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Formalisms for normalization and symmetrization of one-electron Dirac scattered-wave wavefunctions are presented. The normalization integral consists of one-dimensional radial integrals for the spherical regions and an analytic expression for the intersphere region. Symmetrization drastically reduces the size of the secular matrix to be solved. Examples for planar Pb2Se2 and tetrahedral Pd4 are discussed.
An experiment was devised to study the interaction between an underexpanded supersonic gas jet and a flat plate, with the plate located in a region in which the interaction produces shock wave and flow fluctuations. Nitrogen gas at ambient stagnation temperature flowed through a convergent nozzle with exit diameter of 2.03 cm and impinged on a square metal plate normal to the jet. Results revealed local peak pressure fluctuations on the plate at nozzle pressure ratios of about 2 and 4.5, with the latter case producing fluctuations of the same order as the mean pressure on the plate; the frequency of the oscillations was as large as 20 kHz. For choked jet flow at ambient pressure higher than atmospheric, the pressure fluctuations would increase accordingly, and adjacent solid structures would therefore be subjected to proportionately higher normal stresses.
This paper addresses the problem of obtaining numerically maximum-likelihood estimates of the parameters for a mixture of normal distributions. In recent literature, a certain successive-approximations procedure, based on the likelihood equations, was shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, we introduce a general iterative procedure, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. We show that, with probability 1 as the sample size grows large, this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. We also show that the step-size which yields optimal local convergence rates for large samples is determined in a sense by the 'separation' of the component normal densities and is bounded below by a number between 1 and 2.
An inversion procedure to obtain the reflectance of the central region of a satellite's disk from lunar occultation data is presented. The scheme assumes that the limb darkening of the satellite depends only on the radial distance from the center of the disk. Given this assumption, normal reflectances can be derived that are essentially independent of the limb darkening and the diameter of the satellite. The procedure has been applied to our observations of the March 1974 lunar occultation of Tethys, Dione, Rhea, Titan, and Iapetus. In the V passband we derive the following normal reflectances: Rhea (0.97 plus or minus 0.20), Titan (0.24 plus or minus 0.03), Iapetus, bright face (0.60 plus or minus 0.14). For Tethys and Dione the values derived have large uncertainties, but are consistent with our result for Rhea.
A 0.15 scale model of an underfuselage inlet designed for a single-engine fighter airplane was tested. The inlet was a fixed-geometry, normal-shock configuration designed to operate at flight speeds up to Mach 2.0. Peformance data for the basic inlet and several configuration variations are presented as a function of angle of attack, angle of sideslip, and airflow in the 0.6 to 2.0 Mach number range. The configuration variations included boundary-layer diverter height, cowl and splitter-plate modifications, and inlet bleed system variations. Flow-field characteristics at the simulated engine face, at the inlet throat, at the splitter-plate leading edge, and forward of the inlet are presented. The pressure recovery of the inlet is approximately equal to the product of theoretical normal-shock and duct pressure recoverable at cruise angle of attack. Very good performance at high angle of attack was obtained. Pressure distortion and turbulence at the engine face were low, and the inlet remained stable at all engine airflows over the flight maneuver envelope of the aircraft for which the inlet was designed.
The region in the vicinity of the Luna 24 handling site (40-70 deg E; 0-20 deg N) is studied by means of the Apollo orbital fluorescent X-ray Al/Si and Mg/Al data and earth-based normal albedo measurements. By means of unsupervised non-parametric cluster analysis this region is divided into eight surface units. One of these units centered in the Proclus region deviates from the usual trend of increasing normal albedo with increasing Al/Si values and decreasing Mg/Al values
The interaction of a jet entering into a freestream normal to the main flow direction has been studied with particular attention directed to visualization of the large-scale flow interactions and to measurement of the film-cooling performance. Large eddies are apparent downstream of the entering jet even at moderate blowing rate (defined as the ratio of the mass velocity of the jet to the mass velocity of the freestream). At higher blowing rate, there is only intermittent contact between the mass from the jet and the downstream wall. The film cooling downstream from a single normal jet yields a lower centerline effectiveness compared to an inclined jet through a greater lateral spreading. The greater spreading provides a more uniform effectiveness across the span of the downstream wall, in particular at large blowing rate.
The present paper deals with the fundamental issue of whether one can treat waves as normal modes when critical surface, where the phase speed of the wave matches the zonal wind speed, are present. In particular the question of whether a Rossby critical level (such as the zero-wind line for stationary waves) is absorbing or reflecting is raised and subsequently addressed. It is found that the critical level is never totally absorbing; Rossby waves are partially reflected even if the critical layer is dominated by dissipative processes. The relevance of nonlinearity in planetary-scale Rossby wave critical layers is also discussed and it is found to be the dominant mechanism. With the relative magnitudes of nonlinearity versus viscosity relevant to the earth's atmosphere it is found that the steady-state critical level should be almost perfectly reflecting to incident Rossby waves. Consequently, normal-mode solutions can be found; the quantization condition for these waves is also derived.
Chest radiographs in five seated normal volunteers at 1 G and 0 G were made with a view toward comparing human lung shape during normal gravity and weightlessness. Lung shape was assessed by measuring lung heights and widths in upper, middle and lower lung regions. No significant differences were found between any of the 1-G and 0-G measurements, although there was a slight tendency for the lung to become shorter and wider at 0 G. The evidence that gravity causes regional differences in ventilation by direct action on the lung is consistent with the theoretical analysis of West and Matthews (1972).
