Mappings of finite distortion: Hausdorff measure of zero sets

We prove that a for a mapping f of finite distortion , the -Hausdorff measure of any point preimage is zero provided is integrable, with , and the multiplicity function of f is essentially bounded. As a consequence for we obtain that the mapping is then open and discrete. Received: 18 June 2001 / Revised version: 31 January 2002 / Published online: 27 June 2002     

Flat Information Geometries in Black Hole Thermodynamics

The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein–Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.Supported by VR.     

On uniformly Gâteaux smooth norms and normal structure

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure.

     

6-Thia-1-aza-5λ5-phosphabicyclo[3.2.0]hept-3-ene und Isothiocyanatophosphin-sulfide aus (Methylidenamino)phosphinen,Schwefelkohlenstoff und Acetylen- oder Ethylen-Derivaten

6-Thia-l-aza-5λ⁵-phosphabicyclo[3.2.0]hept-3-enes and Isothiocyanatophosphine-sulfides from (Methylidenamino)phosphines, Carbon Disulfide, and Acetylene- or Ethylene Derivatives (Methylidenamino)phosphines react with carbon disulfide to 1,3,2-thiazaphosphete derivatives 2 . Acetylenedicarboxylates give with 2 6-thia-l-aza-5λ⁵-phosphabicyclo[3.2.0]hept-3-enes 4a , b , and acrylates yield isothio-cyanatophosphin-sulfides 5a – c . The structures of 4a and 5a are proven by an X-ray diffraction analysis.     

A note on stability of the Douglas splitting method

In this note some stability results are derived for the Douglas splitting method. The relevance of the theoretical results is tested for an advection-reaction equation.

     

Influence of heat treatment conditions on the porosity changes of sulfonated styrene/divinylbenzene copolymers

Sulfonic cation exchangers with two ion exchange group concentrations (0.5 and 2.4 mmol/g, samples A and B, respectively) were obtained by sulfonation of a porous styrene (S) and divinylbenzene (DVB) copolymer with chlorosulfonic acid. Strong thermal decomposition of the sulfonated copolymer A, accompanied by significant changes in its porous structure, starts at ca. 400°C. The char has no sulfonic groups. After heat treatment at 400°C in steam, a sorbent was obtained (yield 65%) that shows higher phenol sorption than the untreated sample when related to the bed volume. The chlorosulfonic derivatives of the initial copolymer were less thermally resistant than the sulfonic ones obtained by hydrolysis. Pyrolysis of the cation exchanger B, in its H+ and Ca²⁺ forms, was carried out at 900°C (yield of both chars close to 30%). By subsequent steam activation at 800°C to a 50% burn-off of the char, sorbents with well-developed, but distinctly different, porous structures were obtained. The activated char from the sulfonated copolymer in its hydrogen form was highly microporous and indicated an effective surface area of 1180 m²/g. However, because of a low contribution of mesopores, its ability to adsorb phenol from the liquid phase was not very high. The activated char from the calcium-doped copolymer, indicating a smaller surface area (580 m²/g) but characterized by a well-developed mesoporosity, was a better sorbent for phenol. © 1994 John Wiley & Sons, Inc.     

Quantum Nonlocality of Photon Pairs in Interference in a Mach-Zehnder Interferometer

Nonlocal character of entangled photon pairs generated in spontaneous parametric downconversion is demonstrated. One photon from a pair propagates through a Mach-Zehnder interferometer and is detected in coincidence-count measurement with its twin. Width of the coincidence-count interference pattern (measured for various values of path difference in the Mach-Zehnder interferometer) depends on spectral width of the twin as a result of entanglement of photons in a pair. The experimental setup is analyzed for a Gaussian spectral filter and a Fabry-Perot resonator. It is shown that nonlocal interference is much stronger for cw pumping in comparison with femtosecond pulsed pumping for values of parameters commonly used in spontaneous parametric downconversion experiments.