Strong unicity constants for spline functions

In the last years there has been a great deal of interest in strong unicity constants for the space of polynomials of degree at most n. We consider analogous questions for the space of spline functions of degree at most n with k fixed knots. In this case we give formulas and bounds for strong unicity constants and investigate strong unicity for generalized convex functions of order n, i.e. functions whose (n+1)–st derivative is positive.     

Metabolite extraction from adherently growing mammalian cells for metabolomics studies: optimization of harvesting and extraction protocols

Trypsin/ethylenediaminetetraacetic acid (EDTA) treatment and cell scraping in a buffer solution were compared for harvesting adherently growing mammalian SW480 cells for metabolomics studies. In addition, direct scraping with a solvent was tested. Trypsinated and scraped cell pellets were extracted using seven different extraction protocols including pure methanol, methanol/water, pure acetone, acetone/water, methanol/chloroform/water, methanol/isopropanol/water, and acid–base methanol. The extracts were analyzed by GC-MS after methoximation/silylation and derivatization with propyl chloroformate, respectively. The metabolic fingerprints were compared and 25 selected metabolites including amino acids and intermediates of energy metabolism were quantitatively determined. Moreover, the influence of freeze/thaw cycles, ultrasonication and homogenization using ceramic beads on extraction yield was tested. Pure acetone yielded the lowest extraction efficiency while methanol, methanol/water, methanol/isopropanol/water, and acid–base methanol recovered similar metabolite amounts with good reproducibility. Based on overall performance, methanol/water was chosen as a suitable extraction solvent. Repeated freeze/thaw cycles, ultrasonication and homogenization did not improve overall metabolite yield of the methanol/water extraction. Trypsin/EDTA treatment caused substantial metabolite leakage proving it inadequate for metabolomics studies. Gentle scraping of the cells in a buffer solution and subsequent extraction with methanol/water resulted on average in a sevenfold lower recovery of quantified metabolites compared with direct scraping using methanol/water, making the latter one the method of choice to harvest and extract metabolites from adherently growing mammalian SW480 cells.     

A Neuro-Fuzzy Approach to Optimize Hierarchical Recurrent Fuzzy Systems

To simplify the definition of fuzzy systems or to reduce its complexity hierarchical structures can be used. Thus, more transparent rule bases that are also easier to maintain can be designed. Furthermore, it is sometimes necessary to use time delayed input or to reuse time delayed output from the fuzzy system itself to obtain a rule base that describes the analyzed problem appropriately. This leads to hierarchical recurrent architectures that have increased approximation capabilities since they are able to store information of the past. In this article we present a neuro-fuzzy model that can be used to optimize hierarchical recurrent fuzzy rule bases if training data is available. Furthermore, we present an approach to learn initial rule bases from data using rule templates.     

Bivariate segment approximation and splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g., by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently.     

Lagrange and hermite interpolation by bivariate splines

We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.