Generalization of Lagrange's implicit function theorem to n-dimensions
Lagrange implicit functions generalization to n dimensions, developing inversion as power series
Feagin, T.↗
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Results for “generalized functions”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Series representation of generalized temperature functions.
Maclaurin series expansion representation of generalized heat equation
HEAT EQUATION↗
The radial heat polynomials and related functions.
Generalized heat equation solution expansion in terms of radial heat polynomials, Appel transforms and Laguerre polynomials
RADIAL DISTRIBUTION↗
Apollo experience report guidance and control systems
The Apollo guidance and control systems for both the command module and the lunar module are described in a summary report. General functional requirements are discussed, and general functional descriptions of the various subsystems and their interfaces are provided. The differences between the original in-flight maintenance concept and the final lunar-orbital-rendezvous concept are discussed, and the background in philosophy, the system development, and the reasons for the change in concept are chronologically presented. Block diagrams showing the command module guidance and control system under each concept are included. Significant conclusions and recommendations contained in more detailed reports on specific areas of the guidance and control systems are included.
Wilson, R. E., Jr.↗
Expansion of Tabulated Scattering Matrices in Generalized Spherical Functions
An efficient way to solve the vector radiative transfer equation for plane-parallel turbid media is to Fourier-decompose it in azimuth. This methodology is typically based on the analytical computation of the Fourier components of the phase matrix and is predicated on the knowledge of the coefficients appearing in the expansion of the normalized scattering matrix in generalized spherical functions. Quite often the expansion coefficients have to be determined from tabulated values of the scattering matrix obtained from measurements or calculated by solving the Maxwell equations. In such cases one needs an efficient and accurate computer procedure converting a tabulated scattering matrix into the corresponding set of expansion coefficients. This short communication summarizes the theoretical basis of this procedure and serves as the user guide to a simple public-domain FORTRAN program.
scattering matrix↗
Nonstandard Analysis and Shock Wave Jump Conditions in a One-Dimensional Compressible Gas
Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. This report presents a fairly comprehensive tutorial on nonstandard analysis for physicists and engineers with many examples applicable to generalized functions. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a one-dimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservations laws, smooth pre-distributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, smooth pre-distributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in nonconservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical Rankine-Hugoniot jump conditions for an inviscid shock wave. Moreover, non-monotonic entropy jump conditions are obtained for both inviscid and viscous flows. The report shows that products of generalized functions may be defined consistently using nonstandard analysis; however, physically meaningful products of generalized functions must be determined from the physics of the problem and not the mathematical form of the governing equations.
Baty, Roy S.↗
Functional design to support CDTI/DABS flight experiments
The objectives of this project are to: (1) provide a generalized functional design of CDTI avionics using the FAA developd DABS/ATARS ground system as the 'traffic sensor', (2) specify software modifications and/or additions to the existing DABS/ATARS ground system to support CDTI avionics, (3) assess the existing avionics of a NASA research aircraft in terms of CDTI applications, and (4) apply the generalized functional design to provide research flight experiment capability. DABS Data Link Formats are first specified for CDTI flight experiments. The set of CDTI/DABS Format specifications becomes a vehicle to coordinate the CDTI avionics and ground system designs, and hence, to develop overall system requirements. The report is the first iteration of a system design and development effort to support eventual CDTI flight test experiments.
Goka, T.↗
The Kirchhoff Formulas for Moving Surfaces in Aeroacoustics - The Subsonic and Supersonic Cases
One of the active areas of computational aeroacoustics is the application of the Kirchhoff formulas to the problems of the rotating machinery noise predictions. The original Kirchhoff formula was derived for a stationary surface. In 1988, Farassat and Myers derived a Kirchhoff Formula obtained originally by Morgans using modem mathematics. These authors gave a formula particularly useful for applications in aeroacoustics. This formula is for a surface moving at subsonic speed. Later in 1995 these authors derived the Kirchhoff formula for a super-sonically moving surface. This technical memorandum presents the viewgraphs of a day long workshop by the author on the derivation of the Kirchhoff formulas. All necessary background mathematics such as differential geometry and multidimensional generalized function theory are discussed in these viewgraphs. Abstraction is kept at minimum level here. These viewgraphs are also suitable for understanding the derivation and obtaining the solutions of the Ffowcs Williams-Hawkings equation. In the first part of this memorandum, some introductory remarks are made on generalized functions, the derivation of the Kirchhoff formulas and the development and validation of Kirchhoff codes. Separate lists of references by Lyrintzis, Long, Strawn and their co-workers are given in this memorandum. This publication is aimed at graduate students, physicists and engineers who are in need of the understanding and applications of the Kirchhoff formulas in acoustics and electromagnetics.
Farassat, F.↗
Electromagnetic instability in high-beta plasmas with generalized distribution functions
The electrostatic instabilities in an anisotropic plasma have been studied quite extensively, but the electromagnetic instabilities in high-beta plasmas, where they play an important role, have not been thoroughly investigated. Recently, Davidson and Wu (1970) looked into the ordinary mode electromagnetic instability which can arise in high-beta bi-Maxwellian plasmas. Here, the magnetic instability is discussed in (A, beta) space (A is the anisotropy in the temperature of the plasma species and beta is the ratio of the kinetic pressure to the magnetic pressure), which can occur in generalized non-Maxwellian plasmas with an inverted population of different species.
