Unsaturated Molecules Containing Main Group Metals

Molecules in which there are neighboring electrophilic and nucleophilic centers are unusually reactive. Oligomerization can be prevented only by bulky groups attached to the main group metal atom that would act as electron pair acceptor, or to the basic non-metal atom. The basic and the acid centers behave as a single unit in chemical reactions; the system is similar to a “double bond” whose π-electron density is largely concentrated at one atom. The unsaturated nature of these molecules can be seen in (for example) their addition reactions with hydrogen compounds of non-metals, or in reactions that are distantly related to cycloadditions at homopolar double bonds. The selection of suitable reaction partners leads to polycyclic, cage-like molecules containing metal atoms. If these atoms possess lone pairs (as is usual in the lower oxidation states of the third and fourth main groups), these can be utilized to form bonds to further (Lewis acid) metal centers. In some cases large assemblies can be built up from polycyclic systems in this way; a characteristic of these assemblies is a one-dimensional array of metal atoms. Commonly occurring structural features of the polycyclic species are tetrahedra, trigonal bipyramids and cubes.     

希尔伯特空间有限区域成像积分算子的谱

本文讨论了希尔伯特空间中有限区域成像积分算子的本征值和本征函数,研究了非对称核的情形,并用微扰和有限秩方法计算了衍射受限和离焦情况下的本征值和本征函数。     

Time-dependent natural convection in a square cavity: Application of a new finite volume method

A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of 0-71 and Rayleigh numbers equal to 10³, 10⁴ and 10⁵. Both 11 × 11 and 21 × 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.     

Lehmer's problem for compact Abelian groups

We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.

     

On the 𝔉‐hypercentre of a finite group

The main objective of this paper is to study and describe the hypercentre of a finite group associated with saturated formations, in terms of some subgroup embedding properties related to permutability. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

分次环的有限正规扩张

本文研究了群分次环的有限正规分次扩张问题．利用经典环论方法，得到一个群分次环与其有限正规分次扩张环之间关于分次Jacobson根和分次素根的关系，同时，给出了分次情形的Cutting down定理和Lying over定理．     

Local forms of molecular mobility in copolymethacrylates containing side‐chain nonlinear optical chromophores

Local forms of molecular mobility of comb‐like random copolymethacrylates, containing side‐chain nonlinear optical chromophores and amyl‐ or octafluoroamyl‐groups, are studied. In the glassy state, two relaxation processes are found, which are due to reorientation of terminal polar groups (γ₂‐process) and rotation of chromophore groups about their long axes (β‐process). For octafluoroamyl‐containing copolymers, the γ₂‐process is superposition of the two molecular motions due to reorientation of terminal chromophore groups and ? CF₂H groups. The intensity and relaxation times in the range of the γ₂‐process depend on the chemical structure and concentration of chromophores in a copolymer. For the β‐process, the molecular mobility weakly depends on the chromophore structure, being mainly determined by the relative lengths of the rigid chromophore and flexible side chains. On the global spectrum of depolarization current it was observed the γ₁‐process due to reorientation of (CH₂)_n spacers between chromophore groups and copolymer backbone. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1488–1496, 2008     

Photoreactions of a polyamic acid containing 2,2-dinitrodiphenylmethane segments at symmetrical positions in the main chain

A polyamic acid containing 2,2^′-dinitrodiphenylmethane segments in the main chain was synthesized from 4,4^′-diamino-2,2^′-dinitrodiphenylmethane and pyrromellitic dianhydride. Irradiation of the polyamic acid was done in solution (in a protic solvent, isopropanol, in isopropanol in the presence of a triplet exciter, benzophenone and in acidified isopropanol) and as thin film. The irradiations were done in an immersion type photoreactor using a 125 W high-pressure mercury vapour lamp. The average molecular weights of the polymers were determined by gel permeation chromatography. The polydispersity index of the polyamic acid and the photoproduct showed the absence of photodegradation. The photoproducts were identified by comparing the spectra of polymer photoproducts with that of low molecular weight 2,2^′-dinitrodiphenylmethane derivatives under identical conditions. A possible mechanism for the phototransformations is suggested. Evaluation of photoresist behaviour of the polyamic acid was also done.     

On automorphisms of finite Abelian <Emphasis Type="Italic">p</Emphasis>-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A _p . For a finite abelian p-group A of type (k ₁, ..., k _n ), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k ₁, k ₂) is analyzed. This work has begin during the visit of the second author to the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University during the period July 31–August 13, 2005. This visit was supported by the Nicolaus Copernicus University and a grant from Cnpq.     

Unconditionally Stable Z-Transform ADI-PML Algorithm for Lorentzian Double Negative Meta-Materials

Unconditionally stable formulations of the anisotropic perfectly matched layer (APML) are presented for truncating double negative (DNG) meta-material finite difference time domain (FDTD) grids. In the proposed formulations, the Z-transform theory is employed in the alternating direction implicit FDTD (ADI-FDTD) scheme to obtain update equations for the field components in the DNG meta-material domains. Numerical examples carried out in one dimensional Lorentzian type DNG meta-material domains are included to show the validity of the proposed formulations.