LEE- Matematik Mühendisliği Lisansüstü Programı

Bu topluluk için Kalıcı Uri

http://hdl.handle.net/11527/19382

Gözat

Classical yang-baxter equationfrom duality covariant formulation of string theory

(Graduate School, 2024-01-12) Çırak Tunalı, Seçil ; Özer, Aybike ; 509152210 ; Mathematical Engineering

The aim of the thesis is to study the homogeneous Yang-Baxter (YB) deformation proposed in the physics literature for a generic Green-Schwarz sigma model from a geometric point of view. It has been shown that these kind of deformations are generated by a certain kind of non-constant O(d,d) transformation, called β transformation, which acts as solution generating transformations in string theory. We study the construction of such an O(d,d) transformation from a bi-vector field related to the Poisson structure on the manifold. It is a well-known fact that there is a Lie algebroid structure on the cotangent bundle of the manifold when there is a Poisson structure on the manifold. Moreover, this Lie algebroid structure is compatible with the standard Lie algebroid structure on the tangent bundle, so that there is a Courant algebroid structure on the direct sum of the tangent and cotangent bundle (called the generalized tangent bundle) of the manifold. We also study Courant algebroid structures in order to understand and to generalize the transformation and the YB deformation. Given a Lie algebra with a non-degenerate inner product, if there exists an endomorphism R, which satisfies the classical Yang-Baxter equation (CYBE), then the direct sum of the Lie algebra and its dual has a natural Drinfel'd structure. Such an endomorphism can be extended to the tangent bundle of the integral Lie group by the help of the adjoint action. In this way, an automorphism called the dressed R-matrix can be constructed, which satisfies the CYBE since the adjoint action is an automorphism of the Lie bracket. It is possible to build a Poisson bi-vector field on the manifold from the dressed R-matrix. It can be shown that the Schouten-Nijenhuis bracket of the bi-vector field with itself vanishes following directly from the fact that the dressed R-matrix satisfies CYBE. The Lie algebroid structure on the cotangent bundle induced from the Poisson structure is compatible with the standard Lie algebroid structure on the tangent bundle. Then the tangent and cotangent bundles with the stated Lie algebroid structures form a Lie bialgebroid, which is an example of a triangular Lie bialgebroid. The Drinfel'd double of the resulting triangular Lie bialgebroid is a Courant algebroid with transversal Dirac structures. This geometrical structure plays a prominent role in the solution generating mechanism stated above. The dynamical fields in the universal sector of the low energy effection action of string theory are the Riemannian metric, a 2-form field called the B-field and a scalar field called the dilaton field. The first two of these fields become the constituents of the generalized metric, which is a tensor on the generalized tangent bundle T M ⊕ T^{∗}M that transforms naturally under O(d,d). There is a O(d,d) covariant version of string theory, called Double Field Theory (DFT), which is written in terms of the generalized metric and the generalized dilaton field. DFT provides a suitable framework to demonstrate the fact that YB deformation preserves the solutions of string theory. From a geometric point of view, the existence of a generalized metric is equivalent to the existence of a subbundle of the generalized tangent bundle on which the inner product is positive definite. If one starts with a generalized metric of a specific form that solves the field equations of DFT in the limit in which it reduces to the field equations of supergravity and transforms it with the O(d,d) matrix generating the YB deformation, the resulting generalized metric also solves the field equations of DFT in the same limit. In the physics literature, the proof of this is based on comparing the "fluxes" before and after the transformation and showing that these fluxes do not change. From a geometrical point of view the fluxes are just the "structure functions" of the Courant algebroid structure on the generalized tangent bundle, when a specific basis is chosen for the sections of tangent and cotangent bundles. In order to understand this "flux preservation" principle from a geometrical point of view, we also study the axioms defining a Courant algebraid in local coordinates. We also work out in detail the case where the anchor of the Courant algebroid is determined by a bi-vector field associated by the YB deformation.
Elgamal algoritması ve arnold dönüşümüne dayalı biyometrik görüntü kriptolojisi

