INSTITUTE OF MATHEMATICS Section of applied algebra and geometry 1 professor, 3 associate...

INSTITUTE OF MATHEMATICSINSTITUTE OF MATHEMATICS

Section of applied algebra and geometry 1 professor, 3 associate professors, 6 lecturers

Section of computer graphics and geometry1 professor, 1 associate professor, 5 lecturers

Section of applied analysis1 professor, 3 associate professors, 7 lecturers

Section of stochastic and optimization methods1 professor, 2 associate professors, 4 lecturers

SectionsSections

TeachingTeaching

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


ResearchResearch

INSTITUTE OF MATHEMATICSSection 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

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




CooperationCooperation

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)



Samples of problems

investigated

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.



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.



Samples of problems

investigated

Samples of problems

investigated

INSTITUTE OF MATHEMATICSSection 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

ResearchResearch

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





Université Pierre et Marie Curie, Paris (France)

Chalmers University of Technology, Göteborg (Sweden)

Technische Universität Hamburg-Harburg (Germany)

Universiteit Gent (Belgium)




Numerical solution corresponding to two-component periodic composite material with sixteen circular “reinforcements”

Corresponding numerical homogenized solution pertaining to homogeneous material

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

Samples of problems

investigated

The temperature field of the piston of a Diesel engine (finite-element method)



Samples of problems

investigated

Samples of problems

investigated

INSTITUTE OF MATHEMATICSSection of Stochastic and Optimization

Methods


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

ResearchResearch

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


Methods


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.


Methods


Methods


Hellinger quasi-norm estimate

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8

Center of class

Fre

qu

en

cie

s Orig.

K=0

K=1

K=2

K=3

K=4

Shannon quasi-norm estimate

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8

Center of class

Freq

uen

cie

s

Orig.

K=0

K=1

K=2

K=3

K=4

Estimates of discrete probability distributions using Hellinger’s and Shannon’s quasinorms under secondary moment conditions determined from real statistical data


Methods


Methods

ResearchResearch

Program for generating fuzzy reliability characteristics of a fuzzy Weibull type B distribution


Methods


Methods

ResearchResearch

Non-hierarchicalclustering of fuzzy

objects

Hierarchicalclustering of fuzzy

objects


Methods


Methods

ResearchResearch

0 50 100 150 200 250-20

0

20

40 temperature

0 50 100 150 200 2505000

6000

7000

8000

9000

10000RED - predict of load BLACK - real load MAPE=3.2989 MAX=28.4194

Predicting time-dependent energy consumption using a fuzzy model of a time series


Methods


Methods

ResearchResearch

60

80

100

120

140

160

180

1 11 21 31 41 51

Real

Pred.BJ

Pred.NN

Pred FL

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

Approximating the test strength for determining a

sufficient sample size in an

experiment plan [n, U, n] to determine object reliabilities


Methods


Methods

ResearchResearch

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

INSTITUTE OF MATHEMATICSSection of Computer Graphics and

Geometry


Geometry

ResearchResearch

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


Geometry


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)


Geometry


Geometry


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


Geometry


Geometry

Samples of problems

investigated

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.


Geometry


Geometry

Samples of problems

investigated

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


Geometry


Geometry

Samples of problems

investigated

Samples of problems

investigated

New numerical methods for analyzing pictures of human

retina


Geometry


Geometry

Samples of problems

investigated

Samples of problems

investigated

Numerical methods for identifying and analyzing bone splinters in meat foodstuff products


Geometry


Geometry

Samples of problems

investigated

Samples of problems

investigated

software reconstruction

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.


Geometry


Geometry

Samples of problems

investigated

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


Geometry


Geometry

Samples of problems

investigated

Samples of problems

investigated

Using measure-theory-based algorithms, the fractal dimension of microscopic samples can be determined with increasing precision


Geometry


Geometry

Samples of problems

investigated

Samples of problems

investigated


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INSTITUTE OF MATHEMATICS Section of applied algebra and geometry 1 professor, 3 associate...

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