Related papers: Lie group computation of finite difference schemes

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference…

The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension…

By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear…

I prove the "folklore" result that the functional equation for a Lie group homomorphism can be solved by solving the corresponding differential equation.

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion…

In this paper we propose a (non-linear) smoothing algorithm for group-affine observation systems, a recently introduced class of estimation problems on Lie groups that bear a particular structure. As most non-linear smoothing methods, the…

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a…

We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the highest weight of the representation. The proof uses equivariant cohomology techniques,…

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities.…

Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential…

A system of PDE describing bilayers amphiphilic membranes is studied by Lie group analysis. This algorithmic approach allows us to show all the symmetries of the system, to determine all possible symmetry reductions, to recover the…

This paper uses Lie symmetry methods to calculate certain expectations for a large class of It\^{o} diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form $E_x(e^{-\lambda…

For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…

We show how one can associate to a given class of finite type G-structures a classifying Lie algebroid. The corresponding Lie groupoid gives models for the different geometries that one can find in the class, and encodes also the different…

The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…

We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois…

This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…

We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the…