Formalized proof, computation, and the construction problem in algebraic geometry

An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory within a ZFC-like environment

Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page

On group theory for quantum gates and quantum coherence

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M20, the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.Comment: 10 pages, to appear in J Phys A: Math and Theo (Fast Track Communication

Universal behaviour of a wave chaos based electromagnetic reverberation chamber

In this article, we present a numerical investigation of three-dimensional electromagnetic Sinai-like cavities. We computed around 600 eigenmodes for two different geometries: a parallelepipedic cavity with one half- sphere on one wall and a parallelepipedic cavity with one half-sphere and two spherical caps on three adjacent walls. We show that the statistical requirements of a well operating reverberation chamber are better satisfied in the more complex geometry without a mechanical mode-stirrer/tuner. This is to the fact that our proposed cavities exhibit spatial and spectral statistical behaviours very close to those predicted by random matrix theory. More specifically, we show that in the range of frequency corresponding to the first few hundred modes, the suppression of non-generic modes (regarding their spatial statistics) can be achieved by reducing drastically the amount of parallel walls. Finally, we compare the influence of losses on the statistical complex response of the field inside a parallelepipedic and a chaotic cavity. We demonstrate that, in a chaotic cavity without any stirring process, the low frequency limit of a well operating reverberation chamber can be significantly reduced under the usual values obtained in mode-stirred reverberation chambers

The Burbea-Rao and Bhattacharyya centroids

We study the centroid with respect to the class of information-theoretic Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence by measuring the non-negative Jensen difference induced by a strictly convex and differentiable function. Although those Burbea-Rao divergences are symmetric by construction, they are not metric since they fail to satisfy the triangle inequality. We first explain how a particular symmetrization of Bregman divergences called Jensen-Bregman distances yields exactly those Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao divergences, and show that skew Burbea-Rao divergences amount in limit cases to compute Bregman divergences. We then prove that Burbea-Rao centroids are unique, and can be arbitrarily finely approximated by a generic iterative concave-convex optimization algorithm with guaranteed convergence property. In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or "diagonal" multivariate Gaussians. To illustrate the performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present experimental performance results for $k$-means and hierarchical clustering methods of Gaussian mixture models.Comment: 13 page

Tarski's influence on computer science

The influence of Alfred Tarski on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is the work of Tarski on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, modeltheoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up

Long Proteins with Unique Optimal Foldings in the H-P Model

It is widely accepted that (1) the natural or folded state of proteins is a global energy minimum, and (2) in most cases proteins fold to a unique state determined by their amino acid sequence. The H-P (hydrophobic-hydrophilic) model is a simple combinatorial model designed to answer qualitative questions about the protein folding process. In this paper we consider a problem suggested by Brian Hayes in 1998: what proteins in the two-dimensional H-P model have unique optimal (minimum energy) foldings? In particular, we prove that there are closed chains of monomers (amino acids) with this property for all (even) lengths; and that there are open monomer chains with this property for all lengths divisible by four.Comment: 22 pages, 18 figure