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Probabilistic models for the solution of the boundary problems for the equations of elliptic and parabolic types
The urgency and relevance of the dissertation topic.
At the present time, In the word, an expand the field of application of algorithms of the method Monte Carlo for various problems of mathematical physics, especially nonlinear boundary value problems is one of the important tasks. Numerical solution of such nonlinear problems is usually connected with considerable difficulties. Design, development and use of statistical modeling techniques, along with the deterministic methods is an actual problem and allows to obtain numerical results in the solution of applied tasks corresponding to increasingly complex models of the theory of gas dynamics, financial mathematics, biology and other fields. Research conducted in the aforementioned areas, confirm the relevance of the topic of the thesis
The aim of the research work is to construct and substantiate probabilistic models for solving boundary problems for nonlinear equations and systems of equations of elliptic and parabolic types in partial derivatives of the second order.
The scientific novelty of the research work is as follows:
numerical methods based on the probabilistic model for solving boundary value problems for nonlinear equations of parabolic type with constant and variable coefficients in the form of an infinite power series arc developed;
numerical methods based on the probabilistic model for solving the first and second boundary value problems for nonlinear equations of elliptic type arc developed;
numerical methods based on the probabilistic model for solving boundary value problems for systems of equations of elliptic and parabolic types arc developed.
Potential theory on m-subharmonic functions
Importance and relevance of the dissertation topic. Classical potential theory is based on Laplace operator and class of subharmonic functions. Built in the 80s of last century, the pluripotcntial thcoryis related with nonlinear Monge -Ampcre operator and plurisubharmonic functions. The pluripotential theory is intensively developing and has a numerous applications in the geometry of manifolds, in Einstein’s theory of relativity, in particular, to prove the existence of Einstein metrics in the theory of PDE. Naturally, there is a need to study the original extension of the class of phirisubharmonic functions and construction of potential theory for such extensions is actual direction of the complex analysis.
The class of plurisubharmonic functions is a subclass of subharmonic functions. It is naturally to study original extensions of the class of plurisubharmonic functions and construction of potential theory for such extensions.
To construct a theory which covers both classical potential theory and pluripotcntial theory it was expected using operators in hessians which generalize both Laplace operator and nonlinear Monge-Ampcrc operator. However, it was not known until recently a class of functions which expected potential theory will based on. In studying Dirichlet problem for equation in hessians was introduced a notion of class of m-subharmonic functions which was suitable class for constructing potential theory, it plays same role in the solution of equations in hessians as the class of plurisubharmonic functions for Monge-Ampcrc equation. Therefore, it is important deep investigation of the class of m-subharmonic functions, also the class of weakly m-subharmonic functions, in particular, establishment of potential-capacity properties of these classes.
The relevance of the scientific direction of the dissertation is also characterized by the fact that, in the dissertation justified the potential theory, which based on the operators in hessians, developed a method of solution of Dirichlet problem in the class of m-subharmonic and weakly m-subharmonic functions, proved m-subharmonicity of supremum of m-subharmonic functions and (m-1) subharmonicity of restriction of m-subharmonic functions on the hypcrplanc. Definition of weakly m-subharmonic functions which subharmonic in complex planes, proof of their potential-capacity properties, quasi-continuity of m-subharmonic functions, comparison principle, continuity of operators in hessian for a standard approximations and proof of other fundamental theorems arc important results of the dissertation.
The estimates of main characteristic functions of Nevanlinna’s theory, a simple description of m-convex hull in Riemann geometry, an application of the theory of m-subharmonic functions in establishing criteria of pluriharmonicity (analogue of Lelong’s theorem) and in series of applications of theories of m-subharmonic and weakly m-subharmonic functions in multidimensional complex analysis indicates importance and relevance of the dissertation topic.
Aim of research is constructing potential theory on m-subharmonic functions, proving potential properties of weakly m-subharmonic functions and demonstrating of applications of constructed theory to problems of multidimensional complex and harmonic analysis.
