Showing papers on "Antisymmetric relation published in 1974"

Cosmic-ray streaming perpendicular to the mean magnetic field

Abstract: Starting from a quasi-linear approximation for the ensemble-averaged particle distribution function in a random magnetic field, the complete diffusion tensor is derived. This is done by assuming a simple form for the ensemble-averaged distribution function, explicitly retaining all components of the streaming flux. This derivation obtains the antisymmetric terms in a natural manner. The necessary dropping of higher-order terms gives a criterion for the lower-energy limit of validity of the perpendicular and antisymmetric diffusion coefficients. The limit for the assumed distribution function is about 0.8 GV rigidity in the interplanetary field near 1 AU.

Propagation of pulse trains through a random medium

Abstract: Applying the parabolic approximation, the equations for two-frequency symmetric and antisymmetric mutual intensity functions for waves propagating through a random medium are derived, including the multiple scattering effects. These functions are applied to derive the general formulas for the covariance functions of narrowband pulses. They are used to compute the signal intensities for pulse trains passing through an ionospheric irregularity slab.

Standard polynomials in matrix algebras

Abstract: Let Mn(F) be an n x n matrix ring with entries in the field F, and let Sk(X1,.. -, X) be the standard polynomial in k variables. AmitsurLevitzki have shown that S2n(Xi *,X2n) vanishes for all specializations of X ,***,X2n to elements of M (F). Now, with respect to the transpose, let Mn(F) be the set of antisymmetric elements and let Mn(F) be the set of symmetric elements. Kostant has shown using Lie group theory that for n even S2n2(X 1' * *' X2" 2) vanishes for all specializations of X, * ... I X2n 2 to elements of M-(F).By strictly elementary methods we have obtained the following strengthening of Kostant's theorem: S2n2(X I I * * , X2n 2) vanishes for all specializations of X19 *I X2n2 to elements of Mn(F), for all n. S2n 1(X *... ,X2n 1) vanishes for all specializations of Xl,* IX2n2 to elements of Mn(F) and of X to an element of M (F), for all n. S2n 2(X,... ,X2n 2) vanishes for all specializations of X1, X2n3 to elements of M_(F) and of X to an element of M+(F), for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2. Introduction. The object of this paper is to prove the results described in the abstract. The method of proof is to exploit certain properties of the trace (given in ? 1) in connection with an undirected graph whose edges correspond to elementary symmetric and antisymmetric matrices (see ? 2). ??3-6 consist of manipulations of the graph to prove the main theorem (Theorem 1). Although Theorem 1 is sharp in characteristic 0 (as shown via counterexamples in.?8), more results can be obtained in characteristic 2 and, at times, in odd characteristic (viz. ?7). The sharpness of these results is also explained in ?8. In ?9 the relationship of Theorem 1 and the theory of identities of rings with involution is given. In recent months two other people, Joan Hutchinson and Frank Owens, indePresented to the Society, January 25, 1973 under the title Standard identities for matrix rings with involution; received by the editors March 23, 1973. AMS (MOS) subject classifications (1970). Primary 15A24, 15A57, 16A28, 16A38; Secondary 05C30, 05C35.

Buckling in Segmented Shells of Revolution Subjected to Symmetric and Antisymmetric Loads

Abstract: The behavior and stability of shells of revolution subjected to nonuniform loads have been investigated by many workers. Despite the intensive study of the buckling process in shells of revolution, little attention has been paid to buckling under circumferentially varying loads. This paper presents an analysis of buckling in segmented shells of revolution subjected to symmetric and antisymmetric loads. In the buckling analysis, the stability criterion is based on the principle of minimum potential energy. Energy formulation was used in conjunction with finite differences and trigonometric series to derive equations for the buckling loads. By using matrix methods the total potential energy is expressed in quadratic form, which is ideally suited for programing digital computers. The paper also presents a number of numerical results for cylindrical shells. c e? ee> ese H i j K^, Ke, K^e

SCF equations for pure spin states with many-particle interactions

Abstract: SCF equations for any pure spin state are given for a spin-free system with many-particle interactions. The equations are very simple and explicit. Due to the use of different antisymmetric requirement our equations are different from some of the other methods. In our method, the abstract group theory formalism is converted into some explicit and straightforward equations which makes the many-particle interaction problem easier to handle.

Gravidynamics of linear spin two neutral fields

Abstract: The dynamics of a gravity spin two field system is analysed when linear antisymmetric formulations for the ‘matter’ field is assumed and the minimal prescription for the interaction is made. It is shown that this interacting system is equivalent to a non-minimally coupled spin-two field when it is given in terms of the symmetric first-order representation. The relevance of considering the full interacting system (instead of treating gravity as external) is stressed throughout.

The full symmetry of the photoelastic tensor of elastic waves

Abstract: It is shown that the most complete way of treating the antisymmetric contributions to the photoelastic tensor of shear elastic waves is by considering the photoelastic tensor as a mode property of the wave which must conform to the symmetry of the wave in the particular medium. This means that one must take the gradients of the elastic-wave coordinates consistently with the spatial symmetry of the crystal. Neumann's principle of crystal physics is thus extended to include dynamical crystal phenomena. The present approach applies to all mode properties of elastic shear waves.

