Results 1  10
of
20
Configuration Controllability of Simple Mechanical Control Systems
 SIAM Journal on Control and Optimization
, 1995
"... In this paper we present a definition of "configuration controllability" for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is also derived. ..."
Abstract

Cited by 97 (18 self)
 Add to MetaCart
(Show Context)
In this paper we present a definition of "configuration controllability" for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is also derived.
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
(Show Context)
This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Hopf Downup Algebras
"... Given a field k and elements α, β, γ ∈ k, Benkart and Roby [BR] defined the downup algebra A(α, β, γ) to be the algebra generated by two generators u and d subject to the two relations: d2u = αdud+ βud2 + γd, ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Given a field k and elements α, β, γ ∈ k, Benkart and Roby [BR] defined the downup algebra A(α, β, γ) to be the algebra generated by two generators u and d subject to the two relations: d2u = αdud+ βud2 + γd,
RESTRICTED SIMPLE LIE ALGEBRAS AND THEIR INFINITESIMAL DEFORMATIONS
, 2007
"... Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Rest ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. In the first two sections, we review the BlockWilsonPremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Restricted Lie algebras We fix a field F of characteristic p> 0 and we denote with Fp the prime field with p elements. All the Lie algebras that we will consider are of finite dimension over F. We are interested in particular class of Lie algebras, called restricted (or pLie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a pLie algebra) if there exits a map (called pmap), [p] : L → L, x ↦ → x [p], which verifies the following conditions: (1) ad(x [p]) = ad(x) [p] for every x ∈ L. (2) (αx)[p] = αpx [p] for every x ∈ L and every α ∈ F. (3) (x0 + x1) [p] = x [p] 0 + x[p] 1 + ∑ p−1 i=1 si(x0, x1) for every x, y ∈ L, where the element si(x0, x1) ∈ L is defined by si(x0, x1) = − 1 ∑ adxu(1) ◦ adxu(2) ◦ · · · ◦ adxu(p−1)(x1), r u the summation being over all the maps u: [1, · · · , p − 1] → {0, 1} taking rtimes the value 0. Example. (1) Let A an associative Falgebra. Then the Lie algebra DerFA of Fderivations of A is a restricted Lie algebra with respect to the pmap D ↦ → Dp: = D ◦ · · · ◦ D. (2) Let G a group scheme over F. Then the Lie algebra Lie(G) associated to G is a restricted Lie algebra with respect to the pmap given by the differential of the homomorphism G → G, x ↦ → xp: = x ◦ · · · ◦ x. One can naturally ask when a FLie algebra can acquire the structure of a restricted Lie algebra and how many such structures there can be. The following criterion of Jacobson answers to that question. Proposition 1.2 (Jacobson). Let L be a Lie algebra over F. Then (1) It is possible to define a pmap on L if and only if, for every element x ∈ L, the pth iterate of ad(x) is still an inner derivation. (2) Two such pmaps differ by a semilinear map from L to the center Z(L) of L, that is a map f: L → Z(L) such that f(αx) = α p f(x) for every x ∈ L and α ∈ F.
GEOMETRIC STRUCTURES MODELED ON AFFINE HYPERSURFACES AND GENERALIZATIONS OF THE EINSTEIN WEYL AND AFFINE HYPERSPHERE EQUATIONS
, 909
"... Abstract. An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsionfree connection representing the projective structure the completely tracefree part of the covariant derivative of any metric representing the conformal st ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsionfree connection representing the projective structure the completely tracefree part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a nondegenerate cooriented hypersurface in flat affine space by its second fundamental form together with either the projective structure induced by the affine normal or that induced by the conormal Gauß map. There are proposed notions of Einstein equations for AH structures which for Weyl structures specialize to the usual Einstein Weyl equations and such that the AH structure induced on a nondegenerate cooriented affine hypersurface is Einstein if and only if the hypersurface is an affine hypersphere. It is shown that a convex flat projective structure admits a metric with which it generates an Einstein AH structure, and examples are constructed on mean curvature zero Lagrangian submanifolds of certain paraKähler manifolds. The rough classification of Riemannian Einstein Weyl structures by properties of the scalar curvature is extended to this setting. Known estimates on the growth of the cubic form of an affine hypersphere are partly generalized. The
On the group of Lieorthogonal operators on a Lie algebra, Methods Funct. Anal. Topology 17
, 2011
"... Abstract. Finite dimensional Lie algebras over the field of complex numbers with a linear operator T : The group of such nondegenerative linear operators on L is considered. Some properties of this group and its relations with the group Aut(L) in the general linear group GL(L) are described. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Finite dimensional Lie algebras over the field of complex numbers with a linear operator T : The group of such nondegenerative linear operators on L is considered. Some properties of this group and its relations with the group Aut(L) in the general linear group GL(L) are described.