scholarly journals Features of Fundamental Plane for Early-Type Galaxies by Clausius' Virial theory

2006 ◽  
Vol 2 (S235) ◽  
pp. 134-134
Author(s):  
L. Secco
Keyword(s):  
Order Approximation ◽  
Early Type ◽  
Density Profiles ◽  
Fundamental Plane ◽  
First Order ◽  
Thermodynamical Properties ◽  
Cosmological Scenario ◽  
Stellar Component ◽  
First Order Approximation ◽  
Good Agreement

AbstractThe theory of the Fundamental Plane (FP) proposed by Secco (2005) is based on the existence of a maximum in the Clausius' Virial (CV) potential energy of a stellar component when it is completely embedded inside a dark matter (DM) halo. At the first order approximation the theory was developed by modeling the two-components with two power-law density profiles and it produces some expectations in fairly good agreement with the observations. We add other predictions of the theory at the same level of approximation about the Zone of Exclusion (ZOE) in k-space and its possible relationship with cosmological scenario. Some of the consequences of the thermodynamical properties of CV maximum are also taken into account.

On a New Analytic Theory of the Moon's Motion II: Orbit and Length of Months

2020 ◽  
Vol 58 ◽  
pp. 13-54
Author(s):  
Ramon González Calvet ◽  
Keyword(s):  
Order Approximation ◽  
Polar Coordinates ◽  
The Sun ◽  
The Moon ◽  
Lagrange Equations ◽  
Moon System ◽  
First Order ◽  
Relative Coordinates ◽  
First Order Approximation ◽  
Good Agreement

The differential equation in polar coordinates of the Moon's orbit is outlined from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed with relative coordinates and accelerations. The orbit of the Moon calculated this way is similar to Clairaut's modified orbit and has better parameters than those previously published. An improvement to this orbit is proposed based on theoretical arguments. With help of this new orbit, the variations in the draconic, synodic and anomalistic months are also computed showing very good agreement with observations.

Unified Second-Order Stochastic Averaging Approach

1997 ◽  
Vol 64 (2) ◽  
pp. 281-291 ◽  
Author(s):  
M. Hijawi ◽  
N. Moschuk ◽  
R. A. Ibrahim
Keyword(s):  
Order Approximation ◽  
Stochastic Averaging ◽  
Nonlinear Damping ◽  
Second Order ◽  
Unified Approach ◽  
Order Analysis ◽  
First Order ◽  
Different Types ◽  
First Order Approximation ◽  
Good Agreement

First-order stochastic averaging has proven very useful in predicting the response statistics and stability of dynamic systems with nonlinear damping forces. However, the influence of system stiffness or inertia nonlinearities is lost during the averaging process. These nonlinearities can be recaptured only if one extends the stochastic averaging to second-order analysis. This paper presents a systematic and unified approach of second-order stochastic averaging based on the Stratonovich-Khasminskii limit theorem. Response statistics, stochastic stability, phase transition (known as noise-induced transition), and stabilization by multiplicative noise are examined in one treatment. A MACSYMA symbolic manipulation subroutine has been developed to perform the averaging processes for any type of nonlinearity. The method is implemented to analyze the response statistics of a second-order oscillator with three different types of nonlinearities, excited by both additive and multiplicative random processes. The second averaging results are in good agreement with those estimated by Monte Carlo simulation. For a special nonlinear oscillator, whose exact stationary solution is known, the second-order averaging results are identical to the exact solution up to first-order approximation.

On the Huggins constant of dilute polymer solutions

1969 ◽  
Vol 308 (1495) ◽  
pp. 497-515 ◽  
Keyword(s):  
Polymer Solutions ◽  
Relaxation Spectrum ◽  
Order Approximation ◽  
Polymer Concentration ◽  
Numerical Constant ◽  
First Order ◽  
Dilute Polymer Solutions ◽  
Huggins Constant ◽  
First Order Approximation ◽  
Good Agreement

As is well known, the stress in a polymer solution can be expressed in terms of the characteristic relaxation spectrum. Here the effect of polymer concentration on the relaxation spectrum is evaluated to the first-order approximation. It is then possible to calculate the Huggins constant k H (in the viscosity equation ɳ sp. / c = [ɳ] + k H [ ɳ ] 2 c ) from this spectrum. It is found that k H is related to its value k Ɵ at the Ɵ-point, thus: k H = k Ɵ α ɳ -4 + C 0 z α ɳ -5 where C 0 is a numerical constant, z is the customary excluded volume parameter, and α ɳ is the expansion coefficient (approximately measured by the intrinsic viscosity). This equation is compared with recent experiments by Nagasawa and by Inagaki, and good agreement is obtained. Thus two different polymer solutions give plots of adequate linearity for α 4 ɳ k H against M 1/2 /α ŋ (since z is proportional to M 1/2 ) with approximately the same intercept ( k Ɵ ~ 0·45).

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

FIRST-ORDER APPROXIMATION OF THE NUMBER-PROJECTED SO(2N) TAMM-DANCOFF EQUATION AND ITS REDUCTION BY THE SCHUR FUNCTION

1999 ◽  
Vol 08 (05) ◽  
pp. 461-483
Author(s):  
SEIYA NISHIYAMA
Keyword(s):  
Wave Function ◽  
Order Approximation ◽  
Approximate Equation ◽  
Variation Principle ◽  
Schur Function ◽  
Soliton Theory ◽  
First Order ◽  
Fermion Systems ◽  
Group Manifold ◽  
First Order Approximation

First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group manifold.

Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

1995 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj
Keyword(s):  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Amplitude Equations ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Quadratic Nonlinearities ◽  
First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

scholarly journals Analytical quality assessment of iteratively reweighted least-squares (IRLS) method

2014 ◽  
Vol 20 (1) ◽  
pp. 132-141 ◽  
Author(s):  
Jianfeng Guo
Keyword(s):  
Least Squares ◽  
Order Approximation ◽  
Diagnostic Tools ◽  
Iteratively Reweighted Least Squares ◽  
First Order ◽  
Iterative Reweighting ◽  
First Order Approximation ◽  
Detectable Bias ◽  
Theoretical Analyses ◽  
Exact Analytical Method

The iteratively reweighted least-squares (IRLS) technique has been widely employed in geodetic and geophysical literature. The reliability measures are important diagnostic tools for inferring the strength of the model validation. An exact analytical method is adopted to obtain insights on how much iterative reweighting can affect the quality indicators. Theoretical analyses and numerical results show that, when the downweighting procedure is performed, (1) the precision, all kinds of dilution of precision (DOP) metrics and the minimal detectable bias (MDB) will become larger; (2) the variations of the bias-to-noise ratio (BNR) are involved, and (3) all these results coincide with those obtained by the first-order approximation method.

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