Department of General Mathematics

The team of the Department of General Mathematics performs a fundamental research in different areas of the differential equations’ theory. Important results have been obtained in the theory of distributed systems’ control, for the problem of finding boundary controls that transfer a distributed system from an arbitrary given initial state to an arbitrary given final state and for the boundary control optimization. The obtained results were recognized twice (in 2001 and 2007) as the most important achievements of the Russian Academy of Sciences.

The study of spectral problems in the mathematical physics is one of traditional areas of research. A special attention is focused on non-selfadjoint problems related to processes with dissipation. The following topics are also considered: the stability of nonlinear oscillations, problems of p-adic and resurgent analysis, problems of computational and computer algebra, of geometric modeling, computational geometry and topology.

The Department of General Mathematics provides teaching of basic mathematical disciplines at the CMC Faculty.

Staff members:

• Vlasov Viktor, Professor, Dr.Sc.
• Ikramov Saidkhakim, Professor, Dr.Sc.
• Lomov Igor, Professor, Dr.Sc.,
• Korovina Maria, Professor, Dr.Sc.
• Khapaev Michael, Professor, Dr.Sc.
• Fomenko Tatyana, Professor, Dr.Sc.
• Budak Alexander, Associate Professor, PhD, Deputy Dean for publishing
• Denisov Vasiliy, Associate Professor, Dr.Sc.
• Domrina Alexandra, Associate Professor, PhD
• Kim Galina, Associate Professor, PhD
• Kritskov Leonid, Associate Professor, PhD
• Leontyeva Tatyana, Associate Professor, PhD
• Panferov Valery, Associate Professor, PhD
• Sadovnichaya Inna, Associate Professor, PhD
• Sazonov Vasily, Associate Professor, PhD, Deputy Dean, Head of the Training Centre of CMC MSU
• Tikhomirov Vasily, Associate Professor, PhD, Vice Dean for methodological work
• Falin Anatoly, Associate Professor, PhD
• Khoroshilova Elena, Associate Professor, PhD
• Komarov Michael, Associate Professor, PhD
• Nikitin Alexey, Associate Professor, PhD
• Kuleshov Alexander, Assistant Professor, PhD
• Smirnov Ilya, Assistant Professor, PhD, Scientific Secretary of the Department
• Nesterenko Yury, Assistant Professor
• Govorov Valery, Leading Mathematician, PhD
• Zhmykhova Albina, Mathematician

Regular courses:

• Mathematical analysis, 1st and 2nd semesters.
• Mathematical analysis, 3rd and 4th semesters.
• Theory of functions with a complex variable, 4th semester.
• Algebra and geometry, 1st and 2nd semesters.
• Extended course on partial differential equations, 7th semester.

Special courses:

• Generalized functions (distributions) by Prof. Lomov.
• Asymptotic methods by Prof. Khapaev.
• Extended course on abstract algebra by Prof. Ikramov.
• Introduction to the computational geometry by Assoc. Prof. Sazonov.
• Differential geometry and geometric modeling by Assoc. Prof. Sazonov.

Special scientific seminars:

• Spectral theory of differential operators and actual problems of mathematical physics by Prof. Il’in and Prof. Moiseev.
• Theory of nonlinear differential equations by Dr. Komarov.

Main Scientific Directions

Study of spectral characteristics for various problems of mathematical physics

(Academician V.A. Il’in, Professors V.V. Vlasov, I.S. Lomov, Assoc. Professors A.B. Budak, V.N. Denisov, L.V. Kritskov, V.S. Panferov, I.V. Sadovnichaya, and V.V. Tikhomirov)

Many problems of mathematical physics lead to finding eigenvalues and eigenfunctions of the corresponding operators and expanding a function in a series or in an integral over the system of these eigenfunctions. For instance one always encounters these topics applying the Fourier method to solve the mixed problem for a differential equation with partial derivatives. In control problems certain criteria for controllability are also connected with the basis property of root functions of the related differential operators. The method of regularizing the singular perturbations for an exact analysis of the solution’s singularities uses the spectrum of the limit operator while the right-hand parts of equations are expanded in the series with respect to the system of root functions of this limit operator.

