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Source specification: GA AV CR – Grant Agency of the Academy of Sciences of the Czech Republic; GA CR – Grant Agency of the Czech Republic; GA UK – Grant Agency of the Charles University; MSMT CR – Ministry of Education, Youth, and Physical Training of the Czech Republic.
An annotated list of the individual grants follows. Most annotations are presented as appeared in grant applications since no other text is available.
When closed, the grants are evaluated according to their results and graded as excellent, successful, and unsuccessful. No grants finished in the Institute have been rated unsuccessful, most of the grants have been classified excellent.
205/93/0090 B. Balcar (investigator J. Palous): Galaxies: the dissipative dynamical systems, 1993-1995
The project is concerned with our Galaxy, M31 Galaxy, and further close galaxies that are studied as dissipative dynamical systems. Three approaches are employed: 1. Dissipation of energy in interstellar medium. 2. HI structures and their linkage to OB associations. 3. Propagation of the formation of stars in galaxies.
A119401 B. Balcar: Continuous structures and Boolean algebras, 1994-1996
The subject of our research is the interplay of convergence with additional structures. We intend to investigate Boolean algebras in the context of dynamical systems. The universal minimal dynamical system for a semigroup S determines a complete Boolean algebra B(S), a group G(S) of automorphisms of the system, and an effectiveness congruence relation on S. We are concerned with structural properties of this subject with emphasis on Abelian, amenable, and free semigroups.
Socrates-Comenius I CA1-97-II-40 M. Baresova: European mathematical development through exchange and education, 1998-2001
The project is devoted to the exchange of views and experience about the practice and theory of mathematics teaching in nursery, primary and secondary school. The main aims of the project are principally:
To develop mathematical environments programs and activities for the early mathematical learning. To study how to teach in mixed ability classes. To help pupils with special needs and also gifted pupils in further development. To develop students’ problem solving strategies and to find more meaningful problems than we usually meet in our textbooks (”Real Problems with Real World Mathematics”). To develop good strategies in mental computation at different levels. To encourage creative and manipulative way of teaching at all levels including the utilisation of games in the teaching.
406/99/D080 M. Baresova: Pedagogical interactions in the classroom and mathematics education, 1999-2001, postdoctoral
Present mathematics education is devoted mostly to the performance of students, it does not develop in adequate extent the cognitive skills and personal capabilities of students nor their prioritisation of life values. The didacticians realise the necessity of changes especially by decreasing the instructional character of education and increasing its constructive character. In accordance with these requirements we prepare and realise scenarios for project teaching by undertaking the topic ”distance” (the second stage of research is ammended by other topics). We investigate (a) how pedagogical interactions in the classroom influence the process of acquiring and understanding concrete pieces of knowledge, (b) in what way the process of acquiring and understanding concrete pieces of knowledge influences pedagogical interactions in the classroom. The results are summarized in studies and articles addressed to both didacticians and teachers.
201/98/0528 J. Chleboun (investigator I. Marek): Solving problems with large scale sparse systems of linear and nonlinear algebraic equations, 1998-2000
Based on some mathematical and mostly nonlinear models originating in fluid dynamics, continuum mechanics, electrical engineering and electronic as well as physiology and biology, the project is directed towards solving the resulting problems of numerical algebra. It includes numerical analysis of some nonlinear phenomena as bifurcation and the effect of uncertain input data. The approximation and linearization of the original problems give rise to systems of linear algebraic equations. The core of the project focuses on effective methods for solving such systems and exploiting the special structure of the matrices implied by the model and its discretization.
A1019902 O. Dosly: Oscillation and asymptotic properties of solutions of linear differential equations, 1999-2001
Investigation of qualitative properties of solutions of linear differential equations in the following directions:
Oscillation and spectral properties of even order self-adjoint differential operators. Transformation, oscillation and asymptotic properties of solutions of nth order differential equations. Asymptotic properties of solutions of first order delayed differential equations.
C1019601 M. Englis: Function theory and operator theory in the Bergman space, 1996, postdoctoral
The problem of the limit behaviour of the weighted Bergman kernels has been solved, and new results have been obtained concerning the positivity of Green functions of weighted biharmonic operators when the weight is a derivative of a Blaschke product.
A1019701 M. Englis: Function theory and operator theory in the Bergman space, 1997-1999
A detailed description of the asymptotic behaviour of weighted Bergman kernels and of the Berezin transform in strictly pseudoconvex domains has been obtained, and their relationship with geometric invariants of the domain such as the curvature tensor has been clarified. A new mean value theorem for functions on bounded symmetric domains has been established, along with new results on Toeplitz and Hankel operators and their generalizations (transvectants), the Lu Qi-Keng conjecture, model operators for commuting operator tuples, and Green functions of weighted biharmonic operators.
A1019502 M. Fabian: Asplund and weak Asplund spaces and their relationship to optimization and topology, 1995-1996
1. Clarifying of the role of Asplund spaces in optimization and in nonlinear multivalued analysis. Finding those statements which are equivalent with the Asplundness. 2. Finding smooth variational principles for classes of Banach spaces that are not covered by principles of Stegall, Borwein-Preiss or Deville-Godefroy-Zizler. 3. Finding new large subclasses of Banach spaces which belong to the class of weak Asplund spaces. Investigating topological and geometrical properties of these classes. Methods used in items 1 and 2 are a separable reduction plus standard techniques from convex analysis. In 3 we focus on the spaces of type C(K) and we assume that combinatorial methods play a nonnegligible role here.
A1019702 M. Fabian: Geometrical aspects of Asplund and weakly compactly generated spaces, 1997-1999
Characterizing WCG spaces with help of Gateaux smoothness. Extending a theorem on smooth partitions of unity from preduals of WCG spaces to preduals of Vasak spaces. To sqeeze up from weak uniform rotundy more information than what is known, that is, that such a space is Asplund. Characterizing uniform Eberlein compacta by injecting them into spaces with a norm uniformly rotund in every direction. To try to construct LUR norm from a weakly LUR norm and some additional information. Solving the so called three space problem for uniformly Gateaux smooth renorming.
201/98/1449 M. Fabian: Banach spaces and nonlinear analysis, 1998-2000
The project is a continuation of the long standing research of the applicant and the coapplicant. We plan: To study, with help of martingales, Banach spaces admitting a C1-smooth function with bounded support. To study extending Frechet smooth norms from a subspace to the whole space. To solve some open questions concerning James boundary. To investigate the sequential continuity of polynomials on Banach spaces. To answer some questions about the smoothness of the space c0(\Gamma). To construct smooth partitions of the unity in some nonseparable spaces. To study the interrelationship between biorthogonal systems and weakly compactly generated spaces. To solve the problem of three spaces for some types of rotund norms. To clear up the role of Asplundness in nonlinear analysis and subdifferential calculus. To classify asymptotically Hilbert spaces. To study the density of C2-smooth norms on a space.
