Diszkrét és kombinatórikus geometriai kutatások = Topics in discrete and combinatorial geometry

A most lezárult OTKA grant, 8 résztvevő diszkrét geometriai kutatását támogatta. Itt a témák ilusztrálására kiemelünk néhányat az elért 72 publikációból. 1. Jelentős eredmények születtek (8 cikk) gráfok síkba rajzolhatóságáról, például az úgynevezett metszési számról. 2. Többek között sikerült igazolni Katchalski és Lewis 20 éves sejtését, mely szerint diszjunkt egységkörökből álló rendszereknél ha bármely három körnek van közös metsző egyenese akkor van olyan egyenes, amely legfeljebb 2 kör kivételével valamennyit metsz. 3. Littlewood (1964) problémájaként ismert volt az a kérdés, hogy hány henger érintheti kölcsönösen egymást? Viszonylag alacsony felső korlátot találtunk és egy régóta ismert elhelyzés valótlanságát igazoltuk. 4. Többszörös fedések egyszerű fedésekre való szétbontását vizsgáltuk és értünk el lényeges előrelépést. 5. A Borsuk-féle darabolási problémanak azt a variánsát vizsgáltuk, amelyben a darabolást u. n. hengeres darabolásra korlátozták. 6. Bebizonyítottuk, hogy ''nem nagyon elnyúlt'' ellipszisek esetében a sík legritkább fedésének meghatározásánál el lehet tekinteni az u.n. nem-keresztezési feltételtől. 7. A sejtetthez nagyon közeli korlátot találtunk arra a problémára, hogy az n-dimenziós térben legfeljebb hány homotetikus konvex test helyezhető el úgy, hogy bármely kettő érintse egymást. | Discrete geometry in Hungary flourished since the sixties as a result of the work of László Fejes Tóth. The supported research of 8 participant also belongs to this area. Here we illustrate the achieved 72 publications by mentioning a few results. 1. Important theorems (8 papers) were proved concerning graph drawing. 2. Among others, a 20 year old problem of Katchalsky was proved, stating that in a packing of congruent circles, if any three has a common transversal, then there is a line, which avoids at most two of the circles. 3. Concerning a conjecture of Littlewood we found a small upper bound for the number of infinite cylinders which mutually touch each other. 4. We studied decomposability of multiple coverings into single coverings. 5. We studied that variant of the famous Borsuk problem where the partitions are restricted to cylindrical partitions. 6. We proved that in case of ellipses which are not ''too long'' at determining the thinnest covering one can omit the usually needed noncrossing condition. 7. A bound close to the conjectured bound was found concerning the number of n-dimensional homothetic convex solids which mutually touch each other

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

Overview of progress in European medium sized tokamaks towards an integrated plasma-edge/wall solution

Integrating the plasma core performance with an edge and scrape-off layer (SOL) that leads to tolerable heat and particle loads on the wall is a major challenge. The new European medium size tokamak task force (EU-MST) coordinates research on ASDEX Upgrade (AUG), MAST and TCV. This multi-machine approach within EU-MST, covering a wide parameter range, is instrumental to progress in the field, as ITER and DEMO core/pedestal and SOL parameters are not achievable simultaneously in present day devices. A two prong approach is adopted. On the one hand, scenarios with tolerable transient heat and particle loads, including active edge localised mode (ELM) control are developed. On the other hand, divertor solutions including advanced magnetic configurations are studied. Considerable progress has been made on both approaches, in particular in the fields of: ELM control with resonant magnetic perturbations (RMP), small ELM regimes, detachment onset and control, as well as filamentary scrape-off-layer transport. For example full ELM suppression has now been achieved on AUG at low collisionality with n  =  2 RMP maintaining good confinement ${{H}_{\text{H}\left(98,\text{y}2\right)}}\approx 0.95$ . Advances have been made with respect to detachment onset and control. Studies in advanced divertor configurations (Snowflake, Super-X and X-point target divertor) shed new light on SOL physics. Cross field filamentary transport has been characterised in a wide parameter regime on AUG, MAST and TCV progressing the theoretical and experimental understanding crucial for predicting first wall loads in ITER and DEMO. Conditions in the SOL also play a crucial role for ELM stability and access to small ELM regimes

