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INSTITUTE OF MATHEMATICS

INSTITUTE OF MATHEMATICS. Section of applied algebra and geometry 1 professor, 3 associate professors, 6 lecturers Section of computer graphics and geometry 1 professor, 1 associate professor, 5 lecturers Section of applied analysis 1 professor, 3 associate professors, 7 lecturers

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INSTITUTE OF MATHEMATICS

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  1. INSTITUTE OF MATHEMATICS • Section of applied algebra and geometry 1 professor, 3 associate professors, 6 lecturers • Section of computer graphics and geometry 1 professor, 1 associate professor, 5 lecturers • Section of applied analysis 1 professor, 3 associate professors, 7 lecturers • Section of stochastic and optimization methods 1 professor, 2 associate professors, 4 lecturers Sections

  2. INSTITUTE OF MATHEMATICS • Basic mathematical courses offered at the Faculty of Mechanical Engineering including • linear algebra • analytic geometry • calculus • numerical methods • constructive geometry and computer graphics • differential equations • infinite series • probability and statistics • Special field of Mathematical Engineering • for engineers with extended theoretical background based on mathematics and computer science • Other disciplines of applied mathematics • for individual specializations of Master’s degree programmes at the Faculty of Mechanical Engineering Teaching

  3. INSTITUTE OF MATHEMATICS Section of applied algebra and geometry • Applied linear algebra • Differential geometry and its applications • Coalgebras and their applications • Digital topology • Local algebras • Methods of commutative algebra in differential geometry • General algebraic systems and their applications in computer science • Topological structures on categories Research

  4. INSTITUTE OF MATHEMATICS Section of applied algebra and geometry • GAČR Grant GA201/00/1466 Continuous and Set-Theoretic Methods in Topological and Algebraic Structures • GAČR Grant GA201/03/0933 Set-Theoretic and Categorial Methods in Topological and Algebraic Structures • MŠMT Research Plan MSM0021630518 Simulation Modelling of Mechatronic Systems • GAČR Grant GA201/02/0225 (2002-2004) Elongation of Geometric structures • GAČR Grant GA201/05/0523 (2005-2007) Geometric Structures on Fibred Varieties • GAČR Grant GA201/04/0381 (2004-2006) Methods of Theory of Numbers Projects implemented

  5. INSTITUTE OF MATHEMATICS Section of applied algebra and geometry • Masaryk University in Brno • Technische Universität Wien (Austria) • Universität Potsdam (Germany) • Universita degli Studi dell'Aquila (Italy) • Dalhousie University in Halifax (Canada) • Science University of Tokyo (Japan) Cooperation

  6. INSTITUTE OF MATHEMATICS Section of applied algebra and geometry Samples of problems investigated Modelling the structuring of a digital plane using a new Alexandrov topology – closed points are encircled. This topology offers a range of Jordan curves which is wider than that based on the commonly used Khalimsky topology. In particular, it lets curves bend at an angle of /4.

  7. INSTITUTE OF MATHEMATICS Section of applied algebra and geometry Samples of problems investigated Finding the minimum route on a smooth variety with respect to a given metric is among the basic variational problems. This is closely related to parallel transfer of vectors (see the figure) and eventually leads to the definition of a negative connection.

  8. INSTITUTE OF MATHEMATICS Section of Applied Analysis • Mathematical problems in modelling composite materials using a method of homogenization • Adapting the method of reliable solution (method of the worst scenario) for problems of homogenization with uncertain data • Quasi-periodic and random structures, generating, analysis • Developing tools of functional analysis and theory of function spaces for analysing mathematical models – method of two-scale convergence • Devising numerical methods for flow calculation • Investigating qualitative properties of the solution of functional-differential equations, differential and difference equations with a lag • Theory of dynamic equations on time scales and their applications • Solution of selected problems of optimal control Research