Scientific novelty of the research. The dissertation work is a new scientific direction. In it:
m-subharmonicity of supremum in the class of m-subhanninic functions and (m-l)-subharmonicity of restriction on complex hypcrplanes were proved;
A complete construction of potential theory based on hessian operator which includes well-known classical and complex potential theory was given;
Important potential-capacity properties of subharmonic on complex planes weakly m-subharmonic functions were introduced and studied;
The methods of solution of Dirichlet problem in the class of m-subharmonic and weakly m-subharmonnic functions was developed;
Quasicontinuity and comparing principle for m-subharmonic functions were proved;
Continuity of operators in hessians and other fundamental theorems of potential theory in the class of m-subharmonic functions were proved.
CONCLUSION
The main obtained results of the investigation arc following:
1. The m - subhannonicity of supreme in the class of m - subharmonic functions and m - subhannonicity of restriction to the complex hypcrplane were proved;
2. The notion of condenser capacity in the class of m - subharmonic functions was introduced and a series of important properties of capacity were proved;
3. Quasicontinuity and comparing principle for m - subhannonic functions were proved;
4. Convergence of currents for a standard approximations and fundamental theorems of the potential theory in the class of m - subharmonic functions were proved;
5. The class of weakly m - subharmonic functions was defined and a series of potential properties of this class were proved;
6. The method of application of the class of m - subharmonic functions in multidimensional complex analysis and potential theory was developed. In particular, in Nevanlinna’s theory - to estimate characteristic functions, in convex geometry — to describe m-convex hulls, in theory of pluriharmonic functions — to set pluriharmonicity of functions (analogue of Lelon’s theorem).
In general, the obtained results allow us to speak about achieving the goals of research of dissertation work. Constructed potential theory in the class of m -subharmonic functions is a new research direction which has an important application in Nevanlinna’s theory, in complex projective space, in theory of nonlinear elliptic equations and etc.
Possible differential equations that can reduce the order
equation into a low-order equation by substituting variables into it
Pluripolarity of graphs of quasianalytic functions
The aim of research work is to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar; to prove pluripolarity of graphs of quasianalytic functions in the sense of Gonchar; to show pluripolarity of graphs of algcbroid functions; to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Dcnjoy; to define a class of quasiharmonic functions and prove a theorem about thinness of graphs.
Scientific novelty of the research work. All the results obtained in the dissertation arc new and consist of the following:
- Uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar is proved;
- Pluripolarity of graphs of quasianalytic functions in the sense of Gonchar is proved;
- Pluripolarity of graphs of algcbroid functions is proved;
- Pluripolarity of graphs of quasianalytic functions of several variables in the sense of Dcnjoy is proved;
- Pluripolarity of graphs of functions from Gevrey class is proved;
- The class of quasiharmonic functions is defined and the theorem about thinnes of graphs of functions from this class is proved.
Pluripolarity of graphs of quasianalytic functions
The aim of research work is to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar; to prove pluripolarity of graphs of quasianalytic functions in the sense of Gonchar; to show pluripolarity of graphs of algcbroid functions; to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Dcnjoy; to define a class of quasiharmonic functions and prove a theorem about thinness of graphs.
Scientific novelty of the research work. All the results obtained in the dissertation arc new and consist of the following:
- Uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar is proved;
- Pluripolarity of graphs of quasianalytic functions in the sense of Gonchar is proved;
- Pluripolarity of graphs of algcbroid functions is proved;
- Pluripolarity of graphs of quasianalytic functions of several variables in the sense of Dcnjoy is proved;
- Pluripolarity of graphs of functions from Gevrey class is proved;
- The class of quasiharmonic functions is defined and the theorem about thinnes of graphs of functions from this class is proved.
Periods of development of mathematics
done today to develop the field of mathematics
Periods of development of mathematics
done today to develop the field of mathematics
O‘ng tomoni noma’lum bo‘lgan ikkinchi tartibli chiziqli oddiy differensial tenglama uchun nolokal shartli masalalar
On weak periodic states and Gibbs measures of some models on a Cayley tree
Subject of inquiry: Weak periodic Gibbs measures of the Ising model and weak periodic ground states of Ising model with competing interactions.
