M.S. Theses

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On the geometric objects of same type as christoffel symbols of Ehresmann Ë-connections
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 1995., 1995.) Abadoğlu, Ender.; Ortaçgil, Ercüment.
In this thesis the higher order Ehresmann a-connections are studied as geometric objects. We prove that, any two Lie subgroups of the rth order jet group G~ which are isomorphic to G~ are conjugate. It follows from this result that the geometric objects defined by such subgroups are equivalent to the Ehresmann E-connections.
Theory of generating functions and their applications
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 1999., 1999.) Bağrıaçık, A. İlker.; Akyıldız, Yılmaz.
The generating functions are important instruments for solving the enumerative problems in combinatorial analysis and in number theory. Enumerative problems arise when we need to be explicit about the number of ways of choosing particular elements from a finite set. The application of generating functions in this situation consists of establishing a correspondence between the elements of the set and the terms of the products of some series; the solution of enumerative problem is reduced, in fact, to finding a suitable method for the multiplication of these series.The method of generating functions can be effectively applied to enumerative problems of graph theory, that is, problems arising when counting graphs with specific properties. In number theory, the generating functions can be used to prove some identities. In this thesis, we understand the benefits of the generating functions and discuss many identities that come from 'Partitions of Integers', and 'Stirling Numbers'. We see how we can easily prove these identities by using generating functions.
Applications of graph theory to error correcting codes
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2001., 2001.) İmamoğlu, Yeşim.; Oral, Haluk.
Graph Theory has applications in many different fields, especially in combinatorics. In this study, we investigate the methods developed for obtaining error-correcting codes using graphs. First, the codes obtained from cycle and cut-set spaces of a graph are considered. After constructing the codes and giving the decoding schemes, methods for increasing the dimensions of these codes are examined. Then decoding schemes for these new codes are given. Next, a method for obtaining self-dual codes using cubic planar bipartite graphs is examined. The last method covered is to obtain perfect one error-correcting codes using some graphs that are constructed from the Tower of Hanoi Puzzle.
Integration of the deterministic functions with respect to fractional Brownian motion
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2006., 2006.) Yıldırım, Gökhan.; Eden, Alp,
In this thesis, definition and the characteristic properties of fractional Brownian motion are presented and the general idea for the integration of deterministic functions is discussed with a specific class of integrands. First, some notions and facts from probability theory are introduced. The definition and basic properties of Gaussian random variables and processes are discussed and their relation with the self similar, stationary processes is given. Moreover, covariance function of the self similar Gaussian processes with stationary increments is characterized as in Embrechts and Maejima’s book. Next, we give two representations of fractional Brownian motion. One is defined as a stochastic integral with respect to Brownian motion as in Embrechts and Maejima’s book and the other with the fractional integral as Pipiras and Taqqu do. Then we consider a class of deterministic integrands for the case H > 1/2 which is given by Kleptsyna, LeBreton and Roubaud, and we discuss its completeness. Finally, an example of a complete class of integrands for the case H < 1/2 is introduced as Pipiras and Taqqu do.
Connections between adjoint functors and limits
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2006., 2006.) Sözübek, Serdar.; Kanuni, Müge.
The concept of an adjoint functor is one of the most important concepts in category theory. Their close relation with universal arrows and limits makes them indispensable. In this thesis connections between adjoint functors and limits are explored. Firstly the general theory of adjoint functors is presented. In this respect characterization of adjunctions by universal arrows and also by units and counits are given. Secondly the notion of a limit and construction of limits by products and equalizers are presented. As the final step, general and special adjoint functor theorems are proven. These important theorems characterize the existence of a left adjoint to a functor in terms of limits and illuminate the adjoint functor-limit relation most. Also specific examples of adjunctions and applications of adjoint functor theorems in different fields of mathematics are presented.
Reciprocity law of quadratic extensions
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2006., 2006.) Tümel, Filiz.; Feyzioğlu, Ahmet K.
In the first chapter, basic definitions and results which will be used in the following chapters of this thesis are presented. In the following chapter, ideal classes and classes of quadratic forms are reviewed. Then the relationship between the ideal classes of the quadratic field Q(pD) with discriminant and the classes of quadratic forms having discriminant is established. It is proved that if two forms are equivalent, then they are constructed by two equivalent ideals and conversely equivalent ideals construct equivalent forms. The next chapter aims to present one of the proofs of the quadratic reciprocity law which is based on the theory of quadratic number fields. Instead of developing the theory of binary quadratic forms, a proof using the ideal theoretic approach is given since the relation between ideals and forms is discussed in the previous chapter. The Hilbert’s symbol for quadratic number fields is defined in this chapter and it is compared with Legendre symbol. Then genus is defined by using character sets and the quadratic reciprocity law is proved. Furthermore, the number of genera is found. The following chapter again aims to prove the quadratic reciprocity law by using the theory of quadratic number fields. But for this chapter, we will first discuss how the strict sense equivalence change the class number. Then, we will find the number of genera by using exact sequences. It is easier than the previous section since considering strict equivalence brings all cases into one case. With these results, again a proof of the quadratic reciprocity law is given. In addition, genus character and genus field with their properties is presented. In the last chapter, quadratic reciprocity law over Q(i) is presented. The proof is based on the theory of Dirichlet number fields. The relative Hilbert symbol is defined for quadratic number fields over Q(i) and the number of genera of a Dirichlet number field is found by using the parallel arguments in Chapter 4. The number of genera again leads us to prove the quadratic reciprocity law over Q(i).
