On the Number of Tubes Touching a Sphere or a Tube

A problem is formulated about how many unit-radius tubes can touch a ball of given radius from the outside and from the inside. Upper bounds for the maximum numbers of contacts are obtained for both interior and exterior contacts. It is also shown that the maximum number of unit-radius tubes touching the same orthogonal cross-section of a particular tube of radius P is [π (arcsin(P+1)⁻¹)⁻¹] and if the number of contacts takes on its maximum, then all tubes are locally aligned.     

Quasiconformal mappings and periodic spectral problems in dimension two

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem. This work is supported by The Royal Society.     

Variational principle in canonical variables,Weber transformation,and complete set of the local integrals of motion for dissipation-free magnetohydrodynamics

The intriguing problem of the “missing” MHD integrals of motion is solved in this paper; i.e., analogues of the Ertel, helicity, and vorticity invariants are obtained. The two latter have been discussed earlier in the literature only for specific cases, and the Ertel invariant is presented for the first time. The set of ideal MHD invariants obtained appears to be complete: to each hydrodynamic invariant corresponds its MHD generalization. These additional invariants are found by means of the fluid velocity decomposition based on its representation in terms of generalized potentials. This representation follows from the discussed variational principle in Hamiltonian (canonical) variables, and it naturally decomposes the velocity field into the sum of “hydrodynamic” and “ magnetic” parts. The “missing” local invariants are expressed in terms of the “ hydrodynamic” part of the velocity and therefore depend on the (nonunique) velocity decomposition; i.e., they are gauge-dependent. Nevertheless, the corresponding conserved integral quantities can be made decomposition-independent by the appropriate choice of the initial conditions for the generalized potentials. It is also shown that the Weber transformation of MHD equations (partial integration of the MHD equations) leads to the velocity representation coinciding with that following from the variational principle with constraints. The necessity of exploiting the complete form of the velocity representation in order to deal with general-type MHD flows (nonbarotropic, rotational, and with all possible types of breaks as well) in terms of single-valued potentials is also under discussion. The new basic invariants found allow one to widen the set of the local invariants on the basis of the well-known recursion procedure.     

Polymerization of methyl methacrylate by metallocene imido complexes and tris(pentafluorophenyl)alane

The polymerization of methyl methacrylate (MMA) was investigated with tris(pentafluorophenyl)alane [Al(C₆F₅)₃] and four metallocene imido complexes that varied in the complex symmetry/chirality, metal, and R group in the ?NR moiety, as well as a zirconocene enolate preformed from the imido zirconocene and MMA. This study examined four aspects of MMA polymerization: the effects of the metallocene imido complex structure on the polymerization activity and polymer tacticity, the degree of polymerization control, the elementary reactions of the imido complex with Al(C₆F₅)₃ and MMA, and the polymerization kinetics and mechanism. There was no effect of the imido complex symmetry/chirality on the polymerization stereochemistry; the polymerization followed Bernoullian statistics, producing syndiotactic poly(methyl methacrylate)s with moderate (～70% [rr]) to high (～91% [rr]) syndiotacticity, depending on the polymerization temperature. Polymerization control was demonstrated by the number‐average molecular weight, which increased linearly with an increase in the monomer conversion to 100%, and the relatively small and insensitive polydispersity indices (from 1.21 to 1.17) to conversion. The reactions of the zirconocene imido complex with Al(C₆F₅)₃ and MMA produced the parent base‐free imido complex and the [2 + 4] cycloaddition product (i.e., zirconocene enolate), respectively; the latter product reacted with Al(C₆F₅)₃ to generate the active zirconocenium enolaluminate. The MMA polymerization with the metallocene imido complex and the alane proceeded via intermolecular Michael addition of the enolaluminate to the alane‐activated MMA involved in the propagation step. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 3132–3142, 2003     

Crystal structure and synthesis of 17alpha-(5-chlorovaleroyloxy)-16beta-methylpregna-4,6-diene-3,20-dione.

C27H37O4Cl is orthorhombic, P2(1)2(1)2(1). The unit-cell dimensions at 293 K are a = 7.1388(15), b = 12.9836(14), c = 26.665(10)A, V = 2471.5(10)A3, Dx = 1.239 g/cm3, and Z = 4. The R value is 0.070 for 1458 observed reflections. The A, B, C and D rings occur in distorted envelope, distorted half-chair, chair, and envelope and a half-chair conformations. The molecules in the crystal are packed at the normal van der Waals distances.     

An efficient algorithm for calculating whole-profile functions in crystal structure solution from powder diffraction data

A method for calculation of the whole-profile powder diffraction R-factor R_wp, adapted specifically for use in direct-space structure solution, is shown to be faster by a factor of approximately 20 than standard methods for calculating R_wp.     

Econophysics: Scaling and its breakdown in finance

We discuss recent empirical results obtained by analyzing high-frequency data of a stock market index, the Standard and Poor’s 500. We focus on the scaling properties and on its breakdown of the index dynamics. A simple stochastic model, the truncated Lévy flight, is illustrated. Successes and limitations of this model are presented. A discussion about similarities and differences between the scaling properties observed in financial markets and in fully developed turbulence is also provided.