Showing papers on "Antisymmetric relation published in 1972"

An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder

Abstract: A high‐speed computer was used to investigate the problem of wave propagation in an isotropic elastic cylinder. Dispersion curves corresponding to real, imaginary, and complex propagation constants for the symmetric and the first four antisymmetric modes of propagation are given. The radial distributions of axial and radial displacements and of shear and normal stresses are given for the symmetric mode. By using a finite number of modes of propagation, an approximate solution is found for the problem of the L(0,1) mode impinging on a traction‐free interface. The reflection coefficient is determined in this way and the accompanying generation of higher order modes at the interface is shown to cause a high‐amplitude end resonance. Experimental results obtained by using the resonance method in conjunction with a long rod are presented to substantiate the calculated reflection coefficient and the frequency of end resonance. Phase velocities, based on measurements of the wavelength of standing waves and resonance frequencies, were obtained for the symmetric and first two antisymmetric modes. These measurements extend into the frequency range of more than one propagating mode. The rms deviation between theoretical and experimental results is in general less than 0.2% with the exception of the dispersion curve for the L(0,2) mode which deviates by 0.7%.

Reduction of wave drag by antisymmetric arrangement of wings and bodies.

Abstract: The wave interference effects for bodies or wings in a mirror-symmetric arrangement, and in an antisymmetric arrangement are discussed. It is shown that while in the case of a mirror-symmetric arrangement large adverse interference effects can be observed, antisymmetric arrangements provide comparatively much smaller wave drags. The single continuous wing panels also adapt themselves more readily to varying angles of obliquity, and hence, to varying flight speeds. A detailed review is presented of the previous work on the aerodynamic properties and flight stability of oblique elliptic wing combinations. A possible mode of application of these combinations to transport aircraft operating at moderate supersonic speeds is suggested.

Performance of antisymmetric pseudorandom signals in the measurement of 2nd-order Volterra kernals by crosscorrelation

Abstract: The paper is concerned with the measurement of the 2nd-order kernel in a Volterra-series representation of a nonlinear system by continuous or discrete crosscorrelation using an antisymmetric pseudorandom input signal derived from an m sequence. It is shown that the crosscorrelation measurements are related to the corresponding kernel values by a set of equations which may be structured into a number of independent subsets. The m-sequence properties determine how the maximum order of the subsets for off-diagonal values is related to the upper bound of the arguments for nonzero kernel values, which is used as an index of performance. The performance of signals derived from binary, ternary and quinary m sequences is investigated, and the characteristic polynomials and performance indexes of signals with superior performance are tabulated. Comparison of the results obtained demonstrates the advantages of ternary signals in this application, and an example is used to illustrate the solution of a typical problem.

Matrix Elements and Density Matrices for Many-Electron Spin Eigenstates Built from Orthonormal Orbitals

Abstract: Publisher Summary This chapter discusses matrix elements and density matrices for many-electron spin eigenstates built from orthonormal orbitals. For an important class of many-electron problems, namely, those governed by spin-free or nearly spin-free Hamiltonians, it is practically and theoretically useful to formulate many-electron wavefunctions that, in addition to being antisymmetric, are also eigenfunctions of the total spin operators. The most widespread method of constructing antisymmetric wavefunctions is by an expansion in terms of Slater determinants of orthonormal orbitals. Thereby quantum mechanical problems are transformed into matrix problems and the matrix elements are integrals involving two Slater determinants and certain dynamical operators. Slater determinants have, however, one shortcoming: In general, they are not eigenfunctions of the total spin operator. A second route can be considered as originating with Dirac's vector model. A third route is to construct spin eigenfunctions with the help of projection operators that are not derived from group theory. In all of these methods, the construction of wavefunctions with the desired characteristics is the simpler task. It is in the evaluation of the expectation values and matrix elements of the many-electron operators that complexities and complications arise.

