scholarly journals A Integrable Generalized Super-NLS-mKdV Hierarchy, Its Self-Consistent Sources, and Conservation Laws

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Hanyu Wei ◽  
Tiecheng Xia
Keyword(s):  
Conservation Laws ◽  
Lie Superalgebra ◽  
Hamiltonian Form ◽  
Self Consistent ◽  
Set Up ◽  
Mkdv Hierarchy

A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra B(0,1); the resulting supersoliton hierarchy is put into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistent sources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.

scholarly journals Entropic measure unveils country competitiveness and product specialization in the World trade web

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Gianluca Teza ◽  
Michele Caraglio ◽  
Attilio L. Stella
Keyword(s):  
Iterative Scheme ◽  
Entropy Function ◽  
Bipartite Networks ◽  
Complexity Measures ◽  
Wide Range ◽  
Self Consistent ◽  
Set Up ◽  
Technical Features ◽  
Country Competitiveness ◽  
Entropic Measure

AbstractWe show how the Shannon entropy function can be used as a basis to set up complexity measures weighting the economic efficiency of countries and the specialization of products beyond bare diversification. This entropy function guarantees the existence of a fixed point which is rapidly reached by an iterative scheme converging to our self-consistent measures. Our approach naturally allows to decompose into inter-sectorial and intra-sectorial contributions the country competitivity measure if products are partitioned into larger categories. Besides outlining the technical features and advantages of the method, we describe a wide range of results arising from the analysis of the obtained rankings and we benchmark these observations against those established with other economical parameters. These comparisons allow to partition countries and products into various main typologies, with well-revealed characterizing features. Our methods have wide applicability to general problems of ranking in bipartite networks.

scholarly journals A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws

2019 ◽  
Vol 19 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Sergey V. Dolgov
Keyword(s):  
Conservation Laws ◽  
Tensor Decomposition ◽  
Time Discretization ◽  
Approximation Accuracy ◽  
Stochastic Simulation Algorithms ◽  
Right Hand ◽  
Large Linear System ◽  
Linear Ode ◽  
Set Up ◽  
The Right

AbstractWe propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes a reduced TT model of the solution, but in contrast to traditional offline-online reduction schemes, solving the original large problem is never required. Instead, the method solves a sequence of reduced Galerkin problems, which can be set up efficiently due to the TT decomposition of the right-hand side. The reduced system allows a fast estimation of the time discretization error, and hence adaptation of the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the Euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.

scholarly journals Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yu-Qing Li ◽  
Bao-Shu Yin
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Integrable Lattice ◽  
Self Consistent ◽  
Matrix Spectral Problem

A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

Modeling the Systematic Uncertainty of Photogrammetric Tracking of Human Motion

1997 ◽  
Vol 119 (2) ◽  
pp. 358-361 ◽  
Author(s):  
S. B. Bortolami ◽  
P. O. Riley ◽  
D. E. Krebs
Keyword(s):  
Uncertainty Modeling ◽  
Human Locomotion ◽  
Human Motion ◽  
Tracking Data ◽  
Total Uncertainty ◽  
Rotation Matrices ◽  
Self Consistent ◽  
Consistent Manner ◽  
Set Up ◽  
Derivatives Of

We address bias errors of photogrammetric tracking of four SELSPOT-II® cameras using active marker photogrammetry in a 2 m × 2 m × 2 m viewing volume for human locomotion measurements. We present uncertainty modeling regarding the first stage of equipment set up, which provides the camera frame to global frame rotation matrices and the distances among cameras. We also characterize the uncertainty due to the camera distortions of the bare system as compared to published performances achieved with a camera correction procedure. The particular approach is to qualify performances of photogrammetric tracking during routine operation and to identify the nature and magnitude of the uncertainty due to equipment set up and camera distortions as part of the total uncertainty in a self-consistent manner. We found that uncertainty of the camera frame to global frame rotation matrices produced rotation of the image and uncorrected camera hardware uncertainty produced dilatation or compression of the image twice the magnitude of that seen with camera correction. However, camera resolution remains as an equally important factor limiting the accuracy of photogrammetric tracking that can not be easily reduced numerically. In conclusion, the analysis elucidates how uncertainty propagates to numerical derivatives of the tracking data and prepares the groundwork for future development.

Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

2015 ◽  
Vol 5 (3) ◽  
pp. 256-272 ◽  
Author(s):  
Huanhe Dong ◽  
Kun Zhao ◽  
Hongwei Yang ◽  
Yuqing Li
Keyword(s):  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Lie Superalgebra ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Scientific Applications ◽  
Soliton Theory ◽  
De Vries ◽  
Pair Method ◽  
Mkdv Hierarchy

AbstractMuch attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

SOME CONSIDERATIONS REGARDING THE PRINCIPLE OF PHASE INVARIANCE

1955 ◽  
Vol 33 (8) ◽  
pp. 436-440
Author(s):  
F. A. Kaempffer
Keyword(s):  
Conservation Laws ◽  
Complex Field ◽  
Conservation Of Energy ◽  
Self Consistent ◽  
Energy And Momentum ◽  
Ether Theories

Taking the view that "particles" are in fact excitations of the motion of an all-pervading medium (or "ether"), it is shown that the conservation laws characterizing the ether, which are different from the well-known laws of conservation of energy and momentum, flow from a single principle, the principle of phase invariance, provided a complex field is used to describe the ether. There are at least two different self-consistent types of Lorentz-invariant ether theories which satisfy the principle of phase invariance.

Sign in / Sign up

Export Citation Format

Share Document