Normalized information-based divergences

This paper is devoted to the mathematical study of some divergences based on the mutual information well-suited to categorical random vectors. These divergences are generalizations of the "entropy distance" and "information distance". Their main characteristic is that they combine a complexity term and the mutual information. We then introduce the notion of (normalized) information-based divergence, propose several examples and discuss their mathematical properties in particular in some prediction framework.Comment: 36 page

Representing three-dimensional cross fields using 4th order tensors

This paper presents a new way of describing cross fields based on fourth order tensors. We prove that the new formulation is forming a linear space in $\mathbb{R}^9$. The algebraic structure of the tensors and their projections on \mbox{SO}(3) are presented. The relationship of the new formulation with spherical harmonics is exposed. This paper is quite theoretical. Due to pages limitation, few practical aspects related to the computations of cross fields are exposed. Nevetheless, a global smoothing algorithm is briefly presented and computation of cross fields are finally depicted

Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler's Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a Jordan Curve Theorem is stated and proven for any planar hypermap

Spencer Operator and Applications: From Continuum Mechanics to Mathematical physics

The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is to prove that the use of the Spencer operator constitutes the common secret of the three following famous books published about at the same time in the beginning of the last century, though they do not seem to have anything in common at first sight as they are successively dealing with elasticity theory, commutative algebra, electromagnetism and general relativity: (C) E. and F. COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for studying control identifiability and of (C)+(W) for the group theoretical unification of finite elements in engineering sciences, recovering in a purely mathematical way well known field-matter coupling phenomena (piezzoelectricity, photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and though disturbing it could be, we shall prove that these unavoidable new diferential and homological methods contradict the mathematical foundations of both engineering (continuum mechanics,electromagnetism) and mathematical (gauge theory, general relativity) physics.Comment: Though a few of the results presented are proved in the recent references provided, the way they are combined with others and patched together around the three books quoted is new. In view of the importance of the full paper, the present version is only a summary of the definitive version to appear later on. Finally, the reader must not forget that "each formula" appearing in this new general framework has been used explicitly or implicitly in (C), (M) and (W) for a mechanical, mathematical or physical purpos

A Random Difference Equation with Dufresne Variables revisited

The Dufresne laws (laws of product of independent random variables with gamma and beta distributions) occur as stationary distribution of certain Markov chains $X_n$ on $R$ defined by: \begin{equation} X_n = A_n ( X_{n-1} + B_n ) \end{equation} where $X_0 , (A_1,B_1),...,(A_n,B_n)$ are independent and the $(A_i,B_i)'$s are identically distributed. This paper generalizes an explicit example where $A$ is the product of two independent $\beta_{a,1} , \beta_{b,1}$ and $B \sim \gamma_1$ or $\gamma_2$. Keywords: beta, gamma and Dufresne distributions,Markov chains, stationary distributions, hypergeometric differential equations, Poisson process.Comment: 11 pages, 2 tables, 1 figur

Some properties of the range of super-Brownian motion

We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $\epsilon$-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of $X_t$ is capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to $[0,1]^4$ in $\R^d$, $d\geq 5$

The lineage process in Galton--Watson trees and globally centered discrete snakes

We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node $u$ is the vector indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children and for which $u$ is a descendant of the $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived.Comment: Published in at http://dx.doi.org/10.1214/07-AAP450 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org