On the kinetics of the first order phase transitions

An attempt is made to analyse the kinetics of the first order structural phase transitions on the basis of the Landau-Ginzburg equation. Two simple solutions are proposed: One in the form of one-dimensional solitary wave which can be considered as a planar interface separating two phases and travelling along direction normal to the interface. Any direction of propagation is possible. The other solution has a form of a onedimensional kink-antikink couple which under supercooling conditions reveals properties of a critical nucleus. A short discussion of the free energy density form for one- and multidimensional order parameters, and of the symmetry of the quadratic gradient term is included.     

A transplantation theorem for Fourier-Bessel coefficients

We prove a transplantation theorem for Fourier-Bessel coefficients. Theorems of such type were proved byAskey andWainger [1] andAskey [2] for ultraspherical and Jacobi coefficients, respectively. Our theorem can be also seen as a dual result to a transplantation theorem for Fourier-Bessel series which was proved byGilbert [3].     

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ₂ growth condition, (X ₁,Y ₁), (X ₂,Y ₂),…,(X _n ,Y _n ) be arbitrary independent random vectors such that for any given i either Y _i =X _i or Y _i is independent of all the other variates. The purpose of this paper is to develop an approximation of

Complicated dynamics of parabolic equations with simple gradient dependence

Let be a smooth bounded domain. Given positive integers , and , , ..., , consider the semilinear parabolic equation

where and are smooth functions. By refining and extending previous results of Polácik we show that arbitrary -jets of vector fields in can be realized in equations of the form (E). In particular, taking we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on .

     

CO substitution in Ru3(CO)12 by 2,3-bis(diphenylphosphino)-N-phenyl-maleimide (bppm). X-ray diffraction structure of Ru2(CO)6(bppm)

The cluster Ru₃(CO)₁₂ reacts with the diphosphine ligand 2,3-bis(diphenylphosphino)-N-phenyl-maleimide (bppm) in toluene solution at 100C by fragmentation to give a 2:1 mixture of the donor–acceptor complex Ru₂(CO)₆(-bppm) and the phosphido-bridged species Ru₂(CO)₆ , respectively. The new diruthenium compounds have been isolated by chromatography and characterized in solution by IR and NMR (¹H and ³¹P) spectroscopies, in addition to X-ray diffraction analysis in the case of Ru₂(CO)₆( ₂ -bppm). Ru₂(CO)₆( ₂ -bppm), which exists as two crystallographically independent molecules in the unit cell, crystallizes in the orthorhombic space group Pbca, a = 18.8441(9), b = 21.352(2), c = 36.510(2) Å, V = 14,690(1) ų, z = 16, D _calc = 1.649 g/cm³; R = 0.0356, and R _w = 0.0812 for 17732 observed reflections. The reactivity of the bis(diphenylphosphino)-N-phenyl-maleimide ligand with Ru₃(CO)₁₂ under thermolysis conditions is contrasted with the related diphosphine ligands 2,3-bis(diphenylphosphino)maleic anhydride (bma) and 4,5-bis(diphenylphosphino)-4-cyclopenten-1,3-dione (bpcd).     

Cage-annulated thiacrown ethers and thiacryptands

A series of cage-annulated sulfur-containing crown ethers and cryptands have been synthesized as possible specific metal host systems. The synthesis and structures of seven compounds are described, including a thiacryptand complex with Hg(II). The trishomocubane cage is essentially spherical except for a methylene group, which imparts no steric restrictions, and two disordered superimposed orientations occur in most structures. This superposition of four- and five-membered rings usually cannot be resolved into separate entities, resulting in distorted distances, angles, and thermal parameters for the cage. (3) I4₁ cd, a = 13.207(3) Å, b = 13.207(3) Å, c = 35.876(12) Å; (10) C2, a = 14.551(2) Å, b = 10.028(1) Å, c = 10.491(1) Å, = 107.108(2)°; (14) P2₁/n, a = 10.6277(8) Å, b = 9.8488(7) Å, c = 21.822(2) Å, = 97.945(2)°; (19) P2₁/c, a = 15.381(3) Å, b = 6.667(1) Å, c = 18.158(3) Å, = 94.838(4)°; (25) C2/c, a = 34.386(4) Å, b = 11.318(1) Å, c = 13.409(2) Å, = 110.044(2)°; (28) , a = 10.4487(8) Å, b = 11.5677(9) Å, c = 13.354(1) Å, = 71.042(1)°, = 87.344(1)°, = 65.839(1)°; (29) P2₁/c, a = 10.8138(5) Å, b = 16.4949(8) Å, c = 22.054(1) Å, = 96.087(1)°.     

A representation of local time for Lipschitz surfaces

Summary Suppose thatD ⁿ ,n2, is a Lipschitz domain and letN _t (r) be the number of excursions of Brownian motion insideD with diameter greater thanr which started before timet. ThenrN _t (r) converges asr0 to a constant multiple of local time on D, a.s. and inL ^p for allp<. The limit need not exist or may be trivial (0 or ) in Hölder domains, non-tangentially accessible domains and domains whose boundaries have finite surface area.The first author was supported by NSF Postdoctoral FellowshipThe second and third authors were supported in part by NSF grants DMS 8701073 and DMS 8702620     

Semigroups of{\cal I} - density continuous functionscontinuous functions

This paper is concerned with the classes of continuous functions and continuous functions as semigroups with composition as the operation. We also analyze some of their subsemigroups. It is shown that the groups of automorphisms of these semigroups and several of their subsemigroups have the inner automorphism property. Communicated by M. W. Mislove     

Phosphole, pyrrole, and their tetrahydro derivatives: A theoretical study of their properties

A theoretical analysis of four molecules (phosphole, phospholane, pyrrole, and pyrrolidine) in their ground and transition states has been carried out. The transition state of phosphole is highly aromatic and a partition of its energy into atomic contributions reveals that it is the phosphorus atom that is responsible for this observation.