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Item Open Access Mathematical modeling of infectious diseases and impact of vaccination(Faculty of Engineering and Natural Sciences, 2023) Bolatova D.The world is changing and evolving these days, and the various germs and viruses that surround us are also changing and evolving. For this reason, it is essential to choose the best strategy for combating these infections while preserving the health as much as possible. This coursework uses a fundamental mathematical term in bio-mathematics which is a basic reproduction number R0. It is a significant epidemiological term illustrating the number of infected people by one sick individual, hence giving the idea of a possible epidemics. Another thing to investigate in this coursework is a numerical approximation of the epidemic model under various scenarios to examine the model from a mathematical perspective. The following project covers the following topics: vaccination strategy, stability theory, mathematical biology, modeling of epidemics, optimization problems, and Python simulation. According to the epidemic model, one of the best strategies for immunizing the populace is pulse vaccination. The Next Generation matrix approach and the Hartman-Grobman method are two linearization techniques.Item Open Access A symmetric analysis in LP space in two dimensions(Faculty of Engineering and Natural Sciences, 2023) Adilkhanova Zh.The p-Laplacian equation is a nonlinear partial differential equation of third order that arises as Euler-Lagrange equation of the gradient of function in L p norm which was first studied by Gunnar Aronsson in the late 80s [1]. Since then many explicit classical solutions and their generalisations are found. In this paper we find only the classical solutions of p-Laplacian equation in spatial dimensions, i.e. the p-harmonic functions in two dimensions. The p-harmonic functions are found by the use of Lie symmetry analysis method which deals with invariant solutions under some transformations of the solution of the partial differential equations. We obtain Lie algebra generators of the p-Laplacian equation, and the corresponding symmetry reductions of the p-Laplacian equation to ordinary differential equations. Finally, we use the Lie symmetries to construct invariant solutions of p-Laplacian that already known and some new ones in explicit form. Moreover, by using the Lie symmetries we can construct new solutions from known solutions of the p-Laplacian equation. In this article we use Lie symmetry analysis to find two-dimensional a new solution to Laplace’s p-equation. The p-Laplace equation is nonlinear partial differential equation (PDE), arising as an equation Euler-Lagrange expression of the gradient of a function in the L p norm. Using symmetry Lie, we reduce PDEs to ordinary differential equations. We find already known and new solutions of p-Laplace. It turns out using symmetric Lie analysis, we find the symmetry of the given u. We have 8 cases where we ended up getting rid of x,y,u. And in the end what remains is g and s. We were able to find the symmetry for u. In this research article, we employ the Lie symmetry analysis technique to discover two-dimensional solutions of the p-Laplacian equation. The p-Laplacian equation is nonlinear partial differential equation (PDE) that arises as EulerLagrange equation of the gradient of function in L p norm. By using Lie symmetries we reduce PDEs to ordinary differential equations. We find solutions of p-Laplacian that already known and some new. Lie symmetry analysis is a powerful mathematical tool used to investigate symmetries and simplify the solutions of differential equations. In this paper, the authors apply Lie symmetry analysis to the p-Laplacian equation, which is a nonlinear PDE that arises in the context of gradient optimization problems. iv The p-Laplacian equation can be written as: ∇ · (|∇u| p−2∇u) = 0 where ∇ represents the gradient operator and u is the unknown function. The parameter p determines the nonlinearity of the equation. By applying Lie symmetry analysis, the authors are able to identify symmetries of the p-Laplacian equation, which correspond to transformations that leave the equation invariant. These symmetries allow the authors to reduce the PDE to a system of ordinary differential equations (ODEs), which are often easier to solve. Using this approach, the authors are able to find two-dimensional solutions of the p-Laplacian equation. They also compare their results with known solutions in the literature and find some new solutions. Overall, this paper demonstrates the effectiveness of Lie symmetry analysis in simplifying and solving nonlinear PDEs, specifically the p-Laplacian equation. The identified solutions can have various applications in fields such as physics, engineering, and mathematical modeling.Item Open Access Variety of Bicommutative Algebra defined by identities(Faculty of Engineering and Natural Sciences, 2023) Akhmetova Zh.One of the key inquiries in modern algebra is the investigation of algebras that satisfy specific identities.Within the domain of polynomial identities we have 2 questions. The first is to describe an algebra by a defined identity. The second is to describe identities in algebra. The study of identities will help us in the construction of basis of free algebra, in the study of the Hilbert sequence, the Specht problem and problems of the finite basis. In this work, we used two differ- ent research methods. First, the theory of representations of symmetric groups. Second, the theory of representations of linear groups. In this research paper, we have completely described the subvariety of