Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder , where , with a simply connected bounded domain. The equations considered are of the type , with modeled on . We consider a solution satisfying the boundary conditions: for , and , as . We show that the overdetermined co-normal condition for , with 0$"> for at least one value , forces the spherical symmetry of the ground domain and of the solution.

     

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka–Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only if it is flat.     

The Use of an 8X Beam Condenser with the Beckman IR-11 and IR-12

A new 8X beam condenser has been designed by the Beckman Instruments, Inc., for use with the supersize IR-11 and IR-12. This letter will report on results obtained with this accessory.     

A corresponding states principle-based equation for the surface tension of Alkenes

This study presents a new formula for the surface tension prediction of alkenes. As a first step, an analysis of the available data of the experimental surface tension data for alkenes was performed. The experimental data were collected, after a careful literature survey, for the following pure fluids: propene, 1-hexene, 1-heptene, 1-octene, 1-decene, 1-tetradecene, and 1-pentadecene. Then, the experimental data were regressed with the most reliable semi-empirical correlating methods based on the corresponding state theory existing in the literature. As a final step, an analysis of the available data of the experimental surface tension data for alkenes was performed starting from the two recently proposed equations for the prediction of the surface tension of refrigerants based on the corresponding states principle. To minimize the deviation between the predicted data and the experimental data and to find the optimal equation coefficients for experimental data regression, a (μ + λ)-evolution strategy was adopted. The analysis showed that the equation that gave the best results for the prediction of the surface tension of alkenes was the one with a very limited number of parameters. The finally proposed equation is very simple and gives a noticeable improvement with respect to the existing equations. It is based on the corresponding state principle, containing the acentric factor, the critical temperature, and pressure.     

Peptide sequence mediated self-assembly of molybdenum blue nanowheel superstructures

The precise control over the formation of complex nanostructures, e.g. polyoxometalates (POMs), at the sub-nanoscale is challenging but critical if non-covalent architectures are to be designed. Combining biologically-evolved systems with inorganic nanostructures could lead to sequence-mediated assembly. Herein, we exploit oligopeptides as multidentate structure-directing ligands via metal-coordination and hydrogen bonded interactions to modulate the self-assembly of POM superstructures. Six oligopeptides (GH, AH, SH, G₂H, G₄H and G₅H) are incorporated into the cavities of Molybdenum Blue (MB) POM nanowheels. It is found that the helicity of the nanowheel can be readily switched (Δ to Λ) by simply altering the N-terminal amino acid on the peptide chain rather than their overall stereochemistry. We also reveal a delicate balance between the Mo-coordination and the hydrogen bonds found within the internal cavity of the inorganic nanowheels which results in the sequence mediated formation of two unprecedented asymmetrical nanowheel frameworks: {Mo₁₂₂Ce₅} and {Mo₁₂₆Ce₄}.

Peptide sequence can be used to control the self-assembly and structures of nanoscale molybdenum blue polyoxometalate (POM) wheel-shaped clusters.