201/94/1063 E. Feireisl: Asymptotic behaviour of solutions of nonlinear wave equations on unbounded domains, 1994-1995
The suggested project consists in the investigation of the asymptotic behaviour of the solutions of the nonlinear wave equation with damping on a spatially unbounded domain. In this case the behaviour of solutions differs considerably from the problem considered on a bounded domain with suitable boundary conditions.
The problem can be divided into several parts: to investigate bounded trajectories and their convergence to equilibria, to find a suitable phase space, to determine conditions on the nonlinearity (dependence of the asymptotic behaviour on the critical exponent which defines the growth of the nonlinear function) and to study the problem with a nonlinear damping where the standard methods cannot be used.
A1019503 E. Feireisl: Long-time behaviour of solutions to conservation equations with memory effects, 1995-1996
In the present project, we propose to investigate the asymptotic behaviour of solutions to nonlinear conservation laws with memory terms.
The main goal is to obtain new results on existence and uniqueness of solutions, their regularity and long time behaviour. The problems of stabilization and compactness of trajectories in a suitable phase space are emphasized.
The methods of modern functional analysis are used, in particular, the kinetic formulation of conservation laws and the concept of regularized solutions.
201/96/0432 E. Feireisl: Nonlinear phenomena resulting from a dynamic balance between dissipation and nonlinearity in equations of evolution, 1996-1997
In the present project, we propose to investigate the long time behaviour of solutions to nonlinear evolution equations and systems on unbounded spatial domains.
The main goal is to obtain new results concerning the asymptotic behaviour of finite energy (bounded) solutions. The problems of stabilization and convergence to stationary states and soliton-like solutions are emphasized.
The methods of modern functional analysis are used, in particular, the theory of concentrated and compensated compactness in combination with recent results concerning the related stationary problems.
A1019703 E. Feireisl: Mathematical aspects of energy dissipation in nonlinear systems, 1997-1999
In the present project, we propose to study qualitative properties of solutions to nonlinear equations and systems describing processes with energy dissipation.
The aim is to obtain new theoretical results on the existence, uniqueness, regularity and asymptotic behaviour of the solutions.
The main tool are the methods of modern functional analysis, in particular, the kinetic formulation of conservation laws, the theory of compensated compactness, viscosity solutions and abstract Fourier analysis.
201/98/1450 E. Feireisl: Asymptotic behaviour of solutions to nonlinear evolution equations, 1998-2000
The aim of the present research project is to study the qualitative properties of solutions to evolutionary partial differential equations. The main topics include:
1. Nonlinear conservation laws. 2. The equations of hydrodynamics. 3. Functional equations. 4. Degenerate equations of parabolic type. 5. Domain dependence of solutions.
The goal of the project are new theoretical results, in particular, existence and uniqueness theorems for problems with low regularity of solutions, stability with respect to perturbations and the description of the long-time behaviour of solutions.
The methods are based on the latest results of modern functional analysis as, for instance, the weak convergence methods, the kinetic formulation and the theory of nonlinear semigroups.
201/94/1064 I. Havel: Algorithmic structures and hypercubes, 1994-1996
In computer science as well as applied mathematics, a great effort has been devoted to the study and design of parallel algorithms recently. The emphasis is, in particular, on (1) the research of classical data structures in parallel environment, (2) the simulation and optimization of already existing algorithms for various parallel architectures, and (3) the design of new algorithms for solving particular problems on particular parallel architectures.
The project is based on these facts and uses the recent results, assumptions, and abilities of all the three investigators. The goal of the proposal is to reach, in mutual cooperation, new results in the above mentioned areas. In applications, we emphasize graph theory, computational geometry and cluster analysis, and the hypercube architecture is to be investigated primarily.
201/98/1451 I. Havel: Special graph, hypergraph and algorithmic structures, 1998-2000
In the project, selected graph, hypergraph, algorithmic, and data structures are investigated with the help of methods of contemporary discrete mathematics and computer science. Primarily we solve graph and hypergraph problems, and focus on proper models of communication networks and possible applications, in particular on parallel computation. Mutual relations of various graph classes, construction of efficient algorithms for solving graph and hypergraph problems, and relations of various classes of combinatorial and algorithmic problems are investigated.
LB 98250 M. Jarnik (investigator A. Vitek): The creation, linking, sharing, and availability of new information sources in fundamental research, 1998-2000
The project is concerned with the information system of the Academy of Sciences of the Czech Republic, its contents, properties, organization, and availability.
A1075707 J. Jarusek (investigator J. Outrata): Nonsmooth analysis in problems of continuum mechanics, 1997-1999
The goal of the project was the investigation of some particular complex problems in the field of continuum mechanics. Their common feature was the occurrence of nonsmooth or multivalued mappings or terms and/or the necessity to apply tools of the nonsmooth analysis. The obtained results answer a wide spectrum of questions: the existence of (possibly generalized) solutions, optimality conditions, numerical analysis up to numerical solutions of problems from the technical practice. The existence of solutions both to static and dynamic contact problems with Coulomb friction possibly involving some thermal aspects was proved. Theoretical investigation of approximation of hemivariational inequalities both of elliptic and parabolic type was performed and appropriate numerical methods using optimization methods were developed. The results were published in more than 60 papers or proceedings contributions and represent an essential content of two published and one prepared monographs.
A1019504 K. John: Tensor products of Banach and topological linear spaces and their connections to ideals of operators, 1995-1997
Possible construction of Banach spaces with the complemented spaces of compact operators, connections with M-ideals of operators.
A1019801 K. John: Spaces of compact operators, 1998-2000
Proofs of the non-complementability of certain spaces of operators with the aim of possible solution of the old hypothesis on non-complementation of compact operators. M-ideals of operators.
201/99/D081 J. Komenda: Functional equations in special function spaces, 1999-2003, postdoctoral
In the last years, new approches to the study of discrete nonlinear systems, based on the use of special algebraic structures, have appeared. For instance, the use of the idempotent semiring (dioid) called (min,+) algebra leads to a linear description of some systems with the synchronization phenomenon and many results in the analysis and control of such systems have been obtained using this linear description. The aim of this grant is to extend some results known for discrete systems into a class of continuous and hybrid systems. First, we want to study the representation of these systems by special functional equations and the underlying Petri net models. Then, some analogies with the discrete case should be established. In particular, the analysis of periodicity, stability and input-output properties is of special interest. Linear input-output relation with a transfer matrix can be applied to the optimal control of these systems.
