A “Multiplicative Coboundary” theorem for some sequences of random matrices

Schmidt(¹⁰) showed that if the family of distributions of the partial sums of a strictly stationary random sequence is tight, then that random sequence is a coboundary. Here an analog of that result is proved for some sequences of random matrices, with partial sums replaced by matrix products.     

Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

Stories about groups and sequences

The main theme of this article is that counting orbits of an infinite permutation group on finite subsets or tuples is very closely related to combinatorial enumeration; this point of view ties together various disparate stories. Among these are reconstruction problems, the relation between connected and arbitrary graphs, the enumeration of N-free posets, and some of the combinatorics of Stirling numbers.Dedicated to Hanfried Lenz on the occasion of his 80th birthday.     

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.     

The permutational power of a priority queue

A priority queue transforms an input permutation of some set of sizen into an output permutation. It is shown that the number of such pairs (, ) is (n + 1) ^n–1. Some related enumerative and algorithmic questions are also considered.Supported by the National Science and Engineering Research Council of Canada under Grant A4219.     

Stereogenicity/Astereogenicity as Global/Local Permutation-Group Symmetry and Relevant Concepts for Restructuring Stereochemistry

Molecules of ligancy 4 that have been derived from an allene, an ethylene, a tetrahedral, and a square-planar skeleton have been investigated to show that their symmetries are dually and distinctly controlled by point groups and permutation groups. Insomuch as the point-group symmetry was exhibited to control the chirality/achirality of a molecule, sphericity in a molecule, and enantiomeric relationship between molecules [S. Fujita, J. Am. Chem. Soc. 112 (1990) 3390], the permutation-group symmetry has been now clarified to control the stereogenicity of a molecule, tropicity in a molecule, and diastereomeric relationship between molecules. To characterize permutation groups, proper and improper permutations have been defined by comparing proper and improper rotations. Thereby, such permutation groups are classified into stereogenic and astereogenic ones. After a coset representation (CR) of a permutation group has been ascribed to an orbit (equivalence class), the tropicity of the orbit has been defined in term of the global stereogenicity and the local stereogenicity of the CR. As a result, the conventional stereogenicity has now been replaced by the concept local stereogenicity of the present investigation. The terms homotropic, enantiotropic, and hemitropic are coined and used to characterize prostereogenicity. Thus, a molecule is defined as being prostereogenic if it has at least one enantiotropic orbit. Since this definition has been found to be parallel with the definition of prochirality, relevant concepts have been discussed with respect to the parallelism between stereogenicity and chirality in order to restructure the theoretical foundation of stereochemistry and stereoisomerism. The derivation of the skeletons has been characterized by desymmetrization due to the subduction of CRs. The Cahn–Ingold–Prelog (CIP) system has been discussed from the permutational point of view to show that it specifies diastereomeric relationships only. The apparent specification of enantiomeric relationships by the CIP system has been shown to stem from the fact that diastereomeric relationships and enantiomeric ones overlap occasionally in case of tetrahedral molecules.     

Laponite clay in homopolymer and tri-block copolymer matrices

Macromolecule/laponite nanomaterials were studied by DSC and X-ray diffraction techniques. The matrices are poly(ethylene) glycols at various molecular masses and poly(ethylene oxides)-poly(propylene oxides)-poly(ethylene oxides) tri-block copolymers. The latter were tuned by modulating the molecular masses, at constant hydrophilic/hydrophobic ratio, and the hydrophilicity. For all the investigated systems, the enthalpy of melting (ΔH _m) is nearly constant up to a given composition thereafter it increases monotonically reaching the value of the pure macromolecule. We proposed a model to interpret the DSC data. Briefly, it was invoked a mechanism of interaction following which some segments of the adsorbed macromolecule are anchored to the laponite (RD) particles and the remaining segments are radiating away from the surface. The portion of the macromolecule in contact with RD does not contribute to ΔH _m whereas that radiating away from the clay does. Once that the RD surface is saturated, the excess of the macromolecule behaves like the pure one. The proposed model allowed to compute successfully the ΔH _m values. The X-ray diffraction experiments ruled out the polymer intercalation between the silicate sheets.     

Quantitative separation of zinc traces from cadmium matrices by solid-phase extraction with polyurethane foam

A system for separation of zinc traces from large amounts of cadmium is proposed in this paper. It is based on the solid-phase extraction of the zinc in the form of thiocyanate complexes by the polyurethane foam. The following parameters were studied: effect of pH and of the thiocyanate concentration on the zinc extraction, shaking time required for quantitative extraction, amount of PU foam necessary for complete extraction, conditions for the separation of zinc from cadmium, influence of other cations and anions on the zinc sorption by PU foam, and required conditions for back extraction of zinc from the PU foam. The results show that zinc traces can be separated from large amounts of cadmium at pH 3.0±0.50, with the range of thiocyanate concentration from 0.15 to 0.20 mol l⁻¹, and the shaking time of 5 min. The back extraction of zinc can be done by shaking it with water for 10 min. Calcium, barium, strontium, magnesium, aluminum, nickel and iron(II) are efficiently separated. Iron(III), copper(II) and cobalt(II) are extracted simultaneously with zinc, but the iron reduction with ascorbic acid and the use of citrate to mask copper(II) and cobalt(II) increase the selectivity of the zinc extraction. The anions nitrate, chloride, sulfate, acetate, thiosulphate, tartarate, oxalate, fluoride, citrate, and carbonate do not affect the zinc extraction. Phosphate and EDTA must be absent. The method proposed was applied to determine zinc in cadmium salts using 4-(2-pyridylazo)-resorcinol (PAR) as a spectrophotometric reagent. The result achieved did not show significant difference in the accuracy and precision (95% confidence level) with those obtained by ICP–AES analysis.     

Chemical applications of topology and group theory

The 60 even permutations of the ligands in the five-coordinate complexes, ML ₅, form the alternating group A ₅, which is isomorphic with the icosahedral pure rotation group I. Using this idea, it is shown how a regular icosahedron can be used as a topological representation for isomerizations of the five-coordinate complexes, ML ₅, involving only even permutations if the five ligands L correspond either to the five nested octahedra with vertices located at the midpoints of the 30 edges of the icosahedron or to the five regular tetrahedra with vertices located at the midpoints of the 20 faces of the icosahedron. However, the 120 total permutations of the ligands in five-coordinate complexes ML ₅ cannot be analogously represented by operations in the full icosahedral point group I _h, since I _his the direct product I×C₂ whereas the symmetric group S ₅ is only the semi-direct product A ₅S₂. In connection with previously used topological representations on isomerism in five-coordinate complexes, it is noted that the automorphism groups of the Petersen graph and the Desargues-Levi graph are isomorphic to the symmetric group S ₅ and to the direct product S ₅×S ₂, respectively. Applications to various fields of chemistry are briefly outlined.     

The computation of the representation matrices of the generators of the unitary group

A method is presented for the efficient computation of the representation matrices of the unitary group, U(n) in the Gelfand—Tsetlin basis (corresponding to the usual spin-symmetry adapted basis for an N electron CI). The present scheme is conceptually and computationally attractive in that it is formulated directly in terms of Weyl tableaux and also that it permits simultaneous basis vector generation and matrix element evaluation. In addition the basis vectors are ordered so that subsequent restriction to the three dimensional rotation group is facilitated. An illustrative example is also presented.Taken in part from a thesis submitted to the University of London in partial fulfilment of the requirements for the degree of PhD.