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.
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.
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: 103/11/0517
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.
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.
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: KJB100190901
from 01/01/2009
to 31/12/2011
Grantor: Akademie věd České republiky
Singulární a maximální operátory na prostorech funkcí
Cílem projektu je zkoumání chování operátorů spojených s Fourierovou transformací na prostorch funkcí.
Projekt základního výzkumu v matematické logice a teoretické informatice. Soustredíme se na omezenou aritmetiku a dukazovou složitost, teorii množin, teorii výpocetní složitosti a teorii algoritmu. Témata výzkumu sahají od oblastí základu matematiky až po algoritmické problémy motivované aplikovaným výzkumem. Výsledky projektu budou publikovány v kvalitních
zahranicních casopisech a sbornících predních výberových konferencí v oboru.
Grant: IAA 100190901
from 01/01/2009
to 31/12/2011
Grantor: Grant Agency of Czech Academy of sciences
Topological and geometrical structures in Banach spaces
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: 201/08/0383
from 01/01/2008
to 31/12/2012
Grantor: Grant Agency of the Czech Republic
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.
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.
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: Czech Science Foundation - GAČR
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: Czech Science Foundation - GAČR
Qualitative analysis and numerical solution of flow problems
Mathematical modelling of fluid flows in different regimes.
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: Czech Science Foundation
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.
Byly studovány vlastnosti a aplikace algebraické klasifikace Weylova tenzoru ve vyšších dimenzích. Bylo dokončeno zobecnění Newmanova-Penroseova formalismu do vyšších dimenzí a tento formalismus byl v různých modelových situacích aplikován
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: IAA1001190606
from 01/01/2006
to 31/12/2008
Grantor: Science Foundation of the 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.
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: 201/05/2033
from 01/01/2005
to 31/12/2007
Grantor: The Grant Agency of the Czech Republic
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.
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.
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: Czech Science Foundation - 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: Czech Science Foundation - GAČR
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: Czech Science Foundation
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: Science Foundation of the 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.
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: Czech Science Foundation - 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.
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.
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