A Wiener-Hopf type factorization for the exponential functional of Levy processes

For a Lévy process ξ=(ξt)t≥0 drifting to −∞, we define the so-called exponential functional as follows: Formula Under mild conditions on ξ, we show that the following factorization of exponential functionals: Formula holds, where × stands for the product of independent random variables, H− is the descending ladder height process of ξ and Y is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of Iξ for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein–Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.

Spectra of a class of non-self-adjoint matrices

We consider a new class of non-self-adjoint matrices that arise from an indefinite self- adjoint linear pencil of matrices, and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity. We prove that the spectrum is qualitatively different when a certain parameter c equals 0, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of c.

Diagonalization and Jordan Normal Form – motivation through Maple

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package Maple®.

Large vibrational variational calculations using 'multimode' and an iterative diagonalization technique

We have favoured the variational (secular equation) method for the determination of the (ro-) vibrational energy levels of polyatomic molecules. We use predominantly the Watson Hamiltonian in normal coordinates and an associated given potential in the variational code 'Multimode'. The dominant cost is the construction and diagonalization of matrices of ever-increasing size. Here we address this problem, using pertubation theory to select dominant expansion terms within the Davidson-Liu iterative diagonalization method. Our chosen example is the twelve-mode molecule methanol, for which we have an ab initio representation of the potential which includes the internal rotational motion of the OH group relative to CH3. Our new algorithm allows us to obtain converged energy levels for matrices of dimensions in excess of 100 000.

Representing geometrical knowledge

This paper introduces perspex algebra which is being developed as a common representation of geometrical knowledge. A perspex can currently be interpreted in one of four ways. First, the algebraic perspex is a generalization of matrices, it provides the most general representation for all of the interpretations of a perspex. The algebraic perspex can be used to describe arbitrary sets of coordinates. The remaining three interpretations of the perspex are all related to square matrices and operate in a Euclidean model of projective space-time, called perspex space. Perspex space differs from the usual Euclidean model of projective space in that it contains the point at nullity. It is argued that the point at nullity is necessary for a consistent account of perspective in top-down vision. Second, the geometric perspex is a simplex in perspex space. It can be used as a primitive building block for shapes, or as a way of recording landmarks on shapes. Third, the transformational perspex describes linear transformations in perspex space that provide the affine and perspective transformations in space-time. It can be used to match a prototype shape to an image, even in so called 'accidental' views where the depth of an object disappears from view, or an object stays in the same place across time. Fourth, the parametric perspex describes the geometric and transformational perspexes in terms of parameters that are related to everyday English descriptions. The parametric perspex can be used to obtain both continuous and categorical perception of objects. The paper ends with a discussion of issues related to using a perspex to describe logic.

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.

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.

Reactions of hydroxyl radicals with trichloroethene and tetrachloroethene in argon matrices at 12 K

Irradiation of argon matrices at 12 K containing hydrogen peroxide and tetrachloroethene using the output from a medium-pressure mercury lamp gives rise to the carbonyl compound trichloroacetyl chloride (CCl3CClO). Similarly trichloroethene gives dichloroacetyl chloride ( CCl2HCClO) - predominantly in the gauche form - under the same conditions. It appears that the reaction is initiated by homolysis of the O-O bond of H2O2 to give OH radicals, one of which adds to the double bond of an alkene molecule. The reaction then proceeds by abstraction of the H atom of the hydroxyl group and Cl-atom migration. This mechanism has been explored by the use of DFT calculations to back up the experimental findings. The mechanism is analogous to that shown by the simple hydrocarbon alkenes.

On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

Recommendations for the viability assessment of probiotics as concentrated cultures and in food matrices

Due to the fact that probiotic cells need to be alive when they are consumed, culture-based analysis (plate count) is critical in ascertaining the quality (numbers of viable cells) of probiotic products. Since probiotic cells are typically stressed, due to various factors related to their production, processing and formulation, the standard methodology for total plate counts tends to underestimate the cell numbers of these products. Furthermore, products such as microencapsulated cultures require modifications in the release and sampling procedure in order to correctly estimate viable counts. This review examines the enumeration of probiotic bacteria in the following commercial products: powders, microencapsulated cultures, frozen concentrates, capsules, foods and beverages. The parameters which are specifically examined include: sample preparation (rehydration, thawing), dilutions (homogenization, media) and plating (media, incubation) procedures. Recommendations are provided for each of these analytical steps to improve the accuracy of the analysis. Although the recommendations specifically target the analysis of probiotics, many will apply to the analysis of commercial lactic starter cultures used in food fermentations as well.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .