Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

The synthesis and structure of potassium cyanotrimethylaluminate

The reaction of KCN with Al(CH₃)₃ to form K[Al(CH₃)₃CN] is greatly facilitated by the presence of an aromatic solvent: for p-xylene a solid complex, K[Al(CH₃)₃CN]·C₆H₄(CH₃)₂, has been isolated. The crystal structure of potassium cyanotrimethylaluminate has been determined from three-dimensional X-ray data measured by counter methods. K[Al(CH₃)₃CN] crystallizes in the monoclinic space group C2/c with cell dimensions a = 19.902(7), b = 9.211(4), c = 9.615(4) Å, β = 107.74(5)°, and p_calcd. = 1.09 g cm^?1 for Z = 8. Least squares refinement gave a conventional weighted R factor of 4.9% for 807 independent reflections. The monomeric [Al(CH₃)₃CN]^? units possess no crystallographic symmetry, and the packing in the unit cell is such that the nitrogen atoms on three such units approach the potassium atom to within 3.11 Å. The average aluminum-methyl carbon bond distance is 1.971 (7) Å, while the aluminum-cyano carbon distance is 2.047 (7) Å. This significant lengthening is attributed to partial electron deficiency in the aluminum-cyano carbon bond.     

The classical solution of the one-dimensional two-phase stefan problem with energy specification

Summary Existence and unicity for the solution of the one-dimensional two-phase Stefan problem with energy specification in the liquid phase is established via potential theoretic arguments employing the Schauder fixed point theorem and a contracting map.This work was supported in part by NSF Grant MCS-7801687.This work was carried out at UT-Austin while on sabbatical from the University of Adelaide.     

Six‐coordinate Co2+ with imidazole,NH3, and H2O ligands: Approaching spin crossover

Octahedral, six‐coordinate Co²⁺ can exist in two spin states: S = 3/2 and S = 1/2. The difference in energy between high spin (S = 3/2) and low spin (S = 1/2) is dependent on both the ligand mix and coordination stereochemistry. B3LYP calculations on combinations of neutral imidazole, NH₃, and H₂O ligands show that low‐spin isomers are stabilized by axial H₂O ligands and in structures that also include trans pairs of equatorial NH₃ and protonated imidazole ligands, spin crossover structures are predicted from spin state energy differences. Occupied Co d orbitals from the DFT calculations provide a means of estimating effective ligand strength for homoleptic and mixed ligand combinations. These calculations suggest that in a labile biological system, a spin crossover environment can be created. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

The kinetics of propene and hexene polymerisation with [(SBI)ZrR]+X-: evidence for monomer-dependent early or late transition states

Whereas the rates of propene polymerisation catalysed by zirconocene ion pairs are strongly anion-dependent, hexene polymerisations are not; the findings demonstrate the existence of very different kinetic regimes for two closely related reactions.     

The bidmensional stefan problem with convection: the time dependent case

This paper considers the time dependent stefan problem with convection in the fluid phase governed by the Stokes equation, and with adherence of the fluid on the lateral boundaries: The existence of a weak solution is obtained via the introduction of a temperature dependent penalty term in the fluid flow equation, together with the application of various compactness arguments.