Analytic expressions for the second-order scalar perturbations in the ΛCDM Universe within the cosmic screening approach

The energy level shifts for an electron in a high-nL Rydberg state of an atom such as helium are well described by an asymptotic potential with terms proportional to the multipole moments of the core. We obtain exact analytic expressions as a function of n and L for second-order perturbation corrections to the energy due to the asymptotic potential. The results are of importance in the analysis of high-precision variational calculations and experimental transition energies for Rydberg states.

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Gauge dependence of gravitational waves generated at second order from scalar perturbations

Physical Review D ◽

10.1103/physrevd.101.083529 ◽

2020 ◽

Vol 101 (8) ◽

Cited By ~ 8

Author(s):

Keitaro Tomikawa ◽

Tsutomu Kobayashi

Keyword(s):

Gravitational Waves ◽

Second Order ◽

Gauge Dependence ◽

Scalar Perturbations

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ANALYTICAL TREATMENT OF NUCLEAR MAGNETIC SHIELDING TENSOR INTEGRALS OVER EXPONENTIAL-TYPE FUNCTIONS

Journal of Theoretical and Computational Chemistry ◽

10.1142/s0219633608004374 ◽

2008 ◽

Vol 07 (06) ◽

pp. 1215-1225 ◽

Cited By ~ 2

Author(s):

LILIAN BERLU ◽

HASSAN SAFOUHI

Keyword(s):

Second Order ◽

Basis Set ◽

Magnetic Shielding ◽

Analytical Treatment ◽

Transform Method ◽

Analytic Expressions ◽

Molecular Integrals ◽

Shielding Tensor ◽

The Fourier Transform ◽

Nuclear Magnetic

The present work concerns the analytical and numerical development of three-center molecular integrals over Slater-type functions (STFs) and B functions of the second order involving [Formula: see text] in the operator. These integrals appear in the analytic expression of the nuclear magnetic shielding tensor. The basis set of STFs is used to represent atomic orbitals. These STFs are expressed in terms of B functions, which are better suited to apply the Fourier transform method thoroughly developed by Steinborn group. Analytic expressions are obtained for the integrals of the second order involved in nuclear magnetic resonance shielding tensor over B functions. These expressions turned out to be similar to those obtained for the usual molecular multi-center integrals. Consequently, the numerical evaluation of the integrals under consideration will benefit from the work previously done on the molecular multi-center integrals.

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Nonlocal Convective PBL Model Based on New Third- and Fourth-Order Moments

Journal of the Atmospheric Sciences ◽

10.1175/jas3474.1 ◽

2005 ◽

Vol 62 (7) ◽

pp. 2189-2204 ◽

Cited By ~ 36

Author(s):

Y. Cheng ◽

V. M. Canuto ◽

A. M. Howard

Keyword(s):

Fourth Order ◽

Potential Temperature ◽

Turbulence Models ◽

Second Order ◽

Order Moment ◽

Third Order ◽

Mean Velocity ◽

Eddy Simulation ◽

Analytic Expressions ◽

Aircraft Data

Abstract The standard approach to studying the planetary boundary layer (PBL) via turbulence models begins with the first-moment equations for temperature, moisture, and mean velocity. These equations entail second-order moments that are solutions of dynamic equations, which in turn entail third-order moments, and so on. How and where to terminate (close) the moments equations has not been a generally agreed upon procedure and a variety of models differ precisely in the way they terminate the sequence. This can be viewed as a bottom-up approach. In this paper, a top-down procedure is suggested, worked out, and justified, in which a new closure model is proposed for the fourth-order moments (FOMs). The key reason for this consideration is the availability of new aircraft data that provide for the first time the z profile of several FOMs. The new FOM expressions have nonzero cumulants that the model relates to the z integrals of the third-order moments (TOMs), giving rise to a nonlocal model for the FOMs. The new FOM model is based on an analysis of the TOM equations with the aid of large-eddy simulation (LES) data, and is verified by comparison with the aircraft data. Use of the new FOMs in the equations for the TOMs yields a new TOM model, in which the TOMs are damped more realistically than in previous models. Surprisingly, the new FOMs with nonzero cumulants simplify, rather than complicate, the TOM model as compared with the quasi-normal (QN) approximation, since the resulting analytic expressions for the TOMs are considerably simpler than those of previous models and are free of algebraic singularities. The new TOMs are employed in a second-order moment (SOM) model, a numerical simulation of a convective PBL is run, and the resulting mean potential temperature T, the SOMs, and the TOMs are compared with several LES data. As a final consistency check, T, SOMs, and TOMs are substituted from the PBL run back into the FOMs, which are again compared with the aircraft data.

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On solutions of second-order matrix polynomial equation of high degree

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501006 ◽

2021 ◽

pp. 2250100

Author(s):

Lu Tan ◽

Xue-Han Cheng ◽

Tong-Song Jiang ◽

Si-Tao Ling

Keyword(s):

Matrix Polynomial ◽

Sufficient Conditions ◽

Polynomial Equation ◽

Second Order ◽

General Solutions ◽

Order Matrix ◽

Algebraic Properties ◽

Analytic Expressions ◽

Matrix Polynomial Equation ◽

High Degree

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of the solutions, as well as the corresponding algorithms for finding the solutions are provided. An example is given so as to verify the theoretical results we have derived.

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Simulation and Nonlinear Dynamics Analysis of Planing Hulls

Journal of Offshore Mechanics and Arctic Engineering ◽

10.1115/1.2826989 ◽

1995 ◽

Vol 117 (1) ◽

pp. 38-45 ◽

Cited By ~ 13

Author(s):

J. D. Hicks ◽

A. W. Troesch ◽

C. Jiang

Keyword(s):

Differential Equations ◽

System Analysis ◽

Equations Of Motion ◽

Path Following ◽

Experimental Tests ◽

Second Order ◽

Design Tool ◽

Continuation Methods ◽

System Response ◽

Analytic Expressions

The high speeds, small trim angles, and shallow drafts of planing hulls produce large changes in vessel wetted surface which, in turn, lead to significant hydrodynamic and dynamic nonlinearities. Due to the complex nonlinearities of this type of craft, naval architects and planing boat designers tend to rely upon experimental tests or simulation for guidance. In order for simulation to be an effective design tool, a fundamental understanding of the system’s dynamic characteristics is required. This paper describes a developing methodology by which the necessary insight may be obtained. A demonstration of the combined use of modern methods of dynamical system analysis with simulation is given in the evaluation of the vertical motions of a typical planing hull. Extending the work of Troesch and Hicks (1992) and Troesch and Falzarano (1993), the complete nonlinear hydrodynamic force and moment equations of Zarnick (1978) are expanded in a multi-variable Taylor series. As a result, the nonlinear integro-differential equations of motion are replaced by a set of highly coupled, ordinary differential equations with constant coefficients, valid through third order. Closed-form, analytic expressions are available for the coefficients (Hicks, 1993). Numerical examples for all first-order and some second-order terms are presented. Once completely determined, the coefficient matrices will serve as input to path following or continuation methods (e.g., Seydel, 1988) where heave and pitch magnification curves can be generated, allowing the entire system response to be viewed. The branching behavior of the solutions resulting from a variation of the center of gravity is examined in detail. These studies of the second-order accurate model show the potential of the method to identify areas of critical dynamic response, which in turn can be verified and explored further through the use of the simulator.

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