Folds on the Solution Manifold of a Parametrized Equation

1986 ◽  
Vol 23 (4) ◽  
pp. 693-706 ◽  
Author(s):  
James P. Fink ◽  
Werner C. Rheinboldt
Keyword(s):  
Solution Manifold

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.

SOLUTION OF THE YANG–BAXTER SYSTEM FOR QUANTUM DOUBLES

1999 ◽  
Vol 14 (19) ◽  
pp. 3029-3058 ◽  
Author(s):  
LADISLAV HLAVATÝ ◽  
LIBOR ŠNOBL
Keyword(s):  
Large Class ◽  
Discrete Symmetries ◽  
Complete List ◽  
Dimension 2 ◽  
Solution Manifold

The Yang–Baxter system related to quantum doubles is introduced and a large class of both continuous and discrete symmetries of the solution manifold are determined. A strategy for solution of the system based on the symmetries is suggested and accomplished in dimension 2. A complete list of invertible solutions of the system is presented.

Generalized Polar Decompositions in Matrix Exponential

2011 ◽  
Vol 130-134 ◽  
pp. 3023-3026
Author(s):  
Yi Min Tian ◽  
Ao Zhang
Keyword(s):  
Differential Equation ◽  
Lie Group ◽  
Analytic Solution ◽  
Matrix Exponential ◽  
Group Method ◽  
Lie Group Method ◽  
The Matrix ◽  
Numeric Solution ◽  
Basic Ideas ◽  
Solution Manifold

Matrix exponential computstion is a difficulty thing when the order of the matrix get big and big after discretion. When we use Lie group method to get numeric solution of a differential equation, we often face this problem.Li group method is a kind of prosperous method, its basic ideas is to keep the numeric solution in a manifold which is less than the Euclid space while bigger than the analytic solution manifold, so we can get more exact numeric solution than other method. So we discussed the generalized polar decompositions method for matrix exponential.

CLASSIFICATION OF THE SPATIAL EQUILIBRIA OF THE CLAMPED ELASTICA: NUMERICAL CONTINUATION OF THE SOLUTION SET

2004 ◽  
Vol 14 (04) ◽  
pp. 1223-1239 ◽  
Author(s):  
MICHAEL E. HENDERSON ◽  
SÉBASTIEN NEUKIRCH
Keyword(s):  
Boundary Conditions ◽  
Solution Set ◽  
Numerical Continuation ◽  
Center Line ◽  
Symmetry Properties ◽  
Equilibrium Configurations ◽  
Continuation Algorithm ◽  
Parameter Values ◽  
Solution Manifold

We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it using a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconnected layers. We show that the border set of the different layers is the well-known solution set of buckled rings.

scholarly journals A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

2021 ◽  
Author(s):  
Gitta Kutyniok ◽  
Philipp Petersen ◽  
Mones Raslan ◽  
Reinhold Schneider
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Networks ◽  
Reduced Basis ◽  
Partial Differential ◽  
Approximation Rates ◽  
Solution Maps ◽  
Low Dimensionality ◽  
Solution Manifold

AbstractWe derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.

The Sadovskii vortex in strain

2017 ◽  
Vol 825 ◽  
pp. 479-501
Author(s):  
Daniel V. Freilich ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Point Vortex ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Vortex Patch ◽  
Bifurcation Points ◽  
General Flow ◽  
The Family ◽  
Hollow Vortex ◽  
Two Parameters ◽  
Solution Manifold

The point vortex is the simplest model of a two-dimensional vortex with non-zero circulation. The limitations introduced by its lack of core structure have been remedied by using desingularizations such as vortex patches and vortex sheets. We investigate steady states of the Sadovskii vortex in strain, a canonical model for a vortex in a general flow. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. We recover previously known limiting cases of the vortex patch and hollow vortex, and examine the bifurcations away from these families. The result is a solution manifold spanned by two parameters. The addition of the vortex sheet to the vortex patch solutions immediately leads to splits in the solution manifold at certain bifurcation points. The more circular elliptical family remains attached to the family with a single pinch-off, and this family extends all the way to the simpler solution branch for the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, but these families do not exist in the vortex sheet limit.

scholarly journals State space analysis of timing: exploiting task redundancy to reduce sensitivity to timing

2012 ◽  
Vol 107 (2) ◽  
pp. 618-627 ◽  
Author(s):  
Rajal G. Cohen ◽  
Dagmar Sternad
Keyword(s):  
Central Nervous System ◽  
Task Analysis ◽  
Performance Task ◽  
Timing Accuracy ◽  
State Space Analysis ◽  
Skilled Performance ◽  
Practice Performance ◽  
Extended Practice ◽  
Solution Manifold

Timing is central to many coordinated actions, and the temporal accuracy of central nervous system commands presents an important limit to skilled performance. Using target-oriented throwing in a virtual environment as an example task, this study presents a novel analysis that quantifies contributions of timing accuracy and shaping of hand trajectories to performance. Task analysis reveals that the result of a throw is fully determined by the projectile position and velocity at release; zero error can be achieved by a manifold of position and velocity combinations (solution manifold). Four predictions were tested. 1) Performers learn to release the projectile closer to the optimal moment for a given arm trajectory, achieving timing accuracy levels similar to those reported in other timing tasks (∼10 ms). 2) Performers develop a hand trajectory that follows the solution manifold such that zero error can be achieved without perfect timing. 3) Skilled performers exploit both routes to improvement more than unskilled performers. 4) Long-term improvement in skilled performance relies on continued optimization of the arm trajectory as timing limits are reached. Average and skilled subjects practiced for 6 and 15 days, respectively. In 6 days, both timing and trajectory alignment improved for all subjects, and skilled subjects showed an advantage in timing. With extended practice, performance continued to improve due to continued shaping of the trajectory, whereas timing accuracy reached an asymptote at 9 ms. We conclude that skilled subjects first maximize timing accuracy and then optimize trajectory shaping to compensate for intrinsic limitations of timing accuracy.

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