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Contents

Curricula, and Programs.

Mathematical Analysis. (The Program of Mathematical Physics
Department)..............................................
Algebra and Analytical Geometry.(The program of (General "Mathematics"
Department)................................
Computer and Programming. (The program of Algorithmic Languages
Department)....................................
The practical work on Computers.( Department of Algorithmic.
Languages)...............................................
Discrete Mathematics. (Department of Mathematical
Cybernetics).............................................
Mathematical Analysis. (Department of General Mathematics)..
Functions of Complex Variable Theory. ......................
Ordinary Differential Equations.(Department of General
Mathematics).............................................
Physics.(Department of Mathematical Physics)................
Introduction to Computing Mathematics. (Department of Research
Automation).....................................
Mathematical Analysis. (Department of General Mathematics)..
Numerical Methods. (Department of Numerical Methods)........
Numerical Methods of Problem Solving of Mathematical
Physics..................................................
Methods of Mathematical Physics. (Department of Mathematical
Physics).................................................
Optimal Control.............................................
The Theory of Probability and Mathematical Statistics.......
Programming Languages. Department of Algorithmic Languages..
Operating Systems. (Department of Computer Systems
Architecture)............................................
Additional Chapters of Equations in Private Derivatives of Mathematical
Physics.(Department of Numerical Methods)...
Mathematical Logic. (Department of Mathematical Cybernetics).
...........................................
Operations Research.........................................
Elements of Cybernetics. (Department of Mathematical
Cybernetics).............................................
Physics.....................................................
Data Organization, data access and data storage methods.....
Extreme tasks Solution Methods.(Optimal Control
Department)..............................................
Methods for Solving Extremal Problems.......................
The Game Theory and Operations Research.....................
The History and Methodology of Applied Mathematics..........
Computers and Programming...................................
Structures of Computing Systems.............................
Mathematical Analysis
(1,2 Semesters; Lectures-136 ac.h.; Seminars-136 ac.h.)

Real Numbers. Decimal fractions set and its ordering. Existence of
supremum (least upper bound) and infinum (greatest lower bound) for bounded
set. Real numbers properties. The Real Numbers Theory (extra questions).
Elements of the Set Theory.
The Theory of Limit. Sequence and its limit. Monotone sequences.
Arbitrary sequences. Function limit. General definition of a base-limit.
Continuity of Function. Concept of continuity of function. Monotone
function properties. Simplest elementary functions. Two wonderful limits.
Discontinuity points. Local and global continuous 'functions' properties.
Differential Calculus. Concept of derivative. Differentiability of
function and its differential. Differentiation of inverse and composite
function. Derivatives and differentials of inverse and composite functions.
Derivatives and differentials of superior order.
Antiderivative of Function and Indefinite Integral. The basic concepts.
The basic methods of integration. Classes of functions, integral within the
class of elementary functions.
The Basic Theorems about Differentiable Functions. Increase and decrease
of function in point. Local extremum. The Theorem about zero of derivative.
Mean value formula and its consequences. Cauchy formula. L'Hospital rule.
Taylor formula. Different forms of remainder term. Various applications.
Analysis of Function Diagram and Search for Extremum Values. Necessary
and sufficient condition for function extremum. Convexity of function
diagram. Points of inflection. Construction of function diagram.
Definite Riemann Integral. Definition of integral. Integrability.
Darboux integration theory. Necessary and sufficient condition for function
Integrability. Basic classes of integrable functions. Definite integral
properties. Estimates of integrals. Basic formula of integral calculus.
Basic rules of integration. "Improper integrals of the first kind. Cauchy
criterion. Sufficient criterions for convergence of improper integrals.
Absolute and conditional convergence of improper integrals. Improper
integrals of the second kind.
Geometric Applications of Definite Integrals. Length of curve arc.
Concept of rectifiable curve. Calculation of the length of curve arc.
Square of flat figure. Square of curvilinear trapezoid and curvilinear
sector. Size of a body in space. Some classes of cubicable bodies.
Approximate Methods of Solutions of Equations and Definite Integral
Calculation. Methods of plug, iteration, chords and tangents. Methods of
rectangles, trapezoids and parabolas.

Several Variables Functions. The concept of several variables function.
Limit of several variables function. Continuity of function and variables.
Basic properties of continuous functions. Partial derivatives.
Differentiability. Differential. Composite function differentiation.
Invariance of the first differential form. Directional derivative and
gradient. Partial derivative of supreme order. Taylor formula. Local
extremum of function and variables.
Implicit Functions. Existence and differentiability of implicit defined
function. Theorem about solvability of function equations system.
Dependence of function. Sufficient conditions of independence. Lagrange
multiplier method. Sufficient conditions.

Gradient Method of Strongly Convex Function Extremum Search.

The program compilers are: Prof. V.A. Iljin;
Prof. V.V. Tichmirov

Algebra and Analytical Geometry.
(1,2 semesters; Lectures-136 ac.h.; Seminars-120 ac.h.)

Analytical Geometry.

Descartes (Cartesian) Coordinate System on Flat Surface and in Space.
Elementary problems of analytical geometry. Polar coordinate system.
Vectors. Addition of vectors. Multiplication of vector and number.
Linear dependence of vector system. Geometrical meaning of linear
dependence. Concept of basis. Affine coordinate system. Vector projection.
Scalar, vector and mixed product of vectors.
Coordinate System Method. Substitution of affine coordinate system.
Transformation of Cartesian coordinate system on flat surface and in space.
Line and surface equation. Algebraic lines and surfaces.
Straight Line and Flat Surface. Straight line on flat surface. Straight
line as an algebraic line of the first degree. Parametric and canonical
line equations. Mutual disposition of straight lines on flat surface.
Straight line bundle. Partition of flat surface by straight line. Straight
line on Euclidean flat surface: vector equation for straight line, normed
equation for straight line, angle between straight lines.
Flat Surface as an Algebraic Surface of the First Degree. Parametric
flat surface equation. Mutual disposition of flat surfaces. Flat surface
bundle. Flat surface connective. Partition of space by flat surface. Flat
surface in Euclidean space.
Ways of Representation of Straight Line in Space. Mutual disposition on
straight line and flat surface. Straight line connective. Straight line in
Euclidean space: angle between lines, angle between line and flat surface,
distance between point and straight line, distance between straight lines.
Lines and Surfaces of the Second Degree. Ellipse, its focuses and
directrixes. Hyperbola, its focuses, directrixes and asymptotes. Parabola,
its focus and directrix. Ellipse, hyperbole and parabola equation in polar
coordinate system. Reduction of two-variable polynomial of the second
degree to the canonical form. Classification of lines of the second degree.
Ellipsoid. Hyperboloid of one sheet and of two sheets. Hyperbolic and
elliptic paraboloid. Linear ruling surfaces. Cones and cylinder of the
second degree. Reduction of three-variable polynomial of the second degree
to the canonical form (without demonstration). Classification of surfaces
of the second degree.

General Algebra Elements.

Matrixes and Determinants. Concept of matrix. Multiplication of matrix
and number. Multiplication of matrixes. Transposed and conjugate matrix.
Square matrix. Concept of determinant. Linear property of determinant.
Asymmetry property. Transposed matrix determinant. Minors and their
cofactors (algebraic additions). Laplace theorem. Determinant of product
of matrixes. Nonsingular matrixes. Reciprocal matrixes.

Rank of matrix. Linear dependence of rows (columns) of matrix. Basic
minor theorem. Matrix elementary transformation.
Linear Equations Systems. Gauss method. Linear equations system
research. Cramer rule. Kronecker- Kapelli theorem. Search for linear
equations system solutions. Homogeneous linear equations system.
Fundamental system of solutions.
Maps. Groups. Rings. Fields. Arbitrary sets maps. Types of maps.
Composition of maps, its associativity. Inverse map. Equivalence map.
Equivalence classes. Factor set.

Inner Composition Law Concept.
Group. Definition and elementary properties. Subgroup. Symmetric group
of permutations. Isomorphism of group concept. Residue classes. Normal
subgroup. Factor group.
Ring. Definition and elementary properties. Zero divisor. Reversible
ring elements group. Ring of one-variable polynomials under number field.
Field. Definition and elementary properties. Field characteristic.
Residue field. Complex numbers field. Trigonometrical form of complex
number. Taking the root of N power.
Polynomials. Division of polynomials. Greatest common divisor.
Euclidean algorithm. Irreducible polynomials. Polynomial roots.
Algebraically closed field concept. «Algebra basic theorem». Canonical
factorization of polynomial under field of real numbers.

Linear Algebra.
Linear Spaces. Definition and elementary properties. Linear spaces
product. Linear space. Linear span. Linear dependence of vectors. Basis.
Dimension. Linear spaces isomorphism. Linear spaces direct sum. Factor
space. Substitution of basis. Real space orientation.
Linear Affine Manifold. Transfer in linear space. Parallel linear affine
manifolds. Affine manifolds intersection.

Convex Sets in Linear Space.
Euclidean and Unitary Spaces. Orthonormalized basis. Gram matrix.
Orthogonal and unitary matrixes. Orthogonal supplement. Euclidean and
unitary spaces isomorphism.
Linear Operator. Definition and elementary properties. Kernel and linear
operator image. Operations with linear operators. Linear operators ring.
Inverse operator. Linear operator matrixes in various basis.
Characteristic polynomial of linear operator. Invariant subspaces.
Linear operator characteristic values and characteristic vector
(eigenvalue and eingenvector) and their calculation complex and real
spaces. Simple structure operator. Linear operator canonical form. Jordan
box. Theorem about linear operator matrix Jordan normal form in complex
space.
Bilinear and Quadratic Forms. Bilinear form. Bilinear form matrixes.
Reduction of quadratic form to the sum of squares in linear space.
Lagrange's method. Jacobi's method. Definite (positive or negative)
quadratic forms. Inertia law. Sesquilinear and Hermitian forms. Multilinear
forms.
Linear Operators in Unitary Space. Linear and sesquilinear form in
unitary space. Conjugate linear operator. Normal, unitary and self-
conjugate linear operators. Reduction of linear form to the main axes in
unitary space. Problem of reduction of two quadratic forms to the main
axes.
Linear operator norm. Self-conjugate linear operator norm. Extreme
properties of self-conjugate linear operator eigenvalues. Spectral
expansion of self-conjugate linear operator. Positive operators. Operation
of taking of root out of an operator. Hermitian and polar expansion of
linear operator.
Linear operators in Euclidean space. Symmetrical operator. Orthogonal
operator and its geometrical meaning.
Hypersurfaces of the second degree in Euclidean space. Expancion of
equation of hypersurfaces of the second degree. Geometric properties of
hypersurfaces of the second degree.
Some Methods of Operator Equation Solution. Fredholm's alternative.
Approximate operator equations. Nonsingular operator perturbation. Normal
solution of linear operator. Pseudo-solutions.
Approximate solution error. Concept of correctly and incorrectly
formulated problems. Method of regularization of normal solution search.

The program compilers are: Prof. Y.V. Shikin;
Prof. V.N. Reshetnikov



Computer and Programming
(1-3 semesters; Lectures-140 ac.h.;
1-7 semesters; practical work on computer -376 ac.h.)

