On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.     

Relations between the Local Chromatic Number and Its Directed Version

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.     

A Family of Geometries Related to the Suzuki Tower

ABSTRACT

Michio Suzuki constructed a sequence of five simple groups G _i , with i = 0,…, 4, and five graphs Δ _i , with i = 0,…, 4, such that Δ _i appears as a subgraph of Δ _i+1 for i = 0,…, 3 and G _i is an automorphism group of Δ _i for i = 0,…, 4. The largest group G ₄ was a new sporadic group of order 448 345 497 600. It is now called the Suzuki group Suz. These groups and graphs form what Jacques Tits calls the Suzuki tower. In a previous work, we constructed a rank four geometry Γ(HJ) on which the Hall-Janko sporadic simple group acts flag-transitively and residually weakly primitively. In this article, we show that Γ(HJ) belongs to a family of five geometries in bijection with the Suzuki tower. The largest of them is a geometry of rank six, on which the Suzuki sporadic group acts flag-transitively and residually weakly primitively.     

A Dynamical Approach to Constrained Nonsmooth Convex Minimization Problem Coupling with Penalty Function Method in Hilbert Space

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item ?(t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of ?(t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article.     

On 3-choosable planar graphs of girth at least 4

Let G be a plane graph of girth at least 4. Two cycles of G are intersecting if they have at least one vertex in common. In this paper, we show that if a plane graph G has neither intersecting 4-cycles nor a 5-cycle intersecting with any 4-cycle, then G is 3-choosable, which extends one of Thomassen’s results [C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Combin. Theory Ser. B 64 (1995) 101-107].     

Ternary codes from graphs on triples

For a set Ω of size n≥7 and Ω^{3} the set of subsets of Ω of size 3, we examine the ternary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω^{3}, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively.     

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

We introduce completely monotonic functions of order r>0 and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.     

Interleaving schemes on circulant graphs with two offsets

Interleaving is used for error-correcting on a bursty noisy channel. Given a graph G describing the topology of the channel, we label the vertices of G so that each label-set is sufficiently sparse. The interleaving scheme corrects for any error burst of size at most t; it is a labeling where the distance between any two vertices in the same label-set is at least t.We consider interleaving schemes on infinite circulant graphs with two offsets 1 and d. In such a graph the vertices are integers; edge ij exists if and only if |i−j|∈{1,d}. Our goal is to minimize the number of labels used.Our constructions are covers of the graph by the minimal number of translates of some label-set S. We focus on minimizing the index of S, which is the inverse of its density rounded up. We establish lower bounds and prove that our constructions are optimal or almost optimal, both for the index of S and for the number of labels.