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## Elements of Self Organisation

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**Elements of Self Organisation**• Groups**Content**• Introduction • Historic Perspective • Related Subjects • Basics • Connection to Self Organisation • Summary and Discussion**Historic Perspective**• Classification • Adam • Linnaeus • Sets • Cantor • Hilbert • Galois • Gödel**James Ussher**(1581-1656) Adam October 23, 4004 BC.**Carl Linnaeus**(1707-1778) Linnaean Taxonomy Modern Kingdoms**Diagonal Argument**Leopold Kronecker (1823-1891) • Kronecker David Hilbert (1862-1943) Georg Cantor (1845-1918)**(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**Esscher**The Incompleteness theorem Kurt Gödel(1906-1978) Continuum Hypothesis**Related Subjects(Classification Systems)**• Dewey Decimal • SIC (NAISC) • Celestial Emporium of Benevolent Knowledge’s Taxonomy • Cladistics • Global Data Model • Dynamic Classification**Melvil Dewey**(1851-1931) Dewey Decimal Classification System**SIC Codes from the perspective of Lambert**NAISC in Relation to SIC Standard Industry Code (and NAISC) Employees per sector**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)**Cladistic Tree of Life**Cladogram**Classified Satellite Imagery**Spam Classification Dynamic Classification Data Mining**Basics**• Sets • Group theory • Clustering**Sets**• Zermelo-Fraenkel Set Theory • Russel’s Paradox • Fuzzy Sets**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.**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)**Adaptive Fuzzy System**Fuzzy Sets Fuzzy Set Union**Group theory**• Definition • Examples**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.**Examples of**Group Theory Rubik’s Cube automat Symmetry in Chemistry Samual Loyd’s 1415 puzzle Statistical physics on complex networks**Clustering**• Distance • Euclidian • Manhattan • Hamming • Jaccard • Algorithms • Hierarchical • Partitional • Agglomerative • Divisive • Dynamic Clustering**Distance**Manhattan distance Hamming distance Jaccard Genetic distance Euclidian distance**Conceptual Clustering: COBWEB**Hierarchical Clustering Algorithms Partitional Clustering: K-Means Clustering Divisive: Top-down Agglomerative: Bottom-up**Dynamic Clustering of Property**Dynamic Clustering Dynamic clustering of sound sources for efficient 3D audio rendering. Senate Network**Connection to Self Org.**• Group Emergence • Speciation • Schismogenesis • Tags • Examples • Dynamic tags • Aggregation • Similarity • Emergence**Banners**Pheromones Signs Flags Antigen Binding Site Examples of Tags Mating Signs Trademarks**Tagging the web**Dynamic Pricing Dynamic Tags Evolving Signs Memes**Volvox**Aggregation: Similarity Hierarchical Network Agent Layers**Ant Hill**Organic forms and the mathematical rules behind them Aggregation: Emergence**Religion**Drosophilia Speciation Winner takes all**Gregory Bateson**(1904-1980) Complementary Schismogenesis: Class Struggle Schismogenesis Idealogical Amplification Symmetrical Schismogenesis: Arms Race**Summary and Discussion**• Sets • Groups • Clustering • Group Emergence • Literature • Discussion of Examples in Practice**Examples in Practice**• Late group evaluation • Dynamic group creation/splitting • Subjective Groups • Evolving Tags