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This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing., Comment: v3: 35 pages, 3 figures; minor corrections

An overview of the recent developments in plurifine potential theory., Comment: 18 pages, Conference on SCV on the occasion of Jozef Siciak's 80'th birthday

Locally supported biorthogonal wavelets are constructed on the unit interval with respect to which second-order constant coefficient differential operators are sparse. As a result, the representation of second-order differential operators on the hypercube with respect to the resulting tensor product wavelet coordinates is again sparse. The advantage of tensor product approximation is that it yields (nearly) dimension-independent rates. An adaptive tensor product wavelet method is applied to solve various singularly perturbed boundary value problems. The numerical results indicate robustness with respect to the singular perturbations. For a two-dimensional model problem this will be supported by theoretical results.

The convex hull of n+1 affinely independent vertices of the unit n-cube Cn is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in Cn can be described by nonsingular 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn of symmetries of Cn. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature for n < 7. Using the computed cycle indices in combination with Polya's theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n x n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler's Tree of Fractions that enumerates the rationals between 0 and 1. Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric matrices., 52 pages, 25 figures

A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nosé-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermostats in the setting of simulating systems with holonomic constraints. The approaches described are easily implemented and facilitate the recovery of correct canonical averages with minimal disturbance of the underlying dynamics. For the purpose of illustrating our results, we examine the numerical application of these methods to a simple atomic chain, where a Fixman term is required to correct the thermodynamic ensemble.

Given a biorthogonal pair of multi-resolution analyses on the interval, by integration or differentiation, we build a new biorthogonal pair of multi-resolution analyses. Using both pairs, isotropic or, as we focus on, anisotropic divergence-free wavelet bases on the hypercube are constructed. Our construction generalizes the one from [16] by P.G. Lemarié-Rieusset (1992) for stationary multi-resolution analyses on R. It turns out that it requires a specific choice of boundary conditions.

Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ℝ3 that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.

In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments with a double well potential corroborate the asymptotic analysis.

The famous Zlamal's minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2d. In this paper we present and discuss its generalization to simplicial partitions in any space dimension.

A simultaneous space-time variational formulation of a parabolic evolution problemis solved with an adaptive wavelet method. This method is shown to converge with the best possiblerate in linear complexity. Thanks to the use of tensor product bases, there is no penalty in complexitydue to the additional time dimension. Special wavelets are designed such that the bi-infinite systemmatrix is sparse. This sparsity largely simplifies the implementation and improves the quantitativeproperties of the adaptive wavelet method. Numerical results for an ODE and the heat equation arepresented.

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics., 15 pages, 4 figures

We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided. © 2011 American Mathematical Society.

Assessment of high-affinity antibody-antigen binding parameters is important in such diverse areas as selection of therapeutic antibodies, detection of unwanted hormones in cattle and sensitive immunoassays in clinical chemistry. Label-free assessment of binding affinities is often carried out by immobilization of one of the binding partners on a biosensor chip, followed by monitoring the binding equilibrium of the other partner. However, for the measurement of high-affinity binding, with dissociation constants in the picomolar range or lower, equilibration times exceed practical limits and one has to resort to the measurement of sorption kinetics. Here we evaluate a new technique, using PEIA(1)-ellipsometty and establishment of equilibrium in solution. Binding parameters are determined for two high-affinity anti-interleukin 6 antibodies, anti-IL6.16 and anti-IL6.8, and compared with values obtained by a bioassay, based on IL6-dependent cell growth, and with values obtained by a standard technique based on SPR(2). The high affinities of both antibodies as found with the bioassay (5 and 50 pM for anti-IL6.8 and anti-IL6.16, respectively), could be conveniently measured by PEIA-ellipsometry. Using SPR, equilibrium measurements indeed proved too time-consuming and analysis of adsorption/desorption kinetics revealed that the binding of the antibodies on the chip caused the appearance of different populations of antibodies with different affinities. (C) 2010 Elsevier B.V. All rights reserved

We study a linear partial integro-differential equation which arises in the modeling of various physical and biological sciences. We analyze numerical stability and numerical convergence of a one step approximation of the problem with smooth and non-smooth initial functions.

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s, Comment: 13 pages

A heteroclinic network for an equivariant ordinary differential equation is called switching if each sequence of heteroclinic trajectories in it is shadowed by a nearby trajectory. It is called forward switching if this holds for positive trajectories. We provide an elementary example of a switching robust homoclinic network and a related example of a forward switching asymptotically stable robust homoclinic network. The examples are for five-dimensional equivariant ordinary differential equations. © 2010 Taylor & Francis.

