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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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 s-cocycles modulo algebraic equivalence on a smooth projective variety.

Algorithms in computational geometry often use the real-RAM model of computation. This model as-sumes that exact real numbers can be stored in mem-

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We consider the problem of estimating the 3D motion of an articulated object, such as a robot arm or a human body, from a monocular sequence of 2D perspective views. We advocate an approach of decomposition. The object under analysis is decomposed into simpler parts, each containing a small number of links. We first estimate the motion of the simplest part(s) and then propagate the analysis to the remaining parts of the object. Human gait is used as an example; however, the approach is general. To use this decomposition approach, we need a repertoire of results for the motion of simple articulated objects: motion estimation algorithms, and uniqueness and number of solutions (especially, how many views are needed for uniqueness). With the help of techniques in algebraic geometry, we have results for a number of cases which are particularly useful for human gait analysis.

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 structure-based 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...

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.

this paper is devoted to an analysis of moment problems in R

this paper we establish the optimal result in all dimensions, except at Q n ;