On the Factorization of Secular Equations by Group Theory
Secular equation factorization in quantum mechanics of atoms, molecules, and crystals by group theory
FACTOR ANALYSIS↗
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On the factorization of secular equations by group theory.
Explicit invariant forms of factor equations for factorization of secular equations by group theory
QUANTUM MECHANICS↗
An application of group theory to the numerical integration of the equations of motion of a conservative dynamical system
Application of group theory to numerical integration of motion equations of conservative dynamical system
NUMERICAL INTEGRATION↗
Renormalization-group theory for the eddy viscosity in subgrid modeling
Renormalization-group theory is applied to incompressible three-dimensional Navier-Stokes turbulence so as to eliminate unresolvable small scales. The renormalized Navier-Stokes equation now includes a triple nonlinearity with the eddy viscosity exhibiting a mild cusp behavior, in qualitative agreement with the test-field model results of Kraichnan. For the cusp behavior to arise, not only is the triple nonlinearity necessary but the effects of pressure must be incorporated in the triple term. The renormalized eddy viscosity will not exhibit a cusp behavior if it is assumed that a spectral gap exists between the large and small scales.
Zhou, YE↗
Application of dimensional analysis and group theory to the solution of ordinary and partial differential equations Final report, 17 Jan. 1966 - 16 Jan. 1967
Dimensional analysis and group theory methods of solving ordinary and partial differential equations
GROUP THEORY↗
Testing the renormalisation group theory of cooperative transitions at the lambda point of helium
The status of high resolution tests of the renormalization group theory of cooperative phase transitions performed near the lambda point of helium is described. The prospects for performing improved tests in space are discussed.
Lipa, J. A.↗
Application of renormalization group theory to the large-eddy simulation of transitional boundary layers
An eddy viscosity model based on the renormalization group theory of Yakhot and Orszag (1986) is applied to the large-eddy simulation of transition in a flat-plate boundary layer. The simulation predicts with satisfactory accuracy the mean velocity and Reynolds stress profiles, as well as the development of the important scales of motion. The evolution of the structures characteristic of the nonlinear stages of transition is also predicted reasonably well.
Piomelli, Ugo↗
Applied group theory applications in the engineering (physical, chemical, and medical), biological, social, and behavioral sciences and in the fine arts
A generalized applied group theory is developed, and it is shown that phenomena from a number of diverse disciplines may be included under the umbrella of a single theoretical formulation based upon the concept of a group consistent with the usual definition of this term.
Borg, S. F.↗
Solid state and molecular theory group Semiannual progress report
Use of scattered wave method to compute molecular wave functions, augmented plane wave method for energy band calculations, and Casimir invariants as invariant operators in Lie groups
Source record↗
Recursive renormalization group theory based subgrid modeling
Advancing the knowledge and understanding of turbulence theory is addressed. Specific problems to be addressed will include studies of subgrid models to understand the effects of unresolved small scale dynamics on the large scale motion which, if successful, might substantially reduce the number of degrees of freedom that need to be computed in turbulence simulation.
Zhou, YE↗
Group-kinetic theory and modeling of atmospheric turbulence
A group kinetic method is developed for analyzing eddy transport properties and relaxation to equilibrium. The purpose is to derive the spectral structure of turbulence in incompressible and compressible media. Of particular interest are: direct and inverse cascade, boundary layer turbulence, Rossby wave turbulence, two phase turbulence; compressible turbulence, and soliton turbulence. Soliton turbulence can be found in large scale turbulence, turbulence connected with surface gravity waves and nonlinear propagation of acoustical and optical waves. By letting the pressure gradient represent the elementary interaction among fluid elements and by raising the Navier-Stokes equation to higher dimensionality, the master equation was obtained for the description of the microdynamical state of turbulence.
Tchen, C. M.↗
Toward a theory of group-decision behavior.
Theory of group decision behavior tested on dyads
INFORMATION PROCESSING↗
System theory on group manifolds and coset spaces.
The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.
