Numerical solution of random differential initial value problems: Multistep methods

This paper deals with the construction of numerical methods of random initial value problems. Random linear multistep methods are presented and sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

The CRT is the scaling limit of unordered binary trees

We prove that a uniform, rooted unordered binary tree (also known as rooted, binary Pólya tree) with n leaves has the Brownian continuum random tree as its scaling limit for the Gromov‐Hausdorff topology. The limit is thus, up to a constant factor, the same as that of uniform plane trees or labeled trees. Our analysis rests on a combinatorial and probabilistic study of appropriate trimming procedures of trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 467–501, 2011     

Topological quantum field theories and Markovian random fields

We discuss an analogy between topological quantum field theories and the theory of Markov processes, which both rely on the combination of a notion of transition and a notion of locality. We assume no prior knowledge of topological quantum field theory (TQFT) and devote the first section to a motivation of its definition, which was originally given by M. Atiyah. We then discuss 1- and 2-dimensional TQFT?s, and a mild generalization of them which incorporates a notion of time and is suited to the parallel with Markov processes.This text is a written version of a talk whose emphasis was on explaining and illustrating ideas. There are few rigorous statements in it, and no proof.     

Practical strategies for generating rank-1 split cuts in mixed-integer linear programming

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.     

Variations of constants and red-shift in 6D cosmology

On the basis of the general class of solutions of 6D vacuum Einstein equations and using well-known results of the 6D geometric model of electroweak and gravitational interactions, the variations of fundamental physical constants and their influence on the observed red-shift of galaxies are investigated. By considering the interpretation of extra components of the 6D metric as reproducing the properties of matter in the 4D curved space-time, the effective quantities of matter are obtained. In particular, the states of dust and radiation are shown. The corresponding 5D models were also investigated.     

137-01137-01137-01-deformed extended Galilei algebra

We apply the pseudoextension mechanism, which in the undeformed case gives the centrally extended Galilei group {ie137-02} as a contraction of a trivial extensionP×U(1) of the Poincaré group, to the case of the-Poincaré algebra. As a result, the four-dimensional {ie137-03}-deformed extended Galilei algebra is obtained.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

Random repeated quantum interactions and random invariant states

We consider a generalized model of repeated quantum interactions, where a system ${\mathcal{H}}$ is interacting in a random way with a sequence of independent quantum systems ${\mathcal{K}_n, n \geq 1}$ . Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between ${\mathcal{H}}$ and ${\mathcal{K}_n}$ . The other involves random quantum states describing each copy ${\mathcal{K}_n}$ . In the limit of a large number of interactions, we present convergence results for the asymptotic state of ${\mathcal{H}}$ . This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble.     

Projections, entropy and sumsets

In this paper we shall generalize Shearer??s entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections.     

b-colouring outerplanar graphs with large girth

A b-colouring of a graph is a colouring of its vertices such that every colour class contains a vertex that has a neighbour in all other classes. The b-chromatic number of a graph is the largest integer $k$ such that the graph has a $b$-colouring with $k$ colours. We show here how to compute in polynomial time the $b$-chromatic number of an outerplanar graph of girth at least $8$. This generalizes the seminal result of Irving and Manlove on trees.     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).