For random diffeomorphisms depending on a parameter, nearly invariant sets are described via an associated deterministic control system. Conditions are provided guaranteeing that the system leaves the support of an invariant measure under small perturbations of the parameter and estimates for the exit times are given. © 2010 Taylor & Francis.

We construct unbounded strange attractors for vector fields in R^3 that are robust transitive under uniformly small perturbations. Their geometry is reminiscent of geometric Lorenz and other singular hyperbolic attractors, but they contain no equilibria.

The one-way wave equation is widely used in seismic migration. Equipped with wave amplitudes, the migration can be provided with the reconstruction of the strength of reflectivity. We derive the one-way wave equation with geometrical amplitude by using a symmetric square root operator and a wave field normalization. The symbol of the square root operator, $\omega\sqrt{1/c(x,z)^2-\xi^2/\omega^2}$, is a function of space-time variables and frequency $\omega$ and horizontal wavenumber $\xi$. Only by matter of quantization it becomes an operator, and because quantization is subjected to choices it should be made explicit. If one uses a naive asymmetric quantization an extra operator term will appear in the one-way wave equation, proportional to $\partial_xc$. We propose a symmetric quantization, which maps the symbol to a symmetric square root operator. This provides geometrical amplitude without calculating the lower order term. The advantage of the symmetry argument is its general applicability to numerical methods. We apply the argument to two numerical methods. We propose a new pseudo-spectral method, and we adapt the 60 degree Padé type finite-difference method such that it becomes symmetrical at the expense of almost no extra cost. The simulations show in both cases a significant correction to the amplitude. With the symmetric square root operator the amplitudes are correct. The $z$-dependency of the velocity lead to another numerically unattractive operator term in the one-way wave equation. We show that a suitably chosen normalization of the wave field prevents the appearance of this term. We apply the pseudo-spectral method to the normalization and confirm by a numerical simulation that it yields the correct amplitude.

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with the theoretical models, as well as with the continuum statistical mechanical theory for ideal fluid flow developed by Ellis et al. (2002) [10]. In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean.

We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold M can meet a pluripolar set. M has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K subset of C. If the graph Gamma(f) (K) is pluripolar, then. partial derivative f/partial derivative z = 0 in the closure of the fine interior of K.

Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that we may lose is independent of the space dimension n. The cost of producing these approximations will be proportional to their length with a constant factor that may grow with n, but only linearly.

Are there infinitely many prime pairs with given even difference? Most mathematicians think so. Using a strong arithmetic hypothesis, Goldston, Pintz and Yildirim have recently shown that there are infinitely many pairs of primes differing by at most sixteen. There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood [G.H. Hardy, J.E. Littlewood, Some problems of 'partitio numerorum'. III: On the expression of a number as a sum of primes, Acta Math. 44 (1923) 1-70 (sec. 3)] on the asymptotic behavior of @p"2"r(x), the number of prime pairs (p,p+2r) with [email protected]?x. Assuming Riemann's Hypothesis (RH), Montgomery and others have studied the pair-correlation of zeta's complex zeros, indicating connections with the PPC. Using a Tauberian approach, the author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta's complex zeros. A certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery's work on pair-correlation, would imply that there is an abundance of prime pairs.

Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples.

With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

Racah and Wilson polynomials with dilated and translated argument are reparametrized such that the polynomials are continuous in the parameters as long as these are nonnegative, and such that restriction of one or more of the new parameters to zero yields orthogonal polynomials lower in the Askey scheme. Geometrically this will be described as a manifold with corners., 29 pages, 8 figures, figures improved, references and remarks added, accepted by Ramanujan Journal

Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N. N. Pham Thi, D. M. Karl, and B. P. Sommeijer, Nature, 439 (2006), pp. 322–325]. This paper is a first step towards understanding the bifurcational structure associated with nonlocal coupled phytoplankton-nutrient models as studied in that paper. Its main subject is the linear stability analysis that governs the occurrence of the first nontrivial stationary patterns, the deep chlorophyll maxima (DCMs) and the benthic layers (BLs). Since the model can be scaled into a system with a natural singularly perturbed nature, and since the associated eigenvalue problem decouples into a problem of Sturm–Liouville type, it is possible to obtain explicit (and rigorous) bounds on, and accurate approximations of, the eigenvalues. The analysis yields bifurcation-manifolds in parameter space, of which the existence, position, and nature are confirmed by numerical simulations. Moreover, it follows from the simulations and the results on the eigenvalue problem that the asymptotic linear analysis may also serve as a foundation for the secondary bifurcations, such as the oscillating DCMs, exhibited by the model.

