scholarly journals Free energy of the three-state τ 2 ( t q ) model as a product of elliptic functions

2006 ◽  
Vol 462 (2075) ◽  
pp. 3323-3341
Author(s):  
R.J Baxter
Keyword(s):  
Free Energy ◽  
Potts Model ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions ◽  
Chiral Potts Model ◽  
Energy Problem ◽  
Q Model

We show that the free energy of the three-state model can be expressed as products of Jacobi elliptic functions, the arguments being those of an hyperelliptic parametrization of the associated chiral Potts model. This is the first application of such a parametrization to the N -state chiral Potts free-energy problem for N >2.

Minimum Energy Control of a Unicycle Model Robot

2021 ◽  
Author(s):  
Youngjin Kim ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Minimum Energy ◽  
Kinematic Model ◽  
Elliptic Functions ◽  
Terminal Point ◽  
Jacobi Elliptic Functions ◽  
Differential Drive ◽  
Terminal Constraints ◽  
Differential Drive Robot ◽  
Point To Point

Abstract Point-to-point path planning for a kinematic model of a differential-drive wheeled mobile robot (WMR) with the goal of minimizing input energy is the focus of this work. An optimal control problem is formulated to determine the necessary conditions for optimality and the resulting two point boundary value problem is solved in closed form using Jacobi elliptic functions. The resulting nonlinear programming problem is solved for two variables and the results are compared to the traditional shooting method to illustrate that the Jacobi elliptic functions parameterize the exact profile of the optimal trajectory. A set of terminal constraints which lie on a circle in the first quadrant are used to generate a set of optimal solutions. It is noted that for maneuvers where the angle of the vector connecting the initial and terminal point is greater than a threshold, which is a function of the radius of the terminal constraint circle, the robot initially moves into the third quadrant before terminating in the first quadrant. The minimum energy solution is compared to two other optimal control formulations: (1) an extension of the Dubins vehicle model where the constant linear velocity of the robot is optimized for and (2) a simple turn and move solution, both of whose optimal paths lie entirely in the first quadrant. Experimental results are used to validate the optimal trajectories of the differential-drive robot.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

2012 ◽  
Vol 08 (05) ◽  
pp. 1257-1270
Author(s):  
M. A. GÓMEZ-MOLLEDA ◽  
JOAN-C. LARIO
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Elliptic Functions ◽  
Number Fields ◽  
Jacobi Elliptic Functions ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Finite Sums ◽  
Finite Sum

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

scholarly journals Optimal Control on the Rotation Group SO (3)

2012 ◽  
Vol 28 (2) ◽  
pp. 321-328
Author(s):  
CLAUDIU C. REMSING ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic Functions ◽  
Rotation Group ◽  
Jacobi Elliptic Functions ◽  
Hamilton Equations ◽  
Extremal Curve

A typical left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived in a simple and elegant manner. These equations are then explicitly integrated by Jacobi elliptic functions.

scholarly journals He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions

2020 ◽  
Vol 39 (4) ◽  
pp. 1216-1223 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Luis Manuel Palacios-Pineda ◽  
Isaac H Jiménez-Cedeño ◽  
Oscar Martínez-Romero ◽  
Daniel Olvera Trejo
Keyword(s):  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Numerical Results ◽  
Jacobi Elliptic Functions ◽  
Analytical Expression ◽  
Analytical Frequency

In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.

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