Low-order Equations Research Articles

Decoupled simulation of asymmetrical three-phase power network transients

A set of power element models are established in an /spl alpha/-/spl beta/-0 reference frame with different degrees of coupling. For untransposed, asymmetrical three-phase transmission lines, the so-called decoupling compensation models can be used. For Y/delta-connected transformers and fault/manipulation interfaces, different sets of low-order algebraic equations are introduced. Thus, by incorporating these models with certain appropriate techniques of bus-ordering, an asymmetrical three-phase power system can be simulated by three loosely interconnected /spl alpha/-, /spl beta/- and 0-phase models. Simulation practice verifies the validity and effectiveness of this approach.

Compton scattering by a pion and off-shell effects.

We consider Compton scattering by a pion in the framework of chiral perturbation theory. We investigate off-shell effects in the s- and u-channel pole diagrams. For that purpose we perform a field transformation which, in comparison with the standard Gasser and Leutwyler Lagrangian, generates additional terms at order ${\mathit{p}}^{4}$ proportional to the lowest-order equation of motion. As a result of the equivalence theorem the two Lagrangians predict the same Compton scattering S-matrix even though they generate different off-shell form factors. We conclude that off-shell effects are not only model dependent but also representation dependent.

An essential ambiguity of quantum theory

It is shown that there is a substantial family of Lagrangians for one and the same ground problem of Newtonian mechanics where linear momentum is conserved. These alternative Lagrangians involve an arbitrary function of conserved momentum, which reduces to a one-parameter family when it is required that the Galilean group be canonically represented. Then the canonical momentum becomes parameter-dependent: and this leads quantally to ambiguous commutation rules and Heisenberg uncertainty principles—for example the coordinate of one particle may be complementary not to its own mechanical momentum (as in standard treatments) but to that of a different particle. One possible way of resolving the ambiguity is by reformulating many-particle dynamics as higher-order one-particle (HOOP) dynamics by decoupling the standard lower-order equations of motion. This also admits relativistic generalization in which the Poincare group is canonically represented.

The variational piezoelectromagnetic equations for elastic dielectrics

The aim of this paper is to extend the variational principles of piezoelectricity [M. C. Dökmeci, IEEE Trans. Ultrason. Ferroelec. Freq. Control UFFC-35(6), 775–787 (1988)] so as to incorporate the magnetic effects. The principle of virtual work is applied to the motions of piezoelectromagnetic medium, and an associated variational principle is derived that yields some of the fundamental equations of the medium and includes its remaining equations as constraints. By use of Lagrange multipliers, the constraints are incorporated into the variational principle and a unified variational principle is derived. Likewise, a variational principle is formulated for an electromagnetic medium with an internal surface of discontinuity. These integral and differential types of variational principles provide a standard basis for generating approximate direct solutions as well as deducing systematically lower order equations of piezoelectromagnetic medium. Also, special cases and the reciprocals of the variational principles are indicated. The results are shown to agree with the earlier variational principles [e.g., M. C. Dökmeci, J. Math. Phys. 19, 109–126 (1978) and P. C. Y. Lee, J. Appl. Phys. 69, 7470–7473 (1991)]. [Work supported in part by The Scientific and Technical Research Council of Turkey.]

Using parallel salinity and temperature profiles for calculations of estuarine fluxes with reference to the baltic proper

A diagnostic advection-diffusion equation system in a ‘natural coordinate’ frame of reference is investigated in order to improved its robustness. The dependence of the leading-order solutions on higher-order features of the hydrographic data has been decreased and a differential equation with and analytical solution is obtained. The properties of the lowest-order equation are compared with previous studies where the undegenerated equations were used to estimate the distribution of both the diapycnal salt flux and the oxygen consumption within the Baltic proper. The agreement found is within the acceptable bounds. Finally a pilot study of a silicate budget of the Baltic proper has been performed.

Improved numerical diffraction coefficients with application to frequency-selective surfaces

A method for numerically determining diffraction coefficients for arbitrary scattering centers is described. In this method finite bodies possessing scattering centers of the type of interest are first analyzed via the moment method. The various contributions to the total scattered fields are then isolated by solving low-order simultaneous equations obtained by writing expressions for the fields in terms of unknown diffraction coefficients. The method yields numerical diffraction coefficients in angular sectors where previous methods fail (e.g., near grazing angles), and can be applied in the context of measured as well as simulated scattering data. Finite frequency-selective surfaces are shown to be amenable to analysis with ray-optics techniques, and several two-dimensional examples are given with comparisons to far- and near-field moment method results. >

Optimal output feedback control of discrete linear, singularly perturbed, stochastic systems

The static output feedback control problem for discrete linear, singularly perturbed, stochastic systems is studied. A recursive algorithm is presented to solve the corresponding coupled non-linear algebraic equations. The algorithm removes the ill-conditioning by decomposing the higher order equations into lower order equations and develops a technique which enables us to obtain an arbitrary accuracy, that is, 0(ej) (j = 1, 2, 3, …)a approximation for this problem. A numerical example is presented to support the theoretical results. Communicated by Professor A. A. Ioannou.

