Potential energies of one-dimensional systems at equilibrium

1981 ◽  
Vol 59 (7) ◽  
pp. 859-862
Author(s):  
Shafique Ahmed
Keyword(s):  
Dynamical Systems ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Bessel Polynomials ◽  
One Dimensional ◽  
The One

The equilibrium positions of some one-dimensional systems coincide with the zeros of Hermite, Laguerre, and Jacobi polynomials. Using the discriminants of these classical orthogonal polynomials it is then possible to calculate the values of the potential energies of the one-dimensional systems at equilibrium. It is also shown that the zeros of Bessel polynomials coincide with the equilibrium positions of certain dynamical systems.

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Spectral statistics of orthogonal and symplectic ensembles

2018 ◽  
Author(s):  
Greg W. Anderson
Keyword(s):  
Orthogonal Polynomials ◽  
Fundamental Theorem ◽  
The Other ◽  
Classical Orthogonal Polynomials ◽  
The Matrix ◽  
Spectral Statistics ◽  
Orthogonal And Symplectic Ensembles ◽  
The One ◽  
Gap Probabilities ◽  
Matrix Kernels

This article describes a direct approach for computing scalar and matrix kernels, respectively for the unitary ensembles on the one hand and the orthogonal and symplectic ensembles on the other hand, leading to correlation functions and gap probabilities. In the classical orthogonal polynomials (Hermite, Laguerre, and Jacobi), the matrix kernels for the orthogonal and symplectic ensemble are expressed in terms of the scalar kernel for the unitary case, using the relation between the classical orthogonal polynomials going with the unitary ensembles and the skew-orthogonal polynomials going with the orthogonal and symplectic ensembles. The article states the fundamental theorem relating the orthonormal and skew-orthonormal polynomials that enter into the Christoffel-Darboux kernels

scholarly journals ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

2016 ◽  
Vol 56 (4) ◽  
pp. 283-290 ◽  
Author(s):  
Jiri Hrivnak ◽  
Lenka Motlochova
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Special Functions ◽  
The Other ◽  
Trigonometric Functions ◽  
The One ◽  
Sine Functions

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 617 ◽  
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

scholarly journals Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

2018 ◽  
Author(s):  
Dmitry Victorovich Dolgy ◽  
Dae San Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon
Keyword(s):  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Linear Combinations ◽  
The Third ◽  
Connection Problem ◽  
Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions ${}_2 F_0, {}_2 F_1$, and ${}_3 F_2$.

scholarly journals OCTONIONIC REALIZATIONS OF ONE-DIMENSIONAL EXTENDED SUPERSYMMETRIES: A CLASSIFICATION

2003 ◽  
Vol 18 (11) ◽  
pp. 787-798 ◽  
Author(s):  
H. L. CARRION ◽  
M. ROJAS ◽  
F. TOPPAN
Keyword(s):  
Dynamical Systems ◽  
Dimensional Reduction ◽  
Spinning Particles ◽  
One Dimensional ◽  
The One ◽  
M Theory

The classification of the octonionic realizations of the one-dimensional extended supersymmetries is here furnished. These are non-associative realizations which, albeit inequivalent, are put in correspondence with a subclass of the already classified associative representations for 1D extended supersymmetries. Examples of dynamical systems invariant under octonionic realizations of the extended supersymmetries are given. We cite among the others the octonionic spinning particles, the N = 8 KdV , etc. Possible applications to supersymmetric systems arising from dimensional reduction of the octonionic superstring and M-theory are mentioned.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART XIV: MORE BERNOULLI στ-SHIFT RULES

2010 ◽  
Vol 20 (08) ◽  
pp. 2253-2425 ◽  
Author(s):  
LEON O. CHUA ◽  
GIOVANNI E. PAZIENZA
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Present Article ◽  
Initial State ◽  
One Dimensional ◽  
The Past ◽  
Time Patterns ◽  
The One ◽  
Science Part

Over the past eight years, we have studied one of the simplest, yet extremely interesting, dynamical systems; namely, the one-dimensional binary Cellular Automata. The most remarkable results have been presented in a series of papers which is concluded by the present article. The final stop of our odyssey is devoted to the analysis of the second half of the 30 Bernoulli στ-shift rules, which constitute the largest among the six groups in which we classified the 256 local rules. For all these 15 rules, we present the basin-tree diagrams obtained by using each bit string with L ≤ 8 as initial state, a summary of the characteristics of their ω-limit orbits, and the space-time patterns generated from the superstring. Also, in the last section we summarize the main results we obtained by means of our "nonlinear dynamics perspective".

UNCONVENTIONAL INVERTIBLE BEHAVIORS IN REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

2008 ◽  
Vol 18 (12) ◽  
pp. 3625-3632
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZÁLEZ HERNÁNDEZ ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
SERGIO V. CHAPA VERGARA ◽  
HAROLD V. McINTOSH
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Temporal Evolution ◽  
Complete Characterization ◽  
Dimensional Case ◽  
Neighborhood Size ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
Reversible Cellular Automata ◽  
The One

Reversible cellular automata are discrete invertible dynamical systems determined by local interactions among their components. For the one-dimensional case, there are classical references providing a complete characterization based on combinatorial properties. Using these results and the simulation of every automaton by another with neighborhood size 2, this paper describes other types of invertible behaviors embedded in these systems different from the classical one observed in the temporal evolution. In particular, spatial reversibility and diagonal surjectivity are studied, and the generation of macrocells in the evolution space is analyzed.

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