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

Cell density dependent stimulation of PAI-1 and hyaluronan synthesis by TGF-[béta] in orbital fibroblasts

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Single parameter galaxy classification: The Principal Curve through the multi-dimensional space of galaxy properties

We propose to describe the variety of galaxies from SDSS by using only one affine parameter. To this aim, we build the Principal Curve (P-curve) passing through the spine of the data point cloud, considering the eigenspace derived from Principal Component Analysis of morphological, physical and photometric galaxy properties. Thus, galaxies can be labeled, ranked and classified by a single arc length value of the curve, measured at the unique closest projection of the data points on the P-curve. We find that the P-curve has a "W" letter shape with 3 turning points, defining 4 branches that represent distinct galaxy populations. This behavior is controlled mainly by 2 properties, namely u-r and SFR. We further present the variations of several galaxy properties as a function of arc length. Luminosity functions variate from steep Schechter fits at low arc length, to double power law and ending in Log-normal fits at high arc length. Galaxy clustering shows increasing autocorrelation power at large scales as arc length increases. PCA analysis allowed to find peculiar galaxy populations located apart from the main cloud of data points, such as small red galaxies dominated by a disk, of relatively high stellar mass-to-light ratio and surface mass density. The P-curve allows not only dimensionality reduction, but also provides supporting evidence for relevant physical models and scenarios in extragalactic astronomy: 1) Evidence for the hierarchical merging scenario in the formation of a selected group of red massive galaxies. These galaxies present a log-normal r-band luminosity function, which might arise from multiplicative processes involved in this scenario. 2) Connection between the onset of AGN activity and star formation quenching, which appears in green galaxies when transitioning from blue to red populations. (Full abstract in downloadable version)Comment: Full abstract in downloadable versio

What turns galaxies off? The different morphologies of star-forming and quiescent galaxies since z~2 from CANDELS

We use HST/WFC3 imaging from the CANDELS Multicycle Treasury Survey, in conjunction with the Sloan Digital Sky Survey, to explore the evolution of galactic structure for galaxies with stellar masses >3e10M_sun from z=2.2 to the present epoch, a time span of 10Gyr. We explore the relationship between rest-frame optical color, stellar mass, star formation activity and galaxy structure. We confirm the dramatic increase from z=2.2 to the present day in the number density of non-star-forming galaxies above 3e10M_sun reported by others. We further find that the vast majority of these quiescent systems have concentrated light profiles, as parametrized by the Sersic index, and the population of concentrated galaxies grows similarly rapidly. We examine the joint distribution of star formation activity, Sersic index, stellar mass, inferred velocity dispersion, and stellar surface density. Quiescence correlates poorly with stellar mass at all z<2.2. Quiescence correlates well with Sersic index at all redshifts. Quiescence correlates well with `velocity dispersion' and stellar surface density at z>1.3, and somewhat less well at lower redshifts. Yet, there is significant scatter between quiescence and galaxy structure: while the vast majority of quiescent galaxies have prominent bulges, many of them have significant disks, and a number of bulge-dominated galaxies have significant star formation. Noting the rarity of quiescent galaxies without prominent bulges, we argue that a prominent bulge (and perhaps, by association, a supermassive black hole) is an important condition for quenching star formation on galactic scales over the last 10Gyr, in qualitative agreement with the AGN feedback paradigm.Comment: The Astrophysical Journal, in press; 20 pages with 13 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)

On the diophantine equation 1k+2k+...+xk+R(x)=yz

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Number systems over general orders

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