Aim of the inquiry: We study weak periodic Gibbs measures of the Ising model and weak periodic ground states of Ising model with competing interactions.
Methods of the inquiry: Methods of Markov random fields and recurrent equations of this theory. Also methods of measure theory and contractive maps, Pirogov-Sinay theory.
The results achieved and their novelty: The main results of work arc the following:
о For Ising model on a Cayley tree under some conditions it is proved that there arc five weak periodic Gibbs measures corresponding to arbitrary normal subgroup of index two.
о In case of normal subgroups of index four under some conditions on parameters of Ising model it is shown that there arc seven weak periodic Gibbs measures.
о An uncountably many new non periodic (and non weak periodic) Gibbs measures arc constructed.
о For Ising model with competing interactions sufficient and necessary conditions arc obtained under which there arc four weak periodic ground states.
о For arbitrary normal subgroup of index r sufficient and necessary conditions arc given under which a configuration is ground state of the Ising model with competing interactions on Cayley tree of order к > 1.
Practical value: the results of the dissertation work have theoretical character. They can be applied in problem of statistical physics.
Sphere of usage: results of the work can be used in measure theory, theory of phase transitions, theory of probability, theoretical and mathematical physcs.
On the essential and discrete spectra of the Schrödinger operator associated to a system of two and three identical particles on lattices
Actuality and demand of the theme of dissertation. Numerous scientific and applied researches conducted around the world show the following fact: throughout physics stable composite objects are usually formed by way of attractive forces, which allow the constituents to lower their energy by binding together. Repulsive forces separate particles in a free space. However, in recent years scientists have proved that in a structured environment such a periodic potential and in the absence of dissipation, stable composite objects can exist even for repulsive interactions. The Bose-Hubbard models, which have been used to describe the repulsive pairs, i.e. the Schrodinger operators on lattices is the theoretical basis for the experimental observations and applications. In this regard, the study of Schrodinger operators, associated to Hamiltonians of systems of particles moving on lattices, which appear in models of solid state physics and lattice field theory, is one of the priority areas of science.
In our country in the years of independence, a great attention has been paid to scientific areas having a practical importance; in particular, a great emphasis has been placed on study of Schrodinger operators associated to Hamiltonian of a system of particles moving on integer lattices. Significant results have been achieved in finding conditions for the existence of bound states and for their number, the energy of which is located outside the essential spectrum, and also to the threshold effects of the essential spectrum for Schrodinger operators, associated to systems of two and three particles on lattices.
Since the spectrum of the family of the Schrodinger operators appears quite sensitive to a change of the quasi-momentum of system, solving problems related to the spectrum of these operators, in particular, to prove the existence of bound states as well as to determine their number depending to the quasi-momentum of system, for three particle discrete Schrodinger operators is of highly importance. In this regard, the implementation of investigations in the following directions is one of the main problems: to investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two identical particles (bosons or fermions) with short-range pair potentials on lattices; to establish the threshold phenomenon of the essential spectrum for these operators; to obtain an asymptotic formula for the number of eigenvalues for the three-particle Schrodinger operator associated to a system of three identical particles on the three-dimensional lattice with a short-range pair interaction; to show the existence of eigenvalues of the three-particle Schrodinger operator associated to a system of three identical particles on lattices of dimensions one and two. Many research activities carried out in the aforementioned scientific areas all around the world exhibit a great interest and motivation to the topic of dissertation.
The research conducted in this thesis corresponds to the tasks specified in the Decree of the President of the Republic of Uzbekistan № PD-436 on August 7, 2006 “On measures to improve coordination and management of the development of science and technology”, No. PD-916 from July 15, 2008 “On additional measures to stimulate innovative projects and technologies” and other normative and legal acts relating to the fundamental sciences.
The aim of the research is studying the essential and discrete spectrum of two and thrcc-particle Schrodinger operators associated to a system of two or three identical particles (bosons or fermions) with short-range pair potentials on lattice.
