Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

Common eigenvectors and quasicommutativity of sets of simultaneously triangularizable matrices

A set of simultaneously triangularizable square matrices over an arbitrary field is considered. If the matrices are also quasicommutative, then they have a common eigenvector for every distinct set of corresponding eigenvalues. Conversely, if the set of matrices has this common eigenvector property hereditarily (i.e., for every set of corresponding blocks in every simultaneous block triangularization), then the matrices are quasicommutative.     

Theory of reality

Bell's theorem is used to guide the formulation of a unified theory of reality that incorporates the basic principles of relativistic quantum theory.Work supported by U. S. Energy Research and Development Administration.     

Optimum Aerodynamic Design Using the Navier–Stokes Equations

This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in viscous compressible flow, modeled by the Navier–Stokes equations. It extends previous work on optimization for inviscid flow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Fréchet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. The cost is kept low by using multigrid techniques, in conjunction with preconditioning to accelerate the convergence of the solutions. The power of the method is illustrated by designs of wings and wing–body combinations for long range transport aircraft. Satisfactory designs are usually obtained with 20–40 design cycles. Received 5 February 1997 and accepted 30 May 1997     

Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

Summary The purpose of this article is to obtainL ² and uniform norm error estimates for the Galerkin approximation of the solution of certain boundary value problems via a comparison with then-norm projection of the solution. In some cases, these estimates constitute an improvement over known results.This research was supported in part by AEC Grant (11-1)-2075.     

Synthesis of polyperfluoroalkylenetriazines

Four new s-triazine polymers were prepared by the mercury-induced coupling of bromodifluoromethyl-substituted ditriazinyl-perfluoroalkane derivatives. Each polymer was composed of CF₃-substituted s-triazin rings alternately joined with ? CF₂CF₂? segments and \documentclass{article}\pagestyle{empty}\begin{document}$\hbox{---}\hskip-5pt({\rm CF}_2 )_8 \hbox{---}$\end{document}, a mixture of ? (CF₃CFO(CF₂)₅OCF(CF₃)? and ? (CF₃)CFOCF₂CF(CF₃)OCF₂)₄? , ? CF₂OCF₂? , or ? CF₂SCF₂? , segments. All polymers were elastomeric and displayed high temperature stability on thermogravimetric analysis. The polymers containing either ? CF₂OCF₂? or ? CF₂SCF₂? segments showed high temperature limits slightly lower than the other two systems.     

Magnetic proximity effect in EuO-Cs interface

In the surface of the ferromagnetic insulator EuO is covered with a small amount of cesium, the photoelectric magnetization curves P(H) show magnetic saturation in contrast to those of the pure surface. The proximity of the Cs film causes the paramagnetism of the clean EuO surface to disappear.     

Chiral prostanoid intermediates from aucubin

Several potentially useful prostanoid intermediates have been prepared from aucubin. Dry DMSO-NBS converts enol ethers to α-bromoesters in one step.     

Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software

This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its availability in classrooms also provides opportunities for positive changes to teaching and learning. Very many examples are documented in the professional and research literature, and in this paper we organize them into 10 types. These are displayed in the form of a ‘pedagogical map’, which further classifies them according to whether the opportunity arises from new opportunities for the mathematical tasks used, change to interpersonal aspects of the classroom or change to the point of view on mathematics as a subject. The map can be used in teacher professional development to draw attention to possibilities for lessons or as a catalyst for professional discussion. For research on teaching, it can be used to map current practice, or to track professional growth. The intention of the map is to summarise the potential benefits of teaching with technology in a form that may be useful for both teachers and researchers.