(Lisansüstü Eğitim Enstitüsü, 2023-09-03) Ünlü, Rabia ; Bilir Çivi, Gülçin ; 509201241 ; Matematik Mühendisliği

Görüntü şifreleme, modern iletişim ve güvenlik alanlarında önemli bir rol oynayan kritik bir teknolojidir. Hassas verilerin, özellikle insan yüzlerinin güvenli bir şekilde saklanması, aktarılması ve doğrulanması, kişisel gizliliğin korunması ve yetkisiz erişime karşı savunmanın sağlanması açısından hayati öneme sahiptir. Bu çalışmada, kişisel verilerin güvenli bir şekilde paylaşılması, veri bütünlüğünün ve gizliliğinin sağlanması için etkili bir homomorfik şifreleme yöntemi olan ElGamal dönüşümü ile Arnold dönüşümünün entegrasyonundan oluşan bir görüntü şifreleme algoritması önerilmektedir. Arnold Dönüşümü, eşit uzunluk ve genişlikteki piksel noktalarından oluşan bir görüntünün piksellerinin konumları üzerinde birden çok matris işlemi gerçekleştiren, klasik kriptografik sistem temelli bir görüntü şifreleme algoritmasıdır. Bu dönüşüm, dijital görüntünün karıştırılıp, tanınmaz bir hale getirilmesini sağlamaktadır. NxN boyutundaki görüntünün piksellerinin x ve y koordinatlarını değiştirerek bir karışıklık oluşturur ve orijinal görüntünün okunmasını zorlaştırır. Yeni x ve y koordinatları, orijinal x ve y koordinatları üzerinde bazı matematiksel işlemler uygulanarak hesaplanır. Bu işlemler, piksellerin yatay konumlarını değiştirerek görüntünün şifrelenmesini sağlar. Arnold dönüşümü, geniş çapta kullanılan önemli bir görüntü şifreleme tekniği olmasına rağmen, güvenlik zayıflıklarına sahiptir ve her boyuttaki görüntü verilerine uygulanması zordur. Bu zayıflıkları aşmak için, mevcut çalışmalar Arnold dönüşümüne entegre edilebilecek çeşitli yaklaşımlar önermektedir. ElGamal şifreleme sistemi ise, 1985 yılında Taher Elgamal tarafından geliştirilen ve Diffie-Hellman anahtar değişimi prensibine dayanan bir genel anahtarlı kısmi homomorfik şifreleme algoritmasıdır. Asimetrik anahtarlı kriptografi kullanarak güvenli bir şifreleme yöntemi sunar. Özel anahtar şifreleme işlemi için gizli tutulurken, genel anahtar çözme işlemi için kullanılır ve genel olarak erişilebilir. Şifreyi oluşturan kişinin şifrelemeyi inkar edemeyeceği bir doğruluk seviyesi sağlar. Bu nedenle sadece şifreleme değil, aynı zamanda görüntü doğrulama için de uygundur. Bu çalışmada, öncelikle simetrik, asimetrik ve hibrit görüntü şifreleme tekniklerinin kapsamlı bir incelemesi yapılmıştır. Daha sonra literatüre giren görünü şifreleme çalışmaları dikkate alınarak, biyometrik görüntülerin, 2D Arnold dönüşümü kullanılarak karıştırılması ardından ElGamal algoritmasıyla şifrelenmesi ve tersine işlem ile orijinal görüntünün elde edilmesi problemi ele alınmıştır. Önerilen yaklaşımı özel kılan nokta Arnold dönüşümü sonrası şifrelenmiş biometrik görüntünün deşifresinin ayrık logaritma hesaplamasının zorluğuna dayalı olmasıdır. Örneklerle açıklanan yaklaşım ile biyometrik görüntülerin depolanması, iletilmesi ve doğrulanması sırasında olabilecek saldırılara karşı etkili ve güvenli olan bir hibrit görüntü kriptolojisi hedef alınmıştır.
Euclid uzaylarındaki hiperyüzeylerin Gauss tasvirinin tipleri ve Cheng Yau operatörü

