Matematik

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A mathematical model of drug delivery by modifying the tumor microenvironment: MMP activation
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2013., 2013.) Kaba, Duygu.; Gürel, Burak.; Ünlü, Mehmet Burçin.
Cancer modeling has become one of the challenging frontiers of applied mathematics mostly for two decades. Drug delivery to solid tumors also attracts both theorists and experimentalists due to its signi cance in cancer therapy. Recent studies have revealed that drug responses of tumor cells are determined both by intrinsic characteristics and by regulation of tumor microenvironment. The irregularity tumor blood vessel structure results in disorganized blood flow within the tumor and increased leakage which causes increased interstitial fluid pressure (IFP). This IFP forms a barrier for drug transport and it could be seen as an apparent obstacle to the delivery of therapeutic molecules. The basis of this thesis is the e ects of the delivery of matrix metalloproteinases (MMPs) to the tumor tissue, which in turn, increased hydraulic conductivity, improved interstitial transport and enhanced distribution of nanotherapeutics. In the light of Mok et. al.'s researches, a mathematical model which associates the effect of MMPs to convective transport and to drug distribution in tumors is constructed. Governing equations in the model with suitable boundary conditions involve the principles for transvascular and interstitial drug transport and conservation laws. Finally, these equations are discretized with nite element method (FEM) and the simulation results are discussed comparing to the literature.
A number theoretical approach to polynomials over finite fields
(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2023., 2023) Girgin Öztürk, Neslihan.; Bassa, Alp, 1982-
Let q be a prime power and Fq be the finite field with q elements. The explicit constructions of irreducible polynomials over Fq of high degree is one of the main problems in the arithmetic of finite fields which has many applications in several areas such as coding theory and cryptography. In general, some recursive methods are preferred to do these constructions using rational transformations. In particular, we are interested in methods that are obtained by using quadratic transformations. For doing this, we will first classify and normalize the rational transformations of degree 2 using the behaviour of the ramified places in the corresponding rational function field extensions over the finite field Fq. Then we will investigate the constructions using Galois theory and some basic observations in group theory. This approach helps to better understand the iterative constructions and gives various generalisations of them. It also enables to determine the requirements put on the initial polynomials.
A study on the Kadison-Singer problem
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2010., 2010.) Arslan, İlker.; Tanbay, Betül.
Let H be a separable Hilbert space and B(H) be the space of all bounded linear operators on H. A state of a C -algebra is a positive linear functional of norm 1. An extreme point of the set of states is called a pure state. The Kadison-Singer problem asks whether every pure state of the space of the diagonal operators on H extends to a unique pure state or not. In this thesis, after understanding the Kadison-Singer problem, the article "A note on the Kadison-Singer problem" is discussed. This article concludes an interesting result that these extensions either lie in a nite dimensional subspace or contains a homeomorphic copy of N.
A survey on endo - trivial modules
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2018., 2018.) Sert, Gözde.; Coşkun, Olcay.
In this thesis, we investigate endo-trivial modules and their classiﬁcation endo trivial modules over p-groups. Endo-trivial modules were introduced by Dade [1] and appear naturally in moduler representation theory. Several contributions towards the general aim of classifying those modules have already obtained and the classiﬁcation of endo-trivial modules was completed in 2004 by Carlson and Thevenaz. Our aim is to see what are the endo-trivials modules over p-groups, especially over abelian groups.
ABC conjecture and its implications
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Koç, Tuğçe.; Özman, Ekin.
In this thesis, our aim is to state the importance of abc conjecture and prove the strong results we obtain with the help of abc conjecture. First, we give necessary notions and tools which are used throughout the thesis. Then we introduce Hall conjecture, Fermat’s last theorem and Mordell conjecture, and their relations with abc conjecture. In particular, we give the eﬀective proof of Mordell conjecture using abc conjecture, given in the article of Noam Elkies, [1], and also get another height bound by combining with a diﬀerent theorem. Finally, we give three examples where we use both of the height bounds. This thesis was supported by T¨UB˙ITAK Project 117F274.
Algebraic overtwised contact structures on 3-sphere
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2020., 2020.) Karadereli, Şeyma.; Öztürk, Ferit.
It is known that all of the complex analytic singularity links and the associated Milnor open books on the 3-sphere correspond to a single contact structure, which is the unique tight structure of the 3-sphere. The main question of this thesis is whether the overtwisted contact structures on the 3-sphere are real algebraic. We will de ne the notion of real algebraicity in the introduction of the thesis. We explicitly construct a family of real algebraic multilinks in the 3-sphere which are the bindings of planar Milnor open book decompositions supporting overtwisted contact structures. Furthermore, we prove that all the overtwisted contact structures with non-negative 3-dimensional invariants are obtained in this family.
Algebro-geometric solutions of the kadomtsev-petviashvili equation
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2014., 2014.) Çiçek, Fatma.; Gürel, Burak.; Boysal, Arzu.
