2013, Bracciali, Cleonice F., Moreno-Balcázar, Juan José

We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.

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.

2010, Bruin, H., Štimac, S.

We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allows us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to simplification of some existing results, including the Ingram Conjecture for Fibonacci-like unimodal inverse limits.

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).

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.

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.

2016, Aron, Richard M., Falcó, Javier, García, Domingo, Maestre, Manuel

Let BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX. In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞(BX), we prove that the fiber in M(H∞(Bc0)) over any point of the distinguished boundary of the closed unit ball B¯ℓ∞ of ℓ∞ contains an analytic copy of Bℓ∞. In the case of Au(BX) we prove that if there exists a polynomial whose restriction to the open unit ball of X is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of D.

2016, Cavalieri, Renzo, Johnson, Paul, Markwig, Hannah, Ranganathan, Dhruv

We explore the explicit relationship between the descendant Gromov–Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov–Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient “graphical user interface” for Okounkov and Pandharipande’s celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.

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)+.

2016, Merino, Bernardo González, Henze, Matthias

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski’s fundamental theorem, and which is strongly linked to the well-known Mahlerconjecture. Based on the covering minima of Kannan & Lovász and a problem posed by Fejes Tóth, we arrange Makai Jr.’s conjecture into a wider context and investigate densities of lattice arrangements of convex bodies intersecting every i-dimensional affine subspace. Then it becomes natural also to formulate and study a dual analog to Minkowski’s second fundamental theorem. As our main results, we derive meaningful asymptotic lower bounds for the densities of such arrangements, and furthermore, we solve the problems exactly for the special, yet important, class of unconditional convex bodies.