Kac–Moody and new infinite-dimensional Lie algebras

It will be shown that general polynomials of the Kac–Moody field can generate either a new infinite-dimensional Lie algebra or W3 -type algebra, at least for the classical field case. For the case of the infinite dimensional Lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations analogous to that of the KdV system. The chiral boson equation emerges as the lowest-order equation of the hierarchy.

The analysis and modelling of dilatational terms in compressible turbulence

It is shown that the dilatational terms that need to be modelled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of the compressible velocity field, which generates these dilatational terms, is explored by asymptotic analysis of the compressible Navier-Stokes equations in the case of homogeneous turbulence. The lowest-order equations for the compressible field are solved and explicit expressions for some of the associated one-point moments are obtained. For low Mach numbers, the compressible mode has a fast timescale relative to the incompressible mode. Therefore, it is proposed that, in moderate Mach number homogeneous turbulence, the compressible component of the turbulence is in quasi-equilibrium with respect to the incompressible turbulence. A non-dimensional parameter which characterizes this equilibrium structure of the compressible mode is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.

Post-Newtonian expansion of an algebraically extended theory of gravity

The post-Newtonian expansion is analyzed for an algebraically extended theory of gravity equivalent to a theory with a real, nonsymmetric metric, previously referred to as Algebraically Extended Hilbert Gravity (AHG). The hermiticity of the algebra-valued covariant and contravariant metrics is found to constrain the form of the expansion of the antisymmetric part of the real metric to one of two possibilities. Both of these display a qualitatively new technical feature: the lowest order equations are nonlinear, depriving the post-Newtonian expansion of its greatest asset. In one case, they lead to a solution with a negative Newtonian mass parameter. In the other case, they lead to a contact-type solution in which some metric components depend directly upon the stress–energy tensor rather than the integral of this distribution. The unphysical nature of these solutions rules out AHG as a physically viable alternative theory of gravitation.

Binding in pair potentials of liquid simple metals from nonlocality in electronic kinetic energy.

The paper presents an explicit expression for the pair potential in liquid simple metals from low-order density-gradient theory when the superposition of single-center displaced charges is employed. Numerical results are presented for the gradient expansion pair interaction in liquid Na and Be. The low-order density-gradient equation for the pair potential is presented.

Redefinition of position variables and the reduction of higher-order Lagrangians

Single-time Lagrangians are treated in this paper, describing the dynamics of systems of point particles, which are given as formal power series in some ordering parameter and which may contain higher time derivatives in all terms but the leading one. An efficient method for eliminating the higher time derivatives directly on the Lagrangian level is presented. This method clarifies the meaning of using the lower-order equations of motion in higher-order terms in a Lagrangian. The method consists of an iterative use of ‘‘contact’’ transformations in the jet prolongation of the extended configurations space and is called ‘‘the method of redefinition of position variables.’’ Several examples from electrodynamics and relativistic gravity are treated explicitly.

Investigation of avoided crossings in the adiabatic hyperspherical potential curves of the helium atom.

Using an asymptotic expansion and the hyperspherical adiabatic approximation, we obtain a representation of the Scrh\"odinger equation for the helium atom and its isoelectronic ions for large values of the hyper-radius. We solve the lowest-order equation for the channel wave functions with boundary conditions imposed by exchange symmetry. Diagonalization of the potential matrix, keeping only terms through second order in the symmetrized basis set, gives potential curves that show avoided crossings essential for the interpretation of atomic dynamics.

Phase and amplitude nonlinear wave equations in polyacetylene and polydiacetylenes

A mean-field model of a polydiacetylene chain backbone is presented. Making use of arguments similar to those used for polyacetylene, the continuum version of the model is established and lowest-order equations of motion for gap parameters are derived. They reveal that the simplest elementary excitations in polydiacetylenes should be of a bipolaric nature.

Modified Tokuyama-Mori theory: Application to quantum diffusion problem

A modified form of the Tokuyama-Mori theory for mean values of operators acting on a system interacting with a reservoir is applied to the quantum diffusion problem with the standard spinboson Hamiltonian. Treating the particle-bath interaction as a perturbation, the lowest-order equations are shown to reproduce the standard picture of relaxation.

