Matematik BölümüMatematik Bölümühttp://hdl.handle.net/11655/222018-11-13T03:01:38Z2018-11-13T03:01:38ZSome Families of Combinatorial Matrices and Their Algebraic PropertiesArıkan, Talhahttp://hdl.handle.net/11655/51922018-10-05T11:16:06Z2018-09-01T00:00:00ZSome Families of Combinatorial Matrices and Their Algebraic Properties
Arıkan, Talha
In this thesis, we will study some properties of certain families of combinatorial matrices. While some of the families will be examined throughout this thesis are new and firstly investigated, the others are the generalizations of some of the previously known matrices. We gather our studies into six different groups. They are non-symmetric band matrices with Gaussian q-binomial entries, generalization of the super Catalan matrix, families of Max and Min matrices, a non-symmetric variant of the Filbert matrix, a nonlinear generalization of the Filbert matrix and some certain Hessenberg matrices. For all matrices will be studied except the Hessenberg matrices, we present explicit formulae for the LU-decompositions, determinants, inverse and LU-decompositions of the inverses of the matrices as well as the Cholesky decompositions when the matrix is symmetric. Additionally, we evaluate some certain Hessenberg determinants via generating function method. We use some new and existing methods to prove our claims. Particularly, we present a new method to evaluate determinants of some Hessenberg matrices whose entries consist of terms of higher order linear recursive sequences.
2018-09-01T00:00:00ZDiçatılara Genelleştirilmiş Bazı Topolojik KavramlarKorkmaz, Esrahttp://hdl.handle.net/11655/49132018-09-13T07:10:11Z2018-01-01T00:00:00ZDiçatılara Genelleştirilmiş Bazı Topolojik Kavramlar
Korkmaz, Esra
The aim of this thesis is to define the notion of diframe as a generalization of ditopological texture spaces and to study the topological concepts such as separation axioms and compactness in diframe setting. This work consists of six chapters. In the first chapter, we give a brief introduction.
In the second chapter, we present some necessary preliminaries including frame theory and ditopological texture spaces.
Chapter three is devoted to the study of coframes. We provide some new definitions and properties dual to those in frame theory.
In chapter four, we establish the category of diframes. We first provide a link between morphisms of the category drTex of texture spaces and the category frames (Frm) and then obtain a full subcategory frTex of Frm. This connection allows us to construct the category diFrm of diframes and diframe homomorphisms. In this chapher, we also give the definitions of base, subbase and subdilocale of a diframe.
In chapter five, we study separation axioms in diframes. In particular, we provide alternative characterizations of these axioms and investigate the connections between them.
The final chapter deals with the compactness and stability in diframes. This chapter is divided into two section. In the first section, we discuss the questions of whether these properties are hereditary, and whether they are preserved by any reasonable kind of homomorphisms. Since stability is a property relating the frame and the coframe parts of a diframe, we replace compactness by stability to obtain diframe versions of topological results relating separation axioms and compactness. We also give a generalization of Alexander's subbase theorem. In the second section, we introduce two main concepts, that of locally compactness and locally stability in diframes. These concepts are defined in terms of suitable binary relations whereas their bitopological versions use the notion of neighbourhood which is a point-based structure. We also show that locally compactness and locally stability are preserved by morphism satisfying appropriate conditions.
2018-01-01T00:00:00ZLineer Olmayan Kısmi Diferansiyel Denklemlerin Backlund DönüşümleriKar, Nurdanhttp://hdl.handle.net/11655/48942018-09-13T07:05:19Z2018-01-01T00:00:00ZLineer Olmayan Kısmi Diferansiyel Denklemlerin Backlund Dönüşümleri
Kar, Nurdan
In this thesis, it is studied how the B¨acklund transformations of nonlinear partial differential
equations are obtained and deriving new solutions from the known solutions by
using these transformations with the superposition formulas obtained via the permutability
conditions of the equations. Particularly, for some nonlinear partial differential
equations which are well known in mathematical physics, the importance of B¨acklund
transformations is emphasized by showing that one can derive N-solitons from 1-soliton
solutions. For this purpose, Hirota D-operator and Hirota method that are used to obtain
B¨acklund transformations are also explained. Within this context Korteweg-de
Vries (KdV), Sine-Gordon (SG) and Boussinesq equations are analyzed in detail.
2018-01-01T00:00:00ZLegendre Düğümlerinin SınıflandırılmasıPekavcılar, Bernahttp://hdl.handle.net/11655/48512018-09-27T07:18:15Z2017-01-01T00:00:00ZLegendre Düğümlerinin Sınıflandırılması
Pekavcılar, Berna
A contact structure on a 3-manifold is a maximally non-integrable 2-plane field distributed
all over the 3-manifold. There are two types of contact structures on 3-manifolds:
tight and overtwisted. Knots that are everywhere tangent to the contact planes are called
Legendrian knots. In this thesis, we study basic techniques used in the classification
Legendrian knots. The aim of this thesis is to examine the techniques used in the classification
of Legendrian knots in tight contact manifolds and the techniques used in the
classification of Legendrian knots that have tight complements in overtwisted contact
manifolds. For this purpose, in this thesis we study the classification of Legendrian
unknots in contact 3-sphere S3 in detail.
2017-01-01T00:00:00Z