Multigraph limit of the dense configuration model and the preferential attachment graph

The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn models.     

A note on β-connectedness

An example is given to show that there exists a generalized topological space which is irreducible but not β-connected. This gives an answer to Question 2.8 in [6]. Besides, some characterizations of β-connectedness are obtained.     

Développement asymptotique q-Gevrey et fonction thêta de Jacobiq-Gevrey asymptotic expansion and Jacobi theta function

Some notions of q-Gevrey asymptotic expansion have been studied in [6,7]. Recently we became interested in a new notion of asymptotic expansion [8]: it is related to a Jacobi theta function and allows one to establish the natural link between the asymptotics of q-difference equations and the theory of elliptic functions. The purpose of this Note is to give some new results related to this notion of asymptotic expansion. To cite this article: J.-P. Ramis, C. Zhang, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 899–902.     

Une suite spectrale de Hochschild–Serre pour l'uniformisation de Rapoport–ZinkAn Hochschild–Serre spectral sequence for the Rapoport–Zink uniformization

In this Note we announce results concerning the first part of a programme intending to generalize the articles [5,7] and thus construct local Langlands correspondences for groups other than GL_n (for example, quasisplit unitary groups) inside the ? adic cohomology of Rapoport–Zink spaces. The method consists in comparing the cohomology of these local objects with that of global objects: Shimura varieties. For this we generalize the spectral sequences constructed in [5] and [4]. A part of these results is quoted in [6]. To cite this article: L. Fargues, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 739–742.     

Poset limits and exchangeable random posets

We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.     

Détermination des images d'un point de base d'une surface rationnelle tensorielle définie par des vecteurs massiquesDetermination of the images of a base point of a tensorial rational surface defined by massic vectors

In this Note, we demonstrate that the image of a base point of a rational surface defined in $\text{R}{}^3$ is a set of rational curves defined in $\text{R}{}^3$. A base point is a parameter value for which the rational parametrization takes the value of $\text{(}\text{0}\text{0}\text{,}\text{0}\text{0}\text{,}\text{0}\text{0}\text{)}$ (see [4,5]). This result was studied by Clebsch in [1]. Here, we use the formalism of massic vectors introduced by Fiorot and Jeannin in [3]. This allows us to give explicitly these rational curves via their massic vectors. These massic vectors are proportional to those whose indices belong to the Newton polygon of the surface. Moreover, it is shown that by using fractional changes of variables, it is possible to obtain any image curve directly without having to apply successive changes of variables as usually done in algebraic geometry. To cite this article: O. Gibaru, J.-C. Fiorot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 283–288.     

Time evolution of dense multigraph limits under edge‐conservative preferential attachment dynamics

We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the notion of convergence of dense graph sequences, defined by Lovász and Szegedy (J. Combin. Theory Ser B 96 (2006) 933–957). We investigate how the limit object evolves under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limit object from its initial state up to the stationary state, which is described in the companion paper (Ráth and Szakács, in press). In our proofs we use the theory of exchangeable arrays, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits subaging. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Riemann Surface Laminations with Singularities

In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in ?² and for generic holomorphic foliations in ?².     

Estimation d'une rupture en dépendance faibleChange point estimation for a weakly dependent sequence

Let $\text{(X}{}_{\mathrm{n}}\text{,Y}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ be a stationary sequence governed by the model Y_n=m(X_n)+σ(X_n)ε_n where $\text{(ε}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is i.i.d. and independent from $\text{(X}{}_{\mathrm{n}}\text{)}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}\text{.}$ The latter sequence satisfy a weak dependence condition proposed by Doukhan and Louhichi in [2]. We provide a Central Limit Theorem for jumps in the regression function. Our method deals with linear local regression described in [4]. We use a variation on Lindeberg–Rio method as in [5]. To cite this article: P. Ango Nze, C. Prieur, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 267–270.     

Crossing Numbers of Periodic Graphs

A graph is periodic if it can be obtained by joining identical pieces in a cyclic fashion. It is shown that the limit crossing number of a periodic graph is computable. This answers a question of Richter [1, Problem 4.2].     

A note on random k-dimensional posets

In 2009, Janson [Poset limits and exchangeable random posets, Institut Mittag-Leffler preprint, 36pp, arXiv:0902.0306] extended the recent theory of graph limits to posets, defining convergence for poset sequences and proving that every such sequence has a limit object. In this paper, we focus on k-dimensional poset sequences. This restriction leads to shorter proofs and to a more intuitive limit object. As before, the limit object can be used as a model for random posets, which generalizes the well known random k-dimensional poset model. This investigation also leads to a definition of quasirandomness for k-dimensional posets, which can be captured by a natural distance that measures the discrepancy of a k-dimensional poset.     

Christoffel functions and universality on the boundary of the ball

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in ? ^d .     

Vertex splitting and the recognition of trapezoid graphs

Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in (Cheah and Corneil, 1996) [3], first augments a given graph G and then transforms the augmented graph by replacing each of the original graph’s vertices by a pair of new vertices. This “splitted graph” is a permutation graph with special properties if and only if G is a trapezoid graph. Recently vertex splitting has been used to show that the recognition problems for both tolerance and bounded tolerance graphs is NP-complete (Mertzios et al., 2010) [11]. Unfortunately, the vertex splitting trapezoid graph recognition algorithm presented in (Cheah and Corneil, 1996) [3] is not correct. In this paper, we present a new way of augmenting the given graph and using vertex splitting such that the resulting algorithm is simpler and faster than the one reported in (Cheah and Corneil, 1996) [3].     

Note on some ideals of associative rings

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies.     

Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph

Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs.     

X-Inner Automorphisms,Automorphisms, and Isomorphisms of Skew Polynomial Rings

Abstract

In this paper, we examine the X-inner automorphisms, automorphisms, and isomorphisms of skew polynomial rings of the form K[x][y;δ], where K is a field of characteristic 0 and δ is a derivation of K [x] such that x ^δ is a polynomial of degree ≥ 1. We 1.  determine the group of X-inner automorphisms of K [x][ y;δ],

2.  analyze the structure of the group of automorphisms of K [x][ y;δ], and

3.  examine the isomorphism classes of skew polynomial rings of the form K [x][ y;δ].

We also provide several examples which indicate the importance of the base field in computing the X-inner automorphisms, automorphisms, and isomorphism classes of K [x][ y;δ].     

A central limit theorem for D(A)-valued processes

Let D(A) be the space of set-indexed functions that are outer continuous with inner limits, a generalization of D[0, 1]. This paper proves a central limit theorem for triangular arrays of independent D(A) valued random variables. The limit processes are not restricted to be Gaussian, but can be quite general infinitely divisible processes. Applications of the theorem include construction of set-indexed Lévy processes and a unified central limit theorem for partial sum processes and generalized empirical processes. Results obtained are new even for the D[0, 1] case.