General theorems concerning the strong consistency of the MLE of exponential mixture parameters are proved. These theorems imply the strong consistency of the MLE of normal mixture parameters when the data is organized into fields each of which is a random sample from one of the component normal distributions.
Some results for the bivariate normal distribution analysis are presented. Computer programs for conditional normal probabilities, marginal probabilities, as well as joint probabilities for rectangular regions are given: routines for computing fractile points and distribution functions are also presented. Some examples from a closed circuit television experiment are included.
The dynamic response of a layered composite under normal and shear impact is analyzed by assuming that the composite contains an initial flaw in the matrix material. One of the objectives was to develop an analytical method for determining dynamic stress solutions which should lead to a numerical method which utilizes Fourier transform for the space variable and Laplace transform for the time variable. The time-dependent angle loading is separated into two parts: a symmetric and a skew-symmetric with reference to the crack plane. By superposition, the transient boundary conditions consist of applying normal and shear tractions to a crack embedded in a layered composite; one phase of the composite could represent the fiber while the other could be the matrix. Mathematically, these conditions reduce the problem to a system of dual integral equations solved in the transform plane for the transform of the dynamic stress-intensity factor.
?jDuring the initial phase of space flight, there is a translocation of fluid from the lower parts of the body to the central vascular compartment with a resultant natriuresis, diuresis, and weight loss. Because water immersion is regarded as an appropriate model for studying the redistribution of fluid that occurs in weightlessness, an immersion study of relatively prolonged duration was carried out in order to characterize the temporal profile of the renal adaptation to central hypervolemia. Twelve normal male subjects underwent an immersion study of 8-h duration in the sodium-replete state. Immersion resulted in marked natriuresis and diuresis which were sustained throughout the immersion period. The failure of that natriuresis and diuresis of immersion to abate or cease despite marked extracellular fluid volume contraction as evidenced by a mean weight loss of -2.2 + or - 0.3 kg suggests that central blood volume was not restored to normal and that some degree of central hypervolemia probably persisted.
A general theorem is given which establishes the existence and uniqueness of a consistent solution of the likelihood equations given a sequence of independent random vectors whose distributions are not identical but have the same parameter set. In addition, it is shown that the consistent solution is a MLE and that it is asymptotically normal and efficient. Two applications are discussed: one in which independent observations of a normal random vector have missing components, and the other in which the parameters in a mixture from an exponential family are estimated using independent homogeneous sample blocks of different sizes.
In order to investigate the mechanism of carbon combustion and to assess the importance of gravitational induced convection on the process, zero and normal gravity experiments were conducted in which spectroscopic carbon rods were resistance ignitied and burned in dry oxygen environments. In the zero-gravity drop tower tests, a blue flame surrounded the rod, showing that a gas phase reaction in which carbon monoxide was oxidized to carbon dioxide was taking place. The ratio of flame diameter to rod diameter was obtained as a function of time. It was found that this ratio was inversely proportional to both the oxygen pressure and the rod diameter. In the normal gravity tests, direct mass spectrometric sampling was used to measure gas phase concentrations. The gas sampling probe was positioned near the circumference of a horizontally mounted 0.615 cm diameter carbon rod, either at the top or at angles of 45 deg to 90 deg from the top, and yielded concentration profiles of CO2, CO, and O2 as a function of distance from the surface. The mechanism controlling the combustion process was found to change from chemical process control at the 90 deg and 45 deg probe positions to mass transfer control at the 0 deg probe position at the top of the rod. Under the experimental conditions used, carbon combustion was characterized by two surface reactions, 2C + O2 yields 2CO and CO2 + C yields 2CO, and a gas phase reaction, 2CO + O2 yields 2CO2.
Certain principal normal curvatures of differential geometry were developed for use in curvature matrices associated with the asymptotic solution of electromagnetic diffraction problems. The effort is directed toward microwave antenna simulations and high speed digital computer analysis of radiometric instruments used to obtain soil moisture, sea state, salinity and temperature data. It is shown that the methods used to develop the principal normal curvatures for paraboloid, hyperboloid, ellipsoid, sphere, and cone can be applied to other radiometer geometries such as the parabolic torus, even though the surface parameterizations are different. It is concluded that deployable offset geometries, distorted by rotational forces and solar loads may be analyzed by similar means given a suitable surface description.
A novel technique for the high-precision isotopic analysis of titanium was applied to three terrestrial rocks and coarse- and fine-grained Allende inclusions. Repeated analyses of the three terrestrial rocks gave excellent agreement with a Ti metal standard. All seven Allende inclusions were previously determined to contain isotopically normal Nd and/or Sm, indicating that none belongs to a small group of peculiar intrusions. The discovery of widespread Ti isotopic anomalies in normal Allende inclusions establishes Ti as the first non-noble gas element studied since oxygen to show such isotopic heterogeneity. A survey of nucleosynthetic origins of Ti isotopes suggests that the dominant Ti-50 excesses in these inclusions are due to the relative enrichment of isotopes synthesized during hydrostatic burning in, or near, the core of a massive star.
An investigation is conducted regarding the influence of mean field variations on the realization of planetary normal modes, taking into account the mode response and structure in the presence of simple background nonuniformities. It is found that mean field variations have the combined effect of depressing, shifting, and broadening the characteristic response of Rossby normal modes. While nonuniformities in both the mean wind and temperature fields contribute to the reduction in peak response, the former are primarily responsible for translation and spectral broadening. An investigation is conducted to determine which modes may be realized in actual atmospheric configurations and which may be identified. For both the equinox and solstice configurations, response peaks corresponding to all of the first four modes of wavenumbers 1, 2, and 3 are readily visible above the noise.