Grants
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Grant: Neuron Impuls Junior
from 01/01/2017
to 30/06/2018
Grantor: Neuron Fund for Support of Science
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Mathematical analysis of hyperbolic conservation laws
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Objectives:
The goal of the project is to deepen present knowledge concerning existence of weak solutions for hyperbolic systems of partial differential equations arising from conservation laws in multiple space dimensions, study of uniqueness and nonuniqueness of entropic weak solutions, analysis of admissibility criteria (maximization of entropy production or inviscid limit) or even designing other suitable admissibility criteria.
A convenient test problem for admissibility criteria is the so called Riemann problem, i.e. problem with initial data consisting of a jump discontinuity separating two constant states. As such initial data are onedimensional, one can use standard theory to find a onedimensional selfsimilar solution which is unique in the class of BV functions. In a broader class of functions (in particular in the class of essentialy bounded functions), other weak solutions may exist as is in the case of compressible Euler equations. In the framework of this project I will study also Riemann problems for various hyperbolic systems concerning uniqueness or nonuniqueness of weak solutions in the class of essentialy bounded functions and in the case of nonuniqueness, admissibility criteria will be tested.
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Grant: Neuron Impuls 24/2016
from 01/01/2017
to 31/12/2019
Grantor: Neuron Fund for Support of Science
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Guaranteed bounds of eigenvalues and eigenfunctions of differential operators
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We will propose new a posteriori error estimates for eigenvalue problems of symmetric elliptic partial differential operators. We will prove their reliability and local efficiency. We will use them in the adaptive finite element method for reliable error estimates of the size of the error in eigenvalues and eigenfunctions.
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Grant: GA16-07378S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)
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Nonlinear analysis in Banach spaces
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Objectives:
We plan to investigate problems concerning uniformly continuous and Lipschitz mappings between Banach spaces and their possible applications in other areas of mathematics, such as theoretical computer science, differential equations etc. This project is devoted to the following aspects of the subject:
a) Uniformly continuous and coarse mappings
b) Lipschitz isomorphism
c) Lipschitz free spaces
d) Linear and descriptive properties
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Grant: GA16-03230S
from 01/01/2016
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)
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Thermodynamically consistent models for fluid flows: mathematical theory and numerical solution
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Objectives:
Mathematical and numerical analysis and numerical solution of fluid flows belong to the most often studied problems of the theory of partial differential equations and their numerical solution. During the last decades, a big progress has been achieved in these fields which enables us to study models of complex fluids including the possibility to consider their dependence on temperature. This project is focused on the study of such models of fluid thermodynamics and mechanics with the aim to extend the knowledge in the field of the theoretical analysis of the corresponding systems of partial differential equations and numerical analysis of the methods for their solution. Computational simulations using specific numerical methods will be performed to support the analytical results concerning the well-posedness of the model problems and qualitative properties of their solutions. The proposed projects assumes a tight collaboration of specialists in these fields which is an important prerequisite for further development of mathematical and computational fluid thermodynamics.
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Grant: GA15-12227S
from 01/01/2015
to 31/12/2017
Grantor: Czech Science Foundation (GAČR)
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Analysis of mathematical models of multifunctional materials with hysteresis
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Objectives:
The project topic is mathematical modeling, analysis, and numerical simulations of processes taking place in multifunctional materials with hysteresis. The results will include (1) a rigorous derivation of systems of ordinary and partial differential equations based on physical principles and experimentally verified constitutive relations, (2) proofs of existence, and possibly also uniqueness and stability of solutions to the equations, and (3) their numerical approximation including error bounds. The main applications will involve piezoelectric and magnetostrictive materials used as sensors, actuators, and energy harvestors, as well as thermoelastoplastic materials subject to material fatigue. The presence of hysteresis makes all these steps challenging, also because hysteresis nonlinearities are non-differentiable, which creates difficulties both in the analysis and in the numerics. New algorithms will have to be developed to treat the problems in maximal complexity.
