•  
•  Grant: GA15-02532S
  from 01/01/2015  to  31/12/2017
  Grantor: Czech Science Foundation (GAČR)
• Modular and Decentralized Control of Discrete-Event and Hybrid Systems with Communication
•  
• Objectives:
  Discrete-event systems represent an important class of dynamical systems with discrete state spaces and event-driven dynamics. For large systems, methods of hierarchical, modular, and decentralized decentralized supervisory control have been proposed. Since a solution to modular and decentralized supervisory control may not exist without communication between controllers, coordination control has been proposed as a form of decentralized control with supervisors communicating via coordinators. In this project we will study computationally efficient solutions to coordination supervisory control of large automata with product structure based on multi-level communication structure. Both logical automata and those stemming from discretizations will be considered. Decentralized supervisory control of automata without a priori known modular (product) structure will also be investigated. The motivations for investigating new efficient methods are that communications are sometimes lost or delayed and the original product structure is often lost after discretization.
• 
•  
•  Grant: GA15-12227S
  from 01/01/2015  to  31/12/2017
  Grantor: Czech Science Foundation (GAČR)
• Analysis of mathematical models of multifunctional materials with hysteresis
•  
• Objectives:
  The project topic is mathematical modeling, analysis, and numerical simulations of processes taking place in multifunctional materials with hysteresis. The results will include (1) a rigorous derivation of systems of ordinary and partial differential equations based on physical principles and experimentally verified constitutive relations, (2) proofs of existence, and possibly also uniqueness and stability of solutions to the equations, and (3) their numerical approximation including error bounds. The main applications will involve piezoelectric and magnetostrictive materials used as sensors, actuators, and energy harvestors, as well as thermoelastoplastic materials subject to material fatigue. The presence of hysteresis makes all these steps challenging, also because hysteresis nonlinearities are non-differentiable, which creates difficulties both in the analysis and in the numerics. New algorithms will have to be developed to treat the problems in maximal complexity.
• 
•  
•  Grant: | 1921-N25
  from 01/01/2015  to  31/12/2017
  Grantor: Czech Science Foundation (GAČR)
• The continuum, forcing and large cardinals
•  
• Project founded based on the Lead Agency cooperation agreement between Czech Science Foundation (GAČR) and Fonds zur Förderung der wissenschaftlichen Forschung (FWF) in Austria.
  
  Objectives:
  The work shall study interactions between large cardinals and forcing and also with regard to properties of the continuum. The work shall be divided into three subtopics: (1) First subtopic is called "Combinatorial properties of cardinals and the continuum function". The focus shall be on the impact of several combinatorial properties on the continuum function (a function which maps kappa to the size of the powerset of kappa). As Easton showed, without extra assumptions, there is very little ZFC can prove about the continuum function on regulars. However, once we  start to consider large cardinals, or in general combinatorial properties often formulated in the context of large cardinals (tree property, square  principles, etc.), suddenly the situation is much more interesting. Subtopic 1 studies these connections. (2) Second subtopic is called"Combinatorial characteristics of the continuum and the Mathias forcing". This subtopic focuses on the real line from the point of certain combinatorial characteristics which shall be studied by means of the Mathias forcing. This forcing is determined by a filter on the powerset of omega. Thus it is natural to study the connections between the combinatorial properties of filters and the properties of the real line. This subtopic combines set theory and topology. (3) The third subtopic is called "New developments in the template iterations." Template iterations is a modern forcing method with the potential to show new results regarding the combinatorial characteristics of the real line. Among the main  problems to be studied are: obtaining cardinal characteristics with singular values, application of template iterations on non-definable posets, and finally the application of the method with the mixed finite and countable support. These new approaches will be useful in obtaining new results.
  
  
  
  
  
• 
•  
•  Grant: GA14-07880S
  from 01/01/2014  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Methods of function theory and Banach algebras in operator theory V.
•  
• The objectives of the present research project are investigations concerning:
  1. orbits of operators, linear dynamics, invariant subspaces;
  2. operator theory in function spaces;
  3. operator positivity in matrix theory.
• 
•  
•  Grant: GB14-37086G
  from 01/01/2014  to  31/12/2018
  Grantor: Czech Science Foundation (GAČR)
• Albert Einstein Center for Gravitation and Astrophysics
•  
• We propose to establish the Albert Einstein Center for Gravitation and Astrophysics, a Project of Excellence that will bring together four leading research teams from the Czech Republic to address outstanding problems in gravitation theory and its astrophysical applications. We will strive to answer questions such as: What are the properties of exact models of gravitational radiation? How will the most important physical processes near rotating black holes change in the presence of large-scale magnetic fields or external sources? What are the mathematical and physical aspects of higher-dimensional relativity, including its implications for other fields of physics? The applying teams have long-term expertise in the relevant areas of theoretical physics, astrophysics, and cosmology. They include internationally recognized leaders as well as young researchers working at the main universities and research institutes in the Czech Republic.
• 
•  
•  Grant: GA14-02067S
  from 01/01/2014  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Advanced methods for flow-field analysis
•  
• The investigation aims at developing advanced methods for flow-field analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortex-identification criterion under development at present is associated with the corotation of line segments near a point and with the so-called residual vorticity obtained after the 'removal' of local shearing motion. The new quantity - the average corotation - is a vector, thus it provides a good starting point for developing both region- and line-type vortex-identification methods, especially predictor-corrector type schemes. Further, the project aims to analyze the strain-rate skeleton so as to draw a more complete picture of the flow. For testing purposes, large-scale numerical experiments based on the solution of the Navier-Stokes equations (NSE) will be performed for selected 3D flow problems using finite element method on parallel supercomputers. Some qualitative properties of the solutions to the NSE will be studied and described in detail.
• 
•  
•  Grant: GA14-01417S
  from 01/01/2014  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Enhancing mathematics content knowledge of future primary teachers via inquiry based education
•  
• Objectives:
  The proposed research will focus on the opportunities to influence professional competences of future primary mathematics teachers through experienced inquiry based mathematics education (IBME). The main project activities are:
  
  1) Clarifying the concept of IBME in the Czech context.
  2) Designing and testing learning environments for future primary teachers in which they experience IBME as pupils.
  3) Analyzing how students are able to evolve the gained experience into strengthening their mathematical SMK.
  4) Analyzing how students are able to evolve IBME environments into own illustrations, examples, explanations and powerful analogies, that is how students are able to transform the SMK into PCK.
• 
•  
•  Grant: GA14-06958S
  from 01/01/2014  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Singularities and impulses in boundary value problems for nonlinear ordinary differential equations
•  
• The main objective of this project is to formulate and prove new principles for existence, uniqueness and characterization of the structure of solution sets for nonlinear problems described by ordinary differential equations and their generalizations like differential equations with impulses, Stieltjes integral equations, equations with fractional derivatives and differential inclusions. A special attention is paid to nonlinear singular problems where the nonlinearities can have singularities in all their variables.
• 
•  
•  Grant: GA13-00522S
  from 01/02/2013  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Qualitative analysis and numerical solution of problems of flows in generally time-dependent domains with various boundary conditions
•  
• Mathematical and computational fluid dynamics play an important role in many areas of science and technology. The project will be concerned with the analysis of qualitative properties of the incompressible and compressible Navier-Stokes equations in fixed or time-dependent domains with various types of, in general nonstandard, boundary conditions. Let us mention, e.g., the existence, uniqueness, regularity and singular limits of their solutions. On the basis of theoretical results, in the numerical part of the project, we shall develop efficient and robust techniques for the solution and validation of theoretically analyzed flow problems and models. The developed numerical methods and their ingredients, as, e.g., adaptivity and hp-methods, will be tested on suitable problems and applied to fluid-structure interaction. With the aid of model problems, theoretical aspects of the worked out methods as stability, convergence and error estimates will be investigated.
