear Schrödinger equation. From the numerical results of the problems, it is believed that this technique can be used to solve other nonlinear and system of nonlinear partial differential equations numerically.Doctoral Thesis Asymptotic Behaviour of Gravity Driven Free Surface Flows Resulting From Cavity Collapse(Izmir Institute of Technology, 2020) Fetahu, Elona; Yılmaz, Oğuz; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, the gravity driven potential flows that result from cavity collapse are studied. Initially, the collapse of a vertical cylindrical cavity of circular cross sections surrounded by a liquid region is examined for two different situations. In the first one the cavity has same depth as the fluid and in the second one the cavity starts from the free surface and has less depth than the fluid. The problem is formulated by using a small parameter that represents the short duration of the stage. The first problem, as the radius and the centre of the cavity approach infinity, reduces to the classical two-dimensional dam break problem solved by Korobkin and Yilmaz (2009). The singularity of the radial velocity at the bottom circle is shown to be of logarithmic type. In the second problem, where the cavity is less deep than the fluid, the flow region is separated into two regions: the interior one, which is underneath the cylindrical cavity and above the rigid bottom, and the exterior one, which is the rest of the flow. The corresponding new problems are solved separately and then the coefficients are found by applying the matching conditions at the interface, where the fluid radial velocities and pressures coincide. On the limiting case, the problem reduces to the two-dimensional dam break flow of two immiscible fluids by Yilmaz et al. (2013a). Singularity at the bottom circle of the cavity is observed, which is of the same type as in the latter paper. Next, a third problem studies the gravity driven flow caused by the collapse of a rectangular section of a vertical plate. During the early stage, the flow is described by the velocity potential. Attention is paid to determining the velocity potential and free surface shapes. The solution follows the Fourier series method in Renzi and Dias (2013) and the boundary element method in Yilmaz et al. (2013a). Singularity is observed at the side edges and lower edge of the rectangular section. The horizontal velocity of the initially vertical free surface along the vertical line of symmetry of the rectangle is the same to the one in the two-dimensional problem Korobkin and Yilmaz (2009). The singularities observed in these problems lead to the jet formation for the initial stage. The methods applied in these computations are expected to be helpful in the analysis of gravity-driven flow free surface shapes. This thesis is a contribution towards the 3-D generalizations of dam break problems.Doctoral Thesis On Relative Projectivity of Some Classes of Modules(Izmir Institute of Technology, 2019) Alagöz, Yusuf; Alagöz, Yusuf; Büyükaşık, Engin; Büyükaşık, Engin; 01. Izmir Institute of Technology; 04.02. Department of Mathematics; 04. Faculty of ScienceThe main purpose of this thesis is to study R-projectivity and max-projectivity of some classes of modules, and module classes related to max-projective modules. A right R-module M is called max-projective provided that each homomorphism f:M → R/I where I is any maximal right ideal, factors through the canonical projection π:R → R/I. We call a ring R right almost-QF (resp. right max-QF) if every injective right R-module is R-projective (resp. max-projective). In this thesis we attempt to understand the class of right almost-QF (resp. right max-QF) rings. Among other results, we prove that a right Hereditary right Noetherian ring R is right almost-QF if and only if R is right max-QF if and only if R = S x T , where S is semisimple Artinian and T is right small. A right Hereditary ring is max-QF if and only if every injective simple right R-module is projective. Furthermore, a commutative Noetherian ring R is almost-QF if and only if R is max-QF if and only if R = A x B, where A is QF and B is a small ring. Moreover, we introduced and studied some homological objects related with max-projective modules.Doctoral Thesis Numerical Methods for Nonlocal Problems(Izmir Institute of Technology, 2018) Kaya, Adem; Kaya, Adem; Tanoğlu, Gamze; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this thesis, numerical methods for nonlocal problems with local boundary conditions from the area of peridynamics are studied. The novel operators that satisfy local boundary conditions were proposed as an alternative to the original nonlocal problems which uses nonlocal boundaries. Peridynamic theory is reformulation of continuum mechanics by integral equations for which it has some advantages over traditional partial differential equations. In peridynamic theory, a point can interact with other points within a certain distance which is called horizon and indicated by the parameter δ. In this thesis, we are particularly interested in role of the parameter δ in numerical methods for the novel problems. More precisely, we aim to show its role in condition number, discretization error and convergence factor of multigrid method.Doctoral Thesis Short Time Behaviour of Dam Break Flow Involving Two Liquids(Izmir Institute of Technology, 2018) Isıdıcı Demirel, Damla; Yılmaz, Oğuz; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe two dimensional dam break problem for wet bed case is investigated. The leading order and the second order problem are stated in nondimensional form. Solution to the leading order problem by using three different methods is given and explained in detail. Both Fourier series method and Galerkin method have difficulties on its own because of the singularity at the triple point. Although the singularity is ignored in Galerkin method, the method does not work except for the interface. Thus conformal mapping techniques is preferred because of the convenience and the strength of the complex analysis. The velocity profiles at whole boundary are obtained by using this conformal mapping. The second order solution of velocities are also obtained by using the same conformal mapping. On the other hand, the domain decomposition method (DDM) is applied for the second order dam break problem of dry bed case. The leading order solution helped to determine the suitable parameters for DDM. The leading order and second order solution of the free surfaces give a more realistic shape using the Lagrangian solution at the upper corner point. We assume this work contains useful and applicable methods in it for gravity driven flows and it will wake up different perspectives in readers mind.Doctoral Thesis Modules Satisfying Conditions That Are Opposites of Absolute Purity and Flatness(Izmir Institute of Technology, 2017) Kafkas Demirci, Gizem; Büyükaşık, Engin; Büyükaşık, Engin; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe main purpose of this thesis is to study the properties which are opposites of absolute pure and flat modules. A right module M is said to be test for flatness by subpurity (for short, t.f.b.s.) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. A left module M is said to be rugged if its flatness domain is the class of all regular right R-modules. Every ring has a t.f.b.s. module. For a right Noetherian ring R every simple right R-module is t.f.b.s. or absolutely pure if and only if R is a right V-ring or R A×B, where A is right Artinian with a unique non-injective simple right R-module and Soc(AA) is homogeneous and B is semisimple. A characterization of t.f.b.s. modules over commutative hereditary Noetherian rings is given. Rings all (cyclic) modules of whose are rugged are shown to be von Neumann regular rings. Over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S × T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical contains no properly nonzero ideals. Connections between rugged and poor modules are shown. Rugged Abelian groups are fully characterized.
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