(Lisansüstü Eğitim Enstitüsü, 2022) Kaya, Furkan ; Turgay, Nurettin Cenk ; 708762 ; Matematik Mühendisliği Ana Bilim Dalı

Chen ve Piccini tarafından ortaya konan "$ \mathbb{E}^{m} $ Euclid uzayının bir alt manifoldunun Gauss tasviri alt manifoldu ne ölçüde belirler?" probleminden sonra sonlu tipten Gauss tasvirine sahip alt manifoldların analizi çok aktif bir araştırma konusu haline gelmiştir. Şimdiye kadar bu probleme bazı faydalı kısmi çözümler sunulmuştur. $ \mathbb{E}^{m} $ Euclid uzayının $ n $ boyutlu bir $ M $ alt manifolduna, eğer $ x $ konum vektörü $ \Delta $ Laplace operatörünün özvektörlerinin sonlu bir toplamı olarak ifade edilebilirse sonlu tiptendir denir. Dolayısıyla $ M $ alt manifoldunun sonlu tipten olması için, $ x=x_0+x_1+x_2 \cdots +x_n$ olmalıdır. Burada $ x_0 $ sabit tasvir ve $ x_1,x_2,\hdots,x_n $ ise $\lambda_i \in \mathbb{R} $ olmak üzere $i=1,2,\hdots,k $ için $ \Delta x_i=\lambda_ix_i$ şartını sağlayan sabit olmayan tasvirlerdir. Eğer $ \lambda_1,\lambda_2,\hdots,\lambda_k $ özdeğerleri birbirinden farklı ise $ M $ alt manifoldu $ k $-tipindendir denir. $ M $, Euclid uzayının bir hiperyüzeyi olsun. Benzer şekilde bir $ \psi: M^{n}\xrightarrow{}E^{n+1} $ düzgün fonksiyonuna, eğer $ M $ hiperyüzeyinin Laplace operatörünün $ k $ tane ayrık özdeğerine karşılık gelen özvektörlerin toplamı olarak yazılıyorsa, $ k $-tipindendir denir. Eğer böyle bir $ k $ değeri varsa, $ \psi $ fonksiyonuna sonlu tiptendir denir. Yukarıda verilen tanımdan dolayı $ M $ hiperyüzeyinin 1-tipinden Gauss tasvirine sahip olması için gerek ve yeter şartın $$ \Delta G=\lambda(G+C) $$ diferansiyel denkleminin bir $ \lambda \in \mathbb{R} $ özdeğeri ve $ C $ sabit vektörü için sağlanması olduğu elde edilir. $ \mathbb{E}^{3} $ Euclid uzayındaki düzlemler, dik silindirler ve küreler 1-tipi Gauss tasvirine sahip yüzeylerdir. Euclid uzayındaki sonlu tipten alt manifoldlar pek çok geometrici tarafından çalışılmış ve önemli sonuçlara ulaşılmıştır. Halen de bu konu ile ilgili pek çok açık problem bulunmakta ve bu açık problemler çözülmeye çalışılmaktadır. Bu problemlerin bazıları da hiperyüzeylerin Gauss tasvirleri ile ilgilidir. Günümüze kadar pek çok geometrici Euclid uzaylarındaki hiperyüzeylerin Gauss