• 
•  
•  Grant: GA13-10042S
  from 01/02/2013  to  31/12/2017
  Grantor: Czech Science Foundation (GAČR)
• Higher dimensional gravity
•  
• Recently our group (with collaborators in Halifax, Canada and Cambridge, UK) has developed a generalization of various fundamental tools of four-dimensional (D=4) general relativity to higher dimensions (Petrov classification, Newman-Penrose and GHP formalisms). In this project we plan to apply these methods to various research problems . i) We will study D>4 Einstein spacetimes , their asymptotic properties and generalization of the Goldberg-Sachs theorem for D>5 and construct exact algebraically special D>4 spacetimes. ii) Following our recent work in quadratic gravity, we plan to find various classes of exact algebraically special vacuum solutions of Lovelock gravity. iii) Recently a duality between solutions of incompressible Navier-Stokes equations in p+1 dimensions with vacuum solutions of Einstein equations in p+2 dimensions (which seem to be algebraically special for p=2 ) has been discussed by Strominger et al. We intend to use our D>4 algebraic classification to study duality between solutions of 3+1 Navier-Stokes equations and 4+1 spacetimes.
• 
•  
•  Grant: GA13-14743S
  from 01/02/2013  to  31/12/2017
  Grantor: Czech Science Foundation (GAČR)
• Function spaces, weighted inequalities, interpolation, integral operators, supremum operators
•  
• The main objective of this project is to find easily verifiable conditions that characterize embeddings between function spaces or boundedness or compactness of linear and quasi-linear operators defined on function spaces and to apply the obtained results. We intend to depart from our results obtained in frame of our previous grants (both domestic and international) and also from results of other leading scientists, and to continue developing this fruitful research. We shall mainly use real-variable methods, results of functional analysis, interpolation and extrapolation theorems, and techniques of harmonic analysis including techniques developed or partly developed by members of the team such as discretization and antidiscretization of weighted inequalities, characterization of weighted inequalities for supremum operators, characterization of optimal partner spaces or iteration methods for characterization of higher-order Sobolev spaces. The project is a natural continuation of the project 201/08/0383.
• 
•  
•  Grant: P201/12/0290
  from 01/01/2012  to  31/12/2016
  Grantor: Czech Science Foundation (GAČR)
• Topological and geometrical properties of Banach spaces and operator algebras
•  
• We would like to study the structure of Banach spaces, spaces of continuous functions, C* algebras and their relationship. Main topics will include quantitative properties of Banach spaces, decompositions of Banach spaces to smaller subspaces, descriptive properties of the weak topology, James' boundaries of compact convex sets, Baire classes of strongly ane functions, noncommutative Choquet theory, weakly compact sets and spaces they generate, problems of Banach-Stone type for spaces of continuous functions, uniformly continuous functions and ane functions, various types of universal Banach spaces, existence of fixed points and approximate fixed points, noncommutative measure theory, structures on the set of all abelian subalgebras of a C* algebra and a Jordan algebra, representations of operator algebras using weights and completely positive maps, new types of orders on operators. We would like to pay special attention to mutual influence of particular structures and to subsequent connecting of dierent areas of functional analysis and clarifying their mutual relationships.
• 
•  
•  Grant: P201/12/0671
  from 01/01/2012  to  31/12/2014
  Grantor: Czech Science Foundation (GAČR)
• Variational and numerical analysis in nonsmooth continuum mechanics
•  
• Modern variational analysis oers powerful tools for creation and treatment of advanced mathematical models. The aim of this project is to use these tools for the investigation of several selected challenging nonsmooth problems coming from continuum mechanics. This will include analysis of existence of solutions, their stability and sensitivity with respect to model data, and their dependence on possible control/design
  variables. A particular attention will be paid to control problems arising from optimal design. New numerical approaches will be developed, tailored to specic classes of large-scale mathematical optimization problems emerging in optimization with PDE constraints or variational inequality constraints, which enable their ecient numerical realization. In all considered problems, nonsmoothness and/or set-valuedness arises either
  in their data or in methods used for their treatment.
• 
•  
•  Grant: P202/12/G061
  from 01/01/2012  to  31/12/2018
  Grantor: Czech Science Foundation (GAČR)
• Centrum excelence - Institut teoretické informatiky (CE-ITI)
•  
• Center of Excellence - Institute for Theoretical Computer Science (CE-ITI) is a research center for theoretical computer science and discrete mathematics. CE-ITI aims at becoming an international leader recognized world-wide, and also a driving force of Czech theoretical computer science and discrete mathematics. To achieve that, CE-ITI includes best professors, young researchers, and students from several top institutes in the Czech Republic, and integrates the following key activities:
  
  1) Conducting research of highest quality, addressing major challenges and open problems, and initiating new lines of research.
  2) Educating a new generation of researchers and active searching for new talents.
  3) Coordinating and fostering international cooperation, strengthening the standing of Czech computer science and mathematics in world-wide context.
• 
•  
•  Grant: CZ.1.07/2.4.00/17.0100
  from 03/05/2011  to  30/04/2014
  Grantor: Ministry of Education, Youth and Sports - MŠMT
• A-Math-Net Applied Mathematics Knowledge Transfer Network
•  
• Objectives:
  
  During the last decades, natural and technical sciences experience an extensive mathematization. Application of mathematical methods in practice becomes an indispensable tool for maintaining the high level of applied research and, in general, for keeping the higher level of competitive ability. In compliance with this trend, the project A-Math-Net aims at developing a new knowledge exchange network in applied mathematics joining together institutions pertaining to higher education and research, private enterprises, and social spheres. The core of the project is to deepen the mutual coordination activities, create new links for information transmission from educational and research institutions to practice, and for an efficient application of mathematical methods. The implementation of these activities will be supported by study visits and researchers' and students' stays at partner institutions and by specialized seminars and workshops. A project office will be created for applications of mathematics and interactive communicational platforms oriented at cooperation in the project implementation and coordination of the students' work.
  
  The project is co-financed by the Europian Social Fund (ESF) and by the state budget of the Czech Republic.
• 
•  
•  Grant: P103/11/0517
  from 01/01/2011  to  31/12/2013
  Grantor: Czech Science Foundation (GAČR)
• Decentralized Control of Timed Automata
•  
• In this research proposal new methodology for supervisory control timed discrete event systems will be elaborated. More specifically, decentralized and modular control of large distributed timed systems will be studied in order to reduce the computational complexity of supervisory control synthesis, which constitutes the major obstacle for application of supervisory control in industry. Weighted automata (e.g. (max,+) automata, interval automata, and other classes of timed automata) model flexible manufacturing systems, computer and information networks with communication protocols, where not only ordering but also timing of discrete events is important. Both cases of local control specifications and more relevant global (indecomposable) specifications will be investigated. Similarly, we will consider the case, where all events are observable as well as the case, where not all events are observable due to the very nature of an event or simply due to economic reasons (too high a cost of a sensor ). Our goal is to find effective methods for supervisory control of this type of systems, in particular decentralized control and coordination control, where local controllers communicate with a coordinator.
• 
•  
•  Grant: P202/11/P028
  from 01/01/2011  to  31/12/2013
  Grantor: Czech Science Foundation (GAČR)
• Decentralized and coordination supervisory control
•  
• As the complexity of man-made systems grows, the risk of a human operator error increases and a correct behavior of complex distributed systems can only be ensured by a supervisory control system. The purpose of this project is to acquire new knowledge in decentralized and coordination supervisory control of DES with a special emphasis on large distributed DES with global specifications using the concept of a coordinator. The project will focus on the basic concepts of coordination control, and on algorithms and methods for determining the minimal size coordinator. The optimal solution is in general a hard problem, therefore principles and methods of approximation algorithms and of game theory, such as Nash equilibrium etc., will be useful. The research will then be generalized to concepts and frameworks of decentralized and coordination control with partial observations with both local and global specifications, and on problems of decentralized control with communication controllers, such as fully decentralized control, fix communication structure, or communication when needed.
• 
•  
•  Grant: P201/11/1304
  from 01/01/2011  to  31/12/2013
  Grantor: Czech Science Foundation (GAČR)
• Flow of fluids in domains with variable geometry
•  
• The goal of the project is to get new relevant results concerning flow in domains with varying geometry. From the viewpoint of theoretical analysis, we will deal with flow of fluids (incompressible and compressible) around a rotating body (existence of weak or very weak solutions, asymptotic behaviour solutions, artificial boundary conditions) in case that the axis of rotation of the body and the velocity at infinity are parallel or not parallel. We will also investigate the related hydrodynamical potential theory. Moreover, we will investigate the case of motion of rigid bodies in viscous fluid (mostrly non-Newtonian incompressible and Newtonian compressible), in several cases we include the changes of temperature. Part of the problems mentioned above will be solved numerically. Finally, we perform the numerical simulation of flow of fluids in domains with complicated geometry corresponding to the flow of blood in healthy veins as well as in cases of cardiovascular diseases.
