Dept. of Mathematics - Master's degree

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https://hdl.handle.net/11693/13897

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Open Access
Synchronization of Kuramoto model with anticipatory agents
(Bilkent University, 2023-08) Dönmez, Bengi
This thesis investigates a system of coupled Kuramoto oscillators on undirected networks comprising of anticipatory agents that try to predict the future states of their neighbors and adjust their states accordingly. The prediction is done using the past behavior of the neighbors and leads to a set of coupled delay differential equations. The study reveals that the anticipatory behavior leads to the emergence of multiple phase-synchronized solutions characterized by distinct collective frequencies and stability properties. An exact criterion for the stability of the phase-synchronized states is derived. It is shown that the system can exhibit multi-stability, where different phase-synchronized solutions can be observed depending on the initial conditions. It is further proved that bipartite graphs can exhibit anti-phase solutions and an exact condition for their stability is provided. Investigation of cycle graphs yields further frequency-synchronized states, in various clustered patterns, depending on the system’s parameter values.
Open Access
Pattern-avoiding permutations: The case of length three, four, and five
(Bilkent University, 2023-06) Akbaş, Zilan
A shorter permutation of length k is said to appear as a pattern in a longer per-mutation of length n if the longer permutation has a subsequence of length k that is order isomorphic to the shorter one. Otherwise, the longer permutation avoids the shorter one as a pattern. We use Sn(τ) to denote the set of permutations of length n that avoid pattern τ. Pattern avoidance induces an equivalence relation on the pattern set Sk. For ρ, τ ∈ Sk, we define the equivalence relation as follows: ρ ∼W τ if and only if |Sn(ρ)| = |Sn(τ)| for all n ≥ 1. The equivalence classes of this relation are called Wilf classes. The main questions are determining the Wilf classes of Sk and enumerating each class. We first study the Wilf classification and enumeration of each class for S3 and S4. We then present some new numerical results regarding the Wilf classification of pairs of patterns of length five. We define a Wilf class as small if it contains only one pair and big if it contains more than one pair. We show that there are at least 968 small Wilf classes and at most 13 big Wilf classes.
Open Access
The method of inverse scattering and Bäcklund transformations
(Bilkent University, 2023-05) Masur, Lütfiye
Solving integrable nonlinear partial differential equations by the method of inverse scattering and Bäcklund transformations are very effective. Both of these methods depend on the existence of Lax pair. The Bäcklund transformations provide us some particular solutions of the integrable nonlinear partial differential equations but the method of inverse scattering solves the initial value problem of these equations. In this work, we use these methods to solve the Korteweg-de Vries equation, the Korteweg-de Vries hierarchy and AKNS system.
Open Access
Martingale representation theorem for diffusion in infinite dimensional spaces and applications
(Bilkent University, 2023-03) Aydın, Uğur
We show that square integrable martingales adapted to the filtration generated by a weak solution of a stochastic differential equation driven by a cylindrical Wiener process on a separable real Hilbert space that has the weak uniqueness property has a martingale representation driven by the martingale part of the stochastic differential equation.
Open Access
Modeling the latent period in compartmental epidemiological models
(Bilkent University, 2023-01) Dizaji, Parvaneh Faraji
Two epidemiological models for the latent period are studied: the SEIR model and the SIR model with delay. Stability analysis of the endemic and the disease-free equilibria is performed for both models, and the basic reproduction number is determined. Using the meningococcal disease datasets of four countries, the best-fitted disease transmission rate is estimated by the least squares method, and the corresponding basic reproduction numbers are calculated for each country.
Open Access
One level density of Hecke L-functions associated with cubic characters at prime arguments
(Bilkent University, 2023-01) Kavalcı, Cazibe
We study the one-level density of low-lying zeros of a family of L-functions associated with cubic Hecke characters defined over the Eisenstein field. We show that this family of L-functions satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in (−1, 1), under the Generalized Riemann Hypothesis.
Open Access
Haar systems on locally compact groupoids
(Bilkent University, 2022-07) Güleken, Ayşe Işıl
Haar systems are generalizations of Haar measures on groups to groupoids. Naturally, important research directions in the field try to generalize the well known existence of a Haar measure on a locally compact group to the existence of Haar systems in different groupoid settings. The groupoid case differs significantly from the group case, evidenced by a result of Deitmar, showing that non-existence is possible even for compact groupoids. We first present the classical theory of locally compact groups and Haar Measures on them. We motivate our investigation by constructing full C∗-algebras on locally compact groups, which uses the existence of Haar measures. Then, we cover the theory of locally compact groupoids and present Renault's result that provides a complete characterization of the existence of Haar systems for the r-discrete locally compact groupoid setting, which are precisely the ones where the range map is a local homeomorphism. We present a question from Williams that investigates if the open range map assumption is redundant for second countable, locally compact and transitive groupoids. Finally, we present Buneci's counter-example that answers this question in the negative.
