We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle QA = ∧topA⊗ ∧topT ∗P . The line bundle QA may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of QA may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H(A) called the modular c...

We calculate the Newtonian gravitational potential and field of a cubic, homogeneous asteroid and we apply it to the orbit of possible satellites. Large astronomical objects such as stars or planets, naturally tend to form spherical shapes due to the dominance of the gravitational forces, but as a thought experiment, we consider the properties of a planet in the form of a perfect cube. We inves...

Poincar6 often insisted existence in mathematics means logical con-sistency, and formal logic ia the sole guarantor of rigor. The paper joins thisto his view of intuition and his own mathematics. It looks at predicativity andthe infinity Poincar^'s early endorsement of the axiom of choice, and Cantor'sset theory versus Zermelo's axioms. PoincarS discussed constructivism sympa-<l...

Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under reasonable conditions the sum of the indices of a vector field equals the Euler characteristic of the manifold, is proven. Some consequences are discussed.

A perturbation method, the Lindstedt-Poincare method, is used to obtain the asymptotic expansions of the solutions of a nonlinear differential equation arising in general relativity. The asymptotic solutions contain no secular term, which overcomes a defect in Khuri’s paper. A technique of numerical order verification is applied to demonstrate that the asymptotic solutions are uniformly valid f...

6 H. Poincare, Meth. Nouv., 1 (1892), chap. III; G. D. Birkhoff, loc. cit.' 6 H. Poincar6, op. cit., 2 (1893), chap. XVII. The figure given by Poincar6 assumes that p is positive. However, the case of a negative p may be reduced to the case of a positive p, as seen from (6) by writing v + 1/27 for v. M. J. 0. Strutt, Math. Ann., 99, 625-628 (1928). 8 Cf. A. Wintner, Bull. Astr., 9, 251-253 (193...

The theory of classical electromagnetism is constructed in terms of two exterior differential systems, F − dA = 0, and J − dG = 0, which act as topological constraints on the variety of independent variables {x, y, z, t}. These two fundamental constraints lead to two other independent concepts of topological torsion, A^F, and topological spin, A^G, which are explicitly dependent upon the potent...

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G y X determines a collection of “peripheral” subgroups H1, . . . , Hk ⊂ G so that the...

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if G ...