Similar problems appear in modeling nuclear reactors, in studying plasma stability, various oscillating processes (including those with dissipation) and supersonic and subsonic gas flows.

The research is accomplished in the spectral theory of selfadjoint elliptic operators, in the spectral theory of non-selfadjoint differential operators, in the theory of multiple Fourier series and Fourier integrals, in the spectral theory of the Schroedinger operators with singular potentials and functional-differential operators. Obtained results are applied for solving urgent problems of the mathematical physics.

The fundamental character is attributed to a universal method that was elaborated for studying spectral expansions which are related to any selfadjoint extensions of elliptic operators. Though the localization and the uniform convergence of spectral expansions have been widely explored, only this method permitted (for an arbitrary domain or manifold and for an arbitrary spectrum) to gain precise conditions for the uniform convergence not only for the spectral expansion itself but for its Riesz means as well.

This method has been fruitfully applied to the selfadjoint extensions of the Schroedinger operator defined over the whole real line or over the whole N-dimensional space in the case when its potential satisfies the Kato condition.

An outstanding scientific impact was formed by the papers on the spectral theory of non-selfadjoint differential operators that appeared in 1975-2011. The preceding papers by M.V.Keldysh on the completeness of root functions for a wide class of differential operators failed to answer whether this system formed a basis, i.e. whether it was possible to expand any function in a bi-orthogonal series with respect to this system.

The core of the developed methods in the spectral theory of non-selfadjoint differential operators forms the refusal to analyze any specific form of the boundary conditions and but rather proposes to consider the root functions only as the regular solutions to the related differential equations with a spectral parameter. Conditions on the boundary conditions are replaced by conditions on the spectrum and the system of root functions which are easy to verify. Thus one can consider various restrictions of the maximal operator. It permits to involve not only the restrictions defined by arbitrary boundary conditions (two-point, multi-point, non-local, integral conditions, conditions that depend on the spectral parameter) but also operators which domain contains discontinuous functions as well.

Necessary and sufficient conditions are obtained for the unconditional and local basis property of the system of root function. Theorems on the equiconvergence of the biorthogonal expansion with the Fourier series are proved locally and uniformly on the segment. The distribution of eigenvalues for a wide class of differential and functional-differential operators is investigated.
Refs:

1. V.A. Il’in, Spectral Theory of Differential Operators // NY, Plenum Publ. Corp., 1995.

Control of the distributed systems

(Acad. V.A. Il’in, Assist. Prof. A.A. Nikitin, and I.N. Smirnov)

The development of methods of the distributed systems’ control is one of the scientific areas of the Department of General Mathematics. The models of these systems are described by equations with partial derivatives – the wave equation and the telegraph equation, the degenerate equations and some other classes of equations.

Methods for solving such problems form a scientific core for constructing controlling and measuring complexes for different complicated technical systems. Thus these methods’ importance in applications maintains the research interest in this area of mathematics. The wave equation describes oscillations of a string, of plane’s wings, of a lifting crane’s beam etc. The telegraph equation characterizes the pressure of oil or gas in pipes, the dynamics of current’s force in the propagation of electromagnetic waves in long lines, free oscillations of the geological stratum (the pressure and shifts in layers). The explicit analytic form of solution helps to reduce calculations in modeling all these processes immensely.

As a result of our research, new methods of distributed systems control are elaborated. They give explicit solutions to the boundary value problems for the equations of gas dynamics, for the wave equation and the telegraph equation. The new methods for the control of distributed systems are based on the system’s discretization and the use of methods of an asymptotic stabilization of discrete systems. These methods give basis for the algorithms for stabilizing oscillating systems and damping oscillation in pipelines.

