Rotation–Vibration Coupling in Diatomic Molecules and the Factorization Method. II. Closed Form Formulas for the Morse–Pekeris Intensities

1974 ◽  
Vol 52 (2) ◽  
pp. 110-119 ◽  
Author(s):  
M. Badawi ◽  
N. Bessis ◽  
G. Bessis ◽  
G. Hadinger
Keyword(s):  
Dipole Moment ◽  
Closed Form ◽  
Explicit Expression ◽  
Factorization Method ◽  
Closed Form Expression ◽  
Matrix Elements ◽  
Moment Matrix ◽  
Form Expression ◽  
Vibration Coupling

It is shown that, by applying an "accelerated" ladder operatorial formalism or an equivalent matrix procedure, one can obtain, easily, for the case of a Morse–Pekeris potential, a closed form expression of the rotation–vibration nuclear dipole moment matrix elements. This explicit expression, which is valid for any degree k of the dipole moment Taylor's expansion, allows the determination of the rotation–vibration intensities for any ΩνJ → Ω′ν′J′ transition.

Rotation–Vibration Coupling in Diatomic Molecules and the Factorization Method. I. Closed Form Formulas for the Vibration Intensities

1973 ◽  
Vol 51 (19) ◽  
pp. 2075-2085 ◽  
Author(s):  
M. Badawi ◽  
N. Bessis ◽  
G. Bessis
Keyword(s):  
Dipole Moment ◽  
Transition Matrix ◽  
Diatomic Molecules ◽  
Factorization Method ◽  
Dipole Matrix Element ◽  
Matrix Elements ◽  
Explicit Formulas ◽  
Overlap Integral ◽  
Transition Matrix Elements ◽  
Vibration Coupling

It is shown that the factorization method followed by an “accelerated" operatorial formalism or an equivalent matrix procedure leads to explicit formulas for calculating transition matrix elements. The calculation has been completely carried out for the pure vibration case of diatomic molecules, for any degree k of the dipole moment Taylor's expansion. The intuitive approximate proportionality between the radial dipole matrix element and an overlap integral is demonstrated.

Conditions for Translation and Rotation of Bodies on Frictional Surfaces

1991 ◽  
Vol 205 (5) ◽  
pp. 311-316
Author(s):  
D Nowell ◽  
D A Hills ◽  
R L Munisamy
Keyword(s):  
Closed Form ◽  
Numerical Technique ◽  
Closed Form Expression ◽  
Frictional Surface ◽  
Form Expression ◽  
Straight Line ◽  
Frictional Surfaces

This paper is concerned with the quasi-static motion of a body resting on a frictional surface when subjected to an arbitrary imposed displacement by a small frictionless finger. A simplified object with three distinct feet is used as an illustration but the method adopted can be generalized to bodies with more feet or distributed contact. A closed-form expression is found which enables the determination of the conditions necessary for a body to rotate about a point of support and a more general numerical technique for determining the instantaneous centre is presented. The indeterminacy that occurs when the points of support lie in a straight line is also discussed.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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