I semester

Introduction. Mathematics and role of its applications. Necessity of
increasing of calculations' speed. Computer and Scientific and technical
progress. The role of the Russian science in creation of computer.
Algorithms. Algorithmization of Calculation Processes. The concept of
algorithm and its role in computer using. Intuitive definition of
algorithm, examples of algorithms and their characteristic features.
Necessity of the algorithm concept improving, formalization of this
concept. Specific features of algorithm creation. Programming "from the
top to the bottom". Ways of program structure description.
Algorithmic Languages. Necessity of formal language for writing
algorithms. Problem-oriented programming languages, requirements for them.
Metalinguistic formulae and syntax diagrams as a formal tool for describing
syntax of a language. Programming language general characteristic. Classes
and types of values. Expressions, their types and assignment Statements and
declarations /description /. Procedures, their definition and using. Data
input and output. Program structure.
Data Structures. Variety of tasks and data structures used in them.
Typical data structures /queues, stacks, lists, tables, trees, arrays /,
their mapping on one-dimension memory and methods of operation with it.

II semester

Computers. The principles of computer construction. Computer structure,
main devices, their assignment and interaction. Computer language and its
features. Set of machine operations, main types of operations. Scheme of
machine program execution. Computer as a formal performer of algorithms.
Variety of real computers. Examples of computer programs of various types.
General and specific features of programming for computers of various
types.
Main Receptions for Computer Programming. Concrete computer
characteristics. Number and command representation. Symbolic coding,
assemblers. Arithmetic operations, programming of formulas. Operations of
conditional and unconditional branches. Programming of branches and loops.
Logical operations. Representation of logical values and characters,
operation with them. Operations with arbitrary codes. Variables with
indexes and address variables. Ways of address variables management. Index
and base registers, their assignment and use. Programming of procedures,
subroutines. Ways of procedure calls and their structure. Concept of
standard subroutines. Subroutine library and its usage. Extension of
machine operations set by program way. Programming language macros.

III semester

Programming Systems. Classification, composition and structure of
programming systems, libraries and packages of application programs.
Multiprogramming. Problem of computer productivity increase. Overlapping
of operations, multiprogram mode. Concept of interruptions, memory
protection, privileged operations.
Operating System Structure. Functions of operating system and main
blocks. Input queue, job planning, planning of die job steps. Response to
interruptions. Input/output management. Resources allocation problem. Job
control language. Multiprogram mode management.

The program compilers are: Prof. L.N. Koroljov;
Prof. N.D. Vasjukova;
Prof. V.G. Baula;
Prof. N.P. Trifonov

Practical Work on Computers.

Assignment of a practical work. The main aim of practical work on
computer is to use in practice all the knowledge, obtained by the students
during lectures on programming and to get practical experience in problems
solution with the help of computer. This practice will be necessary for
them in their future work in the field of computer application.
The main purposes of this practical work are practical mastering of the
main methods and receptions of developing, compilation, debugging and
executing programs on a computer, learning modem mathematical computer
supports and gaining skills in it using, methods and organization work on a
computer.
Practical work begins at the first year, in parallel with the main
course "Computer and programming" and lasts for 7 semesters. The course of
practical work consists of two parts: general practical work which lasts 5
semesters and specialized practical work which lasts 2 semesters (the
specialized practical work can be extended with the culting of the general
practical work).
The General Practical Work. The general practical work is unique for
all the students of all the specializations of the faculty. The main
purposes of this work are the following:
. mastering of one of the programming languages;
. learning and mastering of the main programming methods and receptions;
. acquaintance with the real existing computers;
. acquaintance with the process of task passing on computer by means of
execution of several simple assignments on real computers (it is
desirable to use the computers, which accept the conversational mode,
with immediate involvement of students at the àll stages task passing).
The rest part of the general practical work is mainly connected with
practical work on computer. This part of the practical work consists of
several tasks (usually 2-3 tasks during every semester), independently
executed by the students. It has the following main objectives:
. learning the main components of computer software and practice in its
using, including mastering of the technique of operation with compilers,
file systems, text editor, subroutine libraries, job control languages,
formatted and graphics output tools;
. practice in operating in various modes of computer use;
. practice in numerical solving of standard mathematical tasks and
familiarity with the available subroutine libraries;
. acquisition of skills on software product forming
Usually each task includes one of the themes on computer software and
one of the themes on numerical methods. The subjects of the tasks and their
sequence are maily determined by the requirements of systematic study of
software. The sequence of themes on numerical methods should be coordinated
with the delivered courses.
The specialized practical work. The main purpose of the specialized
practical work is to acquire experience in computer solving of specialized
tasks in particular field, and also to get to know the specialized
mathematical software components. This part of the practical work contains
the tasks, defined by the different departments of the faculty.
Tasks on Mathematical Software. The main aim is to study specialized
mathematical software.
Theme 1. Algorithmic languages and compilers.
Objective: Acquaintance with the stages of task passing on computer,
practice in using of algorithmic languages and appropriate compilers;
acquaintance with elementary methods of testing and debugging of programs.
Theme 2. Batch mode. Job control language.
Objective: Acquaintance with possibilities of the operating system, with
the language of dialogue between a man and an operating system (job
management language) and practice in their use.
Theme 3. Program text libraries and text editors.
Objective: Acquaintance with the ways of library organization, intended
for program texts storage, and the rules of text editor using. Practice in
libraries and text editors use.
Theme 4. Formatted input / output
Objective: Learning of computer input / output devices possibilities,
programming language (or appropriate components) possibilities on output
from machine the obtained results in the form of final documents and
practice in using them.
Theme 5. Modular programming, subroutine library.
Objective: Acquaintance with subroutine library organization; practice
in modular programming and in usage of library units (subroutines).
Theme 6. The conversational mode of computer use.
Objective: Practice in conversational mode of computer usage and work
with terminal.
Theme 7. The assembler language (autocode).
Objective: Acquaintance with the language of the concrete machine and
practice in programming in the machine-oriented languages.
Theme 8. Macros.
Objective: Acquaintance with the ways of machine operation set extension
by means of programming and practice in macro generators use.
Theme 9. Character data processing.
Objective. Learning of machine possibilities of solving
uncomputationable problems and practice in solving the problems, connected
with character data processing.
Note 1. The given subjects can be partially changed and should be
specified according to the available computers and available software.
Note 2. The concrete tasks can either unite some themes or divide them
into separate parts.
The Numerical Methods Included into the Practical Work. The practical
work in the numerical methods includes the following themes:
. methods of elementary function calculations;
. methods of calculation of definite integrals;
. numerical methods of linear algebra and linear programming;
. solving of transcendental equations;
. numerical integration of ordinary differential equations;
. approximation of functions;
. methods of optimization (such as the prompt descent).
Job's Reporting. The report consists of the theme of the task, the
objective, the content of the task, literature, variants of the tasks,
methodical instructions and, probably, calendar schedule of task execution.
After execution of each task a student presents the report in the required
form.

The program compiler is: Prof. N.P. Trifonov

Discrete Mathematics
(2,3 semesters; Lectures-68 ac.h.; Seminars-50 ac.h.)

Introduction. The place of discrete mathematics in the system of
mathematical formation. Discrete mathematics and mathematical cybernetics.
Relation between the discrete and infinite approaches to the study of
various fenomena.

Part I. Functional Systems with Operations. Algebra of Logic. Boolean
functions. Formulae. Realization of functions by formulae, equivalence of
formulae. Properties of elementary functions. The duality principle.
Expansion of Boolean functions on variables. The perfect disjunctive normal
form. Completeness and closure, examples of full systems. The Jegalkin
polynomial. The major closed classes, the theorem of completeness. The
presentation of Post results.
k-symbol Logic. Functions of k-symbol logic; formulas and realization of
functions by the formulas. Some examples of full systems. Algorithm for
recognition of completeness; the theorem of completeness. Some properties
of essential functions, criteria of completeness. Singularities of k-symbol
logic: existence of the closed classes which do not have a basis, and
closed classes with countable basis, representation of functions by
polynomials.
Restrictedly Determined (Automatic) Functions with Operations.
Determined functions. Representation of determined functions with the help
of trees; weight of tree. Restrictedly determined functions, their
representation by Moore diagrams and canonical equations. Operations of
superposition and feed-back, closure of a class of restrictedly determined
functions. Examples of full systems. The relation between the operations of
superposition and feed-back.
Computable Functions. Turing machines. The duality principle for the
programs (machines), two types of composition of Turing machines. A. A.
Ljapunov functional language. Machine codes and their transformations.
Computabl functions. Operations of superposition, primitive recursion and
minimization; classes of partially recursive, recursive and primitively
recursive functions. Closure of the class of computing functions concerning
the operations of superposition, primitive recursion and minimization.
Kleene formula , partial recursiveness of computy functions, examples of
full systems.

Part II. Graphs and Webs. Graphs. Graphs and their geometric
realization, isomorphism of graphs. The theorem about the number of
combinations with recurrings, estimation of the number of graphs.
Webs. Webs and their properties. Evaluation of the number of webs. Two-
object sets bipolar webs and their superposition. Decomposable and non-
decomposable webs. Type of decomposability of web, canonical decomposition,
n- webs and estimation of their number.

Part III. Theories of Coding. Alphabetic Coding. The criterion of
uniqueness of alphabetic decoding. Algorithm for recognition of uniqueness
of alphabetic decoding. Makmillan inequality. Codes with a minimum
redundancy. Self-correcting codes.

The program compiler is: Academician of the Russian Academy of Sciences,
Prof. V.V. Yablonsky

Mathematical analysis.
(3,4 semesters; Lectures-136 ac.h.; Seminars-136 ac.h.)

Series of Numbers. Cauchy criterion of convergence of series. Series of
non-negative terms. Necessary and sufficient condition of convergence.
Criterions of convergence (of comparison, d'Alembert criterion, Cauchy
criterion, Cauchy-Maclaurin criterion). Convergence of series of random
terms. Cauchy and Riemann theorems of rearrangement of terms in series of
numbers. Criterions of convergence (two Abel criterion, Dirichlet-Abel
criterion, Leibniz criterion). Arithmetical operations with convergent
series. Mertns theorem.
Infinite Products. Colligation between convergence of infinite products
and series of numbers. Criterions of convergence of infinite products.
Double and Iterated Series. Colligation between convergence of double
and iterated series. Convergence criterion of double series with non-
negative terms. Absolute convergence of double series. Colligation between
convergence of 4 series: iterated series, double and "single" under
replacement terms to their absolute values.
Generalized Method of Summation of Divergent Series. Poisson-Abel and
Chezaro methods. Linearity, regularity and methods of comparison.
Functional Sequences and Series. Point convergence and convergence on
set. Uniform convergence. Cauchy criterion. Uniform convergence criterion
(two Abel criterions, Weierstrass criterion, Dini criterion). Term-by-term
limit process, continuity of sum of series and of limit sequence's
function. Term-by-term integration and differentiation. Mean convergence.
Colligation with uniform convergence. Arzela-Ascoli theorem. Criterion of
equicontinuity of functional sequence. Power serieses. Cauchy-Hadamard
theorem. Sum continuity, term-by-term integration and differentiation.
Function expansion in a power series. Weierstrass theorem of functions
uniform approximation by algebraic polynomials.
Double and Multiple Integrals. Definition and existence of double
integral. Basic characteristics of double integral. Reduction of double
integral to iterated single. Triple integral and multiple integrals. Change
of variables in multiple integrals. Multiple improper integrals of non-
negative functions. Criterion of convergence. Multiple improper integrals
of alternating functions. Principal value of multiple improper integrals
Curvilinear Integrals. Definitions of curvilinear integrals of the first
and second types, their physical sense. Existence of curvilinear integrals
and their reduction to definite integrals.
Surface Integrals. The concept of surface. Normal vector and tangential
plane to the surface. One-sided and two-sided surfaces. The concept of
surface area. Squareability of smooth surface. Definition and existence of
surface integrals of first and second types.
Basic Operation of the Field Theory. Bases conversions. Invariant of
closable linear operator. Divergence and rotor of closable linear operator.
Concepts of scalar and vector fields. Differentiable vector fields.
Divergence, rotor and directional derivative of vector field. Iterated
operations of field theory. Green-Stokes-Ostrogradsky formula. Independence
condition of line contour integrals of the first and second types on a
surface from the path of integration.
Integrals Depended on Parameters. Non-singular integrals.
Characteristics of continuity, integrability and differentiability of
integrals. Improper integrals of the first and second types. The concept of
uniform convergence. Cauchy criterion, Weierstrass, Dirichlet-Abel and Dini
criterions of uniform convergence of improper integrals. Characteristics of
continuity, integrability of improper integrals. Calculation of Dirichlet's
integral. Euler's gamma- and beta- functions. Their characteristics.
Colligation between Euler integrals. Stirling formula.
Fourier integral and series. The concept of orthonormal systems and
general Fourier aeries. The problem, of Euclidean space element
approximation. Completeness and closure of orthonormal systems. Weierstrass
theorem of uniform approximation of continuous function by trigonometrical
polynomials. Closure of trigonometrical system. Trigonometrical Fourier
series. Conditions of uniform convergence and term-by-term
differentiability. Module of continuity. Holder classes. Dirichlet kernel.
Uniform convergence of Fourier series for function from Holder class. The
principle of localization. Conditions of uniform convergence of
trigonometrical Fourier series on random segment. The theorem about
convergence of Fourier series in a point. Summability of trigonometrical
Fourier series for continuous function by arithmetic means method. Fejer
theorem. Gibbs phenomenon. Fourier integral.

