On equal values of power sums of arithmetic progressions

In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where $a,b,c,d,k,l$ are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its expositio

Mutual information in random Boolean models of regulatory networks

The amount of mutual information contained in time series of two elements gives a measure of how well their activities are coordinated. In a large, complex network of interacting elements, such as a genetic regulatory network within a cell, the average of the mutual information over all pairs is a global measure of how well the system can coordinate its internal dynamics. We study this average pairwise mutual information in random Boolean networks (RBNs) as a function of the distribution of Boolean rules implemented at each element, assuming that the links in the network are randomly placed. Efficient numerical methods for calculating show that as the number of network nodes N approaches infinity, the quantity N exhibits a discontinuity at parameter values corresponding to critical RBNs. For finite systems it peaks near the critical value, but slightly in the disordered regime for typical parameter variations. The source of high values of N is the indirect correlations between pairs of elements from different long chains with a common starting point. The contribution from pairs that are directly linked approaches zero for critical networks and peaks deep in the disordered regime.Comment: 11 pages, 6 figures; Minor revisions for clarity and figure format, one reference adde

The Cdk8 kinase module regulates interaction of the mediator complex with RNA polymerase II

The Cdk8 kinase module (CKM) is a dissociable part of the coactivator complex mediator, which regulates gene transcription by RNA polymerase II. The CKM has both negative and positive functions in gene transcription that remain poorly understood at the mechanistic level. In order to reconstitute the role of the CKM in transcription initiation, we prepared recombinant CKM from the yeast Saccharomyces cerevisiae. We showed that CKM bound to the core mediator (cMed) complex, sterically inhibiting cMed from binding to the polymerase II preinitiation complex (PIC) in vitro. We further showed that the Cdk8 kinase activity of the CKM weakened CKM–cMed interaction, thereby facilitating dissociation of the CKM and enabling mediator to bind the PIC in order to stimulate transcription initiation. Finally, we report that the kinase activity of Cdk8 is required for gene activation during the stressful condition of heat shock in vivo but not under steady-state growth conditions. Based on these results, we propose a model in which the CKM negatively regulates mediator function at upstream-activating sequences by preventing mediator binding to the PIC at the gene promoter. However, during gene activation in response to stress, the Cdk8 kinase activity of the CKM may release mediator and allow its binding to the PIC, thereby accounting for the positive function of CKM. This may impart improved adaptability to stress by allowing a rapid transcriptional response to environmental changes, and we speculate that a similar mechanism in metazoans may allow the precise timing of developmental transcription programs

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD

An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

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)

What makes the prefrontal cortex so appealing in the era of brain imaging? A network analytical perspective

It is thought that the prefrontal cortex (PFC) subserves cognitive control processes by coordinating the flow of information in the cerebral cortex. In the network of cortical areas the central position of the PFC makes difficult to dissociate processing and the cognitive function mapped to this region, especially when using whole brain imaging techniques, which can detect frequently activated regions. Accordingly, the present study showed particularly high rate of increase of published studies citing the PFC and imaging as compared to other fields of the neurosciences on the PubMed. Network measures used to characterize the role of the areas in signal flow indicated specialization of the different regions of the PFC in cortical processing. Notably, areas of the dorsolateral PFC and the anterior cingulate cortex, which received the highest number of citations, were identified as global convergence points in the network. These prefrontal regions also had central position in the dominant cluster consisted exclusively by the associational areas of the cortex. We also present findings relevant to models suggesting that control processes of the PFC are depended on serial processing, which results in bottleneck effects. The findings suggest that PFC is best understood via its role in cortical information processing