2009, Bodnarchuk, Lesya, Drozd, Yuriy, Greuel, Gert-Martin

In 1957 Atiyah classifed simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classifcation by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.

2009, Brauchart, J.S., Dragnev, P.D., Saff, E.B.

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|−s with d−2 s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s = d − 2 yields interesting phenomena, which we investigate in detail. A weak∗ asymptotic analysis is provided as s ! (d − 2)+.

2008, Matessi, Carlo, Schneider, Kristan A.

We consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient system for one locus, multiple alleles and for multiple loci, assuming linkage equilibrium, and its potential is a Lyapunov function; (ii) the LF of the particular models are special cases of the Wildcard Model's LF; (iii) the LF of the Wildcard Model can be derived from a LF previously identified for a model of density- and frequency- dependent selection due to Lotka-Volterra competition, with one locus, multiple alleles, multiple species and continuous-time dynamics (Matessi and Jayakar, 1981). We extend the LF with density and frequency dependence to a multilocus, linkage equilibrium dynamics.

2008, Kerner, Dmitry

We introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate . For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or directionally Newton-non-degenerate a property of singular types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This is proved for curves. For hypersurfaces we give an example of a Newton-non-degenerate hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, the classical formulas for the Milnor number (Kouchnirenko) and the zeta function (Varchenko) of the Newton-non-degenerate singularity are generalized to some classes of directionally Newton-non-degenerate singularities.

2009, Happel, Dieter, Zacharia, Dan

Let Delta be a finite dimensional algebra over an algebraically closed field k. We will investigate homological properties of piecewise hereditary algebras Delta. In particular we give lower and upper bounds of the strong global dimension, show the behavior of the strong global under one point extensions and tilting. Moreover we show that the pieces of mod Delta have Auslander-Reiten sequence.

2008, Edo, Eric, van den Essen, Arno, Maubach, Stefan

We prove that for a polynomial f 2 k[x, y, z] equivalent are: (1)f is a k[z]-coordinate of k[z][x, y], and (2) k[x, y, z]/(f) = k[2] and f(x, y, a) is a coordinate in k[x, y] for some a 2 k. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate f 2 k[x, y, z] which is also a k(z)-coordinate, is a [z]-coordinate. We discuss a method for onstructing automorphisms of k[x, y, z], and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method essentially linking Nagata with a non-tame R-automorphism of R[x], where R = k[z]/(z2).

2008, Kurke, Herbert, Osipov, Denis, Zheglov, Alexander

We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-toone correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field.

2007, Kesseböhmer, Marc, Stratmann, Bernd O.

In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil’s staircases) supported on limit sets of finitely generated conformal iterated function systems in R. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not a-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.

2009, Böcherer, Siegfried, Nagaoka, Shoyu

Using theta series we construct Siegel modular forms of level p which behave well modulo p in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod p to level one modular forms of weight p + 1; in particular, this is true for Yoshidal lifts of level p.

2009, Foss, Sergey, Korshunov, Dmitry, Zachary, Stan

This text studies heavy-tailed distributions in probability theory, and especially convolutions of such distributions. The mail goal is to provide a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions which includes many novel elements and, in particular, is based on the regular use of the principle of a single big jump.