Monday, June 18. 2012
J. Morais2 Freiberg University of Mining and Technology, Institute of Applied Analysis, Prueferstr. 9, 09596 Freiberg, Germany Received 26 November, 2010; accepted in revised form 28 July, 2011 Abstract: During the past few years considerable attention has been given to the role played by monogenic functions in approximation theory. The main goal of the present paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Riesz system over prolate spheroids in R3 . This will be done in the spaces of square integrable functions over R. As a first step towards is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. Some important properties of the system are briefly discussed.
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Monday, June 18. 2012
E. Mainar and Juan Manuel Pena Departamento de Matematica Aplicada, Universidad de Zaragoza, Zaragoza, Spain Received 15 December, 2009; accepted in revised form 18 October, 2011 Abstract: We analyze some properties of bivariate tensor product bases. In particular, we obtain conditions so that a general bivariate tensor product basis is optimally stable for evaluation. Finally, we apply our results to prove the optimal stability of tensor product normalized Bbases.
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Monday, June 18. 2012
C. Dagnino , V. Demichelis Department of Mathematics, Faculty of Scienze Matematiche Fisiche e Naturali, University of Torino, 10100 Torino, Italy Received 30 January, 2009; accepted in revised form 13 October, 2011 Abstract: We propose a new quadrature rule for Cauchy principal value integrals based on quadratic spline quasiinterpolants which have an optimal approximation order and satisfy boundary interpolation conditions. In virtue of these spline properties, we can prove uniform convergence for sequences of such quadratures and provide uniform error bounds. A computational scheme for the quadrature weights is given. Some numerical results and comparisons with other spline methods are presented.
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Monday, June 18. 2012
Luigi Brugnano and Alessandra Sestini Dipartimento di Matematica “U. Dini” Viale Morgagni 67/A, 50134 Firenze, Italy Dedicated to Prof. D. Trigiante on the occasion of his 65th birthday. Received 16 December, 2009; accepted in revised form 24 September, 2011 Abstract: We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of freesurface problems. In particular, we here study their application to the numerical solution of both the (linear) parabolic obstacle problem and the obstacle problem. We propose a class of effective semiiterative Newtontype methods to find the exact solution of such piecewise linear systems. We prove that the semiiterative Newtontype methods have a global monotonic convergence property, i.e., the iterates converge monotonically to the exact solution in a finite number of steps. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.
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Monday, June 11. 2012
Karline Soetaert Centre for Estuarine and Marine Ecology, Netherlands Institute of Ecology, 4401 NT Yerseke, The Netherlands
Thomas Petzoldt Institut fur Hydrobiologie Technische Universitat Dresden 01062 Dresden, Germany Received 21 February, 2011; accepted in revised form 11 March, 2011 Abstract: The opensource problem solving software R [1] has become one of the most widely used systems for statistical data analysis. As it is a powerful interpreted language, it is also very well suited for other disciplines in scientific computing. One of the fields where considerable progress has been made is the solution of differential equations. In this paper we describe a set of recently developed tools, socalled Rpackages, to efficiently solve and analyze initial value problems of differential equations in R. Most of the methods are based on welltested opensource numerical codes, combining the robustness and efficiency of these codes with the flexibility of the R language. We exemplify the use of these tools by several examples. We start by implementing a wellknown test problem for nonstiff solvers, the Arenstorff orbit ordinary differential equations. Next we solve the pendulum problem, a DAE of index 3. A description of a bouncing ball shows how roots and events can be programmed in R. After that we describe how to implement delay differential equations, which we exemplify with a DDE that is subject to an impulse, triggered at specific times. We end with a rather stiff partial differential equation, a combustion problem modeled in 2D. The presented R packages provide additional facilities to efficiently plot the outcome, to compare different scenarios, to estimate summary statistics, or to display execution statistics that help in assessing the performance of a particular method.
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Monday, June 11. 2012
Natasa Djurdjevac , Sharon Bruckner, Tim OF Conrad, Christof Schuette Fachbereich Mathematik und Informatik Institut fuer Mathematik Freie Universitaet Berlin {djurdjev,sharonb,conrad,schuette}@math.fuberlin.de Received 20 February, 2011; accepted in revised form 10 March, 2011 Abstract: Complex modular networks appear frequently, notably in the biological or social sciences. We focus on two current challenges regarding network modularity: the ability to identify (i) the modules of a given network, and (ii) the hub states as nodes with highest importance in terms of the communication between modules. Our approach towards these goals uses random walks as a mean to global analysis of the topology and communication structure of the network. We show how to adapt recent research regarding coarse graining of random walks. The resulting algorithms are based on spectral analysis of random walks and allow (A) an optimal identification of fuzzy assignments of nodes to modules, (B) computation of the fraction of the overall communication between modules supported by certain nodes, and (C) determination of the hubs as the nodes with the highest communi cation load.
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Monday, June 11. 2012
W. Auzinger Institute for Analysis and Scientific Computing, Vienna University of Technology, 1040 Vienna, Austria Received 20 February, 2011; accepted in revised form 10 March, 2011 Abstract: The wellknown technique of defect correction can be used in various ways for estimating local or global errors of numerical approximations to differential or integral equations. In this paper we describe the general principle in the context of linear and nonlinear problems and indicate the interplay between the auxiliary scheme involved and a correct definition of the defect. Applications discussed include collocation approximations to first and second order boundary value problems for nonlinear ODEs and, in particular, exponential splitting approximations for linear evolution equations. We describe the de sign of error estimators and their essential properties and give numerical examples. The theoretical tools for the analysis of the asymptotical correctness of such estimators are described, and references to original research papers are given where the complete analysis is provided.
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Monday, June 11. 2012
Pierluigi Amodio , Giuseppina Settanni Dipartimento di Matematica, Universita di Bari, I70125 Bari, Italy Received 22 February, 2011; accepted in revised form 13 March, 2011 Abstract: We investigate the numerical solution of regular and singular SturmLiouville problems by means of finite difference schemes of high order. In particular, a set of differ ence schemes is used to approximate each derivative independently so to obtain an algebraic problem corresponding to the original continuous differential equation. The endpoints are treated depending on their classification and in case of limit points, no boundary condi tion is required. Several numerical tests are finally reported on equispaced grids show the convergence properties of the proposed approach.
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