scholarly journals Calculation of the Lowest Resonant States of H− and Li by the Complex Absorbing Potential Method

Atoms ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 72
Author(s):  
Sergey O. Adamson ◽  
Daria D. Kharlampidi ◽  
Yurii A. Dyakov ◽  
Gennady V. Golubkov ◽  
Maxim G. Golubkov
Keyword(s):  
Optimal Trajectory ◽  
Single Channel ◽  
Initial Conditions ◽  
Scattering Matrix ◽  
Potential Method ◽  
Optimal Point ◽  
Closed Area ◽  
Resonant States ◽  
Special Cases ◽  
Ambiguous Result

The analysis of the features of the method of complex absorbing potential (CAP) is carried out for a single-channel problem with an explicit parameterization of the scattering matrix. It is shown that there can be several types of CAP trajectories depending on the choice of the initial conditions. In any case, the estimation of the resonance parameters from the position of the optimal trajectory point can lead to a systematic error or an ambiguous result. In special cases, the search for the optimal point can be replaced by the averaging over a closed section of the trajectory. The CAP trajectories constructed in the H− and Li resonance calculations correlate well with the model trajectories, which have a curl around the resonance. The averaging over a closed area of the trajectory leads to better estimates of the energy and width of the resonance in comparison with the technique of searching for the optimal point.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

Explicit asymptotic results for open times in ion channel models

1997 ◽  
Vol 29 (04) ◽  
pp. 947-964
Author(s):  
Valeri T. Stefanov ◽  
Geoffrey F. Yeo
Keyword(s):  
Single Channel ◽  
Exponential Families ◽  
Markov Renewal Process ◽  
Actual Process ◽  
Asymptotic Results ◽  
Normal Distributions ◽  
Special Cases ◽  
The Difference ◽  
Asymptotic Normal ◽  
Receptor Channel

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.

On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

2009 ◽  
Vol 46 (02) ◽  
pp. 363-371 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Growth Process ◽  
Initial Conditions ◽  
Steady State Distribution ◽  
State Distribution ◽  
Stationary Structure ◽  
Stationary Version ◽  
Special Cases ◽  
The Times

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.

scholarly journals Unsupervised Learning for Monaural Source Separation Using Maximization–Minimization Algorithm with Time–Frequency Deconvolution

Sensors ◽  
2018 ◽  
Vol 18 (5) ◽  
pp. 1371 ◽  
Author(s):  
Wai Lok Woo ◽  
Bin Gao ◽  
Ahmed Bouridane ◽  
Bingo Wing-Kuen Ling ◽  
Cheng Siong Chin
Keyword(s):  
Unsupervised Learning ◽  
Single Channel ◽  
Learning Algorithm ◽  
Source Separation ◽  
Nonnegative Matrix ◽  
Least Square ◽  
Separation Performance ◽  
Time Frequency ◽  
Special Cases ◽  
Leibler Divergence

This paper presents an unsupervised learning algorithm for sparse nonnegative matrix factor time–frequency deconvolution with optimized fractional β-divergence. The β-divergence is a group of cost functions parametrized by a single parameter β. The Itakura–Saito divergence, Kullback–Leibler divergence and Least Square distance are special cases that correspond to β=0, 1, 2, respectively. This paper presents a generalized algorithm that uses a flexible range of β that includes fractional values. It describes a maximization–minimization (MM) algorithm leading to the development of a fast convergence multiplicative update algorithm with guaranteed convergence. The proposed model operates in the time–frequency domain and decomposes an information-bearing matrix into two-dimensional deconvolution of factor matrices that represent the spectral dictionary and temporal codes. The deconvolution process has been optimized to yield sparse temporal codes through maximizing the likelihood of the observations. The paper also presents a method to estimate the fractional β value. The method is demonstrated on separating audio mixtures recorded from a single channel. The paper shows that the extraction of the spectral dictionary and temporal codes is significantly more efficient by using the proposed algorithm and subsequently leads to better source separation performance. Experimental tests and comparisons with other factorization methods have been conducted to verify its efficacy.

Response Surface Designs for Estimating the Optimal Point

1981 ◽  
Vol 30 (3-4) ◽  
pp. 145-170 ◽  
Author(s):  
Shoutir Kishore Chatterjee ◽  
Nripes Kumar Mandal
Keyword(s):  
Response Surface ◽  
Minimal Point ◽  
Optimal Point ◽  
Positive Orthant ◽  
Special Cases ◽  
K Factor ◽  
Response Surface Designs ◽  
Optimum Response ◽  
Factor Design ◽  
Least Squares Estimates

The problem considered is that of choosing a k-factor design meant for locating the maximal (or minimal) point of a second order response surface. Using least squares estimates the deficiency of a design is defined as the expected difference between the optimum response and the true response at the estimated optimal point. It is first shown that a design which minimizes this deficiency must belong to a special subclass of designs. Each member of this subclass cam be represented by a distribution over the unit simplex in the k-dimensional positive orthant apportioning the whole mass exclusively between the origin and the outer surface. Using this result the global solution in some special cases and a restricted solution in the general case are obtained.

CONSTRUCTIVE ASPECTS OF STABILITY RESEARCH IN SPECIAL CASES

2016 ◽  
Vol 8 (4) ◽  
pp. 18-26
Author(s):  
Котов ◽  
P. Kotov
Keyword(s):  
Differential Operator ◽  
Numerical Methods ◽  
Initial Conditions ◽  
Special Cases ◽  
Material Equation

The method of stability research of the dynamic status of system described by the material equation resolved concerning derivatives with the linear summary differential operator, limited elements of the initial conditions system and the substantial basis of numerical methods and of methods of basic researches is offered.

Exact solutions for scalar field cosmology in f(R) gravity

2017 ◽  
Vol 32 (30) ◽  
pp. 1750164 ◽  
Author(s):  
S. D. Maharaj ◽  
R. Goswami ◽  
S. V. Chervon ◽  
A. V. Nikolaev
Keyword(s):  
Scalar Field ◽  
Exact Solutions ◽  
Scalar Curvature ◽  
Evolution Equations ◽  
Initial Conditions ◽  
De Sitter ◽  
Airy Functions ◽  
Special Cases ◽  
The Universe ◽  
Current Stage

We study scalar field FLRW cosmology in the content of f(R) gravity. Our consideration is restricted to the spatially flat Friedmann universe. We derived the general evolution equations of the model, and showed that the scalar field equation is automatically satisfied for any form of the f(R) function. We also derived representations for kinetic and potential energies, as well as for the acceleration in terms of the Hubble parameter and the form of the f(R) function. Next we found the exact cosmological solutions in modified gravity without specifying the f(R) function. With negligible acceleration of the scalar curvature, we found that the de Sitter inflationary solution is always attained. Also we obtained new solutions with special restrictions on the integration constants. These solutions contain oscillating, accelerating, decelerating and even contracting universes. For further investigation, we selected special cases which can be applied with early or late inflation. We also found exact solutions for the general case for the model with negligible acceleration of the scalar curvature in terms of special Airy functions. Using initial conditions which represent the universe at the present epoch, we determined the constants of integration. This allows for the comparison of the scale factor in the new solutions with that for current stage of the universe evolution in the [Formula: see text]CDM model.

On the global attractivity and the solution of recursive sequence

2010 ◽  
Vol 47 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence ◽  
The Difference

In this paper we study the behavior of the difference equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x_{n + 1} = ax_{n - 2} + \frac{{bx_n x_{n - 2} }}{{cx_n + dx_{n - 3} }},n = 0,1,...$$ \end{document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

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