On semiprojectivity of -algebras of directed graphs

It is shown that if is a countable, transitive directed graph with finitely many vertices, then is semiprojective.

     

On a Lp-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions

In this paper a special L_p-estimate for the linearized compressible Navier-Stokes in the Lagrangian coordinates for the Dirichlet boundary conditions is obtained. The constant in the estimate does not depend on the length of time interval [0,T]. The result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in L_p-spaces.     

Chaı̂nes de Markov TripletTriplet Markov Chains

The Hidden Markov Chains (HMC) are widely applied in various problems. This succes is mainly due to the fact that the hidden process can be recovered even in the case of very large set of data. These models have been recetly generalized to ‘Pairwise Markov Chains’ (PMC) model, which admit the same processing power and a better modeling one. The aim of this note is to propose further generalization called Triplet Markov Chains (TMC), in which the distribution of the couple (hidden process, observed process) is the marginal distribution of a Markov chain. Similarly to HMC, we show that posterior marginals are still calculable in Triplets Markov Chains. We provide a necessary and sufficient condition that a TMC is a PMC, which shows that the new model is strictly more general. Furthermore, a link with the Dempster–Shafer fusion is specified. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 275–278.     

Asymmetry and structural information in preferential attachment graphs

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd?s‐Rényi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node with m edges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric for m = 1, and there is some nonnegligible probability of symmetry for m = 2. It was conjectured that these graphs are asymmetric when m ≥ 3. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.     

The constrained degree and fixed-point index theory for set-valued maps

In the first part of the paper we give a construction of a topological degree theory for set-valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a Banach space. The obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds. In the second part we propose a fixed-point index for inward maps on arbitrary closed convex sets.     

A generalized Turán problem in random graphs

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H‐free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2‐balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erd?s‐Stone theorem for G(n,p), proved several years ago by Conlon‐Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán‐type problems that we are unable to solve in full generality.     

On Cyclotomic Numbers and the Reduction Map for the K-Theory of the Integers

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K _2n+1() K_{2n+1}(F_p)is nontrivial. Here n is an even, positive integer and F_p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.     

The Conley index, cup-length and bifurcation

A module structure of the cohomology Conley index is used to define a relative cup-length. This invariant is applied then to prove a multiplicity theorem for periodic solutions to Hamiltonian systems.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.