Crowns,cutsets, and valuable partial orders

We consider the problem of recognizing those partial orders which admit a valuation: this is a linear-algebraic condition which arises naturally in an algebraic/geometric context. We show that a partial order has at most one valuation (which is integer-valued) and present various structural conditions which are either necessary or sufficient for a partial order to be valuable. The first main result is a reduction theorem which allows us to restrict attention to those partial orders which do not have a bounded cutset. We use this and a theorem of Kelly and Rival to prove the second main result: that every contraction of a bounded partial order is fibre-valuable if and only if the partial order is a dismantlable lattice. This result has a geometric interpretation.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392.     

Groups with super-exponential subgroup growth

We show that, if the subgroup growth of a finitely generated (abstract or profinite) groupG is super-exponential, then every finite group occurs as a quotient of a finite index subgroup ofG. The proof involves techniques from finite permutation groups, and depends on the Classification of Finite Simple Groups.The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T7441. The second author was partially supported by the Israeli National Science Foundation.     

Scaling laws of complex band random matrices

We study a model of complex band random matrices capable of describing the transitions between three different ensembles of Hermitian matrices: Gaussian orthogonal, Gaussian unitary and Poissonian. Analyzing numerical data we observe new scaling relations based on the generalized localization length of eigenvectors. We show that during transitions between canonical ensembles of random matrices the changes of statistical properties of eigenvalues and eigenvectors are correlated.     

Preparation of a novel HALS polypropylene stabilizer

The radiation induced graft copolymerization of m-isopropenyl-α,α-dimethyl benzyl iso-cyanate (m-TMI) and styrene onto polypropylene was carried out. The extent of grafting increased with increasing amount of styrene and with increased radiation dose. A graft load of 180% was obtained by immersing a 50 kGy pre-irradiated film in a monomer solution containing 25 mol % m-TMI and 75 mol % styrene. The graft copolymer is suitable for covalently binding nonpolymerizable stabilizers with a suitable nucleophilic moiety. In this work the isocyanate moiety of the graft copolymer was allowed to react with 4-amino-2,2,6,6-tetramethyl piperidine, a hindered amine light stabilizer. Fourier trans-formed infrared spectroscopy confirmed the formation of an urea moiety. © 1995 John Wiley & Sons, Inc.     

Preparation and reaction of bis (perfluoroalkanesulfonyl) methyl halides

Halogenation of the potassium or silver salts of bis(trifluoromethanesulfonyl)methane (CF3SO2)2CH2 and its cyclo analogues 1 with N-fluoro-bis(trifluoromethanesulfonyl)imine [(CF3SO2)2-NF], chlorine or bromine gave good yields of the corresponding α-halo disulfone (CF3SO2)2CHX and cyclo analogues 9, 10. The chemical transformation of these fluorinated α-halo-disulfones are described.     

On the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control

Summary. According to the methodology of [6], many measures of distance arising in problems in numerical linear algebra and control can be bounded by a factor times the reciprocal of an appropriate condition number, where the distance is thought of as the distance between a given problem to the nearest ill-posed problem. In this paper, four major problems in numerical linear algebra and control are further considered: the computation of system Hessenberg form, the solution of the algebraic Riccati equation, the pole assignment problem and the matrix exponential. The distances considered here are the distance to uncontrollability and the distance to instability. Received November 4, 1995 / Revised version received March 4, 1996     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Detecting embedded pure network structures in LP problems

Recently, linear programming problems with special structures have assumed growing importance in mathematical programming. It is well known that exploiting network structures within linear programs can lead to considerable improvement of the computational solution of large-scale linear programming problems. A linear program is said to contain an embedded network structure provided that some subset of its constraints can be interpreted as specifying conservation of flow. If a column of the constraint matrix has at most two non-zeros, then it leads to embedded generalized network structure and if these non-zeros are unit elements and of opposite signs, then it leads to embedded pure network structure. In this paper, we are concerned with algorithms for detecting embedded pure network structures within linear programs. The network extraction methods are presented in two groups. The first group covers deletion and addition based algorithms and the second group covers GUB based algorithms. We have extended the GUB based algorithm appearing in the second group by introducing Markowitz merit count approach for exploiting matrix non zeros. A set of well known test problems has been used to carry out computational experiments which show that our extensions to the GUB based algorithms give better results than the algorithms reported earlier.     

Helmholtz operator with a quaternionic wave number and associated function theory. II. Integral representations

A theory of quaternion-valued hyperholomorphic functions (h.h.f.) is being developed which is closely related to the Maxwell equations for monochromatic electromagnetic fields. The main integral formulas are established, and some boundary-value properties are studied.     

Symplectic integration of Hamiltonian wave equations

Summary The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.