varirty of bicommutative alge- bras defined by the identities (ab)c + (ba)c + (ca)b + c(ba) + c(ab) + a(bc) = 0, 7[(ab)e — 2(ba)e + (ca)b] + d[e(ba) — 2c(ab) + a(bc)] = 0.Item Open Access Development of mathematical models of system for determi ing the optimal ob jects with multi-criterial parameters(Faculty of Engineering and Natural Sciences, 2019) Rayev Zh.This thesis is based on the construction of a mathematical:model for determining unknown parameters using multivariate regression analysis. Structured data are given for the derivation and elimination of significant factors and coefficients. Also, machine learning simple regression models are used for modelling. The results have been evaluated and shown for comparative purposes.Item Open Access Method of generalized Laguerre polynomials in problems with a moving boundary for the generalized heat equation(Faculty of Engineering and Natural Sciences, 2019) Yryskeldi Zh.This thesis is about solving Heat equation. exactly. generalized heat equation. which. solved using Laguerre polvnomials. The heat equation is an important partial differential equation (PDE) which describes the distribution of heat (or variation in temperature) in a given region over time It considers heat transferring in electrical contact from one side to another. Simple examples to the Generalized Heat equation are the melting of ice and the freezing of water. Solving the Heat equation, known problem in numerous industrial and technological applications. such as the manufacture of steel. ablation of heat shields. contact melting in thermal storage systems. ice accretion on aircraft. evaporation of water. Gencralized Heat equation with moving boundary have been solved by Laplace Transform. My aim is to solve the same equation by Laguerre polynomials obtained from heat polynomials.Item Open Access REALIZATION OF ALGORITHM OF SELF-GENERATING NEURAL NETWORKS(Faculty of Engineering and Natural Sciences, 2019) Abdikhaliyev Y.The development of various spheres of human activity is associated with the generation and accumulation of a huge amount of data that can contain the most important practical information. However, significant benefit from this information can be extracted only with proper processing and analysis of this data. Recently, there has been an increased interest in the field of artificial intelligence, and methods of automating the extraction of knowledge based on data mining are actively developing. Self-generating neural networks are built on the principle of biological, of course, with a number of assumptions, they have a huge number of simple processes with many connections. Like the human brain, these networks | are capable of learning. Self-generating neural networks find their application in areas such as computer vision, speech recognition, processing of natural language, etc. The thesis provides an‘analysis of the development of the theory of neural networks, their classification and mathematical formulation of the task of recognizing pattern recognition.Item Open Access Creation of effective chatbots based on neural networks(Faculty of Engineering and Natural Sciences, 2023) Yermek D.This paper presents a comprehensive exploration of the development process for a neural network-based chatbot. Given the increasing popularity of neural networks and chatbots in recent years, investigating the creation process of such a chatbot is both pertinent and valuable. The primary objective of this study is twofold: first, to construct an effective chatbot, and second, to identify the most precise model for text classification. Furthermore, the paper delves into the mathematical aspects of creating a chatbot and provides a detailed explanation of neural networks. The initial step in our project involved the creation of a database containing a sufficient number of student questions, which were then categorized into ten distinct categories. ‘To achieve this, we carefully selected the ten most commonly asked questions and generated multiple paraphrased versions of each question. These variations were employed for training and evaluating our models. During the course of our research, we identified three text classification models that proved to be the most suitable for our purposes: Multinomial Logistic Reeression, Naive Bayes, and Neural Network. We conducted extensive tests using these models and documented the results in a table, providing a comprehensive analysis of their performance. Following the successful completion of the database collection and the identification of the optimal text classification model, we proceeded to create the chatbot using DialogF low. Additionally, we integrated our chatbot into the Telegram messenger for wider accessibility and user convenience.Item Open Access Global stability of dynamical systems and Lyapunov functions(Faculty of Engineering and Natural Sciences, 2024) Aitzhanov Y.In this study, we explore the global stability of a novel epidemic model that integrates reported and unreported cases, distinguishing between symptomatic and asymptomatic individuals. Using a Lyapunov function, we demonstrate the model’s stability, highlighting the crucial role of asymptomatic cases in shaping disease dynamics and control effectiveness. Furthermore, we perform a novel hybrid parameter estimation method based on genetic algorithms, utilizing COVID19 data from the UK to better understand the distribution of reported and unreported cases in the early phases of an epidemic. In addition, we employ sensitivity analysis to understand the impact of this division on the fundamental reproduction number. Our findings underscore the importance of accounting for both symptomatic and