668 research outputs found
Whitney algebras and Grassmann's regressive products
Geometric products on tensor powers of an exterior
algebra and on Whitney algebras \cite{crasch} provide a rigorous version of
Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric
products and their relations with other classical operators on exterior
algebras, such as the Hodge operators and the {\it join} and {\it meet}
products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings
of tensor powers and of Whitney algebras in
terms of letterplace algebras and of their geometric products in terms of
divided powers of polarization operators. We use these encodings to provide
simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras
and of two typical classes of identities in Cayley-Grassmann algebras
An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota
We sketch the outlines of Gian Carlo Rota's interaction with the ideas that
Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as
adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota
variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney
algebra of a matroid, and finally to a resolution of the question "What,
really, was Grassmann's regressive product?". This final question is the
subject of ongoing joint work with Andrea Brini, Francesco Regonati, and
William Schmitt.
The present paper was presented at the conference "The Digital Footprint of
Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It
will appear in proceedings of that conference, to be published by Springer
Verlag.Comment: 28 page
SINGULAB - A Graphical user Interface for the Singularity Analysis of Parallel Robots based on Grassmann-Cayley Algebra
This paper presents SinguLab, a graphical user interface for the singularity
analysis of parallel robots. The algorithm is based on Grassmann-Cayley
algebra. The proposed tool is interactive and introduces the designer to the
singularity analysis performed by this method, showing all the stages along the
procedure and eventually showing the solution algebraically and graphically,
allowing as well the singularity verification of different robot poses.Comment: Advances in Robot Kinematics, Batz sur Mer : France (2008
A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization
In symbolic computing, a major bottleneck is middle expression swell.
Symbolic geometric computing based on invariant algebras can alleviate this
difficulty. For example, the size of projective geometric computing based on
bracket algebra can often be restrained to two terms, using final polynomials,
area method, Cayley expansion, etc. This is the "binomial" feature of
projective geometric computing in the language of bracket algebra.
In this paper we report a stunning discovery in Euclidean geometric
computing: the term preservation phenomenon. Input an expression in the
language of Null Bracket Algebra (NBA), by the recipe we are to propose in this
paper, the computing procedure can often be controlled to within the same
number of terms as the input, through to the end. In particular, the
conclusions of most Euclidean geometric theorems can be expressed by monomials
in NBA, and the expression size in the proving procedure can often be
controlled to within one term! Euclidean geometric computing can now be
announced as having a "monomial" feature in the language of NBA.
The recipe is composed of three parts: use long geometric product to
represent and compute multiplicatively, use "BREEFS" to control the expression
size locally, and use Clifford factorization for term reduction and transition
from algebra to geometry.
By the time this paper is being written, the recipe has been tested by 70+
examples from \cite{chou}, among which 30+ have monomial proofs. Among those
outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose
analytic proof is straightforward but very difficult symbolic computing, is
discovered to have a 3-termed elegant proof with the recipe
Coherent states and geodesics: cut locus and conjugate locus
The intimate relationship between coherent states and geodesics is pointed
out. For homogenous manifolds on which the exponential from the Lie algebra to
the Lie group equals the geodesic exponential, and in particular for symmetric
spaces, it is proved that the cut locus of the point is equal to the set of
coherent vectors orthogonal to . A simple method to calculate the
conjugate locus in Hermitian symmetric spaces with significance in the coherent
state approach is presented. The results are illustrated on the complex
Grassmann manifold.Comment: 19 pages, enlarged version, 14 pages, Latex + some macros from Revtex
+ some AMS font
D=10 super-D9-brane
Superfield equations of motion for D=10 type IIB Dirichlet super-9-brane are
obtained from the generalized action principle. The geometric equations
containing fermionic superembedding equations and constraints on the
generalized field strength of Abelian gauge field are separated from the proper
dynamical equations and are found to contain these dynamical equations among
their consequences. The set of superfield equations thus obtained involves a
Spin(1,9) group valued superfield h_\a^{~\b} whose leading component appears
in the recently obtained simplified expression for the kappa-symmetry projector
of the D9-brane. The Cayley image of this superfield coincides (on the mass
shell) with the field strength tensor of the world volume gauge field
characteristic for the Dirichlet brane. The superfield description of the
super-9-brane obtained in this manner is known to be, on the one hand, the
nonlinear (Born-Infeld) generalization of supersymmetric Yang-Mills theory and,
on the other hand, the theory of partial spontaneous breaking of D=10, N=IIB
supersymmetry down to D=10, N=1.Comment: 34 pages, LATEX. Minor corrections. References adde
Singularity Analysis of Lower-Mobility Parallel Manipulators Using Grassmann-Cayley Algebra
This paper introduces a methodology to analyze geometrically the
singularities of manipulators, of which legs apply both actuation forces and
constraint moments to their moving platform. Lower-mobility parallel
manipulators and parallel manipulators, of which some legs do not have any
spherical joint, are such manipulators. The geometric conditions associated
with the dependency of six Pl\"ucker vectors of finite lines or lines at
infinity constituting the rows of the inverse Jacobian matrix are formulated
using Grassmann-Cayley Algebra. Accordingly, the singularity conditions are
obtained in vector form. This study is illustrated with the singularity
analysis of four manipulators
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