tasvirlerinin üzerine çalışmıştır. Diğer taraftan, Euclid uzayındaki bir $ M $ manifolduna, $ G $ Gauss tasviri $$ \Delta G=f(G+C) $$ denklemi düzgün bir $ f$ fonksiyonu ve bir $ C $ sabit vektörü için sağlanırsa, noktasal 1-tipinden Gauss tasvirine sahiptir denir. Eğer bu denklem $ C=0 $ için sağlanırsa Gauss tasviri birinci çeşit noktasal 1-tipinden; $ C\neq0 $ için sağlanırsa ikinci çeşit noktasal 1-tipindendir denir. Örneğin, $ \mathbb{E}^{3} $ Euclid uzayındaki helikoit, katenoid ve dik koni noktasal 1-tipinden Gauss tasvirine sahip yüzeylerdir. Son senelerde bu kavramlar genişletilerek genelleştirilmiş 1-tipinden Gauss tasvirine sahip alt manifold tanımı verilmiştir. Euclid uzayındaki bir $ M $ manifoldunun $ G $ Gauss tasviri $$ \Delta G=f_1G+f_2C $$ denklemi $ f_1,f_2 $ düzgün fonksiyonları ve bir $ C $ sabit vektörü için sağlanırsa genelleştirilmiş 1-tipinden Gauss tasvirine sahiptir denir. Örneğin, $ \mathbb{E}^{3} $ Euclid uzayındaki tüm dönel yüzeyler genelleştirilmiş 1-tipinden Gauss tasvirine sahiptir. Bu tez çalışmasında $ \mathbb{E}^{3} $ uzayındaki yüzeylerin Gauss tasvirlerinin tiplerine göre sınıflandırılmaları ile ilgili bazı teoremler çalışılmıştır. Üçüncü bölümde Cheng-Yau operatörüne göre noktasal 1-tipinden Gauss tasvirine sahip sabit ortalama eğrilikli ve sabit esas eğrilikli yüzeyler ile ilgili bilinen sonuçlar ayrıntılı bir şekilde açıklanmıştır. Sonra Weingarten yüzeyleri incelenmiştir. $ \mathbb{E}^{3} $ Euclid uzayındaki doğrusal Weingarten yüzeyinin Cheng-Yau operatörüne göre ikinci çeşit noktasal 1-tipinden Gauss tasvirine sahip olması için bu yüzeyi düzlemin açık bir parçası olması gerektiği gösterilmiştir. Dördüncü bölümde ise $ \mathbb{E}^{3} $ Euclid uzayındaki minimal yüzeylerin Cheng-Yau operatörüne göre genelleştirilmiş 1-tipinden Gauss tasvirine sahip olması için bazı teoremler elde edilmiştir. Ayrıca, helikal yüzeyler incelenmiş ve $ \mathbb{E}^{3} $ Euclid uzayındaki bir helisoidal yüzeyin $ \square $ noktasal 1-tipinden Gauss haritasına sahip olması için gerek ve yeter şartın o yüzeyin bir dönel yüzey olması veya sabit Gauss eğriliğine sahip olması gerektiği gösterilmiştir.
Exact soliton solutions of cubic nonlinear Schrödinger equation with a momentum term