MSMT CR 93025 J. Krajicek: Mathematical logic: arithmetic, proof theory and complexity theory, 1993-1996
The research concerns the following topics: (i) bounded arithmetic, (ii) complexity theory and (iii) complexity of propositional and first order proof systems. These topics are closely related in aims and techniques, and the major problems in all three areas have the same character as the famous ”P versus NP” problem. Recent developments (some of it by the participating investigators) made these connections quite explicit and we believe they should be further investigated. Specifically, we investigate the following problems: finite axiomatizability of bounded arithmetic, lower bounds to the depth and to the size of Boolean circuits, bounds on the length of Frege, extended Frege and first-order proofs.
ME 103 J. Krajicek: Mathematical logic, complexity theory, and their connections, 1997-2000
The proposed research concerns the following areas of mathematical logic and computational complexity theory: (1) bounded arithmetic, (2) complexity of propositional proof systems, (3) circuit complexity and communication complexity, and (4) cryptography. These topics are closely related not only in an intuitive, heuristic sense, but also in a technical sense. The connections range from similar aims and techniques to deeply interconnected open problems. The most famous problem in this area is the ”P =? NP” problem. We believe that this area is one of the most interesting in mathematical logic and complexity theory and should be vigorously investigated. The goal of this project is to obtain results concerning fundamental open problems in mathematical logic and in complexity theory.
201/96/0431 M. Krbec: Integral operators in harmonic analysis and potential theory, 1996-1998
The main topics studied in the frame of this grant project include boundedness of integral operators on spaces of homogeneous type, applications of the method of integral equations for solutions of the Laplace equation and the heat equation, and boundedness and estimates for approximation numbers of the Hardy operator. The main idea of the project was to study problems reflecting the needs of the theory itself as well as needs of appropriate tools in the PDE’s theory. The developed theory of operators goes far beyond the framework of the Euclidean metric and the importance of the concept of a homogeneous space is given by the extent of its applications. The method of integral equations, besides from setting problems on general domains has made it possible to suggest methods for computing the solutions. Further, the new results also concern manageable sufficient conditions for boundedness of the Hardy operator in weighted spaces together with applications to the strong unique continuation property of the Schrodinger operator.
201/95/0568 P. Krejci: Mathematical models for multidimensional plastic hysteresis, 1995-1996
The project is devoted to mathematical methods for investigation of multidimensional models of plasticity. Classical models of the Prandtl type are not frequently used because of their numerical complexity and insufficient qualitative correspondence with experiments. Recently, the interest in the Mroz model has increased accordingly a need of corresponding mathematical methods. The aim of the project is a response to mathematical problems arising in connection with numerical implantations of the Mroz model, e.g. the invertibility of the Mroz constitutive operator, extension of the classical Mroz model with spherical surfaces of plasticity to a more general geometric class, and specifying a relation between the Mroz and the Prandtl-Ishlinskii models.
A1019601 M. Krizek: Superconvergence phenomena and higher order schemes in the finite element method, 1996-1998
We proved several higher-order error estimates for a class of nonlinear elliptic boundary value problems. A comparison principle for a nonpotential problem of nonmonotone type has been derived. We have also developed an algorithm for local refinements of tetrahedral partitions of a polyhedral domain.
201/98/1452 M. Krizek: Computational aspects of the finite element method, 1998-2000
A detailed numerical analysis of a heat radiation problem with a nonlinear Stefan-Boltzmann boundary condition was done. We have introduced a new refinement algorithm of tetrahedral partitions which produces nonobtuse tetrahedra, which play an important role in deriving a discrete maximum principle. Another algorithm has been developed for generating regular families of partitions of domains with a piecewise curved boundary.
ME 148 M. Krizek: Reliability problems in computational mechanics, 1998-2000
We examine an influence of numerical quadrature formulae in solving elliptic boundary value problems on domains with a curved boundary by the finite element method. An analysis of harmonic and biharmonic finite elements has been performed. The main effort is focused on reliability of numerical calculations in solving a various problems of mechanics and mathematical physics.
201/95/0630 M. Kucera: Bifurcations and stability of stationary and periodic solutions of variational inequalities, 1995-1997
The project represents a part of a long time effort of the principal investigator to build a theory of bifurcations for variational inequalities and related problems. The problems under consideration cannot be linearized, therefore classical methods of the bifurcation theory cannot be used and it is necessary to develop new approaches.
The results about the stability of bifurcating periodic solutions in the three dimensional case were completed. Reaction-diffusion systems with unilateral conditions were studied. The influence of unilateral conditions to bifurcation of stationary spatially nonhomogeneous solutions (spatial patterns) in the case of activator-inhibitor systems was systematically studied. New ideas were found which make possible to localize the bifurcation points obtained. A destabilizing effect of unilateral conditions was proved also in the case of quasivariational inequalities and inclusions.
201/98/1453 M. Kucera: Bifurcation of solutions of variational inequalities and inclusions with applications to mathematical models in natural sciences, 1998-2000
The results obtained in the previous stage of the research for reaction-diffusion systems with boundary conditions described by variational inequalities and inclusions are further developed, completed and unified. Particularly, spatial patterns in reaction-diffusion systems of activator-inhibitor type are discussed in more general situations than before. A new view to former information concerning a destabilizing effect of unilateral conditions to spatial patterns was found. The methods developed for variational inequalities are used also for the investigation of bifurcations for problems with jumping nonlinearities. A character of bifurcating branches for variational inequalities in some particular cases is investigated.
Further, bifurcations of periodic solutions to evolution variational inequalities in finite dimensional spaces are studied. The theory given by the principal investigator and his cooperators concerning the finite-dimensional case is further developed to obtain results applicable to a wider class of concrete examples.
201/94/0008 A. Kufner (investigator P. Drabek): Qualitative analysis of boundary value problems for a certain class of nonlinear ordinary and partial differential equations, 1994-1996
The project is concerned with the study and description of fundamental properties of weak solutions to nonlinear elliptic equations where the basic space is a weighted Sobolev space. Existence theorems are proved and spectra of the corresponding differential operators are described. The degree of mapping is applied to the analysis of existence and asymptotic stability for particular ordinary differential systems, the wave equation, and the equation of beam.
A1019506 A. Kufner: Weighted Sobolev spaces and their applications, 1995-1997
The aim of the project is a detailed description of weighted Sobolev spaces from the following viewpoints: a) the domain of definition (geometric structure, smoothness); b) the weight functions; c) the parameters of the smoothness of the space (fractional order spaces); d) the growth parameters (e.g., spaces with variable growth degree). The aim is to give a full characterization of the ”classical” weighted spaces and to introduce and investigate some extensions (Besov, Orlicz-Sobolev spaces etc.).
