On the chromatic number of random d-regular graphs

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k−1)log(k−1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k−1 or k. If moreover d>(2k−3)log(k−1), then the value k−1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of equitable k-colourings, where a colouring is equitable if the number of vertices of each colour is equal.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Nondivergent elliptic equations on manifolds with nonnegative curvature

We consider a class of second-order linear elliptic operators, intrinsically defined on Riemannian manifolds, that correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From the Harnack inequality, we obtain Alexandroff-Bakelman-Pucci estimates and maximum principles for subsolutions. © 1997 John Wiley & Sons, Inc.     

Regularity of solutions to scalar conservation laws with a force

We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.     

Representing scientific activity: Affordances and constraints of central design and enactment features of a model‐based inquiry unit

This research explores how explaining an anchoring phenomena and engaging students in investigations, as central designs of a model‐based inquiry (MBI) unit, afforded or constrained the representation of scientific activity in the science classroom. This research is considered timely as recent standards documents and scholars in the field have highlighted the significance of identifying what features of scientific activity are important and how these can be represented for students in classrooms. Through taking advantage of qualitative research methods to closely examine the enactment of an MBI unit, both affordances and constraints were identified for each design. More specifically, explaining an anchoring phenomenon provided a context for more authentically framing the work of students, while investigations afforded students insight into the role these play in the refinement of models. Further, the teacher's attempts to support student reasoning and, at times, reasoning for students when they were found struggling were the most salient constraints identified connected to explaining an anchoring phenomenon and engaging students in investigations.     

Perfect matchings and Hamiltonian cycles in the preferential attachment model

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if , then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that . One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are “older” (ie, are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting—sometimes called uniform attachment—in which vertices are added one by one and each vertex connects to m older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.     

Tabu Search Based Procedure for Solving the 0-1 MultiObjective Knapsack Problem: The Two Objectives Case

We consider in this paper the solving of 0-1 knapsack problems with multiple linear objectives. We present a tabu search approach to generate a good approximation of the efficient set. The heuristic scheme is included in a redu tion decision space framework. The case of two objectives is developed in this paper. TS principles viewed into the multiobjective context are discussed. According to a prospective way, several variations of the algorithm are investigate. Numerical experiments are reported and compared with available exact efficient solutions. Intuitive justifications for the observed empirical behavior of the procedure and open questions are discussed.     

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.