• 
•  
•  Grant: P201/11/0345
  from 01/01/2011  to  31/12/2015
  Grantor: Czech Science Foundation (GAČR)
• Nonlinear functional analysis
•  
• The subject of our proposal are abstract problems concerning nonlinear mappings between Banach spaces and their subsets. For easier orientation, it is convenient to divide the project into the following five interdependent areas. 1. General properties of uniform mappings, their reduction to Lipschitz mappings, and their metric properties. 2. Linearization properties of Lipschitz mappings, in particular the existence of derivatives. 3. Structural properties of participating spaces, linear theory. 4. Renormings of Banach spaces. 5. Applications to other areas of mathematics, such as fixed point theory and differential equations. Concrete examples of the proposed problems. Are the classical Banach function spaces linearly isomorphic to their uniformly homeomorphic images? Is the unit ball uniformly homeomorphic to the unit sphere? Are Lipschitz isomorphic separable Banach spaces linearly isomorphic? What are the complemented subspaces of the classical function Banach spaces? Do reflexive Banach spaces have a fixed point property for nonexpansive mappings?
• 
•  
•  Grant: P201/10/2315
  from 01/01/2010  to  31/12/2014
  Grantor: Czech Science Foundation (GAČR)
• Mathematical modeling of processes in hysteresis materials
•  
• Hysteresis, i.e., nonlinear relations exhibiting a complicated input-output behavior in form of nested loops that cannot be described by functions or graphs, occurs in many fields of science, e.g., in ferromagnetism, micromagnetics, solid-solid phase transitions, and elastoplasticity. Hysteretic systems carry a memory of their former states, which renders their input-output mapping both nondifferentiable and nonlocal in time, so that conventional weak convergence techniques for solving evolution systems fail. Therefore, dynamical elastoplastic processes with hysteresis are found in the mathematical literature much less frequently than quasistatic ones, and a substantial progress in this direction is necessary. In a recent breakthrough, it was shown that the three-dimensional single-yield von Mises constitutive law leads, after a dimensional reduction to beams or plates, to a multi-yield Prandtl-Ishlinskii hysteresis operator. It is in fact quite natural that the lower dimensional observer does not see any sharp transition from the purely elastic to the purely plastic regime as in the von Mises model: if a plate is bent then small plasticized zones start forming first near the boundary and then propagate to the interior, which still preserves a partial elasticity. This gradual plasticizing is reflected by the Prandtl-Ishlinskii superposition of single-yield elements that are successively activated. This new groundbreaking theory will be expanded to more complex structures like Mindlin-Reissner plates, and curved rods and shells. Temperature and material fatigue effects will be included. A thermodynamically consistent theory of temperature and fatigue dependent Prandtl-Ishlinskii operators will be developed, along with efficient and reliable numerical methods. Questions of theoretical and numerical stability, and the long time behavior of the system of energy and momentum balance laws are central objectives.
• 
•  
•  Grant: P201/10/1920
  from 01/01/2010  to  31/12/2012
  Grantor: Czech Science Foundation (GAČR)
• Contemporary function spaces theory and applications
•  
• This project can be briefly characterized as investigation of imbedding and trace properties of weighted and anisotropic spaces of Sobolev type, research in the extrapolation theory, and applications to the qualitative theory of differential equations, particularly to the Stokes and Oseen problem and also to the Navier-Stokes systems. Specifically we intend to study imbeddings, traces ad interpolation inequalities in general spaces of Besov and Lizorkin-Triebel type with dominating mixed smoothness in the framework of the Fourier analytic approach to the theory, and some of the related unsolved problems not falling into this general scheme as reduced imbeddings and inequalities in Orlicz spaces. Application areas include weighted estimates for the heat kernel, shape optimization and properties of very weak solutions to Navier-Stokes problems in weighted spaces and various type of domains.
• 
•  
•  Grant: P202/10/0854
  from 01/01/2010  to  31/12/2012
  Grantor: Czech Science Foundation (GAČR)
• Circuit complexity and self-reducibility
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• The project proposes to study two areas of computational complexity: circuits of bounded depth and self-reducibility of functions. In the area of circuits of bounded depth we aim to study the class of functions computed by small-polynomial size circuits, the relationship between depth and size of circuits, and equivalence of circuit classes. In the area of self-reducibility we intend to study downward self-reducibility and instance compression. The main goal of the project is to bring further understanding of classes of functions computed by small depth circuits and their relationship to larger complexity classes.
• 
•  
•  Grant: 201/09/0917
  from 01/01/2009  to  31/12/2013
  Grantor: Czech Science Foundation (GAČR)
• Mathematical and computer analysis of the evolution processes in nonlinear viscoelastic fluid-like materials
•  
• This project proposal focuses on theoretical and computer analysis, and their mutual interplay, related to several classes of evolutionary models that have been recently designed to capture complex behavior of various fluid-like materials within the framework of nonlinear continuum mechanics. The characteristic keywords of these particular classes are implicit constitutive relations, nonlinear rate type fluids, nonlinear integral type fluid-like materials, inhomogeneous incompressible fluids, compressible non-Newtonian fluids, and chemically reacting fluids. Regarding specific applications, we intent to concentrate on unsteady flows of biological liquids and time-dependent processes in geophysical materials. The goal is to develop new methods and tools to solve initial boundary-value problems for large data, both theoretically and numerically.
• 
•  
•  Grant: IAA100190902
  from 01/01/2009  to  31/12/2013
  Grantor: Grant Agency of Czech Academy of Sciences
• Mathematical logic, complexity, and algorithms
•  
• Project of basic research in mathematical logic and theoretical computer science. We focus on bounded arithmetic and proof complexity, set theory, computational complexity theory, and the theory of algorithms. The topics range from foundational areas of mathematics to algorithmic problems motivated by applied research. The results of the project will be published in high quality international scientific journals and in the proceedings of selective conferences.
• 
•  
•  Grant: IAA100190903
  from 01/01/2009  to  31/12/2013
  Grantor: Grant Agency of Czech Academy of Sciences
• Orbits, invariant subspaces and positivity in operator theory
•  
• The objectives of the research project are investigations in the following mutually interconnected areas: 1. orbits of operators, hypercyclic and supercyclic vectors; 2. existence of invariant subspaces and subsets; 3. operator positivity in matrix theory.
• 
•  
•  Grant: 201/09/0473
  from 01/01/2009  to  31/12/2014
  Grantor: Czech Science Foundation (GAČR)
• Methods of function theory and Banach algebras in operator theory IV
•  
• The objectives of the present research project are investigations concerning: 1. existence of invariant subspaces and subsets with given properties; 2. operator theory in spaces of holomorphic functions; 3. orbits of operators, hypercyclic and supercyclic vectors, semigroups of operators; 4. interpolation problems, operator models and commutant lifting theorem.
• 
•  
•  Grant: 201/08/0012
  from 01/01/2008  to  31/12/2012
  Grantor: Czech Science Foundation (GAČR)
• Qualitative analysis and numerical solution of flow problems
•  
• Mathematical modelling of fluid flows in different regimes.
• 
•  
•  Grant: 201/08/0383
  from 01/01/2008  to  31/12/2012
  Grantor: Czech Science Foundation (GAČR)
• Function spaces and weighted inequalities and interpolation
•  
• The main goal of this project is to find easily verifiable conditions which characterize
  embeddings of function spaces and boundedness of linear and quasilinear operators
  acting between function spaces, and to apply obtained results in the theory of real
  interpolation.
  The problems we propose to be studied are central to Mathematical Analysis, in
  particular in the study of PDE’s, integral operators, function spaces, and the real interpolation. In addition to their intrinsic interest and importance they underpin much
  of the work in subjects as diverse as Fluid Mechanics and Mathematical Physics. They
  involve techniques which have been developed a great deal during the last decade and
  the members of our grant group have taken part in their development. The participants
  of the grant project published a number of important results in the field of function
  spaces in well-known academic journals.
• 

•  
•  Grant: P203/10/0749
  from 01/01/2010  to  31/12/2012
  Grantor: Czech Science Foundation (GAČR)
• General Relativity in higher dimensions
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• Study of various aspects of higher-dimensional relativity with main emphasis on the algebraic classification of curvature tensors and applications.
