Singularities of Blaschke normal maps of convex surfaces

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.Comment: 5 pages Version 3; to appear in Comptes Rendus Mathematiqu

Exponential families, Kahler geometry and quantum mechanics

Exponential families are a particular class of statistical manifolds which are particularly important in statistical inference, and which appear very frequently in statistics. For example, the set of normal distributions, with mean {\mu} and deviation {\sigma}, form a 2-dimensional exponential family. In this paper, we show that the tangent bundle of an exponential family is naturally a Kahler manifold. This simple but crucial observation leads to the formalism of quantum mechanics in its geometrical form, i.e. based on the Kahler structure of the complex projective space, but generalizes also to more general Kahler manifolds, providing a natural geometric framework for the description of quantum systems. Many questions related to this "statistical Kahler geometry" are discussed, and a close connection with representation theory is observed. Examples of physical relevance are treated in details. For example, it is shown that the spin of a particle can be entirely understood by means of the usual binomial distribution. This paper centers on the mathematical foundations of quantum mechanics, and on the question of its potential generalization through its geometrical formulation

Quantization of formal classical dynamical r-matrices: the reductive case

In this paper we prove the existence of a formal dynamical twist quantization for any triangular and non-modified formal classical dynamical $r$-matrix in the reductive case. The dynamical twist is constructed as the image of the dynamical $r$-matrix by a $L_\infty$-quasi-isomorphism. This quasi-isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence.Comment: 13 pages, 1 section added (classification of dynamical twists), LaTeX, final version, to appear in Adv. Mat

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.Comment: published versio

Substrate specificity of the streptococcal cysteine protease.

The streptococcal pyrogenic exotoxin B (SpeB) is an important factor in mediating Streptococcus pyogenes infections. SpeB is the zymogen of the streptococcal cysteine protease (SCP), of which relatively little is known regarding substrate specificity. To investigate this aspect of SCP function, a series of internally quenched fluorescent substrates was designed based on the cleavage sites identified in the autocatalytic processing of SpeB to mature SCP. The best substrates for SCP contain three amino acids in the nonprimed position (i.e. AIK in P(3)-P(2)-P(1)). Varying the length of the substrate on the primed side of the scissile bond has a relatively lower effect on activity. The highest activity (k(cat)/K(M) = 2.8 +/- 0.6 (10(5) x m(-1)s(-1)) is observed for the pentamer 3-aminobenzoic acid-AIKAG-3-nitrotyrosine, which spans subsites S(3) to S(2)' on the enzyme. High pressure liquid chromatography and mass spectrometry analyses show that the substrates are cleaved at the site predicted from the autoprocessing experiments. These results show that SCP can display an important level of endopeptidase activity. Substitutions at position P(2) of the substrate clearly indicate that the S(2) subsite of SCP can readily accommodate substrates containing a hydrophobic residue at that position and that some topological preference exists for that subsite. Substitutions in positions P(3), P(1), and P(1)' had little or no effect on SCP activity. The substrate specificity outlined in this work further supports the similarity between SCP and the cysteine proteases of the papain family. From the data regarding the identified or proposed natural substrates for SCP, it appears that this substrate specificity profile may also apply to the processing of mammalian and streptococcal protein targets by SCP

Invariant characterization of Liouville metrics and polynomial integrals

A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals of geodesic flows