The program compilers are: Prof. V.A. Iljin;
Prof. I.S. Lomov

Complex variable function of theory.

Complex numbers. Operations with complex numbers. Geometrical
interpretation. Point at infinity. Stereographic projection. Sequences and
serieses of complex numbers.
Complex variable functions. The concept of region. Complex-value
functions of real variable. Jordan's lines. Complex-value functions of
complex variable. Continuity of functions.
Analytical functions. Differentiability of complex variable functions.
Cauchy-Riemann conditions. Derivative. Geometrical sense of modulus and
argument of derivative.
Integration of complex variable functions. Definitions and properties of
integral. Cauchy theorem. Indefinite integral. Newton- Leibniz formula.
Cauchy integral formula. Average value formula. The principle of absolute
value's maximum for analytical functions.
Representation of analytical functions by means of series. Properties of
power series. Abel theorem. Taylor theorem. Cauchy inequality. Liouville
theorem. Morer theorem. Uniform convergent series of analytical functions.
Weierstrass theorems. Uniqueness theorem. Laurent series. Isolated singular
point's of analytical functions, their classification.
Analytic continuation. Branches of analytical functions, Riemann
surface. Analytic continuation through the boundary. Analytic continuation
by the power series. The concept of full analytical function. Elementary
functions of complex variable.
Residues. Definition of residue. Formulae of calculation of residues.
Cauchy theorem about residues. Integral Cauchy formula for analytical
functions in unbounded areas. Integrals' calculation with the help of
residues. Jordan lemma. Logarithmic residue. Argument principle. Roushe
theorem.
Conformal mappings. The principles of biunique correspondence,
correspondence of boundaries and symmetry. Riemann theorem formulation.
Linear-fractional mapping and their properties. Conformal mappings defined
by elementary functions.
Harmonic function. Definition. Properties. The mean value theorem. The
uniqueness theorem. Extremum principle. Liouville theorem. Invariance
relative to conform all mappings. Dirichlet problem. Reduction to the
circle. Solution of Dirichlet problem for the circle, Poisson integral
problem function source of the first boundary value of construction for
Laplace equation.
Laplace transformation. Definition. Analyticity of Laplace
representation. Properties of representations. Source determination by the
representation.. Mellin formula. Examples of calculation of Mellin
integral. Problem solution for linear differential equations by operations
research method.

The program compiler is: Prof. V.S. Serov


Ordinary Differential Equations.
(3-4 semesters; Lectures-68 ac.h.; Seminars-68 ac.h.)

The Subject of Differential Equations. Mechanics, Physics and other
natural sciences as the source of basic notions of differential equations.
Mathematical model of a process or an effect as the subject of research in
the theory of differential equations. Dialectics of mathematical model
construction.
Notion of approximate integration methods of ordinary differential
equations. Types of equations which are integrable in quadrature.
Existence and uniqueness of Cauchy problem solution theorem for one
equation and for system of equations of the first degree. Continuous
dependence of Cauchy problem solution according to the initial conditions
and to the right part (to the parameter). Differentiation of solution along
argument and parameter. Integral inequality.
Linear Equations. Linear equations with variable coefficients, general
theory, linear dependence and independence of solutions, Wronski
determinant, general solution, variation of parameters method. Linear
equations with constant coefficients, Euler equations. Nonhomogeneous
equations with constant coefficients and with right part in the form of
quasipolynomial. Fundamental solution system construction with the help of
rows. Sturm matching theorem.
Linear Ordinary Equation System. Homogeneous systems general theory.
Nonhomogeneous systems, variation of parameters, source influence function.
Systems of linear differential equations with constant coefficients.
Setting of Boundary-Value Problem. Representation of nonhomogeneous
equation solution with the help of Green function; eigenvalue problems.
Statement of Steklov theorem.
Elements of Qualitative Ordinary Differential Equations Theory. Singular
points classification. Limit cycle. Stability theory. Stability according
to Ljapunov. Ljapunov second method, Ljapunov and Chetaev functions.
Linearization of stability by the first approximation theorem.
Approximate Decomposition. Linear equations with coefficient
peculiarities, approximate decomposition for great x. Regularly perturbed
ordinary differential equation systems. Poincare theorem. Singularly
perturbed equations. A.N. Tichonov theorem. Equation systems in standard
form. N.N. Bogoljubov averaging method.
Linear and quasilinear partial differential equations of the first
degree, characteristic system (eigensystem), Cauchy system. General
solution.
Elements of Calculus of Variations. Concept of functional. Variational
problem for elementary functional. Extremum necessary conditions. Function
and functinal variation. Calculus of variations basic lemma. Euler
equation.
Functionals which contain derivatives of the more than first degree and
which depend on several functions, Many-dimensional variational problems,
Euler-Ostrogradskii equation. Variational principles of mechanics.

Moving boundary problems.

Conditional Extremum Problems, Lagrange indeterminate multiplier method.
Optimal Control Problems. Notion about Pontrjagin maximum principle.

The program compiler is: Prof. M.M. Khapaev;
Prof. V.I. Blagodatskikh



Physics.
(3-6 semesters; Lectures-136 ac.h.; Seminars-104 ac.h.)

Classical mechanics. Matter and movement. Space and time as the forms
of matter existance. The methods of physical research. Physics and
mathematics. The concept of numerical experiment. Modern physics and
computational mathematics.
The subject and the main problems of mechanics. The basic concepts of
kinematics.
The principle of a relativity in the mechanics. Inertial and uninertial
systems of reference. Newton laws. The concepts of force and weight. The
law of world gravitation.
The classical mechanics and mathematical model of the real world.
Opportunities of specification of this model.
Impulse. The law of conservation of impulse. Jet movement. Meschreky
equation. Ciolkovsky research.
Work and energy. Kinetic and potential energy. The law of conservation
of energy in mechanics.
Forward and rotary movement of a rigid body. Center of weights. The
theorem of movement of center of weights.
The equation of the moments. The law of conservation of moment of
impulse. Kepler low Le Verrier discovery.
Movement of a body in uninertial system of reference. Forces of inertia.
The moment of inertia of solid body. Rigid body flat movement. Movement
of a body fixed in point. Euler equation. Gyroscopes.
Systems with connections. Number of degrees of freedom. Lagrange
equations. Lagrange function. Mechanical fluctuations. The problem of small
fluctuations.
Numerical integration of a movement equations.
Molecular and Statistical Physics. Methods of molecular and statistical
physics.
Molecular structure of substance. Brown movement. Rigid, liquid and gas
forms of matter.
The model of ideal gas. Clapeyron equation and its proof based on me
molecular-kinetic theory. Absolute temperature. Bolcman constant.
Statistical laws of set of molecules. Gibbs distribution. Maxwell
distribution by speeds . Âîltzmann law for gas in the field of forces.
Mathematical modeling of behavior of system of many molecules. The
phenomena of carry.
Thermodynamic method. Thermodynamic parameters; pressure, temperature,
volume. The equation of a condition of gas.
Internal energy of gas. Distribution of energy on degrees of freedom.
External work and quantity of heat. The first law of thermodynamics.
Adiabatic processes. The adiabatic equation for ideal gas.
Isotermic and isobar processes. Entalpy as the thermodynamic potential.
Convertible and irreversible processes, Carnot cycle. The principle of
work of thermal machine and its efficiency. The second law of
thermodynamics.

Entropy as the function of condition. The formulation of the second law
of thermodynamics with the help of the entropy. Free energy and connected
energy. Communication (connection) of free energy with the statistical sum.
Expression for the entropy through probabilities. Entopy and free energy of
the ideal gas. The philosophical sense of the word entropy.
Statistical property of the second law of thermodynamics. Fluctuations.
Role of interaction of molecules. Van der Waals gas . Physical sense of
van der Waals constant. Van der Waals isoterms. Critical temperature. Joule-
Thompson phenomena.
Polarization of dialectic in the electrical field as a thermodynamic
process. Account of the statistical sum and the dielectric polarization.
The concept of correlation function the Naqvist formula. Examples of
correlation functions for physical systems.
Electricity and magnetism. Electrical charges and currents. Coulomb law.
Electrical field of motionless charges. Intensity of field and potential.
Polarization of dielectrics. Vector of polarization. Vector of electrical
induction. Ostrogradki-Gauss theorem.
Magnetic field of stationary currents. Ampere law. Vector of induction
of magnetic field. Biot-Savart law. Vector of magnatation of field. Vector
of intensity of magnetic field. The theorem of circulation.
Lorentz force. Faraday law of electromagnetic induction.
Electrical currents in environment. Ohm law. Joule-Lentz law. The
equations of circuits with the concentrated elements. Capacity, inductance,
resistance. Electromagnetic energy.
Alternating current. A method of complex amplitudes.
Current of displacement. Maxwell system of equations. The material
equations of environment. Linear material equations - as one of possible
models of real environments. Nonlinear environments. Boundary conditions.
Scalar and vector potentials. Equations for potentials. Speed of
electromagnetic waves distribution.
The law of energy conservationin electrodynamics. Umov-Pointing vector.
Pressure of electromagnetic field. Lebedev experimant. Impulse of
electromagnetic field.
Lorentz transformations for electromagnetic field. Postulates of the
theory of relativity. Michelson experiment. Lorentz transformations for
coordinates and their consequences. Relative mechanics. Rule of addition of
speeds. Connections between weight and energy.
Wave processes. Generality of wave movement in nature. Transmission of
information and energy by wave.
Description of wave phenomena. Harmonic wave. Longitudinal and cross
waves. Polarization of cross waves.
Linearized equations of acoustics. Sound waves; force of sound.
Electromagnetic waves; electromagnetic wave as a cross wave. Flow of
energy, impedance of wave.
Mathematical problem of wave distribution various environments. Physical
interpretation of stability conditions of differential circuits for wave
equations.