One-dimensional closure models for three-dimensional incompressible viscoelastic free jets: von Kármán flow geometry and elliptical cross-section

In this paper we derive one-space-dimensional, reduced systems of equations (one-dimensional closure models) for viscoelastic free jets. We begin with the three-dimensional system of conservation laws and a Maxwell-Jeffreys constitutive law for an incompressible viscoelastic fluid. First, we exhibit exact truncations to a finite, closed system of one-dimensional equations based on classical velocity assumptions of von Kármán (1921). Next, we demonstrate that the three-dimensional free-surface boundary conditions overconstrain these truncated systems, so that only a very limited class of solutions exist. We then proceed to derive approximate one-dimensional closure theories through a slender-jet asymptotic scaling, combined with appropriate definitions of velocity, pressure and stress unknowns. Our non-axisymmetric one-dimensional slender-jet models incorporate the physical effects of inertia, viscoelasticity (viscosity, relaxation and retardation), gravity, surface tension, and properties of the ambient fluid, and include shear stresses and time dependence. Previous special one-dimensional slender-jet models correspond to the lowest-order equations in the present asymptotic theory by an a posteriori suppression to leading order of some of these effects, and a reduction to axisymmetry. We thereby: (i) derive existing one-dimensional models from the three-dimensional free surface boundary-value problem; (ii) clarify the sense of the one-dimensional approximation; (iii) deduce new one-dimensional closure models for non-axisymmetric viscoelastic free jets.

Estimation of noise variance from the noisy AR signal and its application in speech enhancement

In a number of applications involving the processing of noisy signals, it is desirable to know a priori the noise variance. The author proposes a method of estimating the noise variance from the autoregressive (AR) signal corrupted by the additive white noise. This method first estimates the AR parameters from the high-order Yule-Walker equations, and then uses these AR parameters to estimate the noise variance from the low-order Yule-Walker equations. The method is used in a speech enhancement application where its performance is studied for stationary as well as nonstationary noise conditions. The results are found to be encouraging. >

The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer

Two kinds of small-parameter expansions are used to derive a series of ordinary differential equations governing spatially growing or decaying disturbances of the travelling-wave type in non-parallel laminar boundary layers. Eigensolutions of the lowest-order equation are shown to tend to resemble eigensolutions of the unsteady boundary-layer equation as the Reynolds number Re decreases, and to the Tollmien-Schlichting wave solutions of the Orr-Sommerfeld equation as Re increases. Examination of the next-order approximation reveals that the unsteady boundary-layer equation has eigensolutions only if Re exceeds a critical value depending on the frequency of disturbances, while it remains a partial differential equation for lower Reynolds numbers. Thus the Tollmien-Schlichting wave may be considered to originate at the above critical value of Re. The subsequent development of the wave with fixed frequency can be followed by integrating the spatial growth rate of the eigensolutions with respect to the downstream distance. The amplitude is at first damped but grows even higher than its original value farther downstream, provided the frequency is less than a certain value.

Overtaking collision between two solitary waves

Under the assumptions of long wavelength, small amplitude, and propagation in one direction, it is well known that the water wave equations in the lowest order of approximation with slow variation in time and space can be reduced to the Korteweg–de Vries (KdV) equation. Linear inhomogeneous equations with their coefficients, as well as the inhomogeneous terms depending on the solutions of the lower-order equations, have been obtained in the next order and the third order of approximation. An accurate numerical method is used to integrate these equations for an overtaking interaction between two solitary waves. It is found, up to the second order of approximation, that the collision is elastic, i.e., the solitary waves regain their original form without giving rise to any secondary wave. The results of second-order solution are in good agreement with the analytic solution of Sachs [SIAM J. Math. Anal. 15, 468 (1984)]. However, the numerical computation on the third-order equation shows a secondary wave train trailing behind the smaller wave after the collision.

時間-空間スペクトル大気大循環モデルI.周期外力項を有する低次順圧方程式系の時間-空間スペクトルモデル

For the purpose of studying the atmospheric response (or sensitivity) to the seasonal forcing, we propose a time-space spectral general circulation model. In the present work we use a low-order equivalent Barotropic vorticity equation with forcing and dissipation terms. The original low-order equation consists of three-component in the space-spectral domain: one purely zonal component and two wave components with the same zonal wavenumber but different mode numbers. Further expanding this equation in the space-spectral domain into that in the time-spectral domain, we finally obtain a time-space spectral model, which is a simultaneous nonlinear algebraic equations system.The model used in the present work consists of one steady and one periodic components in the time-domain. The nonlinear equations system is solved by using the "revised Marquardt method" (Levenberg-Marquardt-Morrison method) and a "Continuation method". Addition of the periodic forcing besides the steady forcing brings about a new stable solution together with the stable solutions corresponding to those under the steady forcing only. This multiplicity of solutions with the moderate intensity of steady and periodic forcings is very interesting for understanding climatic change, and suggests that we can expect to obtain useful results, by extending our model to a further larger timespace spectral general circulation model based on the above-mentioned numerical methods.