The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1)– α n2, then one can remove εn2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.

For a real constant α, let $\pi _3^\alpha (G)$ be the minimum of twice the number of K2’s plus α times the number of K3’s over all edge decompositions of G into copies of K2 and K3, where Kr denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices.

The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza (Studia Sci. Math. Hungar.22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput.28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α = 3 that Kn and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

The solution of complex rarefied flows with the BGK equation and the Discrete Velocity Method (DVM) requires a large number of velocity grid points leading to significant computational costs. We propose an adaptive velocity grid approach exploiting the fact that locally in space, the distribution function is supported only by a sub-set of the global velocity grid. The velocity grid is adapted thanks to criteria based on local temperature, velocity and on the enforcement of mass conservation. Simulations in 1D and 2D are presented for different Knudsen numbers and compared to a global velocity grid BGK solution, showing the computational gain of the proposed approach.

This open study evaluates the effect of agomelatine, a melatonergic receptor agonist and 5-HT2C antagonist antidepressant, on sleep architecture in patients suffering from major depressive disorder. Fifteen outpatients with a baseline HAMD score ⩾20 were treated with 25 mg/d agomelatine for 42 d. Polysomographic studies were performed at baseline, day 7, day 14, and day 42. Sleep efficiency, time awake after sleep onset and the total amount of slow-wave sleep (SWS) increased at week 6. The increase of SWS was predominant during the first sleep cycle. The amount of SWS decreased throughout the first four sleep cycles from day 7 and delta ratio increased from day 14 onwards. No change in rapid eye movement (REM) latency, amount of REM or REM density was observed and agomelatine was well tolerated. In conclusion agomelatine improved sleep continuity and quality. It normalized the distribution of SWS sleep and delta power throughout the night.

Various projects to find planets or entire planetary systems around main sequence stars in the solar neighborhood are presently under way. When ELTs will be operational, there will be literally thousands of confirmed planetary systems including spectro-photometric detections. At this point it becomes inevitable to consider the next logical step: the spectroscopic analysis of the atmospheres of these planets. High-resolution spectroscopy, i.e. resolving $v \times \sin (i)$ of these planets, in the wavelength regime of 950-5500nm is a powerful and promising tool. In view of the obvious contrast problems in detecting such planets non-LTE features are specifically targeted. Sensitivity estimates for the detection of the non-thermal OH glow in oxygen-bearing atmospheres are given. With 8m-class telescopes such a search is impossible, but a dedicated spectrograph, e.g. at the projected ESO 100m OWL telescope could detect Earth-like planets at a distance of ${\approx} 10$ parsec. A conceptual design for a dedicated spectrograph, NOCTUA, is presented. In case of ELTs of smaller size the science case changes and the instrument requirements have to be adjusted. Preparatory work with CRIRES, ESO's Cryogenic Infrared Echelle Spectrograph on the VLT at $\frac{\lambda}{\Delta \lambda} \approx 10^5$ as well as other science cases are shortly discussed.