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.     

Kinetic analysis of plasma VLDL-TG and VLDL-remnant-TG turnover in anesthetized rats

We have estimated the turnover and relative pool sizes of nascent-VLDL-TG and VLDL-remnants-TG in anesthetized rats. [1-¹⁴C]Palmitoyl- and [2-³H]glyceryl-labeled “VLDL”-TG (including nascent VLDL-TG and VLDL-remnants-TG) were prepared by injecting labeled palmitate and glycerol into donor rats. Labeled serum from these rats was then injected intravenously into nembutalized male rats and serial blood samples taken for 30 min. Special care was taken to define any early components in the labeled “VLDL”-TG disappearance curves. In other experiments, the donors were rendered functionally hepatectomized 30 min after injection of³H-glycerol and the endogenous labeled VLDL-TG was allowed to circulate 30–60 min before collection of the TG-labeled VLDL-remnants-containing serum. The latter was injected into 4 recipient nembutalized rats and the remnant-TG-turnover measured by serial sampling as above. In two cases,¹⁴C-“VLDL” and³H-VLDL-remnants were injected as a single bolus into ether-anesthetized rats. Despite its complex composition, “VLDL”-TG in most cases disappeared in a single exponential fashion for 30 min with an average half-life of 5.9 min in nembutalized and 2.8 in ether-anesthetized rats. VLDL-remnants-TG showed a more complex behavior, but contained a major rapid component with a mean t_1/2 of ca. 1.5 min in both groups. The data, analyzed by multicompartmental analysis, were fitted to a simple model in which turnover of a larger nascent VLDL-TG pool with formation of a more rapidly turning over smaller pool of VLDL-remnant-TG is the rate-limiting step in overall TG removal from the d<1.006 fraction of rat serum. The data are consistent with our theoretical prediction that under these conditions the kinetics of the VLDL-remnants cannot be resolved from analysis of the total composite “VLDL” (nascent plus remnant) pool.     

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.     

Analysis of solvent extracts of bituminous coal macerals by chromatography, 1H n.m.r. and field-desorption mass spectrometry

Analysis of the carbon disulphide extracts of nine samples of UK coal-maceral concentrates by ¹H highresolution n.m.r. spectroscopy, gas-liquid chromatography and field-desorption mass spectrometry indicates the presence of components with a wide molecular weight range extending up to 1200 amu; these are attributed to n-alkanes up to ≈C₅₀ and highly condensed polynuclear aromatic compounds.     

IR study of the interaction of hydroxyl groups of silica gel with Cr species of Phillips' type catalysts

Spectroscopic evidence for the interaction of hydroxyl groups and chromium ions was obtained using a catalyst prepared from chromyl chloride. A new OH peak, observed at 3705 cm^–1 after pumping away CO gas, is attributed to the direct interaction of OH with the low-valent chromium. This peak shifts to 3590 cm^–1 on contact with O₂ at room temperature and it is assigned to a hydroxyl interacting with the oxidized chromium. New assignments are also proposed for IR bands of CO presorbed on the catalyst. The peak due to CO at 2188 cm^–1 decreases as the OH intensity at 3705 cm^–1 increases, suggesting that the former peak arises from adsorption on Cr(II) species to which two oxygen atoms are attached.     

Effect of eicosa-5,8,11,14-tetraynoic acid on fatty acid composition of selected organs in the rat

Eicosa-5,8,11,14-tetraynoic acid or arachidonic acid or no supplement (controls) was given orally to rats maintained on a fat free diet and the fatty acid composition of total lipids of several organs determined. No changes were noted in the total fatty acid concentrations of the organs examined in the various groups. A decrease in the amount of arachidonic acid, 22∶4ω6 and 22∶5ω6 (as percent of total fatty acids), and an increase in the amount of 20∶3ω6 and linoleate were observed in total lipids of several organs. In the group receiving the arachidonate supplement, there was less linoleate and 20∶3ω6 and more arachidonate than in the controls. Both eicosa-5,8,11,14-tetraynoic acid and arachidonate supplements resulted in a decrease in 20∶3ω9 in most organs studied. Generally, the most marked changes were seen in liver but, of the other organs examined (heart, kidney, testis, brain, and adrenals), only the adrenals failed to show any significant differences between the controls and each of the two supplemented groups. Although the experimental conditions preclude conclusive interpretation of the changes observed, it is suggested that eicosa-5,8,11,14-tetraynoic acid was effective in inhibiting the conversion of linoleate to arachidonate and the conversion of arachidonate to 22∶4 and 22∶5.