Modulated wave. Superposition of harmonic waves, pulsation, ampitude-
modulated wave. Radio broadcasting. Wave package; correspondencebetween
duration and strip of frequencies. Synchronization of styles in laser.
Spectral analysis. Spectra of periodic and single pulses. Damping
harmonic oscillator; natural width of line.
Fourier discrete transformation; Nyquist frequency; imposing of
frequencies; Gibbs oscillation.
Distribution of waves in dispergic environment. Group speed. Normal and
abnormal dispersy. Waves in. chains. Model of wave in ionn crystal;
acoustic and optical branch. The classical theory of light wave dispersy;
spectra of absorption. Dispersy of electromagnetic waves in ionosphere, in
waveguide.
Reflection and refraction of waves. The phenomenon of complete
reflection. Fiber optics.
The phenomenon of interference.. Cogerrent waves. Time and length of
congerrence of waves. Partial interferention of cogerrent waves. Classical
interferention of the circuit. Interferention in roars: interferring
coverings. Standing waves. Michelson interferator. Interference of waves
from chain of cogerrent sources. Antenn lattice, radio telescopes. The
Fabry-Perot standard.
Physical basis of diffraction of waves. principle of Huygenc-Fresnel.
Fresnel zonez, zoned plate. Fresnel diffraction on opening, edge of screen.
Aapproximation of geometric optics. Fraunhoffer diffraction on gap. Lattice
of diffraction.
Spectral analysis of spatial - modulated waves; angular spectrum,
diffractional divergence. Parabolic approximation in the diffraction
theory.
Optical methods of information processing. Fourier transformation in
space. Optical processors.
Notion of golography.
Classical model of optical fluctuations of atom. Radiation of
elementary vibrator. Spontaneous and compelled radiation. Amplification of
optical waves. Laser. Application of lasers in science and technic.
Distribution of waves in nonlinear environments. Formation of shock
wave. Self-influence of waves. Notion of numerical methods of problem
solving related to self-influence of waves.
Physical Basis of Computers. Physical principles of transmission
processes, processing and storage of information. Analog and digital
signals. Integrated microcircuits basis of modem computer.
The principle of work of basic semi-conductor devices which integrated
microcircuits are made of. Semi-conductor diodes and triodes. Devices with
charging connection. Optoelectronic devices.
Basic radio devices which are included into the structure of computer.
Amplifiers of signals. Operational amplifiers. Comparators. Generators of
electrical signals. Multivibrator. Trigger. Logic circuits. Elementary
binary counting circuit. Memory device.
Magnetic elements of a computer. Principles of magnetic record and
processings of information. Application of ferromagnetic materials.
Ferromagnetic domains. Thin-film remembering devices.

Application of optical methods in computer facilities. Optical methods
of data transmission and. Basic opportunities of optical computing devices.
Note. The lectures on "classical mechanics", "molecular and statistical
physics" and "electricity and magnetism" are accompanied by tests (á). On
the subjects "wave processes" and "physical basis of computer work" the
performance of independent tasks is stipulated. The council of the
department can change one of themes of the program with another theme
depending on the specifity of preparation of experts at this particular
university.

Literature:

1) Strelkov S,P. Mechanics» M, 1975.
2) Olchovaki LL «A course of a theoretical phisics for the phisaicians» M,
1975.
3) Stratonovich R.L. Poliakova M.S. «Òhå basis of the molecular ðhósics»
thermodynamics and statistical physics». M, 1975.
4) ReiffF, «Staistical physics». M, 1975.
5) Kalashnickov S.G. «Electricity». M, 1976.
6) Matveff A.N. «Electricity and magnetism» M, 1983.
7) Krauford F. «Waves». M, 1976.
8) Kapteov LN. «Physics of the details of the computer». M.1983.

The pogram compilers are: Prof. S.A. Akhmanov;
Prof. B.B. Bukhovtsev;
Prof. S.S. Chesnokov;
Prof. R.L. Stratonovich;
Prof. V.I. Shmal'gausen;
Prof. V.P. Kandidov;
Prof. L.N. Kaptsov

Introduction to computing mathematics.
(4 semester, Lectures - 32 ac.h. Seminars 32 ac.h.)

Mathematical modelling is a method of knowledge of real world laws and
their use in practical activity.
Mathematical models, primal and inverse problems. The role of numerical
methods, orientated towards ECM, in the research of complex mathematical
models.
Ideas about incorrect problems. The computing experiment. Computing
mathematics and scientific and technical progress.
Function of continuous and discrete argument. Nets, network funtions,
network norms. Projection of function of continuous argument on the net.
Interpolation.
Numerical integration. Formulae of rectangles, trapezoids, Simpson.
Methods of bound of error. Raising of exactness, Runger's rule. Numerical
methods of decigion of linear algebraic systems. Gauss' method. Iteration
methods: method of direct iteration, Zeidel's, upper relaxation.
Numerical decision of Cauchy' s problem for ordinary differential
equations. Methods of Eilier, Runger - Cuitt, Adams. Error of
approximation, convergence, exactness. Bourd of error of decisions on
results of calculations with different steps. Rasing of exactness.
Numerical methods of decision of boundary problems for ordinary
differential equations. Boundary value problem for stationary heat
equation. Construction of difference scheme with integral-interpolating
method (balance method). Homogeneaus conservative schemes. Maximal
principle, existence of unique decision of difference problem. Decision of
difference problem with driving method. Computing stability of driving
method. Formulae of summation by parts, difference formulae of Green. Self-
adjointness and fixed sign of difference operator. Bound of decision of
difference problem. Error of approximation, stability, covergence,
exactness. Boundary problem about characteristic values and it's difference
approximation.
Problem of error in computing mathematics. Error of model, method, input
data and computing process. Badly conditioned and incorrect problems.

Literature:

1. Samarsky A. A. Introduction to numerical methods. M.1982
2. Kalitkin. Numerical methods. M.1978
3. Kucherov, Luckshin. Problems of numerical methods. M.1980


The program compiler is: Academician of the Russian Academy of Sciences,
Prof. D.P. Kostomarov
Mathematical Analysis.
(5-6 semesters, stream A, Lectures 84 ac.h.,
Seminars 16 ac.h.,
7-8 semesters, stream B, Lectures 68 ac.h.,
Seminars 32 ac.h.)

Part I. Theory of functions of the real variable.

THEORY OF SETS. Operations with sets. The principle of duality. Upper
and lower limits of the set. Mapping of sets. Cantor-Burnstein theorem. The
power of the power set.
METRIC SETS. Closed and opened sets. Theorems of unification and
intersection of a countable number of opened and closed sets. Completeness
of the metric set. Contraction mapping principle.
LEBEG MEASURE AND MEASURABLE FUNCTIONS. Measure of the opened set on
the line, its properties. Measure of the open set in a multidimensional
space. Measurable sets. Measurability of a closed set. Countable additivity
of measure. The example of a nonmeasurable set. Measurable functions.
Measurability of the sum, difference, product and quotient of functions.
The limit of measurable functions. Theorems of convergence almost
everywhere and by measure.
LEBEG INTEGRAL. Determination of Lebeg integral for bounded function.
The properties of Lebeg integral. Lebeg theorem of integrability of
measurable and bounded function. Lebeg integral from positive unlimited
function. Its properties. Theorems of absolute continuity and countable
additivity of the integral. Lebeg integral from an arbitrary unlimited
function. Lebeg, Levi and Fatous theorems of a passage to the limit under
the integral sign. H(lder inequality, Minkowski inequality. Completeness of
the space Lp. Theorems about the density in Lp of the set of continuous
functions. Fubbini theorem (without a proof).

Part II. Functional analysis.

TOPOLOGICAL SPACES. Main notions. Compact sets in topological spaces.
Countably compact sets.
METRIC SETS. Hausdorff theorem of a metric space supplement. Baire-
Hausdorff theorem of category. Theorem of a countable compactness of the
totally limited sets. Ascoly-Artzel criterion for compactness in C(K).
Reisz criterion for compactness in Lp.
BANACH SPACES. Banach theorems of boundedness of a continuous
operators. Banach-Steinhaus theorem of proportional boundedness of a
sequence of operators. Hahn-Banach theorem of an extension of a linear
functional. A dual space to Lp. Theorem of a weak completeness in Lp.
Theorem of a weak countable compactness of a solid sphere in Lp.
HILBERT SPACES. Theorem of a least element of a convex limited set in
Hilbert space. B. Levi theorem of an orthogonal projection. Reisz-Frechet
theorem of representation of a linear bounded functional. Schmidt process
of orthogonalization. Reisz-Frechet theorem. Theorem of a compactness of a
totally continuous operators. The first Fredholm theorem. The second
Fredholm theorem. The third Fredholm theorem. Spectrum of a linear
continuous operator. Spectrum of a self-conjugate totally continuous
operator. Hilbert- Schmidt theorem for totally continuous self-conjugate
operators.

Literature

Colmogorov A.N. , Famin S.V. "Elemrnts of the functions theory and a
functional analysis". Moscow 1969.

The program compiler is: Prof. Y.I. Moiseev.

Numerical Methods.
(5-6 semesters; Lectures-68 ac.h.; Seminars-32 ac.h.)

Algebra Numerical Methods.
Gauss method for linear algebraic equation system solving. Compact
scheme for Gauss method. Connection with matrix factorization. Gauss method
with main element choice. Inversion of matrix by means of Gauss-Jordan
method. Square root method.
Convergence research of one-step iterative methods for linear algebraic
equation system solving. Chebyshev polynomial. Explicit and implicit
iterative methods with Chebyshev set of parameters. Conjugate gradients
method.
Power method and inverse iteration method for eigenvalue partial problem
solving. Notion of eigenvalue complete problem solving methods.
Nonlinear equations and equation systems solving. Prime iteration,
Newton, secant methods. Convergence research of prime iteration and Newton
methods.

Function approximation.
Optimisation of points of interpolation distribution. Interpolation with
multiple points. Spline interpolation. Rational function approximation.
Examples of multivariable function interpolation.
Best mean square approximation of functions with continuous and
discrete argument. Function smoothing. Fast discrete Fourier approximation.
Notion of best proportional approximation polynomials. Disjoint linear
algebraic equation system solution by means of method of least squares.

Numerical Integration.
Quadrature formulae of interpolation type. Gauss quadrature formulae.
Calculation of integrals of rapidly oscillating functions. Calculation of
improper integrals. Notion of approximate calculation of multiple
integrals.

Cauchy Problem Solving for Ordinary Differential Equations.
Stability and convergence research of multistep difference methods.
Definition and examples of stiff differential equation systems. Difference
methods for stiff systems integration.




