An optimization problem concerning multiplicative functions

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0

An example in Beurling's theory of generalised primes

In this paper we prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes.

Gál-type GCD sums beyond the critical line

We prove that ∑k,ℓ=1N(nk,nℓ)2α(nknℓ)α≪N2−2α(logN)b(α) holds for arbitrary integers 1≤n1<⋯

Cooperative scattering by cold atoms

We have studied the interplay between disorder and cooperative scattering for the single scattering limit in the presence of a driving laser. Analytical results have been derived and we have observed cooperative scattering effects in a variety of experiments, ranging from thermal atoms in an optical dipole trap, atoms released from a dark MOT and atoms in a BEC, consistent with our theoretical predictions.

Beyond the average: Detecting global singular nodes from local features in complex networks

Deviations from the average can provide valuable insights about the organization of natural systems. The present article extends this important principle to the systematic identification and analysis of singular motifs in complex networks. Six measurements quantifying different and complementary features of the connectivity around each node of a network were calculated, and multivariate statistical methods applied to identify singular nodes. The potential of the presented concepts and methodology was illustrated with respect to different types of complex real-world networks, namely the US air transportation network, the protein-protein interactions of the yeast Saccharomyces cerevisiae and the Roget thesaurus networks. The obtained singular motifs possessed unique functional roles in the networks. Three classic theoretical network models were also investigated, with the Barabasi-Albert model resulting in singular motifs corresponding to hubs, confirming the potential of the approach. Interestingly, the number of different types of singular node motifs as well as the number of their instances were found to be considerably higher in the real-world networks than in any of the benchmark networks. Copyright (C) EPLA, 2009

Modification of radiation pressure due to cooperative scattering of light

Cooperative spontaneous emission of a single photon from a cloud of N atoms modifies substantially the radiation pressure exerted by a far-detuned laser beam exciting the atoms. On one hand, the force induced by photon absorption depends on the collective decay rate of the excited atomic state. On the other hand, directional spontaneous emission counteracts the recoil induced by the absorption. We derive an analytical expression for the radiation pressure in steady-state. For a smooth extended atomic distribution we show that the radiation pressure depends on the atom number via cooperative scattering and that, for certain atom numbers, it can be suppressed or enhanced. Cooperative scattering of light by extended atomic clouds can become important in the presence of quasi-resonant light and could be addressed in many cold atoms experiments.

Ca(2+)-mediated thiolysis of peptide-resin linkage as an option for preparing protected peptide thioesters

A mild new procedure for preparing protected peptide thioesters, based oil Ca(2+)-assisted thiolysis of peptide-Kaiser oxime resin (KOR) linkage, is described. Ac-Ile-Ser(Bzl)-Asp(OcHx)-SR (Ac: acetyl; Bzl: benzyl; cHx: cyclohexyl), model peptide, was readily released from the resin by incubating the peptide-KOR at 60 degrees C in mixtures of DMF with n-butanethiol [R = (CH(2))(3)CH(3)] or ethyl 3-mercaptopropionate [R = (CH(2))(2)COOCHCH(3)] containing Ca(CH(3)COO)(2). After serine and aspartic acid side-chain deprotection under acid conditions, Ac-Ile-Ser-Asp-S(CH(2))(2)COOCH(2)CH(3) was successfully obtained with good quality and high yield. This type of C-terminal modified peptide may act as an excellent acyl donor in peptide segment condensation by the thioester method, native chemical ligation and enzymatic methods. (c) 2008 Elsevier Ltd. All rights reserved.