Deformation coupling of X-ray scattering factors

Abstract: Antisymmetric charge distortions form the basic deformation of the shell lattice dynamics models. The effect of these deformations on the scattering factors of atoms in a solid is investigated through the Born beta coupling coefficients. Particular reference is made to the alkali halides. The Fourier transform coefficients which directly affect the X-ray scattering intensities are calculated for fitted shell models of NaF and NaCl. The theory is extended to define deformation matrices useful for antisymmetric distortions. These matrices are related to the shell-core displacements. Real space coupling parameters are found and values for a microcrystal of atoms surrounding the origin have been calculated. These give some insight into the relative effect of different polarization mechanisms. They also show that second- and third-neighbour coupling parameters are comparable in magnitude to the nearest-neighbour parameter.

Electronic correlation studies. II. Self‐correlated field method. Application to ground state and first 1P, 3P excited states of two‐electron atomic systems

Abstract: An antisymmetric pair function can be built upon two kinds of monoelectronic functions, the former ones being correlated local functions and the second ones nonlocal functions taking external effects into account. This function, brought into the generalised product function procedure by means of the density matrix formulation, makes possible the study of correlation within N-electronic systems. The results of a first application of this method to the fundamental and to 1P and 3P excited states of two-electron systems are given.

Periodic Solutions of Gravity Oriented Axisymmetric Systems

Abstract: quadrature schemes may be used in effecting the numerical integrations. In this sense, specific discrete element "condensation" techniques represent only one of many possible ways for carrying out the required process of integration and summation in the Rayleigh-Ritz method. To further illustrate the alternative ways in which Rayleigh's principle may be applied and the fact that the use of a vibration mode determined from a first iteration does not necessarily lead to best results, the numerical example given by Fried 3 will be considered from the simpler (and more traditional) point of view of the dynamicist. The Rayleigh quotient may be formed from Fried' s Eq. (6) by multiplying the first row of the matrix equation by xl9 the second by x2, etc., and summing the results to give Ml Apart from physical constants which are absorbed in the definition of A, this may be viewed as Rayleigh's quotient for the transverse vibration of a massless string under constant tension with five equal masses equally spaced along the string. In this case the numerator represents the strain energy and the denominator represents the kinetic energy divided by frequency squared. In the traditional approach of the dynamicist, a reasonable (and typical) assumption for the shape of the lowest symmetric mode of this system would 5e a parabola (as in Appendix 1 of Ref. 4). Thus Xj_ = 5, x2 = 8, x3 = 9, x4 — 8, x5 = 5 Substituting these values into Eq. (1) gives A! - 70/259 - 0.2703 This compares to Fried's result for the singly iterated mode, A! - 0.2860, for the doubly iterated mode, ^ = 0.2680, and the exact value, Aj_ = 0.2679. The accuracy of the result obtained here by application of the primitive Rayleigh method is thus much better than the result obtained by Fried using a singly iterated mode and differs from the exact solution by less than i°/ For the an ti symmetrical case, applying a similar assumed parabola to each half of the system leads to A 2 = 1.000 which corresponds to the exact solution. By contrast, Fried obtained /2 = 1.200 for his singly iterated mode and A2 = 1.024 for his doubly iterated mode. Fried's results are a consequence of the fact that his assumed antisymmetric loading for the first iteration destroyed the subsymmetry of the first antisymmetric vibration mode in each half of the system. This example is not, of course, typical of all problems to which Rayleigh's principle might be applied. (For instance, in the bending vibration of cantilever beams of variable properties use of the first iteration for the mode shape in application of Rayleigh's principle to determination of the lowest natural frequency is often a most satisfactory procedure.) It does, however, emphasize that to obtain best results with minimum effort from Rayleigh's principle, it is necessary for the analyst to exercise a degree of judgment based on the geometrical, structural, and dynamic properties of the system under consideration and to choose analytical techniques accordingly.

Aeroelastic loads predictions using finite element aerodynamics

Abstract: Aerodynamic and structural influence coefficients are utilized to determine the load distributions, deflections, and trim parameters for a vehicle in quasi-static aeroelastic equilibrium. A matrix formulation is used to solve the various quasi-static aeroelastic problems. Nonlinearities in the aeroelastic trim equations are accounted for by an iteration of the classical closed form solution. Aerodynamic and structural idealizations are related by a surface spline transformation. Solutions are developed for symmetric, antisymmetric, and asymmetric load conditions on symmetric vehicles of general geometric shapes which may include both lifting surfaces and lifting bodies. INTRODUCTION With the advent of large order matrix solutions for the analysis of complex structures a need has arisen for a complementary approach to the external loads problem. Finite element structural analysis techniques demand that the external loads be distributed over the structure at discrete points. Therefore, shear, moment, and torque distributions along a psuedoelastic axis are no longer sufficient to define the external load distributions required by the stress analysis. A general finite element approach to the problem of the determination of aeroelastic loads on a flexible vehicle flying in a state of quasi-static equilibrium is presented here. The vector point loads available from this solution are directly applicable to matrix structural analyses. Structural and aerodynamic influence coefficients obtained from finite element idealizations of the aircraft are utilized as a basis for the method. The technique is primarily an extension of the method first suggested by ~ e d m a n ( l ) * and later generalized by odde en.(^) This work extends the efforts of the above-mentioned authors by * Numbers in parentheses designate References at end of paper.