293 research outputs found
The approach to consistency in the analytic hierarchy process
AbstractIn his first book on the Analytic Hierarchy Process, T. L. Saaty left open several mathematical questions about the structure of the set of positive reciprocal matrices. In this paper we consider three of these questions: Given an eigenvector and all matrices which give rise to it, can one go from one of them to any order by making small perturbations in the entries? Given two positive column vectors v and w is there a perturbation which carries the set of all positive reciprocal matrices with principal right eigenvector v to the set of positive reciprocal matrices with principal right eigenvector w? Does the set of positive reciprocal n×n matrices whose left and right principal eigenvectors are reciprocals coincide with the set of consistent matrices for n⩾4
Harmonic Manifolds and the Volume of Tubes about Curves
H. Hotelling proved that in the n-dimensional Euclidean or spherical space,
the volume of a tube of small radius about a curve depends only on the length
of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's
theorem to rank one symmetric spaces computing the volumes of the tubes
explicitly in these spaces. In the present paper, we generalize these results
by showing that every harmonic manifold has the above tube property. We compute
the volume of tubes in the Damek-Ricci spaces. We show that if a Riemannian
manifold has the tube property, then it is a 2-stein D'Atri space. We also
prove that a symmetric space has the tube property if and only if it is
harmonic. Our results answer some questions posed by L. Vanhecke, T. J.
Willmore, and G. Thorbergsson.Comment: 17 pages, no figures. This version is different from the journal
versio
Publishing Undergraduate Research Electronically
The College of Arts and Sciences at the University of Pennsylvania has as a goal expanding opportunities for undergraduates to conduct significant research and promoting the products of this research. CUREJ, the College Undergraduate Research Electronic Journal, was developed in collaboration with the Penn Libraries to achieve this goal
Metrics with Prescribed Ricci Curvature near the Boundary of a Manifold
Suppose is a manifold with boundary. Choose a point . We
investigate the prescribed Ricci curvature equation \Ric(G)=T in a
neighborhood of under natural boundary conditions. The unknown here is
a Riemannian metric. The letter in the right-hand side denotes a
(0,2)-tensor. Our main theorems address the questions of the existence and the
uniqueness of solutions. We explain, among other things, how these theorems may
be used to study rotationally symmetric metrics near the boundary of a solid
torus . The paper concludes with a brief discussion of the Einstein
equation on .Comment: 13 page
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