Operator–valued Fourier multiplier theorems on Besov spaces

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ? supp ν on the real line. This class of operators includes a generalization of the well‐known Sturm‐Liouville operator as well as of the measure geometric Laplacian given by . We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß‐Green‐formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.     

Proton relaxation times in human red bone marrow by volume selective magnetic resonance spectroscopy

In hematological diseases the composition of red bone marrow shows alterations. The relaxation timesT ₁ andT ₂ of water and lipids in human hemopoietic bone marrow of 14 normal volunteers and 10 patients with acute leukemia and bone marrow carcinosis are determined using a double spin echo spectroscopy sequencein vivo. The volumes of interest (VOI) of (13 mm)³ in the center of vertebral bodies are examined using different measurement parameters. ForT ₁ measurements an inversion-recovery method is used.T ₂ is evaluated from spectra with differentTE. T ₁ (water) is found in a range between 1000 and 1700 ms,T ₁ (lipids) in a range between 260 and 320 ms in healthy volunteers.T ₂ (water) is determined between 32 and 65 ms. In some cases phase distortions of the water signals occur in the spectra. Water flow within the VOI may be a possible reason.T ₂ (lipids) is evaluated between 73 and 91 ms. The patients with acute leukemia exhibit clearly reduced lipid signals in their spectra. Lipid relaxation times could not be determined in these cases.T ₂ (water) is prolonged in acute leukemia to 51–98 ms.T ₁ (water) was not significantly different from values of healthy volunteers in our measurements. Results are discussed in comparison to relaxometric data from imaging and STEAM spectroscopic methods of other authors.     

Symplectic Yang–Mills theory,Ricci tensor,and connections

A Yang–Mills theory in a purely symplectic framework is developed. The corresponding Euler–Lagrange equations are derived and first integrals are given. We relate the results to the work of Bourgeois and Cahen on preferred symplectic connections.     

Photodecomposition of several compounds commonly used as sunscreen agents

The photolysis reactions of three compounds commonly used as a sunscreen agents, Parsol 1789 (1-[4-(1,1-dimethylethyl)phenyl]-3-(4-methoxyphenyl)-1,3- propanedione), Oxybenzone ((2-hydroxy-4-methoxyphenyl)phenyl-methanone) and Padimate O (2-ethylhexyl-4-(dimethylamino)benzoate), were investigated to provide a chemical background to aid in the understanding of the photosensitization of the sunscreen agents. Photolysis was carried out in cyclohexane for 70–140 h using a mercury vapor lamp (450W) without excluding oxygen.

Irradation of Parsol 1789 in cyclohexane yielded tert-butylbenzene, p-tert-butylbenzoic acid and p-methoxybenzoic acid; products obtained from the combination of the sunscreen with the solvent included the cyclohexyl esters of p-methoxybenzoic acid, p-tert-butylbenzoic acid and methanoic acid; products obtained from the solvent included cyclohexanol, cyclohexanone and dicyclohexyl ether.

Irradiation of Oxybenzone in the cyclohexane for 100 h produced no detectable products by either gas or liquid chromatographic analysis. Oxybenzone was recovered unchanged and no products were observed from the photoinitiated reaction of oxygen with the solvent.

Irradiation of Padimate O in cyclohexane yielded the ethylhexyl esters of p-aminobenzoic acid, p-monomethylaminobenzoic acid and p-dimethylamino (o/m)-methylbenzoic acid, as well as products from the photoinitiated reaction of oxygen with the solvent.     

The π N → e +e−N reaction close to the vector-meson production threshold

The π^? p→e ⁺ e ^? n and π⁺ n→e ⁺ e ^? p reaction cross sections are calculated below and in the vicinity of the vector-meson (?⁰,ω) production threshold. These processes are largely responsible for the emission of e ⁺e^? pairs in pion-nucleus reactions and contribute to the dilepton spectra observed in relativistic heavy ion collisions. They are dominated by the decay of low-lying baryon resonances into vector-meson-nucleon channels. The vector mesons materialize subsequently into e ⁺ e^? pairs. Using πN→?⁰ N and πN→ωN, amplitudes calculated in the center of mass energy interval 1.4 < √s<1.8 GeV, we compute the π^? p→e ⁺ e ^? n and π⁺ n→e ⁺ e ^? p reaction cross sections in these kinematics. Below the vector-meson production threshold, the π⁰?ω interference in the e ⁺ e^? channel appears largely destructive for the π^? p→e ⁺ e ^? n cross section and constructive for the π⁺ n→e ⁺ e ^? p cross section. The pion beam and the HADES detector at GSI offer a unique possibility to measure these effects. Such data would provide strong constraints on the coupling of vector-meson-nucleon channels to low-lying baryon resonances.     

The speed of biased random walk on percolation clusters

 We consider biased random walk on supercritical percolation clusters in ℤ². We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. Received: 20 November 2002 / Revised version: 17 January 2003 Published online: 15 April 2003 Research supported by Microsoft Research graduate fellowship. Research partially supported by the DFG under grant SPP 1033. Research partially supported by NSF grant #DMS-0104073 and by a Miller Professorship at UC Berkeley. Mathematics Subject Classification (2000): 60K37; 60K35; 60G50 Key words or phrases: Percolation – Random walk