Ph.D. Theses
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Item Smoothing properties of initial-boundary value problems(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Başakoğlu, Engin.; Gürel, Burak.This thesis discusses the smoothing properties of dispersive partial differential equations. In the first part of the thesis, we consider the Davey–Stewartson system on R 2 and demonstrate that the nonlinear part of the solution flow is smoother than the initial data. As an application of the smoothing result, we address the dissipa tive Davey–Stewartson system and give a simplified proof of the existence of a global attractor for the system. In the next part, we study well- posedness and regularity properties of the biharmonic Schr¨odinger equation on the half-line. More precisely, we prove local existence and uniqueness and show that the data to solution map is continuous. Moreover, we establish global well- posedness and global smoothing for higher regular spaces by showing that the solution grows at most linearly. As regards to the smoothing result, the derivative gain we obtain for the nonlinear part of the so lution is up to full derivative. The last part of the thesis addresses the Hirota–Satsuma system on the torus. The Hirota–Satsuma system is given by two Korteweg- de Vries equations exhibiting different dispersion relations which is due to the coupling coeffi cient a. The main result demonstrates the regularity level of the nonlinear part of the evolution compared to initial data. The gain in regularity depends very much on the arithmetic properties of the coefficient a. Then, we consider the forced and damped Hirota–Satsuma system and establish the analogous smoothing estimates. By the help of the smoothing estimates, we prove the existence and regularity of a global attractor in the energy space.Item Simple section biset functors(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Muslumov, Ruslan.; Coşkun, Olcay.Let G and H be finite groups and k be a commutative unitary ring. The Burnside group B(G,H) is the Grothendieck group of the category of finite (G,H)-bisets. The biset category kC of finite groups is the category defined over finite groups, whose morphism sets are given by the kB(G,H) groups. A biset functor defined on kC, with values in k-Mod is a k-linear functor from kC to the category of k-Mod. The remarkable results as the evaluation of the Dade group of endopermutation modules of a p- group and finding the unit group of the Burnside ring of a p- group are done using the theory of biset functors. Looking for ring objects in the category of biset functors one gets a more sophisticated structure, which is called a Green Biset Functor. Serge Bouc introduced the slice Burnside ring and the section Burnside ring for a finite group G. He also showed that these two rings have a natural structure of a Green Biset Functor. In our work we classify simple modules over the section Burnside ring of G using the approach of the paper Fibered Biset Functors by Robert Boltje and Olcay Coşkun.Item Composition factors of the functor of the complex characters(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2018., 2018.) Arslan, Mehmet.; Coşkun, Olcay.When we consider CRC as a map sending any finite group G to the complex vector space CRC(G) of complex valued class functions on G, it becomes an A-fibered biset functor for any group A ≤ C×. Its structure is known for tirivial fiber groups A = 1 and A = C×. While it is a direct sum of simple biset functors in the case that A = 1, in the other case it is a simple C×-fibered biset functor. We noticed that as the fiber group grows, some of simple summands of 1-fibered biset functor CRC unite and form new fibered simple summands. In this thesis, we investigate the structure of the functor CRC for two intermediate fiber groups. The first one is a group containing all pn-th roots of unity for any n ∈ N and for any prime number p from a fixed set of primes π. The second one is the group of all pn-th roots of unity for a fixed n ∈ N. For both cases, we identify its new fibered simple summands by determining uniting summands via defining equivalence relations on them.Item On a shell formula of closed curves in riemannian manifolds(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 1986., 1986.) Şimşek, Ahmet.; Aşkar, Attila.Item Vertical distribution problems in Zeta-Function theory(Thesis (Ph.D.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2015., 2015.) Karabulut, Yunus.; Yıldırım, Cem Y.In this thesis, we focus on the vertical distribution problems of the zeros of the Riemann zeta-function and other related functions. In the rst half of our study we modify Montgomery's argument [1] in such a way that we can obtain some analogues of the pair correlation of zeta zeros, which provide some gap and multiplicity results. In the second half of our study we estimate the averages studied in [2] over the zeta maximas on the critical line instead of zeros so that we arrive at a result on the number of distinct zeta zeros.Item On infinite dimensional spherical analysis(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2017., 2017.) Çam, Şermin.; Boysal, Arzu.; Demir, Selçuk.This thesis is concerned with the spherical analysis of two di erent Olshanski pairs, one of which is related to Heisenberg groups, and the other to the automorphism groups of homogeneous trees. The spherical functions of positive type on the in nite dimensional Heisenberg group H(1) which are invariant under the natural action of the in nite dimensional unitary group U(1) are determined. On the other hand, we consider an Olshanski pair which is constructed from the stabilizers of the horicycles of homogeneous trees of nite degree, where the horicycles form a partition of the set of vertices of the tree, and then we nd all spherical functions of this pair. Finally, we give realizations of the corresponding irreducible unitary representations.Item The fibered burnside ring of a fusion system(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Seviniş, Mert.; Coşkun, Olcay.J. S. Reeh, in his PhD thesis [1], described the Burnside ring and its free basis for the fusion-stable sets. In this thesis we will extend his results for fibered sets. Basically we will use the same technics J.S. Reeh developed in his thesis and we will show that these work with little modifications for the fibered case as well. In the end we hope to provide the reader with a similar description of the fıbered Burnside ring and also to write a free basis for fusion-stable fibered sets in general.