The scientific novelty consists of the following:
the conditions for existence of the eigenvalues outside the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential in all dimensions of the lattice is found;
the finiteness of the number of eigenvalues lying outside of the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential on lattice is proved;
the number and location of eigenvalues of Schrodinger operator associated to a system of two particles (fermions), interacting on neighboring sites of lattice for all values of the parameters of the operator is determined;
the asymptotic behavior of eigenvalues lying below of the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice is studied;
the finiteness of the number of eigenvalues lying below the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero is shown;
the existence of eigenvalues of the Schrodinger operator associated to a system of three particles with pair two-particle zero-range potential in the one and two-dimensional lattices is proved. Our result is the first one in the theory of the thrcc-particle Schrodinger operators.
Conclusion
The thesis is devoted to investigate the essential and discrete spectra of two and thrcc-particle Schrodinger operator corresponding to the system of two or three identical particles (bosons or fermions) interacting via short-range pair potentials on lattices.
Basic results of the research arc as follows.
1. We introduce the notion of resonance for the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential on one-dimensional and two-dimensional lattice.
2. We find the conditions for existence of the eigenvalues lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential for all dimensions of the lattice.
3. We determine the number and location of the eigenvalues of the Schrodinger operator corresponding to a system of two particles (fermions), interacting on neighboring sites of latticcfor all parameters of the operator.
4. We obtain an asymptotic formula for the number of eigenvalues lying to the left of the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials in the thrccdimcnsional lattice.
5. We show the finiteness of the number of eigenvalues lyingbclow the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials on thcthrcc dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero.
6. We prove the existence of an eigenvalue lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with pair contact potentials on one dimensional and two dimensional lattice, which is the only result in the theory of discrete Schrodinger operators.
7. Weestablish the finiteness of the number of eigenvalues lying below the bottom of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with zero-range pair potentials on oncand two-dimensional lattices.
On problem for a third-order equation with multiple characteristics
are proved by the method of regularization and energy integrals
Numerical modeling of the non-linear system of a biological population of the Kolmogorov-Fisher type
The aim of the research work. The aim of the study is to develop numerical models describing the processes of multicomponent systems competing biological population quasilinear parabolic equations and their systems in homogeneous and inhomogeneous medium by the method of nonlinear splitting.
The tasks of research:
investigate the properties of two classes of models - models of nonlinear population, and a system of competing populations;
modeling on a computer the processes of one and multi-component systems competing biological population based on die algorithm of nonlinear splitting:
construct lower and upper solutions of die Cauchy problem by the algorithm of nonlinear splitting for multi-component systems competing biological population equation depending on the values of environmental parameters and the dimension of the space;
develop asymptotic expressions for solving systems of parabolic equations describing nonlinear process multicomponent competing biological populations:
create an initial approximation for the application of iterative methods and to construct a numerical scheme in the study of nonlinear processes in multicomponent systems competing biological population:
create algorithms and software for solving the foregoing problems, to determine new effects associated with die nonlinearity, visually present the decision to conduct a computational experiment.
The object of the research work. The object of the study are the nonlinear processes of the biological population, described by nonlinear parabolic equations and their systems.
Scientific novelty of the research work. The scientific novelty of die study is as follows:
developed methods to produce self-similar and approximately self-similar solutions for nonlinear models of multicomponent systems competing biological population based on the algorithm of nonlinear splitting;
identified new properties of a nonlinear madiematical model of the process of multicomponent competing biological population described by a system of Kolmogorov-Fisher type parabolic equations;
developed asymptotic expressions of solutions of self-similar equations and estimates of solutions of the Cauchy problem for multicomponent competitive systems of equations of biological population depending on parameter values the environment and the dimension of the space;
developed methods for constructing lower and upper solutions is needed for computer calculations of multicomponent tasks competing tasks biological populations;
created appropriate initial approximation, which provides the calculations with die required accuracy depending on the numerical values of the parameters using iterative techniques for fast and accurate numerical solution of the nonlinear task of Kolmogorov-Fisher type biological population;
developed computational schemes, algorithms and a software for performing numerical simulation of nonlinear mathematical models.