• 
•  
•  Grant: IAA100190901
  from 01/01/2009  to  31/12/2011
  Grantor: Grant Agency of Czech Academy of Sciences
• Topological and geometrical structures in Banach spaces
•  
• Stability of weak Asplund spaces and Gateaux differentiability spaces, Banach spaces with projectional skeleton, stability of Valdivia compacta, topological characterizations of some classes of Banach spaces, compact convex sets - duality of compact spaces and Banach spaces and topological classification, weak topology of weakly K-analytic Banach spaces, geometric properties of boundaries of compact convex sets, strongly affine functions and Baire classes of Banach spaces, isomorphic properties of L_1-preduals, descriptive properties of ranges of derivatives on Banach spaces, descriptive properties of norm and weak topologies on Banach spaces, binormality in Banach spaces, spaces of continuous functions, renormings and differentiability, weak-star uniformly Kadec-Klee norms, Lipschitz homeomorphisms, Markushevich bases, extensions of special Lipschitz mappings, retracts, absolutely minimal Lipschitz functions, Gurarij space, compacta with extra structure.
• 
•  
•  Grant: KJB100190901
  from 01/01/2009  to  31/12/2011
  Grantor: Grant Agency of Czech Academy of Sciences
• Singular and maximal operators on function spaces
•  
• We propose to study certain classes o operators which play important role in Harmonic analysis. A singular integral operator is a convolution operator with non-integrable kernel. If the kernel does not satisfy any smoothness conditions the operator is called rough. We intend to study rough operators of certain types, linear and bilinear. Maximal operators have many uses, probably the most important is the study o almost everywhere convergence. We wish to study rough maximal operators, which are related to the rough singular integral operators. We also want to focus on maximal operators related to Fourier multiplier operators. Furthermore, we want to work on a problem related to interpolation spaces of certain types.
• 
•  
•  Grant: IAA100190905
  from 01/01/2009  to  31/12/2011
  Grantor: Grant Agency of Czech Academy of Sciences
• Dynamical properties of the Navier-Stokes and related equations
•  
• The project is focused on the study of 1) asymptotic and dynamic properties of solutions of the Navier-Stokes (= N-S) equations, 2) flows of a N-S fluid in a channel and 3) stability of a solution to the N-S equations modelling flow around a compact body. Item 1) concerns the expansion of a solution to modes (frequencies) and asymptotic behaviour of the modes, with the accent to the question which modes overrule the others and in which ratio at certain times (especially time tending to infinity). Item 2) is interesting due to the non-standard boundary condition on the outflow of the channel. We will deal with existence, respectively uniqueness of solutions (mainly strong) to the N-S equations and we will also extend the mathematical model and qualitative results to flows of a heat conductive fluid. In item 3), we will focus on sufficient conditions for stability without restriction on the size of a basic flow, using especially spectral properties of an associated linear operator.
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•  
•  Grant: 201/08/0315
  from 01/01/2008  to  31/12/2011
  Grantor: Grant Agency of Czech Republic (GACR)
• Mathematical analysis of complex systems in the fluid mechanics
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• The main goal of the project is to develop a rigorous mathematical theory of complex systems in fluid mechanics. Such problems arise in models of chemical reactions, astrophysics, biological models, atmosphere and geophysical fluid dynamics. The main challenge here is to handle problems with large data and without any restriction concerning the time scale. The main topics include: Multicomponent problems and mixtures. 2. Equations of magnetohydrodynamics. 3. Atmospheric and geophysical models. 4. Large time behavior of solutions and equilibrium states.
• 
•  
•  Grant: IAA100190805
  from 01/01/2008  to  31/12/2010
  Grantor: Grant Agency of Czech Academy of Sciences
• Bifurcation and dependence on parameters for unilateral boundary value problems and interpretation in natural sciences
•  
• Smoothness of bifurcation branches for a Signorini problem was proved. Location of bifurcation points for reaction-diffusion systems with various unilateral conditions was described. The result is surprising in the case of Signorini-Neumann conditions.
• 
•  
•  Grant: IAA100190804
  from 01/01/2008  to  31/12/2010
  Grantor: Grant Agency of Czech Academy of Sciences
• The motion of rigid bodies in liquid: mathematical analysis, numerical simulation and related problems
•  
• In the framework of the project we will study the steady flow around bodies. We will consider the case when the direction of the angular velocity and of the velocity at infinity are or are not parallel. We will extend the results from the previous project, where the angular and tranlation velocities were parallel. We will study the linear cases and Navier-Stokes equations. We will investigate the existence of solution, asymptotic behaviour, resolvent and spectrum problem. Further, we will study the motion of several bodies in the fluid. We will consider the influence of boundary conditions and possibility of collisions. In this part we will study the existence of weak solution for steady and non-steady cases. We will investigate fluid flows described by Navier-Stokes equations as well as by non-Newtonian models. We will investigate the modeling of blood flow and related cardiovascular cases. Next to it the numerical simulation of severeal models will be performed.
• 
•  
•  Grant: IAA100190803
  from 01/01/2008  to  31/12/2012
  Grantor: Grant Agency of Czech Academy of Sciences
• The finite element method for higher dimensional problems
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• The proposed project is a free continuation of the grant Finite element method for three-dimensional problems IAA1019201, which terminated in 2006 and which was fulfilled with excellent results. The main goal of the new project will be a thorough mathematical and numerical analysis of the finite element method for solving partial differential equations in higher dimensional spaces. The necessity of solving such problems arises, e.g., in theory of relativity, statistical and particle physics, financial mathematics. In particular, we would like to deal with generation of simplicial meshes of polytopic domains. Further, we will investigate the existence and uniqueness of continuous and approximate solutions of problems that are often nonlinear. A special emphasize will be laid also on a priori and a posteriori error estimates, superconvergence, discrete maximum principle, stability of numerical schemes, etc.
• 
•  
•  Grant: IAA100750802
  from 01/01/2008  to  31/12/2011
  Grantor: Grant Agency of Czech Academy of Sciences
• Nonsmooth and set-valued analysis in mechanics and thermomechanics
•  
• Nonsmooth and set-valued analysis represent powerful mathematical tools which make possible to build and to treat complex models. This includes solvability of problems, stability and sensitivity of their solutions with respect to model data, their dependence on possible control variables, and numerical solution. In the proposed project, we plan to apply these tools to six closely related problem areas. Specifically, we will be dealing with rate-independent processes, anisothermal processes, generalized equations with nonpolyhedral multifunctions, various types of contact problems (static and dynamic) and free boundary problems. The planned research encompasses both answers to deep theoretical questions (e.g. investigation of thermally coupled plasticity) as well as the development of new numerical techniques and solution of difficult non-academic test problems (e.g. shape optimization in contact problems with Coulomb friction with a solution-dependent coefficient of friction).
• 
•  
•  Grant: IAA100190802
  from 01/01/2008  to  31/12/2011
  Grantor: Grant Agency of Czech Academy of Sciences
• Function theory and operator theory in Bergman spaces
•  
• Operator theory and function theory in Bergman spaces is a relatively new and very active area of functional analysis which has close ties with other branches of mathematics (complex analysis, partial differential equations, group representations). The proposed project would concentrate on three topics in this area: Berezin transforms associated to spaces of holomorphic and non-holomorphic functions; function spaces and operators on bounded symmetric domains; and boundary singularities of weighted Bergman kernels. All these problems have applications in mathematical physics (quantization on Kähler manifolds), operator theory, complex geometry, and other fields.
• 
•  
•  Grant: 201/08/0397
  from 01/01/2008  to  31/12/2012
  Grantor: Grant Agency of Czech Republic (GACR)
• Algebraic methods in geometry and topology
•  
• The aim of the project is to bring together mathematicians working in diverse but closely related fields (algebra, topology, differential geometry), emphasizing the synthesis that takes place in contemporary mathematics. More concretely, we mean the following topics. (1) Applications of graph complexes to invariant differential operators, with particular attention paid to Riemann and symplectic geometry. (2) Investigation of Cartan connections and parabolic geometries. (3) Description of algebras of symmetries of differential operators and construction of operators of special types. (4) Study of questions related to classification of hypersurfaces in CR-geometry. (5) Construction of higher-dimensional analogs of the Dolbeaut complex as resolutions of the Dirac operator in several variables. (6) Appliacations of homotopy methods to formal solutions of differential relations. (7) Study of forms on low-dimensional manifolds and induced G-structures.