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Grant: GA15-02532S
from 01/01/2015
to 31/12/2017
Grantor: Czech Science Foundation (GAČR)
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Modular and Decentralized Control of Discrete-Event and Hybrid Systems with Communication
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Objectives:
Discrete-event systems represent an important class of dynamical systems with discrete state spaces and event-driven dynamics. For large systems, methods of hierarchical, modular, and decentralized decentralized supervisory control have been proposed. Since a solution to modular and decentralized supervisory control may not exist without communication between controllers, coordination control has been proposed as a form of decentralized control with supervisors communicating via coordinators. In this project we will study computationally efficient solutions to coordination supervisory control of large automata with product structure based on multi-level communication structure. Both logical automata and those stemming from discretizations will be considered. Decentralized supervisory control of automata without a priori known modular (product) structure will also be investigated. The motivations for investigating new efficient methods are that communications are sometimes lost or delayed and the original product structure is often lost after discretization.
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Grant: GA14-06958S
from 01/01/2014
to 31/12/2016
Grantor: Czech Science Foundation (GAČR)
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Singularities and impulses in boundary value problems for nonlinear ordinary differential equations
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Objectives:
The main objective of this project is to formulate and prove new principles for existence, uniqueness and characterization of the structure of solution sets for nonlinear problems described by ordinary differential equations and their generalizations like differential equations with impulses, Stieltjes integral equations, equations with fractional derivatives and differential inclusions. A special attention is paid to nonlinear singular problems where the nonlinearities can have singularities in all their variables.
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Grant: GA14-01417S
from 01/01/2014
to 31/12/2016
Grantor: Czech Science Foundation (GAČR)
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Enhancing mathematics content knowledge of future primary teachers via inquiry based education
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Objectives:
The proposed research will focus on the opportunities to influence professional competences of future primary mathematics teachers through experienced inquiry based mathematics education (IBME). The main project activities are:
1) Clarifying the concept of IBME in the Czech context.
2) Designing and testing learning environments for future primary teachers in which they experience IBME as pupils.
3) Analyzing how students are able to evolve the gained experience into strengthening their mathematical SMK.
4) Analyzing how students are able to evolve IBME environments into own illustrations, examples, explanations and powerful analogies, that is how students are able to transform the SMK into PCK.
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Grant: GA14-02067S
from 01/01/2014
to 31/12/2016
Grantor: Czech Science Foundation (GAČR)
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Advanced methods for flow-field analysis
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Objectives:
The investigation aims at developing advanced methods for flow-field analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortex-identification criterion under development at present is associated with the corotation of line segments near a point and with the so-called residual vorticity obtained after the 'removal' of local shearing motion. The new quantity - the average corotation - is a vector, thus it provides a good starting point for developing both region- and line-type vortex-identification methods, especially predictor-corrector type schemes. Further, the project aims to analyze the strain-rate skeleton so as to draw a more complete picture of the flow. For testing purposes, large-scale numerical experiments based on the solution of the Navier-Stokes equations (NSE) will be performed for selected 3D flow problems using finite element method on parallel supercomputers. Some qualitative properties of the solutions to the NSE will be studied and described in detail.
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Grant: GB14-37086G
from 01/01/2014
to 31/12/2018
Grantor: Czech Science Foundation (GAČR)
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Albert Einstein Center for Gravitation and Astrophysics
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Objectives:
We propose to establish the Albert Einstein Center for Gravitation and Astrophysics, a Project of Excellence that will bring together four leading research teams from the Czech Republic to address outstanding problems in gravitation theory and its astrophysical applications. We will strive to answer questions such as: What are the properties of exact models of gravitational radiation? How will the most important physical processes near rotating black holes change in the presence of large-scale magnetic fields or external sources? What are the mathematical and physical aspects of higher-dimensional relativity, including its implications for other fields of physics? The applying teams have long-term expertise in the relevant areas of theoretical physics, astrophysics, and cosmology. They include internationally recognized leaders as well as young researchers working at the main universities and research institutes in the Czech Republic.
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Grant: GA14-07880S
from 01/01/2014
to 31/12/2016
Grantor: Czech Science Foundation (GAČR)
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Methods of function theory and Banach algebras in operator theory V.
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Objectives:
The objectives of the present research project are investigations concerning:
1. orbits of operators, linear dynamics, invariant subspaces;
2. operator theory in function spaces;
3. operator positivity in matrix theory.
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