(Graduate School, 2024-12-26) Uzunoğlu, Haldun Taha ; Akar Bakırtaş, İlkay ; 509221207 ; Mathematics Engineering

There are various interconnections between the positive sciences, with differential equations serving as a fundamental bridge linking mathematics to other scientific disciplines. Nonlinear wave phenomena have recently gained considerable attention due to their theoretical significance and applied relevance. Nonlinear optical wave equations not only facilitate the development of advanced techniques but also play a crucial role in elucidating natural phenomena across diverse fields, including biology, nonlinear optics, and quantum physics. Among these, solitons—localized nonlinear waves—stand out as valuable tools for understanding complex nonlinear systems. Solitons are widely studied in areas such as plasma physics, nonlinear optics, and quantum mechanics. Optical solitons, in particular, have drawn significant interest due to the inherently interdisciplinary nature of soliton theory, making it a pivotal topic for advancing technologies like high-speed data transmission. The external potential strongly influences the shape and stability of optical pulses. In quantum mechanics and nonlinear optics, potentials with parity-time symmetry (PT -symmetry) are frequently utilized. Numerous studies in the literature examine the stability of nonlinear Schrödinger (NLS) equations with PT -symmetry. These equations admit various nonlinear wave solutions, including solitons, which are localized waves that propagate without distortion. Solitons demonstrate remarkable resilience during collisions, retaining their properties even after interacting with other waves. This work investigates the soliton solutions and their stability in an NLS equation incorporating a momentum term and cubic nonlinearity under an external PT -symmetric potential. The governing equation is expressed as: iu_(z) +αu_(xx) −iΓu_(x) +φ|u|^2u+V_(PT) u = 0. Here, z denotes the scaled propagation distance, u is the differentiable complex-valued slowly varying amplitude, u_(xx) represents diffraction, Γ is the momentum term taken as a constant, and V_(PT) denotes the external potential. The PT -symmetric potential is defined as: VPT = V(x) +iW(x)=V0 +V1sech(x) +V2sech^2(x)+i[W0sech(x)tanh(x) +W1tanh(x)]. Here, V(x) and W(x) represent the real and imaginary components of the potential, where V(x) is an even function and W(x) is an odd function. A detailed introduction to solitons and their interdisciplinary significance is provided in Chapter 1. The NLS equation is introduced, along with its recent developments, including the momentum term and PT -symmetry. The chapter also outlines the research objectives and the thesis hypothesis, emphasizing the importance of the momentum term in the NLS equation. Chapter 2 describes the Spectral Renormalization (SR) Method, an iterative Fourier technique used to numerically solve the NLS equation with a momentum term and a PT -symmetric potential. The method is adapted to the problem at hand, and numerical solutions are obtained. In Chapter 3, the structure of the NLS equation without potential is analyzed to investigate the effect of the momentum term. Variations in soliton structures are examined in relation to changes in the momentum term coefficient, Γ, and the propagation constant, µ. Chapter 4 explores exact solutions of the NLS equation with a momentum term and PT -symmetric potential. Using the ansatz u(x,z) = f(x)e^i(µz+g(x)), where f(x) and g(x) are real-valued functions, analytical solutions are derived. These solutions are compared with the numerical results, which shows excellent agreement. The chapter also verifies the parity-time symmetry properties of the potential, confirming that its imaginary part is odd and its real part is even. Chapter 5 focuses on the stability analysis of soliton solutions. The Split-Step Fourier method is employed to investigate nonlinear stability, while linear stability is examined through the linear spectrum. The results indicate that the solitons become unstable with even slight increases in the momentum term coefficient, Γ. Additionally, enhancing the complex component of the potential increases instability, whereas increasing the real component improves stability. The acquired results are summed up in Chapter 6. Moreover, a brief discussion on potential future research is included. All numerical results were obtained using MATLAB2023®.
Manifolds of generalised G-structures in string compactifications

(Graduate School, 2023-03-22) Diriöz, Emine ; Özer, Aybike ; 509162201 ; Mathematical Engineering