Numerical methods of problem solving of mathematical physics
( 7 semester. 72 ac.h. )

Introductory concepts
Difference approximation of the equations with partial derivatives. The
difference schemes for one-dimensional heat conduction and string
oscillations equations. The theorem of connection between stability and
convergence of the difference scheme. Construction of the difference
schemes by a method of balance. Examples of the economical difference
schemes for a many-dimensional heat conduction equation. Nonlinear heat
conduction equation and its difference schemes.
The elements of the theory of difference methods
Maximum difference schemes principle and its corollaries. Stability and
convergence proof in the uniform metric of Dirichlet difference problem for
Poisson equation. Monotone difference schemes.
Eigenvalues problem for the second difference derivative operator and
for Laplace five-point difference operator. Research by the variables
separation method of stability and convergence of the two-sheeted
difference schemes for one-dimensional and two-dimensional heat conduction
equations. Canonical form and conditions of two-sheeted and three-sheeted
difference schemes stability.
The methods of grid equstion solution
Matrix drive method and its application for Poisson difference equation
solution. Evaluation of sums by Fourier fast transformation method.
Solution of Poisson difference equation with the Fourier fast
transformation. Decomposition method. Application of iterative methods for
solution of grid boundary problems. Simple iteration methods, method with
Chebyshev's variable directions parameter set. Alternately - triangular
iteration method.
Concept of other methods of mathematical physics problem solving
Finite element method. Variation-difference schemes. The solution of
first order equation hyperbolic sets by the method of performances. Methods
of integral equations solution. Fredholm's equations of the second kind.
Mechanical quadratures method. Method of the kernel replacement to
degenerated one. Successive approximations method. Solution of the
Voltairre's second kind equations. First kind integral equations. Method of
incorrect problems regularization.

Literature

SamarskyA.A. «Introduction into Numerical Methods». Moscow, 1982.
SamarskyA.A. «The Theory of Diference Systems». Moscow, 1983.
The program compilers are: Academician A.A. Samarsky;
Prof. A.V. Gulin;
Prof. A.P. Favorsky

Methods of Mathematical Physics.
(5-6 semestors, lectures - 68 ac.h. seminars - 68 ac.h.)

Subject of mathematical physics. Physical fenomena and their
mathematical models. Mathematics as method of research of physical fenomena
and their using in practical activities.
Role E. C. M. ( Electronic - calculating machine ) in the mathematical
physics.
Classification of equations in the private derivatives of the second
oder and reducing it to canonical form.
Equation of hyperbolic reducing type. Physical objects leading to
equation of hyperbolic type. Koshi problems oscillation equations. The
Kirghov-Sobolev formula and spreading of waves in the boundless. Space
solution of Koshi object. Generalized numerical methods of solution.
Statement of boundary problems for the oscillation equation. Energy
inequalities and the theorem of singlularity. The method of variable
division.Own values and own functions of boundary problems. Special
functions.
Equations of parabolic type. Physical objects problems reducing to
equation of parabolic type. The principle of maximum, theorems about
uniqueness. Methods of variable division The basic solution of heat-
conducting equations. Solution of the problems the boundless spaces. The
theorems of singularity. Integral of Puasson. Boundary difference methods
of solution straight of heat conducting equations.
Equations of elliptical type. Phyical problem reducing to elliptical-
type equations. Equation of Laplas, general properties of harmonical
functions. Statement of boundary problems. Method of variable separation.
Basic solutions of Laplas equations. Potential theory. Reducing of boundary
problems to the equations of elliptical type to integral equation. Solution
of the boundary problems by boundless-diference methods. Concept of a prior
and estimation method of analytical lasting. Gelmgolts equation. Extremal
extremal boundary problems and the principle of radiation.
Variation methods of solution of mathematical-physical problems.
Extremal properties of own meanings. Methods of Galerkin and Ritc. Examples
of numerical realization of straight methods for solution of mathematical
physical problems.

Literature:

1.Tihonov A.N.,Samarskiy A.A. Equations ofmatematical
physics.M.,Nauka,1972.
2.Budak B.M.,Samarskiy A.A.,Tihonov A.N. Objects of mathematical
physics.,Nauka, 1972.
ADDITIONAL LITERATURE:
1.Arsenin V.Y.Methods of mathematical physics and spesial
functions.M.,Nauka,1974.
2.Vladimirov V.S.Equations of mathematical
physics.M.NAUKA, 1976. S.Sobolev s.1. Equations of mathematical
physics.M.,Nauka, 1966.

The program compiler
prof Kostomarov D.P., prof Suchko V.G.
Optimal Control.
( 5 semester, Lectures-36 ac.h.)

General setting of optimal control problem. Basic questions of optimal
control theory.
Nonempty compact sets space. Operations of sum of sets and of
multiplication of set and number. Hausdorff distance.
Support function. Geometric meaning of support function. Examples.
Support function properties. Support function continuity along both
arguments. Expression of Hausdorff distance by means of support function.
Multiform mappings. Multiform mapping continuity.
Measurable sets, measurable functions, and their properties. Lebesgue
integral, its properties.
Measurable multiform mappings, their properties.
Integral along multiform mapping, its properties.
Continuity of integral for multiform mapping by upper limit.
Linear speed problem. Basic questions of optimal control theory.
Linear differential equations. Caratheodory theorem.
Set of accessibility, its properties.
Controllability problem. Theorem about controllability.
Lemma about inner point of integral.
Local controllability problem. Theorem about local controllability.
Optimal control existence theorem.
Pontrjagin maximum principle, its geometrical sense.
Necessary optimum conditions theorem.
Application scheme for necessary optimum conditions.
Sufficient optimum conditions theorem.
Sufficient optimum conditions for speed problem in the origin of
coordinates.
Notion of synthesis problem.
Optimal control uniqueness.


The program compiler is: Prof. V.I. Blagodatskikh
Probability Theory and Mathematical Statistics.
(5,6 semesters; Lectures-68 ac.h. Seminars-52 ac.h)

Discrete Probability Space. Probability space. Events independence.
Conditional probability. The formulation of the events' unification
probability. Composite probability formulation. Bajes formulation. Random
variable. Mathematical expectation. Variates expectation. Mathematical
expectation's additivity and multiplicativity. Moments. Additivity of
independant variates' sum's dispertion. Generating function. Binomial
distribution. Puason distribution. Negative binomial distribution.
Binomial distribution. Approximation with Puason distribution. Event's
indicator. Markov inequality. Tne law of huge numbers. Probability's proof
of the theorem about uniform approximation of continuous function on a
segment with polynomials.
Common Probability Space. The definition and construction of probability
space. Probability's space replenishment. Continuous probability space.
Variable definition. Operations with random variables. Variable equivalence
and convergence almost everywhere Variates' independence. Mathematical
expectation and its qualities. Distribution function. Distribution's
density Perfomance functions and elementary qualities. Normal distribution.
Gamma distribution. Beta distribution. Perfomance functions of vectorial
variates and qualities. Multidimentional normal distribution.
Processes.Random process definition, trajectories. Processes'
equivalence. Formulation of Kolmogorov theorem about random process'
existence in the specified family of coordinated finite-dimentional
distributions. Uniform calculating Markov series with discrete continuous
time definition. Examples. Limit distribution of uniform Markov series.
The formulation of the theorems. The LHN for uniform Markov series with
discrete time. The LHN for uniform Markov series with continuous time.
Structure of uniform Markov process with final set of states and continuous
time. Structure of the destruction and multiplication process with final
set of states. Stational distribution of the process. Reconstruction
process. Existence of all moments of reconstruction's numbers.
Reconstruction equation. Formulation of elementary reconstruction theorem:
Formulation of Blackuel theorem and nodal reconstruction theorem.
Regenerating process definition. Formulation of limiting theorem for
regenerating process. Gauss process and it's existence. Wiener process and
qualities of its trajectories. Forms of variables convergence and relations
among them. Construction of ortogonal measurement with values in Hibert
space. Stational random processes in the broad sence. Spectral
disintegration.
Statistics. Formulation of statistical problem. Risc function. Strategy
with uniform minimum risc. Regulation of strategies. Bayes approach.
Minimax approach. Statistical structure. Statistics. Sufficient
statistics. Full and free statistics. Sufficient condition of strategy with
uniform minimum risc existence. Minimum plausibility's estimation (MPE) for
multidimentional normal distribution parameters. Consistency of MPE for
multidimentional normal distribution's parameters. Normal equation of the
method of the least squares (MLS). The first two moments of the MLS
estimation. Unbiased estimation of errors' dispertion in MLS. Gauss-Markov
theorem for MLS estimation. Consistency of MLS estimation. Regression
linear model. MLS in the case of normal distribution of measuring errors.

The program compiler is: Prof. G.P. Klimov





Programming Languages
( 5-6 semesters lectures -102 ac.h. )

Programming Services Production as the Main Purpose of Programming.
Characteristics of synthesised industrial programming. Complex applications
software. Software life-cycle. Main characteristics (reliability,
portability, resource capacity, cognoscibility). Complexity as the main
problem of programming. Language as a technological means of representation
of data and knowledge during all the stages of software life-cycle.
Programming Language as a Sign System for the Planning of Executor's
Conduct. Message passing model. Syntax, semantics, pragmatics. Five main
stages during language examination: technological, author's, semiotic,
mathematical. Abstract-concrete definition method as the main means to
avoid complexity.
Programming Languages Models. Neumann model. Main semantic correlation
(free access and stored information principle and semantic function).
Modified Markov model (refal). Analysis and synthesis as the main
abstraction of this model. Main semantic correlation. Means of development.
Bakus model. Criterion of conceptual clearness and functions of higher
degrees. Main forms of structurasation. Algebra of software and its
application for basing of programme correctness.
Main Abstractions and Principles of Programming Language Construction
(after the example of the basic industrial programming language). Principle
of technological nature of abstraction. Data, operations and binding as the
main abstractions, their unity and relativity. Binding and compilation
theory. Principle of correlation of main abstractions. Modularity as the
materialisation of abstractions. Principle of specification division,
realisation and usage. Principle of abstraction defence. Encapsulation as
the realisation of abstraction defence principle. Data classification as
the basis for prediction and planning of their conduct. Type and concept of
it as a unique object. Crucial technological necessity. Abstract-concrete
definition peculiarities for external language conceptions (notation,
numerical calculation model, input-output, etc.). Main standards.
Main Constructions of the Basic Industrial Programming Language.
Segments. Declarations. Operators. Constructions connected with the
conception of type. Dynamic objects and reference types. Asynchronous
processes and shareable resources. Common model of numerical calculations.
Exceptional cases. Representation control. Separate compilation. External
environment exchange.
Structure of Programming Language Realisation. Realisation components.
Classic compilation scheme. Concept of compiler creation technology.
Projection principle. Language rapprochement principle. Parametric
compilation systems. Principle of syntactic control of compilation.
Syntactic analyser as the example of parametric compilation module.
Portability of compilers and bootstrap principle. Means of realisation
control concept.
Mathematical Models of Language and Demonstrative Programming.
Syntactic, context-syntactic and semantic models. Operational, denotative,
and deductive semantics. Application of deductive semantics for the proof
of programmes' properties.
Problems and Tendencies of Programming Languages Development. Short
overview of the most significant languages. Problem orientation, personal
orientation, applied software packages. System method of approach to the
programming language. Language niche. Programming languages
standardisation. Programming language as a social phenomenon.

The program compilers are: Prof. V.Sh. Kuafmann



Operating systems.
( 5 semester, Lectures - 30 ac.h. )

Structural organization of modern computer systems. Problems of their
work control.
Distribution of functions between hardware and operating system.
Virtual machines. Means of hardware support of multiprogramming mode, time
sharing mode and on-line mode.
Operating systems main functions. Operating systems functional
structure. Technology of operating systems construction. Operating systems
objects.
Organization of process control. Planning of tasks stream passing. Main
types of RAM control. Operating systems' functions for memory control.
Organization of external devices control in computer systems. Operating
systems actions on controlling external devices work and organizing
information input/output. Organization of multiprocessor complexes
operating systems. Functions of multimachine complexes and computer
networks operating systems.

