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1	Fuchsian groups of signature (0 : 2, ... , 2; 1; 0) with rational hyperbolic fixed points Norfleet, Mark Alan 23 October 2013 (has links) We construct Fuchsian groups [Gamma] of signature (0 : 2, ... ,2 ;1;0) so that the set of hyperbolic fixed points of [Gamma] will contain a given finite collection of elements in the boundary of the hyperbolic plane. We use this to establish that there are infinitely many non-commensurable non-cocompact Fuchsian groups [Delta] of finite covolume sitting in PSL₂(Q) so that the set of hyperbolic fixed points of [Delta] will contain a given finite collection of rational boundary points of the hyperbolic plane. We also give a parameterization of Fuchsian groups of signature (0:2,2,2;1;0) and investigate when particular hyperbolic elements have rational fixed points. Moreover, we include a detailed list of the group elements and their killer intervals for the known pseudomodular groups that Long and Reid found; in addition, the list contains a new list of killer intervals for a pseudomodular group not found by Long and Reid. / text Fuchsian groups Rational hyperbolic fixed points Pseudomodular groups
2	Computer-aided Computation of Abelian integrals and Robust Normal Forms Johnson, Tomas January 2009 (has links) This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle. Ordinary differential equations parameter estimation planar Hamiltonian systems bifurcation theory Abelian integrals limit cycles normal forms hyperbolic fixed points numerical integration invariant manifolds 34C07 34C20 37D10 37G15 37M20 37M99 65G20 65L09 65L70. MATHEMATICS MATEMATIK

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Fuchsian groups of signature (0 : 2, ... , 2; 1; 0) with rational hyperbolic fixed points

Computer-aided Computation of Abelian integrals and Robust Normal Forms