All these results are new and of the high world level. Obtained explicit analytic formulae for problems of excitation and damping for the hyperbolic and parabolic equations with a discontinuous coefficient have no analogues. There appeared more than 50 papers in the leading journals in Russia and abroad. These results have been utilized in scientific and technical products of the Institute for the System Analysis of RAS, the Institute of Program Systems of RAS named after A.K.Aylamazyan, the Mathematical Institute of RAS named after V.A.Steklov. Many researchers in distributed systems’ control are inspired by these results.

Differential equations with extinctions

(Prof. M.V. Korovina)

The problem of constructing a semi-bounded selfadjoint extension of the Laplace operator with a specific initial domain is solved. This domain consists of functions that vanish in the neighborhood of a pencil of planes. In order to obtain this construction a special theory which allows one to investigate the properties of translations and to state necessary and sufficient conditions of the Sobolev problem’s solvability on a stratified manifold, is developed.

Asymptotics of solutions to differential equations with conic and edge extinctions are constructed in spaces with asymptotic forms. Differential equations with an arbitrary polynomial extinction in coefficients are investigated. The main result is a theorem which states that solutions of all differential equations with polynomial extinctions in coefficients with resurgent right-hand part are also resurgent. This theorem is formulated both for the ordinary differential equations and for the equations with partial derivatives. Asymptotics for certain classes of similar equations is constructed. The research of all these topics is continuing.

Asymptotic methods in mathematical physics

(Prof. M.M. Khapaev, Assoc. Prof. V.V. Ternovskiy (Dept of Comp. Methods), and A.I. Falin)

A generalization of Lyapunov's second method which reduces both terms of its applicability is proposed. It is focused on the study of stability in critical situations. With the help of this approach the study on the stability of the parameters of the planetary orbits in the three-body problem is accomplished in the frame of the proposed hydrodynamic model of planets which takes into account the size of the planet, the planet's distance from the Sun, its oblateness and inclination of the planet's axis of rotation to the orbital plane. Using averaging the effect of these factors on the evolution of the orbital parameters is studied.

For the first time in the USSR, the mathematical models for thin magnetic films have been constructed, the static and dynamic processes in thin-film materials are studied. Currently, the magnetic processes are being studied in the innovating nanostructures. In order to describe the dynamic processes in magnetic materials, it is proposed to take into account the variability of the magnetization vector’s length, the dependence of this length on the intensity of the exchange interaction.

Papers on the plasma physics were the first to introduce by the singular integral manifolds on which the right-hand sides of differential equations become infinite. It is proposed to use these singular manifolds not only in problems of the constrained optimization and control but in problems of processing the results of observations and experiments on the Gauss least-squares method as well. Embedded into the data processing scheme these singular manifolds permit to take into account the errors of observation and experiment. Meanwhile the scheme becomes complicated and nonlinear.

A theorem on the limit transition for a singularly perturbed system which is analogous to the Tikhonov theorem, is proved in the case when the degenerate system does not have roots on which the right-hand side vanishes but is furnished with the singular manifolds on which the right-hand side is infinite. The attention is drawn to the incorrectness of singular perturbation problems for partial differential equations. The obtained results and the ill-posedness of the problem give an idea of the existence of many solutions with internal boundary layers.

Methods for solving ill-posed problems have been developed in optimal control problems. The control problems are formulated as variational problems on a conditional extremum of the objective functional with integral and local restrictions. We study the problem of controlling a nonlinear pendulum with friction. As for the practical applications of the Fourier series, the problem of reconstructing a periodic function by approximate Fourier coefficients is considered. Instead of using the regularized Fourier series for the function’s search a system of equations of the first kind is numerically solved while a-priori information about the functional class is taken into account.
Refs:

1. M.M. Khapaev, Averaging in the theory of stability. Moscow, Nauka, 1986.

Solving non-linear differential equations

(Senior Lecturer M.V.Komarov)

Problems of the global in time existence and the asymptotic long-time behavior for solutions to nonlinear evolutionary equations and system of these equations are studied.

