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Characterizing Algebraic Invariants by Differential Radical Invariants [chapter]

Khalil Ghorbal, André Platzer
2014 Lecture Notes in Computer Science
The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable.  ...  This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it.  ...  Checking Invariant Varieties by Differential Radical Invariants (Section 3.1) The differential radical characterization allows to check for and falsify the invariance of a variety candidate.  ...

Casimir operators of Lie algebras with a nilpotent radical [article]

J.C. Ndogmo
2007 arXiv   pre-print
We show that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators.  ...  We give a different proof of this fact in the special and well-known case where the radical is abelian.  ...  Since L has a nilpotent radical, we may assume by Lemma 1 that it is algebraic. It has therefore a fundamental set of invariants that consists of rational invariants, by Lemma 2.  ...

Algebraic Solutions to the Hamilton-Jacobi Equation with the Time-Varying Hamiltonian

Yu KAWANO, Toshiyuki OHTSUKA
2013 SICE Journal of Control Measurement and System Integration
If such an ideal is found, an algebraic gradient can be obtained simply by solving a set of algebraic equations.  ...  In this paper, the HJE with coefficients belonging to meromorphic functions is considered, and its solutions with algebraic gradients are characterized in terms of commutative algebra.  ...  If an H-invariant and involutive zero-dimensional radical ideal is found, an algebraic gradient is obtained simply by solving a set of algebraic equations.  ...

On Computing Linearizing Coordinates From Symmetry Algebra [article]

Sajid Ali, Hassan Azad, Said Waqas Shah, Fazal M. Mahomed
2020 arXiv   pre-print
A characterization of the symmetry algebra of the nth order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given.  ...  The procedure is illustrated by several examples from literature.  ...  These conditions characterize the algebras and the differential equations -intrinsically. Theorem 4.1.  ...

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations [chapter]

Khalil Ghorbal, Andrew Sogokon, André Platzer
2014 Lecture Notes in Computer Science
The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities.  ...  The procedure is based on a necessary and sufficient condition that characterizes invariant conjunctions of polynomial equalities.  ...  The contributions of this paper are twofold: • It extends differential radical invariants  to obtain a characterization of invariance for algebraic sets under the flow of algebraic differential equations  ...

Page 2469 of Mathematical Reviews Vol. , Issue 91E [page]

1991 Mathematical Reviews
In particular, it is shown that the radical topological groups G of finite rank can be characterized by means of the lattice L(G) of all closed subgroups of G.  ...  A further application of the Abel transform is that it intertwines invariant differential operators on G/K and Weyl group invariant differential operators on A with constant coefficients.  ...

Page 2837 of Mathematical Reviews Vol. , Issue 83h [page]

1983 Mathematical Reviews
Finally, R denotes the Jacobson radical of Alg 2. A characterization of the radical ® of a nest algebra as an intersection of larger ideals (the diagonal ideals) has been given by J. R.  ...  J. 21 (1971/72), 887-906; MR 45 #2516], say that an algebra of operators is para-refiexive if every operator which leaves invariant all of the algebra’s invariant operator ranges is in the algebra.  ...

Page 471 of Mathematical Reviews Vol. , Issue 81B [page]

1981 Mathematical Reviews
system is characterized by its tangent Lie triple algebra.  ...  This method is a generalisation of a well- known procedure for Lie algebras, whose Casimir invariants can be found by solving systems of first-order linear partial differential equations.  ...

Page 6234 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews
He also gives a unified module-theoretic characterization of supernilpotent radicals and special radicals.  ...  The main aim of the author is to motivate and introduce the notion of a quasitriangular Hopf algebra. This notion was first introduced by  ...

Page 1272 of Mathematical Reviews Vol. 57, Issue 4 [page]

1979 Mathematical Reviews
Another result is that if M is any such class, then the upper radical a defined by M (which is a special radical) is a dual supernilpotent radical and b=a* is a dual subidempotent radical with a and b  ...  The author presents a characterization of “homogeneous systems” induced on an algebra A in a manner analogous to the classical construction of the Lie algebra of A.  ...

Page 6874 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews
The author shows that the subspace of vari- ational functions of the above space is characterized by a simple algebraic condition, namely, the condition of smoothnes of a sys- tem in the derivative variables  ...  This Darboux transformation is characterized by the fact that it connects solutions of two distinct equations.  ...

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

R Campoamor-Stursberg, S G Low
2009 Journal of Physics A: Mathematical and Theoretical
Such operators are said to generate a virtual copy of s in U(g), and allow to compute the Casimir operators of g in closed form, using the classical formulae for the invariants of s.  ...  Given a semidirect product g=sr of semisimple Lie algebras s and solvable algebras r, we construct polynomial operators in the enveloping algebra U(g) of g that commute with r and transform like the generators  ...  left-invariant 1-forms on the simply connected Lie group whose algebra is isomorphic to g allows to define an exterior differential d on g * by dω (X i , X j ) = −C k ij ω (X k ) , ω ∈ g * . (5) This coboundary  ...

Localization and ideal theory in iterated differential operator rings

Allen D Bell
1987 Journal of Algebra
In this paper we study primahty, hypercentrality, simplicity, and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings.  ...  algebra R#U(L) satisfies the second layer condition.  ...  We can now characterize the prime radical of R# U(L). Note that if R has the a.c.c. on ideals and N is the prime radical of R, then N, = (N : L).  ...

Page 631 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 54, Issue 7 [page]

1948 American Mathematical Society. Bulletin of the American Mathematical Society
Kolchin: Existence theorems connected with the Picard- Vessiot theory of homogeneous linear ordinary differential equations. It is shown how a theorem on algebraic differential equations due to J. F.  ...  Soc. vol. 48 (1940) pp. 543-544) yields two results concerning the differential equation L(y)=y+pryy*")+ --- +pay=0 (f1,---, fa in an ordinary differential field F of charactgristic 0 with algebraically  ...

Page 500 of Mathematical Reviews Vol. , Issue 85b [page]

1985 Mathematical Reviews
Lohmus (Tartu) 18 CATEGORY THEORY, HOMOLOGICAL ALGEBRA See also 01098, 18010. 85b:18001 Proceedings of the workshop held in Aarhus, June 1-15, 1983. Edited by A. Kock.  ...  Let G be a Lie algebra over a field k and let G* be its dual space. Suppose that the number of independent polynomial invariants of the coadjoint action K of G in G* is equal to the index of G [J.  ...
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