Few words about us …

The Department of Applied Mathematics was found in 1996. Since 1995 we have considerably contributed to curricula development in transportation sciences with emphasis on project oriented education which combines professional skill of academic staff and young would-be researchers. We understand this form of education can create a new spirit in all de-partments.

MISSION

Perform education of the Bachelor, Master and PhD. stu-dents in various mathematical disciplines with emphasis on analysis, geometry, probability, statistics, algebra, graph theory and discrete mathematics, and physics. Conduct fundamental research, and applied research in transportation sciences, recognised at European level. Cooperate with groups and departments of a similar orientation at the Czech Technical University.

prof. RNDr. Miroslav Vlček, DrSc.
http://vlcek.fd.cvut.cz
Head of Department

RECENT IMPORTANT PROJECTS

• GACR P102/11/1795: Novel selective transforms for non-stationary signal processing
• GACR P108/10/P446 Multipurpose nanostructured carbon-based coatings for tribological applications
• TA 01030603 Novel control methods for urban traffic

RECOGNISED PUBLICATIONS

1. M. Novak, D. Vojtech, T. Vitu: Influence of heat treatment on tribological properties of electroless Ni-P and Ni-P-Al2O3 coatings on Al-Si casting alloy. Applied Surface Science. 2010, vol. 256, no. 9, p. 2956-2960.
2. P. Zahradnik, M. Vlcek: Equiripple Approximation of Half-Band FIR Filters, IEEE Trans. Circuits Syst.II, Vol. 56, No. 12, December 2009, pp. 941 - 945.
3. P. Zahradnik, M. Vlcek, R. Unbehauen: Design of Optimal Comb FIR Filters-Speed and Robustness, IEEE Signal Processing Letters, Vol.16, No. 6, June 2009, pp. 465-468.
4. P. Zahradnik, M. Vlcek: Note on the design of an equiripple DC-notch FIR Filter, IEEE Trans. Circuits Syst.II, Vol.54, No. 2, February 2007, pp. 196-199.
5. T. Polcar, R. Martinez, T.Vitu, L. Kopecký, R. Rodriguez et al.: High temperature tribology of CrN and multilayered Cr/CrN coatings. Surface and Coatings Technology. 2009, vol. 203, no. 20-21, p. 3254-3259.
6. P. Zahradnik, M. Vlcek: An analytical procedure for critical frequency tuning of FIR filters, IEEE Trans. Circuits Syst. II-Express Briefs, Vol. 53, No. 1, January 2006, pp. 72-76.
7. T. Kubart, T. Polcar, L. Kopecky, R. Novak, D. Novako-va, D.: Temperature dependence of tribological properties of MoS2 and MoSe2 coatings. Surface and Coatings Technology. 2005, vol. 193, no. 1-3, p. 230-233.
8. P. Zahradnik, M. Vlcek: Fast Analytical Design Algorithms for FIR Notch Filters, IEEE Transactions on Circuits and Systems, Vol. 51, No. 3, March 2004, pp. 608 - 623.
9. M. Vlcek, P. Zahradnik and R. Unbehauen : Analytic Design of FIR Filters, IEEE Transaction on Signal Pro-cessing, Vol. 48, No. 9, September 2000, pp. 2705 - 2709.
10. M. Vlcek, R. Unbehauen: Zolotarev Polynomials and Optimal FIR Filters, IEEE Transaction on Signal Pro-cessing, Vol. 47, No.3, March 1999, pp. 717 - 730.

AWARDS OF DIPLOMA THESES

• Ing. Pavel Paclík: Siemens award in 1998
  
• Ing. Jiří Prchal: Babuška price in 1999
• Ing. Miroslav Vlček: Siemens award in 2002
• Ing. Tomáš Hradec: Siemens award in 2006

RESEARCH ACTIVITY AND PROJECTS

1. Statistical methods and modelling of a large traffic system

   The project deals with statistical pattern in modelling and controlling of a traffic network. Information about real system (urban traffic micro-area, motorway system, railway network) is obtained through the identification of the measured data. The traffic model is based on an opti-mal control synthesis which is carried out using Bayesian statistics methods.

2. Tribology of thin abrasion resistant coa-tings

   Carbon-based coatings have significantly improved the working performance of the surfaces exposed to friction and wear in various environments. It has been showed that the driving effect of carbon-based coatings friction and wear is the third-body sliding interlayer formation. The aim of this project is to describe systematically friction, wear and the mechanisms leading to surface tribolayer formation. For this purpose, more carbon-based coating structures will be prepared. The research work is focused on interpretation of tribological measurements results obtained in ambient conditions, at elevated temperatures and also in vacuum environment. The main attention will be paid to deep analysis of surface tribolayer formation mechanisms, namelywear debris analysis, morphology, nucleation, growth, mechanical and thermal stability of surface tribolayer and its effect on tribological process.