Periodic problems for the multi-dimensional complex Landau-Ginzburg equation and for a wide class of one-dimensional non-linear equations (such as the Korteweg-de Vries-Burgers equation, the Kuramoto-Sivashinsky equation, the Otto-Sudan-Ostrovsky equation etc.) are considered. The cases of both small and arbitrary initial conditions are studied. Sufficient conditions for solutions’ smoothing are found. Various asymptotics (growing, vanishing and oscillating ones) for the solutions are constructed on the large time-scale.

Long-time asymptotics are found for solutions of the mixed problem on the half-line for the nonlinear nonlocal Schroedinger equation and the Korteweg-de Vries-Burgers equation.

The Cauchy problem for a rather wide class of multi-dimensional (with respect to space variable) systems of nonlinear evolutionary equations with dissipation is studied in the supercritical and critical cases. Problems of local and global in time existence are considered.

Asymptotics for solutions of the Boussinesq system for surface waves and of the two-dimensional Navier-Stokes system are constructed; it is shown that, in the critical case, long-time asymptotics is self-similar.
Refs:

1. I.A. Shishmarev and P.I. Naumkin, Nonlinear nonlocal equations in the theory of waves // AMS Transl. Math. Monographs, vol. 133, 1994.

Research in the Area of Linear Algebra and Matrix Analysis

(Prof. Kh.D. Ikramov)

A wide range of problems in linear algebra, fundamental as well as computational, is examined. They include numerical stability of direct methods for solving systems of linear algebraic equations, problems concerning matrices symmetric in some generalized sense, inverse eigenvalue problems, classification problems of linear algebra, and so on. The purpose of the study is often to develop algorithms that are finite with respect to a certain set of allowed operations. As an example, it is well known that the eigenvalues of a complex matrix (of order at least five) cannot be found by radicals. This is all the more so for the eigenspaces. Nevertheless, it turns out that, for instance, the sum of the eigenspaces of a nondiagonalizable matrix can be found by a finite computation.

The theory of unitary similarity and the development of a parallel theory of unitary congruence are two fields where our contribution is most substantial. For the first time in algebraic literature, we have proposed finite algorithms of the Specht--Pearcy type for solving the following two problems: (a) verifying simultaneous unitary similarity between two finite matrix sets; (b) verifying unitary congruence for a pair of complex matrices.
Refs:

1. V.N. Chugunov and Kh.D. Ikramov, A complete solution of the normal Hankel problem // Linear Algebra Appl., no. 432, pp. 3210-3230, 2010.
2. Kh.D. Ikramov, A note on complex matrices that are unitarily congruent to real matrices // Linear Algebra Appl., no. 433, pp. 838-842, 2010.
3. V.N.Chugunov and Kh.D.Ikramov, There exist normal Hankel (phi, psi)-circulants of any order n // Matrix Methods: Theory, Algorithms, and Applications. Hackensack, USA: Word Scientific Publ., pp. 222-226, 2010.

Computational geometry and geometric modeling

(Assoc. Prof. V.V. Sazonov)

The need to apply special geometric methods frequently appears in many practical problems of natural sciences and technique. Problems of constructing stable geometric algorithms, special approaches to representing surfaces and curves for solving further applied problems - these are the scope of our interest. Work on developing the special software for solving problems related to the controlled and automatic space flights is carried out.

The new effective algorithm which permits to find lighted areas of surfaces in the parallel light flux was proposed. This algorithm was implemented to analyze the uncontrolled motion with respect to the mass center of the spaceship “Photon-11” which performed its orbital flight in 2005. Also this software simulated the motion of a spacecraft with a solar sail.

Works on mathematical modeling of the electromagnetic field around the International Space Station are proceeding.

Investigations in 2011-12 were supported by the Grant of the President of the Russian Federation for young Russian scientists.

Topological methods in the theory of fixed points and coincidences

(Dr.Sc. T.N. Fomenko)

The problems of existence, minimization and approximation of common fixed point sets, coincidence sets, and sets of common preimages of a given closed subspace under a finite number of given mappings between metric spaces are studied.