I. M. Krichever suggested a method to solve nonlinear partial di erential equations in the form of the Zaharov-Shabat Equation [L {u100000} @y;A {u100000} @t] = 0 where L and A are di erential operators including derivatives only with respect to the x variable in 1976 [1]. The method uses so called Baker-Akhiezer functions on Riemann surfaces and provides periodic and conditionally periodic solutions to such nonlinear equations that can be expressed in terms of the so called Riemann -function, a -function de- ned on some n dimensional complex space where the Riemann matrix of the function corresponds to a Riemann surface. In this thesis, we will mainly consider the Kadomtsev-Petviashvili equation (or KP equation) 3 4 uyy = @ @x ut {u100000} 1 4 (6uux + uxxx) which is an example of the Zaharov-Shabat equation. Following the expository paper of B. A. Dubrovin [2], we will present the construction of such solutions to the KP equation given as u(x; y; t) = 2 @2 @x2 log (xU +yV +tW +z0)+c. It was observed that this construction allows one to investigate the su cient conditions on arbitrary vectors U, V , and W that make the above function u(x; y; t) a solution to the KP equation. We explain the answer to this question for Riemann surfaces of small genera, and mention the result for more general Riemann surfaces which are both given in [2].
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.
An application of ergodic theory to Geometric Ramsey theory
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2009., 2009.) Şimşek, Sevim.; Demir, Selçuk.; Budak, Talin.
In this thesis we first present two examples from Geometric Ramsey theory in R2. Then we generalize these results to higher dimensions and construct our main theorem. Then we translate our geometric problem into dynamical form. Finally we prove the main theorem by using methods from ergodic theory.
An application of ergodic theory to Szemerédi's theorem
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2009., 2009.) Vurgun, Demet.; Demir, Selçuk.; Işık, Nilgün.
In this thesis, Szemerédi's theorem has been translated into the ergodic problem. For this purpose, ergodic theory and its tools has been studied. The ergodic version of the theorem is equivalent to Furstenberg Multiple Reccurence Theorem. So the structure of the ergodic systems has been analyzed. Finally, ergodic theoretical proof of the theorem has been given.
An optimal change of variables scheme for single scattering problems
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2015., 2015.) Eruslu, Hasan Hüseyin.; Ecevit, Fatih.
In this work we are concentrated on the direct obstacle scattering problem for convex bodies in two dimensions. In order to calculate the scattered eld, we rst need to compute the normal derivative of the total eld on the object's surface. This quantity is the unique solution of a combined eld integral equation which we solve using Galerkin method wherein the approximation spaces depend on the wave number and the geometry of the scatterer. We are particularly focused on the large wave numbers in which the solution has highly oscillating behavior. In order to analyze this solution accurately, we separate the highly oscillating part of it and then study the derivatives of the acquired function. This derivative study gives us the information about the smoothness of the solution and an idea about how to approximate it. As for the geometry of the scatterer, we divide the boundary of the object into subregions regarding where we expect high oscillations. In each region, in order to achieve improved approximations, we choose di erent polynomial bases. In various scenarios, we examine the polynomial bases such as monomial, Lagrange, and Chebyshev. As the wave number increases, in order to obtain better results one needs to formulate these approximation spaces with higher polynomial degrees. However, it includes enormous computational cost and the condition numbers of Galerkin matrices elevate dramatically. The goal of this research is to optimize the choice of approximation spaces so as to improve accuracy of numerical solutions while keeping the number of degrees of freedom independent of frequency, and reduce the condition numbers of the related Galerkin matrices.
Analysis of convergent integral equation methods for high-frequency scattering
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2012., 2012.) Keserci, Samet.; Ecevit, Fatih.
The main aim of this thesis is to devise numerical methods for the solution of high-frequency scattering problems in 2 dimensional settings by utilizing geometrical optics ansatz and asymptotic properties of solutions for convex obstacles (see [1]). To this end, we formulate the sound soft scattering problem as a well-posed boundary integral equation. Among the numerical methods (Nyström, collocation, Galerkin, two-grid and multi-grid) appropriate for solving integral equations, we focus on the classical but e cacious ones, namely the two- and multi-grid methods. We rst portray the defect correction principle for integral equations of the second kind which constitutes a basis for the two- and multi-grid methods, then we de ne both methods over the defect correction iteration. We also set up these methods to compute the scattering return by the unit circle numerically and compare theoretical and numerical results. By virtue of the geometrical optics ansatz, which expresses the normal derivative of the total eld as a highly oscillating complex exponential modulated by a slowly oscillating amplitude, we construct a new Galerkin method well adapted to the slowly oscillating nature of the unknown function which we approximate by polynomials. We hereby eliminate the serious drawbacks arising from high oscillations for approximating the solutions. As our main convergence result will display, our new algorithm entails that it su ces to increase the degrees of freedom proportional to k (for any > 0) in order to preserve a given accuracy. In contrast with the previous e orts on the problem, we construct our local approximation spaces with particular emphasis on the transition regions to capture the boundary layers around shadow boundaries and utilize approximation spaces in the deep shadow region to incorporate the e ects of grazing rays.