asymptomatic cases in epidemic modeling and control strategies.Item Open Access Hypoelliptic functional inequalities and applications(Faculty of Engineering and Natural Sciences, 2023) Seitkan M.In this thesis we discuss cylindrical extensions of the improved Rellich inequalities on R* x R"-* with the Euclidean norm |- |, on R*. Actually, we show sharp remainders of the Rellich type inequalities yielding the classical Rellich inequality. Moreover, Rellich type identities and inequalities with more general weights are established. In addition, we present horizontal extensions of these results on stratified Lie groups.Item Open Access Distribution of periods of the continued fractions for quadratic irrationals(Faculty of Engineering and Natural Sciences, 2023) Tlepova M.This dissertation thesis focuses on the distribution patterns of the periods of continued fractions for quadratic irrational numbers. The main objective of this thesis is to investigate and examine the patterns and distribution of the periods of continued fractions associated with quadratic irrational numbers of the form a = nVd, where n is a positive integer and d is a square-free integer. We denote the length of the period of the continued fraction as D(a). We establish a relationship between the length of the periodic part of the continued fraction D(a) and the quadratic irrational number a = nvd by introducing an exact and explicit formula for the irrational number a for specific cases when D(a) = 2 and D(a) = 3. By utilizing the equation a = n d and considering cases where the length of the periodic part is 2, we have successfully demonstrated that when n 2 1 is the solution of the Pell’s equation in the form of n? — dz* = 1, the corresponding quadratic irrational a = nvd will indeed have a periodic part of length 2. Through the proof of this theorem, we have established that for any valuc of d greatcr than or equal to 1, there exists an infinite set of positive integers n for which the length of the period D(nv 4) is precisely 2.Item Open Access Grobner-Shirshov bases theory for Zinbiel stperalgebras(Faculty of Engineering and Natural Sciences, 2023) Meiirbek K.This thesis is a collection of 6 chapters .The Grébner-Shirshov basis is an impor- tant mathematical apparatus in algebra and commutative algebra, which is used to study and analyze polynomials and their ideals. The Grdébner-Shirshov basis has a number of important properties that make it a powerful tool for solving various algebraic problems, such as searching for ideals, solving systems of equa- tions and determining the basic invariants of polynomials. In this paper we will construct a Grobner-Shirshov basis for Zinbiel algebras. Algebra with the identity (ab) c = a(bc) + a(cb) is called the Zinbiel algebra. In the process of construct- ing the Grebner-Shirshov basis, two compositions are found and the composition lemma is proved. The method of mathematical induction is used to prove the lemma.Item Open Access Identities in mutations of bicommutative algebras(Faculty of Engineering and Natural Sciences, 2024) Ostemirova M.An algebra with identities a·(b·c) = b·(a·c), (a·b)·c = (a·c)·b is called bicommutative. In this work, we study bicommutative algebras under mutation product and prove that any bicommutative algebra under mutation product satisfies a Lie-admissible identity, which follows from two independent identities of the third degree, and we obtain all identities of the fourth degree.Item Open Access Functional inequalities on Lie groups and applications(Faculty of Engineering and Natural Sciences, 2023) Kalaman M.In this thesis we discuss sharp remainder formulae for the cylindrical extensions of the improved Hardy inequalities. For more general p we obtain cylindrical improved L?-Hardy identities for all real-valued functions f € Cg°(R"\{2x' = 0}), while in L? case we have them for any complex-valued function f € Cg°(R"\{z' = 0}). Moreover, we show cylindrical L?-Hardy inequalities for all complex-valued functions f € Cp°(IR"\{a’ = 0}). As applications, we establish Heisenberg-PaulWeyl type uncertainty principles and Caffarelli-Kohn-Nirenberg type inequalities. In particular cases, these inequalities imply new functional inequalities, which are not covered by the classical Caffarelli-Kohn-Nirenberg inequalities. Furthermore, the thesis contains L* and L? identities with logarithmic type functions on the quasi-ball B(0,R) with R > 0. In addition, we also discuss the results in the setting of homogeneous Lie groups.Item Open Access Fujita hypothesis and birational models in algebraic geometry(Faculty of Engineering and Natural Sciences, 2019) Kunanbaev A.The aim of mv research is to study Fujita hypotheses. which states that if is a smooth projektive varvety of dimension n then: (i) Assume that XV. is.a minimal (i.c.. Ay is nef) varvety of general type.