SummaryIn fluorogenic thrombin generation (TG) experiments, thrombin concentrations cannot be easily calculated from the rate of the fluorescent signal increase, because the calibration coefficient increases during the experiment, due to substrate consumption and quenching of the fluorescent signal by the product. Continuous, external calibration via an in a parallel sample therefore was hitherto required for an accurate calculation of the TG curve. A technique is presented that allows mathematical transformation of experimental fluorescence intensities into "ideal" data, i.e. in the data that would have been obtained if substrate consumption and quenching by the product would not play a role. The method applies to fluorescence intensities up to 90% of the maximal fluorescent signal corresponding to total substrate conversion and thereby covers the entire region of interest encountered in practice. The first derivative of the transformed signal can then be converted into thrombin concentrations via a conventional, fixed calibration factor. This calibration factor can be obtained from a separate experiment but also by measuring the amidolytic activity of the α2macroglobulin-thrombin complex present in the reaction mixture ("serum") after thrombin generation is over. This method halves the amount of sample required per experiment. Keywordscalibration, thrombin generation, fluorogenic methods

A one-dimensional analytical solution is derived for saturated charged compressible porous media. The equations describe infinitesimal deformation of charged porous media saturated with a fluid with dissolved cations and anions. In the one-dimensional case the governing equations reduce to a coupled diffusion equations for electro-chemical potentials. Suitable initial and boundary conditions complete the equations.

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.

We study connectedness in the pluri-fine topology on $\CC^n$ and obtain the following results. If $\Omega$ is a pluri-finely open and pluri-finely connected set in $\CC^n$ and $E\subset\CC^n$ is pluripolar, then $\Omega\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $\Omega$ be a pluri-finely open subset of $\CC^{n}$. If $z$ is any point in $\Omega$, and $L$ is a complex line passing through $z$, then obviously $\Omega \cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component of $z$ in $\Omega\cap L$. Then $\cup_{L\ni z} C_L$ is a pluri-finely connected neighborhood of $z$. As a consequence we find that if $v$ is a finely plurisubharmonic function defined on a pluri-finely connected pluri-finely open set, then $v= -\infty$ on a pluri-finely open subset implies $v\equiv -\infty$., Comment: 13 pages

We develop a fast method for solving the one-dimensional wave equation based on geometrical optics. From geometrical optics (e.g., Fourier integral operator theory or WKB approximation) it is known that high-frequency waves split into forward and backward propagating parts, each propagating with the wave speed, with amplitude that is slowly changing depending on the medium coefficients, under the assumption that the medium coefficients vary slowly compared to the wavelength. Based on this we construct a method of optimal, $O(N)$ complexity, with basically the following steps: 1. decouple the wavefield into an approximately forward and an approximately backward propagating part; 2. propagate each component explicitly along the characteristics over a time step that is small compared to the medium scale but can be large compared to the wavelength; 3. apply a correction to account for the errors in the explicit propagation; repeat steps 2 and 3 over the necessary amount of time steps; and 4. reconstruct the full field by adding forward and backward propagating components again. Due to step 3 the method accurately computes the full wavefield. A variant of the method was implemented and outperformed a standard order (4,4) finite difference method by a substantial factor. The general principle is applicable also in higher dimensions, but requires efficient implementations of Fourier integral operators which are still the subject of current research.

This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an approximation of the appropriate zero of the (m+1)-st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with explicit fast--slow systems, in which there is an explicit small parameter, epsilon, measuring the separation of time scales. We show that, for each m = 0, 1, ..., the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of O(epsilon^m). Moreover, for each m, we identify explicitly the conditions under which the m-th iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems-in which there need not be an explicit small parameter-to which the algorithms also apply.

Let { αn }n = 1∞ be a sequence of points in the open unit disk in the complex plane and letB0 = 1 and Bn (z) = underover(∏, k = 0, n) frac(over(αk, -), | αk |) frac(αk - z, 1 - over(αk, -) z), n = 1, 2, ...,(over(αk, -) / | αk | = - 1 when αk = 0). We put L = span { Bn : n = 0, 1, 2, ... } and we consider the following "moment" problem:. Given a positive-definite Hermitian inner product 〈 ·, · 〉 in L, find all positive Borel measures ν on [- π, π) such that〈 f, g 〉 = ∫- ππ f (ei θ) over(g (ei θ), -) d ν (θ) for f, g ∈ L .We assume that this moment problem is indeterminate. Under some additional condition on the αn we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞. © 2007 Elsevier B.V. All rights reserved. ispartof: Journal of Computational and Applied Mathematics vol:219 issue:2 pages:359-369 ispartof: location:FRANCE, Lille status: published

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral., Comment: 20 pages; added in v3: more references to earlier occurrences of the identity and its multivariable analogue, combinatorial proof of the identity and extension to noninteger m,n, proof of multivariable identity by splitting up Dirichlet's multivariable beta integral

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.