Buti, B.↗
A numerical method for determination of source time functions for general three-dimensional rupture propagation
A method to determine the displacement and the stress on the crack plane for a three-dimensional shear crack of arbitrary shape propagating in an infinite, homogeneous medium which is linearly elastic everywhere off the crack plane is presented. The main idea of the method is to use a representation theorem in which the displacement at any given point on the crack plane is written as an integral of the traction over the whole crack plane. As a test of the accuracy of the numerical technique, the results are compared with known solutions for two simple cases.
Das, S.↗
Transformations of the hodograph flow equation and the introduction of two generalized potential functions
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Crocco, Luigi↗
Elliptic Functions and Integrals with Real Modulus in Fluid Mechanics
Advantage of the elliptic functions and of the more general functions of Schwarz for fluid mechanics. Flows outside and inside polygons. Application to the calculation of an elbow diffuser for a wind tunnel. Properties of the elliptic integrals of the first kind and of the elliptic functions. Properties of the theta functions and decomposition of the elliptic functions into products of theta functions. Properties of the zeta functions. Decomposition of the elliptic functions into sums of zeta functions and calculations of the elliptic integrals. Applications to the calculation of wing profiles, of compressor profiles, and to the study of the vibrations of airplane wings and of compressor vanes. The manuscript of the present paper was checked by Mr. Eichelbrenner who corrected several imperfections and suggested numerous improvements to make reading of the paper easier. However, the limited subject does not permit filling in more than an incomplete knowledge of the properties of analytic functions.
Legendre, Robert↗
On One-Dimensional Stretching Functions for Finite-Difference Calculations
The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids.
Vinokur, M.↗
Nonlinear upper hybrid drift waves for a longitudinal electric field perpendicular to a uniform magnetic field in the Vlasov-Maxwell approximation
Upper hybrid drift waves are found as a special solution to a Vlasov-Maxwell plasma which has a longitudinal electric field and a perpendicular uniform magnetic field. A single-species plasma with a constant-density mobile neutralizing background supports spatially varying disturbances that oscillate at the upper hybrid frequency. The general functional dependences of the electric field, the plasma number density, and the one-particle distribution function for the special case are found from more general Vlasov-Maxwell equations invariant under a Lie group point transformation. The one-particle distribution function for the plasma is a function of the Liouville invariant, which is the energy in the generalized Bernstein-Greene-Kruskal (BGK) reference frame, and the momentum in the drift direction.
Abraham-Shrauner, B.↗
Dispersion of ensembles of non-interacting particles
The dynamics of an ensemble of noninteracting particles dispersing from a common origin and moving in a common force field with an initial distribution of momenta is analyzed using an approach where the particles are considered as a continuum described by a phase-space distribution function. General solutions are obtained for both the distribution function and the associated spatial density function. The linear case of small departures from circular orbits in an axisymmetric gravitational field is treated along with the specific case of particle dispersion from an object in a circular orbit in the same type of field. Numerical results are presented for the latter case, and consideration is given to the inverse problem of determining the initial time and velocity distribution from knowledge of the ensemble structure at a later time. Explicit results are provided for the case of an ellipsoidal distribution of initial momenta, and a numerical procedure is indicated for treating more general cases.
Heard, W. B.↗
Innovation Inspired by Nature: Capabilities, Potentials and Challenges
Through evolution, nature came up with many effective solutions to its challenges and continually improving them. By mimicking, coping and being inspired, humans have been using Nature's solutions to address their own challenges. In recent years, the implementation of nature's capabilities has intensified with our growing understanding of the various biological and nastic mechanisms and processes. Successes include even the making of humanlike robots that perform such lifelike tasks as walking, talking, making eye-contact, interpreting speech and facial expressions, as well as many other humanlike functions. Generally, once humans are able to implement a function then, thru rapid advances in technology, capabilities are developed that can significantly exceed the original source of inspiration in Nature. Examples include flight where there is no species that can fly as high, carry so much mass, has so large dimensions and fly so fast, and operate at as such extreme conditions as our aircraft and other aerospace systems. However, using the capabilities of today's technology, there are many challenges that are not feasible to address in mimicking characteristics of species and plants. In this manuscript, state-of-the-art of biomimetic capabilities, potentials and challenges are reviewed.
Sensors↗
Intelligent Welding Controller
Control system adapts to changing design requirements and operating conditions. Proposed control system for gas/tungsten arc welding requires only that operator specifies such direct parameters of welds as widths and depths of penetration. In control system for robotic welder, components and functions intimately connected with welding process assigned to controller domain. More general functions assigned to supervisor domain. Initial estimate of indirect parameters of welding process applied to system only at beginning of weld (t=0); after start of welding, outputs from multivariable controller takes place of estimate.
Cook, George E.↗
Functional differential systems and extension of lyapunov's method
Solutions of ordinary differential systems extended to general functional differential equations and considering stability and boundedness
FUNCTIONAL ANALYSIS↗