The clausal discovery engine Claudien is presented. Claudien is an inductive logic programming engine that fits in the knowledge discovery in databases and data mining paradigm as it discovers regularities that are valid in data. As such Claudien performs a novel induction task, which is called characteristic induction from closed observations, and which is related to existing formalizations of induction in logic. In characterising induction from closed observations, the regularities are represented by clausal theories, and the data using Herbrand interpretations. Claudien also employs a novel declarative bias mechanism to define the set of clauses that may appear in a hypothesis. Keywords : Inductive Logic Programming, Knowledge Discovery in Databases, Data Mining, Learning, Induction, Semantics for Induction, Logic of Induction, Parallel Learning. 1 Introduction Despite the fact that the areas of knowledge discovery in databases [Fayyad et al., 1995] and inductive logic programmin...

In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves). The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon’s nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux. 1.

We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1991 Mathematics Subject Classifications: Primary 60C05, 05E10, 15A52, 60F05. Research supported by N.S.F. Grant MCS 96-22859 1 Introduction This survey paper treats two themes in parallel. One theme is a purely mathematical question: describe the asymptotic law (probability distribution) of the length of the longest increasing subsequence of a random permutation. This question has been studied by a variety of increasingly technically sophisticated methods over the last 30 years. We outline three, apparently quite unrelated, methods in sections 2 - 4. The other theme is a card game, patience sorting. This gam...

1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /|G|. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number

Schubert polynomials were introduced by Bernstein Gelfand Gelfand and De- mazure, and were extensively developed by Lascoux, Schiitzenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polyno- mial in terms of the reduced decompositions of the permutation w. Using this Supported by the National Physical Science Consortium.

We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[x1,...,xn,y1,...,yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1,...,xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.

Contents. 1. Introduction 2. The symmetric functions P (q; t) 3. Duality 4. Skew P and Q functions 5. Explicit formulas 6. The Kostka matrix 7. Another scalar product 8. Conclusion 9. Appendix 1. Introduction I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these lectures. Let x 1 , : : : , xn be independent indeterminates. The symmetric group Sn acts on the polynomial ring Z[x 1 ; : : : ; xn ] by permuting the x's, and we shall write n = Z[x 1 ; : : : ; xn ] Sn for the subring of symmetric polynomials in x<F8

We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.

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Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials.