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1. Title and identification code of the research plan
Complex Development of All Branches of Mathematics with Regard to the Needs of Physics and Technology
CEZ:A05/98:Z1-019-9-ii
2. Title, identification number and type of research institution
Mathematical Institute of the Academy of Sciences of the Czech Republic, Ident. No. 67985840, a partially budget-funded organization
3. Manager of the research plan
Karel Segeth, Doc., RNDr., CSc.
4. Duration of the research plan
1999-2003 (extended by one year, till 2004)
5. Objectives of research
Annotation
Development of the knowledge basis in real and functional analysis, mathematical and numerical analysis, theory of ordinary and partial differential equations, differential geometry, topology, mathematical statistics, mathematical logic, and complexity theory. A special regard is devoted to the general and complex development that only can guarantee a full and general progress in mathematical cognition. The cultivation of a single mathematical branch cannot play a positive role in the development of mathematics as a science entity.
Some of the specific research topics are stimulated by the progress of investigation in physics and technology and will be directly presented as application outputs. The other topics are concentrated to the inner needs of the complex development of mathematics as an integral science discipline.
Key words
real analysis; functional analysis; mathematical analysis; numerical analysis; theory of differential equations; differential geometry; topology; mathematical statistics; mathematical logic; complexity theory; applications of mathematics
Subject codes according to CEZ
BA, BK
6. A detailed proposal of the research plan
Since the research is purely basic and theoretical and has been successfully carried out for a number of years (with possible applications) no time stages can be planned.
a) Real and probabilistic analysis. Didactics of mathematics
In real analysis, problems of the general theory of nonabsolutely convergent integrals based on classical integral sums are investigated especially from the viewpoint of convergence results and the topological structure of convergence in the space of primitives. Results on integration are applied to problems for ordinary differential equations, and integral equations in a general setting.
The main topics concerning probabilistic analysis are concentrated on stochastic differential equations. Mainly the asymptotic behaviour of probabilistic distributions induced by stochastic equations are studied, problems concerning stability are a special part of this research. Attention is paid to ergodic and adaptive control of stochastic control systems.
In both parts (the real as well as the probabilistic one) special attention will be paid to the case of processes having values in infinite dimensions.
Simultaneously a research is carried out in stochastic geometry. The spatial distribution of random closed set components is examined by means of distance analysis and by means of Voronoi tessellations generated by them. An extensive database of tessellations produced by pseudo-hard-core, hard-core and cluster point fields is exposed on the Internet. Several applications in materials science and life sciences are systematically developed.
Qualitative theory of linear Hamiltonian differential systems with applications in oscillation and spectral theory of Sturm-Liouville higher-order differential operators, discrete quadratic functionals and qualitative theory of symplectic difference systems, oscillation theory of half-linear second order differential equations, functional equations and transformation theory of higher-order linear differential equations, boundary value problems for differential and functional differential equations are the main topics of research in differential equations represented by the Brno branch of the Institute.
A key role is played by the combination of the variational principle and Riccati technique with the generalized reciprocal principle. Combination of these methods enables us to obtain a new characterization of nonegativity of quadratic functionals corresponding to linear Hamiltonian systems and characterization of the spectrum of higher-order Sturm-Liouville differential operators.
Using the discrete version of the Reid roundabout theorem, fundamental facts of oscillation and transformation theory of symplectic differences are obtained.
The recently established half-linear version of the Picone identity combined with certain special methods (asymptotic estimates, a generalized Riccati technique, etc.) leads to new results in oscillation theory of half-linear second order differential equations. A typical method is the combination of some results of the theory of functional equations with the transformation theory of linear differential equations.
Methods of mathematical and functional analysis combined with special techniques give nonimprovable (in a certain sense definite) conditions for (unique) solvability of various boundary value problems.
In the future the investigation in the above mentioned fields will be continued. Particular attention will be paid to the investigation of dynamics for equations on time scales, special cases of which being e.g. differential and difference equations.
In the period 1995-99 many scientific papers were published in the world mathematical scientific periodicals and they are also quoted in these journals. The research carried out by scientists working in this research area follows topical trends and represents a good deal of mathematical research.
The methods used in the research are typical for mathematics. No special devices are needed (except for computers for communication, computation, and publishing). The access to new results published in the mathematical literature plays the most important role in the research of this type.
Beside the experienced scientists of higher category of age (J. Kurzweil, F. Neuman, I. Saxl, S. Schwabik, I. Vrkoc) covering all the above mentioned fields, the outstanding scientists with promising future in this research area are:
The core of the work of a small didactics group (M. Ticha and M. Baresova-Matyasova) is devoted to theoretical questions of didactics of mathematics and to the development of didactics of mathematics as a scientific discipline. The long term research is carried out with teachers of elementary and lower secondary schools and with co-workers from pedagogical faculties (Charles University Praha, University of South Bohemia Ceske Budejovice, University of Education Hradec Kralove).
Orientation of the present research is consistent with main directions of the didactical research carried out in eminent didactical centers abroad.