  9. INSTITUTE OF MATHEMATICS Section of Applied Analysis • GAČR Grant GA201/01/0079 (2001-2003) Qualitative Theory of Solutions of Differential Equations • GAČR Grant GA201/00/0557 (2000-2002) Mathematical Modelling of Some Nonlinear Problems of the Mechanics of Continuum • GAČR Grant GA201/03/0570 (2003-2005) Mathematical Modelling of Some Nonlinear Problems of the Mechanics of Continuum • GA AVČR Grant IAA1163401 (2004-2006) Limit Properties of the Solutions of Differential Equations • MŠMT ČR Research Plan MSM0021630518 (2005-until now) Simulation Modelling of Mechatronic Systems Projects implemented

  10. INSTITUTE OF MATHEMATICS Section of Applied Analysis • Université Pierre et Marie Curie, Paris (France) • Chalmers University of Technology, Göteborg (Sweden) • Technische Universität Hamburg-Harburg (Germany)  • Universiteit Gent (Belgium) Cooperation

  11. INSTITUTE OF MATHEMATICS Section of Applied Analysis Homogenization (replacing strongly heterogeneous material by fictional material homogeneous in terms of identical macroscopic properties) second order elliptic equations in 2D with a homogeneous Dirichlet boundary condition Samples of problems investigated Numerical solution corresponding to two-component periodic composite material with sixteen circular “reinforcements” Corresponding numerical homogenized solution pertaining to homogeneous material

  12. INSTITUTE OF MATHEMATICS Section of Applied Analysis Samples of problems investigated The temperature field of the piston of a Diesel engine (finite-element method)

  13. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods • Stochastic computing models for determining the reliability of systems, statistical models of forming processes, generalized regression linear models and classification quality • Estimates of discrete probability distributions under external conditions, multi-dimensional probability distributions and their properties • Markov homogeneous and non-homogeneous chains in quality • Fuzzy probability, fuzzy random variables, and fuzzy probability distributions, reliability models of elements and systems with vague information, fuzzy clustering and time series prediction in power engineering and finance, fuzzy metric spaces and their applications to databases with vague information • Reliability diagnostic supporting nuclear power station blocks control • Stochastic optimization and its application in mechanical and civil engineering • Optimization algorithms and applications in technology, foundry, and water management • Application of statistical methods in metrology, production process control, materials engineering, and medicine Research

  14. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods • MŠMT ČR Research 1M06047 Centre for the Quality and Reliability of Production (CQR) • MŠMT ČR Research Plan CEZ: J22/98: 261100009 Non-traditional Methods of Investigating Complex and Uncertain Systems • GAČR Grant GA103/05/0292 Optimizing the Design of Progressive Concrete Structures • State project FF-P114 Research and Development of an Advanced Probability Model With Accumulated Knowledge Base of an Experienced Industrial Operator With Focus on the Operation of the VVER-Type Nuclear Power Plants • IGA MZ ČR Grant ND 7134-3 Possibilities of Significantly Affecting the Intra-Abdominal Pressure After Operations of the Aboral and Gastrointestinal Tract By Enteral Feeding Applied through a Three-Luminal Probe • GAČR Grant Project GA101/00/0170 Transferability of the Fracture Strength In Terms of Evaluating the Integrity of Components with Defects • FRVŠ Project 1746/2002 Materials Science – interactive textbook • Cooperation on international projects COST-APOMAT OC526, EUREKA COOP 2716, EUREKA E! 2914 FGM, NAZV EP 096000 6151 Projects implemented

  15. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods • University of Malta (stochastic, dynamic and integer optimization) • Molde University College (optimization algorithms) • Nuclear Research Institute at Dukovany (development of a supporting diagnostic system of the states of a nuclear block) • BUT FCE Institute of Concrete and Masonry Structures (optimization and reliability of progressive concrete structures) • Faculty of Medicine of Masaryk University in Brno (statistical modelling of intra-abdominal pressure after operation) • Cardiochronomics Brno (statistical analysis in cardiology and chronobiology in cooperation with the University of Minnesota) • Institute of Physics of Material, Academy of Sciences ČR (stochastic modelling of fracture strength characteristics) • SC&C Partner (statistical methods in quality control) • Consulting in mathematical applications for the following companies: Alcan Děčín, Alstom Power, Barum Continental, Celestica Czech Republic, Invensys Appliance Controls, LG Philips Displays Czech, METAL Ústí n.L., Schneider Electric, Siemens, Kostal etc. Cooperation

