A Convergence Proof for Constrained Finite Element Solutions

A convergence proof is given for the finite-element solutionof the infinite dimensional quadratic programming problem ofminimizing a quadratic functional subject to linear constraints.The proof for the unconstrained problem is briefly reviewed,and then extended to the constrained case. Only the first partof the proof is given, in which necessary conditions for convergenceare derived for the specific problem and its finite-elementapproximation. The final step of proving that any problem doesobey these conditions will depend on the specific problem, butit is shown that if the finite element formulation is pointwiseconvergent and the unconstrained problem is convergent, thenso too will be the constrained problem.     

Overflow, Underflow, and Severe Loss of Significance in Floating-Point Addition and Subtraction

In this paper it is shown that, under the assumption of thelogarithmic distribution of numbers, floating-point additionand subtraction can result in overflow or underflow with alarmingfrequency—a frequency that increases rapidly with machinespeed unless the range of the exponent is also increased. Ifnumbers are assumed to be distributed in accordance with Sweeney's(1965) experiments, then severe loss of significance occurswith large probability in floating point subtraction. Theseresults have implications for computer design and lead to thesuggestion of a long word format which will reduce the risksto acceptable levels.     

The Factorization of Sparse Symmetric Indefinite Matrices

The Harwell multifrontal code MA27 is able to solve symmetricindefinite systems of linear equations such as those that arisefrom least-squares and constrained optimization algorithms,but may sometimes lead to many more arithmetic operations beingneeded to factorize the matrix than is required by other strategies.In this paper, we report on the results of our investigationof this problem. We have concentrated on seeking new strategiesthat preserve the multifrontal principle but follow the sparsitystructure more closely in the case when some of the diagonalentries are zero.     

The Frictional Unloading Problem on a Linear Elastic Half-Space

Present Address: ICI Agricultural Divison, P.O. Box 1, Billingham, Cleveland, TS23 ILB. A variational formulation is derived for contact problems withfriction. The problem of contact between a rigid, circular,cylindrical indentor and an isotropic, homogeneous, linearlyelastic half-space is then considered. Using the above variationalformulation numerical solutions are obtained to the problemsof frictionless contact, adhesive contact, and frictional loading,for which there are known analytical solutions, as well as thefrictional unloading problem, for which there is no previousknown solution. Comparisons are made with experiment and predictionsfrom physical considerations.     

The Distribution of Leading Significant Digits

Most numbers which are encountered are the result of some arithmeticoperations. The object of this paper is to show why their leadingsignificant digits are not uniformly distributed. It has long been surmised that the distribution of leading significantdigits is logarithmic. Indeed this is already known under theassumption of invariance to scaling. In this paper we establishthis distribution without any such assumption, and thus thatover 30% of decimal numbers can be expected to have leadingsignificant digit 1.     

DEVELOPMENT AND EVALUATION OF A SYSTEM DYNAMICS MODEL FOR INVESTIGATING AGRICULTURALLY DRIVEN LAND TRANSFORMATION IN THE NORTH CENTRAL UNITED STATES

Land transformation from grassland to cropland in the Northern Great Plains (NGP) has become a growing concern among many stakeholders. A growing body of work has sought to determine the amount and rate of land use change with less emphasis on the systemic structures or feedback processes of land use decisions. This paper presents the development of a system dynamics simulation model to integrate ecological, economic, and social components influencing land use decisions, including cattle ranching, cropland production, rural communities, land quality, and public policies. Evaluation indicated that the model satisfactorily predicted historical land, agricultural commodity, and rural community data from the model structure. Reference modes for key variables, including the farmland area, were characterized by a bias correction of 0.999, root mean squared error of prediction of 0.053, R² of 0.921, and concordance correlation coefficient of 0.0959. The model was robust under extreme and varying sensitivity tests, as well as adequately predicting land use under changing system context. The model's major contributions were the inclusion of decision‐making feedbacks from economic and social signals with connectivity to land quality and elasticity values that drive land transformation. Limitations include lack of spatial input and output capabilities useful for visual interfacing.     

Synthesis and Properties of Chiral Molecular Clefts Related to Tröger's Base

The chiral cavities present in 2,3:6,7-dibenzo-9-oxabicyclonona-2,6-diene, dibenzobicyclo[b,f][3.3.1]nona-5a, 6a-diene-6,12-dione, bicyclo[3.3.1]nonane and dibenzobicyclo[b,f],diazocines are reminiscent of Tröger's base, which has been widely used as a molecular cleft in supramolecular chemistry. The synthetic methodology to provide key derivatives for elaboration into new supramolecular structures, efficient resolution methods, the introduction of additional recognition groups and applications in supramolecular chemistry of these structurally related molecular clefts are reviewed.     

Influence of chemical and physical surface heterogeneity on chemical reaction equilibria in carbon micropores

Recent simulation results are presented for the equilibrium yield of the ammonia synthesis reaction in various model microporous carbons. It is found that the reaction equilibria within the micropores is affected by many factors, including pore size, pore shape, connectivity, surface roughness, and surface chemical activation. In order to probe these effects, reactive Monte Carlo simulations of the reaction were performed in several microporous carbon models: smooth slit-shaped carbon pores, a realistic carbon model generated from experimental diffraction data, single-walled carbon nanotubes, and smooth slit-shaped pores activated by carboxyl surface groups. The simulations show that the ammonia conversion is most sensitive to the carbon pore width and to the amount of surface chemical activation. Effects of surface corrugation and pore connectivity on the equilibrium reaction yield are minimal.     

Further Revelations on L.S.D.

In a recent paper the author showed that the distribution ofleading significant digits, l.s.d., resulting from successivemultiplications is logarithmic. In this paper these resultsare extended by establishing, still without any assumptionsof invariance to scaling, that this distribution remains invariantunder all further arithmetic operations—both multiplicativeand additive. The fact that 30% of decimal numbers have l.s.d.1 persists.