Showing papers on "Antisymmetric relation published in 1971"

Korteweg‐de Vries Equation and Generalizations. IV. The Korteweg‐de Vries Equation as a Hamiltonian System

Abstract: It is shown that if a function of x and t satisfies the Korteweg‐de Vries equation and is periodic in x, then its Fourier components satisfy a Hamiltonian system of ordinary differential equations. The associated Poisson bracket is a bilinear antisymmetric operator on functionals. On a suitably restricted space of functionals, this operator satisfies the Jacobi identity. It is shown that any two of the integral invariants discussed in Paper II of this series have a zero Poisson bracket.

Erratum: Kinetic Equations for Orientational and Shear Relaxation and Depolarized Light Scattering in Liquids

Abstract: The application of Mori's theory of fluctuations to the calculation of depolarized light scattering spectra of liquids is described. Two sets of kinetic equations are derived. The first is capable of describing the shear wave doublet observed by Fabelinskii and co‐workers and Stegeman and Stoicheff and reduces to the theory of Leontovich when the antisymmetric part of the stress tensor is ignored. The second set describes the doublet as well as the broad background observed in several laboratories and reduces to the theory of Volterra in the limit where antisymmetric stress is ignored. The angular dependence of the spectrum predicted by the first set of equations is compared with the experimental results of Stegeman and Stoicheff for aniline and quinoline. Good agreement is obtained for quinoline at all angles and for aniline at all but the highest scattering angle at which doublet structure was observed. Our results indicate that the relaxation of the antisymmetric part of the stress tensor may be import...

Antisymmetric Vibrations of a Circular Plate

Abstract: This paper, motivated by the use of “bender” disks with underwater transducers, presents a complete solution of the elastic equations for freely supported thick disks that experience no net radial extension. The solution is complete in that all roots of the frequency equation, both real and complex, are considered when the boundary conditions are satisfied.

Inviscid Instability of Antisymmetric Flows in a Channel

Abstract: Rayleigh equation in the hydrodynamic stability theory is investigated for antisymmetric velocity profiles in a channel. For the disturbance of zero-phase-velocity, it is revealed that there exist only two neutrally stable solutions, one is the spatially antisymmetric mode of zero-wave-number and the other is the symmetric one of finite wave-number.

Wigner Coefficients for the Group E(2)

Abstract: The Wigner coefficients for the faithful unitary representations of the 2‐dimensional Euclidean group are derived from two identities involving Bessel functions. Since the multiplicity in the decomposition of a direct product is two, we find two sets of coefficients which are real and mutually orthogonal and symmetric and antisymmetric, respectively, under interchange of constituent representations. Properties of the coefficients at the ends of the decomposition spectrum are discussed.

Aerial field simulation

Abstract: Any arbitrary antenna array distribution can be represented as the sum of a symmetric and an antisymmetric function. The radiation at any point in the far field of the aerial array can then be simulated by adding together the symmetric and antisymmetric outputs in quadrature with the correct amplitude. This simplifies the construction of subsequent combining units since these reproduce the far field at any point by amplitude wieghting along without the need for differential phasing.

Continuous lattice ordering by Schauder basis cones

Abstract: Let (E, r) be a barrelled Hausdorff space lattice ordered by the cone of an unconditional Schauder basis (xn, fn). It is shown that under such an ordering (E, T) is a locally convex lattice. Necessary and sufficient conditions are given for the lattice operations to be continuous with respect to the weak topologies on E and its topological dual E': the lattice operations are a(E, E')continuous on E if and only if {f.:necoI is a Hamel basis for E'. 1. Preliminaries. Let (E, r) be a locally convex space. Let E' be the space the all r-continuous linear functionals on E. A biorthogonal system in (E, r) is a sequence {Xk } in E and a sequence I fk } in E' with the property that fn(xm) =,.. A Schauder basis for (E, r) is a biorthogonal system in (E, r) such that for each xCE we have x = EkZlfk(x)xk. The series is understood to converge to x in the topology (E, r). If (xk, fk) is a Schauder basis for (E, r) and every rearrangement of Et=lfk(x)xk converges to x, then (xk, fk) is called an unconditional Schauder basis for (E, r). An ordered vector space is a real vector space E equipped with a transitive, reflexive, antisymmetric relation 0}, where 6 is the zero vector of E. The set K has the properties that K+KCK, XKCK for each X?_O, and Kn -K= {0}. A set with the above three properties is called a cone. If K is a cone in a real vector space E, then a relation ? is defined on E by x?y if y-xCK. With this order E is an ordered vector space with positive cone K, i.e. E is ordered by K. The terminology in this paper concerning ordered vector spaces is that of [5]. Presented to the Society, November 8, 1968 under the title Schauder basis cones and continuity of the lattice operations in barreled spaces; received by the editors March 27, 1970 and, in revised form, March 1, 1971. AMS 1969 subject classifications. Primary 4606, 4601; Secondary 0680.

On the backscattering from two arbitrarily located identical parallel wires

Abstract: A study of the backscattering from two identical perfectly conducting, thin wires illuminated by a plane wave at an arbitrary angle of incidence is presented. The theory is based on an integral equation method. By decomposing the induced currents into symmetric and antisymmetric modes, the simultaneous integral equations for the induced currents are converted into independent integral equations similar to the one for a single wire for which the solution has already been carried out. The induced currents on and the backscattering cross sections of the wires are determined. Numerical examples include nonstaggered, staggered, and collinear cases of both half-wave and full-wave wires. Comparisons are made between the calculated and measured values of the echo area. The experimental results are in good agreement with theory.

The optimum projection technique in many‐electron treatments

Abstract: The optimum projection technique is the determination of the best function in the space spanned by a set off f(N, S) linearly independent antisymmetric space-spin eigenfunctions of S2 obtainable from a spatial function made of a product of N-independent orbitals. This is formulated in the spin-free framework. We consider several sets of predetermined orbitals for the lithium 2S state. Both the energy and spin-density are determined for each optimum projected function. The behavior of certain results is explained in terms of the “closeness” of the ls and ls′ split-shell core orbitals.

Dynamic properties of suspension bridges

Abstract: EXPERIMENTAL DETERMINATION OF THE NATURAL FREQUENCIES OF VIBRATION, THE VERTICAL AND TORSIONAL MODE SHAPES, AND THE MODAL DAMPING OF THE NEWPORT BRIDGE, RHODE ISLAND, AND THE WILLIAM PRESTON LANE MEMORIAL BRIDGE, MARYLAND IS BRIEFLY DESCRIBED. THE RESULTS OF THE TWO MEASUREMENTS ARE COMPARED TO SHOW FUNDAMENTAL SIMILARITIES IN THEIR DYNAMIC BEHAVIOR. THE SIMILARITIES INCLUDE THE ORDER THAT THE TYPES OF MOTION OF THE DECK OCCUR. THIS ORDER IS AS FOLLOWS: SYMMETRIC VERTICAL, ANTISYMMETRIC VERTICAL, ANTISYMMETRIC VERTICAL, SYMMETRIC VERTICAL, SYMMETRIC VERTICAL, SYMMETRIC TORSIONAL AND ANTISYMMETRIC VERTICAL. THE RATIOS OF HIGHER ORDER MODAL FREQUENCIES TO THE LOWEST MODAL FREQUENCIES ARE GRAPHED. THE MEAN RATIOS ON THE TWO BRIDGES ARE FOUND TO AGREE WITHIN 10 PERCENT. THESE FACTORS MAY SERVE AS USEFUL RULES OF THUMB DURING THE DESIGN OF SUSPENSION BRIDGES. /AUTHOR/