Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices. Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both - systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed. A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix. (c) 2007 Elsevier Inc. All rights reserved.

Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.

Approximate factorization constraint preconditioners for saddle-point matrices

We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.

Coriolis interactions about x-y axes in symmetric tops

Selection rules and matrix elements are derived for Coriolis interactions between vibrational levels due to rotation about (x, y) axes in symmetric top molecules. The theory is developed in detail for the case of interaction between an A1 and an E species vibrational level in a C3v molecule; perturbations to both the positions and the intensities of the rovibration transitions in the spectrum are considered. A computer program has been written which calculates exactly the perturbed spectrum of two interacting rovibration bands according to this model, the results being presented directly by a graph plotter connected to the computer. This has been used to interpret perturbations observed in two pairs of interacting fundamentals in the spectrum of CH3F (ν2 - ν5 and ν3 - ν6) and one pair in CD3Cl (ν2 - ν5). The resulting analysis of the observed spectrum leads to new values for some vibration-rotation interaction constants and also leads to a unique determination of the sign relationship between the dipole moment derivatives in each pair of interacting normal vibrations. These sign relations are summarized in Figs. 8, 12, and 15.

Selection rules for rovibronic transitions in symmetric top molecules

A simple diagrammatic rule is presented for determining the rotational selection rules governing transitions between any pair of vibronic states in electric dipole spectra of symmetric top molecules. The rule is useful in cases where degenerate vibronic levels with first-order Coriolis splittings occur, because it gives immediately the selection rule for the (+l) and (-l) components in any degenerate state. The rule is also helpful in determining the symmetry species and the effective zeta constants in overtone and combination levels involving degenerate vibrations. Particular attention is devoted to the conventions concerning the signs of zeta constants.

Raman selection rules for vibration-rotation transitions in symmetric top molecules

Symmetry restrictions on Raman selection rules can be obtained, quite generally, by considering a Raman allowed transition as the result of two successive dipole allowed transitions, and imposing the usual symmetry restrictions on the dipole transitions. This leads to the same results as the more familiar polarizability theory, but the vibration-rotation selection rules are easier to obtain by this argument. The selection rules for symmetric top molecules involving the (+l) and (-l) components of a degenerate vibrational level with first-order Coriolis splitting are derived in this paper. It is shown that these selection rules depend on the order of the highest-fold symmetry axis Cn, being different for molecules with n=3, n=4, or n ≧ 5; moreover the selection rules are different again for molecules belonging to the point groups Dnd with n even, and Sm with 1/2m even, for which the highest-fold symmetry axes Cn and Sm are related by m=2n. Finally it is shown that an apparent anomaly between the observed Raman and infra-red vibration-rotation spectra of the allene molecule is resolved when the correct selection rules are used, and a value for the A rotational constant of allene is derived without making use of the zeta sum rule.

Introduction of 4-chloro-alpha-cyanocinnamic acid liquid matrices for high sensitivity UV-MALDI MS

Matrix-assisted laser desorption/ionization (MALDI) is a key ionization technique in mass spectrometry (MS) for the analysis of labile macromolecules. An important area of study and improvements in relation to MALDI and its application in high-sensitivity MS is that of matrix design and sample preparation. Recently, 4-chloro-alpha-cyanocinnamic acid (ClCCA) has been introduced as a new rationally designed matrix and reported to provide an improved analytical performance as demonstrated by an increase in sequence coverage of protein digests obtained by peptide mass mapping (PMM) (Jaskolla, T. W.; et al. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 12200-12205). This new matrix shows the potential to be a superior alternative to the commonly used and highly successful alpha-cyano-4-hydroxycinnamic acid (CHCA). We have taken this design one step further by developing and optimizing an ionic liquid matrix (ILM) and liquid support matrix (LSM) using ClCCA as the principle chromophore and MALDI matrix compound. These new liquid matrices possess greater sample homogeneity and a simpler morphology. The data obtained from our studies show improved sequence coverage for BSA digests compared to the traditional CHCA crystalline matrix and for the ClCCA-containing ILM a similar performance to the ClCCA crystalline matrix down to 1 fmol of BSA digest prepared in a single MALDI sample droplet with current sensitivity levels in the attomole range. The LSMs show a high tolerance to contamination such as ammonium bicarbonate, a commonly used buffering agent.

Small towns as 'sub-poles' in English rural development: Investigating rural-urban linkages using sub-regional social accounting matrices

This paper uses spatial economic data from four small English towns to measure the strength of economic integration between town and hinterland and to estimate the magnitude of town-hinterland spill-over effects. Following estimation of local integration indicators and inter-locale flows, sub-regional social accounting matrices (SAMs) are developed to estimate the strength of local employment and output multipliers for various economic sectors. The potential value of a town as a 'sub-pole' in local economic development is shown to be dependent on structural differences in the local economy, such as the particular mix of firms within towns. Although the multipliers are generally small, indicating a low level of local linkages, some sectors, particularly financial services and banking, show consistently higher multipliers for both output and employment. (c) 2007 Elsevier Ltd. All rights reserved.