• 
•  
•  Grant: 406/08/0710
  from 01/01/2008  to  31/12/2010
  Grantor: Grant Agency of Czech Republic (GACR)
• Development of Mathematical Literacy in Primary Education
•  
• In the project proposal mathematics education is understood as one of the constituents of general primary school education (pupilss aged 6 - 15) whose aim is universal development of a pupil's personality. Following issues will be explored: (a) appropriate conception of mathematics with balanced motivation, operable, application and work factors, (b) work of the teacher who is well aware of his/her role and modifies his/her work with respect to his/her beliefs, particular school and class practice and goals of mathematics education. Pupils' cognition will be treated in its relation to teachers' activities. The result of the project elaboration of studies in the following fields: (a) mathematical literacy of primary school pupils specifiedon the level of content and teaching forms in primary education, (b) characterisation of competences in didactics of mathematics and its relation to teachers´ beliefs, (c) importance of the use of calculus (computation) in problem solving.
• 
•  
•  Grant: IAA100190801
  from 01/01/2008  to  31/12/2010
  Grantor: Grant Agency of Czech Academy of Sciences
• Smoothness in Banach spaces
•  
• We intend to study the possible generalizations of the classical theorems of finite dimensional analysis to the setting of Banach spaces. We are mostly concerned with smooth approximations, in the spirit of the Stone Weierstrass theorem, and the closely connected study of polynomials on Banach spaces. Let us state a few typical problems. Does the existence of a separating polynomial on a Banach space imply the existence of a convex and separating polynomial, or more generaly is there a way to obtain convex higher smooth functions from higher smooth bumps? When are uniform approximations of a function together with its higher derivatives possible? Does Alexandroff theorem hold on a Hilbert space? Are real analytic approximations possible on c_0, and are there very smooth points for convex function threon? Is there a characterization of polyhedrality for Orlicz spaces using the Orlicz function? Most of these problems are well known to the specialists and permeate the literature.
• 
•  
•  Grant: KJB100190702
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Solutions of the Einstein equations in higher dimensions
•  
• One of the most important methods used for finding exact solutions of the Einstein equations in four dimensions (4D) is the Petrov classification. Recent rapid development of higher dimensional gravity was motivated by string theory and large extra dimensions scenarios. In our previous joint work with A. Coley and R. Milson, we developed a generalization of the Petrov classification for higher dimensions. This classification has been already used in differential geometry, high energy physics and higher dimensional gravity but in comparison with the situation in four dimensions, possible applications remain largely unexplored. We plan to focus on these applications. We want to study certain classes of spacetimes (the Kundt class, Kerr-Schild metrics) and generalize various 4D theorems to higher dimensions. We also plan to study algebraically special classes (types N, D). Besides vacuum solutions we also plan to study solutions of the Einstein-Maxwell-Chern-Simons equations.
• 
•  
•  Grant: 102/07/0496
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Republic (GACR)
• Advanced algorithms for solution of coupled problems in electromagnetism
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• The project resulted in internationally recognized results in the development and implementation of advanced algorithms for numerical modeling of coupled problems in the field of heavy current electrical engineering and electrotechnics. These tasks are characterized by an interaction of several physical fields. Characterization of such interactions is essential for reliable and economical design. Members of the research team focused primarily on advanced finite element method of higher order accuracy (hp-FEM) and the selected method of integral and integro-differential equations. The obtained were published in Dolezel, I., Karban, P. Solin, P.: Integral Methods in Low-Frequency Electromagnetics. Wiley, Hoboken, NJ, UA (2009), 388 pages.
• 
•  
•  Grant: IAA100760702
  from 01/01/2007  to  31/12/2011
  Grantor: Grant Agency of the Academy of Sciences of the CR
• Methods of higher order of accuracy for solution of multi-physics coupled problems
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• The design of efficient numerical methods for computer simulations of large nonlinear and associated transient problems belongs among the most recent topics in the sphere of technical and scientific computing. Examples include processing solid and liquid metals by electromagnetic field,
  problems of thermoelasticity and termoplasticity, fluid interaction with solid structures and others. The difficulty of coupled problems stems from the fact that various components of solutions exhibit specific characters, such as boundary layers in fluids or singularities in electromagnetic fields. Efficient and accurate solution of these problems requires the representation of various components by geometrically different meshes. From the mathematical point of view, various solution components belong to different Hilbert spaces and, therefore, their approximations require various types of finite elements. For each solution component we use the modern hp-adaptive version of the finite element method (hp-FEM), which is known for its exponential convergence.
• 
•  
•  Grant: 201/07/0394
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Republic (GACR)
• Infinite dimensional analysis
•  
• We propose to work on problems in several areas of infinite dimensional analysis. Regarding the structure of Banach spaces, does every Banach space contain a monotone basic sequence? What is the optimal value of boundedness constant for an M-basis of a general Banach space. We conjecture the answer is 2. What are the structural consequences of a long Schauder basis. In connection with renorming theory, find an example of a LUR renormable space with M-basis, but no strong M-basis. Clarify the relationship between the separable complementation property and PRI. Clarify the relationship between Szlenk index and dentability index, for which there exists a nonconstructive proof of dependence. Try to generalize the Szlenk index technique in the context of topologies framented by a metric and beyond. In particular, as an application, does there exist a universal Corson compact space?
• 
•  
•  Grant: 201/07/P276
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Republic (GACR)
• Computational and communication complexity of Boolean functions, and derandomization
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• The project proposes to study three related areas of computational complexity: circuit and branching program lower bounds, multi-party communication complexity and derandomization. In the first area we propose to study the size of bounded-depth counting circuits and the depth of Boolean circuits needed to compute explicit functions. Furthermore, we propose to study the recently introduced variants of branching programs---incremental and tight branching programs. In the area of multiparty communication complexity we want to focus on the relationship among deterministic, nondeterministic and randomized protocols. In the area of derandomization we want to consider several key problems related to derandomization of space-bounded computation.
• 
•  
•  Grant: IAA100190701
  from 01/01/2007  to  31/12/2010
  Grantor: Grant Agency of Czech Academy of Sciences
• Investigating manifolds with special structures from topolog
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• Manifolds with special structures are presently actively investigated by mathematicians and physicists. They are related to outstanding theoretical questions in geometry and appear as models in the string theory. Our aims are: 1. Finding necessary and sufficient conditions for the existence of a closed G2-structure (resp. a flat G2-structure among a given cohomology class) on a 7-manifold. 2. Finding global invariants of closed G2-structures. 3. Investigating the above problems for multi-symplectic forms in dimensions 6 and 8. 4. Finding necesary and sufficient conditions for the existence of symplectic or Kaehler submanifolds realizing a given homology class. 5. Develop techniques to deal with above problems in a general framework. Our main approach to this project has been guided by recent observations that it is not only possible to apply methods presently known. It must be here a bigger framework which unifies these problems and these methods.
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•  
•  Grant: IAA100190702
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Summation integration and generalized differential equations
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• The project is devoted to further development of the theory of the summation integration and its application to generalized ordinary differential equations (ODE) and consequently to various types of classical differential equations. Our aim is to continue the study of summation integrals of the Henstock-Kurzweil type of functions with values in a Banach space and to create a new unified theory of generalized ODE's which should cover both the cases of fastly oscillating right hand sides of an ODE and the case of an ODE of Carathéodory type. The signification of our project consists in the further development of the qualitative theory of ordinary and functional differential equations. We expect new results concerning the general summation integral and its applications in the theory of differential and integral equations. Our goal is to continue and to unify in many aspects the results obtained until now. We are planning a new book devoted to generalized ODE's in Banach spaces.
• 
•  
•  Grant: IAA100190703
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Singular nonlinear boundary value problems
•  
• The aims of the submitted project are the following: 1. Formulation and proofs of new principles for existence, uniqueness or multiplicity of solutions of nonlinear regular and singular boundary value problems with quasilinear differential operators. In the singular case nonlinearities in differential equations can have singularities in all variables. 2. Application of obtained principles to singular problems arising in physics and technical practice, for example in the theory of shallow elastic membrane caps. 3. Description of global properties of solutions of functional-differential equations with state dependent deviation. In particular, existence and continuation of solutions, a structure of the set of complete solutions and asymptotic properties will be studied.