Literature

1. L. N. Korolev. "Structures of computers and mathematical software for
them", M., "Nauka", 1978 (2nd edition). (Ë. Í. Êîðîëåâ. "Ñòðóêòóðû ÝÂÌ è
èõ ìàòåìàòè÷åñêîå îáåñïå÷åíèå", 2 èçäàíèå, M., "Íàóêà", 1978.)
2. D. Cikritzis, F. Bernstein. "Operating systems", M., "Mir", 1977. (Ä.
Öèêðèòçèñ, Ô. Áåðíñòàéí. "Îïåðàöèîííûå ñèñòåìû", M,, "Ìèð", 1977.)
3. J. Donovan. "System programming", M., "Mir", 1975. (Äæ. Äîíîâàí.
"Ñèñòåìíîå ïðîãðàììèðîâàíèå", M., "Ìèð", 1975.)
4. "Supervisor of OS ofES computer", M., 1975. ("Ñóïåðâèçîð ÎÑ ÅÑ ÝÂÌ", M.,
1975.)
5. "System of a mathematical software for ES computers", M., "Statistica",
1974. ("Ñèñòåìà ìàòåìàòè÷åñêîãî îáåñïå÷åíèÿ ÅÑ ÝÂÌ", M., "Ñòàòèñòèêà",
1974.)

The program compiler is: Prof. A.N. Tomilin.

Additional chapters of equations of mathematical physics.
(7,8 semesters, stream A, Lectures - 86 ac.h. seminars - 32 ac.h.)

Modeling of some physical phenomena in terms of equations in partial
derivatives. Equations of the theory of elasticity. Navier-Stock equations.
Equations of the theory of relativity.
The main types of equations in partial derivatives and problems
correctly formulated for them. Concept of the characteristic form and
division into types of equations (generally speaking, nonlinear) in partial
derivatives. Classification of linear equations in partial derivatives.
Linear problems correctly formulated for three main types of linear
equations (elliptic, hyperbolic and parabolic). Cauchy problem for equation
of Cauchy-Kovalevskaya type. Equations of mixed type.
Method of integral equations of solution linear equations in partial
derivatives. Idea of parametrix. Elementary (fundamental) solutions of
linear equations in partial derivatives. Methods of theory of functions of
complex variable.
Generalized solutions of classical problems of mathematical physics.
Equations the smoothness degree of variational problems solutions and the
idea of derivative. Different extensions of generalization idea of solution
of equations.
Methods of separation of variables. The common method of separation of
variables. Method of integral transformation. Fouier method. Spectral
properties of some problems for equations of mathematical physics.
Some classes of nonlinear equations in partial derivatives. Dirichlert
problem for some classes of nonlinear equations of elliptic type. Monge-
Ampere equation of Becklund transformation. Classes of nonlinear equations,
which can not be linearized. The notion with the help of replacement of the
functions in search .

Literature.

1. Adamar . Cauchey's problem for linear equations with partial derivatives
of hyperbolic type. M.1978
2. Bitsadze. Some classes of equations in partial derivatives. M.1981
3. Vladimirov. Equations of mathematical physics. M.1976
4. Curant R. Equations with partial derivatives. M.1964
5. Ladizenskaya 0. A. Boundary problems of mathematical physics. M.1973
6. Miranda K. Equations with partial derivatives of elliptic type. M.1957
7. Sobolev S. L. Some applications of functional analysis in mathematical
physics. Novosibirsk 1962.
8. Tichonov A. N. Equations of mathematical physics. M.1966

The program compiler is: Academician of the Russian Academy of Scieces,
Prof. A.V. Bitsadze
Mathematical Logic
( 7 semester, Lectures - 36 ac.h.)

INTRODUCTION. Basic propositions and laws of classical formal logic.
Usage of mathematical logic in contemporary science
BASIC CONCEPTS AND SYMBOLS OF A FORMAL LOGIC THEORY. The alphabet of the
theory, object variables, object constants, predicate and functional
symbols, copulas and quantors. Prepositional variables. The signature of
the theory. Metasymbols. Individual predicate and functional symbols. Term,
elementary formula, and formula of the given signature. The field of action
of the quantor. Free and constrained variables in the formula, closed
formula, Substitution of terms and formulae.
AXIOMS AND DEDUCTION RULES OF THE FORMAL THEORY. Axioms and schemes of
axioms. Rules of substitution, conclusion and generalization. Term, free
for the variable in the given formula. Axiomatic descriouonof calculation
ofstatements and narrow calculation 9f predicates.
CONCLUSION IN A FORMAL THEORY. Definition of conclusion. Elementary
properties of conclusion. Theorem in a formal theory. Examples of theorems
in narrow calculation of predicates. The theorem of deduction for narrow
calculation of predicates (NCP). and for (classical) calculation of
statements (CCS). Metatheorema in formal theories. Conclusion of some
theorems and the rules of conclusion in NCP (the rule of simple syllogism,
the rule of adding and removal of a double negation, the law of
contradiction in different forms, the law of identity, distribution of
quantors into members of an implication, etc.). The C-rule and C-
conclusion. The theorem about C-conclusion. Usage of the theoremabout C-
conclusion for the proof of a number of theorems of NCP.
NORMAL FORMS FOR FORMULAE OF NCP. Forestalled normal form, The theorem
about reduction of NCP formulae to forestalled normal form. Scolem normal
form. The theorem about reduction of formulae of formulae NCP to Scolem
normal form.
CALCULATIONS OF STATEMENTS. Different descriptions of classical
calculation of statements (with copulas-negation and implication, - and
with copulas-conjunction, disjunction, negation and implication). The idea
of intuitionistic calculation of statements (ICS). Comparing ICS with CCS.
The systems of natural conclusion for CCS, ICS and NCP.
NON-CONTRADICTIVITY OF FORMAL THEORIES. Non-contradictivity in the
absolute sense, in the relative sense arid in Post sense. The tautology of
the theorems of CCS. The non-contradictivity of CCS and NCP.in the staled 3
senses.
COMPLETENESS OF FORMAL THEORIES.IN 3 SENSES. The theorem about the
completeness of CCS in the 3 senses fthe absolute, the relative. Post
sense"). The non-cprnDletenesa of NCP in the narrow sense.
INTERPRETATION OF FORMAL THEORIES. Definition of interpretation.
Execution and significance of the formulae with the given interpretation
and in the given field of interpretation. Logical execution and
significance. Tautologies. The theorem about the therems of NCP tautology.
Model for a formal theory. The statemert about the relative non-
contradictivity of a substantial non-contradictory theory. The substantial
non-contrudictivity of NCP. Spreading Scolem theorem (about reduction to
Scolem normal form) to the case of tautologies.
GODEL THEOREM AND ITS CONSEQUENCES. Godel theorem. The connection
between the significance of a formula of NCP on the natural line, its
concludability and tautology. Levengame theorem (about the connection
between the significance in an account-infinite area and the tautology).
The statement about the execution of a formula of NCP in a finite area and
in an account-infinite area. Examples of the NCP formulae, which are
significant in any finite area and which are not tautological, and of the
formulae, which are executable only in account-infinite areas,
Substantiations of the restrictions, that were put on some axioms and
auxiliary rules of conclusion (the axiom of removal of the quantor of
generality-restriction on the term, which substitutes for the variable; the
axiom about the distribution of a quantor into members of the implication-
restriction, connected with the lack of free entry of the variable in
thepremiseof the implication; the theorem of deduction-restriction on
application of the rule of generalization).
THE IDEA OF ALGEBRAIC SYSTEMS. EXAMPLES OF FORMAL THEORIES. Algebraic
system, its connection with interpretations. AJgebra and model (relative
system). Classes of algebraic systems with a given signature. The
elementary theory of a class of algebraic systems. Possibility of the
organization of an axiom system in a class of algebraic systems.
Calculation of predicate» with equality. The theory of semi-group, the
theory of strict partial order,
THE PROBLEM OF SOLUTION OF A FORMAL THEORY. The formulation of the
problem of solution. The solvability of the problem of solution for CCS.
The immersion of NCP in the formal theory N, connected with the natural
line. Godelization of the objects of the theory N. Description of
predicates of a special kind, reflecting the concludability of some objects
of the theory N from others and from the empty plurality (the property "to
be a theorem") , recourcing - enumerated and recoursive pluralities and
predicates (definitions and elementary propertyes). The formulation of the
theorem about the exbressiveness of general - recoursive functions through
the formulae of the theory N. Chearch theorem (about the non-solvability of
the problem solution of NCP): the proof of the non-solvability of the
problem of solution for the theory N and the reduction of the problem of
solution for NCP to the corresponding problem for the theory N.
INDEPENDENCE OF AXIOM SYSTEMS AND RULES OF CONCLUSION. The statement of
the task. The proof of the independence of the axioms and the rules of
conclusion of NCP (using different interpretation, which use multi-ciphered
(functional) logic, and using special reflections of a class of formulae of
NCP on itself).

THE LIST OF LITERATURE

1 .Gawilov G. P. «Mathematical Logic» M,, 1976
2.Mendelson E. «Introduction to Mathematical Logic» M., 1976
3. Noviko. P. S. «The Elements of Mathematical Logic» M, 1973

ADDITIONAL LITERATURE

l.Cleanie S. C. «Mathematical Logic» M., 1973
Z.Maltscv A. L «Algorithms and Recoursive Functions» M., 1965
3.Chearch A «Introduction to Mathematical Logic» Vol. 1, M, 1960
4.ShenfieldJ. «Mathematical Logic» M., 1975

The program compiler is: Prof. G. Ð. Gavrilov
Operations Research
(7 semester; Lectures - 42 ac.h.)

