Invariants of matrix pairs over discrete valuation rings and Littlewood-Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)?(PMQ^-1,QNT^-1) possesses a discrete invariant in the form of Littlewood-Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood-Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood-Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood-Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.     

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.     

Niche width and scale in organizational competition: A computational approach

In some organizational applications, the principle of allocation (PoA) and scale advantage (SA) oppose each other. While PoA implies that organizations with wide niches get punished, SA holds that large organizations gain an advantage because of scale efficiencies. The opposition occurs because many large organizations also possess wide niches. However, analyzing these theoretical mechanisms implies a possible trade-off between niche width and size: if both PoA and SA are strong, then organizations must be either focused or large to survive, resulting in a dual market structure, as proposed by the theory of resource partitioning. This article develops a computational model used to study this trade-off, and investigates the properties of organizational populations with low/high SA and low/high PoA. The model generates three expected core “corner” solutions: (1) the dominance of large organizations in the strong SA setting; (2) the proliferation of narrow-niche organizations in the strong PoA setting; and (3) a bifurcated or dual market structure if both SA and PoA are present. The model also allows us to identify circumstances under which narrow-niche (specialists) or wide-niche (generalists) organizations thrive. We also use the model to examine the claim that concentrated resource distributions are more likely to generate partitioned or bifurcated populations. We also investigate the consequences of environments comprised of ordered versus unordered positions.     

Technical Report: The Fastest Relativistic Jets from Quasars and Active Galactic Nuclei

This workshop was held to gather scientists interested in exploiting beamlines I06 and I10 of the Surface and Interfaces Village at Diamond Light Source from June 10–11, 2009. Sarnjeet Dhesi introduced the meeting with a short explanation of the village structure at Diamond. This village includes the Nanoscience beamline (I06), catering for soft X-rays for Photo-Emission Electron Microscopy (PEEM) and X-ray Magnetic Circular and Linear Dichroism (XMCD and XMLD), and the Beam Line for Advanced Dichroism Experiments (BLADE, beamline I10), which is a polarized soft X-ray beam for XMCD, XMLD, and soft X-ray diffraction. I06 has been operational for over two years, while I10 is scheduled to come on-line in late 2010. In addition, there are two surface science beamlines (I07 and I09) in the village dedicated to surface diffraction and X-ray standing waves.     

Asymptotic Sequences over a Module

Let (R, 𝔪) be a Noetherian local ring and M be a submodule of the free module F = R ^r with height(I _r (M)) > 0. Asymptotic sequences over M will be defined analogous to Rees’ definition of asymptotic sequences over an ideal. It is then shown that all maximal asymptotic sequences over M have the same length. This length gives a bound on the analytic spread of M. Namely, if s is the length of a maximal asymptotic sequence over M then l(M) ≤dim R + rank M ? 1 ? s. Equality holds if R is quasi-unmixed.     

Electron channelling contrast imaging of dislocations in a conventional SEM

Abstract

Dislocations in shock loaded tantalum single crystals were imaged using both transmission electron microscope (TEM) and electron channelling contrast image (ECCI) in a scanning electron microscope with a conventional backscattered electron detector. The results were compared with backscattered electron intensity profiles across dislocations calculated via the dynamic theory of electron diffraction. A one-to-one correspondence between ECCI and TEM is established. High voltage and low index reflections should be used to obtain the highest dislocation contrast and greatest imaging depth.     

Emission spectrum of a dressed exciton-biexciton complex in a semiconductor quantum dot

The photoluminescence spectrum of a single quantum dot was recorded as a secondary resonant laser optically dressed either the vacuum-to-exciton or the exciton-to-biexciton transitions. High-resolution polarization-resolved measurements using a scanning Fabry-Pérot interferometer reveal splittings of the linearly polarized fine-structure states that are nondegenerate in an asymmetric quantum dot. These splittings manifest as either triplets or doublets and depend sensitively on laser intensity and detuning. Our approach realizes complete resonant control of a multiexcitonic system in emission, which can be either pulsed or continuous wave, and offers direct access to the emitted photons.     

Facilitating active learning within green chemistry

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Studies on the stereoselective synthesis of C-allyl glycosides

An investigation was carried out to explore the use of sulfoxide donors as common precursors to stereoisomeric C-glycoconjugates of glycoprotein and glycolipid tumor antigens. A study focusing on the effects of reaction conditions and substrate structure on the stereoselectivity of allylation was carried out. Although conditions were realized to selectively afford alpha-allylation products in good to excellent yields, the search for conditions favoring beta-selectivity proved less successful.