Community Resilience Engineering: Reflections and Preliminary Contributions

An important challenge for human societies is that of mastering the complexity of Community Resilience, namely “the sustained ability of a community to utilize available resources to respond to, withstand, and recover from adverse situations”. The above concise definition puts the accent on an important requirement: a community’s ability to make use in an intelligent way of the available resources, both institutional and spontaneous, in order to match the complex evolution of the “significant multi-hazard threats characterizing a crisis”. Failing to address such re-quirement exposes a community to extensive failures that are known to exacerbate the consequences of natural and human-induced crises. As a consequence, we experience today an urgent need to respond to the challenges of community resilience engineering. This problem, some re-flections, and preliminary prototypical contributions constitute the topics of the present article.

Capabilities of alkali Beam Emission Spectroscopy for density profile and fluctuation measurements

A comprehensive, IDL language BES [1] simulation program has been developed, the primary objective of which is to support the design and development of atomic beam diagnostic measurements. The simulation models the beam-plasma interaction, considering the atomic physical processes according to a Collisional-Radiative model The simulation calculates the light profile, using a magnetic geometry and measured or simulated one dimensional plasma parameter profiles -n e , T e , Z eff , average impurity chargeas a function of flux coordinates. In addition it takes the observation process into account, including the geometrical factors of the set-up, the efficiency of the whole optical system, the characteristics of the filter, the Doppler-shift of the detected light and the covering of the observed region by detector segments. There is also the possibility to calculate with "extended -when the beam is deflected between two or more positions -can also be simulated by the code. The effect of observation It can be concluded from the extended beam calculations that the evolution of the populations of excited atomic states -thus the intensity of the spontaneous emission -follows well the flux surfaces. As a consequence if the lines of sight are not tangential to the flux surfaces (like i

Experimental investigation of the radial structure of energetic particle driven modes

Alfvén eigenmodes (AEs) and energetic particle modes (EPMs) are often excited by energetic particles (EPs) in tokamak plasmas. One of the main open questions concerning EP driven instabilities is the non-linear evolution of the mode structure. The aim of the present paper is to investigate the properties of beta-induced AEs (BAEs) and EP driven geodesic acoustic modes (EGAMs) observed in the ramp-up phase of off-axis NBI heated ASDEX Upgrade (AUG) discharges. This paper focuses on the changes in the mode structure of BAEs/EGAMs during the non-linear chirping phase. Our investigation has shown that in the case of the observed down-chirping BAEs the changes in the radial structure are smaller than the uncertainty of our measurement. This behaviour is most probably the consequence of the fact that BAEs are normal modes, thus their radial structure strongly depends on the background plasma parameters rather than on the EP distribution. In the case of rapidly upward chirping EGAMs the analysis consistently shows shrinkage of the mode structure. The proposed explanation is that the resonance in the velocity space moves towards more passing particles which have narrower orbit widths

Deconvolution-based correction of alkali beam emission spectroscopy density profile measurements

A deconvolution-based correction method of the beam emission spectroscopy (BES) density profile measurement is demonstrated by its application to simulated measurements of the COMPASS and TEXTOR tokamaks. If the line of sight is far from tangential to the flux surfaces, and the beam width is comparable to the scale length on which the light profile varies, the observation may cause an undesired smoothing of the light profile, resulting in a non-negligible underestimation of the calculated density profile. This effect can be reduced significantly by the emission reconstruction method, which gives an estimate of the emissivity along the beam axis from the measured light profile, taking the finite beam width and the properties of the measurement into account in terms of the transfer function of the observation. Characteristics and magnitude of the mentioned systematic error and its reduction by the introduced method are studied by means of the comprehensive alkali BES simulation code RENATE