Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MMP0004	Stochastic Processes and Fractals	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics - in Hungarian	6	2+0+2	speciality	optional	
Computer Science - in Hungarian	6	2+0+2	speciality	optional	
Mathematics-Computer Science - in Hungarian	6	2+0+2	speciality	optional	
Information engineering  - in Hungarian	8	2+0+2		optional	

Teaching Staff in Charge	Assoc.Prof. SOOS Anna, Ph.D., asoos@math.ubbcluj.ro	
Aims	To give to the students the principal notions of the stochastic processes which are necessary in the model process of the economic, social and other fenomens. To introduce the basic notions of fractal theory.	
Content	1. Stochastic processes: definitions and classification. Markoc chane. Transition probabilities. Chapman Kolmogorov relations. Random walk. 
2. Continue Stochastic processes. Markov type processes. Poisson processes. Gaussian processes.
3. Contraction principle. Iterated function systems.
4. Hausdorff measure. Definition and properties.
5. Hausdorff dimension. Definition and properties.
6. Invariant sets, fractal sets. Existence and uniqueness.
7. Invariant measure, fractal measure.
8. Fractal functions.  Fractal interpolation.    
9. Selfsimilariry.
10. Similarity dimension.
11. Stochastic fractals.
12. Applications: Brownian motion. Fractal compression. Virtual reality.	
Bibliography	1. M.F.BARNSLEY:  Fractals Everywhere, Academic Press,1993.
2. K.J.FALCONER: Fractal geometry, mathematical foundations and applications, John Wiley & Sons, 1990.
3. K.J.FALCONER:  Techniques in fractal geometry, John Wiley & Sons, 1997.
4. S. KARLIN, H. TAYLOR: A First Course in Stochastic Processes, Academic Press, 1975.
5. A. SOOS: Contraction Methods in Fractal Theory, Cluj University Press
Printing House,  2002.	
Assessment	Practical works 30%
Presentation 30%
Exam 40%