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Several approximations to the Doppler broadening functions psi(x, theta) and chi(x, theta) are compared with respect to accuracy and speed of evaluation. A technique, due to A. M. Turning (1943), is shown to be at least as accurate as direct numerical quadrature and somewhat faster than Gaussian quadrature. FORTRAN 4 listings are included.
Broadband shortwave and longwave radiance measurements obtained from the Nimbus-7 Earth Radiation Budget scanner were used to develop reflectance and emittance models for the Sahara-Arabian, Gibson, and Saudi Deserts. The models were established by fitting the satellite measurements to analytic functions. For the shortwave, the model function is based on an approximate solution to the radiative transfer equation. The bidirectional-reflectance function was obtained from a single-scattering approximation with a Rayleigh-like phase function. The directional-reflectance model followed from integration of the bidirectional model and is a function of the sum and product of cosine solar and viewing zenith angles, thus satisfying reciprocity between these angles. The emittance model was based on a simple power-law of cosine viewing zenith angle.
Numerical analysis of functions with independent variables - guidance systems
Mathematical analysis and dispersion relations are used to obtain the general functional form of the neutrino charge form-factor. It is shown that if the scale for weak interactions is given by G, where G is the Fermi coupling constant, then for a large range of momentum transfer values, the form-factor may be approximated by a function that is linear in the square of the momentum transfer.
Illustrations of both the relative and absolute accuracy of eight different radiative transfer approximations as a function of optical thickness, solar zenith angle and single scattering albedo are given. Computational results for the plane albedo, total transmission and fractional absorption were obtained for plane-parallel atmospheres composed of cloud particles. These computations, which were obtained using the doubling method, are compared with comparable results obtained using selected radiative transfer approximations. Comparisons were made between asymptotic theory for thick layers and the following widely used two stream approximations: Coakley-Chylek's models 1 and 2, Meador-Weaver, Eddington, delta-Eddington, PIFM and delta-discrete ordinates.
The representation min t s.t. F(I)(x). - t less than or equal to 0 for all i is examined. An active set strategy is designed of functions: active, semi-active, and non-active. This technique will help in preventing zigzagging which often occurs when an active set strategy is used. Some of the inequality constraints are handled with slack variables. Also a trust region strategy is used in which at each iteration there is a sphere around the current point in which the local approximation of the function is trusted. The algorithm is implemented into a successful computer program. Numerical results are provided.
An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.
Carbon nanotubes have attracted much interest in recent years due to their exceptional mechanical properties. Currently, the tensile properties of bulk carbon nanotube-based materials (yarns, sheets, etc.) fall far short of those of the individual nanotube elements. The premature failure in these materials under tensile load has been attributed to inter-tube sliding, which requires far less force than that needed to fracture individual nanotubes.1,2 In order for nanotube materials to achieve their full potential, methods are needed to restrict this tube-tube shear and increase inter-tube forces.Our group is examining covalent crosslinking between the nanotubes as a means to increase the tensile properties of carbon nanotube materials. We are working with multi-walled carbon nanotube (MWCNT) sheet and yarn materials obtained from commercial sources. Several routes to functionalize the nanotubes have been examined including nitrene, aryl diazonium, and epoxide chemistries. The functional nanotubes were crosslinked through small molecule or polymeric bridges. Additionally, electron beam irradiation induced crosslinking of the non-functional and functional nanotube materials was conducted. For example, a nanotube sheet material containing approximately 3.5 mol amine functional groups exhibited a tensile strength of 75 MPa and a tensile modulus of 1.16 GPa, compared to 49 MPa and 0.57 GPa, respectively, for the as-received material. Electron beam irradiation (2.2x 1017 ecm2) of the same amine-functional sheet material further increased the tensile strength to 120 MPa and the modulus to 2.61 GPa. This represents approximately a 150 increase in tensile strength and a 360 increase in tensile modulus over the as-received material with only a 25 increase in material mass. Once we have optimized the nanotube crosslinking methods, the performance of these materials in polymer matrix composites will be evaluated.
