The trigonometric Grassmannian and a difference W-algebra

We introduce a W-algebra which is a central extension of the Lie algebra of difference operators with rational coefficients acting on functions of a discrete variable. We construct its natural fermionic and bosonic representations. We define a module over this difference W-algebra, which characterizes the trigonometric Calogero–Moser spaces.     

The X-Ray Transform for a Generic Family of Curves and Weights

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ^* M. Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.     

Extracting biological information with computational analysis of Fourier-transform infrared (FTIR) biospectroscopy datasets: current practices to future perspectives

Applying Fourier-transform infrared (FTIR) spectroscopy (or related technologies such as Raman spectroscopy) to biological questions (defined as biospectroscopy) is relatively novel. Potential fields of application include cytological, histological and microbial studies. This potentially provides a rapid and non-destructive approach to clinical diagnosis. Its increase in application is primarily a consequence of developing instrumentation along with computational techniques. In the coming decades, biospectroscopy is likely to become a common tool in the screening or diagnostic laboratory, or even in the general practitioner's clinic. Despite many advances in the biological application of FTIR spectroscopy, there remain challenges in sample preparation, instrumentation and data handling. We focus on the latter, where we identify in the reviewed literature, the existence of four main study goals: Pattern Finding; Biomarker Identification; Imaging; and, Diagnosis. These can be grouped into two frameworks: Exploratory; and, Diagnostic. Existing techniques in Quality Control, Pre-processing, Feature Extraction, Clustering, and Classification are critically reviewed. An aspect that is often visited is that of method choice. Based on the state-of-art, we claim that in the near future research should be focused on the challenges of dataset standardization; building information systems; development and validation of data analysis tools; and, technology transfer. A diagnostic case study using a real-world dataset is presented as an illustration. Many of the methods presented in this review are Machine Learning and Statistical techniques that are extendable to other forms of computer-based biomedical analysis, including mass spectrometry and magnetic resonance.     

Biomimetic synthesis, antibacterial activity and structure-activity properties of the pyroglutamate core of oxazolomycin

Biomimetic intramolecular aldol reactions on oxazolidine templates derived from serine may be used to generate densely functionalised pyroglutamates, which are simpler mimics of the right hand side of oxazolomycin. Some of the compounds from this sequence exhibit in vivo activity against S. aureus and E. coli, suggesting that pyroglutamate scaffolds may be useful templates for the development of novel antibacterials, and cheminformatic analysis has been used to provide some structure-activity data.     

Synthesis and photophysical properties of novel amphiphilic ruthenium (II) complexes containing 4,4′‐dialkylaminomethyl‐2,2′‐bipyridyl ligands

The synthesis of a number of new 2,2′‐bipyridine ligands functionalized with bulky amino side groups is reported. Three homoleptic polypyridyl ruthenium (II) complexes, [Ru(L)₃]²⁺ 2(PF₆^?), where L is 4,4′‐dioctylaminomethyl‐2,2′‐bipyridine (Ru4a), 4,4′‐didodecylaminomethyl‐2,2′‐bipyridine (Ru4b) and 4,4′‐dioctadodecylaminomethyl‐2,2′‐bipyridine (Ru4c), have been synthesized. These compounds were characterized and their photophysical properties examined. The electronic spectra of three complexes show pyridyl π → π* transitions in the UV region and metal‐to‐ligand charge transfer bands in the visible region. Copyright © 2005 John Wiley & Sons, Ltd.     

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and, therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.     

Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

For tri-diagonal matrices arising in the simplified Jaynes–Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.     

Discrete orthogonal polynomials and difference equations of several variables

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.