(ie., Ky is big == NY > 9). Then the linear system |nvAx| is free for m > n+ 2; (ii) Let A be an ample invertible sheaf on X. Then the linear system |mJvy + (n +1) Al is free. and JnKy + (n+ 2) Al is verv ample on Y, For surfaces Fujita hypotheses was proved by Igor Reider. In my work I generalize result for X not be a sinooth projektive varyety but F-rational. So if Fujita hypotheses is true for a smooth projektive varyety then it is true for more general case F-rational. I use methods of commutative algebra and definition of tight closure.Item Open Access Numerical methods for matrix completion problem(Faculty of Engineering and Natural Sciences, 2024) Muzdybayeva G.This work addresses the significant challenge of framework completion in machi– ne learning, focusing on enhancing the accuracy and computational productivity of algorithms under conditions of large, noisy, and incomplete datasets. Central to this work are changes to two primary matrix completion techniques: Singular Value Thresholding (SVT) and Collaborative Filtering (CF). These methods were systematically progressed to handle common real-world data issues such as noise and sparsity, and were thoroughly tried over different applications, demonstrating significant execution enhancements. Through a detailed theoretical analysis, this research contributes robust frameworks for the convergence behaviors of these algorithms, giving a solid foundation for their application in practical scenarios. Improved SVT calculation, in particular, shows considerable reductions in Mean Absolute Error (MAE) and Mean Squared Error (MSE), indicating a superior performance over conventional methods. Besides, the refined CF approach presently integrates novel matrix factorization procedures, improving its utility in dynamic, personalized recommendation systems.The thesis underscores the potential of these refined algorithms in diverse fields, from advanced media to educational analytics, and sets a course for future investigate that incorporates integrating deep learning models and expanding into new data structures like tensors.Item Open Access Realization of Monoplan, NetLines, NetSphere algorithms(Faculty of Engineering and Natural Sciences, 2019) Bazatbekov B.In this work, I consider a learning algorithmis for classification tasks, called Monoplane, NetLines and NetSphere, which are all adapted for binary real input patterns. Algorithms generally helpful in classification data by self-constructing neural network, it generates new neurons to fix previous errors and stop new compilations when the output will have minimum error percent. To make realization of algorithms, that automatically construct neural networks by using appropriate methods, like backpropagation, gradient boosting, perceprtons, adaptive boosting and e.t.c To use algorithms (Monoplane, NetLines, NetSphere) to make classification of data by self-constructing neural networks.Item Open Access Categories of dialgebras(Faculty of Engineering and Natural Science, 2019) Sartayev B.A category (sometimes called an abstract category to distinguish it from a concrete category) is a set of "objects" that are connected by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.Item Open Access Adaptive traffic control(Faculty of Engineering and Natural Sciences, 2024) Aitureyeva B.Adaptive traffic control is a key aspect of modern transport systems, aimed at optimizing the flow of vehicles, increasing safety and reducing congestion on the road infrastructure. This research abstract examines the concepts, methods and technologies of adaptive traffic control and their application to improve traffic efficiency. The paper discusses the basic principles of adaptive control, including the use of real-time traffic data, traffic flow forecasting, as well as methods for optimizing traffic light control and dynamic changes in road speed limits. Modern technologies, such as smart city systems and autonomous vehicles, and their impact on the development of adaptive traffic management are also discussed. The results of the study are of practical interest to city authorities, transport organizations and engineers seeking to make traffic more efficient, safe and sustainable. During the research, an article was published in an international collection on the topic “The implementation of adaptive traffic control system for reducing traffic congestion.” Work was carried out on methodology, data collection and analysis. The experiments were carried out in a simulated environment The SUMO simulator for the intersection of K. Tulemetov and T. Utegenova streets, Shymkent. The SUMO simulator – Simulation of Urban MObility is a discretetime platform for modeling traffic flows and is intended for assessing vehicle mobility models in the context of traffic management, and simulating assessment of traffic video surveillance systems. To simulate the transport situation in the simulator, data from video cameras close to the intersection are used.Item Open Access FACTORIZATIONS AND HARDY TYPE INEQUALITIES(Faculty of Engineering and Natural Science, 2024) Apseit K.In this thesis, we derive the Hardy and critical Hardy inequalities using any homogeneous quasi-norm in a unified way. Specifically, we establish a sharp remainder formula for these inequalities. Our identity extend to Hardy and critical Hardy inequalities for the radial derivative operator with any homogeneous quasi-norm, offering enhanced versions of classical results. Our approach is based on the factorization method of differential operators introduced by Gesztesy and Littlejohn. As an application, we show Caffarelli-Kohn-Nirenberg type inequalities with more general weight. Because of the freedom in the choice of any homogeneous quasi-norm, our results give new insights already in both anisotropic R n and isotropic R n . Our results not only generalize existing inequalities but also uncover new perspectives in the study of partial differential equations and functional inequalities. These advances could have significant implications for theoretical research and applications in which anisotropic and isotropic properties are relevant.Item Open Access Mutation of nonassociative algebras(Faculty of Engineering and Natural Science, 2024) Abdrashit A.We consider bicommutative algebras under mutation products. We obtain that any bicommutative algebra under the mutation product satisfies Lie-admissible identity, which follows from two independent identities of degree three. Moreover, we obtain all identities of degree four.