  16. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods Research Estimates of discrete probability distributions using Hellinger’s and Shannon’s quasinorms under secondary moment conditions determined from real statistical data

  17. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods Program for generating fuzzy reliability characteristics of a fuzzy Weibull type B distribution Research

  18. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods Non-hierarchical clustering of fuzzy objects Research Hierarchical clustering of fuzzy objects

  19. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods Predicting time-dependent energy consumption using a fuzzy model of a time series Research

  20. INSTITUTE OF MATHEMATICS Section of Stochastic and Optimization Methods Using fuzzy sets to predict the energy consumption of the city of Olomouc as compared with other predictions using other methods: neural networks and statistical methods Research Approximating the test strength for determining a sufficient sample size in an experiment plan [n, U, n] to determine object reliabilities

  21. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry • Adaptive filters for visualizing images with high dynamic content • Numerical methods of solar spectrum analysis, solar corona images (in cooperation with the Úpice Observatory, the Slovak Academy of Sciences, and the Ondřejov Observatory) • Methods of 3D reconstruction of images from the classic, confocal, and tunnelling microscope (in cooperation with the Faculty of Medicine of Masaryk University in Plzeň) • Fast algorithms of computer graphics • Methods of image registration based on phase correlation and their applications • Applications of digital topology and fractal geometry in image analysis • Numerical methods of human retina analysis • Engineering applications of constructive geometry, multi-valued logic in system control Research

  22. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry • FRVŠ project 374/2002 Multimedia support of the Structure and Composition of Foodstuffs course • FRVŠ project 855/2005 Interactive textbook for the Structure and Composition of Foodstuffs course • FRVŠ project 3323/2006 Electronic support of the Mathematical Maple Calculations course Projects implemented

  23. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry • University of Veterinary and Pharmaceutical Sciences Brno (numerical methods of image analysis – histological examination of foodstuffs) • Faculty of Medicine of Masaryk University in Brno (numerical methods of analysis of the human retina) • Institute of Physics at the BUT Faculty of Civil Engineering (fractal analysis of the fracture surfaces of building materials) • The Úpice Observatory (numerical methods of solar corona analysis) • Astronomic Institute of the Slovak Academy of Sciences at Tatranská Lomnica (numerical methods of image analysis) • Astronomic Institute of the Czech Academy of Sciences at Ondřejov (numerical methods of solar corona analysis) • Biological Institute at the Faculty of Medicine of Charles University in Plzeň (numerical methods of confocal microscopy) Cooperation

  24. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated The solar corona during the total eclipse of 29th March 2006 reconstructed from 51 images and visualised by newly developed adaptive filters. The image significantly exceeds the capacities of the state-of-the-art space probes

  25. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated Dopplergramme of a part of the solar chromosphere obtained by analyzing a series of images of the H-alpha absorption line.

  26. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated A new exact numerical method for determining the absorption of the light radiated by the solar photosphere in the hydrogen H-alpha line

  27. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated New numerical methods for analyzing pictures of human retina

  28. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated Numerical methods for identifying and analyzing bone splinters in meat foodstuff products

  29. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry software reconstruction Samples of problems investigated photograph of a real object Numerical methods developed for processing optical sections obtained form devices with narrow sharpness band. Using software tools based on state-of-the-art numerical methods of image analysis, not only a sharp image can be obtained but also a 3D object almost undistinguishable from the real one.

  30. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated The new methods of processing transparent optical sections from a confocal microscope facilitate a detailed study of the cell morphology and provide a virtual tour of their interiors

  31. INSTITUTE OF MATHEMATICS Section of Computer Graphics and Geometry Samples of problems investigated Using measure-theory-based algorithms, the fractal dimension of microscopic samples can be determined with increasing precision

  32. INSTITUTE OF MATHEMATICS Thank you for your attention

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