Synthesis and Crystal Structure of 3,3''-Bis(2-benzimidazolyl)-2,2''-dipyridine with Hydrated Zinc(Ⅱ) Perchlorate

The reaction of Zn(ClO4)2·6H2O with 3,3'-bis(2-benzimidazolyl)-2,2'-dipyridine (L) gave a mononuclear zinc(Ⅱ) complex: [ZnL2](ClO4)2·2DMF·4H2O, which was structurally characterized by EA, IR, UV, 1H NMR, fluorescence spectrum and single-crystal X-ray diffraction. The crystal (C54H54Cl2N14O14Zn, Mr=1259.38) belongs to the monoclinic system, space group C2/c with a=20.874(2), b=14.9953(16), c=20.462(3) (A), β=101.553(2)°, V=6274.8(13) (A)3, Z=4, Dc=1.333 g/cm3, F (000)=2608, μ(MoKα)=0.548 mm-1, R=0.0682 and Wr=0.1931 for 4984 observed reflections with Ⅰ ＞ 2σ(Ⅰ). The Zn(Ⅱ) is four-coordinated in a slightly distorted tetrahedral geometry through four N atoms from four benzimidazole units of two ligands. In the crystal lattice, the [ZnL2]2 cations are linked to each other by extensive intermolecular hydrogen bonds between nitrogen atoms of benzimidazole rings, water and DMF molecules.     

Cyclic by prime fixed point free action

Let the finite group be acting on a (solvable) group and suppose that no non-trivial element of is fixed under the action of all the elements of . Assume furthermore that . A long standing conjecture is that then the Fitting height of is bounded by the length of the longest chain of subgroups of . Even though this conjecture is known to hold for large classes of groups , it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.

     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.     

On the Continuous Extension of States on the Ccr Algebra

The algebra CCR (E) of the CCR depends sensitively on the choice of the one boson test-function space E. Here the unique extension of certain continuous states, resp. representations, to the extended CCR algebra CCR E corresponding to an enlarged test-function space E is studied. Various applications in statistical physics, QED, quantum optics, solid state physics, and quantum chemistry are pointed out.     

Unitarity of highest weight modules for quantum groups

We determine the highest weights that give rise to unitarity when q is real. We further show that when q is on the unit circle and q ± 1, then unitary highest-weight representations must be finite-dimensional and q must be a root of unity. We analyze the special case of the 'Ladder' representations for su . Finally we show how the quantized Ladder representations and their analogues for other Lie algebras play an important role.     

Construction of Local Quartic Spline Elements for Optimal-Order Approximation

This paper is concerned with a study of approximation order and construction of locally supported elements for the space of (piecewise polynomial) functions on an arbitrary triangulation of a connected polygonal domain in . It is well known that even when is a three-directional mesh , the order of approximation of is only 4, not 5. The objective of this paper is two-fold: (i) A local Clough-Tocher refinement procedure of an arbitrary triangulation is introduced so as to yield the optimal (fifth) order of approximation, where locality means that only a few isolated triangles need refinement, and (ii) locally supported Hermite elements are constructed to achieve the optimal order of approximation.

     

Polymer models and generalized Potts-Kasteleyn models

The well-established relation between Potts models withv spin values and random-cluster models (with intracluster bonding favored over intercluster bonding by a factorv) is explored, but with the random-cluster model replaced by a much generalized polymer model, implying a corresponding generalization of the Potts model. The analysis is carried out in terms a given defined functionR(), an entropy/free-energy density for the polymer model in the casev=1, expressed as a function of the density of units. The aim of the analysis is to determine the analogR _v () ofR() for general nonnegativev in terms ofR(), and thence to determine the critical value of density _vg at which gelation occurs. This critical value is independent ofv up to a valuev _P, the Potts-critical value. What is principally required ofR() is that it should show a certain given concave/convex behavior, although differentiability and another regularizing condition are required for complete conclusions. Under these conditions the unique evaluation ofR _v () in terms ofR() is given in a form known to hold for integralv but not previously extended. The analysis is carried out in terms of the Legendre transforms of these functions, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently andv _P is easily determined. The value ofv _P is 2 under mild conditions onR. Special interest attaches to the functionR ₀(), which is shown to be the greatest concave minorant ofR(). The naturalness of the approach is demonstrated by explicit treatment of the first-shell model.