On derivatives and norms of generalized matrix functions and respective symmetric powers

In recent papers, the authors obtained formulas for directional derivatives of all orders, of the immanant and of the m-th xi-symmetric tensor power of an operator and a matrix, when xi is a character of the full symmetric group. The operator norm of these derivatives was also calculated. In this paper, similar results are established for generalized matrix functions and for every symmetric tensor power.

Analytic approximation of matrix functions in Lp

"Vegeu el resum a l'inici del document del fitxer adjunt."

QRT: A Qr-based tridiagonalization algorithm for nonsymmetric matrices

The stable similarity reduction of a nonsymmetric square matrix to tridiagonal form has been a long-standing problem in numerical linear algebra. The biorthogonal Lanczos process is in principle a candidate method for this task, but in practice it is confined to sparse matrices and is restarted periodically because roundoff errors affect its three-term recurrence scheme and degrade the biorthogonality after a few steps. This adds to its vulnerability to serious breakdowns or near-breakdowns, the handling of which involves recovery strategies such as the look-ahead technique, which needs a careful implementation to produce a block-tridiagonal form with unpredictable block sizes. Other candidate methods, geared generally towards full matrices, rely on elementary similarity transformations that are prone to numerical instabilities. Such concomitant difficulties have hampered finding a satisfactory solution to the problem for either sparse or full matrices. This study focuses primarily on full matrices. After outlining earlier tridiagonalization algorithms from within a general framework, we present a new elimination technique combining orthogonal similarity transformations that are stable. We also discuss heuristics to circumvent breakdowns. Applications of this study include eigenvalue calculation and the approximation of matrix functions.

Functions of the extracellular matrix and matrix degrading proteases during tumor progression

Cell interactions with extracellular matrices are important to pathological changes that occur during cell transformation and tumorigenesis. Several extracellular matrix proteins including fibronectin, thrombospondin-1, laminin, SPARC, and osteopontin have been suggested to modulate tumor phenotype by affecting cell migration, survival, or angiogenesis. Likewise, proteases including the matrix metalloproteinases (MMPs) are understood to not only facilitate migration of cells by degradation of matrices, but also to affect tumor formation and growth. We have recently demonstrated an in vivo role for the RGD-containing protein, osteopontin, during tumor progression, and found evidence for distinct functions in the host versus the tumor cells. Because of the compartmentalization and temporal regulation of MMP expression, it is likely that MMPs may also function dually in host stroma and the tumor cell. In addition, an important function of proteases appears to be not only degradation, but also cleavage of matrix proteins to generate functionally distinct fragments based on receptor binding, biological activity, or regulation of growth factors.

Putative functions of extracellular matrix glycoproteins in secondary palate morphogenesis

Cleft palate is a common birth defect in humans. Elevation and fusion of paired palatal shelves are coordinated by growth and transcription factors, and mutations in these can cause malformations. Among the effector genes for growth factor signaling are extracellular matrix (ECM) glycoproteins. These provide substrates for cell adhesion (e.g., fibronectin, tenascins), but also regulate growth factor availability (e.g., fibrillins). Cleft palate in Bmp7 null mouse embryos is caused by a delay in palatal shelf elevation. In contrast, palatal shelves of Tgf-β3 knockout mice elevate normally, but a cleft develops due to their failure to fuse. However, nothing is known about a possible functional interaction between specific ECM proteins and Tgf-β/Bmp family members in palatogenesis. To start addressing this question, we studied the mRNA and protein distribution of relevant ECM components during secondary palate development, and compared it to growth factor expression in wildtypewild type and mutant mice. We found that fibrillin-2 (but not fibrillin-1) mRNA appeared in the mesenchyme of elevated palatal shelves adjacent to the midline epithelial cells, which were positive for Tgf-β3 mRNA. Moreover, midline epithelial cells started expressing fibronectin upon contact of the two palatal shelves. These findings support the hypothesis that fibrillin-2 and fibronectin are involved in regulating the activity of Tgf-β3 at the fusing midline. In addition, we observed that tenascin-W (but not tenascin-C) was misexpressed in palatal shelves of Bmp7-deficient mouse embryos. In contrast to tenascin-C, tenascin-W secretion was strongly induced by Bmp7 in embryonic cranial fibroblasts in vitro. These results are consistent with a putative function for tenascin-W as a target of Bmp7 signaling during palate elevation. Our results indicate that distinct ECM proteins are important for morphogenesis of the secondary palate, both as downstream effectors and as regulators of Tgf-β/Bmp activity.

