Solvability on manifolds by quadratures permitting only integrals

Quadratures for oscillatory and singular integrals

Numerical methods for strongly oscillatory and singular functions are given in this paper. Beside a summary of standard methods and product integration rules, we consider a class of complex integration methods, as well as Gaussian quadratures with respect to the oscillatory weight w(x) = xe, x ∈ [−1, 1]. Numerical examples are included.

Renormalizing Curvature Integrals on Poincaré-einstein Manifolds

After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure, and obtain, from the renor...

On Chebyshev-Type Quadratures

According to a result of S. N. Bernstein, «-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 or n ä 10. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We...

On Simple Quadratures

Right integrals and invariants of three–manifolds

This paper gives a summary of our approach to invariants of three manifolds via right integrals on finite dimensional Hopf algebras and their relation to the Kirby calculus. It gives the author great pleasure to dedicate this paper to Rob Kirby on his sixtieth birthday. AMS Classification 57N10; 57M50