Let (M, g) be a complete Riemannian manifold with no conjugate points and f : (M, g) → (B, gB) a principal G-bundle, where G is a Lie group acting by isometries and B the smooth quotient with gB the Riemannian submersion metric. We obtain a characterization of conjugate point-free quotients (B, gB) in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle T...

We consider a generalized conjugate gradient method for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations. The method is based on splitting off from the original coefficient matrix a symmetric, positive-definiteonethat corresponds to a more easily solv...

In the Sylow theorems f we learn that if the order of a group 2Í is divisible hj pa (p a prime integer) and not by jo*+1, then 31 contains one and only one set of conjugate subgroups of order pa, and any subgroup of 21 whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of 2Í. These conjugate subgroups may be called the Sylow subgroups of 21. It will be o...

An n-dimensional polyhedral space is a length space M (with intrinsic metric) triangulated into n-simplexes with smooth Riemannian metrics. In the definitions below, we assume that the triangulation is fixed. The boundary of M is the union of the (n− 1)-simplexes of the triangulation that are adjacent to only one (n− 1)-simplex. As usual, a geodesic in M is a naturally parametrized locally shor...

In this paper, we study the Arneodo-Coullet-Tresser map F (x, y, z) = (ax − b(y − z), bx + a(y − z), cx − dx + ez) where a, b, c, d, e are real with bd 6= 0 and k > 1 is an integer. We obtain stability regions for fixed points of F and symmetric period-2 points while c and e vary as parameters. Varying a and e as parameters, we show that there is a hyperbolic invariant set on which F is conjuga...

In this paper, we study the Arneodo-Coullet-Tresser map F (x, y, z) = (ax − b(y − z), bx + a(y − z), cx − dx + ez) where a, b, c, d, e are real with bd 6= 0 and k > 1 is an integer. We obtain stability regions for fixed points of F and symmetric period-2 points while c and e vary as parameters. Varying a and e as parameters, we show that there is a hyperbolic invariant set on which F is conjuga...