LOCALIZATION OF DNA REPAIR FUNCTIONS TO THE NUCLEAR MATRIX

The nucleus of a eukaryotic cell contains both structural and functional elements that contribute to the controlled operation of the cell. In this context, functional components refers to those nuclear constituents that perform metabolic activities such as DNA replication and RNA transcription. Structural nuclear components, designated nuclear matrix, organize the DNA into loops or domains and appear to provide a framework for nuclear DNA organization. However, the boundary between structural and functional components is not clear cut as evinced by reports of associations between metabolic functions and the nuclear matrix. The studies reported here attempt to determine the relationship of another nuclear function, DNA repair, to the nuclear matrix.^ One objective of these studies was to study the initiation of DNA repair by directly measuring the UV-incision activities in human cells and determine the influence of various extractable nuclear components on these activities. The assay for incision activities required the development of a nuclear isolation protocol that produced nuclei with intact DNA; the conformation of the nuclear DNA and its physical characteristics in response to denaturing conditions were determined.^ The nuclei produced with this protocol were then used as substrates for endogenous UV-specific nuclease activities. The isolated nuclei were shown to contain activities that cause breaks in nuclear DNA in response to UV-irradiation. These UV-responsive activities were tightly associated with nuclear components, being unextractable with salt concentration of up to 0.6 M.^ The tight association of the incision activities with salt-extracted nuclei suggested that other repair function might also be associated with salt-stable components of the nucleus. The site of unscheduled DNA synthesis (UDS) was determined in salt-extracted nuclei (nucleoids) using autoradiography and fluorescent microscopy. UDS was found to occur in association with the nuclear matrix following low-doses (2.55 J/M('2)) of ultraviolet light, but the association became looser after higher doses of ultraviolet light (10-30 J/m('2)). ^

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.

Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Fuzzy pole placement based on piecewise Lyapunov functions

This paper presents a controller design method for fuzzy dynamic systems based on piecewise Lyapunov functions with constraints on the closed-loop pole location. The main idea is to use switched controllers to locate the poles of the system to obtain a satisfactory transient response. It is shown that the global fuzzy system satisfies the requirements for the design and that the control law can be obtained by solving a set of linear matrix inequalities, which can be efficiently solved with commercially available softwares. An example is given to illustrate the application of the proposed method. Copyright (C) 2009 John Wiley & Sons, Ltd.

Bias-corrected Pearson estimating functions for Taylor`s power law applied to benthic macrofauna data

Estimation of Taylor`s power law for species abundance data may be performed by linear regression of the log empirical variances on the log means, but this method suffers from a problem of bias for sparse data. We show that the bias may be reduced by using a bias-corrected Pearson estimating function. Furthermore, we investigate a more general regression model allowing for site-specific covariates. This method may be efficiently implemented using a Newton scoring algorithm, with standard errors calculated from the inverse Godambe information matrix. The method is applied to a set of biomass data for benthic macrofauna from two Danish estuaries. (C) 2011 Elsevier B.V. All rights reserved.

Dissolution parameters for sodium diclofenac-containing hypromellose matrix tablet

Sodium diclofenac (SD) release from dosage forms has been studied under different conditions. However, no dissolution method that is discriminatory enough to reflect slight changes in formulation or manufacturing process, and which could be effectively correlated with the biological properties of the dosage form, has been reported. This study sought to develop three different formulae of SD-containing matrix tablets and to determine the effect of agitation speed in its dissolution profiles. F1, F2 and F3 formulations were developed using hypromellose (10, 20 and 30%, respectively for F1, F2 and F3) and other conventional excipients. Dissolution tests were carried out in phosphate buffer pH 6.8 at 37 degrees C using apparatus 11 at 50, 75 or 100 rpm. Dissolution efficiency (DE), T(50) and T(90) were determined and plotted as functions of the variables agitation speed and hypromellose concentration. Regarding DE, F2 showed more sensitivity to variations in agitation speed than F1 and F3. Increasing hypromellose concentration reduced DE values, independent of agitation speed. Analysis of T(50) and T(90) suggests that F1 is less sensitive to variations in agitation speed than F2 and F3. Most discriminatory dissolution conditions were observed at 50 rpm. Results suggest that the comparison of dissolution performance of SD matrix tablets should take into account polymer concentration and agitation conditions. (C) 2009 Published by Elsevier B.V.

Extracellular matrix components and regulators in the airway smooth muscle in asthma

There is an intimate relationship between the extracellular matrix (ECM) and smooth muscle cells within the airways. Few studies have comprehensively assessed the composition of different ECM components and its regulators within the airway smooth muscle (ASM) in asthma. With the aid of image analysis, the fractional areas of total collagen and elastic fibres were quantified within the ASM of 35 subjects with fatal asthma (FA) and compared with 10 nonfatal asthma (NFA) patients and 22 nonasthmatic control cases. Expression of collagen I and III, fibronectin, versican, matrix metalloproteinase (MMP)-1, -2, -9 and -12 and tissue inhibitor of metalloproteinase-1 and -2 was quantified within the ASM in 22 FA and 10 control cases. In the large airways of FA cases, the fractional area of elastic fibres within the ASM was increased compared with NFA and controls. Similarly, fibronectin, MMP-9 and MMP-12 were increased within the ASM in large airways of FA cases compared with controls. Elastic fibres were increased in small airways in FA only in comparison with NFA cases. There is altered extracellular matrix composition and a degradative environment within the airway smooth muscle in fatal asthma patients, which may have important consequences for the mechanical and synthetic functions of airway smooth muscle.