Open Access
Extremal problems on Bergman spaces A¹α and Besov spaces
(Bilkent University, 2022-05) Balcı, Alper
Extremal problems in different function spaces have long been investigated. Ferguson provides a method, using Bergman projections, to solve certain types of extremal problems in Bergman spaces for 1 < p < ∞ in his work [3]. Later the method is extended to weighted Bergman spaces for 1 < p < ∞ in [13]. Now, we extend this method to the p = 1 case. The two cases differ in the structure of Bergman projections and dual spaces. First, we define some function spaces, namely weighted Bergman spaces, the Bloch space, and Besov spaces, and show the usage of Bergman projection on these spaces. Then, we find some conditions to ensure the existence of unique solutions for extremal problems. Later, we use Bergman projection to find a candidate function for the solution in the p = 1 case, and we prove that the candidate function is the solution if it never attains the value 0. Finally, under special conditions, we solve a similar problem in Besov spaces.
Open Access
Finite p-subgroups of Sp(n)
(Bilkent University, 2021-09) Çetin, Mustafa Seyyid
Hikari [1] classiﬁes all ﬁnite p-subgroups of simple algebras, and Banieqbal [2] classi-ﬁes the ﬁnite subgroups of 2×2 matrices over a division algebra of characteristic zero. In this thesis, we give a new proof for the classiﬁcation of the ﬁnite p-subgroups of 2 × 2 matrices over the quaternionic algebra. Using this classiﬁcation, we classify ﬁ-nite p-subgroups of the symplectic group Sp(2). More precisely, for every prime p, we deﬁne a denumerable family of p-subgroups of Sp(2) so that every ﬁnite p-subgroup of Sp(2) lives inside one of the members of this family. To give this classiﬁcation, we proved general results for Sp(n) whenever possible.
Open Access
The geometry of sheaves on sites
(Bilkent University, 2021-01) Parsizadeh, Pejman
In this work, we study doing geometry on sheaves on sites. Categories of our sites consist of objects that are building blocks for a given geometry. Generalized spaces then will be sheaves on these sort of sites. Next we introduce the notion of varieties, and show the relationship between certain class of varieties known as diﬀeologies with the category of smooth manifolds. Along the way, the notion of schemes will be generalized as a variety on symmetric monoidal categories. And we show how a diﬀerential geometric construction on a site can be translated to a construction on generalized spaces.
Open Access
Lebesgue constants on cantor type sets
(Bilkent University, 2020-09) Paksoy, Yaman
The properties of compact subsets of the real line which are in the class of Bounded Lebesgue Constants (BLC) are investigated. Knowing that any such set must have 1-dimensional Lebesgue measure zero and nowhere density, and the fact that there are examples of countable sets both inside and outside of the class BLC, families of Cantor-type sets were focused on. Backed up by numerical experiments (up to degree 128) and analytical results, the conjecture “there exists no perfect set in BLC” was put forward.
Open Access
Constructions and simplicity of the Mathieu groups
(Bilkent University, 2020-08) Karakaş, Mete Han
Of the 26 sporadic finite simple groups, 5 were discovered by E. Mathieu in 1861 and 1873 [1], [2]. These Mathieu groups are the focus of this thesis, where we will prove their simplicity using elementary methods. E. Witt [5] realized a connection between the Mathieu groups and certain combinatorial structures known as Steiner systems. We will follow his construction to define the Mathieu groups as the automorphism groups of certain Steiner systems. Much of the work of the thesis lies in the construction of these Steiner systems, which we achieve by using both methods from finite geometry and the theory of Golay codes.
Open Access
Mackey decomposition for Brauer pairs
(Bilkent University, 2020-08) Okur, Utku
For a ﬁnite group G and an algebraically closed ﬁeld k of characteristic p, a k-algebra A with a G-action is called a G-algebra. A pair (P,c) such that P is a p-subgroup of G and c is a block idempotent of the G-algebra A(P)is called a Brauer pair. Brauer pairs form a reﬁnement of the G-poset of p-subgroups of a ﬁnite group G. We deﬁne the ordinary Mackey category B of Brauer pairs on an interior p-permutation G-algebra A over an algebraically closed ﬁeld k of characteristic p. We then show that, given a ﬁeld K of characteristic zero and a primitive idempotent f ∈ AG, then the category algebra of Bf over K is semisimple.
