A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finitedimensional represent...

The world-line path-integral representation of fermion propagators is discussed. Particular attention is paid to the representation of γ5, which is connected to the realization of manifest world-line supersymmetry. Contribution to Proceedings of 29th Int. Symposium on the Theory of Elementary Particles Buckow (1995) Introduction The path-integral representation of the evolution operator of dyna...

1  Abstract—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...

Let f : Gn,k−→Gm,l be any continuous map between two distinct complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of f is zero provided n,m are sufficiently large and l ≥ 2. If the degree of f is ±1, we show that (m, l) = (n, k) and f is a homotopy equivalence. Also, we prove that the image under f ∗ of elements of a set of algebra generators of H(Gm,...

In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space R8 and the spheres S4, S6. By the spin representation of G(2, 8) ⊂ Spin(8) we show that the Grassmann manifold G(2, 8) can be looked as the set of complex structures on R8. In this way, we show that G(2, 8) and CP 3 can be looked as twistor spaces of S6 and S4 respectively. Then we ...

We study the noncommutative geometry of a two-leaf Parisi{Sourlas supermanifold in Connes' formalism using di erent K-cycles over the Grassmann algebra valued functions on the supermanifold. We nd that the curvature of the trivial noncommutative vector bundle de nes in the simplest case the super Yang{Mills action coupled to a scalar eld. By considering a modi ed Dirac operator and a suitable l...

Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant app...

Two supersymmetric classical mechanical systems are discussed. Concrete realizations are obtained by supposing that the dynamical variables take values in a Grassmann algebra with two generators. The equations of motion are explicitly solved. A genuine Lie group, the supergroup, generated by supersymmetries and time translations, is found to act on the space of solutions. For each system, the s...

Let A be a unital C*-algebra. We describe K-skeleton factorizations of all invertible operators on a Hilbert C*-module HA, in particular on H = l 2, with the Fredholm index as an invariant. We then outline the isomorphisms K0(A) ∼= π2k([p]0) ∼= π2k(GL p r(A)) and K1(A) ∼= π2k+1([p]0) ∼= π2k+1(GL p r(A)) for k ≥ 0, where [p]0 denotes the class of all compact perturbations of a projection p in th...

Boolean, free and monotone cumulants as well relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean were successfully used infinite divisibility via Bercovici--Pata bijection. On other hand, recent years concept infinitesimal has been developed, together with notion which can useful context combinatorial questions. In this ...