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Mathematical analysis technique generalizes a root squaring and root cubing method into a general root powering method. The introduction of partitioned polynomials into this general root powering method simplifies the coding of the polynomial transformations into input data suitable for processing by computer. The method includes analytic functions.
Correct combinations of remote, analytic, and functional methods and measuring devices for detecting extraterrestrial life are elaborated. Considered are techniques and instruments available both on earth and aboard spacecraft and artificial planetary satellites. Emphasis is placed on the abiogenetic synthesis of organic compounds formed in photosynthesis on Mars.
The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct a uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.
We describe a method for computing the history of the luminosity function in a young cluster still forming within a molecular cloud complex. Our method, which utilizes detailed results from stellar evolution theory, assumes that clusters arise from the continuous collapse of dense cloud cores over a protracted period of time. It is also assumed that stars reaching the main sequence are distributed in mass according to a prescribed initial mass function (IMF). We keep track separately of the contributions to the luminosity function from the populations of protostars, pre-main-sequence stars, and main-sequence stars. We derive expressions for the fractional contribution of these populations to both the total number of stars produced and the total cluster luminosity. In our model, the number of protostars rises quickly at first, but then levels off to a nearly constant value, which it maintains until the dispersal of the cloud complex. The number fraction of protostars always decreases with time. Averaged over the life of the parent cloud, this fraction is typically a few percent. The protostar mass distribution can be expressed as an integral over the IMF.
An analytic solution to the heavy ion transport in terms of Green's function is used to generate a highly efficient computer code for space applications. The efficiency of the computer code is accomplished by a nonperturbative technique extending Green's function over the solution domain. The computer code can also be applied to accelerator boundary conditions to allow code validation in laboratory experiments.
The purpose of this research program is to design, build and test (in cooperation with NASA personnel from the NASA Langley Research Center) neural controllers for two different small air-gap magnetic suspension systems. The general objective of the program is to study neural network architectures for the purpose of control in an experimental setting and to demonstrate the feasibility of the concept. The specific objectives of the research program are: (1) to demonstrate through simulation and experimentation the feasibility of using neural controllers to stabilize a nonlinear magnetic suspension system; (2) to investigate through simulation and experimentation the performance of neural controllers designs under various types of parametric and nonparametric uncertainty; (3) to investigate through simulation and experimentation various types of neural architectures for real-time control with respect to performance and complexity; and (4) to benchmark in an experimental setting the performance of neural controllers against other types of existing linear and nonlinear compensator designs. To date, the first one-dimensional, small air-gap magnetic suspension system has been built, tested and delivered to the NASA Langley Research Center. The device is currently being stabilized with a digital linear phase-lead controller. The neural controller hardware is under construction. Two different neural network paradigms are under consideration, one based on hidden layer feedforward networks trained via back propagation and one based on using Gaussian radial basis functions trained by analytical methods related to stability conditions. Some advanced nonlinear control algorithms using feedback linearization and sliding mode control are in simulation studies.
It is shown that when differencing analytic functions using the pseudospectral Fourier or Tchebyshev methods, the error committed decays to zero at an exponential rate.
Using analytic methods, we derived the first-order Green's functions, in terms of the observing angle, for the reflected spectra after a single Compton scattering of relativistic photons by a plane-parallel semiinfinite medium of cold electrons. We also calculated the total Green's functions for incident photons of 511 keV, using Monte Carlo methods. We show that both the first-order and total Green's functions have strong angular dependences and that the first-order Green's functions play a dominant role in determining the properties of the total reflected spectra.
The analytic energy gradient for the single and double excitation coupled-cluster (CCSD) wave function has been reformulated and implemented in a new set of programs. The reformulated set of gradient equations have a smaller computational cost than any previously published. The iterative solution of the linear equations and the construction of the effective density matrices are fully vectorized, being based on matrix multiplications. The new method has been used to investigate the Cl2O2 molecule, which has recently been postulated as an important intermediate in the destruction of ozone in the stratosphere. In addition to reporting computational timings, the CCSD equilibrium geometries, harmonic vibrational frequencies, infrared intensities, and relative energetics of three isomers of Cl2O2 are presented.
A simple kinematically consistent expression for the longitudinal two-point correlation function related to both the integral length scale and the Taylor microscale is obtained. On the inner scale, in a region of width inversely proportional to the turbulent Reynolds number, the function has the appropriate curvature at the origin. The expression for two-point correlation is related to the nonlinear cascade rate, or dissipation epsilon, a quantity that is carried as part of a typical single-point turbulence closure simulation. Constructing an expression for the two-point correlation whose curvature at the origin is the Taylor microscale incorporates one of the fundamental quantities characterizing turbulence, epsilon, into a model for the two-point correlation function. The integral of the function also gives, as is required, an outer integral length scale of the turbulence independent of viscosity. The proposed expression is obtained by kinematic arguments; the intention is to produce a practically applicable expression in terms of simple elementary functions that allow an analytical evaluation, by asymptotic methods, of diverse functionals relevant to single-point turbulence closures. Using the expression devised an example of the asymptotic method by which functionals of the two-point correlation can be evaluated is given.
Risk decomposition and ring theory, lattice techniques and universal algebras, and unary functions are considered.
Convective heat transfer for steady state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature and heat transfer coefficients for flow with Reynolds number of 5.0, 100.0, 1000.0, and 2000.0, and Prandtl number of 0.1, 1.0, and 10.0 with uniform grid sizes are reported for axisymmetric pipe with sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic solution is stable, converges rapidly, and simulates the convection of fluid flow accurately since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values thereby minimizing the false numerical diffusion.
A nonperturbative analytic solution of the high charge and energy (HZE) Green's function is used to implement a computer code for laboratory ion beam transport in multilayered materials. The code is established to operate on the Langley nuclear fragmentation model used in engineering applications. Computational procedures are established to generate linear energy transfer (LET) distributions for a specified ion beam and target for comparison with experimental measurements. The code was found to be highly efficient and compared well with the perturbation approximation.
Computer application to flat earth problem with analytical functions - guidance systems
Demonstration that accurate solutions to the integral equation or to the extremal function can be easily obtained by the variational method in many cases when a stepwise constant function is used for the trial function. The evaluation procedure is simple and straightforward; integrals of the kernel function can be evaluated analytically; the method provides, when the solution involves singularities, the best mean value across the singularity; the results are accurate in both the detailed physical quantities and their averages; the resultant solution can be further integrated analytically over the parameters associated with the problem; and the method can be readily applied to nonlinear integral equations.
Two orbit computation methods were used: (1) numerical method- The solution to the satellite differential equations were solved in a step-by-step manner, using a mathematical algorithm taken from numerical analysis; and (2) analytical method - The solution was expressed by explicit functions of the independent variable. Analytical drag modules, tesseral terms initialization module, second order and long period terms module, and verification testing of the ASOP program were also considered.
A method for determining boundary node displacements for three-dimensional ablation solvers is developed. The approach is applicable to a wide range of mesh motion algorithms used to determine internal node displacements. The method leverages radial basis functions to analytically define the geometry to which the boundary nodes should be constrained. Constrained optimization, elasticity, and spring analogy solvers are utilized to redistribute the nodes on the evolved geometry surface while maintaining mesh quality. The method is implemented in the CHarring Ablator Response (CHAR) code, and a realistic demonstration case is presented on the Boundary Layer Transition (BOLT) flight experiment configuration. The method is shown to be robust for highly-curved geometries with large domain deformations.
Aerodynamic shape design has long persisted as a difficult scientific challenge due its highly nonlinear flow physics and daunting geometric complexity. However, with the emergence of Computational Fluid Dynamics (CFD) it has become possible to make accurate predictions of flows which are not dominated by viscous effects. It is thus worthwhile to explore the extension of CFD methods for flow analysis to the treatment of aerodynamic shape design. Two new aerodynamic shape design methods are developed which combine existing CFD technology, optimal control theory, and numerical optimization techniques. Flow analysis methods for the potential flow equation and the Euler equations form the basis of the two respective design methods. In each case, optimal control theory is used to derive the adjoint differential equations, the solution of which provides the necessary gradient information to a numerical optimization method much more efficiently then by conventional finite differencing. Each technique uses a quasi-Newton numerical optimization algorithm to drive an aerodynamic objective function toward a minimum. An analytic grid perturbation method is developed to modify body fitted meshes to accommodate shape changes during the design process. Both Hicks-Henne perturbation functions and B-spline control points are explored as suitable design variables. The new methods prove to be computationally efficient and robust, and can be used for practical airfoil design including geometric and aerodynamic constraints. Objective functions are chosen to allow both inverse design to a target pressure distribution and wave drag minimization. Several design cases are presented for each method illustrating its practicality and efficiency. These include non-lifting and lifting airfoils operating at both subsonic and transonic conditions.