Methodological Questions Of Operations Research. Operation, operative
side and operation researcher. Criterion of effectiveness, random and
undetermined elements. Mixed strategies.
The Theory Of Two-Person Zero-Sum Games. Maximum functions, their
continuity and convexity. The definition of two-person zero-sum game,
finite games, upper and lower value of the game, inequality between them,
the definition of saddle point. Von Neumann's theorem about the saddle
point. Mixed strategies in finite games, optimal mixed strategies and their
property. Reduction of the simple solution of the finite two-person zero-
sum game. The theorem of prevalence. Reduction of the problem of searching
for optimal mixed strategies fo the task of linear programming. R-
games,reduction of the solution of R-games to the task of convex
programming.
The Theory Of Graphs. Definition of the graph, connectedness of graph.
Tree, the number of the edges of a tree. Vectors of incidence of the archs
and edges of a graph. Incidence matrix of a graph. The theorem of adduction
of the tree's incidence matrix fo the semitriangular form. Necessery and
sufficient condition for graph being a covering tree.
The Foundations Of A Net Planning. Net diagram, events, works. classes
of events, enumeration of the events according to their classes. Value of
the path, the main theorem of a web planning. Scheduling plan. The problem
of the linear programming for a scheduling plan of a minimal period, the
problem dual to the first one and their solution.
Problems Of Distribution Of The Resources On Web Diagrams. Resources,
distributions of the resources. Time, required for a work, as a function of
a definite distribution of the resources. Problem of the minimization of
the time necessary for realization of all works in terms of scheduling
plan, existence of its solution, values of the realization of all works
under given directive time of their realization formulated in terms of the
scheduling plan, conditions of the existence of this task's solution.
Problems of the distribution of the resources on web diagrams in terms of
vectorial incidence of the paths. Problem of minimization of the time
required for realizing all works in the presence of indeterminate factors,
formulated on terms of scheduling plan, and reduction of this task to the
task of nonlinear programming. Existence of the solution of this task. Task
of minimization of the value needed for performing all works in the
presence of indeterminate factors and under the given directive time for
work realization. Reduction of this problem to the problem of nonlinear
programming formulated in terms of the scheduling plan. Condition for an
existence of the solution of the problem. Problems of the distribution of
the resources on web diagrams in the presence of indeterminate factors
formulated in terms of vectorial incidence of the paths and their reduction
to the tasks of nonlinear programming.. Gambling task of the distribution
of the resources on web diagrams and its reduction to R-game.
Problems Of Synthesis Of Communicational Nets. Streams in nets, theorem
about the maximum flow and minimum cutting. Problems of synthesis of the
communicational nets in determined case, Gailean type tasks as problems of
nonlinear programming. Problems of linear separable synthesis of
communicational nets. Tasks of maximization of the maximum flow in a net in
the presence of indeterminate factors and their reduction to the tasks of
mathematical programming. An existence of solution of such tasks. Tasks of
the minimisazion of the value of synthesis of communicational nets in the
presence of indeterminate factors and their reduction to the tasks of
mathematical programming. Gambling task of the synthesis of the communi-
cationel nets and its reduction to the R-game.
Elementary Problems Of The Distribution Of The Resources Of The Nets,
Hibbs Lemma, Equalization Principle Of Hermayer And Their Correlation.
Tasks of the distribution of the resources on the nets leading to Hibbs and
Hermayer tasks. Necessary and sufficient conditions for effectiveness of
Hibbs' task's solution. Monotonicity of the derivative of convex function.
Convexity of indefinite integral from the monotonical function. Hermayer
principal of equalizing. The correlation of Hibbs lemma and Hermayer
principle of equalizing; their economical interpretation.

The List Of Literature

1. U.Â.Hermayer. An introduction to the theory of operations research.
"Science", M., 1971.
2. A.G.Davydov. The methods and the models of two-person zero-sum games.
"Moscow State University", M., 1978.
3. A.G.Davydov. Games, graphs, resources. "Radio, Connection", M., 1981.

The program compilers are: Prof. A.G. Davidov


Elements of Cybernetics
( 7, 8 semesters, Lectures - 64 ac.h)

Examples of Control Systems. Structures from functional elements on the
basis {&,V, }. Formulae in the basis {&,V, }. Disjunctive normal forms
(d.n.f). Contact schemes. Automata in the basis {&, V, } and delay.
Operational .algorithms (on the example of matrices multiplication). The
notion of control system. The main tasks of cybernetics.
Control Systems' Synthesis. The problem formulation. Two approaches in
the synthesis problem. Construction of minimal disjunctive normal forms
(d.n.f) Simplification of d.n.f. Cul-de-sac d. n. f. Geometrical model and
reduced d.n.f. Construction of reduced and cul-de-sac d.n.f. Cvine and
Churavlev methods. Environing algorithms and evaluation of Cvine and
Churavlev algorithms. Synthesis of s.f.e. Some simple methods of the
synthesis (synthesis on the basis of perfect d.n.f. on the basis of
decomposition along the variable, etc.). The correct introduction of
algebraic logic functions. Algorithm of Lupanov synthesis. Lower evaluation
and a symptotic Shennon function for s.f.e. class. Synthesis of contact
scheme. Some simple methods of synthesis (synthesis on the basis of perfect
s.f.e. and the tree. Shennon method). Cascades method. Method of Lupanov
synthesis. Lower evaluation and a symptotic Shennon function for contact
schemes. Invariant classes, their composition and matrix properties.
Complexity of realization of linear function in class schemes - Khrapchenko
theorem.
Equivalent transformations of control systems. The problem formation and
the problem of equivalent transformations. Equivalent transformations of
algebraic logic formulae. Presentation of Lindon results for the closed
classes of algebraic logic. Equivalent transformations of formulae Ê-signed
logic. Construction of final complete system of identities in one basis, if
the final complete system of identities in another basis is known. Lindon
example. Equivalent transformations of contact schemes. Construction of
complete system of identities. Absence of final complete system of
identities. Equivalent transformations s.f.e. in basis {&.V. }.
construction of final complete system of identities Equivalent
transformations of automata. Agreed canonic schemes of two automata.
Construction of final complete system of identities for automata with
restricted, number of delays. Absence of final complete system of
identities in the common case.
Problems of reliability of Control Systems. The problem formulation) and
different cases control systems' reliability. Statistical and combining-
logical approaches to the solution of the reliability problems.
Construction of reliable s.f.e. from unreliable elements the errors' growth
effect, Feliability increase with the help of elements h(x,y,z) the theorem
of any number of reliable systems existance. Synthesis reliable and
selfcorrecting s.f.e. in the basis from unreliable elements {&.V. } and
reliable element K (x,y,z) a symptotics of the corresponding Shennon
functions. Construction of self correcting contact systems, a symptotics of
the corresponding Shennon functions. The problem of control systems'
testing, tests for the tables. Algorithms of all cul-de-sac tests
construction, evaluation of the minimum unconditional length for almost all
tables. Tests for contact schemes Construction of the tests for block
systems on the example of the linear function scheme. Problem of automa
control, Moore theorem.

The program cmpiler is: Academician of the Russian Academy of Sciences,
Prof. S.V.Yablonsky

PHYSICS
(7-8 semesters, 64 àñ.h.).

Speciality "A"

The main Concepts and Gas Dynamics Equations. The main concepts and
models of gas dynamics. Hypothesis of uniformity ideal gas, adiobaticity,
anthropy of ideal gas. Euler and Lagrange approaches to the discription of
uniformal medium dynamics. Integral form of gas dynamics equations and
Euler variables. Differential equations of gas dynamics and Euler
variables. Integral equations of gas dynamics and Lagrange variables.
Differential equations of gas dynamics and Lagrange variables. Circulation,
the theorem of circulation conservation Bernoulli integral.
One-dimentional unstable gas streams. One-dimensional equations of gas
dynamics. Mass Lagrange coordinates. Acoustic approaches. Sound speed.
Properties and characteristic forms of one-dimentional gas dynamics
equations. Dependence area, influence area, the method of characteristics
for one-dimensional equations of gas dynamics. Simple waves. Simple waves
of rarefaction and compression. The centred wave of rarefaction.
Development of the final magnetic perturbation in one-dimensional geometry.
Gradient catastrophy. Explosive solutions, Hugoiniot correlations.
Adiobatic curve of Poisson and adiobatic curve of Hugoiniot in plane (P,V).
Strong shock waves of weak intensity, entropic changes during the motion
through the shock wave front. Inpossibility of shock wave rarefaction
existence. Poisson adiobatic curve and Hugoiniot adiobatic curve in the
plane (P,T). Disintegration of arbitrary explosion.
Ideal Incompressed Fluid. Incompressed fluid equations. The theorem of
circulation conservation. Bernoulli integral. Potential and stream function
of incompressed fluid. Statement of the problem about the plane stream with
potential. Stream near cyclic cylinder. Complex potential of incompressed
fluid. Stream field of the point source. Vortex dipoles. Complex potential
in the problem about the circulation of the cyclic cylinder by the
incompressed fluid stream. Chapligin formula. Zhukovsky formula about the
lifting force. D'Alembert paradox.
Viscous Incompressed Fluid. Tensors' concept. Second-rank tensors. in
three - dimensional Decartes coordinates. Viscosity. Tendor of viscous
stresses for incompressed fluids. Navier-Stokes equations for incompressed
fluids. Poiseuille stream tension. Reynolds critical number. Concept of
turbulance.
Similarity and Dimension Theory. The main concept of dimension. The
theorem of dimension theory. Regular varying condition of one-dimensional
problems of gas dynamics. Dimensional analysis of piston problems moving
with the constant speed dimensional analysis of the accelerating piston;
point explosion. Fluid motion in pipes, bodies' motion in fluids. Theory of
Elasticity. The main equations of elasticity theory. Tensor of the
deformation. Tensor of the stress. Hook law. Speciality "B".
Computational experiment.
Model of uniformal medium. Lagrange and Euler variables. Equations
of.continuity in Lagrange and Euler variables.
Uniformal medium motion equations. Tensor of tension Equations of ideal
fluid motion.
Some information from thermodynamics. The first and the second laws of
the thermodynamics. The equation of the ideal gas state. Ideal gas entropy.
The equation of warmth's carry.
Conservation laws. Equations of impulse carry. Equations of energy
carry.
Sound waves. Wave equation.
One-dimensional equations of gas dynamics. Equations in Lagrange and
Euler forms. Mass coordinates. Hyberbolicity of gas dynamics equations.
Isotropic stream. Invariants of Riemann. The problems about a piston.
Rarefaction wave. Compression wave.
The Concept of self-similar solution. Streaming into vacuum. Adiabatic
and isotermenal streams.
Explosive solutions. Contact explosion. Shock wave. Coordination on the
shock wave. Hugoiniot adiabatic curve. Calculations of the parametres of
the shock wave. Strong shock wave. Reflection of the shock wave from the
wall.
Shock wave structure. Viscocity. Solution of the traveling wave type.
Solution for the linear and quadratic viscosity.
Elements of dimension theory. The concept of dimension. The system of
units. Transition to dimensionless values. The problem of nonlinear carry
of the warmth from the instantaneous source.
The concept of the difference scheme. The mesh. The error of
approximation. Convergence and stability of the difference schemes.
Difference schemes for the carry equations. The scheme with the directed
difference. Laks scheme. Implicit scheme.
Difference schemes for aucustics theorems. The schemes for Riemann
invariants. The scheme "cross". Laks scheme. Ñonservative scheme.
Difference schemes for the heat conductivity equations.
Solutions of one-dimensional equations of gas dynamics. Fully
conservative difference schemes. Realization of difference schemes of gas
dynamics. Newton method. Sequentional sweep method.
Programming of gas dynamics problems.

The program compilers are: Prof. L.M. Degtjarev;
Prof. V.Y. Karpov



Data Organization, Data Access and Data storage Methods
(7-8 semesters, Lectures - 52 ac.h.)

Data organization
External, logical and physical types of data organization. Data
structures. Relations. Data elements, records, sets, files, keys and
indexes. Sorting out. Rows, graphs, trees. File structures.
Hierarchical order. Data storing. Structures. Data representation. Indexing
methods. Operational system, access methods.
Sorting out.
Internal and external sorting. Internal sorting strategies. Straight
selection sort, straight insertion sort, distribution counting sort.
External sorting methods.
Database system architecture
Basic concepts. Database evolution. Three levels of data models:
conceptual, external and internal levels. Database administrator.
Relational approach to database systems creation
Relational, hierarchical and network approaches. Relational data model.
Relations, domains, attributes, keys. Relational algebra and relational
calculus. Languages based on relational calculus, relational algebra
and mapping theory. Factors of assessment and choosing of relational
languages. Four normal forms. Relational database systems.
Hierarchical approach to database systems creation
System architecture. Data model. Physical databases. Hierarchical order.
Logical databases. Communication with the program block. Basic language
operations. Storage structure. Data independence. Databases description,
loading and manipulation. Secondary indexing. Hierarchical database
systems.
Network approach to database systems creation
System architecture. Data model. Data sets construction. Data
description language. Data manipulation language. Data storage description
language. Network database systems.
Information retrieval systems
Relevant answer searching problems and procedures. Factual and
descriptive systems. Information retrieval systems examples.
Database systems efficiency
Efficiency criterion. Factors of assess and choosing database management
systems. Database systems tendencies.

