Strebel differentials and Hamilton sequences

Let T(S) be a Teichmüller space of a hyperbolic Riemann surface S, viewed as a set of Teichmüller equivalence classes of Beltrami differentials on S. It is shown in this paper that for any extremal Beltrami differential μ₀ at a given point τ of T(S), there is a Hamilton sequence for μ₀ formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.     

Strebel differentials on families of hyperelliptic curves

Our paper is devoted to Strebel pairs on families of hyperelliptic curves. We provide a complete proof of the fact that the constructed differentials are Strebel and point out the connections between the introduced constructions and classical objects of complex algebraic geometry. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 121–130, 2007.     

True trees are dense

We show that any compact, connected set K in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, K can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck’s dessins d’enfants.     

Arithmetic cusp shapes are dense

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n-manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n + 1)-orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise in arithmetic orbifolds.      

The complete finitely axiomatized theories of order are dense

We prove a conjecture of Lauchli and Leonard that every sentence of the theory of linear order which has a model, has a model with a finitely axiomatized theory.     

The computable Lipschitz degrees of computably enumerable sets are not dense

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility (Downey et al. (2004) [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. sets are not dense (George Barmpalias, Andrew E.M. Lewis (2006) [2]), however the problem for the computable Lipschitz degrees is more complex.     

On quadratic differentials on multiply connected domains that are perfect squares

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 113–133.     

Differential evolution with generalized differentials

In this paper, we study the mutation operation of the differential evolution (DE) algorithm. In particular, we propose the differential of scaled vectors, called the ‘generalized differential’, as opposed to the existing scaled differential vector in the mutation of DE. We derive the probability distribution of points generated by the mutation with ‘generalized differentials’. We incorporate a vector-projection-based exploratory method within the new mutation scheme. The vector projection is not mandatory and it is only invoked if trial points continue to be unsuccessful. An algorithm is then proposed which implements the mutation strategy based on the difference of the scaled vectors as well as the vector projection technique. A numerical study is carried out using a set of 50 test problems, many of which are inspired by practical applications. Numerical results suggest that the new algorithm is superior to DE.     

Sobolev homeomorphisms are dense in volume preserving automorphisms

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-$(1,p)$ volume preserving homeomorphisms on closed and connected n-dimensional manifolds, $n\ge 3$, for $p<n?1$. We also prove that if $p>n$ this result is not true. More precisely, we obtain the density of Sobolev-$(1,p)$ homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-$(1,p)$ ball with the same radius centered at the identity.     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.     

Periodic billiard orbits are dense in rational polygons

We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of

     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

Meromorphic quadratic differentials with prescribed singularities

We study the singular flat structure associated to any meromorphic quadratic differential on a compact Riemann surface to prove an existence theorem as follows. There exists a meromorphic quadratic differential with given orders of the poles and zeros and orientability or non orientability of the horizontal foliation, iff these prescribed topological data are admissible according to the Gauss-Bonnet Theorem, the Residue Theorem and certain conditions arising from local orientability or non orientablity considerations. Some few exceptional cases remain excluded. Thus, we generalize two previous results. One due to Masur & Smillie, which assumes that poles are at most simple; and a second one due to Muciño-Raymundo, which assumes that the horizontal foliation is orientable.Partially supported by DGAPA-UNAM and CONACYT 28492-E.     

Mahler measures in a field are dense modulo 1

Let K be a number field. We prove that the set of Mahler measures M(α), where α runs over every element of K, modulo 1 is everywhere dense in [0, 1], except when or , where D is a positive integer. In the proof, we use a certain sequence of shifted Pisot numbers (or complex Pisot numbers) in K and show that the corresponding sequence of their Mahler measures modulo 1 is uniformly distributed in [0, 1]. Received: 24 March 2006     

On quadratic differentials on multiply connected domains that are perfect squares. II

Problems on extremal decomposition of a multiply connected domain are considered. Two theorems exhibiting the role in such problems of quadratic differentials that are perfect squares are proved. As an implication, it is shown that the extremal metric approach used in the paper naturally leads to the results recently obtained by Dubinin and Eyrikh (Zap. Nauchn. Semin. POMI, 314, 52–75 (2004)) with the use of the capacity theory for generalized condensers. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 40–51.