This paper uses spaceborne lidar data to study how near-cloud aerosol statistics of attenuated backscatter depend on cloud fraction. The results for a large region around the Azores show that: (1) far-from-cloud aerosol statistics are dominated by samples from scenes with lower cloud fractions, while near-cloud aerosol statistics are dominated by samples from scenes with higher cloud fractions; (2) near-cloud enhancements of attenuated backscatter occur for any cloud fraction but are most pronounced for higher cloud fractions; (3) the difference in the enhancements for different cloud fractions is most significant within 5km from clouds; (4) near-cloud enhancements can be well approximated by logarithmic functions of cloud fraction and distance to clouds. These findings demonstrate that if variability in cloud fraction across the scenes used to composite aerosol statistics are not considered, a sampling artifact will affect these statistics calculated as a function of distance to clouds. For the Azores-region dataset examined here, this artifact occurs mostly within 5 km from clouds, and exaggerates the near-cloud enhancements of lidar backscatter and color ratio by about 30. This shows that for accurate characterization of the changes in aerosol properties with distance to clouds, it is important to account for the impact of changes in cloud fraction.
Floating point computation presents a number of problems for formal verification. Should one treat the actual details of floating point operations, or accept them as imprecisely defined, or should one ignore round-off error altogether and behave as if floating point operations are perfectly accurate. There is the further problem that a numerical algorithm usually only approximately computes some mathematical function, and we often do not know just how good the approximation is, even in the absence of round-off error. ORA has developed a theory of asymptotic correctness which allows one to verify floating point software with a minimum entanglement in these problems. This theory and its implementation in the Ariel C verification system are described. The theory is illustrated using a simple program which finds a zero of a given function by bisection. This paper is presented in viewgraph form.
We present the first comprehensive set of lunar exospheric line width and line width derived effective temperatures as a function of lunar phase (66° waxing phase to 79° waning phase). Data were collected between November 2013 and May 2014 during six observing runs at the National Solar Observatory McMath-Pierce Solar Telescope by applying high-resolution Fabry-Perot spectroscopy (R ~ 180,000) to observe emission from exospheric sodium (5,889.9509 Å, D2 line). The 3-arc min field of view of the instrument, corresponding to ~336 km at the mean lunar distance (384,400 km), was positioned at several locations off the lunar limb; only equatorial observations taken out to 950 km are presented here. We find the sodium effective temperature distribution to be approximately a symmetric function of lunar phase with respect to full Moon. Within magnetotail passage we find temperatures in the range of 2500-9000 K. For phase angles greater than 40deg we find that temperatures flatten out to ~1700 K.
An approximate solution for interlaminar stresses in finite width, laminated composites subjected to uniform extensional, and bending loads is presented. The solution is based upon the principle of minimum complementary energy and an assumed, statically admissible stress state, derived by considering local material mismatch effects and global equilibrium requirements. The stresses in each layer are approximated by polynomial functions of the thickness coordinate, multiplied by combinations of exponential functions of the in-plane coordinate, expressed in terms of fourteen unknown decay parameters. Imposing the stationary condition of the laminate complementary energy with respect to the unknown variables yields a system of fourteen non-linear algebraic equations for the parameters. Newton's method is implemented to solve this system. Once the parameters are known, the stresses can be easily determined at any point in the laminate. Results are presented for through-thickness and interlaminar stress distributions for angle-ply, cross-ply (symmetric and unsymmetric laminates), and quasi-isotropic laminates subjected to uniform extension and bending. It is shown that the solution compares well with existing finite element solutions and represents an improved approximate solution for interlaminar stresses, primarily at interfaces where global equilibrium is satisfied by the in-plane stresses, but large local mismatch in properties requires the presence of interlaminar stresses.
An investigation of two-dimensional rain cell size statistics has analyzed regression relations relating radar-determined rain rates to disdrometer data. This has yielded least-squares fits of radar reflectivity factors, and the application of a contouring program has generated 22,000 contours in which each isopleth belongs to predefined rain-rate intervals. An abundance of total and cell contours were observed belonging to all rain-rate categories. Both the computed number distributions and the conditional cumulative distributions as a function of contour diameter were found to be represented with good approximation by given exponential functions.
CNDOL is an a priori, approximate Fockian for molecular wave functions. In this study, we employ several modes of singly excited configuration interaction (CIS) to model molecular excitation properties by using four combinations of the one electron operator terms. Those options are compared to the experimental and theoretical data for a carefully selected set of molecules. The resulting excitons are represented by CIS wave functions that encompass all valence electrons in the system for each excited state energy. The Coulomb–exchange term associated to the calculated excitation energies is rationalized to evaluate theoretical exciton binding energies. This property is shown to be useful for discriminating the charge donation ability of molecular and supermolecular systems. Multielectronic 3D maps of exciton formal charges are showcased, demonstrating the applicability of these approximate wave functions for modeling properties of large molecules and clusters at nanoscales. This modeling proves useful in designing molecular photovoltaic devices. Our methodology holds potential applications in systematic evaluations of such systems and the development of fundamental artificial intelligence databases for predicting related properties.
New, high-resolution images of the disks of NGC 4038/4039 obtained with the Wide Field Camera of the Hubble Space Telescope (HST) are presented. NGC 4038/4039, nicknamed The Antennae, is a prototypical example of a pair of colliding galaxies believed to be at an early stage of a merger. Down to the limiting magnitude of V approximately 23 mag, the HST images reveal a population of over 700 blue pointlike objects within the disks. The mean absolute magnitude of these objects is M(sub V) = -11 mag, with the brightest objects reaching M(sub V) approximately -15. Their mean apparent color indices ar U - V = -0.7 mag and V - 1 = 0.8 mag on the Johnson UVI passband system, while their mean indices corrected for internal reddening are (u - v)(sub 0) = -1.0 mag and (V - I(sub 0) = 0.5. Their mean effective radius, determined from slightly resolved images, is 18 pc (for H(sub 0) = 50 km/s /Mpc). Based on their luminosities and resolution, most of these objects cannot be individual stars, but are likely young compact star clusters. The brighter ones are similar to the objects found in NGC 1275 and NGC 7252, which appear to be young globular clusters formed during recent galazy mergers. Based on their U - V and V - I colors, the brightest, bluest clusters of NGC 4038/4039 appear to be less than 10 Myr old. Most of these bright clusters are relatively tightly clustered themselves, with typically a dozen individual clusters belonging to a complex identified as a giant H II region from ground-based observations. The cluster luminosity function (LF) is approximately a power law, phi(L)dL proportional to L(exp -1.78+/-0.05)dL, with no hint of a turnover at fainter magnitudes. This power-law shape agrees with the LF of Magellanic Cloud clusters and Galactic open clusters, but differs from the LF of old globular cluster systems that is typically Gaussian with a Full Width at Half Maximum (FWHM) of approximately 3 mag. Besides the blue clusters, we also find about a dozen extremely red objects with V - I greater than 3.0. The highest number density of these red objects is found in the SE quadrant, where star formation appears to be most recent. We propose that these objects may be very young star clusters still embedded in their placental dust cocoons.
The interacting shell model, a configuration-interaction method, is a venerable approach for low-lying nuclear structure calculations, but it is hampered by the exponential growth of its basis dimension as one increases the single-particle space and/or the number of active particles. Recent, quantum-information-inspired work has demonstrated that the proton and neutron sectors of a nuclear wave function are weakly entangled. Furthermore, the entanglement is smaller for nuclides away from N = Z, such as heavy, neutron-rich nuclides. Here, in this study, we implement a weak entanglement approximation to bipartite configuration-interaction wave functions, approximating low-lying levels by coupling a relatively small number of many-proton and many-neutron states. This truncation scheme, which we present in the context of past approaches, reduces the basis dimension by many orders of magnitude while preserving essential features of nuclear spectra.
The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.
The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.