   

3. Air Flow Visualisation using Infrared Camera

   A technology to visualise air flow using infrared camera was developed. The main objective is to recognise laminar-to-turbulent transition and location of primary and secondary vortices over a wing. Results from flight tests which visualised surface flows using oil flow, tufts, or flow cones will be adjust to the infrared measurements.

4. Approximation, higher transcendental function for DSP and digital filter design

   An innovative time-frequency transform has been devel-oped during the last 12 years of fundamental research in the field of function approximations and higher transcen-dental functions. In 1997 we discovered the algebraic form of Zolotarev polynomials refraining from a parametric representation, and developed an extremely efficient algorithm for evaluating them [10]. This method allows computation of expansion coefficients for Zolotarev polynomials of the first kind in terms of power series expansion and expansion into Chebyshev polynomials. In contrast to power series representation, the Chebyshev polynomial approach leads to coefficients valued in an astonishingly small range. The algorithm is of linear complexity with respect to the polynomial order and is robust enough to easily generate tens of thousands of degree polynomials. Since 1999 we have used Zolotarev polynomials for notch FIR filter design [2] – [4], [6], [8], [9]. Recently, (R. Spetik: The Discrete Zolotarev Transform, PhD. Thesis, FEE CTU, 2009), we have devel-oped the same algorithm for the Zolotarev polynomial of the second kind, which completes the set of functions generalizing a complex exponential. This marks the birth of a fundamentally new spectral transform of non-stationary signals. The method is based on signal decomposition of a 1D signal into a set of vectors related to Zolotarev polynomials of the first and second kind. We have named the novel method the Zolotarev Transform in honour of E.I. Zolotarev, who proposed and solved the approximation problems leading to these polynomials.

   State-of-the-art, and beyond As the application of Zolo-tarev polynomials to the analysis of non-stationary signals came to us recently it needs further development beyond the state-of-the-art. The transform that we have developed is naturally reversible, with excellent time-frequency resolution. Moreover, the Zolotarev Transform is signal adaptive and therefore its time-frequency resolution can be continuously matched to the input signal for optimal representation. We have not yet published the algorithms for the selective sine function and selective exponential function, nor have we published the organisation of our transform and its reversibility. All these items are considered as central for our future study, development and potential applications.

   There are several approaches available for addressing non-stationary signals, including the Short Time Fourier Transform (STFT), the Wavelet Transform (WT), and the Hilbert-Huang Transform (HHT). A large group of non-linear multiresolution transforms is also available. However, these approaches suffer from being difficult to interpret. STFT, WT, and HHT are therefore the most frequently used tools for non-stationary signal analysis. Researchers have devoted enormous efforts to the Wavelet Transform and Short Time Fourier Transform. The number of relevant publications confirms that this has been a field of considerable interest for decades. For example, a simple request on Web of Knowledge for "Wavelet Transform" reveals close to 570 publications in 2009. We are aware of a substantial number of these publications, but we will quote only a few selected references in formulating our project. We point out that our work is pioneering research, and the available references to other researchers in related fields do not reflect the efforts that we are proposing for our project.

   From our perspective and to the best of our knowledge, the main properties and limitations of the above-mentioned transforms can be summarized as follows:

   1. The Hilbert-Huang Transform, in spite of its good performance in analysing non–linear, non–stationary signals, has no inversion (e.g. there is no process for reconstructing the original signal).
   2. The Wavelet Transform possesses excellent time resolution, but in principle it has poorer frequency resolution than the Short Time Fourier Transform. The continuous wavelet functions are not generally orthogonal, so the signal reconstruction is ill-defined, while the Discrete Wavelet Transform (DWT) forms orthogonal or biorthogonal bases which do not produce redundant components in signal analysis.
   3. The Short Time Fourier Transform has in principle excellent frequency resolution, with the time resolution varying with the length of the window, which for some non-stationary signals fails to provide information about substantial time events.

   

   The above discussion points to the need for a transform possessing high, simultaneous frequency and time resolution. Moreover, reversibility is highly desirable, as it extends the application field from signal analysis to more general signal processing. A straightforward physical interpretation of the transformed signal would be another advantage. Our selective transform endeavours to address all of these issues, and we will develop its mathematical fundamentals in order to provide proofs. A selective transform of 1-D non-stationary signals produces spectrograms with superb time-frequency resolution. The transform can be applied for speech, for music, and for bio-signals such as EEG or EKG, and also in the analysis of electrical machines, etc. The selective cosine transform of 2-D signals has the potential to create a new paradigm in image compression and content understanding in multimedia applications. The fundamental research work will be driven by potential applications.

5. History of mathematics

   The objective of¨ the research is to enlarge students inter-est in mathematics in untraditional way, to show them historical-mathematical and technical relations and contribute to the cultivation of their technological thinking.

The result of thinking should not be sense, but action.

[Vincent Willem van Gogh]

Potential research cooperation

• Statistical method and modelling of a large traffic system
• Thin abrasion resistant coatings
• Approximation, higher transcendental functions for DSP and digital filter design
• Analysis of HMI
• Infrared Camera Visualisation
• History of mathematics

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