It is well known that the fixed point existence problem is closely connected with one of the mapping degree’s calculation when this degree is assignable. The problem of degree’s calculation is studied for an equivariant mapping of cohomological spheres with given group actions. Its solving uses the theory of cohomological spectral sequences of fiber bundles and, in particular, the equivariance index theory which was earlier suggested by the author. General formulas are obtained for the calculation of the degree of equivariant mappings between integer cohomological spheres with given actions of finite or some compact groups.

A fixed point set minimization theorem is obtained in the case of a continuous equivariant self-mapping of a compact n-dimensioned polyhedron (n>=2) with a given action of a finite group, under some additional conditions. Equivariant analogues of methods of the classical Nielsen fixed point theory are used. Also, the more complicated problem of the coincidence set minimization is considered for two mappings between smooth manifolds in positive codimesion, that is in the case when the dimension of the target-manifold is less than the dimension of the origin one. A constructive algorithm is proposed for a partial minimization of the coincidence set in the case when the number of common values of given two mappings is finite, and their common preimages are smooth submanifolds.

A general iteration principle (so-called, the cascade search principle) is worked out for searching zeros of the so-called search functionals on a metric space. Basing on this principle the cascade search methods are proposed for searching the coincidence points, common fixed points and common preimages of a given closed subspace, under given finite collection of one-valued or multi-valued mappings between metric spaces. The stability of the proposed iteration methods is also considered. These investigations are in progress.

Qualitative theory for solutions to partial differential equations of parabolic and elliptic types

(Assoc. Prof. V.N. Denisov, Dr. Sc.)

The influence of the low-order coefficients of a second-order parabolic equation on the property of stabilizing to zero is studied for the solution of the Cauchy problem related to various classes (including the Tikhonov’s class) of initial functions.

The problem on the effect of initial function’s domain on the stabilization property of the Dirichlet problem solution related to a parabolic equation with the divergent-type operator is solved. A corresponding criterion is obtained in terms of a certain Wiener-type series divergence. The Dirichlet problem for an elliptic equation on the half-space is investigated and a criterion for vanishing of its solution at infinity is obtained.

Recent publications:

• 2013

1. Abdukarimov M.F., Kritskov L.V. Boundary control by the displacement for the telegraph equation with a variable coefficient and the Neumann boundary condition // Caspian J. Appl. Mathematics, Economics and Ecology. 2013. 1. N 1. P. 51-69.
2. Abdukarimov M.F., Kritskov L.V. Boundary control of the displacement at one end with the other end free for a process described by the telegraph equation with a variable coefficient // Doklady Mathematics. 2013. 87. N 3. P. 351-353.
3. Abdukarimov M.F., Kritskov L.V. Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient. The case of control by disp lacement at one endpoint with the other endpoint being fixed // Differential Equations. 2013. 49. N 6. P. 731-743.
4. Abdukarimov M.F., Kritskov L.V. Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient: the case of control by disp lacements at two endpoints // Differential Equations. 2013. 49. № 8. C. 1006-1017.
5. Denisov V.N. Necessary and sufficient condition for a stabilization to the Dirichlet problem for parabolic equation // J. Math. Sciences. 2013. 189. N 3. P. 422-430.
6. Denisov V.N. Necessary and sufficient conditions for stabilization of a solution to the Cauchy problem for the parabolic equation with radial potential // J. Math. Sciences. 2013. 188. N 3. P. 207-218.
7. Denisov V.N. Stabilization of the solution of the Cauchy problem for parabolic equation in nondivergence form with growing lower coefficients // Differential Equations. 2013. 45. N 5. P. 1-14.
8. Denisov V.N. Stabilization of the solution of the Cauchy problem for parabolic equation with growing lower coefficients // Doklady Mathematics. 2013. 87. N 3. P. 348-350.
9. Fomenko T.N. Cascade search for preimages and coincidences: global and local versions // Mathematical Notes. 2013. 93. N 1. P. 3-17.
10. Ikramov Kh.D. Solutions to sesquilinear matrix equations: Conspectral approach // Textos Matematica. 2013. 44. C. 73-79.
11. Lomov I.S., Markov A.S. Estimates of the local convergence rate of spectral expansions for even-order differential operators // Differential Equations. 2013. 49. N 5. P. 529-535.
12. Nikitin A.A. On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions // Differential Equations. 2013. 49. N 5. P. 645-653.
13. Savchuk A.M., Sadovnichaya I.V. Asymptotic formulas for fundamental solutions of the Dirac system with complex-valued integrable potentia // Differential Equations. 2013. 49. N 5. P. 273-284.
14. Nikitin A.A. On an optimal control problem for the wave equation in one space dimension controlled by third type boundary data // Progress in Partial Differential Equations. Springer Proceedings in Mathematics & Statistics. Berlin, Germany: Springer, 2013. P. 223-238.