E1019601 A. Kufner: Nonlinear degenerated partial differential equations of elliptic type, 1996, editorial
Combining abstract methods of functional analysis and properties of weighted Sobolev spaces, the existence of weak solutions to some nonlinear elliptic operators of second as well as of higher order was shown. These operators have - contrary to the ”usual” ones - degeneration and/or singularity in the coefficients. Special attention is paid to the p-Laplace-operator and its generalization to the degenerated or singular case. Basic spectral properties of these special operators are studied, and some important properties are derived – as positivity of the first eigenfunction, bifurcation from the first eigenvalue, maximum principle etc.
MSMT CR A. Kufner (investigator P. Drabek): Centre of Applied Mathematics, 1997-2000
The aim of the project is to establish a research group dealing with the investigation of nonlinear differential equations and systems modelling some stationary and nonstationary phenomena. Results and methods of functional analysis are further developed and applied to investigate the existence, multiplicity and stability of solutions to some model-type nonlinear elliptic equations. Further, problems of bifurcation and stability to strongly nonlinear processes are studied, described by variational inequalities and appearing in biological and ecological models. The main assumption is to build a compact research team with close connections to top mathematical departments of the Academy of Sciences.
201/98/P017 J. Lang: Integral operators in Banach function spaces, weighted inequalities and applications, 1998-2000, postdoctoral
The project concerns two fields: the problem of conditions for the boundedness and compactness of the Hardy and Volterra integral operators in Banach function spaces and generalization of weighted inequalities (in particular, Carleman’s and Harnack’s inequalities) with respect to applications in partial differential equations. These parts of the harmonic analysis and the theory of real functions are currently very rapidly developing and bring new results, which have immediate applications in other fields of mathematics, especially in partial differential equations, potential theory, and in mathematical physics, particularly in the study of the Schrodinger operator.
A2060907 V. Lovicar (investigator M. Severa): Interaction of hard particles with a turbulent flow of a liquid, 1999-2003
It was found that the crucial point for calculation and simulation is a knowledge of the velocity of hindered sedimentation and knowledge of particle drag coefficient in a turbulent flow which is in some cases substantially larger than in a still liquid.
We expect i) to formulate and to derive theoretical backgrounds for the drag coefficient determination of sphere in a turbulent flowing suspension, ii) to verify experimentally the theory proposed using LDA/PDA method, iii) to formulate simulation models and to solve them using PHOENICS or FLUENT for complex two-phase flow in process equipment (mixed vessels, fluidization). For solving the suspension mixing we use the sliding mesh technique.
A1019507 M. Markl: Operads in algebra, geometry and physics, 1995-1996
The aim is to study consequences of the existence of the operad structure on some physically relevant spaces, namely on configuration spaces of various types. The first problem is to find minimal models for those operads and to investigate their formality. The next aim is to describe the homology induced by these models; the motivation here is the hope to solve the commutative algebra cohomology problem. The necessity to understand the algebraic structure of vertex operator algebras leads to the concept of a ”symmetry up to homotopy”. All three problems above are closely related to each other, the unifying ingredient being the notion of an (algebraic or geometric) resolution of an operad. This notion was introduced and studied in the previous work of the applicant.
A1019804 M. Markl: Higher homotopy structures, 1998-1999
Understand the existence of strongly homotopy algebras in terms of homological properties of corresponding operads.
Develop a general theory of ‘loop homotopy algebras’ for algebras over cyclic operads.
Study structures of traces over compactifications of configuration spaces considered as modules over the operad of local configurations.
Develop a theory of ‘homotopy traces’ at least at the ‘tree level’ and study possible generalizations to the ‘loop’ case. Try to formulate and prove an analog of the approximation theorem.
ME 333 M. Markl: Operads in algebra, topology, and mathematical physics, 1998-2000
Study of algebraic and geometric properties of compactifications of configuration spaces.
201/99/0675 M. Markl: Geometric and topological structures in matematical physics, 1999-2001
Growing importance of abstract algebraic and topological methods is a characteristic feature of mathematical physics of the second half of the last century. The problems of our grant are directly related to this development.
(1) Application of the theory of operads to the study of structures arising from (open or closed) string field theory and to the construction of higher products in the category of partial differential operators. (2) Modelling of geometric structures by parabolic subgroups of semisimple Lie groups. (3) Study of geometric structures on vector bundles with applications to those structures which play a role in mathematical physics. (4) Study of properties of solutions of field equations with values in general spin representations and multiplicative properties of solutions of invariant differential equations.
201/95/0629 B. Maslowski: Asymptotic properties of solutions to stochastic evolution equations, 1995-1997
Long-time behaviour of solutions to stochastic evolution equations has been studied with emphasis put on problems of existence and uniqueness of invariant measures, ergodicity and mixing. Also, pathwise behaviour has been investigated, in particular, stability and integral continuity, with applications to averaging of equations with quickly oscillating coefficients.
Problems of stochastic control of linear evolutionary systems were dealt with.
201/98/1454 B. Maslowski: Qualitative theory of stochastic evolution equations, 1998-2000
Markov processes induced by stochastic equations in infinite-dimensional spaces have been studied with particular emphasis put on invariant measures, ergodicity, mixing, and analytical properties of associated Markov semigroups. Nonexplosion and stability in the mean and in probability has been treated. Ergodic and adaptive control problems for nonlinear infinite dimensional systems have been solved by means of corresponding infinite dimensional Hamilton-Jacobi equations. The results have been applied to equations of stochastic hydrodynamics, stochastic reaction-diffusion systems and equations of population dynamics.
A1019705 E. Matouskova: Null sets in topological groups, 1997-1999
The project compares the properties of Lebesgue null sets and their generalization to infinite dimensional Banach spaces in the context of differentiation of convex functions and Lipschitz mappings.
A119106 V. Muller: Applications of the function theory and Banach algebras methods to the operator theory, 1993-1995
The objectives of the research project are investigations concerning:
1. Spectral theory of operators in Banach and Hilbert spaces with emphasis on the Banach algebras methods and questions connected with the invariant subspace problem. 2. Toeplitz and Hankel operators in the Hardy and Bergman spaces and their application to the function theory, harmonic analysis and the approximation theory. 3. Theory of functional models of non-selfadjoint operators, dilations and lifting theory, especially questions concerning the control theory.
201/96/0411 V. Muller: Applications of the function theory and Banach algebras methods to the operator theory, 1996-1998
The objectives of the research project are investigations concerning:
1. Spectral theory of operators in Banach and Hilbert spaces. 2. Operator theory on Hardy and Bergman spaces, its applications to harmonic analysis and theory of quantization. 3. Functional models of non-selfadjoint operators an Hilbert spaces, dilations and lifting theory.
