The role of interfacial energy in zeolite synthesis

The thermodn. factors playing a role in the crystn. of Si-rich zeolites are discussed. The Ostwald ripening model for nucleation and the Ostwald law of successive transformations make it possible to describe the kinetic phenomena in terms of the relative stability of the intermediate phases. The contribution of the interfacial free energy between the large internal surface area in the micropores and the intracryst. liquor to the chem. potential of the zeolitic material plays a crucial role. A model for the elementary steps occurring in pentasil zeolite formation is propose

Quarnet Inference Rules for Level-1 Networks

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set X of species from a collection of trees, each having leaf-set some subset of X. In the 1980s, Colonius and Schulze gave certain inference rules for deciding when a collection of 4-leaved trees, one for each 4-element subset of X, can be simultaneously displayed by a single supertree with leaf-set X. Recently, it has become of interest to extend this and related results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has recently been shown that a certain type of phylogenetic network, called a (unrooted) level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here, we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of X, can all be simultaneously displayed by a level-1 network with leaf-set X. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of X from orderings on subsets of X of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics

An Algorithm for Packing Connectors

Given an undirected graph G = (V; E) and a partition fS; Tg of V , an S--T connector is a set of edges F ` E such that every component of the subgraph (V; F ) intersects both S and T . If either S or T is a singleton, then an S--T connector is a spanning tree of G. On the other hand, if G is bipartite with colour classes S and T , then an S--T connector is an edge cover of G (a set of edges covering all vertices). An S--T connector is a common spanning set of two graphic matroids on E. We prove a theorem on packing common spanning sets of certain matroids, generalizing a result of Davies and McDiarmid on strongly base orderable matroids. As a corollary, we obtain an O((n; m) + nm) time algorithm for finding a maximum number of S--T connectors, where (n; m) denotes the complexity of finding a maximum number of edge disjoint spanning trees in a graph on n vertices and m edges. Since the best known bound for (n; m) is O(nm log(m=n)), this bound for packing S--T connectors ..

A Note on Packing Connectors

Given an undirected graph G = (V; E) and a partition fS; Tg of V , an S-T connector is a set of edges F ` E such that every component of the subgraph (V; F ) intersects both S and T . We show that G has k edge-disjoint S-T connectors if and only if jffi G (V 1 ) [ : : : [ ffi G (V t )j kt for every collection fV 1 ; : : : ; V t g of disjoint nonempty subsets of S and for every such collection of subsets of T . This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph

An Efficient Algorithm for Minimum-Weight Bibranching

Given a directed graph D = (V; A) and a set S ` V , a bibranching is a set of arcs B ` A that contains a v--(V n S) path for every v 2 S and an S--v path for every v 2 V n S. In this paper, we describe a primal-dual algorithm that determines a minimum weight bibranching in a weighted digraph. It has running time O(n 0 (m + n log n)), where m = jAj, n = jV j and n 0 = minfjSj; jV n Sjg. Thus, our algorithm obtains the best known bounds for two important special cases of the problem: bipartite edge cover and r-branching