An existence result for a quadrature surface free boundary problem

The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.     

A Generalized Sampling Theorem with the Inverse of an Arbitrary Square Summable Sequence as Sampling Points

In this article a generalized sampling theorem using an arbitrary sequence of sampling points is derived. The sampling theorem is a Kramer-type sampling theorem, but unlike Kramer's theorem the sampling points are not necessarily eigenvalues of some boundary value problems. The theorem is then used to characterize a class of entire functions that can be reconstructed from their sample values at the points t_n = an + b if n = 0, 1, 2, ... and t_n = an + c if n = 0, -1, -2, ..., where a, b, c are arbitrary constants. The reconstruction formula is derived explicitly in the form of a sampling series expansion. When a = 1, b = 0 = c, the famous Whittaker-Shannon-Kotel'nikov sampling theorem is obtained as a special case.     

Theoretical estimates for proton-NMR spin–lattice relaxation rates of heterometallic spin rings

Heterometallic molecular chromium wheels are fascinating new magnetic materials. We reexamine the available experimental susceptibility data on MCr₇ wheels in terms of a simple isotropic Heisenberg Hamiltonian for M=Fe, Ni, Cu, and Zn and find in that FeCr₇ needs to be described with an iron–chromium exchange that is different from all other cases. In a second step we model the behavior of the proton spin lattice relaxation rate as a function of applied magnetic field for low temperatures as it is measured in nuclear magnetic resonance (NMR) experiments. It appears that CuCr₇ and NiCr₇ show an unexpectedly reduced relaxation rate at certain level crossings.     

Postmodification of poly(9‐alkyl‐9H‐carbazole‐2,7‐diyl)s as a route to new main‐chain‐functionalized carbazole polymers

The postmodification of poly[9‐(2‐hexyldecyl)‐9H‐carbazole‐2,7‐diyl] ( P1 ) upon its reaction with N‐bromosuccinimide affords exclusive and full bromination of the 3,6‐positions of the carbazole repeat units to yield poly[3,6‐dibromo‐9‐(2‐hexyldecyl)‐9H‐carbazole‐2,7‐diyl] ( P2 ). Brominated polymer P2 can be used as a precursor for further functionalization at the 3,6‐positions with the desired functional group to afford other useful polymers. Polymer P2 has hence been reacted with copper(I) cyanide to afford poly[3,6‐dicyano‐9‐(2‐hexyldecyl)‐9H‐carbazole‐2,7‐diyl] ( P3 ). Full substitution of the bromide groups with nitrile‐functional groups has been achieved. The preparation and structural characterization of polymers P2 and P3 are presented together with studies on their electronic conjugation and photoluminescence properties. Cyclic voltammetry studies on polymer P3 indicate that the new polymer is easier to reduce (n‐dope) but more difficult to oxidize than its unsubstituted counterpart ( P1 ) as a result of the introduction of the electron‐withdrawing nitrile‐functional groups at the 3,6‐positions on the carbazole repeat units on the polymer chains. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 3336–3342, 2006     

Fe2P2O7 and Fe2P4O12 studied between 5–800 K

Mössbauer spectra of triclinic Fe₂P₂O₇ indicate the existence of two crystallographic metal positions in the structure. In the paramagnetic region the two Mössbauer doublets are closely overlapping. The magnetic transition takes place at ≈ 21 K and the saturated fields are around 12 tesla for the two positions. In monoclinic Fe₂P₄O₁₂ the two octahedrally coordinated metal positions give quite different quadrupole splittings (1.5 and 3 mm/s at room temperature) and hyperfine field values (42 and 12.5 Tesla at 5 K). The transition temperature is at ≈ 18.5 K.     

Heat “polynomials” for a singular differential operator on (0, ∞)

We generalize the theory of the heat polynomials introduced by P. V. Rosenbloom and D. V. Widder for a more general class of singular differential operator on (0, ). The heat polynomials associated with the Bessel operator and studied by D. T. Haimo appear as a particular case in this paper. In the special cases of second derivative and Bessel operators the heat polynomials are in fact polynomials inx andt, however, this property does not hold in general.Communicated by Tom. H. Koornwinder.     

Catalytic Para‐Xylene Maximization. V. Toluene Methylation with Methanol and Disproportionation of Toluene Using Pt/ZSM‐5 and Pt/Mordenite Catalysts

Toluene was methylated with methanol and disproportionated using catalysts containing different Pt contents (0.2, 0.4 and 0.6%) supported on H‐ZSM‐5 or H‐mordenite (H‐M) zeolites in a fixed‐bed flow‐reactor operated atmospherically at temperatures of 300–500 °C in a flow of hydrogen. Platinum dispersion in the zeolite supports and acid sites strength distribution were evaluated using hydrogen chemisorption (1:1 stoichiometry) and ammonia temperature programmed desorption (TPD) in a differential scanning calorimeter (DSC). Toluene methylation was much faster on all catalysts than toluene disproportionation (DISP). Both reactions were more accelerated using H‐ZSM‐5 containing catalysts than H‐M containing catalysts. The yield of xylenes, and in particular para‐xylene, was significantly influenced by the yield of trimethylbenzenes (TMBs) in product. The selectivities for para‐, ortho‐ and meta‐xylenes production were found largely dependent on the Pt content in the catalysts, particularly when supported on H‐ZSM5‐zeolite. However, using Pt/H‐M catalysts, these selectivities were not strictly controlled by Pt content in the catalysts.