Transformation properties of antisymmetric spin eigenfunctions under linear mixing of the orbitals

Abstract: After recalling the duality between the general linear group GL(m), represented by its N‐fold inner product, and the permutation group SN, we have given a survey of its quantum chemical consequences. It causes the one‐to‐one correspondence between the total spin quantum number and the permutation symmetry of N‐electron spin functions, and, via the Pauli principle which imposes permutation symmetry on the spatial part also, it leads to specific properties of antisymmetric spin eigenfunctions under orbital transformations. Such functions can be classified according to the irreducible representations of GL(m). For special orbital transformations, often occurring in quantum chemistry, which mix only orbitals in different subsets among each other, we have derived how the transformation of the N‐electron wavefunctions simplifies, by a reduction of the representations of GL(m). The theory is illustrated by an example and some applications are discussed.

Die Wechselwirkung zwischen Walsh- und π-Orbitalen im 7-Cyclopropyliden-norbornadien†

Abstract: The qualitative analysis of the PE.-spectrum of 7-cyclopropylidene-norbornadiene (3) shows that the ‘through-space’ interaction of the eA-Walsh orbital with the antisymmetric linear combination of the two π-orbitals in 3 is of the same order as the analogous interaction between π- and the π-orbital of the exocyclic double bond in 7-isopropylidene-norbornadiene (2).

Exchange interactions in rare earth chalcogenides

Abstract: The author obtains explicit formulae for the admixtures of the J=1 level of Sm2+ (4f)6 7F into the J=0 level by exchange coupling to an adjacent Eu2+ (4f)7 8S ion. The problem is formulated using states which are antisymmetric with respect to interchanges of electrons. An unperturbed Hamiltonian is defined which is symmetric with respect to interchanges, and the difference between it and the full Hamiltonian is treated by perturbation theory, taken to second order. It is found that the dominant processes are those in which, in the intermediate states, either two electrons, one from each 4f shell, have been excited into unoccupied orbitals, or two electrons from closed shells have been excited into the 4f shells, one electron to each rare earth ion. By the use of equivalent operator techniques, it is shown that the admixtures can be derived from an equivalent anisotropic exchange interaction, with the important feature that the spin operators which appear are defined in second quantized forms and are not identical with the commonly used spin operators. It is suggested that in phenomenological exchange spin Hamiltonians these new definitions should be used.

Symmetry Labeling of Low Lying Electronic Levels of EuGaG by Means of Light Scattering Techniques

Abstract: A discussion is given of the Raman tensors of the 7F0→ 7F1, 7F0→ 7F2, and the 7F1 intermanifold transitions of the compound EuGaG. Some of these tensors are completely antisymmetric and the correlation is given of tensors, symmetry of the states, and crystal field parameters.

Waves in an elastic plate with an irregular boundary

Abstract: Investigated is the effect of a surface irregularity on the propagation of waves in an isotropic, elastic plate, the method of perturbation being employed to determine the scattered field to the first order in a small parameter descriptive of the height of the irregularity. While the firstorder displacement field in the region under, and in the immediate vicinity of, the disturbed surface is very complicated, the far field in either direction, generated by either a symmetric or an antisymmetric incident wave, consists of a finite number of propagating waves, symmetric and antisymmetric, their number corresponding to the number of real roots of the frequency equation for the given frequency of incident wave.

On Symmetric and Antisymmetric Relations.

Abstract: Let R be an antisymmetric relation on a set X; or equivalently, let (X,/~) be a directed antisymmetric graph. The symmetrization R s of R is defined by R s = R w . R l = { ( x , y ) [ either ( x , y ) eR or (y,x) ~ R}. Conversely, if R is any symmetric relation on X, i. e. (X,/~) an ordinary undirected graph, then an orientation of R is any antisymmetric relation/~0 with (RO) s ~ 1~. Let (X, R) be a relation. We denote by C (X, R) (respectively A (X, 17)) the monoid of all its endomorplfisms (respectively automorphisms). Recall that a mapping f : X + X is an endomorphism of ( X , R ) f f ( f ( x ) , f ( y ) ) e R whenever (x ,y)eR, see [1]. I t is known that for every monoid M there is a symmetric relation (X, R) with C (X, R) -~ M, see [2, 4]. Our aim is to study the orientations of a graph; that is to say, we are interested in what changes of endomorphisms can be made by means of changing arrows in the graph only. Particularly, it follows from Theorem 1 that, given two monoids M,N, there are antisymmetrie relations (X,/~), (X,/~) such that Rs-----_~ s and C ( X , R ) ~_ M, C(X,Y~) ~N. In a special case a stronger result can be proved. Observe that C (X, R ~ is a submonoid of C (X, R) for every symmetric re l~ion R. I t follows from Theorem 2 that given a group G there is a symmetric relation (X, B) such that the sets {A (X,_R0) IRO orientation of R} and {H1H subgroup of G} coincide (up to isomorphism). We prove in fact that these sets considered as lattices are lattice isomorphic.

