Lagrange Interpolation by Bivariate C 1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C ¹-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.     

Local lagrange interpolation with cubic splines on tetrahedral partitions

We describe an algorithm for constructing a Lagrange interpolation pair based on C¹ cubic splines defined on tetrahedral partitions. In particular, given a set of points , we construct a set P containing and a spline space based on a tetrahedral partition whose set of vertices include such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets , and (2) the method provides optimal approximation order of smooth functions.     

Effect of Electrical Pulses on the Mechanical Behavior of Single Crystals of Nickel-Based CMSX-4 Superalloy and the Mobility of Low-Angle Grain Boundary in Aluminum Bicrystals

The electroplastic effect is studied on mechanically loaded single crystals of nickel-based CMSX-4 superalloy and pure aluminum bicrystals. Both objects of study exhibit elevated plasticity under electrical pulses. Traces of the electroplastic deformation that occurs in CMSX-4 single crystals at current densities J higher than 3 kA mm^?2 and aluminum bicrystals at J exceeding 1.6 kA mm^?2 are analyzed using a scanning electron microscope.     

Strong unicity of best approximations : A numerical aspect

The set of functions in C(T) which have a strongly unique best approximation from a given finite-dimensional subspace is denoted by SU(G). Since strong unicity plays an important role in numerical computations and since there the functions are only known up to some error, it is natural to ask what are the functions from the interior of SU(G). A complete characterization of those functions is given and the result is applied to weak Chebyshev and spline subspaces.     

Improved enantiomer resolution and quantification of free D-amino acids in serum and urine by comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry

The potential of comprehensive two-dimensional gas chromatography-time-of-flight mass spectrometry (GC×GC-TOFMS) in the quantitative analysis of amino acid enantiomers (AAEs) as their methyl chloroformate (MCF) derivatives in physiological fluids was investigated. Of the two column sets tested, the combination of an Rt-γDEXsa chiral column with a polar ZB-AAA column provided superior selectivity. Twenty AAEs were baseline resolved including L-Leu and D-Ile, which had failed separation by one-dimensional chiral GC-quadrupole-MS (GC-qMS). Lower limits of quantification (LLOQ) were in the range of 0.03-2 μM. Reproducibility of the analysis of a serum specimen in octaplicate ranged from 1.3 to 16.6%. The GC×GC-TOFMS method was validated by analyzing AAEs in 48 urine and 43 serum specimens, respectively, and by comparing the results with data obtained by a previously validated GC-qMS method. Mean recoveries ranged from 78.4% for D-Leu to 116.4% for D-Pro in urine and 72.2% for L-Thr to 129.4% for L-Ile in serum. The method was applied to the comparison of AAE serum levels in patients suffering from liver cirrhosis to a control group. Significantly increased D-AA concentrations were found for the patient group, whereas L-AA levels were slightly decreased.     

Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.     

Uniqueness and strong uniqueness of best approximations by spline subspaces and other subspaces

A complete characterization is given of those functions in $\text{C¦a, b¦}$ which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots. Also, the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of C₀(T) is investigated.