Stability criteria for n-th order, homogeneous linear differential equations
Stability criteria for n-th order homogeneous linear differential equations for real continuous functions obtained by Liapunov method
LIAPUNOV FUNCTION↗
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Efficient computation of the N-th rank QED polarization tensor: Universal worldline structure of form factors
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73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Lyapunov functions for a class of n-th order nonlinear differential equations.
Liapunov function from quadratic polynomial for n-order nonlinear differential equations
LIAPUNOV FUNCTION↗
Stability criteria for n-th order, homogeneous linear differential equations.
Linear differential equations stability criteria using quadratic Liapunov functions
Infante, E. F.↗
Some efficient methods for obtaining infinite series solutions of n-th order linear ordinary differential equations
The use of the theta-operator method and generalized hypergeometric functions in obtaining solutions to nth-order linear ordinary differential equations is explained. For completeness, the analysis of the differential equation to determine whether the point of expansion is an ordinary point or a regular singular point is included. The superiority of the two methods shown over the standard method is demonstrated by using all three of the methods to work out several examples. Also included is a compendium of formulae and properties of the theta operator and generalized hypergeometric functions which is complete enough to make the report self-contained.
Allen, G.↗
The effects of bandpass limiters on n-phase tracking systems
The combination of a bandpass limiter and an n-th power tracking loop is considered. Performance of the combination does not depend on the form of the nonlinear function used to create the n-th harmonic signal as long as some energy in that zone is produced. The order in which the limiter and n-th power nonlinearity occur is unimportant so that the results apply equally well to the n-phase Costas type of implementations of the n-th power loops. Closed form expressions for the signal suppression factors are obtained. Coherent and noncoherent SNR's out of the n-th power-limiter combination can be expressed in terms of two of these suppression factors. The loop SNR of an n-th power-limited phase locked loop is estimated by simply calculating the suppression factors for n-th and 2n-th power loops. Finally, n-th power phase locked loops without limiters are compared with similar loops with limiters; the limiter can actually enhance output SNR at moderate to large input SNR.
Butman, S. A.↗
Compression of transmission bandwidth requirements for a certain class of band-limited functions.
A study of source-encoding techniques that afford a reduction of data-transmission rates is made with particular emphasis on the compression of transmission bandwidth requirements of band-limited functions. The feasibility of bandwidth compression through analog signal rooting is investigated. It is found that the N-th roots of elements of a certain class of entire functions of exponential type possess contour integrals resembling Fourier transforms, the Cauchy principal values of which are compactly supported on an interval one N-th the size of that of the original function. Exploring this theoretical result, it is found that synthetic roots can be generated, which closely approximate the N-th roots of a certain class of band-limited signals and possess spectra that are essentially confined to a bandwidth one N-th that of the signal subjected to the rooting operation. A source-encoding algorithm based on this principle is developed that allows the compression of data-transmission requirements for a certain class of band-limited signals.
Smith, I. R.↗
Geophysical properties of the ionospheric irregularities responsible for radio scintillation
The properties of F-region ionospheric irregularities are described based on in-situ measurements of the actual waveforms of ion concentration. The spectral properties of the irregularities are discussed. In high, middle and low latitudes most of the irregularities observed fall into a single 'noiselike' category having power spectra which can be approximated by f to the negative n-th power and S to the n-th power, where S is the irregularity scale size and n is approximately 2. Thus the spectral components have a maximum gradient which is almost independent of their size. Other categories of irregularities are also observed occasionally.
Mcclure, J. P.↗
Parallel and pipeline computation of fast unitary transforms
The letter discusses the parallel and pipeline organization of fast-unitary-transform algorithms such as the fast Fourier transform, and points out the efficiency of a combined parallel-pipeline processor of a transform such as the Haar transform, in which (2 to the n-th power) -1 hardware 'butterflies' generate a transform of order 2 to the n-th power every computation cycle.
Fino, B. J.↗
Position Operators in Terms of Converging Finite-Dimensional Matrices and Their Intertwining with Geometry, Transport, and Gauge
The position operator 𝑟̂ appears as 𝑖∂ 𝑝 in wave mechanics, while its matrix form (e.g., under a Bloch basis) is well known diverging in diagonals, causing difficulties in basis transformation, observable yielding, etc. We aim to find a convergent r-matrix (CRM) to improve the existing divergent r-matrix (DRM), and investigate its influence at both the conceptual and the application levels. A key modification is increasing the familiar substitution of 𝑟̂ by 𝑖∂ 𝑝 to 𝑖∑ 𝑗 ∂ 𝑘 𝑗 , namely the N-th Weyl algebra. Resolving the divergence makes r-matrix rigorously defined, and we are able to show r-matrix is distinct from a spin matrix in terms of its defining principles, transformation behavior, and the observable it yields. Conceptually, the CRM fills the logical gap between the r-matrix and the Berry connection (this unremarked vagueness has caused the diagonal divergence). In application, we focus on transport, and discover that the Hermitian matrix is not identical with the associative Hermitian operator, i.e., 𝑟 𝑚,𝑛 = 𝑟$^∗_{𝑛,𝑚}$ ⇎ 𝑟̂ = 𝑟̂ † , which subtly affects the celebrated Berry curvature formula for adiabatic current. We also discuss how such a non-representation CRM can contribute to building a unified transport theory.
