Stability of Adams-Type Formulae for Second-Order Ordinary Differential Equations

The stability of methods for systems of second-order equationsis discussed. Stability regions are obtained for a single equationand the existence of stable step-sizes is shown for systems.An example is used as an illustration of the effect of the usualorder selection strategies on stability and accuracy.     

An Implicit Matching Principle for Global Element Calculations

A variational method is described which allows elliptic boundaryvalue problems with complex domains to be solved as a set ofcoupled problems over simple subdomains (global elements); thetrial functions used need not satisfy any of the boundary conditions. For smooth problems the method retains the rapid convergenceof the global variational approach; a major advantage howeveris that rapid convergence should also be attainable for singularproblems. In many cases the method will be simpler to use thanthe finite element method.     

Approximation Methods in the Computer Numerically Controlled Fabrication of Optical Surfaces, Part 1: Finite Dimensional Material Removal Profile Spaces

An analysis of the process of grinding and polishing of lensesand mirrored surfaces using a computer numerically controlledmachine is presented. This analysis provides the motivationfor the study of L₁ approximation of continuous functions bynonstandard approximants. Under suitable conditions, L₁ convergenceis established for a space of material removal profiles basedon a piecewise continuous material removal rate. A linear programmingalgorithm is used to find the best discrete L₁ approximation.Numerical examples are included.     

THE MAXIMUM EFFICIENCY OF PHOTOSYNTHESIS *

Within the ideal assumptions: (1) two Photosystems for photosynthetic fixation of CO₂, (2) all solar photons with Λ≥ 700 nm are absorbed, (3) the photon requirement is 8 for each CO₂ molecule fixed and O₂ molecule evolved and (4) the principal stable product of photosynthesis is d-glucose, the theoretical maximum efficiency of conversion of light to stored chemical energy in green-plant type (oxygen-evolving) photosynthesis in bright sunlight is calculated to be 13.0%. Thermodynamic arguments are presented which indicate that a photosynthetic system with one Photosystem would be highly unlikely to be able to drive each electron from water to evolve O₂ and reduce CO₂. The practical maximum efficiency of photosynthesis under optimum conditions is estimated to be 8–9%.     

Approximation methods in the computer numerically controlled fabrication of optical surfaces, Part 2: mollifications

The process of grinding and polishing optical surfaces usinga Computer Numerically Controlled machine produces a machinematerial removal profile. The profiles achievable by the machinedepend on the nature of the tool used in the process, and thetool center motions. In this paper machine material removalprofiles are developed as mollifications of given workpieceprofiles for a variety of tool configurations. The form of themollilication, in effect, defines the tool center motion. Convergenceof the machine's material removal profile to the given workpieceprofile as the support of the tool goes to zero is establishedunder mild assumptions. Numerical examples are included. This work supported in part by AFOSR Grant No. AFOSR 900094.     

Families of regular solutions of singular systems

The seminal 1969 paper of W.A. Harris Jr., Y. Sibuya, and L. Weinberg provided new proofs for the Perron-Lettenmeyer theorem, as well as several other classical results, and has stimulated renewed consideration of families of regular solutions of certain singular problems. In this paper we give some further applications of the method developed there and, in addition, examine some connections between the Lettenmeyer theorem and an alternative theorem which addresses a problem posed by H.L. Turrittin that dates back to an 1845 example of Briot and Bouquet     

A Finite Element that Satisfies Natural Boundary Conditions Exactly

Blending function interpolation is used to construct finiteelements that match both essential and natural boundary conditionsexactly along the entire boundary. The convergence of the methodis analyzed as well as the effect on the energy functional offorcing the admissible functions to satisfy the natural conditions.     

On an Exact Boundary Technique

In Marshall & Mitchell (1973) a finite element was introducedwhich matches Dirichlet boundary data exactly and was comparedwith the standard bilinear finite element which matches suchdata at only a finite number of points on the boundary. Ourpurpose here is to compare these same two elements with respectto the energy norm and the process of minimizing the energyfunctional.