Algebraic composition operators

The focus of this paper is on the following problem. Given a linear space F of complex-valued functions on a set X and a polynomial p(z), is there an algebraic composition operator on F whose characteristic polynomial equals p(z)? We show that the supply of all the polynomials p(z) for which the answer to this question is affirmative depends heavily on the structure of the space F.     

P-harmonic obstacle problems

We prove an extension theorem for Sobolev functions which plays an important role in the partial regularity theory of vector valued functions which minimize degenerate variational integrals subject to some nonlinear side conditions in the image space.     

Book reviews

Ohne Zusammenfassung

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Book reviews

     

Separation of rat pancreatic secretory proteins by cation-exchange fast protein liquid chromatography.

Rat pancreatic secretory proteins were separated by an automated liquid chromatography system utilizing a Mono S cation-exchange column. Optimal resolution was obtained with a multistep salt and pH gradient (0.01-2 M LiCl, pH 5.3-63). A total of fourteen well-separated peaks, as well as several minor peaks, were detected by UV absorption. The main pancreatic enzymes were resolved (two amylases, two chymotrypsinogens, two trypsinogens, proelastase, lipase, prophospholipase A2, procarboxypeptidase A, procarboxypeptidase B, and ribonuclease). In addition, proteins without enzymic activity, such as lithostathine and pancreatitis-associated protein, were identified. Activation of proenzymes did not occur during the separation. At a flow-rate of 0.5 ml/min, ca. 250 micrograms to 5 mg of protein could be applied with equal resolution. The reproducibility of retention volumes and peak areas was high (less than 1% or 5% variation, respectively). When radiolabeled proteins were separated, a comparable pattern of peaks was obtained. The technique described is, therefore, not only useful for analytical and preparative separation of pancreatic proteins but can additionally serve for quantitative determination of the pancreatic isoenzyme pattern.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.