Some remarks on an equivalence theorem for an interacting massive spin one particle in quantum field theory

Abstract: The quantization of a massive spin one particle field, satisfying the first order system of equations due to Proca, is summarized. The interaction of such a field with an arbitrary field is then considered. Equivalent theories in terms of vector and antisymmetric second rank tensor spin one fields are then constructed in a simple manner, providing a generalization of an equivalence theorem due to the author. In addition, this approach, involving the use of first order systems of equations, allows a simple extension to the case of equivalence theorems for arbitrary integral spin. The nature of this extension is indicated.

Tensor approach to spin-one mesons. iii. magnetic dipole moment and electric quadrupole moment.

Abstract: In previous papers the massive spin-one mesons were described by means of an antisymmetric second-rank tensor field. In the present paper their free Lagrangian is modified in such a way that when the electromagnetic interactions are introduced by the minimal substitution the mesons get an arbitrary magnetic dipole moment. The addition of other terms in the Lagrangian allows the spin-one mesons to also have an arbitrary electric quadrupole moment. The covariance of the $S$ matrix to order ${e}^{2}$ is achieved by the addition of counterterms.

The use of explicitly correlated, partially antisymmetric wave functions in atomic and molecular calculations

Abstract: Trial wave functions, written as the sum of a configuration interaction expansion and an explicitly correlated term which is not antisymmetric, are proposed for use in calculating the electronic properties of atoms and molecules. A variational principle, modified to allow the use for such partially antisymmetric wave functions, is developed. It is shown that the consequences of partial antisymmetry on calculated expectation values can be estimated. The method avoids difficult three-electron integrals which arise in other theories.

Some Aspects of Group Theory

Abstract: In the discussion offered in Chapters I and II, no mention has been made of the impact of the symmetry of the scattering entity, e.g. atom, ion or molecule, on the properties of the scattered light. Yet, intuitively it can be felt that the properties of, for instance, the Raman radiation of a molecule like CCl4 could be different from that of the CHCl3, not because a hydrogen atom replaces a chlorine atom of CCl4 but because the symmetry of the latter molecule is different from that of the former. The symmetry difference is reflected, for example, through the absence of a three-fold rotational axis around any of the C-Cl bonds of the CHCl3 molecule, while four of such rotational axes are present for the CCl4 molecule. The effect of symmetry on the Rayleigh and Raman processes should, of course, be reflected in the scattering tensor — not to such extent that a separation of symmetric and antisymmetric tensors may occur, (they are associated with the type of Rayleigh and Raman process) — but more importantly, it determines which of the six elements of the symmetric tensor and three elements of the anti-symmetric tensor are different from zero. It may be remarked here that the trend is that the scattering tensor will become more simple (more elements will be equal to zero) if the symmetry of the scattering entity becomes higher. The interesting aspect of application of group theory is that a considerable amount of information on the tensor is obtained without going into details, about the mathematical formulation of the elements of the scattering tensor.

Extension of the Curie principle and constitutive relations for fluids with antisymmetric stress

Abstract: In recent years several papers [Grad(9), Condiff and Dahler(5), Baronowski and Romatowski(2), Eringen(8), Allen and de Silva(i)] have been written dealing with the possibility of antisymmetric stress in a fluid and its relationship with the internal micro-structure of the fluid. In the classical fluid dynamics of the Navier–Stokes equations, the molecules of the fluid element are treated as material points devoid of any internal spin motion and the only type of angular motion that the macroscopic elements of the fluid possess is the usual vorticity ½ curl v, the velocity of the fluid being v. In the case of fluids whose molecules are not regarded as material points, but are treated as micro-structures having internal spin, the total angular velocity of a macroscopic volume element will be equal to the vector sum of spins of the micro-structures and the part due to the vorticity.