The program compiler are: Prof. V.S. Minaev and others
Extreme Tasks Solution Methods
(8 semesters, Lectures - 42 ac.h.)

Statement of minimization and maximization tasks. Examples.
Function of one parameter minimization methods: division the segment in
half, golden section, broken lines, tangent lines.
The Veyershtrass theorem about reaching the lowest plane and its
generalizations. The Veyershtrass theorem application to square functional
minimization tasks on the solution of linear system of the ordinary
differential equations and to the task of the optimal heat of the core.
Convex functional. The criteria of functional convexity. Examples.
Functional gradient. Examples. Necessary and sufficient condition of the
optimality for smooth and convex minimization tasks. The rule of Lagrunge
multipliers. The Kuna-Tacker theorem. Applications. Gradient in the task of
the optimal control with the free right end of trajectory.
Hard convex functional, its properties. The criteria of functional hard
convexity. Projection of the point on the multitude, its properties.
Examples.
Function of multiple parameters and functionals minimization methods:
method of gradients, gradient projection, possible directions, conditional
gradient, Nuton's method and coordinate-by-coordinate descent.
Applications.
Correctly stated extreme tasks. Examples.
Incorrect extreme tasks. A.N.Tichonov's regularization methods.
Applications.
Statement of the general task of the optimal control. L.S.Pontriagin's
principle of maximum. Boarder task of principle of maximum.
Dynamic programming.

Literature:

1. Vasiliev P.P. Numerical methods of extreme tasks solution. - M.:
Science, 1980.
2. Vasiliev F.P. Extreme tasks solution methods. - M.: Science, 1980.
3. Karmanov V.G. Mathematical programming. - M.: Science, 1979.
Additional literature:
1. Pontriagin L.S., Boltiansky V.G., Gumkrelidze R.V., Mishenko E.F.
Mathematical theory of optimal requests. Science, M., 1976.
2. Tichonov A. N., Arsenin V.Y. Incorrect tasks solution methods. - M.:
Science, 1979.
3. Moiseev N.N. Elements of optimal systems theory. Science, M., 1975.
4. Moiseev N.N., Ivanilov Y.P., Stoliarova E.M. Optimization methods.
Science, M., 1978.
5. Pshenichny B.N., Danilov Y.M. Numerical methods in extreme tasks.
Science, M., 1975.
6. Evtushenko Y.G. Extreme tasks solution methods and their application in
optimization systems. Science, M., 1982.

The program compiler is: Prof. F.P.Vasiliev.
Methods for solving extremal problems.
(8 semester, Lectures - 42 ac.h.) spec. "B"

Introduction. Statement of the optimization problem. Classification and
examples of optimization problems.
Numerical methods for one-dimensional optimization. Methods for
minimization of the unimodal functions. Methods for minimization ofmulti-
extremum functions. The concept of the optimal algorithms for searching the
extremum.
Elements of the convex analysis. Convex sets. Separability theorems and
their implications. Convex functions.
Numerical methods for unconditional optimization. The gradient method.
Newton's method. The conjugate gradient methods. Quasi-Newtonian methods.
Optimality conditions for the mathematical programming problems. The
optimality conditions for the convex-set minimization problem. Differential
optimality conditions for the mathematical programming problem: necessary
conditions, sufficient conditions. Necessary and sufficient optimality
conditions for the convex programming problem. Theory of duality in linear
and convex programming.
Numerical methods for linear programming. Extreme solutions to the
linear programming problem. Simplex-method.
Numerical methods for nonlinear programming problem. Penalty functions
method. Projected gradient method. Linearization problem. Methods for
solving the quadratic programming problems.
Methods for discrete optimization. Methods for solving integer and mixed
integer linear programming problems. Branch and bound algorithm. Dynamic
programming methods.
Elements of the theory of optimal control. Pontrjagin maximum principle.
Synthesis problem and methods for solving it.

Literature:

1. Vasil'ev F.P. "Numerical methods for solving extremal problems"
2. Karmanov V.G. "Mathematical programming".

Additional literature:

1. Ashmanov S.A. "Linear programming".
2. Korbut A.A., Finkenstein U.U. "Discrete Programming".
3. Moiseev N.N., Ivanov U.P., Stoljarova E.M. "Optimization methods".
4. Pontrjagin L.S., Boltjanski V.G., Gamkrelidze P.V, Mischenko E.F.
"Mathematical theory of optimal processes".
5. Pshenichni B.N., Danilin U.M. "Numerical methods for solving extremal
problems".

The program compiler is: Prof. A.G. Sukharev.
The Game Theory and Operations Research

1. Formalization and basic principles of operations research.
Mathematical model of operation. Operating side and operation
researcher. Controllable, indeterminate and stochastic factors. The
criterion of efficiency and strategies.
Models with vectorial criteria. Simple methods for their convolution
into nonvector ones. Bases of vectorial optimization.
The principal of guaranteed result. The estimation of strategy's
efficiency depending upon our information about indeterminate and
stochastic factors. The extremum of function's passive search strategy. The
example of the estimation of this strategy's efficiency. The estimation of
mixed strategy's efficiency.
2. Main conceptions of optimality in operations research.
Optimal and absolutely-optimal strategies. Formulas of the best
guaranteed results for different types of strategies.
Saddle points and antagonistic games. Existence conditions of saddle
points. Bases of antagonistic games' theory. Duel model research. The
solution of method of matrix games.
Many-person coalition-free games. Equilibrium situation of a game-
optimal strategy in some auxiliary operation.
Multistage antagonistic games. Zermelo theorem about a solution of
multistage games with complete information.
Germever's hierarchical two-person games. The best guaranteed results of
the first player in two-person hierarchical games.
3. Optimal strategies' search methods.
Necessary conditions for optimal strategies. The function of minimum
directional derivative. Necessary optimality conditions for strategies -
interval points. Germeyer's compensation principle in one resource
allocation problem. The examination of "defense and attack" model in pure
and mixed strategies.
Optimal srategies' search methods. The reduction of the maximin search
problem to the maximum search problem. Clearing of constraints. Gaming
sense of Langrangian multipliers. Penalty function method.
The approximation of models of games and operations. The solution of
continuous game's existence theorem by matrix games approximations. The
instability of hierarchical games' solving and its regularization.
Some network problems. Bases of scheduling. Network flows.


The program's compiler: prof Morozov V.V.
The History and Methodology of Applied Mathematics

The main purpose of studying the history of science is the discovering
of the laws of its development. The influence of society framework, its
world outlook, level of technology on the rate of science development.
Significance of studying the classical legacy for modern science.
The most important stages of mathematic's development. The
mathematization of science is the typical feature of its up-date
development. Mathematics and scientific - technical progress.
The struggle of materialism and idealism in mathematics. Mathematical
modelling as a method of cognition of the real world. Deterministic and
nondeterministic models. The main sources of the mathematical development.
The structure of modern mathematics.
The history of approximation theory development. Interpolation. Gregory
and Newton and their part played in applied mathematics. The development of
mid - quadratic approximation by Legendre and Gauss. Chebyshev and Marcov
and their role in the approximation theory. The up-date problems in the
approximation theory.
The history of development of the ordinary differential derivation
equation. Euler and his role in the progress of applied mathematics. Adams'
development of calculus. Runge method and its development. The influence of
computers' technology development on the calculus methods of ordinary
differential equation's derivation. Modern problems in the calculus theory
of ordinary differential equation derivation.
Physical problems leading to the incorrect mathematical problems.
Methods of incorrect problems' solution.
Some mathematical models in natural science, engineering, economics and
linguistics.
The use of computers in controlling the manufacturing, national economy
and its branchs.

The program compiler is: Prof. N.N. Ghitkov

Computers and Programming
(9 semester, Lectures - 36 academic hours)

1. Introduction. Architecture of modern computing complexes and
software.
2. Superpowerful computers ( Cray -1 as an example ). The main
characteristics of the machine. Particuarlies of hardware realization.
Functional devices. Vector registers. Scalar and address registers.
Buffers. Memory. Memory protection. Input - output organization.
Processing the vector values on machine Cray - 1. Commands on vector
values and their execution Reservation of functional devices and vector
registers . Main line execution of operations.
Peculiarities of programming at vector registers. Compiler from Fortran.
Parallel calculations. Specialized languages for vector machines.
3. Personal computers. Destination. Structure. Provision by external
devices. Typical features. Software. Interfaces.
3. Computer Networks. Distributed data processing. Topology of networks
and control; star-shaped, ring, channel, mixed. Channels controlling,
channelling, access. Channel and batch data transfer. Means of message
passing, cable, coaxial cable, optical means. Network protocols. Standards.
Examples of networks. Local and regional networks.
4. Main Concepts of the Modern Programming Languages. The programming
language as the way of thinking. Syntax, semantics, pragmatics. Simple and
composite values. The arrays, records. Sorts; introduction and definition
of new sorts. A block structure. Variables and their scopes. Operations on
data. Operators; the conditional statements, control operators. Operation
with the references. The generator of values. Procedures. Call of
parameters. Introduction of operations on data. Files and operators working
with them.
5. Databases. Definition of databases. Evolution of concept of database.
Logical base, condentual scheme, physical base. Types of databases.
Hierarchical, network, relational. Languages of inquiries.

The Literature

1. Ý.À. ßêóáàéòèñ. Architecture of computer networks. Statistics, M., 1980.
2. Ò. Ïðàòò. The programming languages: development and implementation. The
world, 1979.
3. Äæ. Ìàðòèí. Organisation of databases in computing systems. The world,
M., 1978

The program's compiler is: Prof. I.E. Pedanov
Structures of Computing Systems

History of Development of Computer Facilities. A brief history of the
origin of digital computers. A principle of operation and main nodes of the
computer? Evolution of the computer. Generations of the computer, their
differences on elementary bases and mathematical programs, ways of usage,
hardware and software possibilities.
Hardware and Software Structure of the Computer. The cpu, memory,
peripheral units. Operational systems, programming systems, special program
systems (information retrieval, tool, dialogue, databases, application
packages). Languages of dialogue with the computer.
Classification of Computing Tools. Principles of the classification and
structural features of various sets. Sets of great, average and small
computers. Mini and micro computers, personal computers. Multimachine and
multiprocessing systems. Computer networks.
Perspectives of Development of Computer Facilities. Development of
element base, structure and software of the computer. Parallel
calculations, homogeneous and stream computer, computer with a floppy
structure.

The Literature

1. Êîðîë¸â Ë.Í. «A Structure of the computer and their mathematical
software », «Science», M., 1978.
2. Ãîëîâêèí Á.À. « Parallel computing systems », «Science», M., 1980
3. Áðóñíåöîâ Í.Ï. «Minicomputers», «Science», M., 1979
4. Ìàíî M. « Architecture of computing systems », «Ñòðîéèçäàò», M., 1980
5. Áàëàøåâ Å.Ï., Ïóçàíêîâ Ä.Â. « Microprocessors and Microsystems », « a
Wireless and link », M" 1981

The program's compilers are: Prof. L.N. Koroljov;
Prof. N.D. Vasjukova
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