Weyl algebras↗
The solution of a system of n-th-order differential equations using Lie series
Solution to system of n-th order differential equations using Lie series
LIE GROUP↗
On the calculation of panel flutter boundaries.
Methods are described for the complete automation of flutter boundary calculations when the aerodynamic forces are derived from linear three-dimensional unsteady potential flow theory. The usual process of visual curve fairing in the mass ratio versus structural damping plane is replaced by numerical procedures for ordering the eigenvalues in such a way that the n-th eigenvalue is always associated with the same flutter boundary. The mass ratio versus structural damping curves are interpolated via parametric cubic spline functions to produce the desired plots in the stiffness-parameter/mass-ratio plane. The entire process is accomplished in a single computer run.
Gaspers, P. A., Jr.↗
Degenerate R-S perturbation theory
A concise, systematic procedure is given for determining the Rayleigh-Schrodinger energies and wave functions of degenerate states to arbitrarily high orders even when the degeneracies of the various states are resolved in arbitrary orders. The procedure is expressed in terms of an iterative cycle in which the energy through the (2n+1)st order is expressed in terms of the partially determined wave function through the n-th order. Both a direct and an operator derivation are given. The two approaches are equivalent and can be transcribed into each other. The direct approach deals with the wave functions (without the use of formal operators) and has the advantage that it resembles the usual treatment of nondegenerate perturbations and maintains close contact with the basic physics. In the operator approach, the wave functions are expressed in terms of infinite order operators which are determined by the successive resolution of the space of the zeroth order functions.
Hirschfelder, J. O.↗
Optimal compensator structure for linear time-invariant plant with inaccessible states
The problem is considered of designing an optimal linear time-invariant dynamic compensator for the regulation of an n-th order linear time-invariant plant with m independent outputs. The initial plant state is characterized by its first and second moments, and the cost is usual quadratic infinite-time penalty on the state and control, averaged over the initial plant and compensator states. The compensator is based on a minimal-order Luenberger observer and consequently has fixed dimension n-m. Necessary and sufficient conditions are derived for optimality of the compensator gains. The optimal compensator is shown to be unique if the plant has a particular canonical form, and, in general, for any arbitrary plant, the class of all optimal compensators is precisely determined.
Blanvillain, P. J. P.↗
Degenerate RS perturbation theory
A concise, systematic procedure is given for determining the Rayleigh-Schroedinger energies and wave functions of degenerate states to arbitrarily high orders even when the degeneracies of the various states are resolved in arbitrary orders. The procedure is expressed in terms of an iterative cycle in which the energy through the (2n + 1)-th order is expressed in terms of the partially determined wave function through the n-th order. Both a direct and an operator derivation are given. The two approaches are equivalent and can be transcribed into each other. The direct approach deals with the wave functions (without the use of formal operators) and has the advantage that it resembles the usual treatment of nondegenerate perturbations and maintains close contact with the basic physics. In the operator approach, the wave functions are expressed in terms of infinite-order operators which are determined by the successive resolution of the space of the zeroth-order functions.
Hirschfelder, J. O.↗
Interplanetary diffusion coefficients for cosmic rays
Information on the cosmic-ray diffusion coefficient, kappa, derived from near-earth observations of the solar modulation of galactic electron fluxes and from the near-earth power spectra of the interplanetary magnetic field, has been used to study the heliocentric radial dependence of kappa, and to derive limits on the spatial extent of the solar modulation region. Representing kappa, as a separable function of radius r and rigidity, and assumming kappa(r) proportional to r to the n-th power, we can place a limit on the power law exponent, n not greater than 1.2. The distance of the modulation boundary is a function of n, and, e.g., for n = 0, falls into the range of 6-25 AU.
Cummings, A. C.↗
On the formulation of the gravitational potential in terms of equinoctial variables
Explicit expressions in equinoxial elements are derived for the n-th term in the disturbing potential for three-body and zonal harmonics. A method for recursive computation of the disturbing potential and the partial derivatives is proposed. Preliminary results for an arbitrary geopotential term are obtained.
Cefola, P. J.↗
Convective heat transfer during dendritic solidification
Experiments on succinonitrile are described in which the dependence of dendritic growth velocity is studied as a function of orientation with respect to gravity. Growth rate measurements were carried out at a relatively small supercooling, requiring high specimen purity as well as extreme thermal stability and precision temperature measurement. The normalized growth velocity showed a dependence on orientation described by the ratio of observed growth velocity to that expected for convection-free growth being equal to 3.52 times the n-th power of Cos half the orientation angle, where n lies between 0.5 and 0.75.
Glicksman, M. E.↗