An algorithm to compute odd orders and ramification indices of cyclic actions on compact surfaces

In this paper we get an effective algorithm to compute all odd orders and ramification indices of homeomorphisms acting on compact surfaces, orientable or not. E. Bujalance and A. F. Costa were partially supported by DGICYT PB 89-201 and Science Plan N SC1-CT91-0716, J. M. Gamboa was partially supported by DGICYT PB 89-379 and Science Plan N SC1-CT91-0716, and J. Lafuente was partially supported by Science Plan N SC1-CT91-0716.     

Maximum orders of extendable actions on surfaces

We determine the maximum order E_g of finite groups G acting on the closed surface Σ_g of genus g which extends over(S~3, Σ_g) for all possible embeddings Σ_g → S~3, where g 1.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

Analytic actions on compact surfaces and fixed points

 Consider an effective real analytic action of a connected Lie group G on a compact connected surface of Euler characteristic χ≠0. We show that if the action has no fixed point then χ≥1 and the Lie algebra 𝒢 of G is isomorphic either to a subalgebra of the affine algebra of ℝ², which is the extension of the ideal of constant vector fields by an irreducible linear subalgebra, or to sl(2,ℝ), o(3), sl(2,ℂ) and sl(3,ℝ). Received: 7 August 2001 Published online: 24 January 2003     

On extendability of group actions on compact Riemann surfaces

The question of whether a given group which acts faithfully on a compact Riemann surface of genus is the full group of automorphisms of (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group in terms of a concrete partial presentation for associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where is a non-cyclic abelian group.

     

On purely non-free finite actions of abelian groups on compact surfaces

A finite group of self-homeomorphisms of a closed orientable surface is said to act on it purely non-freely if each of its elements has a fixed point; we also call it a gpnf-action. In this paper we observe that gpnf-actions exist for an arbitrary finite group and we discuss the minimum genus problem for such actions. We solve it for abelian groups. In the cyclic case we prove that the minimal gpnf-action genus coincides with Harvey’s minimal genus.     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.     

An effective algorithm to compute Mandelbrot sets in parameter planes

In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.     

An algorithm to compute a sparse basis of the null space

Summary LetA be a realm×n matrix with full row rankm. In many algorithms in engineering and science, such as the force method in structural analysis, the dual variable method for the Navier-Stokes equations or more generally null space methods in quadratic programming, it is necessary to compute a basis matrixB for the null space ofA. HereB isn×r, r=n–m, of rankr, withAB=0. In many instancesA is large and sparse and often banded. The purpose of this paper is to describe and test a variation of a method originally suggested by Topcu and called the turnback algorithm for computing a banded basis matrixB. Two implementations of the algorithm are given, one using Gaussian elimination and the other using orthogonal factorization by Givens rotations. The FORTRAN software was executed on an IBM 3081 computer with an FPS-164 attached array processor at the Triangle Universities Computing Center and on a CYBER 205 vector computer. Test results on a variety of structural analysis problems including two- and three-dimensional frames, plane stress, plate bending and mixed finite element problems are discussed. These results indicate that both implementations of the algorithm yielded a well-conditioned, banded, basis matrixB whenA is well-conditioned. However, the orthogonal implementation yielded a better conditionedB for large, illconditioned problems.The research by these authors was supported by the U.S. Air Force under grant No. AFOSR-83-0255 and by the National Science Foundation under grant No. MCS-82-19500The research by these authors was supported by the Applied Mathematical Sciences Program of the U.S. Department of Energy, under contract to Martin Marietta Energy Systems, Inc.     

Algebraic deformations of compact Kähler surfaces II

In the first version of this paper, a short proof was given of Kodaira’s result that every compact Kähler surface is a deformation of an algebraic surface under the extra assumption that the infinitesimal deformations of the surface were unobstructed. In this paper, the extra assumption is removed.     

Morse indices of critical manifolds generated by min-max methods with compact lie group actions and applications

We derive some upper and lower bounds for Morse indices of critical manifolds generated by min-max principles for functionals invariant under a general compact Lie group or a finite group action. The results generalize the similar results in the nonequivariant (no group action) case. In doing so, we also generalize the extension theorem of Dugundji type in the nonequivariant case to the equivariant (group action) case. As an application, we obtain a precise growth estimate for the whole sequence of critical values given by the min-max procedure for some superquadratic second-order differential equations. It is well-known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed Hamiltonian systems and equations.     

Homomorphic actions of ℍ on a compact groupon a compact group

The way the additive semigroup [0, ∞) carrying the usual topology can act as a semigroup of homomorphisms on a compact group is studied in this paper. It is shown that such actions of [0, ∞) on a compact group G can be completely described in terms of continuous homomorphisms from [0, ∞) into G. Furthermore, it is proven that each such action of [0, ∞) on a compact group G has a unique extension to an action of the additive group of all real numbers endowed with the usual topology on G so that a similar characterization is also valid for actions of the real numbers on a compact group.