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Review of Symbolic Software for Lie Symmetry Analysis
 CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS, VOLUME 3: NEW TRENDS IN THEORETICAL DEVELOPMENT AND COMPUTATIONAL METHODS, CHAPTER 13
, 1995
"... Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are brifley outlined. Examples illustrate the use of the symbolic software. ..."
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Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are brifley outlined. Examples illustrate the use of the symbolic software.
Algorithms for Symmetric Differential Systems
, 2001
"... Over determined systems of partial dierential equations may be studied using differentialelimination algorithms as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are eectively intractable by these methods due to the ex ..."
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Over determined systems of partial dierential equations may be studied using differentialelimination algorithms as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are eectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a large rotation group, so that there is no natural choice of term ordering in the elimination and reduction processes. This article
Differential Invariant Algebras of Lie Pseudo–Groups
, 2012
"... The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepse ..."
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Cited by 15 (9 self)
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The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepseudogroup action, wedevelop computational algorithms for locating a finite generating set of differential invariants, a complete system of recurrence relations for the differentiated invariants, and a finite system of generating differential syzygies among the generating differential invariants. In particular, if the pseudogroup acts transitively on the base manifold, then the algebra of differential invariants is shown to form a rational differential algebra with noncommuting derivations. The essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the usual symbol module associated with the infinitesimal determining system of the pseudogroup, and a new “prolonged symbol module” constructed from the symbols of the annihilators of the prolonged pseudogroup generators. Modulo low order complications, thegenerating differential invariants and differential syzygies are in onetoone correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely oncombining the movingframe approach developed inearlier papers with Gröbner basis algorithms from commutative algebra.
Rankings of Partial Derivatives
 in: W. Kuchlin, Proc. ISSAC '97 (ACM
, 1998
"... Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives a ..."
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Let m be a nonnegative integer, n a positive integer, N = f0; 1; 2; :::g and Nn = f1; : : : ; ng. A ranking is a total order of N m Nn such that (a; i) (b; j) implies (a + c; i) (b + c; j) for a, b, c 2 N m and i; j 2 Nn . We describe an approach to such rankings and a theorem which gives an explicit construction of an arbitrary ranking using nite real data. The case n = 1 corresponds to termorderings of monomials which are crucial inputs for Buchberger's Grobner Basis algorithm for polynomial rings. The case n > 1 corresponds to rankings of partial derivatives which are inputs in algorithms in dierential algebra and Buchberger's algorithm for free modules over polynomial rings. A subclass of such rankings determined by nite integer data is given which is suÆcient for eective implementation of such rankings. This has been implemented in the symbolic language Maple. The rankings considered by Riquier are a special case of those considered here. Examples including appl...
Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems
, 1999
"... We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an alg ..."
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We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an algorithmic fashion. The resulting reduced involutive form (rif 0 form) produced by our rif 0 algorithm is a generalization of the classical form of Riquier and Janet, and that of Cauchy Kovalevskaya. We weaken the assumption of linearity in the highest derivatives in those approaches to allow for systems which are nonlinear in their highest derivatives. A new formal development of Riquier's theory is given, with proofs, modeled after those in Grobner Basis Theory. For the nonlinear theory, the concept of relative Riquier Bases is introduced. This allows for the easy extension of ideas from the linear to the nonlinear theory. The essential idea is that an arbitrary nonlinear system can ...
Cartan structure of infinite Lie pseudogroups
 IN: GEOMETRICAL APPROACHES TO DIFFERENTIAL EQUATIONS, P.J. VASSILIOU AND I.G
, 2000
"... Since Chevalley's seminal work [12], the definition of Lie group has been universally agreed. Namely, a Lie group G is an analytic manifold G on which ..."
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Since Chevalley's seminal work [12], the definition of Lie group has been universally agreed. Namely, a Lie group G is an analytic manifold G on which
The MAPLE Package “Janet”: I. Polynomial Systems
 In the Proccedings of Computer Algebra in Scientific Computing CASC 2003
, 2003
"... Abstract. The MAPLE package “Janet ” ⋆ comes in two parts, the first dealing with polynomials and commutative algebra, the second with linear PDEs. Here the first part, called “Involutive”, is described. Amongst others it contains a MAPLE and a C++ implementation of the involutive technique for pol ..."
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Cited by 12 (7 self)
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Abstract. The MAPLE package “Janet ” ⋆ comes in two parts, the first dealing with polynomials and commutative algebra, the second with linear PDEs. Here the first part, called “Involutive”, is described. Amongst others it contains a MAPLE and a C++ implementation of the involutive technique for polynomial modules as an alternative for conventional Gröbner basis techniques. 1
MaurerCartan Forms and the Structure of Lie Pseudo–Groups
, 2005
"... This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations fo ..."
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Cited by 11 (8 self)
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This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudogroup. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudogroup on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudogroup actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.
Fast Differential Elimination in C: The CDiffElim Environment
, 2000
"... We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in different ..."
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Cited by 9 (7 self)
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We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include lowlevel memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a lowlevel C implementation, seem much more difficult to implement in highlevel general purpose computer algebra systems. A differential elimination algorithm written in this environment is applied to the determination of symmetry properties of classes of n+1dimensional coupled nonlinear partial differential equations of form iut+r2u+ i a(t)jxj2 + b(t) \Delta x + c(t) + djuj 4n j u = 0; where u is an mcomponent vectorvalued function. The resulting systems of differential equations for the symmetries have been made available on the web, to be used as benchmark systems for other researchers. The new differential elimination algorithm in C, runs on the test suite an average of 400 times faster than our RifSimp algorithm in Maple.
Geometric Completion of Differential Systems using NumericSymbolic Continuation
 SIGSAM Bulletin
, 2002
"... Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and underdetermined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases ..."
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Cited by 8 (6 self)
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Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and underdetermined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases the application of exact or numerical integration methods. Motivated to avoid expression swell of pure symbolic approaches and with the desire to handle systems with approximate coefficients, we propose the use of homotopy continuation methods to perform the differentialelimination process on such nonsquare systems. Examples such as the classic index 3 Pendulum illustrate the new procedure. Our approach uses slicing by random linear subspaces to intersect its jet components in finitely many points. Generation of enough generic points enables irreducible jet components of the differential system to be interpolated. 1