A G-structure on a differentiable manifold M of dimension n can be described as a reduction of the linear frame bundle L(M) of M to a Lie subgroup G of $GL(n,\mathbb{R})$. Such a reduction is equivalent to the existence of certain geometric structures on M, depending on what the subgroup G is. For example, an O(n)-structure corresponds to the existence of a Riemannian metric g. Similarly, by the existence of an almost complex structure J, the structure group reduces to $GL(n/2,\mathbb{C})$. If a Riemannian metric and an almost complex structure are compatible and the metric is hermitian then the structure group reduces to SU(n/2). In a similar fashion, a generalized G-structure can be described as a reduction of the structure group of the principal bundle associated with the generalized tangent bundle $TM\oplus T^*M$. The natural structure group of $TM\oplus T^*M$ is O(n,n). The generalized G-structures also correspond to the existence of certain geometrical objects. For example, the reduction of the structure group from O(n,n) to $O(n)\times O(n)$ corresponds to the existence of a generalized metric. Similarly, on an even-dimensional real manifold $M$ a generalized almost complex structure is given by a reduction of the structure group from O(n,n) to U(n/2,n/2). A generalized almost complex structure is defined by the existence of a pure spinor which is a section of the exterior bundle $\bigwedge^\bullet T^* M$. The SU(n/2,n/2)-structure is equivalent to the existence of a globally defined pure spinor of non-vanishing norm. Furthermore, $SU(n/2)\times SU(n/2)$-structure is given by the existence of two compatible pure spinors. The main theme of this thesis is the study of manifolds of generalized G-structure relevant to string compactifications. Superstring theory is a quantum theory of gravity consistent in 10 dimensions. There are five consistent superstring theories and the low energy dynamics of massless space-time fields are governed by ten-dimensional supergravity theories. The supergravity field equations are nonlinear partial differential equations that can be regarded as a generalization of field equations of Einstein's theory of general relativity (GR). In a supersymmetric compactification of Type II string theory down to 4 dimensions, it is required that the structure group of the generalized tangent bundle $TM \oplus T^*M$ of the six-dimensional internal manifold M is reduced from SO(6,6) to $SU(3) \times SU(3)$. This is equivalent to the existence of two globally defined compatible pure spinors $\Phi_1$ and $\Phi_2$. Furthermore, these pure spinors should satisfy certain first-order differential equations, namely supersymmetry equations. We show that these equations are covariant under certain Pin(d,d) transformations. We also show that Non-Abelian T-duality (NATD) which is generated by a coordinate-dependent Pin(d,d) transformation is a particular solution generating transformation for these pure spinor equations. Our method is demonstrated by studying the NATD of a specific class of geometries with SU(2) isometry and SU(3)-structure. Some of the manifolds belonging to this class are $AdS_5\times T^{1,1}$, $AdS_5\times Y^{p,q}$ and $AdS_5\times S^5$. It is interesting to note that in each case, the internal manifold is a Sasaki-Einstein manifold. We show that the transformed pure spinors are associated with an SU(2)-structure. The plan of the thesis is as follows: in section 2, we study principal fiber bundles, vector bundles, and linear frame bundles. Then, we study the concept of the reduction of the structure groups. We also give familiar examples of G-structures in detail. In section 3, we briefly review the relation between G-holonomy and torsion-free G-structures. In section 4, we study the basic concepts regarding the geometry of the generalized tangent bundle $TM\oplus T^*M$. This leads us to the definition of a generalized G-structure. Since our main interest is in $SU(3)\times SU(3)$-structures we give in a separate subsection the description of $SU(3)\times SU(3)$-structures and the associated pure spinors in detail. In section 5, we focus on the differential equations to be satisfied by the pure spinors for preservation of ${\cal{N}}=1$ supersymmetry. We study the covariance of these equations under constant and non-constant Pin(d,d) transformations. Then, we study Non-Abelian T-duality (NATD) transformations in detail, and we show the invariance of pure spinor equations under NATD. In section 6, we consider a specific class of geometries. We transform the pure spinors associated with the SU(3)-structure and show that the resulting pure spinors determine an SU(2) structure. We also study the NATD transformation of the metric, the B field, and the Ramond-Ramond fields.
On geodesic mappings of Riemannian manifolds

(Graduate School, 2022-01-07) Çoraplı, Ahmet Umut ; Canfes, Elif ; 509181210 ; Mathematical Engineering
Parameter optimization for mathematical modeling

(Graduate School, 2023-06-09) Tunçel, Mehmet ; Duran, Ahmet ; 509132057 ; Mathematical Engineering

Mathematical modeling is used to explain and forecast complex systems, and parameter optimization methods have a crucial role to find the optimal set of parameters obtained by minimizing an objective function. Also, the management of computational resources is essential for handling big models in real-time scenarios. A. Duran and G. Caginalp (2008) propose a hybrid parameter optimization forecast algorithm for asset prices via asset flow differential equations. In this thesis, we propose a new mathematical method for an inverse problem of parameter vector optimization in asset flow theory. For this purpose, we use quasi-Newton (QN) and Monte Carlo simulations to optimize the function F[K] for each selected event and initial parameter vector. We present grid and random methods and conclude that the grid approach is better than the random approach in the unconstrained optimization problem. This study also presents a parallel numerical parameter optimization algorithm for dynamical systems used in financial applications. It achieves speed-up for up to 512 cores and considers more extensive financial market situations. Moreover, it also evaluates the convergence of the model parameter vector via nonlinear least squares error, and maximum improvement factor. In this thesis, we also examine the performance, scalability, and robustness of OpenFOAM on the GPGPU cluster for bio-medical fluid flow simulations. It compared the CPU performance of iterative solver icoFoam with direct solver SuperLU_DIST 4.0 and hybrid parallel codes of MPI+OpenMP+CUDA versus MPI+OpenMP implementation of SuperLU_DIST 4.0. Results showed speed-up for large matrices up to 20 million x 20 million. Besides that, we investigate the usage of eigenvalues to examine the spectral effects of large matrices on the performance of scalable direct solvers. Gerschgorin's theorem can be used to bound the spectrum of square matrices, and behaviors such as disjoint, overlapped, or clustered Gerschgorin circles can give clues. We define the minimum number of cores and show that it depends on the sparsity level and size of the matrix, increasing slightly as the sparsity level decreases and the order increases. In sum, this thesis presents new methods for initial parameter selection and a new algorithm for parallel numerical parameter optimization. Also, we define new metrics and show that the importance of right matching for computational systems and the optimal minimum number of cores are important in mathematical modeling and simulation.
Some results on the sums of unit fractions

(Graduate School, 2025-03-26) Altuntaş, Çağatay ; Yareneri, Ergün ; Göral, Haydar ; 509182201 ; Mathematics Engineering

A unit fraction is a rational number having $1$ in its numerator and any positive integer in its denominator. This thesis is devoted to the investigation of various aspects of sums of unit fractions, a topic that covers a wide range of problems and techniques. An elementary example of such sums is the harmonic numbers. Given a positive integer $n,$ the $n^{th}$ harmonic number is defined as $$H_n = 1 + \frac{1}{2} + \dots + \frac{1}{n}.$$ We begin by presenting a generalization of the harmonic numbers called the Dedekind harmonic numbers. In order to define them, we take a number field $K$ and then consider the sum of reciprocals of norms of ideals of $\mathcal{O}_K$, the ring of integers of this number field $K$, whose norms are bounded by a given positive integer $n$. We first show that these numbers are not integers after a while. Then, we provide this specific upper bound for some quadratic number fields to guarantee that they are non-integer. Furthermore, under the Riemann hypothesis, we obtain the non-integerness of differences of these numbers together with uniform bounds for quadratic number fields and derive an asymptotic result. We then continue with another example of the sums of unit fractions called the hyperharmonic numbers. In his paper, Mez\H o proposed that these numbers are never integers, except for the trivial case $1$, and this conjecture remained unresolved for an extended period. Another question is also asked in the same paper: Can two hyperharmonic numbers of different indices and different orders be equal? A partial answer to a more generalized version of this question is given in this thesis, via a geometric approach with the help of related problems in arithmetic geometry. Afterwards, an analytic approach is followed and we deduce that the differences of distinct hyperharmonic numbers are almost never an integer. For any given prime number $p$, the set denoted by $J(p)$ was introduced by Eswarathasan and Levine. This set consists of the indices of the harmonic numbers whose numerators are divisible by this prime $p$ in their lowest terms. The size of this set for several prime numbers was calculated by several authors and some upper bounds for a counting function for this set were given. We generalize the set $J(p)$ to the generalized harmonic numbers. The generalized harmonic numbers are sums of unit fractions where they have some positive integer power $s$ of the positive integers in their denominators. We define the generalizations $J(p,s)$ and $J(p^s,s)$ of $J(p)$, deduce some finiteness results, provide congruence relations and eventually obtain an upper bound for the counting function for $J(p,s)$. Moreover, we provide an explicit criterion that implies the finiteness of our set, together with computational results, and then point out the subjects that may reveal more about the finiteness of $J(p,s)$, by introducing Bernoulli and Euler numbers together with the irregular primes.
The generalized fractional Benjamin Bona Mahony equation: Analytical and numerical results

(Lisansüstü Eğitim Enstitüsü, 2021) Oruç, Göksu ; Mihriye Muslu, Gülçin ; Borluk, Handan ; 692763 ; Matematik Mühendisliği

