The Prym and Ganning Vector Bundles over the Teichmueller Space

We introduce and study the Prym vector bundle P of holomorphic Prym differentials and the Ganning cohomology bundle G over the Teichmueller space of compact Riemann surfaces of genus g2 and over the Torelli space of genus g2. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus g and a special domain in the complex manifold C ^2g/Z ^2g.     

Hyperelliptic Surfaces and their Moduli

The hyperelliptic portion of the moduli space of compact Riemann surfaces of genus g2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with the lattice of maximal g-hyperelliptic group actions (classified up to topological equivalence). The resulting stratification of the hyperelliptic moduli space exhibits regularities which depend on the parity of g and can be detected at the level of groups of order 8.     

Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node

In this article, we consider a family of compact Riemann surfaces of genus q 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.     

Integral Representation of Harmonic Functions Defined Outside a Compact Set in a Harmonic Space

Given here is an integral representation for any harmonic function u0 defined outside a compact set in a Brelot harmonic space with or without positive potentials by means of signed measurers on . This generalizes the Bôcher theorem on positive harmonic singularities in ; ⁿ , n2.     

On Ovals of Riemann Surfaces of Even Genera

Let X be a Riemann surface of genus g2. A symmetry of of X is an antiholomorphic involution acting of X. A classical theorem of Harnack states that the set Fix () of fixed points of is either emplty or it consists of g+1 disjoint simple closed curves called, following Hilberts terminology, the ovals of . A Riemann surface admitting a symmetry corresponds to a real algebraic curve and nonconjugate symmetries correspond to different real models of the curve. The number of ovals of the symmetry equals the number of connected components of the corresponding real model. It is well known that two symmetries of a Riemann surface of genus g have at most 2g+2 ovals, and the bound is attained for every genus and just for commuting symmetries. Natanzon showed that three and four nonconjugate symmetries of a Riemann surface of genus g have at most 2g+4 and 2g+8 ovals respectively, and these bounds are attained for every odd genus and for commuting symmetries. Natanzon found that a Riemann surface of genus g has at most 2( +1) nonconjugate symmetries and, again, this bound is attained for infinitely many of g. Recently we have showed that a Riemann surface of even genus g admits at most four symmetries. Our aim here is to show, using NEC groups and combinatorial methods, that three nonconjugate symmetries of a surface of even genus g has at most 2g+3 ovals and, surprisingly, if such a surface admits four nonconjugate symmetries then its total number of ovals does not exceed 2g+2. Furthermore, we show that this last bound is sharp for every even genus g and for surfaces with automorphism group D _n × Z₂, for each n dividing 2g.     

Arithmetic Properties of Summands of Partitions

Let d d, d2 2. We prove that for almost all partitions of an integer the parts are well distributed in residue classes mod d. The limitations of the uniformity of this distribution are also studied.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Strong-Cut Enumerative procedure for Extreme point Mathematical Programming Problems

Summary In this paper a Strong-Cut Enumerative Procedure for solving Extreme Point Mathematical Programming Problem:MaxC X subject toAX=b and thatX is an extreme point ofDX=d, X is developed. The procedure will avoid the investigation of many of the extreme points ofDX=d,X 0 and also alternative optimas of different best extreme points ofDX=d, X 0 will not be needed. The algorithm is expected to work very efficiently.

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Zusammenfassung In dieser Arbeit Strong-Cut Enumerative Procedure for Extrem Point Mathematical Programming Problem wird ein sehr effizientes enumeratives Verfahren zur Lösung des ProblemsMax {cx}¦Ax=b;x Extremzahl vonDx=d, x 0entwickelt. Extrempunkte vonDx=d, x 0 werden in systematischer Weise abgesucht, bis Zulässigkeit inAx=b erreicht ist. Die dabei benutzten Kriterien vermeiden die Untersuchung vieler nichtoptimaler Extrempunkte und die Bestimmung alternativer Optimalpunkte vonDx=d, x 0.

     

Über die Dimension des singulären Ortes des Modulraumes Mg

The modulus space M^g of compact Riemann surfaces of genus g2 is a normal complex space of dimension 3g–3. In this paper we give necessary and sufficient conditions for the singular locus S^g of M^g to contain points of dimension 0, i. e. isolated singularities, and points of dimension 1. Further we give examples for Riemann surfaces representing those singular points.     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

Stability of the p-Curvature Positivity under Surgeries and Manifolds with Positive Einstein Tensor

We establish the stability of the class of manifolds with positive p-curvature under surgeries in codimension p + 3. As a consequence of this result, we first obtain the classification of compact 2-connected manifolds of dimension 7 with positive Einstein tensor; and secondly the existence of metrics with positive Einstein tensor on any compact, simply connected, non-spin manifold of dimension 7 whose second homotopy group is isomorphic to Z₂.     

Rates of convergence for the stability of large order statistics

Forr1 and eachnr, letM _nr be therth largest ofX ₁,X ₂, ...,X _n , where {X _n ,n1} is an i.i.d. sequence. Necessary and sufficient conditions are presented for the convergence of for all >0 and some –1, where {a _n } is a real sequence. Furthermore, it is shown that this series converges for all >–1, allr1 and all >0 if it converges for some >–1, somer1 and all >0.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let be the fundamental group of a closed orientable surface of genus g 1, and let R(, G)/G be the space of conjugacy classes of representations of into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map ¯ on R(, G)/G and investigate whether the fibres of ¯ are isotropic with respect to the natural symplectic structure on R(, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of ¯ are also isotropic, but we give an example to show that in general they are not.     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

A sphere theorem for non-reversible Finsler metrics

For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility = max {F(–X)|F(X)=1}1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n3 with reversibility and with flag curvature is homotopy equivalent to the n-sphere.Mathematics Subject Classification (2000):53C60; 53C20, 53C22