scholarly journals The Statistical Properties of theq-Deformed Dirac Oscillator in One and Two Dimensions

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Abdelmalek Boumali ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Dynamics ◽  
Relativistic Quantum ◽  
Small Deformation ◽  
Two Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dirac Oscillator ◽  
One Dimensional ◽  
Pure Phase ◽  
The One

We study the behavior of the eigenvalues of the one and two dimensions ofq-deformed Dirac oscillator. The eigensolutions have been obtained by using a method based on theq-deformed creation and annihilation operators in both dimensions. For a two-dimensional case, we have used the complex formalism which reduced the problem to a problem of one-dimensional case. The influence of theq-numbers on the eigenvalues has been well analyzed. Also, the connection between theq-oscillator and a quantum optics is well established. Finally, for very small deformationη, we (i) showed the existence of well-knownq-deformed version of Zitterbewegung in relativistic quantum dynamics and (ii) calculated the partition function and all thermal quantities such as the free energy, total energy, entropy, and specific heat. The extension to the case of Graphene has been discussed only in the case of a pure phase (q=eiη).

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

SOLVING THE TWO-DIMENSIONAL INVERSE FRACTAL PROBLEM WITH THE WAVELET TRANSFORM

Fractals ◽  
1996 ◽  
Vol 04 (04) ◽  
pp. 469-475 ◽  
Author(s):  
ZBIGNIEW R. STRUZIK
Keyword(s):  
Wavelet Transform ◽  
Scale Invariance ◽  
Wavelet Decomposition ◽  
Two Dimensions ◽  
The Self ◽  
Continuous Wavelet ◽  
Two Dimensional ◽  
One Dimensional ◽  
Affine Functions ◽  
The One

The methodology of the solution to the inverse fractal problem with the wavelet transform1,2 is extended to two-dimensional self-affine functions. Similar to the one-dimensional case, the two-dimensional wavelet maxima bifurcation representation used is derived from the continuous wavelet decomposition. It possesses translational and scale invariance necessary to reveal the invariance of the self-affine fractal. As many fractals are naturally defined on two-dimensions, this extension constitutes an important step towards solving the related inverse fractal problem for a variety of fractal types.

scholarly journals Revisiting the 1D and 2D Laplace Transforms

2020 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
José Tenreiro Machado
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Linear Systems ◽  
Laplace Transforms ◽  
Dimensional Case ◽  
Initial Condition ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One

This paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. General two-dimensional linear systems are introduced and the corresponding transfer function defined.

A Two-Dimensional Model of the Fluid Dynamics of an Air Reverser

1998 ◽  
Vol 65 (1) ◽  
pp. 171-177 ◽  
Author(s):  
S. Mu¨ftu¨ ◽  
T. S. Lewis ◽  
K. A. Cole ◽  
R. C. Benson
Keyword(s):  
Design Guidelines ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Web Handling ◽  
One Dimensional ◽  
Air Supply ◽  
Two Dimensional Model ◽  
The One ◽  
The Web

A theoretical analysis of the fluid mechanics of the air cushion of the air reversers used in web-handling systems is presented. A two-dimensional model of the air flow is derived by averaging the equations of conservation of mass and momentum over the clearance between the web and the reverser. The resulting equations are Euler’s equations with nonlinear source terms representing the air supply holes in the surface of the reverser. The equations are solved analytically for the one-dimensional case and numerically for the two-dimensional case. Results are compared with an empirical formula and the one-dimensional airjet theory developed for hovercraft. Conditions that maximize the air pressure supporting the web are analyzed and design guidelines are deduced.

A Comparison of Diffusion Flame Stability in One and Two Spatial Dimensions Near Cold, Inert Surfaces

2001 ◽  
Author(s):  
Robert Vance ◽  
Indrek S. Wichman
Keyword(s):  
Two Dimensions ◽  
Low Flow ◽  
Dimensional System ◽  
Flame Stability ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Cellular Flames ◽  
The One ◽  
Inert Surfaces

Abstract A linear stability analysis is performed on two simplified models representing a one-dimensional flame between oxidizer and fuel reservoirs and a two-dimensional “edge-flame” between the same reservoirs but above a cold, inert wall. Comparison of the eigenvalue spectra for both models is performed to discern the validity of extending the results from the one-dimensional problem to the two-dimensional problem. Of primary interest is the influence on flame stability of thermal-diffusive imbalances, i.e. non-unity Lewis numbers. Flame oscillations are observed when Le > 1, and cellular flames are witnessed when Le < 1. It is found that when Le > 1 the characteristics of flame behavior are consistent between the two models. Furthermore, when Le < 1, the models are found to be in good agreement with respect to the magnitude of the critical wave numbers. Results from the coarse mesh analysis of the two-dimensional system are presented and compared to the one-dimensional eigenvalue spectra. Additionally, an examination of low reactant convection is undertaken. It is concluded that for low flow rates the behavior in one and two dimensions are similar qualitatively and quantitatively.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

1997 ◽  
Vol 07 (07) ◽  
pp. 1451-1496 ◽  
Author(s):  
André Barbé
Keyword(s):  
Cellular Automata ◽  
Three Dimensional ◽  
Coarse Graining ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Complex Order ◽  
Simple Order ◽  
Problems And Solutions ◽  
The One

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

scholarly journals BELLS GALORE: OSCILLATIONS AND CIRCLE-MAP DYNAMICS FROM SPACE-FILLING FRACTAL FUNCTIONS

Fractals ◽  
2008 ◽  
Vol 16 (04) ◽  
pp. 367-378 ◽  
Author(s):  
CARLOS E. PUENTE ◽  
ANDREA CORTIS ◽  
BELLIE SIVAKUMAR
Keyword(s):  
Two Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Circle Map ◽  
Space Filling ◽  
Affine Mappings ◽  
Oscillatory Pattern ◽  
Fractal Functions ◽  
The Many ◽  
The One

The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings' parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.

Some asymptotic properties of a two-dimensional periodogram

1971 ◽  
Vol 8 (4) ◽  
pp. 841-847 ◽  
Author(s):  
M. Pagano
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Properties ◽  
Two Dimensions ◽  
Stationary Time Series ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Regular Lattice ◽  
Homogeneous Random Field ◽  
One Dimensional ◽  
Distributional Properties

The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory for p-dimensional periodograms (p > 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for the p-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.

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