51 research outputs found
Double bracket structures on Poisson manifolds
On a Poisson manifold endowed with a Riemannian metric we will construct a
vector field that generalizes the double bracket vector field defined on
semi-simple Lie algebras. On a regular symplectic leaf we will construct a
generalization of the normal metric such that the above vector field restricted
to the symplectic leaf is a gradient vector field with respect to this metric
Newton Algorithm on Constraint Manifolds and the 5-electron Thomson problem
We give a description of numerical Newton algorithm on a constraint manifold
using only the ambient coordinates (usually Euclidean coordinates) and the
geometry of the constraint manifold. We apply the numerical Newton algorithm on
a sphere in order to find the critical configurations of the 5-electron Thomson
problem. As a result, we find a new critical configuration of a regular
pentagonal type. We also make an analytical study of the critical
configurations found previously and determine their nature using Morse-Bott
theory. Last section contains an analytical study of critical configurations
for Riesz s-energy of 5-electron on a sphere and their bifurcation behavior is
pointed out
A note on stability of nongeneric equilibria for an underwater vehicle
We study the Lyapunov stability of a family of nongeneric equilibria with
spin for underwater vehicles with noncoincident centers. The nongeneric
equilibria belong to singular symplectic leaves that are not characterized as a
preimage o a regular value of the Casimir functions. We find an invariant
submanifold such that the nongeneric equilibria belong to a preimage of a
regular value that involves sub-Casimir functions. We obtain results for
nonlinear stability on this invariant submanifold
The stability problem and special solutions for the 5-components Maxwell-Bloch equations
For the 5-components Maxwell-Bloch system the stability problem for the
isolated equilibria is completely solved. Using the geometry of the symplectic
leaves, a detailed construction of the homoclinic orbits is given. Studying the
problem of invariant sets for the system we discover a rich family of periodic
solutions in explicit form
Hessian Operators on Constraint Manifolds
On a constraint manifold we give an explicit formula for the Hessian matrix
of a cost function that involves the Hessian matrix of a prolonged function and
the Hessian matrices of the constraint functions. We give an explicit formula
for the case of the orthogonal group by using only Euclidean
coordinates on . An optimization problem on is
completely carried out. Its applications to nonlinear stability problems are
also analyzed
Equivalence of energy methods in stability theory
We will prove the equivalence of three methods, the so called energy methods,
for establishing the stability of an equilibrium point for a dynamical system.
We will illustrate by examples that this result simplifies enormously the
amount of computations especially when the stability can not be decided with
one of the three methods.Comment: 10 pages, no figures, minnor correction
Steepest descent algorithm on orthogonal Stiefel manifolds
Considering orthogonal Stiefel manifolds as constraint manifolds, we give an
explicit description of a set of local coordinates that also generate a basis
for the tangent space in any point of the orthogonal Stiefel manifolds. We show
how this construction depends on the choice of a submatrix of full rank.
Embedding a gradient vector field on an orthogonal Stiefel manifold in the
ambient space, we give explicit necessary and sufficient conditions for a
critical point of a cost function defined on such manifolds. We explicitly
describe the steepest descent algorithm on the orthogonal Stiefel manifold
using the ambient coordinates and not the local coordinates of the manifold. We
point out the dependence of the recurrence sequence that defines the algorithm
on the choice of a full rank submatrix. We illustrate the algorithm in the case
of Brockett cost functions
A New Training Method for Feedforward Neural Networks Based on Geometric Contraction Property of Activation Functions
We propose a new training method for a feedforward neural network having the
activation functions with the geometric contraction property. The method
consists of constructing a new functional that is less nonlinear in comparison
with the classical functional by removing the nonlinearity of the activation
function from the output layer. We validate this new method by a series of
experiments that show an improved learning speed and better classification
error
Asymptotic Stability of Dissipated Hamilton-Poisson Systems
We will further develop the study of the dissipation for a Hamilton-Poisson
system introduced in \cite{2}. We will give a tensorial form of this
dissipation and show that it preserves the Hamiltonian function but not the
Poisson geometry of the initial Hamilton-Poisson system. We will give precise
results about asymptotic stabilizability of the stable equilibria of the
initial Hamilton-Poisson system
Openness and convexity for momentum maps
The purpose of this paper is finding the essential attributes underlying the
convexity theorems for momentum maps. It is shown that they are of topological
nature; more specifically, we show that convexity follows if the map is open
onto its image and has the so called local convexity data property. These
conditions are satisfied in all the classical convexity theorems and hence they
can, in principle, be obtained as corollaries of a more general theorem that
has only these two hypotheses. We also prove a generalization of the
"Lokal-global-Prinzip" that only requires the map to be closed and to have a
normal topological space as domain, instead of using a properness condition.
This allows us to generalize the Flaschka-Ratiu convexity theorem to
non-compact manifolds.Comment: 25 pages, 2 figures. Updated version with added details and minor
correction
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