3,552 research outputs found
Rational invariants of even ternary forms under the orthogonal group
In this article we determine a generating set of rational invariants of
minimal cardinality for the action of the orthogonal group on
the space of ternary forms of even degree . The
construction relies on two key ingredients: On one hand, the Slice Lemma allows
us to reduce the problem to dermining the invariants for the action on a
subspace of the finite subgroup of signed permutations. On the
other hand, our construction relies in a fundamental way on specific bases of
harmonic polynomials. These bases provide maps with prescribed
-equivariance properties. Our explicit construction of these
bases should be relevant well beyond the scope of this paper. The expression of
the -invariants can then be given in a compact form as the
composition of two equivariant maps. Instead of providing (cumbersome) explicit
expressions for the -invariants, we provide efficient algorithms
for their evaluation and rewriting. We also use the constructed
-invariants to determine the -orbit locus and
provide an algorithm for the inverse problem of finding an element in
with prescribed values for its invariants. These are
the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application,
refinement of Definition 3.1. To appear in "Foundations of Computational
Mathematics
Rational, Replacement, and Local Invariants of a Group Action
The paper presents a new algorithmic construction of a finite generating set
of rational invariants for the rational action of an algebraic group on the
affine space. The construction provides an algebraic counterpart of the moving
frame method in differential geometry. The generating set of rational
invariants appears as the coefficients of a Groebner basis, reduction with
respect to which allows to express a rational invariant in terms of the
generators. The replacement invariants, introduced in the paper, are tuples of
algebraic functions of the rational invariants. Any invariant, whether
rational, algebraic or local, can be can be rewritten terms of replacement
invariants by a simple substitution.Comment: 37 page
Polynomial Time Nondimensionalisation of Ordinary Differential Equations via their Lie Point Symmetries
Lie group theory states that knowledge of a -parameters solvable group of
symmetries of a system of ordinary differential equations allows to reduce by
the number of equation. We apply this principle by finding dilatations and
translations that are Lie point symmetries of considered ordinary differential
system. By rewriting original problem in an invariant coordinates set for these
symmetries, one can reduce the involved number of parameters. This process is
classically call nondimensionalisation in dimensional analysis. We present an
algorithm based on this standpoint and show that its arithmetic complexity is
polynomial in input's size
Differential invariants of a Lie group action: syzygies on a generating set
Given a group action, known by its infinitesimal generators, we exhibit a
complete set of syzygies on a generating set of differential invariants. For
that we elaborate on the reinterpretation of Cartan's moving frame by Fels and
Olver (1999). This provides constructive tools for exploring algebras of
differential invariants.Comment: Journal of Symbolic Computation (2008
Differential Invariants of Conformal and Projective Surfaces
We show that, for both the conformal and projective groups, all the
differential invariants of a generic surface in three-dimensional space can be
written as combinations of the invariant derivatives of a single differential
invariant. The proof is based on the equivariant method of moving frames.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
The conjugacy problem in extensions of Thompson's group F
The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-016-1403-9We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Gonçalves in [4] showing that F has property R8, and which can be extended to show that Thompson’s group T also has property R8.Peer ReviewedPostprint (author's final draft
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