Equitable Δ-coloring of graphs

Consider a graph $G$ consisting of a vertex set $V\left(G\right)$ and an edge set $E\left(G\right)$. Let $\Delta \left(G\right)$ and $\chi \left(G\right)$ denote the maximum degree and the chromatic number of $G$, respectively. We say that $G$ is equitably $\Delta \left(G\right)$-colorable if there exists a proper $\Delta \left(G\right)$-coloring of $G$ such that the sizes of any two color classes differ by at most one. Obviously, if $G$ is equitably $\Delta \left(G\right)$-colorable, then $\Delta \left(G\right)\ge \chi \left(G\right)$. Conversely, even if $G$ satisfies $\Delta \left(G\right)\ge \chi \left(G\right)$, we cannot guarantee that $G$ must be equitably $\Delta \left(G\right)$-colorable. In 1994, the Equitable $\Delta$-Coloring Conjecture $\left(E\Delta \mathrm{CC}\right)$ asserts that a connected graph $G$ with $\Delta \left(G\right)\ge \chi \left(G\right)$ is equitably $\Delta \left(G\right)$-colorable if $G$ is different from $K_{2n+1,2n+1}$ for all $n\ge 1$. In this paper, we give necessary conditions for a graph $G$ (not necessarily connected) with $\Delta \left(G\right)\ge \chi \left(G\right)$ to be equitably $\Delta \left(G\right)$-colorable and prove that those necessary conditions are also sufficient conditions when $G$ is a bipartite graph, or $G$ satisfies $\Delta \left(G\right)\ge \frac{\left|V\left(G\right)\right|}{3}+1$, or $G$ satisfies $\Delta \left(G\right)\le 3$.     

The Bohnenblust–Spitzer algorithm and its applications

The familiar bijections between the representations of permutations as words and as products of cycles have a natural class of “data driven” extensions that permit us to use purely combinatorial means to obtain precise probabilistic information about the geometry of random walks. In particular, we show that the algorithmic bijection of Bohnenblust and Spitzer can be used to obtain means, variances, and concentration inequalities for several random variables associated with a random walk including the number of vertices and length of the convex minorant, concave majorant, and convex hull.     

A classification of some regular p-groups and its applications

In this paper we classify regular p-groups with type invariants (e, 1,1,1) for e≥2 and (1,1,1,1,1). As a by-product, we give a new approach to the classification of groups of order p5, p ≥ 5 a prime.     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

The Hoffmann-Jørgensen inequality of NA random variables and its applications to the logarithm law

In this paper, the Hoffmann-Jørgensen inequality for negatively associated (NA) random variables is derived. As an important tool, it will be applied to the establishing for the logarithm law of NA arrays, and the results of Su, Hu and Liang {xc[15]} are extended.     

A predual of a predual of B_σ and its applications to commutators

A predual of B_σ-spaces is investigated. A predual of a predual of B_σ-spaces is also investigated,which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation(BMO) and the singular integral operators.What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of B_σ-spaces and B_σ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to B_σ-spaces.     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

The Bakry–Emery Ricci tensor and its applications to some compactness theorems

Let (M, g) be a complete and connected Riemannian manifold of dimension n. By using the Bakry–Emery Ricci curvature tensor on M, we prove two theorems which correspond to the Myers compactness theorem.     

The Γ-algorithm and some applications

In this paper the power of the Γ-algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the dual cone when new generators are incorporated. This algorithm, which is based on the duality concept, allows one to solve many problems in linear algebra, such as determining whether or not a vector belongs to a cone, obtaining the minimal representations of a cone in terms of a linear space and an acute cone, obtaining the intersection of two cones, discussing the compatibility of linear systems of inequalities, solving systems of linear inequalities, etc. The applications are illustrated with examples.     

Generalized Cameron–Storvick theorem and its applications

The goal of this paper is to establish a more generalized Cameron–Storvick theorem with respect to the conditional generalized integral transform (CGIT) and the generalized first variation. Many results and formulas for Cameron–Storvick theorems established in previous papers are included. We first introduce a CGIT and a generalized first variation of functionals by using the concept of the Gaussian process and bounded linear operators. We then give the existence of them for the exponential functionals on function space. We next establish a generalized Cameron–Storvick theorem. Finally, we describe some applications.     

Convergence of scaled vector fields and local approximation theorem on Carnot-Carathéodory spaces and applications

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