The outline of the thesis. The volume of the thesis is 105 pages typewritten text, illustrated by X drawings and 1 tables.
Numerical modeling of processes heat conductivity with gradient nonlinearity
Subjects of research: quasilincar parabolic equations with the gradient nonlinearity, describing nonlinear processes heat conductivity.
Purpose of work: the research of qualitative properties solutions of nonlinear mathematical models, in homogeneous and heterogeneous environment, when coefficient heat conductivity depends on a gradient of temperature, in view of absorption or source.
Methods of research: algorithm of nonlinear splitting, technique of comparison of the solutions, iterative numerical methods, method of variable directions and proracc method.
The results obtained and their novelty: for the quasilincar parabolic equations with the gradient nonlinearity of the second order develops of asymptotic theory based on a method of nonlinear splitting. The estimations of the decisions problem of Cauchy by algorithm of nonlinear splitting for the nonlinear heat conductivity equation with strong absorption divergent and non divergent types arc received. Asymptotic behavior of the solution in a critical ease is investigated. The conditions of occurrence of the unlimited solutions for the equation with a nonlinear source in heterogeneous media are received. Basing on the received estimations of the solutions and fronts, the computing experiments with use MathCad is carried out.
Practical value: results of the dissertation have theoretical character.
Degree of embed and economic efficiency: the results of the dissertation can be used in modeling nonlinear problems of mathematical physics and further in developing on the theory of nonlinear parabolic equations.
Field of application: the results of the dissertation can be used in modeling nonlinear processes heat conductivity, filtration, diffusion and on the base of achieved results, the special courses for students can be tcachcd.
Non-commutative integration with respect to traces Maharam and OrliczKantorovich space »
Subject of the inquiry: Non-commutative /Л-spaces of measurable operators associated with a Maharam trace, Orlicz-Kantorovich space.
Aim of the inquire: Construction theory non-commutative integration for Maharam trace with the values in a complex Dedekind complete Riesz spaces. Description non-commutative Lp -spaces associated with a Maharam trace. Construction theory of Orlicz-Kantorovich lattices.
Method of inquire: Methods of functional analysis, theory of operator algebras and theory of measurable Banach bundles are used.
The results achieved and their novelty: A complete description of Maharam trace on von Neumann algebra with the values in a complex Dedekind complete Riesz spaces is given; non-commutative integration for Maharam trace is constructed; new class Banach-Kantorovich space - non-commutative Lp-spaces associated with a Maharam trace - is defined and their dual spaces is described; new class Orlicz-Kantorovich lattices associated with a disjointly decomposable Z,0-valued measure is constructed; explained condition, in which they re flexed; some version of ergodic theorems for positive contractions in Orlicz-Kantorovich lattices is established; the class a complete Boolean algebras with disjointly decomposable L°-valued measure, representable as a measurable bundle of continuous (respectively, atomic) Boolean algebras is divided.
Practical value: The work has a theoretical character.
Degree of embed and economic effectivity: Results and methods introduced in the thesis can be used in reading special courses on functional analysis and theory of operator algebras.
Sphere of usage: Theory of operator algebras, the vector-valued measure theory, the ergodic theory, and theory of Banach-Kantorovich space.
Newton's method of solving a system of nonlinear equations
equation and the concept of iterative processes are given and their application is shown in the examples. The problem of numerical solution of a number of practical problems consisting of a system of nonlinear equations is considered. There are a number of approximate computational methods for solving systems of nonlinear equations, including Newton's method. Using these methods, a number of specific practical problems were solved, a computational algorithm and a block diagram were developed. An approximate method of finding the true roots of a system of nonlinear equations is given, based on examples, graphs are used in the form of results,
and appropriate conclusions are drawn
Models and algorithms of decision making in management on technological process of underground leaching
Subjects of the research: creation of object-oriented software programs for forecasting and optimum control of underground leaching.
