Brody curves omitting hyperplanes

A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex line to the complex projective space of dimension n, such that the family of its translations is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n=1.Comment: 8 page

Dynamics of a higher dimensional analog of the trigonometric functions

We introduce a higher dimensional quasiregular map analogous to the trigonometric functions and we use the dynamics of this map to define, for d>1, a partition of d-dimensional Euclidean space into curves tending to infinity such that two curves may intersect only in their endpoints and such that the union of the curves without their endpoints has Hausdorff dimension one.Comment: 12 page

On the mean square of the zeta-function and the divisor problem

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then we obtain the asymptotic formula $$\int_0^T (E^*(t))^2 {\rm d} t = T^{4/3}P_3(\log T) + O_\epsilon(T^{7/6+\epsilon}),$$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.Comment: 10 page

Period Matrices Of Accola-Maclachlan And Kulkarni Surfaces

We compute the period matrices of the Riemann surfaces given by the equations w2 = z2g+

ON SOME CLASSES OF TREE AUTOMATA AND TREE LANGUAGES

Abstract. In this paper we give a structural characterization of three classes of tree automata. Namely, we shall homomorphically represent the classes of nilpotent, definite, and monotone tree automata by means of quasi-cascade-products of unary nilpotent and unary definite tree automata in the first two cases, and by means of products of simpler tree automata in the third case

GENERALIZED HECKE GROUPS AND HECKE POLYGONS

Abstract. In this paper, we study certain Fuchsian groups H (p1,...,pn), called generalized Hecke groups. These groups are isomorphic to ∏ ∗ n j=1Zpj. Let Γ be a subgroup of finite index in H (p1,...,pn). By Kurosh’s theorem, Γ is isomorphic to Fr ∗ ∏ ∗ k i=1Zmi,whereFris a free group of rank r,andeachmidivides some pj. Moreover, H2 /Γ is Riemann surface. The numbers m1,...,mk are branching numbers of the branch points on H2 /Γ. The signatureofΓ is (g; m1,...,mk; t), whereg and t are the genus and the number of cusps of H2 /Γ, respectively. A purpose of this paper is to consider two problems. First, determine the necessary and sufficient conditions for the existence of a subgroup of finite index of a given type in H (p1,...,pn). We also extend this work to extended generalized Hecke groups H ∗ (p1,...,pn) whichareisomorphic to Dp1 ∗Z