201/95/1484 J. Nedoma (investigator J. Rohn): Linear problems with inexact data, 1995-1997
Systems of column-separated systems of vague linear equations: structure of the solution set; systems with restricted-rank error matrices. Properties of positively regular vague matrices.
201/98/0222 J. Nedoma (investigator J. Ramik): Linear algebra and its applications to optimization problems with inexact data, 1998-2000
Geometrical interpretation of vague matrices; separation and support properties of the respective set families; a method for finding the interval hull of vague linear equation system; systems with pattern-shaped vague columns; duality of systems of vague linear equations and inequalities.
201/93/0452 F. Neuman: Ordinary linear differential equations, qualitative aspects and the formal theory, 1993-1995
Global properties of solutions of linear differential equations are studied, boundary value problems of Cauchy-Nicoletti type are considered. By methods of differential geometry differential systems are considered.
A119404 F. Neuman: Global behaviour of solutions of ordinary differential equations, 1994-1996
Global properties of solutions of ordinary differential equations are studied and described. Analytic, geometric and topological methods and the theory of functional equations are the main tools of this investigation.
201/96/0410 F. Neuman: Differential and functional-differential equations, 1996-1998
Qualitative properties of solutions of systems of functional-differential equations are studied. Namely their asymptotic behaviour.
201/99/0295 F. Neuman (investigator M. Bartusek): Qualitative theory of differential equations, 1999-2001
Qualitative behaviour of linear and nonlinear differential equations, functional-differential equations and their systems are studied.
201/94/1066 B. Opic: Integral operators and embeddings in function spaces, 1994-1996
The main goal of the project is to establish efficient conditions which characterize important properties (continuity, boundedness, compactness, etc.) of integral operators on function spaces, and to study embeddings of function spaces.
201/97/0744 B. Opic: Weighted inequalities for integral operators, 1997-1999
The main goal of this project is to investigate the behavior of linear and quasilinear integral operators on function spaces and apply the results thereby obtained to prove interpolation and extrapolation theorems and to study function spaces and their mutual relations.
201/94/0069 J. Pelant: Geometrical and topological properties of Banach spaces and their applications in nonlinear functional analysis, 1994-1996
The project of Banach space research concentrates to the study of differentiability, generic properties of monotone type maps, covering properties (namely on spaces of continuous functions), renormations and some problems connected to the theory of optimization.
In the differentiability theory, we focus on the research of weak Asplund spaces and their relationship to covering properties of their duals, derivatives and generalized derivatives of Lipschitz functions and the theory of exceptional sets, related to generic differentiability. For spaces of continuous functions, beside the research mentioned above, Lipschitz structure of these spaces is studied. Those parts of the project dealing with smooth variational principles, set convergence and abstract theory of approximation (properties of metric projections and distance functions) are related to the optimization theory.
201/97/0216 J. Pelant: Mappings and covering properties of topological structures, 1997-1999
The research project deals mainly with topics from general topology. Main aspects, which become important and which combine in various ways, are a combinatorial approach and a categorical one. These aspects apply in a natural way and relationships to fields like topological dynamics, Boolean algebras, functional analysis, measure theory, and models of set theory are very often employed. Properties of the Cech-Stone compactification of integers, finitely additive measures without lifting and combinatorics of the first uncountable cardinal number are investigated.
Problems of the descriptive character of various topologies are related to functional analysis. They are connected with natural problems like geometric properties of Banach spaces or productivity of various classes of structure from topology and algebra, which originate from theorems due to Mazur or Mackey.
202/93/2383 P. Prikryl (investigator R. Cerny): Pulsed-laser induced processes in thin metal and insulator films on semiconductor surfaces, 1993-1995
A mathematical model of pulsed-laser induced nonequilibrium melting and solidification of metal-insulator-semiconductor systems with chemical reactions was developed (moving boundary problem). The phenomenology of the silicide growth and the specific atomic mechanisms of phase formation at nonequilibrium conditions was analyzed for the first time. The comparison of the computational and experimental results made it possible to evaluate the thermophysical and optical properties of thin metal and insulator films. The temperature of the reaction and melting at the metal-semiconductor interface was estimated.
103/93/0395 P. Prikryl (investigator V. Havlik): Diffusion and convection of pollutants from contaminated water in waste disposal sites, 1993-1995
A theoretical physico-chemical analysis of transport phenomena in waste disposal sites with radioactivelly polluted water was performed. A computational model of fluid flow based on this analysis was formulated using methods of continuum physics. Numerical simulations using the model developed were carried out. Based on the computational results, an analysis of water management and environmental protection of surface and ground waters was done.
202/96/1188 P. Prikryl (investigator P. Hoschl): Growth of high quality II-VI single crystals (technology and thermodynamic model), 1996-1998
The aim was to improve the quality of substrates based on CdTe (CdZnTe) which can significantly contribute to better properties of quantum structures based on II-VI compounds. The materials studied have a prospective technological importance (infrared detectors, gamma rays detectors). As a model system for preparation of further materials having a high segregation coefficient between the solid and liquid phases the processes of HgCdTe solidification were studied and modeled. Computational modeling was performed to optimize the experimental setup (construction of furnaces). The mathematical model was based on a moving boundary problem with convection in the melt.
A1010719 P. Prikryl (investigator V. Chab): Laser processing of compound semiconductors and semiconducting alloys 1997-1999
The aim of the project was to determine the segregation coefficients and kinetic phase diagrams under nonequilibrium conditions as functions of the energy density and the time structure of the applied laser pulses. A mathematical model of pulsed-laser induced nonequilibrium melting and solidification of the compound semiconducting alloys was developed (moving boundary problem). The model was used to simulate the experimental time-resolved reflectivity (TRR) spectra. The segregation coefficient was evaluated by statistical methods depending on the solidification conditions. The morphological stability of solidification was studied, melting and solidification kinetics was monitored. The resulting structure, chemical composition, and orientation dependence of segregation coefficient were determined.
202/99/1646 P. Prikryl (investigator P. Hoschl): Experimental study of defects and numerical modeling of growth processes leading to an optimization of growth and properties of CdZnTe single crystals, 1999-2001
This project follows up the project 202/96/1188, which showed that the researchers developed a prospective method for preparing CdZn and CdZnTe single crystals of high quality. It has shown, however, that a detailed study of the defect creation and annihilation during the growth process or during the following annealing experiments has to be performed. This is a complex problem that has not yet been solved in all its respects. The work is devoted to experiments closely connected with computational modeling of growth and annealing processes using the finite element method.