Purpose of work: creation of direction’s models, methods and program means for analyses and decision making in the control of ore mines technological process UL.
Methods of research: numerically-drawn near and approximate-analytical methods, methods of idle time and stream running, methods of variable directions, monotonous schemes, characteristic target function, computer methods of decision making in control.
The results obtained and their novelty:
• the two measured mathematical models direction and algorithms calculation for the decision making in the control of technological process UL, were accepted;
• dynamics of concentration changes corresponding to different values of parameters, which could change technological processes, influenced to the current of technological process of UL;
• program processes for the realization of calculating experiment and calculation of parameters decision making at controlling of UL process control and visualizing calculation results were accepted.
Practical value: Correctness of worked out mathematical models and algorithms was approved on the bases of information from real mines and the reference was got about giving approbation from OSC “Andijonneft”.
Degree of embed and economic effectivity: On the bases of historical information from the real deposit mines 3 blocks of 5-ore uranium getting direction which belongs to Navoi, mountain-metallurgical factory was corroborated authenticity and availability of getting research information. The results of the thesis accepted for the use in the process of gas production control “GissarNefteGaz”.
Field of application: Software is possible to use for calculation of the concentration of the useful component decision making in control on process of underground leaching.
Modeling dynamic processes in heterogeneous environments for supporting to make technological decisions (on the example of underground leaching)
Subjects of research: orc deposit, exploited methods of UL on the conditions of using horizon mining system.
Purpose of work: developing computer model of UL in heterogeneous environments in the realization of horizon mining systems for the analysis and decision making support in the control of technological processes of UL.
Methods of research: methods of control theory, mathematical modeling, finite-difference methods and computational experiment.
The results obtained and their novelty: mathematical model of controlling UL processes was developed on the conditions of horizon mining systems; dynamics of pressure change and the values of reagent concentration in the various values of outcome parameters, influencing on the behavior of technological process of UL on the conditions of horizon mining systems, was studied; computer model for carrying out computational experiments and visualizing the results in two dimensional and three dimensional graphics was developed; software package of UL process in the conditions of horizon mining systems for supporting making technological decisions in the control of mine workings was developed.
Practical value: developed computational algorithms and computer model can be applied for the analysis, prognosis of the parameters of leaching process and decision making for controlling its parameters on the purpose of optimal extraction of minerals from real deposit mines on the conditions of horizon mining systems of mine workings.
Degree of embed and economical effectivity: obtained results were applied in the mine North Bukinai NMSK, the act of applications was taken. Developed software was registered by Government Patent Committee of Uzbekistan.
Field of application: mineral deposit mines exploited by the method of UL.
Modeling dynamic processes in heterogeneous environments for supporting to make technological decisions (on the example of underground leaching)
Subjects of research: orc deposit, exploited methods of UL on the conditions of using horizon mining system.
Purpose of work: developing computer model of UL in heterogeneous environments in the realization of horizon mining systems for the analysis and decision making support in the control of technological processes of UL.
Methods of research: methods of control theory, mathematical modeling, finite-difference methods and computational experiment.
The results obtained and their novelty: mathematical model of controlling UL processes was developed on the conditions of horizon mining systems; dynamics of pressure change and the values of reagent concentration in the various values of outcome parameters, influencing on the behavior of technological process of UL on the conditions of horizon mining systems, was studied; computer model for carrying out computational experiments and visualizing the results in two dimensional and three dimensional graphics was developed; software package of UL process in the conditions of horizon mining systems for supporting making technological decisions in the control of mine workings was developed.
Practical value: developed computational algorithms and computer model can be applied for the analysis, prognosis of the parameters of leaching process and decision making for controlling its parameters on the purpose of optimal extraction of minerals from real deposit mines on the conditions of horizon mining systems of mine workings.
Degree of embed and economical effectivity: obtained results were applied in the mine North Bukinai NMSK, the act of applications was taken. Developed software was registered by Government Patent Committee of Uzbekistan.
