# Communication cascade processes with Markov chains

### Abstract

The work executed within the framework of cascade-probability method, the essence of which is to obtain and further use of cascade-probability functions (CPF) for the different particles. CPF sense the probability that a particle generated at a certain depth h’ reaches a certain depth h after the n-th number of collisions. We have considered the interaction of ions with solids and communication processes of radiation defect with Markov processes and Markov chains. Displaying obtain recurrence relations for the simplest of CPF Chapman-Kolmogorov equations. In this case the particle after the collision does not change its direction of movement, the flow rate is independent of time, and hence the penetration depth. recurrence relations are also obtained for the RAF taking into account losses of energy to the ions of the Chapman-Kolmogorov equation, the intensity of the flow depends on the depth of penetration.### References

1. Boss E.G., Kupchyshyn A.I. The solution of physical problems cascaded probabilistic method. – Alma-Ata: Science, 1988. – 112 p.

2. Boss E.G., Kupchyshyn A.I. The solution of physical problems cascaded probabilistic method. – Alma-Ata: Science, 1998. – 144 p.

3. Guter R.S., Ovchinsky B.V. Fundamentals of Probability Theory. – M .: Education , 1967. – 159 p.

4. Kolmogorov A.N. Basic concepts of probability theory. – M .: Science, 1974. – 119 p.

5. Feller W., An Introduction to Probability Theory and Its Applications. – M .: Mir, 1984. – 527 p.

6. Wentzell A.D, Course of the theory of random processes. – M .: Nauka, 1996. – 400 p.

7. Kupchishin A.I Interaction of radiation with matter. Cascaded probabilistic method (methodical development for students of physics and mathematics). – Almaty, 1986. – 68 p.

8. Boos E.G., Kupchishin A.A., Kupchishin A.I., Shmygalev E.V, Shmygaleva T.A.. Cascaded probabilistic method, the solution of the radiation-physical problems, Boltzmann equations. Communication with Markov chains. Monograph. – Almaty.: KazNPU named after Abai, research institutes and KazNU named after Al-Farabi. – 2015. – 388 p.

9. Kutchukian P., Lou D., Shakhnovich E. "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules occupying Druglike Chemical" // Journal of Chemical Information and Modeling. – 2009. – Vol. 49, No 7. – P. 1630–1642.

10. Kopp V.S., Kaganer V.M., Schwarzkopf J., Waidick F., Remmele T., Kwasniewski A., Schmidbauer M. "X-ray diffraction from nonperiodic layered structures with correlations: Analytical calculation and experiment on mixed Aurivillius films" // Acta Crystallographica Section a Foundations of Crystallography 68. – 2011. 148.doi:10.1107/S0108767311044874. edit

11. Prasad N.R., Ender R.C., Reilly S.T., Nesgos G. "Allocation of resources on a minimized cost basis" // 1974 IEEE Conference on Decision and Control including the 13th Symposium on Adaptive Processes 13. – 1974. – Vol. 402, No 3.

12. Hamilton J. "A new approach to the economic analysis of nonstationary time series and the business cycle" // Econometrica. – 1989. – Vol. 57, No. 2.

2. Boss E.G., Kupchyshyn A.I. The solution of physical problems cascaded probabilistic method. – Alma-Ata: Science, 1998. – 144 p.

3. Guter R.S., Ovchinsky B.V. Fundamentals of Probability Theory. – M .: Education , 1967. – 159 p.

4. Kolmogorov A.N. Basic concepts of probability theory. – M .: Science, 1974. – 119 p.

5. Feller W., An Introduction to Probability Theory and Its Applications. – M .: Mir, 1984. – 527 p.

6. Wentzell A.D, Course of the theory of random processes. – M .: Nauka, 1996. – 400 p.

7. Kupchishin A.I Interaction of radiation with matter. Cascaded probabilistic method (methodical development for students of physics and mathematics). – Almaty, 1986. – 68 p.

8. Boos E.G., Kupchishin A.A., Kupchishin A.I., Shmygalev E.V, Shmygaleva T.A.. Cascaded probabilistic method, the solution of the radiation-physical problems, Boltzmann equations. Communication with Markov chains. Monograph. – Almaty.: KazNPU named after Abai, research institutes and KazNU named after Al-Farabi. – 2015. – 388 p.

9. Kutchukian P., Lou D., Shakhnovich E. "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules occupying Druglike Chemical" // Journal of Chemical Information and Modeling. – 2009. – Vol. 49, No 7. – P. 1630–1642.

10. Kopp V.S., Kaganer V.M., Schwarzkopf J., Waidick F., Remmele T., Kwasniewski A., Schmidbauer M. "X-ray diffraction from nonperiodic layered structures with correlations: Analytical calculation and experiment on mixed Aurivillius films" // Acta Crystallographica Section a Foundations of Crystallography 68. – 2011. 148.doi:10.1107/S0108767311044874. edit

11. Prasad N.R., Ender R.C., Reilly S.T., Nesgos G. "Allocation of resources on a minimized cost basis" // 1974 IEEE Conference on Decision and Control including the 13th Symposium on Adaptive Processes 13. – 1974. – Vol. 402, No 3.

12. Hamilton J. "A new approach to the economic analysis of nonstationary time series and the business cycle" // Econometrica. – 1989. – Vol. 57, No. 2.

How to Cite

SHMYGALEVA, T. A. et al.
Communication cascade processes with Markov chains.

**International Journal of Mathematics and Physics**, [S.l.], v. 7, n. 1, p. 10-15, june 2016. ISSN 2409-5508. Available at: <http://ijmph.kaznu.kz/index.php/kaznu/article/view/153>. Date accessed: 19 june 2018. doi: https://doi.org/10.26577/ijmph.v7i1.153.
Section

Mathematical Modeling

### Keywords

cascade-probability, ions, defect formation, Markov chain, Markov processes.