Matematik Bölümü Tez Koleksiyonuhttp://hdl.handle.net/11655/3082022-05-24T08:42:36Z2022-05-24T08:42:36ZT0-metrikimsi Uzayların Simetrisizliğine Yaklaşım TeorileriJavanshır, Nezakathttp://hdl.handle.net/11655/261192022-04-04T06:19:09Z2021-12-28T00:00:00ZT0-metrikimsi Uzayların Simetrisizliğine Yaklaşım Teorileri
Javanshır, Nezakat
The aim of this thesis is to construct various original metric approach theories speci c
to the asymmetric environment for the asymmetry of T0-quasi-metrics, non-metrics
and also known as asymmetric distance functions, that is, to determine how close or
far T0-quasi-metrics are from being a metric.
In the rst chapter of the thesis, which consists of six chapters, the main ideas on
which it is based are mentioned and an introduction to the subject of the thesis is
made.
Some of the basic features of T0-quasi-metrics and various asymmetric structures
developed in this environment are reminded in the rst part of the second chapter,
after that the new results obtained from these structures are presented in the second
part. The last part of this chapter is devoted to various examples of T0-quasi-metric
spaces, are studied in detail which we will use throughout the thesis.
Considering the symmetry feature of the metric, the previously de ned symmetricantisymmetric
connectedness theories, which enable the approximation of the distances
of the points of the T0-quasi-metric space to each other, through the symmetricantisymmetric
paths established with the other points between these points, form the
basis of this thesis. In the third chapter of the thesis, rstly the details of these theories
are reminded, and in the second part, new results and examples that we have obtained
within the framework of the relevant theories are mentioned.
In the fourth chapter, as another original work; the theories of symmetric and
iii
antisymmetric connection extensions are established for a T0-quasi-metric space. In
particular, it is proved that every bounded T0-quasi-metric space has a symmetrically
connected one-point extension, and every metric space has an antisymmetrically connected
one-point extension. Also, \Does every T0-quasi- metric space have an antisymmetrically
connected extension?" question is investigated, and the positive answers are
given to this question by means of (counter)examples as well as theorems involving
di erent conditions.
As another new approach to asymmetry, the topological approach is discussed in
the fth chapter. In this framework, local symmetric and local antisymmetric connectedness
theories, which are natural localizations of symmetric and antisymmetric
connectedness theories according to the symmetrization topology of T0-quasi-metric,
are constructed. All the properties of locally (anti)symmetrically connected spaces such
as their relations with other structures, their inheritance in subspaces, products, etc.
have been investigated in detail in the rst two subsections, and many useful results
have been reached with the help of examples.
In the last part of the fth chapter, asymmetric norm theory, which is a milestone in
their development in asymmetric topology by producing T0-quasi-metrics, is considered
as another alternative working environment in order to approach to the asymmetry of
T0-quasi-metrics.
The thesis is completed with the last chapter, in which the ndings obtained in the
thesis and open questions that could be the subject of future study are presented.
2021-12-28T00:00:00ZCombınatorıal Solutıons For Consensus Algorıthms And Blockchaın ShardıngJameel, Marwanhttp://hdl.handle.net/11655/260862022-04-04T06:30:08Z2021-01-01T00:00:00ZCombınatorıal Solutıons For Consensus Algorıthms And Blockchaın Shardıng
Jameel, Marwan
The scalability problem in blockchain technology seems to be the essential issue to be solved. It is known that the choice of a compromised algorithm is critical for the practical solution of this important problem. Usually, Byzantine Fault Tolerance (BFT)
methods based on the public blockchain networks have been most widely applied to solve scalability.
In this thesis, we formulate two possible cases to scale the blockchain. Instead of the frequently used proof-of-work or stake methods to form the consensus committee, allowing BFT-based methods, we propose a new model. This new model calculates the reputation value for the nodes that want to join the leader (trust) committee using particle swarm optimization (PSO). It is a computational method for optimizing a problem by improving a candidate solution against a specified quality metric. It solves the problem by populating the search space, so-called particles and moving these particles around according to a simple mathematical formula over the particle's position and velocity. To discard the misbehaving nodes from the trust leader committee, new nodes with high reputation values are selected. Since this study focuses on creating the consensus committee, a simulation test the proposed model more effectively. The results show that the proposed model successfully selects the nodes with high confidence to the consensus committee instead of the malicious nodes.
To select an updated trustworthy committee and then allow all network users to join at any time to protect the blockchain network's security is in general insufficient. However, suspicious nodes must be avoided at all costs. We utilize a straightforward strategy inspired by bio-dynamic systems to deflect the trust committee's focus from the assaulting nodes. Removing poorly-tailored nodes increases the selection of honest nodes or participants. We propose an unsupervised machine learning to solve the current challenge by applying a Grey Wolf Optimization (GWO) technique.
In addition, blockchain studies have recently been splitting the blockchain to address the scalability problem focused on sharding.
Sharding is a helpful technique for exploring fundamental computational challenges in blockchain technology, such as consensus, Byzantine fault tolerance, and self-stabilization. The sharding method creates a small, segmented blockchain network. Rather than creating a more extensive network, networks with fewer nodes are established. Additionally, successful sharding can be applied to various areas, resulting in significantly speedier processes. Our solution will give a safe and dependable use of blockchain components by analyzing the system and fitting the shard size using the Topological Data Analysis (TDA) with the help of an unsupervised machine learning technique. In order to achieve our goal, the Linear Programming Problem (LPP) is constructed and solved using the Dual-Simplex approach to determine the best shard size.