Irregular sampling in shift-invariant spaces
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2006., 2006.) Özkaya, Sadık Görkem.; Eden, Alp,
This thesis is an exposition of the concept of localization of frames in the problem of irregular sampling in shift-invariant spaces. The given definition of the localization of a frame will appear to be equivalent to an off-diagonal decay of the matrix corre sponding to the frame operator. The proofs of some inverse-closedness theorems of certain classes of matrices having an off-diagonal decay will be given. These theorems imply the localization of the dual frame. Under these localization conditions, the Hilbert space theory can be extended to the family of associated Banach spaces. If the generator of a shift-invariant space satisfies necessary decay conditions, then it will be seen that its reproducing kernel frame will be a localized frame, and the theory will be applicable.
Cohomology groups of mapping class groups
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2006., 2006.) Kara, Yasemin.; Öztürk, Ferit.
The mapping class group of an orientable surface of genus g is the group of all orientation preserving piecewise linear homeomorphisms of the surface up to isotopy. In this thesis it is shown that the mapping class group of an orientable surface of genus g is generated by Dehn twists about nonseparating simple closed curves [15]. Then the notion of cohomology groups of a group is introduced following [18]. The first cohomology groups of the mapping class groups of orientable surfaces of genus g greater than one are shown to be trivial [16].
Local topological structure in the LUC compactification of a locally compact group
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Elgün, Elçim.; Budak, Talin.
In this thesis we first construct the LUC-Compactification of a topological group. We present GLUC with two different approaches, first as the set of multiplicative means on the space of LUC functions on G, and when G is locally compact, as a quotient space of the set of ultrafilters on G. Then local topological structure of GLUC is investigated and a neighborhood basis for elements of GLUC is characterized. Results on the injectivity property of multiplication on GLUC are obtained, and a special condition on G, under which injectivity property can be extended is also examined. Finally a subclass, the slowly oscillating functions, of LUC-functions is defined to decompose a special subspace of GLUC. Then the decomposition is extended to discrete cancellative semigroups.
Proof of the weinstein conjecture for overtwisted closed contact 3-manifolds
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Sağlam, Murat.; Öztürk, Ferit.
In this thesis, we study the proof of the Weinstein conjecture for 3-dimensional closed manifolds equipped with an overtwisted contact structure. The method of filling by pseudoholomorphic disks and the bubbling-off analysis are the main tools that are used in this proof.
Incidence algebras and coalgebras
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Sütlü, Serkan.; Kanuni, Müge.
The main objective of this thesis is to study incidence coalgebras and incidence Hopf algebras. To this end, we first introduced algebras and coalgebras in a unified manner and investigated the the duality between them. Then by introducing the concepts of incidence algebra and reduced incidence algebra, we studied incidence coalgebra and incidence Hopf algebra in detail. By concluding, we gave a brief discussion of Galois connection for incidence Hopf algebras.
Variational methods for nonlinear elliptic partial differential equations with nonlocal terms
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Topaloğlu, İhsan Ata.; Eden, Alp,
In this thesis, existence of standing waves for the DaveyStewartson (DS) and generalized DaveyStewartson (GDS) systems are established using variational methods. Since both the DS system and the GDS system reduce to a non-linear Schr¨odinger (NLS) equation with the only difference in their non-local term, arguments used in this thesis apply to a larger class of equations which include the DS and GDS systems as special cases. Existence of standing waves for an NLS equation is investigated in two ways: by considering an unconstrained minimization problem and a constrained minimization problem. These two variational methods apply to the GDS system as well and here the sufficient conditions on the existence of standing wave solutions for the GDS system which are imposed by these methods and the minimizers obtained are investigated in comparison.
Radicals of incidence algebras
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2007., 2007.) Çanakçı, İlke.; Kanuni, Müge.
The incidence algebra of a locally finite partially ordered set X; with the partial ordering "≤", over a ring with identity T is defined as the set of all mappings f : X x X ---T where f(x; y) = 0 for all x; y 2 X with x 6· y and denoted by I(X; T): The operations on I(X; T) are given by (f + g)(x; y) = f(x; y) + g(x; y) (f ¢ g)(x; y) = X x·z·y f(x; z) ¢ g(z; y) (r ¢ f)(x; y) = rf(x; y) for f; g 2 I(X; T); r 2 T and x; y 2 X: When the ring R is commutative, the ring I(X;R) becomes an algebra. The aim of this study is to investigate some special radicals of incidence algebras and determine the necessary and sufficient conditions characterizing elements of these radicals by using the very definition of the strong product property.
