The nature of the electronic states in disordered one-dimensional systems

1963 ◽  
Vol 274 (1359) ◽  
pp. 529-545 ◽  
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Analytical Calculation ◽  
Homogeneous Boundary Condition ◽  
High Electron ◽  
One Dimensional ◽  
Computer Calculations ◽  
A Chain ◽  
Zero Potential

We consider an electron moving in the field of a one-dimensional infinite chain of identical potentials separated by regions of zero potential, the lengths s of these regions being distributed according to a probability density function p(8) . If we define the reduced phase of a real solution of the wave equation as the principal value of arctan ( — ψ'/kψ ) and є i as the reduced phase at the point x i immediately to the left of the i th atomic potential, it is shown for all bounded p(s) and sufficiently high electron energies that the є i are distributed according to a probability density function which depends on the direction of integration from a specified homogeneous boundary condition. This result is shown to imply that the eigenfunctions for such systems are localized in the sense that the envelope of such a function decays on average in an exponential manner on either side of some region. An analytical calculation for a random chain of δ-functions gives the decay of the nodes explicitly for high energies, and numerical calculations of the decay for a liquid model are presented. Further support for the theory is provided by computer calculations of some of the eigenfunctions of a chain of 1000 randomly placed δ-functions.

MEHRAB MAPS: ONE-DIMENSIONAL PIECEWISE NONLINEAR CHAOTIC MAPS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250127 ◽  
Author(s):  
SHAHRAM ETEMADI BORUJENI ◽  
MOHAMMAD ESHGHI ◽  
MAHDI SAFARNEJAD BOROUJENI
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Lyapunov Exponents ◽  
Uniform Density ◽  
Initial Condition ◽  
One Dimensional ◽  
Return Maps ◽  
Horizontal Symmetry ◽  
The Right

In this paper, we propose a new one-dimensional, two-segmental nonlinear map by combining tent, triangle and parabola curve functions. We call the proposed map, Mehrab map since its return maps shape is similar to an altar (which we call it "Mehrab"). Definition and properties of Mehrab map along with orbit diagrams, Lyapunov exponents, and its histograms are considered. To generate more uniform density function maps, two modified versions of the proposed Mehrab map are also defined. In the first modification of Mehrab map (FMM), vertical symmetry and transformation to the right are used. Sensitivity to initial condition and total chaotic range of FMM are medium. Probability density function of FMM map is uniform and its histograms show this uniformity. In the second modification of Mehrab (SMM) map, vertical and horizontal symmetry and transformation to the right are used. According to the orbit diagrams and Lyapunov exponents, the sensitivity to initial condition and the total chaotic range of SMM map are large. This property gives more chaotic region to the map. Its histograms prove that the probability density function of SMM is also uniform.

scholarly journals Improved Chebyshev inequality: new probability bounds with known supremum of PDF

2018 ◽  
Author(s):  
Tomohiro Nishiyama
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Chebyshev Inequality ◽  
Discrete Probability ◽  
One Dimensional ◽  
Probability Bounds ◽  
Mean And Variance ◽  
Discrete Probability Distributions

In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known.This result holds for one-dimensional or multivariate continuous probability distributions with finite mean and variance (covariance matrix).We also show that the similar result holds for specific discrete probability distributions.

Exploratory orientation data analysis with ω sections

2004 ◽  
Vol 37 (5) ◽  
pp. 683-697 ◽  
Author(s):  
K. Gerald van den Boogaart ◽  
Helmut Schaeben
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Unit Disk ◽  
Three Dimensional ◽  
Computer Code ◽  
Orientation Data ◽  
Mean Values ◽  
One Dimensional ◽  
Totally Geodesic Submanifolds

Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere S^2 \subset I\!R^3 onto the unit disk D \subset I\!R^2. The familiar crystallographic `pole figures' are actually mean values of the spherical Radon {\cal R}_1 transform. The mathematical Radon {\cal R}_1 transform associates a real-valued functionfdefined on a sphere with its mean values {\cal R}_{1}f along one-dimensional circles with centre {\cal O}, the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines ω sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displaysfcompletely,i.e.iffis given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display.

scholarly journals Mathematical approach to human reaction

2020 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
The Other ◽  
Mathematical Approach ◽  
One Dimensional ◽  
Human Reaction ◽  
Fixed Distribution

I thought about how to get the magnitude from the event and the reaction of the other party. Evaluating the values of events and opponents' reactions using a one-dimensional random walk shows that the probability density function of the values of events and opponents' reactions has a fixed probability distribution. Similarly, I have shown that the functions that determine the magnitude of events and reactions are also represented by a fixed distribution. Therefore, I also showed that when individuals gather to form a group, the functions that determine the magnitude of events and reactions as a group are also represented by a fixed distribution. Also, as an application example of this model, I described how to show my reaction and what to do when the magnitude of the event is large.

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