Additionally, we segmented the blockchain network using our system. The test results show that reputation values boosted the parties' reliability. Then the likelihood of any piece collapsing and harming the entire blockchain decreases.
2021-01-01T00:00:00ZHalka Yapısının Sonlu Sıfırlanan Modüller Üzerinde BelirlenmesiÇağlar, Deniz Halimhttp://hdl.handle.net/11655/256372021-11-26T07:17:19Z2021-01-01T00:00:00ZHalka Yapısının Sonlu Sıfırlanan Modüller Üzerinde Belirlenmesi
Çağlar, Deniz Halim
This thesis is based on work on modules that satisfy the H-condition, also known as "finitely
annihilated modules" in the theory of modules on unitary rings. Modules that satisfy
the H-condition have taken an important place in ring theory and attracted attention by many
mathematicians because of their emergence and effective use in topics such as Homological
Algebra and localization in non-commutative rings. The H-condition, believed to have been
proposed by P. Gabriel [8] in the literature, allows a transition between the structure of the
ring and the structure of the module on it. The purpose of this thesis is to reveal the structure,
examples and importance of finitely annihilated modules, to examine the ring structure
consisting of finitely annihilated modules on some module classes.
The first chapter of this thesis, which consists of five chapters, consists of information
about the historical development and importance of the thesis topic. The second chapter
includes the basic definitions and theorems required in the next chapters. In the third chapter,
finite annihilated modules are defined and the basic properties they provide are examined.
In the fourth chapter, Artinian Rings are characterized by being finite annihilated of each
module on it, and the concept of "weak H-condition" is defined. In the last chapter, the effects
of semisimple modules, uniform modules, and injective modules to satisfy the H-condition
on the ring structure are examined.
Keywords: Ring, Module, Finitely Annihilated Module, H-condition, Artinian Ring, Semisimple
Module, Uniform Module, Injective Module, Singular module
2021-01-01T00:00:00ZSimitli Çeşitlem Üzerinde Üzerinde Parametrik Kodlar Ve Sıfırlayan İdeallerBaran Özkan, Esmahttp://hdl.handle.net/11655/255432021-11-04T11:17:45Z2021-01-01T00:00:00ZSimitli Çeşitlem Üzerinde Üzerinde Parametrik Kodlar Ve Sıfırlayan İdealler
Baran Özkan, Esma
Let X be a complete simplicial toric variety over a finite field with a split torus T_X.
This thesis is on parameterized codes obtained from the subgroups of the torus T_X
parameterized by matrices. It is very important to find the generators of the vanishing
ideals of these subgroups to compute basic parameters of these codes. In the introduction
part of the thesis, the significance and a literature review of toric codes are given.
The results obtained in the thesis are summarized. The second chapter includes some
background of affine varieties required for the affine toric varieties.
In the third chapter, after defining torus, the concepts of character and one parameter
subgroup of a torus are presented, and are associated with lattices. After giving the
definition of toric variety, the different constructions of affine toric varieties are explained.
The basic topics of rational polyhedral cones are given and, their connections its
with affine toric varieties is explained.
In the fourth chapter, by gluing affine varieties with isomorphisms, abstract varieties
other than affine or projective varieties are constructed. This chapter starts with
projective varieties for a better understanding of these varieties. How to glue affine toric
varieties corresponding to elements in a finite collection of strong rational polyhedral
cones, called fan, is described, and so general toric varieties are constructed. The main
and final purpose of this section is to show that the points of a general toric variety
can be expressed with homogeneous coordinates as in projective space.
For a given matrix Q, denote by T_{X,Q} the subgroup of the torus T_X parameterized by
the columns of Q. In the fifth chapter, 3 algorithms are given to determine a generating
set of the vanishing ideal of T_{X,Q}. Elimination theory is used in the first algorithm
developed. A Macaulay2 code is written to implement the algorithm. Another method
for finding the generators of the same ideal using the base of the lattice describing this
ideal is obtained. An algorithm for finding the lattice L such that I(T_{X,Q}) = I_L and a
procedure implementing this algorithm in the Macaulay2 program is presented. Thus,
it is easily checked whether the vanishing ideal I(T_{X,Q}) is a complete intersection or not.
In this section, finally, a method for conceptually determining the lattice L is obtained
and a Nullstellensatz Theorem is proven on a finite field under some conditions.
The sixth chapter constitutes the heart of the thesis and includes parameterized
codes constructed by calculating homogeneous polynomial functions in the set T_{X,Q}.
For this purpose, firstly, basic topics of linear codes are explained. Since the dimension
of parametric code C_{\alpha_Q} is calculated with multigraded Hilbert function of toric set
T_{X,Q}, some properties of multigraded Hilbert functions are given. Using parametric
definition of the T_{X,Q}, an algorithm directly computing the number of elements of the
subgroup T_{X,Q} which is equal to the length of the code and a lower bound for the
minimum distance of the code is obtained. As an application, the basic parameters of
the parameterized codes obtained from the torus of the Hirzebruch surface are calculated.
Finally, examples illustrating the advantage of passing from projective space to
arbitrary toric variety, in addition to working with parameterized toric set T_{X,Q} instead
of the torus T_X are given.
2021-01-01T00:00:00Z