scholarly journals Power Law Duality in Classical and Quantum Mechanics

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 409
Author(s):  
Akira Inomata ◽  
Georg Junker
Keyword(s):  
Quantum Mechanics ◽  
Green Function ◽  
Power Law ◽  
Ad Hoc ◽  
Fractional Power ◽  
Power Laws ◽  
Energy Coupling ◽  
Arbitrary Power ◽  
Dual Symmetry ◽  
The Green Function

The Newton–Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton’s characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling exchange maps required as part of the duality operations that take one system to another lead to an energy formula that relates the new energy to the old energy. The transformation property of the Green function satisfying the radial Schrödinger equation yields a formula that relates the new Green function to the old one. The energy spectrum of the linear motion in a fractional power potential is semiclassically evaluated. We find a way to show the Coulomb–Hooke duality in the supersymmetric semiclassical action. We also study the confinement potential problem with the help of the dual structure of a two-term power potential.

scholarly journals Emergence of power laws in the pharmacokinetics of paclitaxel due to competing saturable processes.

2008 ◽  
Vol 11 (3) ◽  
pp. 77 ◽  
Author(s):  
Jack A Tuszynski ◽  
Rebeccah E. Marsh ◽  
Michael B. Sawyer ◽  
Kenneth J.E. Vos
Keyword(s):  
Power Law ◽  
Fractional Power ◽  
Area Under The Curve ◽  
Power Laws ◽  
Dose Dependence ◽  
Power Exponent ◽  
Least Squares Regression ◽  
Time Data ◽  
Concentration Time ◽  
Wide Range

Purpose: This study presents the results of power law analysis applied to the pharmacokinetics of paclitaxel. Emphasis is placed on the role that the power exponent can play in the investigation and quantification of nonlinear pharmacokinetics and the elucidation of the underlying physiological processes. Methods: Forty-one sets of concentration-time data were inferred from 20 published clinical trial studies, and 8 sets of area under the curve (AUC) and maximum concentration (Cmax) values as a function of dose were collected. Both types of data were tested for a power law relationship using least squares regression analysis. Results: Thirty-nine of the concentration-time curves were found to exhibit power law tails, and two dominant fractal exponents emerged. Short infusion times led to tails with a single power exponent of -1.57 ± 0.14, while long infusion times resulted in steeper tails characterized by roughly twice the exponent. The curves following intermediate infusion times were characterized by two consecutive power laws; an initial short slope with the larger alpha value was followed by a crossover to a long-time tail characterized by the smaller exponent. The AUC and Cmax parameters exhibited a power law dependence on the dose, with fractional power exponents that agreed with each other and with the exponent characterizing the shallow decline. Computer simulations revealed that a two- or three-compartment model with both saturable distribution and saturable elimination can produce the observed behaviour. Furthermore, there is preliminary evidence that the nonlinear dose-dependence is correlated with the power law tails. Conclusion: Assessment of data from published clinical trials suggests that power laws accurately describe the concentration-time curves and non-linear dose-dependence of paclitaxel, and the power exponents provide insight into the underlying drug mechanisms. The interplay between two saturable processes can produce a wide range of behaviour, including concentration-time curves with exponential, power law, and dual power law tails.

scholarly journals How rare are power-law networks really?

2020 ◽  
Vol 476 (2241) ◽  
pp. 20190742
Author(s):  
I. Artico ◽  
I. Smolyarenko ◽  
V. Vinciotti ◽  
E. C. Wit
Keyword(s):  
Power Law ◽  
Ad Hoc ◽  
Null Distribution ◽  
Power Laws ◽  
Testing Procedure ◽  
Statistical Testing ◽  
Power Law Distribution ◽  
Scale Free ◽  
Common Structure ◽  
Power Of The Test

The putative scale-free nature of real-world networks has generated a lot of interest in the past 20 years: if networks from many different fields share a common structure, then perhaps this suggests some underlying ‘network law’. Testing the degree distribution of networks for power-law tails has been a topic of considerable discussion. Ad hoc statistical methodology has been used both to discredit power-laws as well as to support them. This paper proposes a statistical testing procedure that considers the complex issues in testing degree distributions in networks that result from observing a finite network, having dependent degree sequences and suffering from insufficient power. We focus on testing whether the tail of the empirical degrees behaves like the tail of a de Solla Price model, a two-parameter power-law distribution. We modify the well-known Kolmogorov–Smirnov test to achieve even sensitivity along the tail, considering the dependence between the empirical degrees under the null distribution, while guaranteeing sufficient power of the test. We apply the method to many empirical degree distributions. Our results show that power-law network degree distributions are not rare, classifying almost 65% of the tested networks as having a power-law tail with at least 80% power.

scholarly journals The shape of dendritic tips

2020 ◽  
Vol 378 (2171) ◽  
pp. 20190243 ◽  
Author(s):  
Dmitri V. Alexandrov ◽  
Peter K. Galenko
Keyword(s):  
Experimental Data ◽  
Power Law ◽  
Dendritic Growth ◽  
Binary Systems ◽  
Fractional Power ◽  
Power Laws ◽  
Theme Issue ◽  
Special Case ◽  
Dendritic Shape ◽  
Good Agreement

The present article is focused on the shapes of dendritic tips occurring in undercooled binary systems in the absence of convection. A circular/globular shape appears in limiting cases of small and large Péclet numbers. A parabolic/paraboloidal shape describes the tip regions of dendrites whereas a fractional power law defines a shape behind their tips in the case of low/moderate Péclet number. The parabolic/paraboloidal and fractional power law shapes are sewed together in the present work to describe the dendritic shape in a broader region adjacent to the dendritic tip. Such a generalized law is in good agreement with the parabolic/paraboloidal and fractional power laws of dendritic shapes. A special case of the angled dendrite is considered and analysed in addition. The obtained results are compared with previous experimental data and the results of numerical simulations on dendritic growth. This article is part of the theme issue ‘Patterns in soft and biological matters’.

scholarly journals On the Green function in visco-elastic media obeying a frequency power-law

2011 ◽  
Vol 34 (7) ◽  
pp. 819-830 ◽  
Author(s):  
E. Bretin ◽  
L. Guadarrama Bustos ◽  
A. Wahab
Keyword(s):  
Green Function ◽  
Power Law ◽  
Elastic Media ◽  
The Green Function ◽  
Frequency Power

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

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