Brockett, R. W.↗
Group-kinetic theory of turbulence
The two phases are governed by two coupled systems of Navier-Stokes equations. The couplings are nonlinear. These equations describe the microdynamical state of turbulence, and are transformed into a master equation. By scaling, a kinetic hierarchy is generated in the form of groups, representing the spectral evolution, the diffusivity and the relaxation. The loss of memory in formulating the relaxation yields the closure. The network of sub-distributions that participates in the relaxation is simulated by a self-consistent porous medium, so that the average effect on the diffusivity is to make it approach equilibrium. The kinetic equation of turbulence is derived. The method of moments reverts it to the continuum. The equation of spectral evolution is obtained and the transport properties are calculated. In inertia turbulence, the Kolmogoroff law for weak coupling and the spectrum for the strong coupling are found. As the fluid analog, the nonlinear Schrodinger equation has a driving force in the form of emission of solitons by velocity fluctuations, and is used to describe the microdynamical state of turbulence. In order for the emission together with the modulation to participate in the transport processes, the non-homogeneous Schrodinger equation is transformed into a homogeneous master equation. By group-scaling, the master equation is decomposed into a system of transport equations, replacing the Bogoliubov system of equations of many-particle distributions. It is in the relaxation that the memory is lost when the ensemble of higher-order distributions is simulated by an effective porous medium. The closure is thus found. The kinetic equation is derived and transformed into the equation of spectral flow.
Tchen, C. M.↗
A study of the current group evaporation/combustion theories
Liquid fuel combustion can be greatly enhanced by disintegrating the liquid fuel into droplets, an effect achieved by various configurations. A number of experiments carried out in the seventies showed that combustion of droplet arrays and sprays do not form individual flames. Moreover, the rate of burning in spray combustion greatly deviates from that of the single combustion rate. Such observations naturally challenge its applicability to spray combustion. A number of mathematical models were developed to evaluate 'group combustion' and the related 'group evaporation' phenomena. This study investigates the similarity and difference of these models and their applicability to spray combustion. Future work that should be carried out in this area is indicated.
Shen, Hayley H.↗
On the theory of group generation of stars
The hypothesis proposed is that topology of a rotating gaseous cloud can be variable in the contraction process. Due to rotation an originally spherical cloud is transformed into a toroidal body. The contraction of a thin torus is considered with different suppositions on cooling the gas. In the determined time the torus will become gravitationally unstable. The excitation of Jeans' waves is shown to result in the disintegration of the torus into fragments. The number of the fragments and their mass distributions are calculated. The proposed hypothesis on toroidal stages in stellar evolution can remove some difficulties in the theory of structure and evolution of stars, such as absence of limitary stars, distribution of rotation velocities of early-type stars, origin of poloidal magnetic fields and decline rotators with the magnetic axis orthogonal to the axis of rotation.
Zhilyayev, B. Y.↗
U/12/ and time reversal violation in the weak interactions
Time reversal violation in weak interactions with two U/8/ subgroups of U/12/ symmetries - group theory
GROUP THEORY↗
The Joint NASA/Goddard-University of Maryland Research Program in Charged Particle and High Energy Photon Detector Technology
The Univ. of Maryland portion investigated the following areas. The Space Physics Group performed studies of data from the AMPTE/CCE spacecraft CHEM experiment and found that the ratio of solar wind to photospheric abundances decreased rather smoothly with the first ionization potential (FIP) of the ion with the low FIP ion being about a factor of two overabundant. Carbon and hydrogen fit this trend particularly well. Several occurrences were analyzed of field aligned beams observed when CCE was upstream of the Earth's bow shock. Also using CHEM data, ring current intensity and composition changes during the main and recovery phases of the great geomagnetic storm that occurred in February 1986 was examined in detail. Still using CHEM data, ring current characteristics were examined in a survey of 20 magnetic storms ranging in size from -50 nT to -312 nT. A study was done of energetic ion anisotropy characteristics in the Earth's magnetosheath region using data from the UMD/MPE experiment on ISEE-1. The properties were analyzed of approx. 30 to 130 keV/e protons and alpha particles upstream of six quasi-parallel interplanetary shocks that passed by the ISEE-3 spacecraft during 1978 to 1979. Work from NASA-Goddard include studies from the High Energy Cosmic Ray Group, Low Energy Cosmic Ray Group, Low Energy Gamma Ray Group, High Energy Astrophysics Theory Group, and the X ray Astronomy Group.
Ipavich, F. M.↗