The normalized wavefunctions of the Hartmann potential and explicit expressions for their radial average values

2002 ◽  
Vol 305 (6) ◽  
pp. 341-348 ◽  
Author(s):  
Chang-Yuan Chen ◽  
Cheng-Lin Liu ◽  
Dong-Sheng Sun
Keyword(s):  
Radial Average ◽  
Hartmann Potential ◽  
Explicit Expressions

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

scholarly journals Third derivatives of the integrable part of an asteroid Hamiltonian

2007 ◽  
pp. 53-60 ◽  
Author(s):  
R. Pavlovic
Keyword(s):  
Third Order ◽  
Geometrical Condition ◽  
The Third ◽  
Derivatives Of ◽  
Quasi Convexity ◽  
Explicit Expressions

To apply the theorem of Nekhoroshev (1977) to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .

scholarly journals Relationship between Bondi–Sachs quantities and source of gravitational radiation in asymptotically de Sitter spacetime

2018 ◽  
Vol 27 (04) ◽  
pp. 1850046 ◽  
Author(s):  
Xiaokai He ◽  
Jiliang Jing ◽  
Zhoujian Cao
Keyword(s):  
Cosmological Constant ◽  
Gravitational Wave ◽  
Gravitational Radiation ◽  
The Other ◽  
Perturbation Approach ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Positive Cosmological Constant ◽  
Sitter Spacetime ◽  
Explicit Expressions

Gravitational radiation plays an important role in astrophysics. Based on the fact that our universe is expanding, the gravitational radiation when a positive cosmological constant is presented has been studied along with two different ways recently, one is the Bondi–Sachs (BS) framework in which the result is shown by BS quantities in the asymptotic null structure, the other is the perturbation approach in which the result is presented by the quadrupoles of source. Therefore, it is worth to interpret the quantities in asymptotic null structure in terms of the information of the source. In this paper, we investigate this problem and find the explicit expressions of BS quantities in terms of the quadrupoles of source in asymptotically de Sitter spacetime. We also estimate how far away the source is, the cosmological constant may affect the detection of the gravitational wave.

scholarly journals The Modified Hartmann Potential Effects on γ-rigid Bohr Hamiltonian

2018 ◽  
Vol 1011 ◽  
pp. 012085
Author(s):  
A Suparmi ◽  
C Cari ◽  
Beta Nur Pratiwi
Keyword(s):  
Hartmann Potential

scholarly journals Explicit expressions for the minimum efficiency and most penetrating particle size of Nuclepore filters

2016 ◽  
Vol 100 ◽  
pp. 108-117 ◽  
Author(s):  
Sheng-Chieh Chen ◽  
Yaorui Hu ◽  
David Y.H. Pui ◽  
Jing Wang
Keyword(s):  
Particle Size ◽  
Explicit Expressions

THE INHOMOGENEOUS HEISENBERG CHAIN WITH THE INTERACTION OF CLASSICAL AND QUANTUM SPINS

2000 ◽  
Vol 14 (11) ◽  
pp. 1179-1185
Author(s):  
B. I. SADOVNIKOV ◽  
N. G. INOZEMTSEVA ◽  
V. I. INOZEMTSEV
Keyword(s):  
Free Energy ◽  
Specific Heat ◽  
Heisenberg Chain ◽  
Spin Correlations ◽  
Quantum Spins ◽  
Explicit Expressions

The thermodynamics of an inhomogeneous 1D Heisenberg chain with alternating classical and quantum spins is studied. The explicit expressions for the free energy, specific heat, thermal spin correlations and linear susceptibility are found.

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