The research is devoted to the problems of mathematical education of pupils aged 5-15 years. Its goal is the understanding of processes which are going on during learning and teaching mathematics, and application of this knowledge to optimization of mathematical education. The possibility to decrease the instructional character of mathematics education and to increase its constructive character is emphasized. Therefore, one of the main topics of the research is the study of mechanism of grasping a situation connected with problem posing and problem solving.
Other areas of the research are:
b) Evolution differential equations
In the past, the scientific activities in this research area were mainly concerned with the qualitative properties of solutions to evolution partial differential equations. There has been a long tradition of research oriented towards the time-periodic solutions which culminated in the monograph O. Vejvoda et al.: Partial Differential Equations: Time periodic solutions, Amsterdam, North-Holland 1982. In the late eighties, P. Krejci started to develop a mathematical theory of hysteresis phenomena which since then has become another of the main topics of research. Besides, the equations of hydrodynamics, in particular the Navier-Stokes equations, both in the incompressible and compressible cases have also been studied.
The present research follows basically three main directions:
Outstanding scientists:
Function spaces and differential equations and inequalities have been widely investigated. The research in these mutually connected topics should be systematically continued.
The Mathematical Institute is one of the places where function spaces and related operators are systematically studied. This dynamically developing field reflects the needs of the PDE’s and has been carried out in close collaboration with top centers in Germany, U.K., Italy, Spain, U.S.A., Georgia etc. The role of the group (M. Krbec, A. Kufner, J. Lang, B. Opic, J. Rakosnik, until 1999 also L. Pick) has been emphasized by the recognized series of international conferences organized here (1978, 1982, 1986, 1990, 1994, 1998). On the basis of important results obtained by the group it is intended to continue the research in the following directions: structure and mutual relations of function spaces (inclusions, bounded and compact embeddings), behaviour of linear and quasilinear operators on function spaces, estimates of characteristic quantities such as operator norm, measure of non-compactness, s-numbers etc., and interpolation and extrapolation theory. The applications will be focused on qualitative properties of singular and degenerated differential equations. The group employs a broad spectrum of methods of real and functional analysis, and of real methods of harmonic analysis.
Two topics concerning variational inequalities have been studied: bifurcations for unilateral problems and the existence and regularity of solutions to contact problems with friction. The former topic has a tradition here since 1977 when an original method for the study of existence and location of bifurcations of such problems was found (M. Kucera), the latter one has been developed since 1997 when the group was enlarged by J. Jarusek. While a complete theory of bifurcations for variational inequalities with potential operators was developed by German and Italian schools, only few authors in the world essentially contributed to the case of nonpotential operators which was studied for the first time here. Particularly, the influence of unilateral conditions on the bifurcation of spatial patterns in reaction-diffusion systems has been investigated (J. Eisner, M. Kucera). The mathematical models of contact problems studied intensively in the world are usually simplified (smoothed or replaced by some penalized version). Here, some basic questions were solved for models preserving the unilateral character of the contact condition existence of solutions to several types of contact problems with the original (nonsmooth) Coulomb friction (J. Jarusek). In the future, the former topic should be developed also towards the interpretation in science, the latter one towards plastic materials.
Boundary integral method has a long tradition in the Mathematical Institute; J. Kral was one of the two founders of this theory for open sets with no restriction on the boundary. In the last decade the interest in this method has increased with respect to its connection with some numerical methods (boundary integral method, collocation method). After proving applicability to rectangular domains (R. L. Angel, R. E. Kleinman, J. Kral, W. L. Wendland) and polyhedral domains (N. V. Grachev, V. G. Maz’ya, A. Rathsfeld), the interest concentrated to the question whether this method is applicable to domains with piecewise-smooth boundary. This problem was successfully solved here (D. Medkova). In future, it is planed to apply the method of boundary integrals to more general open sets, for further boundary value problems and for more general PDE’s and systems.
Among other experienced researchers, the following outstanding scientists guarantee a successful continuation of the research:
Most researchers in this research area have dealt with mathematical and numerical analysis of nonlinear physical fields by the finite element method and optimization techniques. The nonlinear problems concerned are mathematically described by partial differential equations and inequalities of elliptic or parabolic type. In particular, finite element approximation of a nonlinear heat equation in anisotropic and inhomogeneous media has been investigated. Existence and uniqueness theorems, comparison and maximum principles, convergence without any additional regularity assumptions, a priori error estimates (the optimal rate of convergence), and a posteriori error estimates have been proved. A special attention was paid to variational crimes including numerical integration and approximation of a curved boundary in the three-dimensional space. The results obtained have technologically important applications in practice, where linear models are often not adequate.
Another direction of our research consisted in construction of three-dimensional meshes. An efficient algorithm that produces local refinement of tetrahedral partitions has been proposed. Moreover, generated tetrahedra do not become flat when the discretization parameter tends to zero. Another algorithm that produces refinements consisting only of acute type tetrahedra has been developed, too. For such partitions a discrete maximum principle was proved. Later, this result was generalized and a sufficient condition for a weakened discrete maximum principle has been established.