• 2012

1. Abdukarimov M.F. On a boundary control problem for forced string oscillations // Azerbaijan J. Mathematics. 2012. 2. N 2. C. 105-116.
2. Denisov V.N. Exact stabilization conditions for the solutions of the Cauchy problem for a nondivergent parabolic equation with dercasing lower-order coefficiants // Doklady Mathematics. 2012. 86. N 2. P. 654-656.
3. Ikramov Kh.D. Antilinear operators and symplectic matrix algebra // Doklady Mathematics. 2012. 85. N 3. P. 369-371.
4. Ikramov Kh.D. Effective algorithms for decomplexifying a matrix by unitary similarities or congruences // Mathematical Notes. 2012. 92. N 6. P. 148-153.
5. Ikramov Kh.D. Equations of the form X \bar X = A with skew-Hamiltonian matrices A // Doklady Mathematics. 2012. 85. N 3. P. 388-390.
6. Ikramov Kh.D. Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm // Comput. Math. and Math. Phys. 2012. 52. N 1. P. 1-3.
7. Ikramov Kh.D. Unitary congruence of mutually transposed matrices // Doklady Mathematics. 2012. 85. N 1. P. 5-7.
8. Ikramov Kh.D., Abdikalykov A.K. On unitary transposable matrices of order three // Mathematical Notes. 2012. 91. N 4. P. 528-534.
9. Ikramov Kh.D., Vorontsov Yu.O. The matrix equation X + AX^TB = C: Conditions for unique solvability and a numerical algorithm for its solution // Doklady Mathematics. 2012. 85. N 2. P. 265-267.
10. Khoroshilova E.V. Extragradient method of optimal control with terminal constraints // Automation and Remote Control. 2012. 73. N 3. P. 517-531.
11. Kuleshov A.A. Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod and for the equation of transverse vibrations of a heterogeneous string consisting of two segments with different densities and elasticities // Doklady Mathematics. 2012. 85. N 1. P. 98-101.
12. Kuleshov A.A. Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod with a free or fixed right end consisting of two segments with different densities and elasticities // Doklady Mathematics. 2012. 85. N 1. P. 80-82.
13. Kuleshov A.A., Il’in V.A. Generalized solutions of the wave equation in the classes L p and with W p 1 p >= 1 // Doklady Mathematics. 2012. 86. N 2. P. 657-660.
14. Smirnov I.N. Oscillations of the process described by the telegraph equation for a system consisting of two segments with different densities and elasticities // Doklady Mathematics. 2012. 85. N 1. P. 63–67.
15. Vlasov V.V., Rautian N.A. Correct solvability of integro-dfferential equations arising in the theory of heat transfer and acoustics // Functional Differential Equations. 2012. 19. N 1-2. C. 213-230.
16. Vorontsov Yu.O., Ikramov Kh.D. Conditions for unique solvability of the matrix equation AX + X^*B = C // Comput. Math. and Math. Phys. 2012. 52. N 5. P. 665-673.