• 
•  
•  Grant: IAA900090703
  from 01/01/2007  to  31/12/2010
  Grantor: Grant Agency of Czech Academy of Sciences
• Dynamic Formal Systems
•  
• The aim of the proposed project is to develop a bunch of methods exploiting dynamic approaches to non-classical logics and formal systems in general, to be applied in various disciplines including computer science, analytic philosophy, and linguistics. The project follows two main branches of research: first, we explore general tools regarding: a) dynamic non-classical logics, b) game-theoretical semantics and informational independence, c) proof-theoretical characterizations, complexity aspects, and decidability. Second, we apply these tools in specific discourses: d) interrogative discourse (questions in logic and linguistics), e) deontic and cognitive discourse (applications in artificial inteligence and knowledge representation) f) inferential discourse (applications in philosophy of language and computer science).
• 
•  
•  Grant: KJB100190701
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Asymptotics, oscillation, and quadratic functionals in theory of dynamic equations
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• This project is focused on the study of asymptotic and oscillatory properties of linear, half-linear, and nonlinear equations, symplectic and Hamiltonian systems, and associated quadratic functionals. The results will be connected with optimality conditions for nonlinear problems in the calculus of variations and optimal control, since the second variation of these problems is a quadratic functional. An important aspect of this project is the inclusion of the theory of dynamic equations on time scales, which allows to study differential and difference equations and systems within one theory, to explain and understand the discrepancies between these theories, and at the same time to generalize them to other time scales. We will focus e.g. on symplectic systems without normality or on the theory of conjugate and coupled points for these systems. Furthermore, we will study effective criteria for the existence of solutions with prescribed properties and open problems arising in this area.
• 
•  
•  Grant: KJB700190701
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Didactic Aspects of Language Structures in Mathematics Education
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• Language structure is understood as a complex of semiotic systems of representation and rules which guide their construction, interpretation and application. Current research shows that the quality of language structures is the key factor of mathematical literacy. The goal of the project is to investigate didactically interesting aspects of pupils' language structures within evocation and reflection of a mathematical topic and within problem posing and problem solving. Research of cognitive processes will be based on the semiotic analysis of communication between subjects of the teaching process via observation, video recordings of lessons and pupils' written work. Observed phenomena will be elaborated by methods of qualitative research, mainly by coding of grounded theory. The result of the project will be a set of phenomena characterising important aspects of language structures, usable as a source for other research in mathematics education and also as recommendations for practice.
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•  
•  Grant: 201/07/0237
  from 01/01/2007  to  31/12/2009
  Grantor: Grant Agency of Czech Republic (GACR)
• Infinite dimensional stochastic systems
•  
• The project is aimed at investigating qualitative properties of stochastic infinite dimensional systems (in particular, of solutions to infinite dimensional stochastic equations) and at research in infinite dimensional stochastic control theory. The moreimmediate aims are: 1. Investigation of existence, uniqueness and pathwise properties of solutions to infinite dimensional stochastic equations, in particular of the path regularity and stability, especially for geometric wave equations, equations in Banach spaces and equations driven by fractional noises. 2. Large time behaviour and ergodicity for Markovian and non-Markovian stochastic equations. This research will be focused on methods applicable to stochastic wave and beam equations and (in the non-Markovian case) on equations where the driving process is a fractional Brownian motion. 3. Infinite dimensional stochastic control and the associated semilinear Hamilton-Jacobi-Bellman equations in infinite dimensions: ergodic and adaptive control.
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•  
•  Grant: 201/06/0254
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Republic (GACR)
• Functional differential equations in Banach spaces
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• The project was aimed at the study of topics related to various kinds of boundary value problems for functional differential equations in a Banach space X. A particular attention was paid to the cases where X is the space of continuous functions on a bounded interval or X has finite dimension. For systems of ordinary and functional differential equations, we established efficient conditions guaranteeing the existence and uniqueness of a solution of the initial-value, periodic, and general non-local boundary value problem, as well as the validity of various theorems on differential inequalities, both in the linear and non-linear cases. In the case of higher-order scalar functional differential equations, efficient solvability conditions were established for the periodic boundary value problem, which are also new for ordinary differential equations. Properties of linear hyperbolic partial differential equations with discontinuous right-hand side, including the Fredholm property of the Darboux and Cauchy problems and the continuous dependence on initial conditions and parameters, were also studied. For certain classes of boundary value problems, we studied possibilities of application of the successive approximation method for the proof of the existence, approximate computation of a solution, and error estimation of the approximation. Some of the topics studied in the project are now investigated to such an extent that one can speak of a kind of completeness of the corresponding part of the theory, and the related groups of results can be published in the form of monographs.
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•  
•  Grant: IAA100190606
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Academy of Sciences
• Asymptotic analysis of infinite dimensional dynamical systems
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• The goal of the project is to obtain new qualitative results concerning the asymptotic behavior of infinite dimensional dynamical systems arising especially in the theory of viscous compressible fluids. The main topics include compactness of solutions, global existence, convergence towards equilibria and problems with rapidly oscillating boundaries.
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•  
•  Grant: LC06052
  from 01/01/2006  to  31/12/2011
  Grantor: Ministry of Education, Youth and Sports - MŠMT
• Nečas Center for Mathematical Modeling - part IM
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• The general goal of the Nečas Center for Mathematical Modeling is to establish a significant scientific team in the field of mathematical properties of models in continuum mechanics and thermodynamics, developed by an intensive collaboration of five important research teams at three Prague affiliations and their goal-directed collaboration with top experts from abroad. The research projects of the center include: 1) Nonlinear theoretical, numerical and computer analysis of problems of continuum physics. 2) Heat-conductive and deforming processes in compressible fluids, incompressible substances of fluid type, and in linearly elastic matters. 3) Interaction of the substances. 4) Biochemical procedures in substances. 5) Passages between models, dimensional analysis.
• 
•  
•  Grant: 201/06/0018
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Republic (GACR)
• Topological structures in functional analysis
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• Banach spaces and their generalizations (locally convex spaces) are one of the main tool in modern analysis. They serve as a framework for differential calculus and solving differential equations (including partial ones) and provide a great variety of questions concerning their structure. We plan to focus on topological, geometrical and algebraic structures of these spaces and interaction between these structures. The main areas include topological characterizations of important classes of Banach spaces, dual classes of compact spaces, spaces of continuous functions, properties of compact convex sets, differentiability of convex functions, relations between different weak topologies, special subsets of Banach spaces, descriptive properties of sets and operators. The nature of the project is a theoretical research in the above mentioned areas. The results will be published in scientific journals and presented at international conferences.
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•  
•  Grant: 201/06/0128
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Republic (GACR)
• Methods of function theory and Banach algebras in operator theory III.
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• The objectives of the present research project are investigations concerning: 1. existence of invariant subspaces and subsets with given properties; 2. operator theory in spaces of holomorphic functions; 3. orbits of operators, hypercyclic and supercyclic vectors, semigroups of operators; 4. interpolation problems, operator models and commutant lifting theorem.
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•  
•  Grant: 201/06/0400
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Republic (GACR)
• Modern methods in function spaces and applications
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• The project continues the research of the applicant in last years and it is focused on topical problems in function spaces relevant for applications in differential equations. The major goals are: extrapolation properties of Lebesgue and Orlicz spaces, including their weighted clones, differentiability properties of operators in them, associated extrapolation behaviour of traces of functions in Sobolev spaces, imbedding properties of anisotropic spaces of Sobolev type via real and Fourier analysis techniques with emphasis on spaces with dominating mixed smoothness.
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•  
•  Grant: IAA100190610
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Academy of Sciences
• Geometry of weakly Lindelöf determined spaces
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• Renorming theorems in weakly Lindelöf determined Banach spaces for the so called uniformly Kadec-Klee smoothness and its variants. The use of Shlenk and dentability index for these renormings. Study of l_p generated Banach spaces for p>2. For a given Banach space with some geometrical property, finding a space with the same property and moreover having an unconditional basis, which is densely embedded in the original space. Investigation of the geometry of Banach spaces possessing an unconditional basis in the spirit of our recent results. Study of long Schauder bases.