Classification of nonsymmetric Riemannian manifolds using holonomy groups
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2008., 2008.) Ferlendez, Bora.; Değer, Nihat Sadık.
In this thesis, Simons’ proof of Berger’s classification of nonsymmetric irreducible Riemannian manifolds with respect to their holonomy groups is studied and Berger’s classification is discussed. The main tools will be principal fibre bundles and vector bundles. Using them, the Ambrose-Singer theorem is investigated, which relates the geometric meaning of curvature to holonomy groups and forms the basis of Simons’ proof.
First steps into heegard floer homology
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2008., 2008.) Özlem, Semih.; Öztürk, Ferit.
Heegard Floer homology is a topological invariant for closed 3-manifolds equipped with a spinc-structure. Construction of Heegard Floer homology in the case when first Betti number is 0 is explained. The tools required in the construction are pseudo holomorphic disks, symmetric product space, Chern class, Maslov index, and spinc-structures. These tools are studied.
Almost cubic nonlinear schrödinger equation: existence, uniqueness and scattering
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2008., 2008.) Kuz, Elif.; Eden, Alp,
In this thesis, a uni ed treatment is given for a class of nonlinear non-local 2D elliptic and hyperbolic Schrödinger equation which includes the 2D nonlinear Schrödinger (NLS) equation with a purely cubic nonlinearity, Davey-Stewartson (DS) system in the hyperbolic-elliptic (HE) and elliptic-elliptic (EE) cases and the generalized Davey- Stewartson (GDS) system in the hyperbolic-elliptic-elliptic (HEE) and elliptic-ellipticelliptic (EEE) cases. Local in time existence and uniqueness of solutions are established for the Cauchy problem when initial data is in L2(R2), H1(R2), H2(R2) and in = H1(R2) \ L2(jxj2 dx) and the maximal time of existence for the solutions all agree. Conserved quantities corresponding to mass, momentum, energy are derived, as well as scale and pseudo-conformal invariance of solutions. Virial identity is also established and its relation to pseudo-conformal invariance is discussed. Various routes to global existence of solutions are also explored in the elliptic case, namely, for small mass solutions in L2(R2); in the defocusing case for solutions in H1(R2) and nally in the focusing case for H1(R2)-solutions with subminimal mass. In all such cases the scattering of such solutions in L2(R2) and topologies are discussed. Moreover, in the focusing case when initial energy is negative, it is shown that solutions in blow-up. The existence and uniqueness results are also considered for more general nonlinearities.
Some cases of generalized fermat equation
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2008., 2008.) Erdoğan, Altan.; Yıldırım, Cem Y.
This thesis presents a classification of some cases of generalized Fermat equation and applications of elementary methods to these equations in order to find integer solutions to these equations. Applications of some more recent and advanced methods for some equations where elementary methods do not work are also presented. The classification consists of three cases. The solution sets of equations in the first two cases are completely determined. The third case where Fermat equation is included still contains unsolved problems. Applications of elementary methods to some of the equations in this case and some open problems related to this case are presented in the last chapter of the thesis.
On minimal defining sets of full designs
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2009., 2009.) Demirkale, Fatih.; Yazıcı, Emine Şule.; Oral, Haluk.
A defining set of a t-(v; k; ̧) design is a subset of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subsets is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v; k; ̧) design D = (V; B) is called a full design if B is the collection of all possible k-subsets of V . Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a de ning set of the corresponding t-design. With this motivation, in this thesis, the full designs are studied when the block size is 3 and several families of non-isomorphic minimal de ning sets of full designs are given. Also, it is proven that there exists some sizes in the spectrum of the full design on v elements such that the number of non-isomorphic minimal de ning sets on each of that sizes goes to infinity as . Moreover, the lower bound on the size of the defining sets of the full designs is improved with finding the size of the smallest defining sets of the full designs on 8 and 9 points. Also, all smallest defining sets of the full designs on 8 and 9 points are classified.
On the integrability of the generalized Davey-Stewartson system
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2009., 2009.) Çolak, İlker Evrim.; Gürel, Burak.
A method developed by V. E. Zakharov and E. I. Shul'man for understanding the integrable cases of a given system of di erential equations having some certain Hamiltonian structure is represented. Then an application of this method to the Zakharov- Shul'man system which has been performed by Shul'man is explained in detail. Finally the same method is applied to the generalized Davey-Stewartson system and some conclusions on its integrability are made.
Some mean values related to Dirichlet L-functions
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2009., 2009.) Kaptan, Deniz Ali.; Yıldırım, Cem Y.
In this work, mean values of derivatives of the Riemann zeta-function and Dirichlet L-functions at the zeros of Dirichlet L-functions have been computed.

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