A119107 P. Pudlak: Metamathematics of arithmetic, complexity theory and set theory, 1993-1995
This is a project of basic research in the following areas: bounded arithmetic, complexity of propositional calculus, circuit complexity, and alternative set theory. The main problems on which we want to work are: separation of fragments of bounded arithmetic, lower bounds to the size of proofs in propositional calculus, lower bounds to the size of bounded depth circuits, combinatorial and algebraic characterization of complexity, developing mathematics in alternative set theory, and investigating interpretability of the theory.
A1019602 P. Pudlak: Arithmetic, proof theory, and complexity theory, 1996-1998
This is a project of basic research in complexity theory and logical foundations of arithmetic. We plan to work on open problems in the following areas: complexity theory, bounded arithmetic and propositional logic, higher order arithmetic and set theory. Problems of particular interest are: lower bounds for communication complexity, relation of cryptographical conjectures to arithmetic, independence results for bounded arithmetic, lower bounds for propositional proof systems, models of alternative set theory.
A1019901 P. Pudlak: Mathematical logic and computational complexity, 1999-2003
We work in the following related areas: the complexity of propositional calculus, bounded arithmetic, circuit complexity, communication complexity, efficient algorithms, on-line algorithms, higher-order arithmetic and set theory, and model theory of fields. The areas both overlap in results and use same or similar methods. The main aim is to gain better understanding of the phenomenon of complexity. The output are scientific publications.
201/93/2172 I. Saxl (investigator J. Rataj): Anisotropy characteristics of random geometrical structures, 1993-1995
The relations have been derived among second order characteristics of non-isotropic random measures, the volume of dilation (found by a suitably rotated convex probe), and the surface measure of a set belonging to the convex ring. The theoretical formulae for random distances in various Newman-Scott point cluster fields have been found. The theoretical results have been used for the examination of anisotropy characteristics of random closed sets (RCS) of various type in a d-dimensional Euclidean space and for study of statistical properties of the estimators of intensities of anisotropic RCS’s. A detailed study was made of characterizing given point patterns by means of distance methods and initiated a similar study using polyhedral (polygonal) methods (based on the dual representation of a point pattern by the Voronoi tessellation generated by it). All results have been applied in the materials science.
201/96/0226 I. Saxl (investigator J. Rataj): Spatial distribution of random set components, 1996-1998
Theoretical (probabilistic) part was concerned mainly with the generalization of some important results for the sets of positive reach and with the variances of stereological estimators. The results obtained follow from the distributions of projection measures expressed by means of the corresponding Palm distribution. Simulations of Voronoi tessellations generated by various point processes of the germ-grain model have compared two opposite cases of germs being either a point lattice or Poisson point process; the grains have been point clusters of different kind. The bimodality or multimodality of the distributions of cell characteristics have been shown to be closely connected with the presence of inner and outer cells. Direct applications concerned the high temperature creep of metals (relations between grain boundary sliding and void formation) and the stereology of grains with a special attention to the standard industrial methods.
201/99/0269 I. Saxl (investigator J. Rataj): Integral geometry and statistical analysis of the random closed set components, 1999-2001
Frequent models of random closed sets in the Euclidean space are unions of processes (stationary random fields) of particles (points, fibres, convex bodies etc.). The particular attention is devoted to integral-geometric relations (of the translative or general kinematics type) for curvature integrals of individual particles as well as of the RCS as a whole. The results are applied to the variance analysis of particular estimators of length and area densities of particles. For the simulation of fiber and surface processes, the recent experience with point processes are used. The development of distance and polyhedral methods for the point pattern recognition are continued with a particular attention to the hard-core and pseudo-hard-core processes.
201/94/1068 S. Schwabik: Summation integral and its applications in equation theory, 1994-1996
The research was devoted to the development of the integration theory based on integral sums and its applications to differential and integral equations. Special attention was paid to the definitions and properties of summation integrals suitable for integration over more dimensional domains and conditions for density of step functions in the space of all functions integrable in this sense. Furthermore, the members of the team continued the study of the properties of the Perron-Stieltjes integration with respect to regulated functions as well as its extension to integration with respect to functions with values in Banach spaces. This enabled us to obtain new fundamental results for generalized differential and integral equations with regulated solutions.
201/97/0218 S. Schwabik: Summation integral and its applications in equation theory, 1997-1999
The primary aims of the project are the following:
a) To collect and integrate the known facts concerning the definitions and properties of summation integrals suitable for integration over more-dimensional domains, in particular in connection with the study of convergence theorem. b) To find a suitable notion of integral based on integral sums and conditions for density of step functions in the space of all functions integrable in this sense. This result would link the elementary ”geometrical” concept of integration with the advanced and powerful integration theory developed by the participants in the project. c) To extend the results known for Volterra-Stieltjes integral equations to general linear causal equations in the space of regulated functions.
201/94/1067 K. Segeth: Methods of nonlinear finite element analysis, 1994-1996
The project is concerned with methods of nonlinear finite element analysis applied to the investigation of physical fields. In the theory of the finite element method for nonlinear problems there exist a number of unsolved problems. In contrast to linear models, there is no general approach to modelling nonlinear fields. Particular methods have to be applied to individual classes of nonlinear problems with regard to their specific features, and the treatment of each problem has to be split into several consecutive steps. Several classes of nonlinear problems, whose analysis employs common mathematical and numerical tools, have been chosen for the project. The problems are described by partial differential equations of elliptic or parabolic type with appropriate boundary conditions. All these problems have important applications to technology and engineering (in particular, to phenomena whose description by rough linear models is inadequate).
KSK1019601 K. Segeth: Contemporary problems of mathematics and mathematical physics, 1996-2000, Programme of development of fundamental research in key areas of science
The project is concentrated on developing and solving contemporary problems of mathematical analysis, mathematical logic, and mathematical physics. The contribution of the project is the derivation of a number of exact results in the fields mentioned as well as the development of new mathematical methods and procedures, the creation of new models and theories, and the investigation of their properties. The results contribute to solving the fundamental problems of today’s physics of particles and solid state physics, too.
103/96/1710 K. Segeth (investigator J. Vaska): Transport processes in a catchment-reservoir system, 1996-1998
Improvement of surface water resources quality, mostly of reservoirs, requires a decrease in the input of pollutants from catchments and an implementation of a reservoir operation system that is in agreement with water quantity requirements. The objective of the project is to analyze the main transport processes of pollutants in a catchment-reservoir system. The transport of solid particles and dissolved substances within the catchment of surface water resources (mostly from diffuse sources), and the hydrodynamic, chemical and biological processes in thermal stratified reservoirs are studied. The project is based on the application of mathematical simulation models and monitoring of input data for their calibration and verification. The theoretical results are applied to a selected model system ”catchment-reservoir”, e.g. the Pastviny reservoir (Eastern Bohemia) or the Rimov reservoir (Southern Bohemia).