Open Access
Analysis on self-similar sets
(Bilkent University, 2020-08) Kesimal, Hayriye Sıla
Self-similar sets are one class of fractals that are invariant under geometric similarities. In this thesis, we study on self-similar sets. We give the deﬁnition of a self-similar set K and present the proof the existence theorem of such a set. We deﬁne the shift space. We deﬁne a relation between the shift space and K. We show the self-similarity of the shift space. We deﬁne overlapping set, critical and post-critical set for a self-similar set. We give the characterization of K by the periodic sequences in the shift space. We give the notion of a self-similar structure and deﬁne a self-similar set purely topologically. We give its local topology. We deﬁne isomorphism between selfsimilar structures so that we can have a classiﬁcation of self-similar structures. We point out that the critical set for a self-similar structure provides us with a characterization for determining the topological structure of a self-similar structure. We deﬁne the notion of minimality for a self-similar structure and give a characterization theorem for investigating the minimality of a self-similar structure. We deﬁne a post-critically ﬁnite self-similar structure.
Open Access
Effect of time delays on the convergence speed of consensus dynamics
(Bilkent University, 2020-01) Alhassan, Mohammed Kamil
We discuss consensus problems under time delays. The presence of time delays results in an inﬁnite-dimensional system rather than a system of ordinary differential equations. It has been shown that information transmission delays do not inﬂuence whether the system converges to a consensus value; however, further eﬀects of delays are unknown. We show that time delays in most graphs decreases the convergence speed; while somewhat surprisingly, they can improve convergence in certain special graphs. We discuss the structure of graphs for which such improvement is possible.
Open Access
Which algebraic K3 surfaces doubly cover an enriques surface: a computational approach
(Bilkent University, 2019-02) Yörük, Oğuzhan
The relationship between K3 Surfaces and Enriques Surfaces is known to mathematicians for the last 30 years. We examined this relationship from a lattice theoretical point of view by looking at transcendental lattice of a K3 surface in the case of Picard number 18 and 19. We established a better way of attacking this problem with the help of a computer assistance.
Open Access
Characteristic bisets and local fusion subsystems
(Bilkent University, 2018-09) Tokmak, Mustafa Anıl
Fusion systems are categories that contain the p-local structure of a finite group. Bisets are sets endowed with two coherent group actions. We investigate the relation between fusion systems and bisets in this thesis. Fusion systems that mimic the inclusion of a Sylow p-subgroup of a finite group are called saturated. Similarly, if S is a Sylow p-subgroup of G, then G regarded as an (S, S)-biset has special properties, which make it a characteristic biset for the p-fusion of G. These two concepts are linked in that a fusion system is saturated if and only if it has a characteristic biset. We give a proof for this result by following the work in [1] and [2]. Fusion systems have a notion of normalizer and centralizer subsystems, mimicking the notion for finite group theory. This thesis reviews a proof by Gelvin and Reeh [3] of a result of Puig [2] asserting that normalizer and centralizer fusion subsystems of a saturated fusion system are saturated. This result comes from the connection between saturation of fusion systems and the existence of characteristic bisets.
Open Access
Which algebraic K3 surfaces cover an enriques surface
(Bilkent University, 2018-09) Sonel, Serkan
We partially determine the necessary and su cient conditions on the entries of the intersection matrix of the transcendental lattice of algebraic K3 surface with Picard number 18 (X) 19 for the surface to doubly cover an Enriques surface.
Open Access
The pandemic fusion system for endomorphism algebras of p-permutation modules
(Bilkent University, 2018-09) Nika, Andi
During the 1980's Puig developed a new approach to modular representation theory, introducing new p-local invariants and thereby extending Green's work on G-algebras. We investigate the Puig category, commenting on its local structure and then introduce a new notion, namely pandemic fusion, which extends the Puig's axioms globally on the G-algebra. Finally we give a sketch of the proof on the existence of some p-permutation FG-module realizing the minimal pandemic fusion system.
Open Access
Chain maps between Gruenberg resolutions
(Bilkent University, 2018-07) Fidan, Müge
Let G be a finite group. For a given presentation of G = hF|Ri, Gruenberg gives a construction of a projective resolution for Z as a ZG-module. This resolution, which is called Gruenberg resolution, only depends on the ideals IF := ker{ZF → Z} and J := ker{ZF → ZG} (see [1]). We write standard resolution as a Gruenberg resolution by following the construction of Gruenberg [2]. We get an explicit chain map formula between Gruenberg resolution for standard presentation and the Gruenberg resolution for the usual presentation of a cyclic group. Then we write an explicit chain map formula between any two Gruenberg resolutions. We also give some calculations with Gruenberg resolution.