Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

Produktintegration mit nicht-äquidistanten Stützstellen

Summary For the numerical evaluation of , 0<<1 andx smooth, product integration rules are applied. It is known that high-order rules, e.g. Gauss-Legendre quadrature, become normal-order rules in this case. In this paper it is shown that the high order is preserved by a nonequidistant spacing. Furthermore, the leading error terms of this product integration method and numerical examples are given.

     

Impedance eduction from energy dissipation measurements with flow

An optimization method for the calculation of the wall impedance of a perforate from measurements under grazing flow conditions is presented. It is in particular characterized by the simultaneous eduction of the parameter set defining the impedance as function of frequency, which allows interpolation and extrapolation. The Extended Helmholtz Resonator is applied as impedance model and the simultaneous eduction is performed using a time–domain CAA method. Furthermore, the method uses processed energy data from experiments to define the objective function. The results give an insight in the effect of bias flow on the wall impedance under grazing flow conditions. It proves to be an important acoustic design parameter. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The structure of some virtually free pro- groups

We prove two conjectures on pro- groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro- group which has an open free pro- subgroup of index is a free pro- product , where the are free pro- of finite rank and the are cyclic of order . The second says that if is a free pro- group of finite rank and is a finite -group of automorphisms of , then is a free factor of . The proofs use cohomology, and in particular a ``Brown theorem' for profinite groups.

     

Regularity of the solution to a class of weakly singular fredholm integral equations of the second kind

Continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived. The equations studied in this paper arise from e.g. potential problems or problems of radiative equilibrium. Under reasonable assumptions it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end-points.     

Multiscale Considerations for Sound Generation in Low Mach Number Flows

Classical approaches in aeroacoustics are mainly based on analytical solutions of the linear wave equations which are valid in the far field. The sound generation is approximated by source terms obtained from a flow simulation. This procedure designated as the ‘acoustic analogy’ was initiated in the classical work of Lighthill. In order to tackle the sound generation problem at low Mach numbers, we consider a multiple scale asymptotic analysis. As we deal with a fluid flow generating the sound itself, the asymptotic ansatz uses one time scale given by the flow convection, but two space scales due to the difference in fluid and sound velocity. The insight given by this analysis is used to obtain source terms describing the sound generation and perturbation equations for the sound propagation. Numerical results are shown for the example of a co‐rotating vortex pair.     

Über die Reaktion von W3O mit lod. Darstellung,Kristallstruktur und Eigenschaften von WOI3

On the Reaction of W₃O with Iodine: Preparation, Crystal Structure, and Properties of WOI₃ The novel tungsten oxide iodide WOI₃ is obtained from the reaction of the metalrich tungsten oxide W₃O (also known as ‘β-tungsten’) with iodine in a sealed glass tube placed in a temperature gradient (440–400°C). In the low-temperature region WOI₃ is formed as mouse-grey intergrown needle-like crystals. WOI₃ crystallizes in the tetragonal NbOCl₃ structure type, space group P4₂/mnm, with a = 12.346(2), c = 3.765(1) Å (20°C). The crystal structure at – 130°C was determined from single-crystal diffractometer data (R = 0.023). Results of magnetic susceptibility and vibrational spectroscopic measurements are given together with the mass spectrum of WOI₃ obtained on heating.