Field of application: mineral deposit mines exploited by the method of UL.
Minimax approach to data structural analysis based on core clusterization method
Subjects of research: matrixes of proximity, images, directed and undirected graphs
Purpose of work: the development and study of minimax model of layered clusterization based on conception of monotonous proximity function
Methods of research: the methods of discrete mathematics, linguistic data analysis, classification, cluster analysis, image processing. Software was realized on C++, MATLAB, C#.
The results obtained and their novelty:
- development of core clusterization parametric model. Based on conception of monotonous proximity function;
- the method of image segmentation based on core clusterization, and its’ efficiency in comparing with method of normalized cut and k-mcans;
- the procedure of protein sequences multiple alignment based on core clusterization was evaluated in comparing with known software such as CLUSTAL and DIALIGN.
Practical value: developed software can be used in different applications in area of data and image processing, particularly in bioinformatics problems solution.
Degree of embed and economic effectivity: the results of dissertation were used as software package for image processing and teaching needs in information system department of Murom Institute of Vladimir state University University and as software for computing diagnostics of cholecystitis different forms in Republican Scientific Center of Emergency Medicine, Uzbekistan Ministry of Healthcare. Efficiency is social impact on research, treatment of patients, and education.
Field of application: Data Mining and decision-making systems in biology and healthcare.
METHODS OF DEVELOPING STUDENTS’ TEACHING SKILLS IN SYNCHRONOUS AND ASYNCHRONOUS COMMUNICATION OF SCHOOL GEOMETRY WITH PHYSICS
In this article, five-stage types of connections for teaching students school geometry in synchronous and asynchronous connection with physics are developed, and at each stage, connections through facts, connections through knowledge, and generalized skills are analyzed. In this approach, such aspects as concretizing the concepts of geometry and physics, revealing the processes and phenomena of one science using the concepts of another science, establishing connections between conclusions based on general concepts, and the ability to form connections between the concepts of different sciences when expressing one’s opinion are considered.
Methods and algorithms of operational management and decision-making analysis in social systems
Subject of research: gas supply systems which operate in normal working mode and under uncontrollable random disturbance influence.
Purpose of work: development of methods and algorithms of efficient management and decision-making analysis for technical and economical gas supply objects factors.
Methods of research: have been used: mathematical programming, probability theory, information systems construction and computing experiment methods.
The results obtained and their novelty: for the first time have been developed methodology and algorithms of interpretation of graphic predetermined functions. Also have been stated and summarized principles of definition and estimation of models of structured identification of gas supply objects accuracies. Devised and researched the information-logical model of management and operational decision-making automation on regulation of work of gas produce, transport and supply objects. Has been developed informational-logical model of automation of technological and economical computation of factors of objects of gas production and supply. Has been shaped mathematical model of gas distribution network as determined serving system, based on scheduling theory conception.
Practical value: have been developed algorithms and software tools that make possible efficiently to manage and regulate gas supply system, which may operate as in normal working mode and as under uncontrollable random disturbance influence.
Degree of embed and economic effectivity: developed software tools were accepted and adopted by state direction of “Samarkand Gas” (which now is SG “Samarkand Gas Supply”) and became component of different scientific research activities of “SCST RUz №-14.3”. Economical effect of cunent work’s adoption is about 9220000 sums per annum. Obtained three certificates of the State Patent Office of the Republic of Uzbekistan: №DGU 01936, №DGU 02011, №DGU2181.
Field of application: objects of production, transport and gas supply, and also in planned-economical departments of industrial factories.
METHODOLOGY OF AN INTEGRATIVE APPROACH IN THEORETICAL AND PRACTICAL MATHEMATICS LESSONS FOR GENERAL EDUCATION SCHOOL PUPILS
This article provides a detailed analysis of teaching mathematics in general education schools based on the methodology of integrative approaches. It examines the effectiveness of integrative methods in enabling students to master theoretical and practical knowledge, as well as forming and consolidating their sustainable understanding. The article highlights concepts such as integration, the method of integrated lessons, challenges of integrative classes, the idea of integrative education, and the integration course, emphasizing their importance in the educational process. Special attention is given to the potential of these approaches to enhance students’ interest in mathematics and develop their mathematical thinking. The study demonstrates the pedagogical efficiency of incorporating modern integrative methods into the learning process.