For the analysis of multi-grid methods applied to finite difference discretizations, two definitions of orders of intergrid transfer operators are being used. The order is either defined in terms of the symbol of the transfer operator, i.e., its Fourier transform, or as the order of polynomials being preserved. In [J. Comput. Appl. Math. 32 (3) (1990) 423-429], Hemker showed that the second definition is stronger than the first one. In this note, we show that both definitions are even equivalent.

In this note we examine several regularity criteria for families of simplicial finite element partitions in R^d,[email protected]?{2,3}. These are usually required in numerical analysis and computer implementations. We prove the equivalence of four different definitions of regularity often proposed in the literature. The first one uses the volume of simplices. The others involve the inscribed and circumscribed ball conditions, and the minimal angle condition.

A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. The method consists of 3 nested loops. The outermost loop consists of an adaptive finite element method for solving the pressure from the (elliptic) Schur complement system that arises by eliminating the velocity. Each of the arising finite element problems is a Stokes-type problem, with the pressure being sought in the current trial space and the divergence-free constraint being reduced to orthogonality of the divergence to this trial space. Such a problem is still continuous in the velocity field. In the middle loop, its solution is approximated using the Uzawa scheme. In the innermost loop, the solution of the elliptic system for the velocity field that has to be solved in each Uzawa iteration is approximated by an adaptive finite element method.

An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n+1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra., Comment: 18 pages, minor corrections and updated references

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas., changed title, references updated, minor changes matching the version to appear in Ramanujan J.; 22 pp

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D.

In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg-Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg-Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x → ±∞. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction-diffusion systems can be studied need to be extended. This results in an 'algebraic NLEP approach', which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al 2004 J. Nonlin. Sci. 14 237-78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account.Mathematics Subject Classification: 35B35, 37L15, 35B32, 35K57, 76E30

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task.Mathematics Subject Classification. 76S05, 74B05, 74F10Key words: Mixture theory, porous media, hydrated soft tissue.

In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method.

This paper deals with numerical approximation of the two-dimensional Poincaré equation that arises as a model for internal wave motion in enclosed containers. Inspired by the hyperbolicity of the equation we propose a discretisation particularly suited for this problem, which results in matrices whose size varies linearly with the number of grid points along the coordinate axes. Exact solutions are obtained, defined on a perturbed boundary. Furthermore, the problem is seen to be ill-posed and there is need for a regularisation scheme, which we base on a minimal-energy approach.

This paper addresses global error estimation and control for initial value problems forordinary differential equations. The focus lies on a comparison between a novel approach based onthe adjoint method combined with a small sample statistical initialization and the classical approachbased on the first variational equation. Control is achieved through tolerance proportionality. Bothapproaches are found to work well and to enable estimation and control in a reliable manner.Key words.Numerical integration for ODEs, global error estimation, global error control, defectsand local errors, tolerance proportionality, adjoint method, small sample statistical initialization

The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory [J.M. Huyghe and J.D. Janssen, Int. J. Engng. Sci. 35 (1997) 793-802; K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modeling of incompressible charged porous media. ESAIM: M2AN 41 (2007) 661-678]. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain accurate approximations of the fluid flow and ions flow. Such accurate approximations can be determined by the mixed finite element method. The solid displacement, fluid and ions flow and electro-chemical potentials are taken as degrees of freedom. In this article the lowest-order mixed method is discussed. This results into a first-order nonlinear algebraic equation with an indefinite coefficient matrix. The hybridization technique is then used to reduce the list of degrees of freedom and to speed up the numerical computation. The mixed hybrid finite element method is then validated for small deformations using the analytical solutions for one-dimensional confined consolidation and swelling. Two-dimensional results are shown in a swelling cylindrical hydrogel sample.Mathematics Subject Classification. 35K55, 35M10, 65F10, 65M60.Key words: Hydrated soft tissue, nonlinear parabolic partial differential equation, mixed hybrid finite element.