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•  
•  Grant: IAA100190612
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Academy of Sciences
• Regularity and other qualitative properties of solutions to the Navier-Stokes and related equations, transition to turbulence
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• Following our previous results, we will study regularity and related qualitative properties of solutions to the Navier-Stokes equations and other equations which express conservation of momentum in an incompressible fluid. We wish to focus especially on these questions: regularity of a weak solution and validity of the generalized energy inequality up to the boundary at various boundary conditions, the choice of initial conditions leading to a global strong solution, geometry of vorticity in the transition region between laminar and turbulent flows. In comparison with usual Dirichlet-type boundary conditions, we will pay more attention to conditions involving especially the rotation of velocity.
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•  
•  Grant: KJB100190609
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Academy of Sciences
• Supervisory control of large distributed discrete event systems
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• This project aims at studying logical and timed discrete event systems using methods from universal coalgebra and idempotent algebra with special focus on supervisory control of large distributed systems. Methods of idempotent algebra enable linear representation on suitable idempotent semirings useful for quantitative (timing) aspect of control, while coalgebra is useful for qualitative aspects of control. A combination of these techniques will be applied to the decentralized and modular supervisory control in order to reduce the computational complexity and make our results applicable to control of large distributed systems.
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•  
•  Grant: 201/06/0302
  from 01/01/2006  to  31/12/2008
  Grantor: Grant Agency of Czech Republic (GACR)
• Applied probability - stochastic geometry and metric tools
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• In applied probability the theory of random sets (stochastic geometry) will be developed; this will lead to new estimators of intrinsic volume densities both in stationary and non-stationary case. The quality of estimators of geometrical characteristics will be described by means of probability distances. The field of probability distances will be improved theoretically, some applications in biology are expected. Stochastic simulations present a complementary tool in stochastic geometry, their development will be directed to random tessellations. Interdisciplinary use of such models will lead to the characterization of the microstructure of nanomaterials. Five PhD students will participate at the solution of the project and an international conference will be organized.
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•  
•  Grant: DML-CZ
  from 01/01/2005  to  31/12/2009
  Grantor: Academy of Sciences of the Czech Republic
• DML-CZ: The Czech Digital Mathematics Library
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• The aim of the project is to investigate, develop and apply techniques, methods and tools allowing to create proper infrastructure and conditions for implementation of the Czech Digital Mathematical Library (DML-CZ) containing the relevant part of special mathematical literature which has been published in the Czech lands and for its incorporation into the World Digital Mathematical Library (WDML).
  The solution will include launching the digitization process and providing access to the digitized material for end users. In this connection research of advanced technologies for search in mathematical documents will start as well as inclusion of born-digital materials.
  The project involves a design and an implementation of solutions of interconnected problem circles in the following five regions: acquisition of selected materials to be digitized and handling the IPR, digitization and provision of metadata, creation of structured digital documents, creation of the digital library and its incorporation into the WDML.
  The DML-CZ should primarily contain specialized journals of international level published by Czech institutions, such as the Czechoslovak Mathematical Journal and Applications of Mathematics published by the Mathematical Institute AS CR, Kybernetika published by the Institute of Automation and Information Theory AS CR and others. Next, conference proceedings published by the Czech universities and research institutes, selected monographs, textbooks, dissertation theses, research reports etc. will be included. Measures will be taken to complement the digital library with materials which have been digitized earlier (e.g. Commentationes Mathematicae Universitatis Carolinae, Mathematica Bohemica, Archivum Mathematicum digitized by the Göttinger DigitalisierungsZentrum within the project DIEPER). According to preliminary estimate the core of the DML-CZ should contain about 300 000 pages.
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•  
•  Grant: 201/05/0164
  from 01/01/2005  to  30/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Mathematical analysis in the thermodynamics of fluids
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• The aim of the present research project is to establish a coherent mathematical theory of viscous heat conducting fluids based on a suitable variational formulation of the problem consistent with the second law of thermodynamics. The main topics include: 1. The existence of solutions on arbitrarily large time intervals with no restriction on the size of data. 2. The questions of uniqueness, boundedness, and stability of solutions with respect to the initial conditions and other parameters as the case may be. 3. The long time behavior, convergence towards equilibria, and attractors. 4. Sensitivity analysis with respect to the shape of the underlying spatial domain.
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•  
•  Grant: 201/05/0005
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Mathematical theory and numerical simulation of problems in the fluid mechanics
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• The goal of this project is to investigate various models of the fluid mechanics from the theoretiacl point of view (exitence, uniqueness, regularity) and to use the theoretical results to improving various numerical methods in the fluid flow modeling.
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•  
•  Grant: IAA100190506
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Academy of Sciences
• Bifurcation and dependence on parameters for variational inequalities with interpretation in natural sciences
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• Smoothness, direction and stability of bifurcating branches for variational inequalities on nonconvex sets, global bifurcation for reaction-diffusion systems with nonstandard conditions and smooth dependence on data for the Signorini problem were given.
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•  
•  Grant: IAA100190505
  from 01/01/2005  to  01/12/2007
  Grantor: Grant Agency of Czech Academy of Sciences
• Mathematical modelling of motion of bodies in Newtonian and non-Newtonian fluids and related mathematical problems
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• Investigation of properties of models describing motion of rigid bodies in viscous fluid. Existence of weak and strong solutions, asymptotic behaviour, attainability, numerical analysis and solution of selected models.
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•  
•  Grant: 201/05/2033
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Function spaces and weighted inequalities for integral operators
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• The main goal of this project is to find easily verifiable conditions which characterize embeddings of function spaces and boundedness of linear and quasilinear operators acting
  between function spaces, and to apply obtained results in the theory of real interpolation.
  The problems we propose to be studied are central to Mathematical Analysis, in particular in the study of PDE's, integral operators, function spaces, and the real interpolation. In addition to their intrinsic interest and importance they underpin much of the work in
  subjects as diverse as Fluid Mechanics and Mathematical Physics. They involve techniques which have been developed a~great deal during the last decade and the members of our grant group have taken part in their development. The participants of the grant project published a~number of important results from the given field in well-known academic journals.
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•  
•  Grant: 201/05/0124
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Online algorithms, randomness, and extremal problems
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• This project proposes basic research in theoretical computer science. Its goal is to obtain new theoretical results in the following two areas: (i) design and analysis of online and approximation algorithms for problems including scheduling, k-server problem, and their variants and (ii) investigation of problems related to randomness (including pseudorandomness and Kolmogorov randomness) and extremal combinatorics. Both of these areas belong to important and active research areas in current computer science. This project is intended as a continuation of the project 201/01/1195 of GA CR and the international cooperative grant KONTAKT ME476, which both ended in 2003. We expect that the proposed project will bring new theoretical results that will be published in high quality international journals and conference proceedings and at the same time it will contribute to teaching advanced courses and advising students.
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•  
•  Grant: 201/05/2117
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Algebraic methods in topology and geometry
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• The project aims at bringing together mathematicians working in diverse but closely related fields (algebra, topology, differential geometry), thus reflecting the synthesis which takes place in modern mathematics. More specifically, we propose: 1) Studytransfers of strongly homotopy Lie structures, with an attention paid to minimal models, properties of the moduli space of solutions of the Mauer-Cartan equation and deformation theory. 2) Investigate invariant differential operators for parabolic geometries, in particular in the case when fields correspond to representations with singular character. Apply the Lie theory to the geometry of manifolds with a given parabolic structure. Study local invariants of pseudo-convex CR manifolds. 3) Describegeometry and topology of orbits of 3-forms with respect to the action of the general linear group. Find necessary and sufficient conditions for the existence of 3-forms on low-dimensional manifolds.
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•  Grant: 406/05/2444
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Republic (GACR)
• Mathematics Classroom Culture
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• By mathematics classroom culture, we mean a set of characteristics describing processes going on during the teaching of mathematics at school. The research will focus on the following areas: (a) characterisation of processes which constitute the mathematics classroom culture, (b) study of mutual influences between the mathematics classroom culture and the process of concept acquisition, (c) study of processes going on in the teaching of mathematics aimed at the way the mathematics classroom culture is built through interaction and communication between a teacher and pupil(s), (d) study of possibilities of changing and developing the culture of teaching via teacher education and co-operation between teachers and researchers.
