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.     

The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph

The minimal weight of a spanning tree in a complete graph K_n with independent, uniformly distributed random weights on the edges is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph.     

Average-case results on heapsort

A new algorithm for rearranging a heap is presented and analysed in the average case. The average case upper bound for deleting the maximum element of a random heap is improved, and is shown to be less than [logn]+0.299+M(n) comparisons, *) whereM(n) is between 0 and 1. It is also shown that a heap can be constructed using 1.650n+O(logn) comparisons with this algorithm, the best result for any algorithm which does not use any extra space. The expected time to sortn elements is argued to be less thann logn+0.670n+O(logn), while simulation result points at an average case ofn log n+0.4n which will make it the fastest in-place sorting algorithm. The same technique is used to show that the average number of comparisons when deleting the maximum element of a heap using Williams' algorithm for rearrangement is 2([logn]–1.299+L(n)) whereL(n) also is between 0 and 1, and the average cost for Floyd-Williams Heapsort is at least 2nlogn–3.27n, counting only comparisons. An analysis of the number of interchanges when deleting the maximum element of a random heap, which is the same for both algorithms, is also presented.     

Long and short paths in uniform random recursive dags

In a uniform random recursive k-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S _n is the shortest path distance from node n to the root, then we determine the constant σ such that S _n /log n→σ in probability as n→∞. We also show that max _1≤i≤n S _i /log n→σ in probability.     

Ionic bonding in free nanoscale NaCl clusters as seen by photoelectron spectroscopy

The free neutral nanoscale NaCl clusters have been produced in a beam and studied with x-ray photoelectron spectroscopy. High resolution spectra simultaneously containing cluster and molecular-monomer, featuring in both the valence and core-level Na 2p and Cl 2p regions, have been obtained. Cluster-level energy shifts of around 3 eV toward lower binding energy for Na 2p and ≈1 eV toward higher binding energy for Cl 2p relative to the monomer levels have been unambiguously established. To rationalize the core-level energy shifts of the nanoscale NaCl clusters, the ionic model taking into account all charge-charge and polarization interactions has been developed and implemented. A satisfactory agreement between the experimental and model results has been obtained. The model calculations have also shed additional light on the size- and site-specific cluster responses.     

Duality in inhomogeneous random graphs,and the cut metric

The classical random graph model G(n, c/n) satisfies a “duality principle”, in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 399–411, 2011     

Correlations for paths in random orientations of G(n,p) and G(n,m)

We study random graphs, both G( n,p) and G( n,m), with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combine probability space, of the events $\{a\to s\}$ and $\{s\to b\}$ . For G(n,p), we prove that there is a $pc = 1/2$ such that for a fixed $p < pc$ the correlation is negative for large enough n and for $p > pc$ the correlation is positive for large enough n. We conjecture that for a fixed $n \ge 27$ the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/({{n}\atop {2}})$ , there is a critical pc that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any $\ell$ directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute \input amssym $\Bbb{P}(a\to s)$ and \input amssym $\Bbb{P}(a\to s, s\to b)$ . We also briefly discuss the corresponding question in the quenched version of the problem. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Generalized Stirling permutations, families of increasing trees and urn models

Bóna (2007) [6] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley (1978) [13]. Recently, Janson (2008) [17] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bóna. Here we will consider generalized Stirling permutations extending the earlier results of Bóna (2007) [6] and Janson (2008) [17], and relate them with certain families of generalized plane recursive trees, and also (k+1)-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Pólya urn models using various methods.     

Advanced Research Using New X-ray Sources

The Nordic summer school in synchrotron radiation science, Advanced Research Using New X-ray Sources, was held from June 13–21, 2011 in Sweden. The school was organized by MAX-lab with economic support from the Nordic Academy for Advanced Studies (NorFA) and the Swedish German Röntgen-Ångström consortium. The school was divided into two parts. The first lecture part was held at Odengården, a cosy conference facility in the countryside of Skåne, Sweden's most southern province, followed by practical exercises at the MAX-lab synchrotron radiation laboratory.