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Elements of Self Organisation

Elements of Self Organisation

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Elements of Self Organisation

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  1. Elements of Self Organisation • Groups

  2. Content • Introduction • Historic Perspective • Related Subjects • Basics • Connection to Self Organisation • Summary and Discussion

  3. Historic Perspective • Classification • Adam • Linnaeus • Sets • Cantor • Hilbert • Galois • Gödel

  4. James Ussher (1581-1656) Adam October 23, 4004 BC.

  5. Carl Linnaeus (1707-1778) Linnaean Taxonomy Modern Kingdoms

  6. Diagonal Argument Leopold Kronecker (1823-1891) • Kronecker David Hilbert (1862-1943) Georg Cantor (1845-1918)

  7. (Don't cry, Alfred! I need all my courage to die at twenty.) Notes Évariste Galois (1811-1832) Group Theory Notes Quintic Equation X⁵ +px +q

  8. Esscher The Incompleteness theorem Kurt Gödel(1906-1978) Continuum Hypothesis

  9. Related Subjects(Classification Systems) • Dewey Decimal • SIC (NAISC) • Celestial Emporium of Benevolent Knowledge’s Taxonomy • Cladistics • Global Data Model • Dynamic Classification

  10. Melvil Dewey (1851-1931) Dewey Decimal Classification System

  11. SIC Codes from the perspective of Lambert NAISC in Relation to SIC Standard Industry Code (and NAISC) Employees per sector

  12. In The Analytical Language of John Wilkins (El idioma analítico de John Wilkins), Jorge Luis Borges describes "a certain Chinese encyclopedia," the Celestial Emporium of Benevolent Knowledge, in which it is written that animals are divided into: 1. those that belong to the Emperor, 2. embalmed ones, 3. those that are trained, 4. suckling pigs, 5. mermaids, 6. fabulous ones, 7. stray dogs, 8. those included in the present classification, 9. those that tremble as if they were mad, 10. innumerable ones, 11. those drawn with a very fine camelhair brush, 12. others, 13. those that have just broken a flower vase, 14. those that from a long way off look like flies. George Luis Borges (1823-1891)

  13. Cladistic Tree of Life Cladogram

  14. AND Global Data Model

  15. Classified Satellite Imagery Spam Classification Dynamic Classification Data Mining

  16. Basics • Sets • Group theory • Clustering

  17. Sets • Zermelo-Fraenkel Set Theory • Russel’s Paradox • Fuzzy Sets

  18. Zermelo-Fraenkel Set Theory (ZFC). An important feature of ZFC (including 10, choice) is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.The ten axioms of ZFC : 1. Axiom of extensionality: Two sets are the same if and only if they have the same elements. 2. Axiom ofempty set: There is a set with no elements. We will use {} to denote this empty set. 3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements. 4. Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x. 5. Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}. 6. Axiom schema of separation(or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds. 7. Axiom schema of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements. 8. Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x. 9. Axiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets. 10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

  19. Russell's paradox: construct the set S := {A : A is not in A} of all sets that do not belong to themselves. (If S belongs to itself, then it does not, giving a contradiction, so S must not belong to itself. But then S would belong to itself, giving a final and absolute contradiction.) Bertrand Russell(1872-1970)

  20. Adaptive Fuzzy System Fuzzy Sets Fuzzy Set Union

  21. Group theory • Definition • Examples

  22. The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing. James Newman Group Theory A group (G,*) is a set G (the underlying set) closed under a binary operation satisfying three axioms: 1. The operation is associative. 2. The operation has an identity element 3. Every element has an inverse element.

  23. Examples of Group Theory Rubik’s Cube automat Symmetry in Chemistry Samual Loyd’s 1415 puzzle Statistical physics on complex networks

  24. Clustering • Distance • Euclidian • Manhattan • Hamming • Jaccard • Algorithms • Hierarchical • Partitional • Agglomerative • Divisive • Dynamic Clustering

  25. Distance Manhattan distance Hamming distance Jaccard Genetic distance Euclidian distance

  26. Conceptual Clustering: COBWEB Hierarchical Clustering Algorithms Partitional Clustering: K-Means Clustering Divisive: Top-down Agglomerative: Bottom-up

  27. Dynamic Clustering of Property Dynamic Clustering Dynamic clustering of sound sources for efficient 3D audio rendering. Senate Network

  28. Connection to Self Org. • Group Emergence • Speciation • Schismogenesis • Tags • Examples • Dynamic tags • Aggregation • Similarity • Emergence

  29. Banners Pheromones Signs Flags Antigen Binding Site Examples of Tags Mating Signs Trademarks

  30. Tagging the web Dynamic Pricing Dynamic Tags Evolving Signs Memes

  31. Volvox Aggregation: Similarity Hierarchical Network Agent Layers

  32. Ant Hill Organic forms and the mathematical rules behind them Aggregation: Emergence

  33. Religion Drosophilia Speciation Winner takes all

  34. Gregory Bateson (1904-1980) Complementary Schismogenesis: Class Struggle Schismogenesis Idealogical Amplification Symmetrical Schismogenesis: Arms Race

  35. Summary and Discussion • Sets • Groups • Clustering • Group Emergence • Literature • Discussion of Examples in Practice

  36. Examples in Practice • Late group evaluation • Dynamic group creation/splitting • Subjective Groups • Evolving Tags

  37. Literature