Related papers: Lie families: theory and applications
In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of…
Systems of nonlinear ordinary differential equations are constructed, for which the general solution is algebraically expressed in terms of a finite number of particular solutions. Expressions of that type are called the nonlinear…
A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use…
Our paper illustrates how the theory of Lie systems allows recovering known results and provide new examples of piecewise deterministic processes with phase-type jumps for which the corresponding first-time passage problems may be solved…
This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we…
A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised…
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, a so-called (nonlinear) superposition rule of a finite number of particular solutions…
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
Default logic can be regarded as a mechanism to represent families of belief sets of a reasoning agent. As such, it is inherently second-order. In this paper, we study the problem of representability of a family of theories as the set of…
We define and analyse the properties of contact Lie systems, namely systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of…
We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which…
In a major advance and simplification of this field, we show that A Local Resolution of the Problem of Time - also viewable as A Local Theory of Background Independence - can at the classical level be described solely by of Lie's…
Zalcman's Lemma makes significant applications in normal families, complex dynamics and related problems in complex analysis. In the present paper, we are devoted to generalizing the classical Zalcman's lemma to complex Lie groups by means…
In this paper, we obtain sufficient conditions for the permanence of a family of nonautonomous systems of delay differential equations. This family includes structured models from mathematical biology, with either discrete or distributed…
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher…
In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…