On the constrained mock-Chebyshev least-squares

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.Comment: 17 pages, 9 figure

Adjustment curves for binary responses associated to stochastic processes

Functional Data, Random Multiplicative Cascade, Adjustment Curve, Stochastic Process

Pharmacological blockade of the WNT-beta-catenin signaling: a possible first-in-kind DMOAD.

International audienc

Polynomial approximation of derivatives by the constrained mock-Chebyshev least squares operator

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea is to improve the mock-Chebyshev subset interpolation, where the considered function $f$ is interpolated only on a proper subset of the uniform grid, formed by nodes that mimic the behavior of Chebyshev--Lobatto nodes. In the mock-Chebyshev subset interpolation all remaining nodes are discarded, while in the constrained mock-Chebyshev least squares interpolation they are used in a simultaneous regression, with the aim to further improving the accuracy of the approximation provided by the mock-Chebyshev subset interpolation. The goal of this paper is two-fold. We discuss some theoretical aspects of the constrained mock-Chebyshev least squares operator and present new results. In particular, we introduce explicit representations of the error and its derivatives. Moreover, for a sufficiently smooth function $f$ in $[-1,1]$, we present a method for approximating the successive derivatives of $f$ at a point $x\in [-1,1]$, based on the constrained mock-Chebyshev least squares operator and provide estimates for these approximations. Numerical tests demonstrate the effectiveness of the proposed method.Comment: 17 pages, 23 figure

Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values

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Culture expansion in low-glucose conditions preserves chondrocyte differentiation and enhances their subsequent capacity to form cartilage tissue in three-dimensional culture.

Culture conditions that preserve a stable chondrocyte phenotype are desirable in cell-based cartilage repair to maximize efficacy and clinical outcome. This study investigates whether low-glucose conditions will preserve the chondrocyte phenotype during culture expansion. Articular chondrocytes were culture-expanded in media supplemented with either low (1 mM) or high (10 mM) glucose. The metabolic phenotype, reactive oxygen species generation, and mRNA expression of markers of differentiation or catabolism were assessed by reverse-transcription quantitative polymerase chain reaction after four population doublings (PDs) and subsequent tissue formation capacity determined using pellet cultures. Continuous monolayer culture was used to determine the population doubling limit. After expansion in monolayer for four PDs, chondrocytes expanded in low-glucose conditions exhibited higher expression of the differentiation markers SOX9 and COL2A1 and reduced expression of the catabolic metalloproteinase matrix metallopeptidase 13. When chondrocytes expanded in low glucose were cultured in micropellets, they consistently generated more cartilaginous extracellular matrix than those expanded in high glucose, as evaluated by wet weight, sulfated glycosaminoglycan content, and hydroxyproline assay for collagen content. The same pattern was observed whether high or low glucose was used during the pellet culture. During expansion, chondrocytes in high-glucose generated 50% more reactive oxygen species than low-glucose conditions, despite a lower dependence on oxidative phosphorylation for energy. Furthermore low-glucose cells exhibited >30% increased population doubling limit. These data suggests that low-glucose expansion conditions better preserve the expression of differentiation markers by chondrocytes and enhance their subsequent capacity to form cartilage in vitro. Therefore, low glucose levels should be considered for the expansion of chondrocytes intended for tissue engineering applications.This study was funded by the Medical Research Council/Engineering and Physical Sciences Research Council (EPSRC) discipline bridging initiative grant PPA026, EPSRC Platform Grant EP/E046975/1; Human Frontier Science Program Grant RGP0025/2009-C and Arthritis Research U.K. grants 19654 and 19344

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates

Activation of WNT and BMP signaling in adult human articular cartilage following mechanical injury

Peer reviewedPublisher PD