Showing papers on "Antisymmetric relation published in 1970"

The number of finite topologies

Abstract: The logarithm (base 2) of the number of distinct topologies on a set of n elements is shown to be asymptotic to n2/4 as n goes to infinity. Let X be a set with n elements, and let T(X) be the set of all topologies that can be defined on X. Then we set T.= I T(X) I. The number T. has been determined for certain small values of n [4]. Tn has been estimated by several authors [2], [3], [8]. We present here an asymptotic estimate for the logarithm of Tn. We do this by considering an equivalent problem. Namely, if P(X) denotes the set of all partial orders that can be defined on a set X with n elements, and if we set Pn = I P(X) |, then we estimate Pn. The sizes of certain special subsets of P(X) have been determined, and thus provide lower bounds for Pn [6 ], [7 ]. We begin by presenting several enumeration problems which are equivalent to one another in a certain sense (see Lemma 1 below). Let Tn and Pn be defined as above. Similarly, for X a set with n elements, let To(X) be the set of all To-topologies that can be defined on X (i.e., if a, b are two elements of X, then there is some open set containing one but not both of them [5]), and let O(X) be the set of all preorders that can be defined on X (i.e., reflexive and transitive, but not necessarily antisymmetric). Then set Tn,o = I To(X) , and On = O(X) I. Let Tn, P', T' o, O' denote the numbers of isomorphism classes in T(X), P(X), To(X), O(X) respectively. From [2] we know that (1) ~~~~~Tn > 2n24 (Alternatively, we have Pn> 2n2 4 which we observe trivially after the introduction of "diagrams" below.) LEMMA 1. Tn = On, Tn,o=Pn, T' = 0', T' o=P. All eight of these quantities have logarithms which are asymptotically equal as n tends to infinity. Received by the editors, August 27, 1969. A MS Subject Classifications. Primary 0565.

Capillary Stability of a Hollow Inviscid Cylinder

Abstract: The application of Rayleigh's treatment of capillary stability to the case of an inviscid liquid bounded by two concentric circular cylindrical surfaces leads to a characteristic equation from which the following predictions can be made The system is stable for disturbances that do not have cylindrical symmetry It is stable (unstable) for disturbances with cylindrical symmetry when the wavelength of the disturbance in axial direction is less (greater) than the circumference of the inner surface There are always two modes of deformation In the special case of a thin‐walled cylinder these two modes are symmetric and antisymmetric, respectively When the wavelength of the disturbance is varied in such a manner that there is a transition from stable to unstable conditions, then at the same time there is a transition from symmetric to antisymmetric mode or vice versa It is shown, further, that when one mode is stable and the other unstable, the stable mode is a deformation for which the total surface area of the system is increased by the disturbance, while the unstable mode corresponds to decreased surface area

Constraints on the asymptotic behaviour of symmetric and antisymmetric amplitudes from positivity and unitarity

Abstract: We consider physical amplitudes constructed from linear combinations of symmetric and antisymmetric amplitudes. The positivity properties of these physical amplitudes and the unitarity conditions imply some connections between the high-energy behaviour of the symmetric and antisymmetric amplitudes. These constraints are investigated.

Note on the space part of anti-symmetric wave functions in the many-electron problem

Abstract: It is pointed out that if a many-electron antisymmetric wave function is expanded as a sum of spin-product functions, each multiplied by a function of coordinates, the resulting functions of coordinates have many of the same useful features found with the symmetric and antisymmetric functions representing singlet and triplet states in a two-electron system. For finding the energy, or any function of coordinates only, in the approximation in which spin-orbit interaction is neglected, one such function of coordinates can be used, the spins being disregarded. Simple procedures allow one to find matrix components of such operators as S2 and L . S from the functions of coordinates. These procedures are much easier to visualize than the use of projection operators, the permutation group, or other methods in current use. The general procedures are illustrated by application to the three-electron problem of the lithium atom, as treated by Lunell, Kaldor, and Harris, and their application to the contact hyperfine structure is pointed out.

Invariant algebras on compact groups

Abstract: The main result is a description of homogeneous (i. e. invariant relative to left and right shifts) algebras with uniform convergence on a compact group. As a corollary we obtain a generalization of a theorem of Rider: let the real annihilater of a homogeneous antisymmetric algebra be separable in the topology of the definite norm in the conjugate space. Then the connected component of the identity of the group is commutative and . Rider proved that if , then is commutative and connected.Bibliography: 3 references.

Matrix Elements of Spin‐Dependent Operators over Total Molecular Wavefunctions

Abstract: There is given a complete formulation of the evaluation of matrix elements of spin‐dependent operators over total wavefunctions for diatomic molecules. General account is taken of the nonseparability of position and spin spaces for polyelectronic problems, but consideration is restricted to orbital product electronic wavefunctions. The analysis has two phases. The first phase makes a complete reduction (including transformation of the operators to the molecular axis fixed coordinate system) of the N‐electron two‐nuclei coupled angular‐momentum problem using Racah tensor operator techniques. A full derivation is given for Hund's case b coupling scheme. Transformations to other coupling schemes are given in an appendix. The second phase extends the Slater‐Lowdin matrix element analysis for antisymmetric functions to include spin‐dependent operators.

