scholarly journals TRI–HAMILTONIAN VECTOR FIELDS, SPECTRAL CURVES AND SEPARATION COORDINATES

2002 ◽  
Vol 14 (10) ◽  
pp. 1115-1163 ◽  
Author(s):  
L. DEGIOVANNI ◽  
G. MAGNANO
Keyword(s):  
Vector Fields ◽  
Geometric Approach ◽  
Casimir Function ◽  
Bivariate Polynomials ◽  
Spectral Curves ◽  
Hamiltonian Vector Fields ◽  
Lax Equations ◽  
Symplectic Leaves ◽  
The Common ◽  
Inverse Procedure

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

scholarly journals Hamiltonian Vector Fields on Weil Bundles

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Norbert Mahoungou Moukala ◽  
Basile Guy Richard Bossoto
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Fields

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

scholarly journals A COMPARISON BETWEEN TWO OLS-BASED APPROACHES TO ESTIMATING URBAN MULTIFRACTAL PARAMETERS

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850019 ◽  
Author(s):  
LIN-SHAN HUANG ◽  
YAN-GUANG CHEN
Keyword(s):  
Ordinary Least Squares ◽  
Regression Method ◽  
Parameters Estimation ◽  
Spectral Curves ◽  
Fractal Parameters ◽  
Multifractal Theory ◽  
Scale Range ◽  
The Common ◽  
Urban Problems ◽  
Parameter Values

Multifractal theory provides a new spatial analytical tool for urban studies, but many basic problems remain to be solved. Among various pending issues, the most significant one is how to obtain proper multifractal dimension spectrums. If an algorithm is improperly used, the parameter spectrums will be abnormal. This paper is devoted to investigating two ordinary least squares (OLS)-based approaches for estimating urban multifractal parameters. Using empirical study and comparative analysis, we demonstrate how to utilize the adequate linear regression to calculate multifractal parameters. The OLS regression analysis has two different approaches. One is that the intercept is fixed to zero, and the other is that the intercept is not limited. The results of comparative study show that the zero-intercept regression yields proper multifractal parameter spectrums within certain scale range of moment order, while the common regression method often leads to abnormal multifractal parameter values. A conclusion can be reached that fixing the intercept to zero is a more advisable regression method for multifractal parameters estimation, and the shapes of spectral curves and value ranges of fractal parameters can be employed to diagnose urban problems. This research is helpful for scientists to understand multifractal models and apply a more reasonable technique to multifractal parameter calculations.

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