Showing papers on "Projection (relational algebra) published in 1970"

Density Autocorrelation Function in a Classical Fluid from Initial Correlations

Abstract: A complete set of time-independent orthogonal phase functions ${{\ensuremath{\Psi}}_{s}}$, $s=0,1,2,\dots{}$, is generated via the Schmidt process and used to represent the Fourier coefficient ${R}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}(t)$ of the time-dependent microscopic density function. The projection of ${R}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}(t)$ on ${\ensuremath{\Psi}}_{0}$ is essentially the density autocorrelation function. The equation of motion of the coefficients of this expansion is found and formally solved to yield the Laplace-Fourier transform of the density autocorrelation function as a ratio of infinite determinants, closely related to Mori's continued-fraction expansion. A non-Markovian memory function is then readily defined in the same terms. These exact results are illustrated by explicit calculations for the ideal gas. Finally, a perturbation expansion of the memory function is developed, leading to practical approximations.

Effects of strength and speed training on power in the projection of light and heavy objects

Abstract: Hie primary purpose of this study was to determine the effects of six training programs on the development of power in projecting light and. heavy objects in an underhand throw. A second purpose was to investi­ gate the effects of the six training programs on endurance in the pro­ jection of light and heavy objects. An incidental purpose was to analyze the development of strength by all participating groups. Finally, the study sought to analyze the relationship between strength and velocity. lhe study included ninety male freshmen enrolled in physical education weight-training classes at the University of Southwestern Louisiana, Lafayette, Louisiana. They were randomly divided into six groups of fifteen subjects meeting three times weekly for five weeks. All groups performed their routines while strapped into specially designed chairs. Group I practiced thirty simulated underhand throws with no resistance; Group II also performed this routine plus forty-five softball throws; Groups III and V both executed thirty simulated throws with five and ten-pound resistance, respectively, on pulley weights; Groups IV and VI performed thirty simulated throws with five and tenpound resistance, respectively, on pulley weights in addition to per­ forming forty-five softball throws. Subjects were preand post-tested for strength and for softball and weighted-ball velocity over seventyfive throws. The data were analyzed for significant gains in each group for each variable by correlated t-tests. Analysis of variance with orthog­ onal comparisons was used to analyze the training effects on strength xiii and weighted-ball velocity, and softball velocity was analyzed by covariance. In addition, analysis of variance and orthogonal regression were employed to analyze endurance performance. Correlation coeffi­ cients were computed to determine relationships among the variables. The results of the study indicated that: 1. Training by simulating throwing against resistance or by throwing with or without supplementary simulated throws against resistance will bring about improvements in velocity and endurance in throwing light and heavy objects. 2. Training involving actual throwing is more effective in developing velocity iu throwing a light or heavy object than training by simulating throwing. However, the use of resistance while simulating throwing appears to be as effec­ tive in Improving velocity as throwing with and without supplementary strength training. 3. The greatest endurance is achieved in throwing light and heavy objects by simulating throwing against a large resistance and by throwing. Greater gains are made in strength when the resistance used in training is increased. 5A higher relationship exists between strength and velocity when initial performances are correlated than when gains in performance resulting from training are correlated.

Argand diagrams, cross-section behavior, and regge poles.

Abstract: Schmid's problem on the interpretation of the anticlockwise Argand diagram obtained from Regge-pole exchanges as the direct-channel resonance is studied. It is shown in the idealized models that the anticlockwise Argand diagram alone cannot guarantee the existence of a resonance, and that it is absolutely necessary to compute the cross section in order to establish the existence of a resonance. The partial-wave projection of the Regge-pole exchanges in pion-nucleon scattering is performed. The Argand diagram, real part of phase shift, absorption factor, partial-wave total cross section ${\ensuremath{\sigma}}^{\mathrm{tot}}$, elastic cross section ${\ensuremath{\sigma}}^{\mathrm{el}}$, ${k}^{2}{\ensuremath{\sigma}}^{\mathrm{tot}}$, and ${k}^{2}{\ensuremath{\sigma}}^{\mathrm{el}}$ are computed. It is found that all ${\ensuremath{\sigma}}^{\mathrm{tot}}$ and ${\ensuremath{\sigma}}^{\mathrm{el}}$ have no cross-section peak at all, despite the existence of the anticlockwise Argand loops in certain partial waves. They show only the monotonically decreasing behavior characteristic of the Regge-pole model as energy increases, except in the unitarity-violating low-energy region. The above facts indicate that the variation of the anticlockwise Argand loop with respect to energy is too weak to produce a cross-section peak by overcoming the variation of the barrier factor $\frac{1}{{k}^{2}}$ in ${\ensuremath{\sigma}}^{\mathrm{tot}}$ and ${\ensuremath{\sigma}}^{\mathrm{el}}$. It is concluded that Schmid's interpretation is incorrect. Several physical implications of this result are discussed.

$sup 28$Si NUCLEUS IN THE PROJECTED HARTREE--FOCK MODEL.

Abstract: The structure of the $^{28}\mathrm{Si}$ nucleus is studied using a variational procedure. The method of angular-momentum projection from a deformed intrinsic state is applied, and each ${J}^{\ensuremath{\pi}}$ state is projected from a determinant which is variationally "best" for that state. This more general variational procedure includes important vibrational correlations, which in $^{28}\mathrm{Si}$ have hexadecapole character. The level spacings in the energy spectrum improve considerably compared to those in the Hartree-Fock method followed by angular momentum projection, and this leads to a much better agreement with the experimental spectrum. The $E2$ transition probabilities, except for the ${6}^{+}$\ensuremath{\rightarrow}${4}^{+}$ transition, are also in good agreement with the experiments.