The research was also focused on various post-processing techniques giving superconvergence properties for derivatives of the solution of elliptic problems. A weighted averaging formula which recovers the gradient of linear elements on two- and three-dimensional unstructured simplicial partitions has been derived. Optimal interior error estimates on nonuniform triangulations were obtained as well. A new method for constructing divergence-free finite element fields was developed. A simple post-processing algorithm that accelerates the convergence of classical iterative methods for large sparse systems of linear algebraic equations that arise from the use of the finite element method has been investigated, too.
Numerical and analytical methods for solving systems of nonlinear ordinary and partial differential equations describing the n-body problem, gravitational potentials, and other related physical quantities have been developed. Nonlinear problems of gravity are mostly modeled by a system of Einstein’s partial differential equations of hyperbolic type. Investigations were focused especially on solutions of Einstein’s equations with symmetries and problems where the gravity is coupled with other physical fields.
Computational modeling of phase-change processes in pure materials and binary alloys has been considered, too. The models describe melting, evaporation, resolidification, and crystal growth from the melt. Models including chemical reactions and fluid flow in the melt were investigated, too.
In future, the derivation of some a posteriori error estimates for nonlinear elliptic boundary value problems, in particular, for a steady-state heat conduction problem in anisotropic and inhomogeneous media is planned. The development of the worst scenario method for elasto-plastic materials and for the control in obstacle-pseudoplate problems with friction including approximate optimal design will begin, too. A special attention will be paid to exact results in the finite element method, to the three-dimensional mesh generation, and to the reliability of numerical computations.
Outstanding scientist:
The research is concentrated on functional analysis, topology, and mathematical methods in physics.
The research in this field has a long tradition which is based on an exceptional research impact of Czech mathematicians E. Cech, M. Katetov, V. Ptak, and Z. Frolik.
Main research subareas:
In the theory of function spaces the Lipschitz structure of spaces of continuous functions equipped with the supnorm is going to be investigated. In the theory of Banach spaces a special attention will be paid to the class of weak Asplund spaces and there are many natural problems arising in this connection.
Further, the convexity properties of isotropic stored energies will be studied. The research will use results already obtained to develop efficient iterative procedures for finding the rank 1 convex hulls of isotropic stored energies.
An outstanding scientist among other experienced researchers:
Main research subareas:
To solve problems in this field, we will primarily use methods of combinatorics and proof theory. In addition to these finite methods, the use of the theory of models seems very promising.
An interaction of linear algebra with theory of systems is a very topical and intensively studied field in the world. The results of Z. Vavrin, M. Van Barel (Leuven), and V. Ptak are very close to the formulae that mutually transform the individual classes of structured matrices into each other and allow for finding stable fast (of order O(n2)) and superfast (of order O(nlog2n)) algorithms.
The theory of vague matrices is a continuation of interval analysis and corresponds to some results of semiinfinite linear programming. These fields have been developing in various research centers. The principal research tool is just the use of motivations from applications in pure mathematical theory, sometimes also a contribution to applications based on general theoretical results.
Outstanding scientists involved:
The long term scientific achievements of the group in the research area of real and probabilistic analysis are incorporated in the following books:
Besides the theoretical studies many practical work was done in connection with writing textbooks and helping the mathematics teachers at the elementary level.
In the research area of evolution differential equations, the existence of the time-periodic solutions of the Navier-Stokes equations of compressible fluid was proved under the main hypothesis that the fluid is isentropic and the adiabatic constant satisfies certain restrictions. The results hold in three space dimensions with no restriction on the magnitude of the driving force which is supposed to be measurable and bounded.
The results are described in the following paper:
Optimal embeddings of Sobolev and Besov type spaces were described. See:
Applications of the method have been given to some problems in heat conduction, elasticity and plasticity. Several papers on this topic were published by I. Hlavacek and J. Chleboun. The worst scenario method appeared, e.g., in the following papers:
It has been known for some 25 years that a Banach space whose dual unit ball is a uniform Eberlein compact admits an equivalent uniformly Gâteaux smooth norm. The paper below proves the reverse implication. Thus, in particular, a compact K is uniform Eberlein if and only if the space C(K) of continuous functions on K admits an equivalent uniformly Gateaux smooth norm. See:
A method for demonstrating lower bounds for lengths of proofs in propositional calculus has been created, the method of efficient interpolation (J. Krajicek). By its application, the following lower bounds have been proved: (a) for resolution (a new proof of a known Haken's result, J. Krajicek), (b) for cutting planes system (P. Pudlak - the most important application of the method), (c) for some generalizations of these two systems (J. Krajicek), and (d) for an algebraic system based on Nullstellensatz (P. Pudlak, J. Sgall). Further, relations to communication complexity and hypotheses forming theoretical foundations of crypto- graphy have been found (J. Krajicek, P. Pudlak). See:
