The radial equation for hydrogen-like atoms. The polynomial versus the factorization technique

1975 ◽  
Vol 52 (2) ◽  
pp. 92 ◽  
Author(s):  
Carl Peterson
Keyword(s):  
Radial Equation ◽  
Factorization Technique ◽  
Polynomial Versus

scholarly journals Augmenting matrix factorization technique with the combination of tags and genres

2016 ◽  
Vol 461 ◽  
pp. 101-116 ◽  
Author(s):  
Tinghuai Ma ◽  
Xiafei Suo ◽  
Jinjuan Zhou ◽  
Meili Tang ◽  
Donghai Guan ◽  
...  
Keyword(s):  
Matrix Factorization ◽  
Factorization Technique ◽  
Matrix Factorization Technique

A UNIFIED APPROACH TO RESOLVENT EXPANSIONS AT THRESHOLDS

2001 ◽  
Vol 13 (06) ◽  
pp. 717-754 ◽  
Author(s):  
ARNE JENSEN ◽  
GHEORGHE NENCIU
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Unified Approach ◽  
Zero Energy ◽  
Real Potential ◽  
Factorization Technique

Results are obtained on resolvent expansions around zero energy for Schrödinger operators H=-Δ+V(x) on L2(Rm), where V(x) is a sufficiently rapidly decaying real potential. The emphasis is on a unified approach, valid in all dimensions, which does not require one to distinguish between ∫V(x)dx=0 and ∫V(x)dx≠0 in dimensions m=1,2. It is based on a factorization technique and repeated decomposition of the Lippmann–Schwinger operator. Complete results are given in dimensions m=1 and m=2.

scholarly journals Lyapunov exponent, ISCO and Kolmogorov–Senai entropy for Kerr–Kiselev black hole

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Monimala Mondal ◽  
Farook Rahaman ◽  
Ksh. Newton Singh
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Lyapunov Exponent ◽  
Circular Orbit ◽  
Real Root ◽  
Angular Frequency ◽  
Geodesic Motion ◽  
Radial Equation ◽  
Stable Circular Orbit ◽  
Innermost Stable Circular Orbit

AbstractGeodesic motion has significant characteristics of space-time. We calculate the principle Lyapunov exponent (LE), which is the inverse of the instability timescale associated with this geodesics and Kolmogorov–Senai (KS) entropy for our rotating Kerr–Kiselev (KK) black hole. We have investigate the existence of stable/unstable equatorial circular orbits via LE and KS entropy for time-like and null circular geodesics. We have shown that both LE and KS entropy can be written in terms of the radial equation of innermost stable circular orbit (ISCO) for time-like circular orbit. Also, we computed the equation marginally bound circular orbit, which gives the radius (smallest real root) of marginally bound circular orbit (MBCO). We found that the null circular geodesics has larger angular frequency than time-like circular geodesics ($$Q_o > Q_{\sigma }$$ Q o > Q σ ). Thus, null-circular geodesics provides the fastest way to circulate KK black holes. Further, it is also to be noted that null circular geodesics has shortest orbital period $$(T_{photon}< T_{ISCO})$$ ( T photon < T ISCO ) among the all possible circular geodesics. Even null circular geodesics traverses fastest than any stable time-like circular geodesics other than the ISCO.

scholarly journals Fractal fluctuations in the cardiovascular dynamical system : from the autonomic control to the central nervous system influence

2021 ◽  
Author(s):  
Asif Hasan Sharif
Keyword(s):  
Central Nervous System ◽  
Heart Rate ◽  
Nervous System ◽  
Autonomic Nervous System ◽  
Dynamic Feature ◽  
Scale Free ◽  
Autonomic Nervous ◽  
Factorization Technique ◽  
The Central Nervous System

The fractal component in the complex fluctuations of the human heart rate represents a dynamic feature that is widely observed in diverse fields of natural and artificial systems. It is also of clinical significance as the diminishing of the fractal dynamics appears to correlate with heart disease processes and adverse cardiac events in old age. While the autonomic nervous system directly controls the pacemaker cells of the heart, it does not provide an immediate characterization of the complex heart rate variability (HRV). The central nervous system (CNS) is known to be an important modulator for various cardiac functions. However, its role in the fractal HRV is largely unclear. In this research, human experiments were conducted to study the influence of the central nervous system on fractal dynamics of healthy human HRV. The head up tilt (HUT) maneuver is used to provide a perturbation to the autonomic nervous system. The subsequent fractal effect in the simultaneously recorded electroencephalography and beat-to-beat heart rate data was examined. Using the recently developed multifractal factorization technique, the common multifractality in the data fluctuation was analyzed. An empirical relationship was uncovered which shows the increase (decrease) in HRV multifractality is associated with the increase (decrease) in multifractal correlation between scale-free HRV and the cortical expression of the brain dynamics in 8 out of 11 healthy subjects. This observation is further supported using surrogate analysis. The present findings imply that there is an integrated central-autonomic component underlying the cortical expression of the HRV fractal dynamics. It is proposed that the central element should be incorporated in the fractal HRV analysis to gain a more comprehensive and better characterization of the scale-free HRV dynamics. This study provides the first contribution to the HRV multifractal dynamics analysis in HUT. The multivariate fractal analysis using factorization technique is also new and can be applied in the more general context in complex dynamics research.

One-Dimensional Jost Solutions: The S-Matrix Revisited

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Boundary Conditions ◽  
Transmission Coefficient ◽  
Radial Equation ◽  
Scattering Problems ◽  
One Dimensional ◽  
S Matrix ◽  
Jost Functions ◽  
Jost Solutions ◽  
Square Well ◽  
Matrix Problems

This chapter examines the properties of one-dimensional Jost solutions for S-matrix problems. It first considers how the left–right transmission and reflections coefficients can be expressed in terms of the elements of the S-matrix for one-dimensional scattering problems on, focusing on poles of the transmission coefficient. It then uses the radial equation to revisit the problem of the square-well potential from the perspective of the Jost solution, with Jost boundary conditions at r = 0 and as r approaches infinity. It also presents the notations for the Jost functions and the S-matrix before discussing the problem of scattering from a constant spherical inhomogeneity.

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