The expected complexity of analytic tableaux analyses in propositional calculus — II

The article continues a study of the complexity of an analytic tableaux algorithm for SAT. The main result is that the average number of branches in analytic tableaux of formulae of lengthn is((1.09988...) ⁿ ). The maximum number of branches is also studied. Both the average and worst case complexity measures are used to compare analytic tableaux and truth tables.For the average case result the precise number of consistent branches and the precise number of inconsistent branches (for formulae of lengthn) are each expressed as multiply indexed sums whose terms involve factorials and Stirling numbers of the second kind. The asymptotic behavior of these sums is determined by adapting a classical technique for determining the asymptotic behavior of singly indexed sums whose terms involve factorials.     

An analytical and numerical study of the two-dimensional Bratu equation

Bratu's problem, which is the nonlinear eigenvalue equationu+ exp(u)=0 withu=0 on the walls of the unit square and as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance ofsymmetry: because of invariance under the C₄ rotation group and parity in bothx andy, one can slash the size of the basis set by a factor of eight and reduce the CPU time by three orders of magnitude. Second, the pseudospectral method is ananalytical as well as a numerical tool: the simple approximation3.2A exp(–0.64A), whereA is the maximum value ofu(x, y), is derived via collocation with but a single interpolation point, but is quantitatively accurate for small and moderateA. Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions—five decimal places—on amicrocomputer inbasic. Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu's equations fall off exponentially rather than algebraically withv untilv is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in thelimit v, are so weak as to be irrelevant, and replacing Bratu's problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more.     

An Active Feedback System for Stabilizing Visual Images

In normal vision, eye movements cause the image to move on the retina. Special apparatus can stabilize the image on the retina so it cannot move, and vision fades away. Previous methods for stabilizing the image were either optical systems or complex computer-controlled feedback systems. A simpler feedback system is presented which detects eye motion and rotates a mirror through which the target is viewed, to exactly compensate for eye motion. Unlike previous optical systems, this new system has no lenses in either the viewing or the image-forming path to limit the spatial resolution.     

The Future of High Energy Accelerators

This paper begins with a review of those high energy accelerators which are now in operation. Accelerators under construction are discussed and very high energy machines, still far in the future, are listed and described. Possible uses of cryogenic and superconducting magnets are mentioned. Colliding beam projects in operation and planned are presented. High energy physics has become very expensive. The total investment in Brookhaven's 33-GeV accelerator complex is now over $60 million and is still rising. Merely to operate the accelerator and its experimental areas requires the services of over 400 people. Our power bill is about $1 million per year. All of this is to produce new knowledge, very fundamental in nature but without evident immediate application. With the budget of the United States in a state of unbalance and with growing sentiment that money spent should show immediate results, is it wise or even sensible to think of new and more expensive steps along the route to the solution of the many mysteries of the nucleus and its components? I think that it is wise and that frontier fields like that of high energy physics must be pursued vigorously if we are to maintain our intellectual and technical position in the world. Sooner or later our experimental results will have important impacts on our life, probably in quite unforeseeable fashions. Nuclear physics began its history before the turn of the century; only in the last two decades has it resulted in a new power industry.     

Toughening by Metallic Lamina in Nickel/Alumina Composites

Tape casting and hot-pressing were used to produce metal/ alumina laminate composites. The mechanical behavior of these laminates was characterized. The strength and toughness of the laminates were greatly improved in comparison to that of monolithic alumina. Indentation strength values of the laminates remained approximately constant for indent loads between 10 and 100 N. Thus, this type of laminate appears to be a good candidate for damage-tolerant composite material design.     

Direct Observation of Triplet Reactions of Anthraquinone Vat Dyes Using Nanosecond Laser Photolysis

The triplet-triplet spectra of three commercial anthraquinone vat dyes (C.I. 67300, C.I. 59100 and C.I. 60515) were studied in solution at room temperature using laser photolysis. The triplet states of these dyes react with oxygen, and the rate constants for the quenching process were measured, together with the rates of energy transfer from the triplet state to anthracene. These observations are considered in relation to the photoreactivity of these dyes.     

Flow of fluidized solids and bubbles in standpipes and risers

Equations are presented for the densities of fluidized solid-bubble mixtures in downward and upward vertical flow. Data from a large commercial standpipe are in close agreement with calculations. Flow instability is predicted when downward solid flow is sufficient to hold a gas bubble stationary in a standpipe. Commercial experience of such instability is presented indicating the existence of slugs in a 45-in.-diam. standpipe.     

Preparation of lot samples of nut meats for mycotoxin assay

A procedure has been devised for preparing lot samples of mycotoxin-contaminated nut meats so that a representative analytical sample may be removed. The sample is rapidly reduced to coarse size. A relatively large portion (about 1/10 of total sample) of subsample is then split out and further comminuted to a fine particle size with the aid of a fat solvent (meat-solvent, w/v, 3:2). The analytical sample is removed from this mixture. The procedure was tested with shelled almonds and shelled walnuts using radioactive nuts to simulate the mycotoxin contamination and provide a simple, precise measure of the contaminated nut meat distribution. The pooled coefficient of variation was 18% for the subsamples and 4.4% for the analytical samples. Considering the dilution factors used (1.50 and 2.14 contaminated nuts/10⁴ nuts) and the low degree of reliability of the lot sample, the sample preparation methods tested appear to be practical and reliable.     

NMR chemical shift reagents in structural determination of lipid derivatives: IV. Methylcis- andtrans-9,10-epoxystearate and methylerythro- andthreo-9,10-dihydroxystearate

Chemical shift reagents were used to expand the amount of structural information obtainable from NMR studies of derivatives of methyl oleate and elaidate:methylcis-9,10-epoxystearate, methyltrans-9,10-epoxystearate, methyl erythro-9,10-dihydroxystearate, and methyl threo-9,10-dihydroxystearate. Chemical shift reagent studies of methyltrans-9,10-epoxystearate and methyl threo-9,10-dihydroxystearate afforded the most information. Chemical shift reagent studies of methylcis-9,10-epoxystearate and methyl erythro-9,10-dihydroxystearate were decidedly inferior. The series of complementary interpretive techniques previously developed during chemical shift reagent studies of monofunctional fatty esters and model polyfunctional fatty esters were found to be applicable in the current study. However, to avoid ambiguity in several proton assignments, supplementary spin decoupling experiments were necessary.