Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given $k\geq 4$ and $L\geq 3$ there are only finitely many arithmetic progressions of the form $(x_0^{l_0},x_1^{l_1},...,x_{k-1}^{l_{k-1}})$ with $x_i\in{\Bbb Z},$ gcd$(x_0,x_1)=1$ and $2\leq l_i\leq L$ for $i=0,1,...,k-1.$ Furthermore, we show that, for L=3, the progression $(1,1,...,1)$ is the only such progression up to sign.Comment: 16 page

On conjectures and problems of Ruzsa concerning difference graphs of S-units

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.Comment: 15 page

A heuristic quantum theory of the integer quantum Hall effect

Contrary to common belief, the current emitted by a contact embedded in a two-dimensional electron gas (2DEG) is quantized in the presence of electric and magnetic fields. This observation suggests a simple, clearly defined model for the quantum current through a Hall device that does not invoke disorder or interactions as the cause of the integer quantum Hall effect (QHE), but is based on a proper quantization of the classical electron drift motion. The theory yields a quantitative description of the breakdown of the QHE at high current densities that is in agreement with experimental data. Furthermore, several of its key points are in line with recent findings of experiments that address the dependency of the QHE on the 2DEG bias voltage, results that are not easily explained within the framework of conventional QHE models.Comment: 20 pages, 6 figure

New skeletal tuberculosis cases in past populations from Western Hungary (Transdanubia)

The distribution, antiquity and epidemiology of tuberculosis (TB) have previously been studied in osteoarchaeological material in the eastern part of Hungary, mainly on the Great Plain. The purpose of this study is to map the occurrence of skeletal TB in different centuries in the western part of Hungary, Transdanubia, and to present new cases we have found. Palaeopathological analysis was carried out using macroscopic observation supported by radiographic and molecular methods. A large human osteoarchaeological sample (n = 5684) from Transdanubian archaeological sites ranging from the 2nd to the 18th centuries served as a source of material. Spinal TB was observed in seven individuals (in three specimens with Pott's disease two of which also had cold abscess) and hip TB was assumed in one case. The results of DNA for Mycobacterium tuberculosis were positive in seven of the eight cases identified by paleopathology, and negative in the assumed case of hip TB. However, the molecular results are consistent with highly fragmented DNA, which limited further analysis. Based on the present study and previously published cases, osteotuberculosis was found in Transdanubia mainly during the 9th–13th centuries. However, there are no signs of TB in many other 9th–13th century sites, even in those that lie geographically close to those where osteotuberculous cases were found. This may be due to a true absence of TB caused by the different living conditions, way of life, or origin of these populations. An alternative explanation is that TB was present in some individuals with no typical paleopathology, but that death occurred before skeletal morphological features could develop