A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.     

A Homeomorphism Invariant for Substitution Tiling Spaces

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of ech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.     

Tolerance graphs

Tolerance graphs arise from the intersection of intervals with varying tolerances in a way that generalizes both interval graphs and permutation graphs. In this paper we prove that every tolerance graph is perfect by demonstrating that its complement is perfectly orderable. We show that a tolerance graph cannot contain a chordless cycle of length greater than or equal to 5 nor the complement of one. We also discuss the subclasses of bounded tolerance graphs, proper tolerance graphs, and unit tolerance graphs and present several possible applications and open questions.     

关于一类复合幂级数展开式的收敛性定理

本文研究了一类复合型幂级数展开式,证明了一个收敛性定理并举例说明其应用.在注记中指出了可进一步研究的问题.     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

Hub location–allocation in intermodal logistic networks

Within the context of intermodal logistics, the design of transportation networks becomes more complex than it is for single mode logistics. In an intermodal network, the respective modes are characterized by the transportation cost structure, modal connectivity, availability of transfer points and service time performance. These characteristics suggest the level of complexity involved in designing intermodal logistics networks. This research develops a mathematical model using the multiple-allocation p-hub median approach. The model encompasses the dynamics of individual modes of transportation through transportation costs, modal connectivity costs, and fixed location costs under service time requirements. A tabu search meta-heuristic is used to solve large size (100 node) problems. The solutions obtained using this meta-heuristic are compared with tight lower bounds developed using a Lagrangian relaxation approach. An experimental study evaluates the performance of the intermodal logistics networks and explores the effects and interactions of several factors on the design of intermodal hub networks subject to service time requirements.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.