On a parabolic equation in a triangular domain

We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space L^p(T). The method of operators sums in the non-commutative case is referred to.     

Solution of the Equations of Motion for Einstein’s Field in Fractional <Emphasis Type="Italic">D</Emphasis> Dimensional Space-Time

As a continuation of Sadallah et al. work (M. Sadallah, S. Muslih and D. Baleanu, Equations of motion for Einstein’s field in non-integer dimensional space. Czechoslov. J. Phys. 56:323, 2006), the fractional action function S is given as an integration over fractional spatial dimension D _s and fractional time D _t dimension. The variational principle which minimize S leads to Euler-Lagrange equations of motion in D _s +D _t fractional dimensions. As an example we extend our study to obtain the equations of motion for Einstein’s field in fractional D _s +D _t fractional dimensions of N+1 space-time coordinates. It is shown that the time dependent solutions are single valued for only D _s =4 dimensional space. Also the angular solutions are convergent for any value of D _s .     

Synthese eines funktionalen Aluminiumalkinids,Me3C‐C≡C‐AlBr2, und dessen Reaktionen mit der sperrigen Lithiumverbindung LiCH(SiMe3)2

Synthesis of a Functional Aluminium Alkynide, Me₃C‐C≡C‐AlBr₂, and its Reactions with the Bulky Lithium Compound LiCH(SiMe₃)₂ Treatment of aluminium tribromide with the lithium alkynide (Li)C≡C‐CMe₃ afforded the aluminium alkynide Me₃C‐C≡C‐AlBr₂ ( 1 ) in an almost quantitative yield. 1 crystallizes with trimeric formula units possessing Al₃C₃ heterocycles and the anionic carbon atoms of the alkynido groups in the bridging positions. A dynamic equilibrium was determined in solution which probably comprises trimeric and dimeric formula units. Reaction of 1 with one equivalent of LiCH(SiMe₃)₂ yielded the compound [Me₃C‐C≡C‐Al(Br)‐CH(SiMe₃)₂]₂ ( 2 ), which is a dimer via Al‐C‐Al bridges. Two equivalents of the lithium compound gave a mixture of four main‐products, which could be identified as 2 , Li[Me₃C‐C≡C‐Al{CH(SiMe₃)₂}₃] ( 3 ), Me₃C‐C≡C‐Al[CH(SiMe₃)₂]₂ ( 4 ), and Al[CH(SiMe₃)₂]₃. The lithium atom of 3 is coordinated by the C≡C triple bond and an inner carbon atom of one bis(trimethylsilyl)methyl group. Further interactions were observed to C‐H bonds of methyl groups.     

Equations of motion for Einstein’s field in non-integer dimensional space

Equations of motion for Einstein’s field in fractional dimension of 4 spatial coordinates are obtained. It is shown that time dependent part of Einstein’s wave function is single valued for only 4-integer dimensional space.     

Simulation of unsteady flows by the DSMC macroscopic chemistry method

In the Direct Simulation Monte-Carlo (DSMC) method, a combination of statistical and deterministic procedures applied to a finite number of ‘simulator’ particles are used to model rarefied gas-kinetic processes. In the macroscopic chemistry method (MCM) for DSMC, chemical reactions are decoupled from the specific particle pairs selected for collisions. Information from all of the particles within a cell, not just those selected for collisions, is used to determine a reaction rate coefficient for that cell. Unlike collision-based methods, MCM can be used with any viscosity or non-reacting collision models and any non-reacting energy exchange models. It can be used to implement any reaction rate formulations, whether these be from experimental or theoretical studies. MCM has been previously validated for steady flow DSMC simulations. Here we show how MCM can be used to model chemical kinetics in DSMC simulations of unsteady flow. Results are compared with a collision-based chemistry procedure for two binary reactions in a 1-D unsteady shock-expansion tube simulation. Close agreement is demonstrated between the two methods for instantaneous, ensemble-averaged profiles of temperature, density and species mole fractions, as well as for the accumulated number of net reactions per cell.     

Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces

In this paper, we give new results about existence, uniqueness and regularity properties for solutions of Laplace equation $$\Delta u = h \quad {\rm in} \, \Omega$$ where Ω is a cusp domain. We impose nonhomogeneous Dirichlet conditions on some part of ?Ω. The second member h will be taken in the little Hölder space ${h^{2 \sigma}(\bar{\Omega})}$ with ${\sigma \, \in \, ]0, \, 1/2[}$ . Our approach is based essentially on the study of an abstract elliptic differential equation set in an unbounded domain. We will use the continuous interpolation spaces and the generalized analytic semigroup theory.     

Polarographic behaviour of some organotin(IV) compounds in dimethyl sulphoxide

The differential pulse polarographic behaviour in dimethyl sulphoxide (DMSO) of 14 organotin(IV) compounds having the general formula R₃SnX (R = Me, Ph; X^? = NCS^?, N₃^?, N₃^?, NO₃^?, OH^?, NCO^? and OAc^?) and ⁿBu₃SnCl and ⁿBu₂SnCl₂ has been studied. The peak potential was found to depend markedly on the organic group and to a lesser extent on the nature of the anion X. The phenyltin compounds were reduced at lower potentials than the corresponding methyltin compounds. The data obtained could be used for trace determination of these compounds. Linear calibration curves were obtained over the concentration range of 2.8 × 10^?4 to 1.9 × 10^?6 mol dm^?3.     

Two Different Structural Motifs Observed for Dimeric Dialkylaluminum and Dialkylgallium Alkynides [R2E‐C≡C‐C6H5]2

The dialkylaluminum and dialkylgallium alkynides [R₂E‐C≡C‐R′]₂ (R = Me, CMe₃; E = Al, Ga; R′ = Ph) containing C≡C triple bonds attached to their central aluminum or gallium atoms are easily obtained by the reactions of dialkylelement chlorides with lithium alkynides or by treatment of the corresponding alkyne R‐C≡C‐H with dialkylaluminum or dialkylgallium hydrides. The first reaction is favored by the precipitation of LiCl, the second one by the formation of elemental hydrogen. All products form dimers in which the carbanionic carbon atoms of the alkynido groups adopt bridging positions, but, interestingly, different types of molecular structures were observed depending on the steric demand of the substituents terminally attached to the aluminum or gallium atoms. The small methyl substituents gave structures in which the aluminum or gallium atoms seemed to be side‐on coordinated by the C≡C triple bonds of almost linear E‐C≡C groups. In contrast, the more bulky tert‐butyl groups forced an arrangement in which the C≡C triple bonds were perpendicular to the E‐E axis of the molecules. Different bonding modes result, which were analyzed by quantum‐chemical calculations.