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Discrete Mathematical Analysis: theory and geophysical applications

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.

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Discrete Mathematical Analysis: theory and geophysical applications

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  1. Discrete Mathematical Analysis: theory and geophysical applications

  2. 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.

  3. Construction scheme of DMA

  4. Fuzzy comparisons

  5. Measures of nearness

  6. Measures of nearness

  7. Nearness measures: nearness point to set

  8. General concept of finite limit in FMS

  9. Construction scheme of DMA

  10. Density as measure of limitness

  11. Algorithm “Choosing of foundation”

  12. 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

  13. CRYSTAL description CRYSTAL block-scheme:

  14. 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)

  15. 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

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

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

  18. Examples  =1.2 =1.0 =2.3

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

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

  21. Geophysical applications

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

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

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

  25. 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:

  26. 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

  27. RODIN overview Algorithms of RODIN family:

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

  29. RODIN overview

  30. Geophysical applications Saint Malo Region

  31. Geophysical applications Saint Malo Region

  32. Geophysical applications Saint Malo Region

  33. 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

  34. Etna. Interferrogramm

  35. Smoothness of displacement

  36. Etna. Smooth points

  37. Etna. Monolith (1-stiteration)

  38. Etna. Monolith (4-thiteration)

  39. Etna. Monolith (boundary)

  40. Etna. General solution

  41. Tracing

  42. Tracing

  43. Construction scheme of DMA

  44. Gravitation limits of time series

  45. Gravitation limits of time series

  46. Algorithm “Equilibrium”

  47. Algorithm “Equilibrium”

  48. Algorithm “Equilibrium” Dependence from l

  49. Algorithm “Equilibrium”

  50. Algorithm “Equilibrium”

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