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Discrete Mathematical Analysis: theory and geophysical applications. DMA definition. Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based on modeling of limit in a finite situation, realized in a series of algorithms. I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. DMA definition Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based on modeling of limit in a finite situation, realized in a series of algorithms. The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas and judgments. That was reason why they became technical foundation of DMA.

2. Construction scheme of DMA

3. Fuzzy comparisons

4. Measures of nearness

5. Measures of nearness

6. Nearness measures: nearness point to set

7. General concept of finite limit in FMS

8. Construction scheme of DMA

9. Density as measure of limitness

10. Algorithm “Choosing of foundation”

11. CRYSTAL description • CRYSTAL goal: to identify -dense subsets against a general background • Definition 1. Subset AX is dense against the background X if • Definition 2. Subset AX is -dense against the background X if

12. CRYSTAL description CRYSTAL block-scheme:

13. CRYSTAL description “Growth I” block: • – the current version of i-th crystal Ki • for • , =1  (1,n)  1 “Struggle” between and in : • If then has “won”, and we proceed to the “Foundation” block • If then has “won”, but Kin+1={Kni, x} should remain dense against the background X in all its points (the “Growth II” block)

14. CRYSTAL description “Growth II” block (1): • Statement. If the “Growth I” condition is fulfilled then Kn+1={Kn, xn+1} will be -dense only if the “Growth II” condition is fulfilled: • =1  • If there is no such xn+1, we proceed to the “Foundation” block for choosing another foundation for the next crystallization • If there are several such xn+1 ( ), then

15. CRYSTAL description “Growth II” block (2): • Recalculation of the densities and in for their further “struggle” in the “Growth I” block:

16. CRYSTAL description “End” block: • Identification of the final crystal versions K1, …, KI, I =I(F,).

17. Examples  =1.2 =1.0 =2.3

18. Examples =2.5 =4.0 blue – foundations; red – crystallised points

19. Examples (1) (2) =2.6 blue – foundations; red – crystallised points

20. Geophysical applications

21. Geophysical applications Real data – region Hoggar (Algeria): definition of magnetization T N=3

22. Geophysical applications Real data – region Hoggar (Algeria): definition of magnetization

23. RODIN overview • The cluster definition: cluster in X, if  A – cluster in X, if

24. RODIN overview Let A be a cluster in X and xAX: • The cluster quality: • Measure of separability of x in A: where The cluster separability:

25. RODIN overview Block-scheme of the Global RODIN:  – given level of quality,  – given level of separability (clusterness), K0 – initial version of cluster, Kn – current version of cluster

26. RODIN overview Algorithms of RODIN family:

27. RODIN overview The examples of Global RODIN clustering with different levels of quality:

28. RODIN overview

29. Geophysical applications Saint Malo Region

30. Geophysical applications Saint Malo Region

31. Geophysical applications Saint Malo Region

32. Algorithm “Monolith” X – multi-dimensional massif, xX, A – subset inX monAx – monolithnessAinx measure of limitnessAinx MonA X– subsep points inX with large A-monolithness (A-foundations) CrossingAA (1) =MonA X – “multi-dimensional topological” smoothingA in X Algorithm“Monolith”: parameters  radius of monolithnessr  weight of monolithness  parameter of choosing  number of iterationsi Algorithm“Monolith”: block-scheme

33. Etna. Interferrogramm

34. Smoothness of displacement

35. Etna. Smooth points

36. Etna. Monolith (1-stiteration)

37. Etna. Monolith (4-thiteration)

38. Etna. Monolith (boundary)

39. Etna. General solution

40. Tracing

41. Tracing

42. Construction scheme of DMA

43. Gravitation limits of time series

44. Gravitation limits of time series

45. Algorithm “Equilibrium”

46. Algorithm “Equilibrium”

47. Algorithm “Equilibrium” Dependence from l

48. Algorithm “Equilibrium”

49. Algorithm “Equilibrium”