The electronic structure of free aluminum clusters: metallicity and plasmons

The electronic structure of free aluminum clusters with ～3-4 nm radius has been investigated using synchrotron radiation-based photoelectron and Auger electron spectroscopy. A beam of free clusters has been produced using a gas-aggregation source. The 2p core level and the valence band have been probed. Photoelectron energy-loss features corresponding to both bulk and surface plasmon excitation following photoionization of the 2p level have been observed, and the excitation energies have been derived. In contrast to some expectations, the loss features have been detected at energies very close to those of the macroscopic solid. The results are discussed from the point of view of metallic properties in nanoparticles with a finite number of constituent atoms.     

Double core hole creation and subsequent Auger decay in NH3 and CH4 molecules

Energies of the hollow molecules CH(4)(2+) and NH(3)(2+) with double vacancies in the 1s shells have been measured using an efficient coincidence technique combined with synchrotron radiation. The energies of these states have been determined accurately by high level electronic structure calculations and can be well understood on the basis of a simple theoretical model. Their major decay pathway, successive Auger emissions, leads first to a new form of triply charged ion with a core hole and two valence vacancies; experimental evidence for such a state is presented with its theoretical interpretation. Preedge 2-hole-1-particle (2h-1p) states at energies below the double core-hole states are located in the same experiments and their decay pathways are also identified.     

A regular decomposition of the edge-product space of phylogenetic trees

We investigate the topology and combinatorics of a topological space called the edge-product space that is generated by the set of edge-weighted finite labelled trees. This space arises by multiplying the weights of edges on paths in trees, and is closely connected to tree-indexed Markov processes in molecular evolutionary biology. In particular, by considering combinatorial properties of the Tuffley poset of labelled forests, we show that the edge-product space has a regular cell decomposition with face poset equal to the Tuffley poset.     

Random cutting and records in deterministic and random trees

We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random and the case when we condition on the tree. The proofs are based on Aldous' theory of the continuum random tree. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Parity Splits by Triple Point Distances in X-Trees

At the conference Dress defined parity split maps by triple point distance and asked for a characterisation of such maps coming from binary phylogenetic X-trees. This article gives an answer to that question. The characterisation for X-trees can be easily described as follows: If all restrictions of a split map to sets of five or fewer elements is a parity split map for an X-tree, then so is the entire map. To ensure that the parity split map comes from an X-tree which is binary and phylogenetic, we add two more technical conditions also based on studying at most five points at a time. Received August 27, 2004     

Dense Packing of Patterns in a Permutation

We study the length L _k of the shortest permutation containing all patterns of length k. We establish the bounds e ⁻² k ² < L _k ≤ (2/3 + o(1))k ². We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k ² containing almost all patterns of length k. Received January 2, 2007     

Patterns in random permutations avoiding the pattern 321

We consider a random permutation drawn from the set of 321 ‐avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by n^{m + ?} where m is the length of σ and ? is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.     

Pattern Avoidance in Alternating Sign Matrices

We generalize the definition of a pattern from permutations to alternating sign matrices. The number of alternating sign matrices avoiding 132 is proved to be counted by the large Schr?der numbers, 1, 2, 6, 22, 90, 394, .... We give a bijection between 132-avoiding alternating sign matrices and Schr?der paths, which gives a refined enumeration. We also show that the 132-, 123-avoiding alternating sign matrices are counted by every second Fibonacci number. Received January 2, 2007     

Statistical molecular design of peptoid libraries

Statistical experimental design provides an efficient approach for selecting the building blocks to span the structural space and increase the information content in a combinatorial library. A set of renin-inhibitors, hexapeptoids, is used to illustrate the approach. Multivariate quantitative structure-activity relationships (MQSARs) were developed relating renin inhibition to the peptoid sequences variation, parametrized by the z-scales. By using the information from the models, the number of building block sets could be reduced from six to three. Using a statistical molecular design (SMD) reduces the number of compounds from more than 100 000 down to 90. A second SMD was used for comparison, based on less prior knowledge. This gave a reduction from over 2 billion to 120 compounds.