Low-rank updates of matrix functions II: Rational Krylov methods

This work develops novel rational Krylov methods for updating a large-scale matrix function ƒ(A) when A is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence analysis is based on an exactness result that is connected to work by Bernstein and Van Loan on rank-one updates of rational matrix functions. We demonstrate the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method for the exponential function and Markov functions. Low-rank updates of the matrix sign function require additional attention; we develop and analyze a combination of our methods with a squaring trick for this purpose. A curious connection between such updates and existing rational Krylov subspace methods for Sylvester matrix equations is pointed out

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and its Exact Solutions

The analytic solutions of the one-dimensional Schroedinger equation for the trigonometric Rosen-Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications. Instead we here solve above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanic superpotential.Comment: 16 pages 7 figures 1 tabl

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Primary Dendrite Distribution and Disorder During Directional Solidification of Pb-Sb Alloys

Pb-2.2 wt pct Sb and Pb-5.8 wt pet Sb alloys have been directionally solidified from a single-crystal seed with its [100] orientation parallel to the growth direction, to examine the primary dendrite distribution and disorder of the dendrite arrays. The dendrite distribution and ordering have been investigated using analysis techniques such as the Gauss-amplitude fit to the frequency distribution of nearest and higher-order spacings, minimum spanning tree (MST), Voronoi polygon, and Fourier transform (FT) of the dendrite centers. Since the arrangement of dendrites is driven by the requirement to accommodate side-branch growth along the (100) directions, the FT images of the fully developed dendrite centers contain spots which indicate this preferred alignment. A directional solidification distance of about three mushy-zone lengths is sufficient to ensure a steady-state dendritic array, in terms of reaching a constant mean primary spacing. However, local dendrite ordering continues throughout the directional solidification process. The interdendritic convection not only decreases the mean primary spacing, it also makes the dendrite array more disordered and reduces the ratio of the upper and lower spacing limits, as defined by the largest 5 pct and the smallest 5 pct of the population

VEGF expression by mesenchymal stem cells contributes to angiogenesis in pancreatic carcinoma

Little is known about the factors that enable the mobilisation of human mesenchymal stem cells (MSC) from the bone marrow into the blood stream and their recruitment to and retention in the tumour. We found specific migration of MSC towards growth factors present in pancreatic tumours, such as PDGF, EGF, VEGF and specific inhibitors Glivec, Erbitux and Avastin interfered with migration. Within a few hours, MSC migrated into spheroids consisting of pancreatic cancer cells, fibroblasts and endothelial cells as measured by time-lapse microscopy. Supernatant from subconfluent MSC increased sprouting of HUVEC due to VEGF production by MSC itself as demonstrated by RT-PCR and ELISA. Only few MSCs were differentiated into endothelial cells in vitro, whereas in vivo differentiation was not observed. Lentiviral GFP-marked MSCs, injected in nude mice xenografted with orthotopic pancreatic tumours, preferentially migrated into the tumours as observed by FACS analysis of green fluorescent cells. By immunofluorescence and intravital microscopic studies, we found the interaction of MSC with the endothelium of blood vessels. Mesenchymal stem cells supported tumour angiogenesis in vivo, that is CD31+ vessel density was increased after the transfer of MSC compared with siVEGF-MSC. Our data demonstrate the migration of MSC toward tumour vessels and suggest a supportive role in angiogenesis

Phloroglucinol Inhibits the Bioactivities of Endothelial Progenitor Cells and Suppresses Tumor Angiogenesis in LLC-Tumor-Bearing Mice

Background: There is increasing evidence that phloroglucinol, a compound from Ecklonia cava, induces the apoptosis of cancer cells, eventually suppressing tumor angiogenesis. Methodology/Principal Findings: This is the first report on phloroglucinol’s ability to potentially inhibit the functional bioactivities of endothelial progenitor cells (EPCs) and thereby attenuate tumor growth and angiogenesis in the Lewis lung carcinoma (LLC)-tumor-bearing mouse model. Although Phloroglucinol did not affect their cell toxicity, it specifically inhibited vascular endothelial growth factor (VEGF) dependent migration and capillary-like tube formation of EPCs. Our matrigel plug assay clearly indicated that orally injected phloroglucinol effectively disrupts VEGF-induced neovessel formation. Moreover, we demonstrated that when phloroglucinol is orally administered, it significantly inhibits tumor growth and angiogenesis as well as CD45 2 /CD34 + progenitor mobilization into peripheral blood in vivo in the LLC-tumorbearing mouse model. Conclusions/Significance: These results suggest a novel role for phloroglucinol: Phloroglucinol might be a modulator of circulating EPC bioactivities, eventually suppressing tumorigenesis. Therefore, phloroglucinol might be a candidat

Cell-surface sensors for real-time probing of cellular environments

Author Manuscript 2012 August 1.The ability to explore cell signalling and cell-to-cell communication is essential for understanding cell biology and developing effective therapeutics. However, it is not yet possible to monitor the interaction of cells with their environments in real time. Here, we show that a fluorescent sensor attached to a cell membrane can detect signalling molecules in the cellular environment. The sensor is an aptamer (a short length of single-stranded DNA) that binds to platelet-derived growth factor (PDGF) and contains a pair of fluorescent dyes. When bound to PDGF, the aptamer changes conformation and the dyes come closer to each other, producing a signal. The sensor, which is covalently attached to the membranes of mesenchymal stem cells, can quantitatively detect with high spatial and temporal resolution PDGF that is added in cell culture medium or secreted by neighbouring cells. The engineered stem cells retain their ability to find their way to the bone marrow and can be monitored in vivo at the single-cell level using intravital microscopy.National Institutes of Health (U.S.) (Grant HL097172)National Institutes of Health (U.S.) (Grant HL095722)National Institutes of Health (U.S.) (Grant DE019191)National Institutes of Health (U.S.) (Grant NIAID 5RC1AI086152)Charles A. Dana FoundationAmerican Heart Association (Grant 0970178N)National Science Foundation (U.S.) (Graduate Fellowship