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It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced…
These lecture notes provide an introduction to the theory and application of symmetry methods for ordinary differential equations, building on minimal prerequisites. Their primary purpose is to enable a quick and self-contained approach for…
Answering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model…
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general cubic algebra and we present specific…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
In this paper, we focus on homogeneous spaces which are constructed from two strongly isotropy irreducible spaces, and prove that any geodesic orbit metric on these spaces is naturally reductive.
We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This…
Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are…
We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations. For an autonomous first-order algebraic ordinary difference equations, we give an upper bound for the degrees…
In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--R\"{o}hrle, the usual notion of $G$-complete reducibility can be…
This paper explores the problem of analytically approximating the orbital state for a subset of orbits in a rotating potential with oblateness and ellipticity perturbations. This is done by isolating approximate differential equations for…
A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the…
In this paper we examine various properties/constructions which are known for reductive groups and we do some experiments to see to what extent they generalize to symmetric spaces.
In this paper we have investigated enumeration orders of elements of r.e. sets enumerated by means of Turing machines. We have defined a reducibility based on enumeration orders named "Enumeration Order Reducibility" on computable functions…
We describe a natural generalization of irreducibility in order lattices with arbitrary metrics. We analyse the special cases of valuation metrics and more general metrics for lattices. This article is mainly based on a part of the author's…
Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics are determined by a Lagrangian or Hamiltonian function. For hybrid systems which possess a continuous-time dynamics determined by a…
We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly…