Properties of balanced and perfect matrices

In this paper we define wheel matrices and characterize some properties of matrices that are perfect but not balanced. A consequence of our results is that if a matrixA is perfect then we can construct a polynomial number of submatricesA _I,,A _n ofA such thatA is balanced if and only if all the submatricesA ₁,,A _n ofA are perfect. This implies that if the problem of testing perfection is polynomially solvable, then so is the problem of testing balancedness.Partial support under NSF Grants DMS-8606188, ECS-8800281 and DDM-8800281.     

Catalysis in Organic Solvents with Lipase Immobilized by Sol-Gel Technique

A series of immobilized lipases were obtained by sol-gel process, using silica prepolymers prepared from tetramethoxysilane, methyltrimethoxysilane, propyltrimethoxysilane and (3-aminopropyl)triethoxysilane. The activities of these biocatalysts were compared with the lipase adsorbed on poly(methylhydroxysiloxane) and encapsulated into a silicone rubber, lipase entrapped in nanoporous silica matrix and commercial sol-gel lipase. Model reactions were the esterification of stearic acid and Corey lactone bisalcohol (an intermediate of prostaglandin synthesis). The positive effect of hydrophobic-hydrophilic interface, created by the addition of organosilanes, on the activity of biocatalysts was partially reduced by decreasing specific surface of mesopores. Hydrophobic solvents increased the activity of the lipase entrapped in tetramethoxysilane–methyltrimethoxysilane prepolymer in the sequence acetone < toluene < benzene < decane < hexane. The activity of silicone rubber-encapsulated biocatalysts was proportional to polymer swelling in organic solvents (hexane > toluene > acetone).     

Preparation and characterization of bismaleimide/1,3-dicyanatobenzene modified epoxy intercrosslinked matrices

An intercrosslinked network of cyanate ester (CE)-bismaleimide (BMI) modified epoxy matrix system was made by using epoxy resin, 1,3-dicyanatobenzene and bismaleimide (N,N^′-bismaleimido-4,4^′-diphenyl methane) with diaminodiphenylmethane as curing agent. BMI-CE-epoxy matrices were characterised using differential scanning calorimetry (DSC), thermogravimetric analysis (TGA) and heat deflection temperature (HDT) analysis. The matrices, in the form of castings, were characterised for their mechanical properties such as tensile strength, flexural strength and unnotched Izod impact test as per ASTM methods. Mechanical studies indicated that the introduction of cyanate ester into epoxy resin improves the toughness and flexural strength with reduction in tensile strength and glass transition temperature, whereas the incorporation of bismaleimide into epoxy resin influences the mechanical and thermal properties according to its percentage content. DSC thermograms of cyanate ester as well as BMI modified epoxy resin show an unimodal reaction exotherm. Electrical properties were studied as per ASTM method and the morphology of the BMI modified epoxy and CE-epoxy systems were studied by scanning electron microscope.     

Unital affine semigroups

Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.     

ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

The (mixed finite element) discretization of the linearized Navier–Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ‘grad‐div’ (GD) stabilization, as well as on the discrete side where it is known as an ‘augmented Lagrangian’ (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of ??‐LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of ??‐LU factorization in the preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.     

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coe?cient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.     

Split‐type octonion matrix

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form , where e_m's have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.