Languages recognized by programs over the two-element monoid

Communicated by N. Reilly     

On the stability of a family of finite element methods for hyperbolic problems

We consider a family of tensor product finite element methods for hyperbolic equations in , , which are explicit and generate a continuous approximate solution. The base case (an extension of the box scheme to higher order) is due to Winther, who proved stability and optimal order convergence. By means of a simple counterexample, we show that, for linear approximation with , the corresponding methods are unstable.

     

The Flooding Time in Random Graphs

Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class G _p (N) of random graphs, the corresponding flooding time is investigated. The flooding time T _N (p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the flooding time T _N (p) converges in distribution to the two-fold convolution ^(2*) of the Gumbel distribution function (z)=exp (–e ^–z ), when the link density p _N satisfies Np _N /(log N)³ if N .     

On the covariance of the level sizes in random recursive trees

In this paper we study the covariance structure of the number of nodes k and l steps away from the root in random recursive trees. We give an analytic expression valid for all k, l and tree sizes N. The fraction of nodes k steps away from the root is a random probability distribution in k. The expression for the covariances allows us to show that the total variation distance between this (random) probability distribution and its mean converges in probability to zero. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 519–539, 2002     

On H-Linked Graphs

For a fixed multigraph H, possibly containing loops, with V(H)={h₁, . . . , h_k}, we say a graph G is H-linked if for every choice of k vertices v₁, . . . , v_k in G, there exists a subdivision of H in G such that v_i represents h_i (for all i). This notion clearly generalizes the concept of k-linked graphs (as well as other properties). In this paper we determine, for a connected multigraph H and for any sufficiently large graph G, a sharp lower bound on δ(G) (depending upon H) such that G is H-linked.     

Iterative patching and the asymmetric traveling salesman problem

Although Branch-and-Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp–Steele procedure are the fastest, and that ‘iterative’ Modified Karp–Steele patching generates the smallest search trees.     

Further advances on low-energy lunar impact dynamics

We extend the analysis, started in a previous work [1], concerning the formation of lunar impact craters due to low-energy trajectories. First, we adopt the Circular Restricted Three-Body Problem and consider different choices of initial conditions inside the stable invariant manifold associated with the central invariant one in the neighborhood of the L₂ equilibrium point in the Earth-Moon system. Then we move to the Bicircular Restricted Four-Body Problem to study the effect of the Sun on the distribution of impacts on the Moon’s surface.     

Parallel connections over symmetric spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.     

Practical strategies for generating rank-1 split cuts in mixed-integer linear programming

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of these variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. An extensive computational evaluation of these cuts points to the following two conclusions. The first is that our rank-1 cuts improve significantly on existing split cut generators (Gomory cuts from single tableau rows, MIR, Reduce-and-Split, Lift-and-Project, Flow and Knapsack cover): on MIPLIB instances, these generators close 24% of the integrality gap on average; adding our cuts yields an additional 5%. The second conclusion is that, when incorporated in a Branch-and-Cut framework, these new cuts can improve computing time on difficult instances.