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267
On the mathematical foundations of learning
 Bulletin of the American Mathematical Society
, 2002
"... The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton ..."
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Cited by 227 (12 self)
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The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton
Split States, Entropy Enigmas, Holes and Halos
, 2007
"... We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute e ..."
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Cited by 95 (11 self)
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We investigate degeneracies of BPS states of Dbranes on compact CalabiYau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial Dbrane systems, and to clarify the subtle relation of DonaldsonThomas invariants to BPS indices of stable D6D2D0 states, realized in supergravity as “hole halos. ” We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are “split ” D6antiD6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor g −2 top e−K and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of “swing states ” — D6 states which exist at large radius but do not form stable D6antiD6 bound states. We point out a likely breakdown of the OSV conjecture at small gtop (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge ΛΓ supporting single centered black holes of entropy ∼ Λ2S(Γ) also admits twocentered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.
Duality Relating Spaces of Algebraic Cocycles and Cycles
"... In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms L s H k (X) → Ln−s ..."
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Cited by 33 (13 self)
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In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms L s H k (X) → Ln−sH2n−k(X) for 2s ≤ k which carry over via natural transformations to the Poincaré duality isomorphism H k (X; Z) → H2n−k(X; Z). More generally, for smooth projective varieties X and Y the natural graphing homomorphism sending algebraic cocycles on X with values in Y to algebraic cycles on the product X ×Y is a weak homotopy equivalence. Among applications presented are the determination of the homotopy type of certain algebraic mapping complexes and a computation of the group of algebraic scocycles modulo algebraic equivalence on a smooth projective variety.
Computer Algebra Methods for Studying and Computing Molecular Conformations
, 1997
"... A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecu ..."
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Cited by 28 (4 self)
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A relatively new branch of computational biology has been emerging as an effort to supplement traditional techniques of large scale search in drug design by structurebased methods, in order to improve efficiency and guarantee completeness. This paper studies the geometric structure of cyclic molecules, in particular the enumeration of all possible conformations, which is crucial in finding the energetically favorable geometries, and the identification of all degenerate conformations. Recent advances in computational algebra are exploited, including distance geometry, sparse polynomial theory, and matrix methods for numerically solving nonlinear multivariate polynomial systems. Moreover, we propose a complete array of computer algebra and symbolic computational geometry methods for modeling the rigidity constraints, formulating the problems in algebraic terms and, lastly, visualizing the computed conformations. The use of computer algebra systems and of public domain software is illustrated...
CRITICAL SETS OF SOLUTIONS TO ELLIPTIC EQUATIONS
"... P bjDju = 0 that the critical set jruj,1 f0g has locally Let u 6 const satisfy an elliptic equation L0u P aijDiju + with smooth coefficients in a domain in R n. It is shown finite n, 2 dimensional Hausdorff measure. This implies in particular that for a solution u 6 0 of (L 0 + c)u =0,with c 2 C 1 ..."
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Cited by 28 (2 self)
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P bjDju = 0 that the critical set jruj,1 f0g has locally Let u 6 const satisfy an elliptic equation L0u P aijDiju + with smooth coefficients in a domain in R n. It is shown finite n, 2 dimensional Hausdorff measure. This implies in particular that for a solution u 6 0 of (L 0 + c)u =0,with c 2 C 1, the singular set u,1 f0g \jruj,1 f0g has locally finite n  2 dimensional Hausdorff measure.
A rational rotation method for robust geometric algorithms
 In Proc. 8th Annu. ACM Sympos. Comput. Geom
, 1992
"... Algorithms in computational geometry often use the realRAM model of computation. This model assumes that exact real numbers can be stored in mem ..."
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Cited by 26 (1 self)
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Algorithms in computational geometry often use the realRAM model of computation. This model assumes that exact real numbers can be stored in mem
On the TimeSpace Complexity of Geometric Elimination Procedures
, 1999
"... In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new ge ..."
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Cited by 24 (17 self)
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In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input.
Solving Moment Problems By Dimensional Extension
, 1999
"... this paper is devoted to an analysis of moment problems in R ..."
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Cited by 23 (3 self)
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this paper is devoted to an analysis of moment problems in R