A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W _cof the Coxeter group W. Specifically, W _cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W _cand compute the cardinality of W _cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W _cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H.     

Curvature and finite domination

Upper bounds obtained by Gromov on the Betti numbers of certain closed Riemannian manifolds are shown to be upper bounds on the minimum number of cells in --spaces dominating such manifolds.

     

On representations of affine Kac-Moody groups and related loop groups

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type for some . Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

     

A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

Some remarks on R-contact flows

Let (M, ) be an R-contact manifold, then the set of periodic points of the characteristic vector field is a nonempty union of closed, totally geodesic odd-dimensional submanifolds. Moreover, the R-metric cannot have nonpositive sectional curvature. We also prove that no R-contact form can exist on any torus.     

Lattice and continuum wavelets and the block renormalization group

We obtain a resolution of the identity operator, for functions on a latticeZ ^d, which is derived from the block renormalization group. We use eigenfunctions of the terms of the decomposition to form a basis forl ₂(Z^d) and show how the basis is generated from lattice wavelets. The lattice spacing is taken to zero and continuum wavelets are obtained.     

A graphical method to construct a phylogenetic tree

A 3D graphical representation of DNA sequences, which has no circuit or degeneracy, is derived for mathematical denotation of DNA sequence. Based on this graphical representation, we propose a new sequence distance measure. We make use of the corresponding similarity matrix to construct a phylogenic tree by virtue of the fuzzy theory. The examination of phylogenic tree belong to eight species illustrates the utility of our approach. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

New 2D graphical representation of DNA sequences

We consider a 2D graphical representations of DNA sequences, which avoids loss of information associated with crossing and overlapping of the corresponding curve. We outline an approach, which is based on the construction of a three-component vector whose components are the normalized leading eigenvalues of the L/L matrices associated with DNA. The examination of similarities/dissimilarities among the coding sequences of the first exon of beta-globin gene of different species illustrates the utility of the approach.     

Sphericities of Double Cosets,Double Coset Representations,and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a _d , c _d , or b _d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.     

Thermodynamic issues associated with combined cyclic voltammetry and wafer curvature measurements in electrolytes

A thermodynamic framework has been provided for the interpretation of combined cyclic voltammetry and surface stress measurements, the latter being obtained from wafer curvature or beam deflection measurements of a solid electrode as a function of applied potential (so-called voltstressograms). Firstly, the derivation of electrocapillarity equations for solid electrodes has been critically reviewed by starting from the Gibbs adsorption equation appropriate for solid–electrolyte interfaces. This allowed us to demonstrate the critical importance of elastic surface strain in the thermodynamic boundary conditions of the partial derivatives intervening in the interpretation of voltstressograms. From these considerations, it was shown for the first time that the electrocapillarity equations for solid electrodes are not appropriate for describing the variation of surface stress with potential obtained from wafer curvature measurements, because such measurements are intrinsically incompatible with the constant strain condition implied in the electrocapillarity equations. An alternative explanation is provided for the experimentally observed proportionality between the current density, measured in cyclic voltammograms, and the first derivative of surface stress with respect to potential, obtained from voltstressograms.