In this thesis study we consider the generalized fractional Benjamin-Bona-Mahony (gfBBM) equation u_t+ u_x + \frac{1}{2}(u^{p+1})_x+ \frac{3}{4}D^{\alpha} u_{x}+ \frac{5}{4}D^{\alpha} u_{t}=0, where $x$ and $t$ represents spatial coordinate and time, respectively. This equation is derived to model the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The gfBBM equation has a general power-type nonlinearity and two fractional-type terms. Thanks to these properties, the gfBBM equation is noticed as a satisfactory and interesting model in the literature. The aim of this thesis study is to perform various mathematical and numerical analyses for the gfBBM equation and to understand the influence of nonlinearity and fractional dispersion on the dynamics of solutions. The thesis study is organized in the following way: In the first chapter, we briefly introduce the general background on the fractional type nonlinear partial differential equations with lower dispersion such as fractional Korteweg de Vries (fKdV) and fractional Benjamin-Bona-Mahony (fBBM) and gfBBM equations. Then, we propose derivation and some properties of the gfBBM equation. We also state the analytical and numerical methods used to solve this equation. Furthermore, the literature overview on gfBBM and related equations is given in this chapter. The second chapter is devoted to the analytical results for the gfBBM equation. In the first section of this chapter we recall the preliminaries. This section contains useful definitions related to functional analysis, lemmas and theorems used in the thesis. In the second section, we derive conserved quantities of the gfBBM equation. We also find constraints on the order $\alpha$ of the fractional term. The aim of the third section is to prove the local well-posedness of the Cauchy problem for the gfBBM equation together with the initial condition u(x,0)=u_0 (x). For the case $1 \leq \alpha \leq 2$, we prove the local well-posedness of the solutions by using contraction mapping principle. On the other hand, for the case $0 < \alpha < 1$, we use the approaches given for the fBBM equation by He and Mammeri (2018). Therefore, we consider the regularization of the Cauchy problem for the gfBBM equation and then use the convergence of regularized solutions to the solutions of main problem. The section 4 presents the conditions for the non-existence of solitary wave solutions to the gfBBM equation. Existence and uniqueness of solitary wave solutions are obtained by using the result of Frank and Lenzmann (2013). We also consider the restrictions on the $\alpha$ and speed of wave $c$ so that the gfBBM equation admits positive or negative solitary waves. Finally, we derive exact solitary wave solutions to the gfBBM equation for the special cases $\alpha=1$ and $\alpha=2$ when $p=1$. In the last section of this chapter we discuss the stability properties of solitary wave solutions associated to the gfBBM equation. We first give the Hamiltonian formulation of the equation. Then, we prove the orbital stability of solitary wave solutions by using approach given by Grillakis Shatah Strauss (GSS) (1987) and for the stability we obtain following conditions when $1 \leq p \leq 4$: 1) $\frac{p}{p+2}<\alpha < \frac{p}{2}$ and $c>c_{1,p}>1$, 2) $\frac{p}{2}<\alpha < 2$ and $c>1$ or $\frac{3}{5}>c>c_{2,p}$, with $c_{1,p}=\frac{6\alpha + 2p + 3 \alpha p + \sqrt 2 p \sqrt{2 \alpha - p + \alpha p} }{5(2 \alpha + \alpha p)}$ and $c_{2,p}=\frac{6\alpha + 2p + 3 \alpha p - \sqrt 2 p \sqrt{2 \alpha - p + \alpha p} }{5(2 \alpha + \alpha p)}$. In the last chapter, we present the numerical results for the gfBBM equation. We first state efficient numerical algorithms for gfBBM equation and then carry out various numerical experiments. The Petviashvili method is proposed for the generation of the solitary wave solutions that cannot be obtained analytically. We numerically investigate the effects of the relation between the nonlinearity and the dispersion on the solutions. The evolution of generated wave profiles in time is investigated numerically by Fourier pseudo-spectral method. The efficiency of the methods will be demonstrated by various numerical simulations.

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