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Item Open Access Development of a mathematical model of the system for collecting and analyzing data(Faculty of engineering and natural sciences, 2019) Suleizhan T.This thesis is aimed at developing practical skills of building mathematical svstems for data collection and analvsis. The peculiarity of the work is to learn to self-decision-making. to develop research skills. which is especially iniportant in a dynamic world. Data collection is carried out. in almost all areas of science and technology. Over the past few vears. data collection techniques have been applied to applications and new corporations have emerged to commercialize the technology. Many business and financial problems are successfully solved with the use of big data. Problems of management, classification. pattern recognition, forecasting, inherent in almost all application areas. such as medicine. military Affairs. aviation and space, construction, are increasingly solved with the use of this technology. In this regard. vou will see a fundamental understanding of the basic concepts and models of big data, as well as learn how to apply this knowledge - in practice.Item Open Access Analysis and the development of a mathematical model of the children mortality(Faculty of engineering and natural sciences, 2019) Zhumabek D.The mortality rate depends on many different factors: the socio-economic development of the country, the environmental situation. the well-being of the population. the level of stress and much more. After fertility. it takes the second place in its importance in the processes of reproduction of the population. has a serious impact on the population size. its structure. and is closely interconnected with all socio-demographic processes. The causes of mortality in Kazakhstan are classified by the main groups: infectious discases. diseases of the respiratory system. circulatory system, neoplasms. accidents, poisonings and injuries. Mortality of the population is a mirror reflection of the level of socio-economic development of society.Item Open Access Integral power balance method in heat problems with free boundary(Faculty of engineering and natural sciences, 2019) Kassabek DinaOne of the important areas of application of the free boundary problems is the mathematical modelling of phenomena in the low-temperature plasma of an electric arc and in contacts of electrical devices. Analysis of solutions makes it possible to verify the obtained theoretical results, to test the effectiveness of the developed algorithms for specific evolutionary processes in electrical apparatuses, and to interpret the available experimental data. The evolution of contact bridge and arcing processes is so fast (nano- and microsecond range) that their experimental study is very difficult. In some cases, only mathematical modeling can give an idea of their dynamics. Thus, the need for modeling is required not only for optimization of the experiment, but also due to the impossibility of using a some different approach. One of the most effective methods of solving heat problem is the method of heat potentials, which reduces the initial boundary value problems to integral equations. However, in the case of regions degenerating at the initial time, additional difficulties arise releted to the singularity of these integral equations. These difficulties are compounded in the case when an unknown function appears not only in the boundary condition, but also in the coefficients of the equation. This method enables us to obtain an approximate solution with desirable degree of accuracy and to evaluate the approximation error, using the maximum principle. Analytical methods for solution of heat and mass transfer problems have recently received a new stimulus to their futher development due to the growing need to solve multicriteria problems for which numerical methods are unable to estimate the influence of a large number of input parameters on the behaviour of the solution and especially on its dynamics. In particular, an integral thermal balance method, a perturbation method, and a number of other methods are widely used to solve problems of the Stefan type with a free boundary, describing heat transfer with phase transitions. The main problem with the use of this method is the estimation of the approximation error, which, as a rule, is replaced for applied problems by comparison of the analytical solution with the experimental data.Item Open Access Laguerre polynomials in axisymmetric heat problems with a free boundary(Faculty of engineering and natural sciences, 2019) Jabbarkhanov Kh.The aim of the thesis is to consider solving heat equation with free boundaries using by heat polynomials method, in particular, using by Laguerre polynomials. There are two problems were considered. It is spherical inverse and direct problems the mcthod of thermal polynomials is appropriate. As exactly as the approximate solutions. The inverse two-phase spherical Stefan problem for unknown boundary heat. flux is solved by the method of the heat polynomials. Side by side with exact solution two methods for the approximate solution, collocation and variational methods, convenient for engineering applications are presented and compared.Item Open Access Some properties of ordered algebraic structures(Faculty of engineering and natural sciences, 2019) Dauletiyarova A.Quantifier elimination is one of the most important tools in model theory. Indeed, if a theory allows quantifier elimination, then this theory is complete, and the description of all definable subsets can be reduced to describing only those subsets that are defined by a quantifier-free formula. One of the most important mathematical structures is the linearly ordered set of real numbers. On it, you can set an ordered group and field. It is known that the elementary theory of these structures admits quantifier elimination, and since these theories are computably axiomatizable, quantifier elimination implies their solvability.Item Open Access On existence of a solution of a class of second order differential equations(Faculty of engineering and natural sciences, 2019) Zhosybayeva B.The paper was devoted to the study of issues on the existence and compactness of a resolvent, as well as estimates of the distribution function of singular numbers (s-numbers) of a differential operator of mixed type in a noncompact domain. We used the embedding theorems for Sobolev-type weighted spaces, the method of a priori estimates, the uniform localization method. In this paper, besides the above methods, an approach is proposed that allows one to find two-sided estimates of singular numbers (approximation numbers, s-numbers) of differential operators containing a sign-changing parameter. As a result of the study, the following results were obtained for a class of mixed-type differential operators defined in a noncompact domain: a theorem on the existence of a bounded inverse operator was proved: a theorem on the compactness (discreteness of the spectrum) of the resolvent of a mixed-type operator was proved; a conditions to the two-sided estimate for the distribution function of singular numbers (s-numbers) of the resolvent of a mixed-type differential operator given in an unbounded domain are obtained.Item Open Access The analysis of indexes quality of a healthcare in Kazakhstan(Faculty of Engineering and Natural Sciences, 2019) Khuandykh A.In this study considered analysis for healthcare service quality. Study considered quantity of hospital facilities for every 10000 population. As well as considered the average nlunber of medical staff per hospital.The results of study can . be used in the shade of the number of medical institutions and the training of the necessary number of medical personnel in the healthcare svstem. For these used regression method and as software used SAS. Python and R programing languages. These methods study were used on the example of two cities of Kazakhstan Almaty and Nursultan.Item Open Access Numerical estimation of hausdorff dimension of self-similar sets and open set condition(Faculty of Engineering and Natural sciences, 2019) Keulimzhayev A.This work is based on two of my scientific articles [10],[7]. When writing this work, I singled out the thesis into two sections. In the first section, I perform computer calculations, in the second section, I confirmed the theorem. That is, in the first part of the "Sierpinski triangle and the open set condition" I study the size of the triangle (Sierpinski triangle) in the fractal system, composed by Vaclav Sierpinski, in different conditions. These studies are conducted by two different methods of computer computing. Comparing the results obtained, I conclude on two methods for calculating the size and triangle of the Serpin in the amount of 729 pieces that we consider. That is, I will define both methods as effective and flawed, as well as some uncertainty of the methods and reasons for this in the form of a table. In the second section, working together with the supervisor, I study a closed Cantor set with parameter A and consider that these fractal sets are completely identical to themselves. In particular, I consider the fractal system Ey constituting specific lines, the iterative functional system fi(z) = 2/3, fa(z) = (2 +A)/3 and fo(x) = (x + 2)/3 consisting of three functions. That is when passing 1 < A <2as\=1+3°" for an integer n, I determine and prove that the fractal set F) is self-similar.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.