201/97/0217 K. Segeth: Numerical analysis of nonlinear boundary value problems, 1997-1999
The project is concerned with the mathematical and numerical analysis of nonlinear boundary value problems. In contrast to linear problems, there is no general approach to solving nonlinear boundary value problems. Particular methods have to be applied to individual classes of nonlinear problems with regard to their specific features, and the treatment of each problem has to be split into several consecutive steps. The finite element method has been chosen for the analysis of the problems considered since it represents a very efficient technique, in particular, for problems of mathematical physics and engineering. However, there exist a number of unsolved problems in the theory of the finite element method for nonlinear problems. Several classes of nonlinear problems, whose analysis employs common mathematical and numerical tools, have been chosen for the project. The problems are described by partial differential equations of elliptic or parabolic type with appropriate boundary conditions. All these problems have important applications to technology and engineering. Particular attention is paid to the modeling of phase-change processes.
201/97/P038 J. Sgall: On-line algorithms and communication complexity, 1997-2000, postdoctoral
This is a project of basic research in the theory of algorithms and the complexity theory. In the theory of algorithms we focus on particular variants of on-line scheduling and related problems. In the complexity theory we study the relations between communication complexity, the rank of the matrix of the computed function, and the Boolean complexity. In both areas the project has continuity with the current research of the involved researchers.
A119109 Z. Sidak: Updating of the book Hajek, Sidak: Theory of Rank Tests, 1993-1995
In 1967, Publishing House Academia, Prague and Academic Press, New York have published in coproduction the book Hajek, Sidak: Theory of Rank Tests, which became apparently the most important and most widely used source in scientific work at an advanced level in the theory of statistical rank tests. Even now, after 25 years, in the world literature there exists no newer book of similar contents and level, though, of course, scientific research in this area has made a substantial progress. The aim of the project is an updating of the contents of this book according to the current state of knowledge.
A119110 M. Silhavy: Thermodynamics, stability and dissipation inequalities for plastic materials, 1993-1995
The research project investigates the dissipation inequalities in the theory of plasticity. This includes the dissipation inequalities coming from the second law of thermodynamics as well as the dissipation inequalities resulting from Il’yushin’s condition. These inequalities were shown to hold only for small processes (in a specific sense). The research answered the question whether these inequalities could be generalized to large processes and to rate-dependent materials and clarified their consequences on the constitutive equations.
A2019603 M. Silhavy: Convexity properties of stored energies, phase transitions and instabilities in materials, 1996-1998
The goal of the project is to find realistic mathematical and physical properties of the free energies of nonlinear materials. Modern methods of the calculus of variations are used. Specifically, one goal of the project is to find conditions for rank 1 convexity in isotropic materials. These and related results are then used to examine the nonelliptic free energies in the presence of phase transitions and instabilities in solids.
A1019807 J. Simsa: Functional and differential equations for functions with finite decompositions, 1998-2000
In many applications, functions in several variables appear as results of a finite number of arithmetic operations, applied to functions in a smaller number of variables. The main purpose of the project is to characterize such functions of some prescribed forms by means of functional or differential equations and, in this way, to continue in fruitful investigations of F. Neuman and J. Simsa of the last two decades. Particularly, a special effort is devoted to a decomposition problem of J. Falmagne, which remains to be open for more than 15 years. The recent research of J. Simsa suggests that a progress in solving this problem is attainable.
101/93/0261 Z. Sobotka: Structural, mechanical, and mathematical relations and models in nonclassical rheology and plasticity, 1993-1995
The subject of the project is the interdisciplinary synthesis of rheological relations, mathematical methods, and structural properties of rheological materials and mechanical systems. Special attention is paid to the long-term deformation, anisotropy, nonlinearity, and mechanical unsymmetry that cause a different mechanical behavior in tension, pressure, shear, and torsion. From the mathematical point of view, the project presents new ways of tensor expansions, nonlinear functional equations, analogies between mathematical and rheological structures, and methods for the evaluation of results of rheological tests of viscoelastic materials. Selected constitutive relations are applied to viscoelastic plates and shells.
106/96/0938 Z. Sobotka (investigator M. Petrtyl): Principle of remodelation of the corticalis composite macrostructure, 1996-1998
Corticalis belongs to composites with controlled properties. Its behaviour and properties in elastic and inelastic domain have not been described satisfactorily yet. The project is concentrated on the formulation of a general theory of remodelation of corticalis composite material.
201/93/2177 I. Straskraba (investigator M. Feistauer): Mathematical modelling and numerical solution of problems in fluid mechanics, 1993-1995
Qualitative and quantitative behavior of solutions to Navier-Stokes equations for viscous fluids was investigated.
101/96/0832 I. Straskraba (investigator W. Kolarcik): Hysteretic model of pump, 1996-1997
Hysteresis phenomena in pumps were experimentally detected and a mathematical model for their description suggested and analyzed.
201/96/0313 I. Straskraba (investigator M. Feistauer): Mathematical theory and numerical solution of problems of fluid mechanics, 1996-1998
The search for existence results and global behavior of solutions to Navier-Stokes equations for viscous fluid was continued.
201/99/0267 I. Straskraba (investigator M. Feistauer): Qualitative theory and numerical analysis of problems in fluid dynamics, 1999-2001
Evolution equations of dynamics of fluids are currently being analyzed with the emphasis to large data and global behavior results.
101/99/0654 I. Straskraba (investigator W. Kolarcik): Parametric model of pumps, 1999-2001
Mathematical description of fluid flows inside pumps is under investigation together with necessary experimental confirmation of analytical and numerical results.
406/93/0974 M. Ticha (investigator M. Hejny): Mathematical education of students aged 11-15, 1993-1995
The aim of the research was to prepare the offer of various levels of mathematical education of students aged 11 to 15 and to elaborate in details the project of teaching mathematics for those students who continue their studies at some of upper secondary schools. It is necessary to view mathematical education primarily through educational structure of mathematics and only secondarily through logical one. The education has to arise from the student’s own work and his/her experience; to know mathematics includes/means to do mathematics. The quality of education depends in a decisive manner on the teacher. Results: Elaboration of studies on approaches to the mathematical education; on generation of concepts; on grasping of word problems and problem solving strategies; on creation of teaching units. Application of results: creation of curricula for students from 11 to 15; creation of textbooks; work with in-service teachers; doctoral study.
406/96/1186 M. Ticha (investigator M. Hejny): Mathematical education of pupils from 5 to 15, 1996-1998
Results of previous research (in equations, fractions, visualization, teaching units, constructive and instructive attitudes and modelling the cognitive net) open new areas of interesting and promising problems. First and foremost, we aimed to get deeper knowledge of these areas to long-term development of all the five elements of the cognitive net. Therefore the age group of pupils was extended from 5 to 15.
