Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Accurate substrate modelling of RF CMOS

The losses within the substrate of an RF IC can have significant effect on performance in a mixed signal application. in order to model substrate coupling accurately, it is represented by an RC network to account for both resistive and dielectric losses at high frequency (> 1 GHz). A small-signal equivalent circuit model of an RF IC inclusive of substrate parasitic effect is analysed in terms of its y-parameters and an extraction procedure for substrate parameters has been developed. By coupling the extracted substrate parameters along with extrinsic resistances associated with gate, source and drain, a standard BSIM3 model has been extended for RF applications. The new model exhibits a significant improvement in prediction of output reflection coefficient S-22 in the frequency range from 1 to 10 GHz in device mode of operation and for a low noise amplifier (LNA) at 2.4 GHz. Copyright (C) 2006 John Wiley & Sons, Ltd.

K-Theory of non-linear projective toric varieties

We define a category of quasi-coherent sheaves of topological spaces on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.

Improved fault diagnosis in multivariate systems using regression-based reconstruction

Subspace monitoring has recently been proposed as a condition monitoring tool that requires considerably fewer variables to be analysed compared to dynamic principal component analysis (PCA). This paper analyses subspace monitoring in identifying and isolating fault conditions, which reveals that the existing work suffers from inherent limitations if complex fault senarios arise. Based on the assumption that the fault signature is deterministic while the monitored variables are stochastic, the paper introduces a regression-based reconstruction technique to overcome these limitations. The utility of the proposed fault identification and isolation method is shown using a simulation example and the analysis of experimental data from an industrial reactive distillation unit.

Linear quadratic and tumour control probability modelling in external beam radiotherapy

The standard linear-quadratic (LQ) survival model for external beam radiotherapy is reviewed with particular emphasis on studying how different schedules of radiation treatment planning may be affected by different tumour repopulation kinetics. The LQ model is further examined in the context of tumour control probability (TCP) models. The application of the Zaider and Minerbo non-Poissonian TCP model incorporating the effect of cellular repopulation is reviewed. In particular the recent development of a cell cycle model within the original Zaider and Minerbo TCP formalism is highlighted. Application of this TCP cell-cycle model in clinical treatment plans is explored and analysed.

Characterisation of fluidised bed granulation processes using in situ Raman spectroscopy

Previous work by the authors Walker et al. [2007b. Fluidised bed characterisation using Raman spectroscopy: applications to pharmaceutical processing. Chemical Engineering Science 62, 3832–3838] illustrated that Raman spectroscopy could be used to provide 3-D maps of the concentration and chemical structure of particles in motion in a fluidised bed, within a relatively short (120 s) time window. Moreover, we reported that the technique, as outlined, has the potential to give detailed in-situ information on how the structure and composition of granules/powders within the fluidised bed (dryer or granulator) vary with the position and evolve with time. In this study we extended the original work by shortening the time window of the Raman spectroscopic analysis to 10 s, which has allowed the in-situ real-time characterisation of a fluidised bed granulation process. Here we show an important new use of the technique which allows in-situ measurement of the composition of the material within the fluidised bed in three spatial dimensions and as a function of time. This is achieved by recording Raman spectra using a probe positioned within the fluidised bed on a long-travel x–y–z stage. In these experiments the absolute Raman intensity is used to provide a direct measure of the amount of any given material in the probed volume, i.e. a particle density. Particle density profiles have been calculated over the granulation time and show how the volume of the fluidised bed decreases with an increase mean granule size. The Raman spectroscopy analysis indicated that nucleation/coalescence in this co-melt fluidised hot melt granulation system occurred over a relatively short time frame (t<30 s). The Raman spectroscopic technique demonstrated accurate correlation with independent granulation experiments which provided particle size distribution analysis. The similarity of the data indicates that the Raman spectra accurately represent solids ratios within the bed, and thus the techniques quantitative capabilities for future use in the pharmaceutical industry.

SPECK-Based Lossless Multispectral Image Coding

This letter proposes an efficient extension of the set partitioning embedded block (SPECK) algorithm to lossless multispectral image coding. Such a wavelet-based coder is widely referred to in the literature, especially for lossless image coding, and is considered to be one of the most efficient techniques exhibiting very low computational complexity when compared with other state-of-the-art coders. The modification proposed in this letter is simple and provides significant improvement over the conventional SPECK. The key idea is to join each group of two consecutive wavelet-transformed spectral bands during the SPECK coding since they show high similarities with respect to insignificant sets at the same locations. Simulation results, carried out on a number of test images, demonstrate that this grouping procedure considerably saves on the bit budget for encoding the multispectral images.

sscMap: An extensible Java application for connecting small-molecule drugs using gene-expression signatures

Background
Connectivity mapping is a process to recognize novel pharmacological and toxicological properties in small molecules by comparing their gene expression signatures with others in a database. A simple and robust method for connectivity mapping with increased specificity and sensitivity was recently developed, and its utility demonstrated using experimentally derived gene signatures.

Results
This paper introduces sscMap (statistically significant connections' map), a Java application designed to undertake connectivity mapping tasks using the recently published method. The software is bundled with a default collection of reference gene-expression profiles based on the publicly available dataset from the Broad Institute Connectivity Map 02, which includes data from over 7000 Affymetrix microarrays, for over 1000 small-molecule compounds, and 6100 treatment instances in 5 human cell lines. In addition, the application allows users to add their custom collections of reference profiles and is applicable to a wide range of other 'omics technologies.

Conclusion
The utility of sscMap is two fold. First, it serves to make statistically significant connections between a user-supplied gene signature and the 6100 core reference profiles based on the Broad Institute expanded dataset. Second, it allows users to apply the same improved method to custom-built reference profiles which can be added to the database for future referencing. The software can be freely downloaded from http://purl.oclc.org/NET/sscMap