Mathematical models and effective numerical algorithms of filtration processes in oil-gas and waterbearing stratums
The aim of the research work. The aim of the research is to develop and improve mathematical models, numerical algorithms and software for filtration processes in oil-, gas- and water-bearing beds.
The scientific novelty of the research work is as follows:
the mathematical model of the process of gas filtration in porous media was improved by taking into account various boundary conditions and the computational algorithm for solving the corresponding problem was developed on the basis of the finite difference method;
the mathematical model of the filtration process in the case of piston displacement was improved by taking into account the factor of oil production from the liquid phase region and the computational algorithm for solving this problem was developed on the basis of method of rectifying the phase fronts;
the mathematical model of the process of joint fluid and gas filtration was improved on the basis of the model of interconnected phases and the computational algorithm for solving this problem was developed on the basis of the variable direction method;
the effective numerical algorithm for solving the problem of gas filtration in porous media by the method of physical splitting was developed;
the parallel computational algorithm was developed to solve the problem of gas filtration in porous media for an arbitrary filtering region.
Mathematical models and algorithms of optimum governing an operation of earning one's living robots on rolling base
Subject of research: earned one's living (PR) on the rolling base.
Purpose of work: Define and value a dynamic inexactness of motion path and develop mathematical models and algorithms of optimum management, allowing enlarge a speed and PR positional accuracy on the rolling base.
Methods of research: methods of mathematical modeling of technological processes, theories of probability, mathematical statistics, algebras, theories and theory of optimum management.
The results obtained and their novelty: Determined mistake existing motion models PR on the rolling base, newly built equation of its motion, on the base which designed mathematical models, algorithms and software programs, allowing enlarge a speed and PR positional accuracy.
Practical value: Software programs of optimum governing the explored robots, due to increasing a speed and positional accuracy, can be used in all branches of public facilities, which provided with by the system that promotes minimization of general time of production and spare an energy facility.
Degree of embed and economical effectivity: Developed on the base mathematical optimum management models PR on the rolling base algorithms and software programs are introduced in the Join-stock company "Technologist". On the example of introduction in the process of assembly of units proved that arc vastly enlarged speed and positional accuracy operated by PR; their possible use mechanical processing in the process of. Annual cost-performance of introduction on one PR forms 535 thousand (on prices 2009).
Field of application: Developing mathematical models and algorithms can be used under optimum governing the different branches of public facilities, which provided with by systems.
Mathematical models and algorithms for solving physically nonlinear rod problems under spatially variable loading.
The aim of the research work is to develop the mathematical models, effective computational algorithms and software for the processes of physically nonlinear strain of rods under spatially variable loading taking into account the damageability of materials.
Scientific novelty of the research work is a follows:
on the basis of the refined theory of V.K..Kabulov and the variation principle, mathematical models are developed for solving physically nonlinear rod problems under the influence of complex external forces, taking into account the damageability of materials;
the multi-parameter mathematical models in the form of a system of nine nonlinear differential equations of the second order are developed with natural boundary conditions for studying the stress state of rods in the case of spatially repeated loading in current and dummy coordinate systems;
by the A.A.Ilyushin method of elastic solution the computational algorithms for solving physically nonlinear problems of rods with different approximations are developed based on the central difference scheme and the modification of A.A.Samarsky-I.V.Fryazinov (MSF) method of finite differences;
the effective computational algorithms providing fast approximation to a stable solution, a high degree of accuracy, directed to the numerical calculation of some physically nonlinear rod problems, described by mathematical models of multi-parameter differential equations are developed;
an automated system has been created on the computer that allows the formation and solution of physically nonlinear rod problems for various variable loads and the planes with geometric, static and mixed boundary conditions.