Lattice Dynamics of a Linear Chain Model of an Imperfect Molecular Crystal

Abstract: The lattice dynamics are discussed, using the method of Green's functions, for a linear chain of homonuclear diatomic molecules, one of which has a different force constant from the others. The model can represent a molecular crystal in which one molecule is electronically excited. The frequencies of the antisymmetric normal modes can be deduced from the frequencies of the symmetric modes of a chain of alternating masses, in which one of the lighter atoms is replaced by an isotopic impurity, and vice versa. The alternation of symmetry arises because, in the present model, the defect consists of two atoms instead of one. A localized mode in the bandgap or above the optical band is obtained.

The N+2 spin-orbital approximation to the N-body antisymmetric wave function

Abstract: For an N-fermion system, using M one-particle basic functions to represent the wave function, it is already well known that the case M=N leads to a single determinant (or Hartree-Fock) form for the wave function. In a recent article, two of the authors showed by numerical experiment that the case M=N+2 also leads to a particularly simple and elegant form of the N-fermion wave function, one which is a certain linear combination of only (1/2N+1) ((p) identical to integral part of p) Slater determinants. The present paper gives an analytical derivation of this useful result.

On the rational foundations of the non-equilibrium thermodynamics of ordinary fluid mixtures

Abstract: Despite the arbitrary and ambiguous character of certain fundamental parts of the usual theory of non-equilibrium thermodynamics of mixtures, it has been successful in describing a variety of experimental situations. The goals of the research partially reported in this paper have been (1) to strengthen the foundations of the usual, practical theory and (2) to extend its usefulness to experimental situations hitherto regarded as too complicated to analyse. The principles and methods of rational mechanics are used to deduce balance equations which take full account of the kinetic energy of each component and of the partial stress tensor of each component Similar equations have been obtained previously by similar techniques. The new results reported here stem from a particular choice of independent variables; namely, for a v-component, non-reacting mixture of fluids, the independent variables are the v component densities and their gradients, the temperature and its gradient, the v — 1 independent diffusive velocities, and the v symmetric and v — 1 antisymmetric parts of the gradients of component velocity. For the special but very important case of an ordinary fluid mixture, whose constitutive relations are linear in the diffusion velocities and the independent gradients, the source term in the entropy balance equation is a bi-linear form in the diffusion velocities and the independent gradients. This bi-linear form for the entropy source term, which has eluded previous workers, has two immediate consequences: (1) Coupled with the second law, it leads to unambiguous conditions for equilibrium; namely, for equilibrium with respect to all processes in a v-component, non-reacting fluid mixture, it is necessary and sufficient that the following shall all vanish: the temperature gradient, the diffusion velocities, and the symmetric and antisymmetric parts of the gradients of component velocity. (2) It permits full comparison with the practical theory, wherein the entropy source plays a central role. It also opens new experimental avenues since it contains terms which arise from component kinetic energy and component stress tensor.

Some remarks forπ-N scattering towards nearly constant bounds for high-energy total cross-sections and the Pomeranchuk theoremscattering towards nearly constant bounds for high-energy total cross-sections and the Pomeranchuk theorem

Abstract: Some rigorous results of local field theory suggest further restrictive assumptions about the crossing symmetric and antisymmetric parts of theπ-N scattering amplitude in so far as their high-energy, small-momentum-transfer behaviour are concerned, thus ensuring the existence of some weighted averages of both the real and imaginary parts in each case. One finds first that the total cross-sectionsσ pπ+ (orσ pπ− ) cannot saturate high-energy bounds of the type (logs)e, (loglogs)e, ,p integer, e>0, and secondly for the Pomeranchuk theorem considered in some average sense one obtains that is integrable whens goes to infinity. These results are respectively obtained from the study of the symmetric and the antisymmetric parts of the amplitude. If these results are not satisfied there exist within our framework two possibilities. On one hand the amplitudes could be indefinitely oscillating. If not, there would appear, for the symmetric amplitude or for the antisymmetric amplitude, particular zeros in linear combinations of the real and imaginary parts of these amplitudes. These zeros would always be present in the nearly forward asymptotic domain with energy as large and transfer as small as one wants.

Wave Propagation in Three‐Layered Plates—Application to Ultrasonic Delay Lines

Abstract: : The propagation of trains of straight crested harmonic waves in infinitely long isotropic, traction free, three-layered plates is investigated on the basis of the three-dimensional linear theory of elasticity. It is established, that as in the case of waves traveling in homogeneous plates, the face shear motion and the longitudinal - flexural motion exist uncoupled. However, independent symmetric and antisymmetric modes exist only in the case of three-layered symmetric plates; that is, plates having identical outer layers. The frequency equations have been evaluated numerically. The possibility is discussed of utilizing thin three-layered symmetric strings as delay media in guided wave ultrasonic delay lines having a linear delay versus frequency relation. (Author)

Symmetric and Antisymmetric Regular Expressions

Abstract: In this paper the “dual” of a binary regular expression is defined as the same expression with 0’s and 1’s interchanged. The “duals” of tapes and events are defined analogously, and various properties and interrelations associated with duality are developed. On the basis of these definitions, two classes of regular expressions are introduced—the “symmetric” and the “antisymmetric” regular expressions (these classes are shown to be infinite). A symmetric expression is one whose dual represents the same event, while an antisymmetric expression is one whose dual represents the complement of the same event. Thus, the acceptor of an event represented by a symmetric expression is invariant under input complementation, while the acceptor of an event represented by an antisymmetric expression is invariant under simultaneous complementation of input and output. It is shown that the class of symmetric expressions is closed under regular operations, while the class of antisymmetric expressions is closed only under s...