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.
Asymptotically anti-de Sitter spacetimes in three dimensions
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2021., 2021.) Deral, Ceren Ayşe.; Değer, Nihat Sadık.
In this thesis, we reviewed several aspects of asymptotically anti-de Sitter (AAdS) spacetimes in three dimensional Einstein gravity by following some important historical work. Starting with a brief introduction to anti-de Sitter (AdS) spacetimes where also the BTZ black hole solution is given we de ned Noether-Wald charges using Noether theorems. Next, we compared di erent de nitions of AAdS spacetimes. Here, we adopted the Fefferman-Graham coordinates and solved Einstein equations order by order to prove that the Fefferman-Graham expansion of AAdS spacetimes terminates at second order in three dimensions, as first shown by Skenderis and Solodukhin. Lastly, we considered two sets of boundary conditions and presented their asymptotic symmetry algebras and charge algebras. Imposing Brown-Henneaux boundary conditions we arrived at Banados metric, which is the most general metric for AAdS spacetimes under these conditions. Then we showed that the asymptotic symmetry algebra is two copies of the Virasoro algebra. Under the Compere-Song-Strominger boundary conditions, we calculated the most general metric and showed the charge algebra is a semidirect sum of Virasoro and Kac-Moody algebras. We concluded with some comments and future research directions.
Banach uzaylarında izdüşüm kavramının bir genelleştirilmesi; |lineer olmayan denklem ve varyasyonel eşitsizlikler ile ilgili bazı sonuçlar
(Thesis (Assoc. Prof.)- Bogazici University. Institute for Graduate Studies in Social Sciences, 1982., 1982.) Ülger, Ali.
Bazı matris gruplarının kohomolojisi hakkında
(Thesis (Assoc. Prof.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 1980., 1980.) Küsefoğlu, Ayşe.
Bernstein center for GL(n, Qp)
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2010., 2010.) Seviniş, Mert.; Demir, Selçuk.; Kanuni, Müge.
In this thesis we will present J. N. Bernstein's description of the centre of the category of smooth modules over l-groups for the case of GL(n;Qp). Firstly we decompose this category into two parts: quasi-cuspidal and non-quasi-cuspidal components. Then we analyse the center of each component separately. In the rst part, we follow the line of reasoning in the article \Le Centre de Bernstein" for the general case of l-groups. In the second part, we will analyse the centre of non-quasi-cuspidal components for the special case of GL(n;Qp).
Cascading behavior in infinite networks
(Thesis (M.S.)-Bogazici University. Institute for Graduate Studies in Science and Engineering, 2012., 2012.) Özdemir, Alperen Yaşar.; Eden, Alp,
The aim of this master thesis is to analyze the underlying mathematical structure of the infinite network models of cascading behavior. Graph theoretical tools are essential to understand the structure of the network and game theoretical tools are employed for the dynamics of the model. It is tried to determine under what conditions on the structure of the graph or on the parameters of the game, cascading is possible. We also consider the optimization problem of choosing the initial set from which cascading behavior spreads through the network. For this purpose, we use the theory of submodular functions. Submodularity condition provides close approximations to the optimal value when the initial set is selected by Greedy Algorithm.
Central-firing of type A2n with initial weight 0
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2020., 2020.) Cin, Uğur.; Boysal, Arzu.
In this thesis a variant of the chip- ring game introduced by Hopkins, McConville and Propp in [1], called the labeled chip- ring on Z, is studied. We will rst explore the basic properties and examples of this game. We will then show, how one can see this game as a binary relation on the weight lattice of Type A root system. It is then a natural step to generalize it to other root systems, which was done by Galashin, Hopkins, McConville and Postnikov in [2] and [3]. After reviewing the basics of central- ring introduced in these papers, we examine the central- ring of type A2n with initial weight 0 in Chapter 5. Finally, we study the restrictions in Lemma 12 of [1] in more detail, and conjecture that the number of permutations with maximum number of inversions allowed by this lemma is given by the Catalan numbers.
Class number one problem
(Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Kır, Harun.; Özman, Ekin.
In 1801, Gauss conjectured that there are exactly nine imaginary quadratic number ﬁelds with class number one, namely: Q(√−1), Q(√−2), Q(√−3), Q(√−7), Q(√−11), Q(√−19), Q(√−43), Q(√−67) and Q(√−163). This conjecture is wellknown as class number one problem. In 1952, K. Heegner ﬁrst solved the problem and he showed that Gauss was right about the assumption in Diophantische analysis und modulfunktionen. In this thesis, we will present a modern approach to the proof of Heegner as in D.A.Cox’s book, Primes of the Form x2 + ny2.

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