Grant: 7AMB12FR003
from 01/01/2012
to 31/12/2013
Grantor: Ministry of Education, Youth and Sports - MŠMT
Non-linear analysis in Banach spaces
The goal of this project is to contribute to a better understanding of the topics described in section 2. "Představení projektu", to answer (at least partially) the open problems mentioned there, and to formulate new problems that will arise from the ongoing research. The results will be published in international peer reviewed journals and presented at international conferences.
A-Math-Net Applied Mathematics Knowledge Transfer Network
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: P201/11/0345
from 01/01/2011
to 31/12/2015
Grantor: Grant Agency of Czech Republic (GACR)
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: MEB021108
from 01/01/2011
to 31/12/2012
Grantor: Ministry of Education, Youth and Sports - MŠMT
Holomorphic functions spaces and their operators
Study of holomorphic functions spaces and related operators (Toeplitz, Hankel, Bergman projection, etc.) and their interplay with the Berezin transform. Where appropriate, extensions to spaces of harmonic functions may also be investigated.
Grant: MEB091101
from 01/01/2011
to 31/12/2012
Grantor: Ministry of Education, Youth and Sports - MŠMT
Spectral theory of linear operators and reflexivity
Research in the field of spectral theory of linear operators.
Grant: P201/11/1304
from 01/01/2011
to 31/12/2013
Grantor: Grant Agency of Czech Republic (GACR)
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: P202/11/P028
from 01/01/2011
to 31/12/2013
Grantor: Grant Agency of Czech Republic (GACR)
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: P103/11/0517
from 01/01/2011
to 31/12/2013
Grantor: Grant Agency of Czech Republic (GACR)
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: EuDML
from 01/02/2010
to 31/01/2013
Grantor: EC Information Society and Media Directorate-General
The European Digital Mathematics Library
In the light of mathematicians reliance on their discipline's rich published heritage and the key role of mathematics in enabling other scientific disciplines, the European Digital Mathematics Library strives to make the significant corpus of mathematics scholarship published in Europe available online, in the form of an authoritative and enduring digital collection, developed and curated by a network of institutions.
National efforts have led to the digitisation of large quantities of mathematical literature, primarily by partners in this project. Publishers produce new material that needs to be archived safely over the long term, made more visible, usable, and interoperable with the legacy corpus on which it settles. In EuDML, these partners will join together with leading technology providers in constructing the Europe-wide interconnections between their collections to create a document network as integrated and trans-national as the discipline of mathematics itself. They will future-proof their work by providing the organisational and technical infrastructure to accommodate new collections and mathematically rich metadata formats, and will work towards truly open access for the whole European Community to this foundational resource, thereby retaining Europe's leadership in the provision, accessibility and exploitation of electronic mathematical content.
EuDML will design and build a collaborative digital library service that will collate the currently distributed content by the diversity of providers. This will be achieved by implementing a single access platform for heterogeneous and multilingual collections. The network of documents will be constructed by merging and augmenting the information available about each document from each collection, and matching documents and references across the entire combined library. In return for this added value, the rights holders agree to a moving wall policy to secure eventual open access to their full texts.
Grant: MEB021024
from 01/01/2010
to 31/12/2011
Grantor: Ministry of Education, Youth and Sports - MŠMT
Variational inequalities in equilibrial optimization
1. Development of tools of variational analysis suitable for study of local properties of solutions of variational inequalities. 2. Analysis of the stability and sensibility of multi-valued mappings which assign suitable sets of solutions to the data of problems or to perturbing parameters. 3. Application of obtained results on selected equilibrial problems of continuum mechanics and of mathematical economy.
Grant: MEB051006
from 01/01/2010
to 31/12/2011
Grantor: Ministry of Education, Youth and Sports - MŠMT
Set theory and its applications
Cooperation in the subject of descriptive set theory and forcing. Finishing the book "Ideals and Equivalences", coothored by Jindřich Zapletal and Marcin Sabok. Organizing bilateral conferences in Prague and Hejnice on the subject of infinite games, set theory, and forcing. Lectures of Jindřich Zapletal at the Institute of Mathematics in Wroclaw.
Grant: MEB0810045
from 01/01/2010
to 31/12/2011
Grantor: Ministry of Education, Youth and Sports - MŠMT
Unilateral dynamic contact problems for thin structures
To prove solvability of dynamic contact problems for different models (e.g. Reisne-Mindlin etc.) of plates and shells. Publication of results in the form of scientific papers and then in a monograph about the topic.