Following domains were analysed: pedagogical thinking of teachers, styles of thinking of students, investigative teaching, sense of mathematical education, methodology. Students’ conception of number and fraction, stages of grasping situations and problem solving were classified and characterised. To the main results, the description of possibilities how to influence teacher’s work belongs.
Research was performed in collaboration with teachers; doctoral students were engaged in it. Results of the research were published in reviewed journals and proceedings.
GA UK 303/1998/A PP/PedF M. Ticha, M. Baresova: Activating role of projects in mathematics education, 1998-2000
The project contributes to the increase and development of constructive character of mathematics education by utilization of projects. The main aims of the project are: To specify the concept of project from the point of view of requirements of mathematics education with regard to the constructive approaches to the formation of notions in the student’s structure of knowledge. To elaborate the methodology of formation of projects which supports and enables to change current receptive approaches to the cognition. To prepare, to implement in the concrete teaching process and to evaluate long-term research devoted to the study of activating role of projects in the education. To summarize the results of the work on research project inclusive of suggestions of its application in the school practice.
406/99/1696 M. Ticha (investigator M. Hejny): The parallels of gaining knowledge and education processes in mathematics, 1999-2001
The didactics of mathematics realizes the need to find ways of changing the present school, supporting creative teaching and inhibiting the long-standing instructive teaching. In connection with the previous research, the project focuses on: (1) regularities which control mathematical cognition (models of the mathematical part of a cognitive net; learning styles; concept creation processes; problem solving, mainly the phase of grasping a problem situation; the process of structuring; mental representation of concepts and processes; investigative teaching), (2) the possibilities of implementing research results into practice (direct work and co-operation in the research with practising teachers and the study of future teachers education), (3) methodology of both the theory and its implementation (developing research methods of atomic analysis, simulation, genetic parallel, etc.). The research results are used in a new conception of future mathematics and elementary teachers education.
201/93/2433 J. Tuma (investigator J. Trlifaj): Modern algebraic structures, 1993-1995
The project continues the further development of the basic research carried out by Czech, Italian, French, and German investigators. It is a continuation of the research whose results have been published in learned journals. It concentrates to the three following areas: 1. modulus categories, 2. binary systems, 3. union theory.
A1019508 J. Tuma: Combinatorical and logical problems of general algebraic structures, 1995-1996
We investigate intervals in subgroup lattices of finite simple groups, especially the sporadic ones, to exclude certain lattices of length 2 as possible examples of these intervals. These results combined with the recent results of A. Lucchini and R.W. Baddeley should bring great improvement in the knowledge of the structure of general finite algebras. We also study the congruence extension property for finite lattices that played recently an important role in the solution of various problems about the structure of congruence lattices of algebras of a fixed similarity type.
201/95/0632 J. Tuma: Structure of finite algebras, 1995-1996
The subject of the project is the study of structure of finite algebras. Finite algebras are investigated from three different aspects: 1. Unions of congruences of finite algebras and intervals in unions of subgroups of finite groups. 2. Equation theories of finite algebras and classes of algebras with a finite basis for equation theories. 3. Unions of equation theories widening a given equation theory.
201/93/2178 J. Vanzura (investigator V. Soucek): Modern geometric methods of mathematical physics, 1993-1995
After a longer period of more or less separate evolution, last decades brought a new golden period of interaction between mathematics and physics. Several topics in geometry inspired by recent progress in this area are chosen for further study:
1. A study of conformally invariant operators, both their local properties and their global behaviour. The main topics are the function theory for solutions of conformally invariant equations, the twistor description of these solutions, problems connected with global existence of relevant structures and a description of conformally invariant operators on curved spaces. 2. Noncommutative geometry founded by A. Connes was recently used in constructions of models of unified description of interaction in elementary particle physics. We study further possible models for the description of gravitation and gauge field theories, both in dimension 2 and 4.
201/96/0310 J. Vanzura (investigator V. Soucek): Geometric structures and invariant operators, 1996-1998
A mutual influence of geometry and mathematical physics has been very fruitful during last decades. Several topics from this area are studied:
1. Applications of tools from the theory of natural operators and representation theory for a discussion of invariant operators on manifolds with a distinguished geometric structure and a study of an analytic version of the Zuckermann translation principle. 2. A generalization of basic results from function theory for Dirac operator to solutions of other invariant operators acting on fields with values in more general representations and to manifolds, a study of properties of solutions of such invariant equations for other geometrical structures. 3. A study of noncommutative version of the Swinger model and regularizations od supersymmetric field theories based on the concept of chirality. 4. An application of operadic approach for study of quantum groups.
A130407 Z. Vavrin (investigator M. Fiedler): Structured matrices, 1994-1996
Principal directions of research:
Theoretical background and properties of some classes of structured matrices (Hankel, Toeplitz, Vandermonde, Cauchy, Loewner, Bezout, companion, etc.) and generalizations of these classes to the confluent cases and infinite-dimensional cases.
Algorithmic problems and complexity considerations for structured matrices.
Special subclasses of structured matrices.
Applications of structured matrices.
A1030701 Z. Vavrin (investigator M. Fiedler): Extensions of linear-algebraic problems, 1997-1999
Extensions were considered in three dimensions: for problems the coefficients of which are matricial blocks, or belong to a noncommutative ring, or in the third direction, generalizations of problems involving positive definite quadratic forms to positive multiquadratic forms were studied. The first direction is important for special classes of matrices such as M-matrices, or matrices occurring in linear control systems, such as Hankel, Loewner, companion matrices of polynomials etc., the second e.g. for totally positive matrices, the third could have prospective use in technical and theoretical-physics problems in case where positive definite quadratic forms do not suffice in the modelling.
A119101 E. Vitasek: Higher order approximations to the solution of linear and nonlinear elliptic and parabolic problems, 1993-1995
The project is concerned with developing such schemes in the finite element and the finite difference methods which lead to higher-order approximations or which have some superconvergence properties, i.e., which converge faster than it can be expected from general investigations. The study of methods having these properties is very important from the point of view of applications in technology and engineering since the algorithms based on them give the solution substantially faster (and, thus, cheaper) than the standard algorithms. There is a very vast class of problems entering this frame. The study of superconvergence of averaged gradients or irregular nets, the generalization of the maximum angle condition for finite elements of higher degrees, and the construction of higher order methods for solving heat-conduction equation have been chosen for the project.
A230106 J. Vondracek (investigator K. Eben): Mathematical modeling of metastatic activity of cancer cells, 1993-1995
The risk factors of metastases in cancer patients were studied and methods for early detection of metastases have been developed, based mainly on statistical analysis of tumor marker measurements.
