Representation of finite graphs as difference graphs of S-units, I

In part I of the present paper the following problem was investigated. Let G be a finite simple graph, and S be a finite set of primes. We say that G is representable with S if it is possible to attach rational numbers to the vertices of G such that the vertices v_1,v_2 are connected by an edge if and only if the difference of the attached values is an S-unit. In part I we gave several results concerning the representability of graphs in the above sense.In the present paper we extend the results from paper I to the algebraic number field case and make some of them effective. Besides we prove some new theorems: we prove that G is infinitely representable with S if and only if it has a degenerate representation with S, and we also deal with the representability with S of the union of two graphs of which at least one is finitely representable with S.p, li { white-space: pre-wrap; }</style

Observed versus modelled u,g,r,i,z-band photometry of local galaxies - Evaluation of model performance

We test how well available stellar population models can reproduce observed u,g,r,i,z-band photometry of the local galaxy population (0.02<=z<=0.03) as probed by the SDSS. Our study is conducted from the perspective of a user of the models, who has observational data in hand and seeks to convert them into physical quantities. Stellar population models for galaxies are created by synthesizing star formations histories and chemical enrichments using single stellar populations from several groups (Starburst99, GALAXEV, Maraston2005, GALEV). The role of dust is addressed through a simplistic, but observationally motivated, dust model that couples the amplitude of the extinction to the star formation history, metallicity and the viewing angle. Moreover, the influence of emission lines is considered (for the subset of models for which this component is included). The performance of the models is investigated by: 1) comparing their prediction with the observed galaxy population in the SDSS using the (u-g)-(r-i) and (g-r)-(i-z) color planes, 2) comparing predicted stellar mass and luminosity weighted ages and metallicities, specific star formation rates, mass to light ratios and total extinctions with literature values from studies based on spectroscopy. Strong differences between the various models are seen, with several models occupying regions in the color-color diagrams where no galaxies are observed. We would therefore like to emphasize the importance of the choice of model. Using our preferred model we find that the star formation history, metallicity and also dust content can be constrained over a large part of the parameter space through the use of u,g,r,i,z-band photometry. However, strong local degeneracies are present due to overlap of models with high and low extinction in certain parts of color space.Comment: MNRAS accepted. 18 pages, incl. 15 figure

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Upper bounds for the degrees of decomposable forms of given discriminant

1. Introduction. In our paper [5] a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author [3] to decomposable forms in n ≥ 2 variables. The bound obtained in [3] depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish an improvement of the bound of [3] which is already best possible for every n ≥ 2. Moreover, all the squarefree decomposable forms in n variables over ℤ will be determined for which our bound cannot be further sharpened. In the proof we shall use some results and arguments of [5] and [3] and two theorems of Heller [6] on linear systems with integral valued solutions