Albert T. Lundell

Nilpotent bases for distributions and control systems

Abstract Let V(M) be the Lie algebra (infinite dimensional) of real analytic vector fields on the n-dimensional manifold M. Necessary conditions that a real analytic k-dimensional distibution on M have a local basis which generates a nilpotent subalgebra of V(M) are derived. Two methods for sufficient conditions are given, the first depending on the existence of a solution to a system of partial differential equations, the second using Darbouxs theorem to give a computable test for an (n − 1)-dimensional distribution. A nonlinear control system in which the control variables appear linearly can be transformed into an orbit equivalent system whose describing vector fields generate a nilpotent algebra if the distribution generated by the original describing vector fields admits a nilpotent basis. When this is the case, local analysis of the control system is greatly simplified.

A divisibility property for stirling numbers

Abstract In this paper we calculate which prime powers p s divide Δ n , m = g.c.d.{ k ! S ( n , k )| m ≤ k ≤ n } for s p . Here S ( n , k ) is a Stirling number of the second kind.

On the denominator of generalized Bernoulli numbers

Abstract The generalized Bernoulli numbers Ak(x) are polynomials with rational coefficients defined by ( t (e t −1) ) x = Σ k≥0 A k (x)( t k k! ) . We compute the least common multiples of the denominators of the coefficients of Ak(x).

Global anomalies and algebraic topology

The algebraic topology aspect of the global pure gauge anomaly calculation is investigated. In particular, the use of a cohomology sequence clarifies the method initiated by Witten [Nucl. Phys. 223, 422, 433 (1983)] and Elitzur and Nair [Nucl. Phys. B 243, 205 (1984)]. Examples in SU(N), Sp(N), and SO(N) are discussed.

Explicit construction of nontrivial elements for homotopy groups of classical Lie groups

Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stable special unitary, orthogonal, and symplectic groups are constructed using Clifford algebras; (b) the values for ‘‘winding numbers’’ for stable SU, SO, and Sp are calculated for generators of homotopy groups; and (c) representatives of generators of homotopy groups Πn−2(O(n−1)), Π2n−2(U(n−1)), Π4n−2(Sp(n−1)) are given.

Combinatorial Cell Complexes

In this chapter we state the basic definitions and propositions about combinatorial cell complexes and work out some examples in detail. We study certain maps of cell complexes that preserve “enough” of the combinatorial structure and construct the product, quotient, and adjunction complexes.

The Singular Homology of CW Complexes

Throughout this chapter R will denote a commutative ring with unit 1 ~O. The purpose of this chapter is to discuss some special properties of the singular homology functor with coefficients in R on the category W of CW complexes. We are particularly interested in those properties that reflect the geometry of the cell structure. We assume the reader is familiar with the fact (and proof) that the singular homology functor H * satisfies the standard six axioms of Ellenberg-Steenrod [10]. We note the continuity property which is valid for singular homology [30J and which is equivalent to the following.

Regular and Semisimplicial CW Complexes

In this chapter we will study two special kinds of CW complexes. Regular complexes, the first topic, are interesting because they are very useful in homological calculations. As we shall see in Chapter V, they share with simplicial complexes the virtue of allowing a relatively easy calculation of the “algebraic boundary” of a cell. At the same time, they share with more general CW complexes the advantage that often a space can be represented as a regular CW complex with many fewer cells than in a simplicial decomposition. A regular CW complex can be subdivided into a simplicial complex: in this sense it is a simplicial complex in which the simplexes are more efficiently combined into closed cells. For example, the ⊗ product of any two regular complexes is again a regular complex. If they are simplicial complexes, then the cells of the product complex are subdivided (in a nonunique way) to give the standard product simplicial complex.

Homotopy Type of CW Complexes

In this chapter we will study conditions on a space that will cause it to have the homotopy type of a CW complex and prove a theorem of J. H. C. Whitehead which gives a criterion that a map between two such spaces be a homotopy equivalence. We note in passing that there is a method using elaborate algebraic techniques (the analysis of Postnikov systems) for determining, in principle, whether two such spaces have the same homotopy type, but this is beyond the scope of this book.