scholarly journals Fractal fits to Riemann zeros

2007 ◽  
Vol 85 (4) ◽  
pp. 345-357 ◽  
Author(s):  
P B Slater
Keyword(s):  
Potential Model ◽  
Sine Series ◽  
Dominant Term ◽  
One Dimensional ◽  
Smooth Potential ◽  
Smooth Part ◽  
Least Squares Fit ◽  
The One ◽  
Higher Order Corrections ◽  
03.65 Sq

Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded — as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) — that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,γ). Setting d = 3/2, we estimate the frequency parameter (γ), plus an overall scaling parameter (σ) that we introduce. We search for that pair of parameters (γ,σ) that minimizes the least-squares fit Sn(γ,σ) of the lowest n eigenvalues — obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) — to the lowest n Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set σ = 1. The fits obtained are compared to those found by using just the smooth part of the Wu–Sprung potential without any fractal supplementation. Some limited improvement — 5.7261 versus 6.392 07 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) — is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of γ = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann–von Mangoldt formula (beyond the leading, dominant term) into the smooth potential. PACS Nos.: 02.10.De, 03.65.Sq, 05.45.Df, 05.45.Mt

scholarly journals Bound states of a Klein–Gordon particle in the presence of a smooth potential well

2020 ◽  
Vol 35 (23) ◽  
pp. 2050140
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Bound State ◽  
Potential Well ◽  
One Dimensional ◽  
Smooth Potential ◽  
Klein Gordon ◽  
The One ◽  
Potential Parameters ◽  
Klein Gordon Equation

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

scholarly journals Canonical Transformation of Potential Model Hamiltonian Mechanics to Geometrical Form I

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1009
Author(s):  
Yosef Strauss ◽  
Lawrence P. Horwitz ◽  
Jacob Levitan ◽  
Asher Yahalom
Keyword(s):  
Canonical Transformation ◽  
Symplectic Geometry ◽  
Potential Model ◽  
Standard Form ◽  
Geodesic Deviation ◽  
One Dimensional ◽  
Model Hamiltonian ◽  
Geometrical Form ◽  
The Stability ◽  
The One

Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geodesic deviation as a test for local stability, shown in recent previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We apply our methods to a simple one-dimensional harmonic oscillator and conclude with a discussion of the relation of bounded domains in the two representations for which Morse theory would be applicable.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

Adiabatic large polarons in anisotropic molecular crystals

2011 ◽  
Vol 35 (1) ◽  
pp. 15-27
Author(s):  
Zoran Ivić ◽  
Željko Pržulj
Keyword(s):  
Energy Transfer ◽  
Effective Mass ◽  
Molecular Crystals ◽  
Single Chain ◽  
Proper Understanding ◽  
One Dimensional ◽  
Interchain Coupling ◽  
Large Polaron ◽  
The One

Adiabatic large polarons in anisotropic molecular crystals We study the large polaron whose motion is confined to a single chain in a system composed of the collection of parallel molecular chains embedded in threedimensional lattice. It is found that the interchain coupling has a significant impact on the large polaron characteristics. In particular, its radius is quite larger while its effective mass is considerably lighter than that estimated within the one-dimensional models. We believe that our findings should be taken into account for the proper understanding of the possible role of large polarons in the charge and energy transfer in quasi-one-dimensional substances.

Modelling Techniques in Applied Glaciology: Numerical Modelling of Avalanches

1983 ◽  
Vol 4 ◽  
pp. 297-297
Author(s):  
G. Brugnot
Keyword(s):  
Finite Difference Method ◽  
Transverse Velocity ◽  
Longitudinal Velocity ◽  
Momentum Conservation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Modelling Techniques ◽  
Reference Grid ◽  
The One ◽  
Near Future

We consider the paper by Brugnot and Pochat (1981), which describes a one-dimensional model applied to a snow avalanche. The main advance made here is the introduction of the second dimension in the runout zone. Indeed, in the channelled course, we still use the one-dimensional model, but, when the avalanche spreads before stopping, we apply a (x, y) grid on the ground and six equations have to be solved: (1) for the avalanche body, one equation for continuity and two equations for momentum conservation, and (2) at the front, one equation for continuity and two equations for momentum conservation. We suppose the front to be a mobile jump, with longitudinal velocity varying more rapidly than transverse velocity.We solve these equations by a finite difference method. This involves many topological problems, due to the actual position of the front, which is defined by its intersection with the reference grid (SI, YJ). In the near future our two directions of research will be testing the code on actual avalanches and improving it by trying to make it cheaper without impairing its accuracy.

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