Grant: P202/10/0854
from 01/01/2010
to 31/12/2012
Grantor: Grant Agency of Czech Republic (GACR)
Circuit complexity and self-reducibility
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: P201/10/1920
from 01/01/2010
to 31/12/2012
Grantor: Grant Agency of Czech Republic (GACR)
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: P203/10/0749
from 01/01/2010
to 31/12/2012
Grantor: Grant Agency of Czech Republic (GACR)
General Relativity in higher dimensions
Study of various aspects of higher-dimensional relativity with main emphasis on the algebraic classification of curvature tensors and applications.
Grant: P201/10/2315
from 01/01/2010
to 31/12/2014
Grantor: Grant Agency of Czech Republic (GACR)
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: 201/09/0473
from 01/01/2009
to 31/12/2014
Grantor: Grant Agency of Czech Republic (GACR)
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: MEB090905
from 01/01/2009
to 31/12/2010
Grantor: Ministry of Education, Youth and Sports - MŠMT
Spectral theory of linear operators and reflexivity
Study of questions of spectral theory of linear operators concerning the reflexivity.
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.
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: MEB060909
from 01/01/2009
to 31/12/2010
Grantor: Ministry of Education, Youth and Sports - MŠMT
Set theory and its applications
Exploring the interaction between set theory on one hand, and abstract analysis and combinatorics on the the other.
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: 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: 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: 201/09/0917
from 01/01/2009
to 31/12/2013
Grantor: Grant Agency of Czech Republic (GACR)
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.
Distributed Supervisory Control of large plants (DISC)
The objective of DISC is the design of supervisors and fault detectors exploiting the concurrency and the modularity of the plant model. Coordinated controllers should preferably be designed using only local plant behaviour models, and requiring only limited information exchange between the different local controllers.
We plan to use several techniques to reduce the computational complexity of solving the above mentioned problem for distributed plants: modularity in the modelling and control design phases; decentralized control with communicating controllers;
modular state identification, distributed diagnosis and modular fault detection based on the design of partially decentralized observers;
fluidisation of some discrete event dynamics to reduce state space cardinality.
The expected outcome of this project are: new methodologies for applying the above described techniques for embedded controllers to distributed plants;
new tools for the modelling, simulation and supervisory control design that will be part of an integrated software platform; the application of these methodologies to a few cases of industrial relevance using the developed tools; the dissemination of the results.
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: 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: 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: 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: 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: 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
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: 201/08/0383
from 01/01/2008
to 31/12/2012
Grantor: Grant Agency of Czech Republic (GACR)
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: 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: 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: 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
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: 201/08/0012
from 01/01/2008
to 31/12/2012
Grantor: Grant Agency of Czech Republic (GACR)
Qualitative analysis and numerical solution of flow problems
Mathematical modelling of fluid flows in different regimes.
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
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.
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
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: 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: 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: IAA100190702
from 01/01/2007
to 31/12/2009
Grantor: Grant Agency of Czech Academy of Sciences
Summation integration and generalized differential equations
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: IAA100190701
from 01/01/2007
to 31/12/2010
Grantor: Grant Agency of Czech Academy of Sciences
Investigating manifolds with special structures from topolog
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.
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
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: 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: 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
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: 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
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: 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: 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
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.
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
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.
Grant: IAA100190610
from 01/01/2006
to 31/12/2008
Grantor: Grant Agency of Czech Academy of Sciences
Geometry of weakly Lindelöf determined spaces
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.
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
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.
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.
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: 201/06/0018
from 01/01/2006
to 31/12/2008
Grantor: Grant Agency of Czech Republic (GACR)
Topological structures in functional analysis
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.
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
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: IAA100190606
from 01/01/2006
to 31/12/2008
Grantor: Grant Agency of Czech Academy of Sciences
Asymptotic analysis of infinite dimensional dynamical systems
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.
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
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.
Grant: 1M0545
from 01/01/2005
to 31/12/2011
Grantor: Ministry of Education, Youth and Sports - MŠMT
Institute for Theoretical Computer Science
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.
Grant: 1P05ME749
from 01/01/2005
to 31/12/2008
Grantor: Ministry of Education, Youth and Sports - MŠMT
Fermant numbers and their application
Fundamental research on Fermant numbers and their practical application.
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
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.
Grant: IAA100190502
from 01/01/2005
to 31/12/2007
Grantor: Grant Agency of Czech Academy of Sciences
The structure of Banach spaces
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.
Grant: 406/05/2444
from 01/01/2005
to 31/12/2007
Grantor: Grant Agency of Czech Republic (GACR)
Mathematics Classroom Culture
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
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
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.
Grant: 201/04/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
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.
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
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.
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
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.
Grant: 201/02/0684/A
from 01/01/2002
to 31/12/2004
Grantor: Grant Agency of Czech Republic (GACR)
Mathematical and numerical analysis of problems in fluid mechanics
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.
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
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.
Kvalitativní teorie systémů funkcionálních diferenciálních rovnic
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.
EN