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•  Grant: IAA100190502
  from 01/01/2005  to  31/12/2007
  Grantor: Grant Agency of Czech Academy of Sciences
• The structure of Banach spaces
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• The aim of the project is to contribute to the solution of some of the fundamental question regarding the structure of Banach spaces. In the contextof separable spaces it is especially the containment of copies of cO into quotients of polyhedral spaces. The question on the existence of smooth separating functions on asplund spaces. Problems of the density of smooth renormings and the existence of lipschitz retractions for certain C(K)spaces. The boundary problem, ie. generalization of Rainwaters theorem to the case of arbitrary boundary and dropping the sequentiality assumption. Structure of biorthogonal systems and Markushevich basas in nonseparable Banach spaces.
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•  Grant: LC505
  from 01/01/2005  to  31/12/2011
  Grantor: Ministry of Education, Youth and Sports - MŠMT
• Eduard Čech Center for Algebra and Geometry
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• The Eduard Cech Center for Algebra and Geometry forms an institutional background for long term post-doc stays, stays of leading foreign experts, and direct collaboration of a large group of Czech and foreign institutes. The research will focus on interaction between algebraic and geometric approaches in Differential Geometry, Geometric Analysis, Category Theory, Algebraic Topology, Algebraic Number Theory, Coding Theory, and Proof Complexity. There will be international open calls for the post-doc positions and short time stays of small groups of leading experts will be organized. The activities of the center will be monitored and governed by the international Steering Board. Indicators The numbers of post-docs and their publications,The measurable outputs Public databases of publications,Critical conditions The interest in the opened positions.
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•  Grant: 1P05ME749
  from 01/01/2005  to  31/12/2008
  Grantor: Ministry of Education, Youth and Sports - MŠMT
• Fermant numbers and their application
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• Fundamental research on Fermant numbers and their practical application.
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•  Grant: 1M0545
  from 01/01/2005  to  31/12/2011
  Grantor: Ministry of Education, Youth and Sports - MŠMT
• Institute for Theoretical Computer Science
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• Subjects of the research are methods, algorithms, and stuctures of theoretical computer science and their applications in information technologies, e.g. problems of large networks (WWW), problems of security and reliability, and optimalization of concrenecessary condition for further growth. From the point of view of the quality of included researchers and industrial partners it is a unique project, which is connecting the intelectual capacity of world famous scientists and economic predacity and stronapplication of new results in real life and also forming of new goals and priorities for basic research and training of young scientists. The center is connecting leading research communities in Prague, Brno and Pilsen, whose research interests and experOrganization, management and development of the center will use the experience acquired by research center ITI (project LN00A056 supported in 2000-2004). Through 5 years of its existence this center became an important center in Europe and in the world.
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•  Grant: IAA100110502
  from 01/01/2005  to  31/12/2009
  Grantor: Grant Agency of Czech Academy of Sciences
• Spatial statistics of 3D confocal images of capillary network
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• The aim of the project is quantitave evaluation of length density and spatial arrangement of brain microvessels from stacks of optical sections captured by a laser scanning confocal microscope. We will use image enhancement by deconvolution, horizontal and vertical image composition of adjacent fields of view. We will develop and test several methods for capillary detection, measure the length automatically and compare the results with state of art interactive techniques of length measurement. The detection methods will include use of top-hat filters, path extraction methods by minimal path and probabilistic methods using Markov random fields of directions. Spatial arrangement will be characterized by topology, correlation functions, sphere contact distribution function and rose of directions of microvessels.
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•  Grant: 201/04/P183
  from 01/01/2004  to  31/12/2006
• Kvalitativní teorie systémů funkcionálních diferenciálních rovnic
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• V rámci projektu byly studovány kvalitativní vlastnosti systémů funkcionálních diferenciálních rovnic, zejména otázka platnosti vět o diferenciálních nerovnostech pro lineární systémy a otázka existence a jednoznačnosti řešení Cauchyovy úlohy pro lineární i nelineární systémy. Byla dokázána řada postačujících podmínek na pravou stranu lineárního systému zaručujících platnost věty o diferenciálních nerovnostech, nebo-li nezápornost příslušného Cauchyova operátoru. Na základě těchto výsledků byla nalezena nová efektivní kritéria řešitelnosti a jednoznačné řešitelnosti Cauchyovy úlohy pro systémy funkcionálních diferenciálních rovnic jak v lineárním tak i nelineárním případě. Podrobněji byly také vyšetřovány systémy diferenciálních rovnic s deformovanými argumenty, pro které byla nalezená kritéria dále upřesněna. Získané podmínky zahrnují výsledky dobře známé pro systémy obyčejných diferenciálních rovnic. Většina nalezených podmínek je v jistém smyslu nezlepšitelná, což je ukázáno na sestrojených protipříkladech. Zvlášť byly studovány některé otázky týkající se dvoudimenzionálních systémů a diferenciálních rovnic druhého řádu.
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•  Grant: IAA1019302
  from 01/01/2003  to  01/01/2005
  Grantor: Grant Agency of Czech Academy of Sciences
• Compatibility of dynamics and statics in multicomponent dissipative systems
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• The main topics of the project is to study the asymptotic behaviour of solutions to partial differential equations arising in multicomponent systems modelling. The long time behaviour of solutions as well as the problem of stabilization towards stationary state will be investigated. Specifically, we shall investigate: 1. The equations describing the motion of one or several rigid bodies in a viscous fluid. 2. The solid-liquid phase fields models. 3. Dynamical solid-solid phase transition models.
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•  Grant: P201/04/P021
  from 01/01/2003  to  31/12/2006
  Grantor: Grant Agency of Czech Republic (GACR)
• Mesh adaptivity for numerical solution of parabolic partial differential equations
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• We analyzed adaptive methods for numerical solution of partial differential equations. We concentrated on the hp-version of the finite element method (hp-FEM) and on the problem of hp-adaptivity. One of the studied aspects were the a posteriori error estimates. We developed a new guaranteed error estimate, which enables to compute an approximate solution with guaranteed accuracy. We also optimized the hp-FEM basis functions in order to improve the conditioning properties of the resulting matrices. Another aspect of the project was the analysis of the discrete maximum principles. We developed a simple conditions that guarantee the nonnegativity of the hp-FEM solutions. Within the project we also participated on the development of the hp-FEM software project Hermes.
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•  Grant: 201/02/0684
  from 01/01/2002  to  31/12/2004
  Grantor: Grant Agency of Czech Republic (GACR)
• Mathematical and numerical analysis of problems in fluid mechanics
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• The project includes several topics: 1) Regularity of weak solutions to the Navier-Stokes equations. 2) Existence, stability and long time behaviour of solutions to compressible Navier-Stokes equations. 3) Interaction of fluids with rigid bodies. 4) Qualitative theory of higher-degree fluids. 5) Higher order schemes for compressible flows. 6) Adaptive methods of mesh refinement near discontinuities. 7) Extensions of finite volume - finite element methods from 2D to 3D problems.
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•  Grant: IAA1019202
  from 01/01/2002  to  31/12/2004
  Grantor: Grant Agency of Czech Academy of Sciences
• Bifurcation and stability for variational inequaities with applications to mathematical models in biology
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• A smooth dependence of solutions and contact sets on parameters and existence of smooth bifurcation branches for certain classes of variational inequalities was proved. In particular cases, even direction, stability and global behaviour of bifurcation branches were described.
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•  Grant: 201/00/D058
  from 01/09/2000  to  31/08/2003
  Grantor: Grant Agency of Czech Republic (GACR)
• Qualitative theory of functional differential equations of non-Volterra's type
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• Two-point boundary value problems for the first order scalar functional differential equations were studied in the framework of the project. Efficient criteria guaranteeing the existence, uniqueness, and non-negativity (resp. non-positivity) of solutions to the problems considered were established. The results were successively published in scientific journals. The research was crowned by a monograph where the mentioned results were generalized, made more complete, and presented in the comprehensive form. The other questions of the qualitative theory of functional differential equations (like oscillation and asymptotic behaviours of solutions, solvability of boundary value problems for systems of functional differential equations, etc.) can be investigated using the techniques and methods developed in the framework of the project.
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last update: 05/06/2015

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