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.     

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.

     

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.     

Adaptation of a fast Fourier transform-based docking algorithm for protein design

Designing proteins with novel protein/protein binding properties can be achieved by combining the tools that have been developed independently for protein docking and protein design. We describe here the sequence-independent generation of protein dimer orientations by protein docking for use as scaffolds in protein sequence design algorithms. To dock monomers into sequence-independent dimer conformations, we use a reduced representation in which the side chains are approximated by spheres with atomic radii derived from known C2 symmetry-related homodimers. The interfaces of C2-related homodimers are usually more hydrophobic and protein core-like than the interfaces of heterodimers; we parameterize the radii for docking against this feature to capture and recreate the spatial characteristics of a hydrophobic interface. A fast Fourier transform-based geometric recognition algorithm is used for docking the reduced representation protein models. The resulting docking algorithm successfully predicted the wild-type homodimer orientations in 65 out of 121 dimer test cases. The success rate increases to approximately 70% for the subset of molecules with large surface area burial in the interface relative to their chain length. Forty-five of the predictions exhibited less than 1 A C(alpha) RMSD compared to the native X-ray structures. The reduced protein representation therefore appears to be a reasonable approximation and can be used to position protein backbones in plausible orientations for homodimer design.     

Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by _m (L)=(Inq/E)L ^–d +O(L ^–2d ) with respect to the infinite-volume transition point _t . We also propose an alternative definition of the finite-volume transition temperature _t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections from _t .     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.