A general nonlinear regularity model for a set-valued mapping $F:X\times R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping $F:X\rightrightarrows Y$.

Phan Q. Khanh Alexander Y. Kruger Nguyen H. Thao

We investigate the nature of the topological phase transition of the antiferromagnetic Kitaev model on the honeycomb lattice in the presence of a magnetic field along the [111] direction. The field opens a topological gap in the Majorana fermion spectrum and leads to a sequence of topological phase transitions before the field polarised state is reached. At mean field level the gap first closes at the three $M$ points in the Brillouin zone, where the Majorana fermions form Dirac cones, resulting in a change of Chern number by three. An odd number of Dirac fermions in the infrared is unusual and requires Berry curvature compensation in the UV, which occurs via topological, ring-like hybridisation gaps with higher-energy bands. We perform a renormalisation-group analysis of the topological phase transition at the three $M$ points within the Yukawa theory, allowing for intra- and inter-valley fluctuations of the spin-liquid bond operators. We find that the latter lead to a breaking of Lorentz invariance and hence a different universality compared to the standard Ising Gross-Neveu-Yukawa class.

S. Thiagarajan C. Watson T. Yzeiri H. Hu B. Uchoa F. Krüger

We present an omnidirectional matter wave guide on an atom chip. The rotational symmetry of the guide is maintained by a combination of two current carrying wires and a bias field pointing perpendicular to the chip surface. We demonstrate guiding of thermal atoms around more than two complete turns along a spiral shaped 25mm long curved path (curve radii down to 200$\mu$m) at various atom--surface distances (35-450$\mu$m). An extension of the scheme for the guiding of Bose-Einstein condensates is outlined.

X. Luo P. Krüger K. Brugger S. Wildermuth H. Gimpel M. W. Klein S. Groth R. Folman I. Bar-Joseph J. Schmiedmayer

We present the first detector capable of recording high-bandwidth real time atom number density measurements of a Bose Einstein condensate. Based on a two-color Mach-Zehnder interferometer, our detector has a response time that is six orders of magnitude faster than current detectors based on CCD cameras while still operating at the shot-noise limit. With this minimally destructive system it may be possible to implement feedback to stabilize a Bose-Einstein condensate or an atom laser.

C. Figl L. Longchambon M. Jeppesen M. Kruger H. A. Bachor N. P. Robins J. D. Close

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms.

Alexander Y. Kruger D. Russell Luke Nguyen H. Thao

We analyze the low-energy dynamics of quasi one dimensional, large-$S$ quantum antiferromagnets with easy-axis anisotropy, using a semi-classical non-linear sigma model. The saddle point approximation leads to a sine Gordon equation which supports soliton solutions. These correspond to the movement of spatially extended domain walls. Long-range magnetic order is a consequence of a weak inter-chain coupling. Below the ordering temperature, the coupling to nearby chains leads to an energy cost associated with the separation of two domain walls. From the kink-antikink two-soliton solution, we compute the effective confinement potential. At distances large compared to the size of the solitons the potential is linear, as expected for point-like domain walls. At small distances the gradual annihilation of the solitons weakens the effective attraction and renders the potential quadratic. From numerically solving the effective one dimensional Schr\"oedinger equation with this non-linear confinement potential we compute the soliton bound state spectrum. We apply the theory to CaFe$_{2}$O$_{4}$, an anisotropic $S=5/2$ magnet based upon antiferromagnetic zig-zag chains. Using inelastic neutron scattering, we are able to resolve seven discrete energy levels for spectra recorded slightly below the N\'eel temperature $T_\textrm{N}\approx 200$~K. These modes are well described by our non-linear confinement model in the regime of large spatially extended solitons.

H. Lane C. Stock S. -W. Cheong F. Demmel R. A. Ewings F. Krüger

Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. We verify that the limit of the entropy density of triangulations is independent of genus and orientability and are able to determine the next-to-leading and the next-to-next-to-leading order terms. We conjecture for the number of surface triangulations the asymptotic behavior \begin{equation*} N(m,h) \rightarrow (170.4 \pm 15.1)^h m^{-2(h - 1)/5} \left( \frac{256}{27} \right)^{m / 2}\;, \end{equation*} what might guide a mathematicians proof for the exact asymptotics.

Benedikt Krüger Klaus Mecke

The European X-ray Free Electron Laser (XFEL) is a new research facility currently under construction in Hamburg, Germany. With a pulse length of less than 100 fs and an extremely high luminosity of 27000 flashes per second the European XFEL will have a unique time structure that demands the development of new detectors tailored to the requirements imposed by the experiments while complying with the machine specific operation parameters. The Adaptive Gain Integrating Pixel Detector (AGIPD) is one response to the need for large 2D detectors, able to cope with the 4.5 MHz frame rate, as well as with the high dynamic range needed by XFEL experiments ranging from single photons to more than 10$^4$ 12 keV photons per pixel per pulse. In addition it has to withstand doses of up to 1 GGy over three years.

J. Becker L. Bianco P. Göttlicher H. Graafsma H. Hirsemann S. Jack A. Klyuev S. Lange A. Marras U. Trunk R. Klanner J. Schwandt J. Zhang R. Dinapoli D. Greiffenberg B. Henrich A. Mozzanica B. Schmitt X. Shi M. Gronewald H. Krüger

In Part I of this paper a modified BCS mechanism of Cooper pair formation was proposed. The present Part III gives a physical interpretation of this mechanism in terms of spin-flipping processes in superconducting bands.

Ekkehard Krüger

We report on dust measurements obtained during the seventh orbit of the Galileo spacecraft about Jupiter. %which had a close flyby at Ganymede. The most prominent features observed are highly time variable dust streams recorded throughout the Jovian system. The impact rate varied by more than an order of magnitude with a 5 and 10 hour periodicity, which shows a correlation with Galileo's position relative to the Jovian magnetic field. This behavior can be qualitatively explained by strong coupling of nanometer-sized dust to the Jovian magnetic field. In addition to the 5 and 10 h periodicities, a longer period which is compatible with Io's orbital period is evident in the dust impact rate. This feature indicates that Io most likely is the source of the dust streams. During a close (3,095 km altitude) flyby at Ganymede on 5 April 1997 an enhanced rate of dust impacts has been observed, which suggests that Ganymede is a source of ejecta particles. Within a distance of about 25 $\rm R_J$ (Jupiter radius, $\rm R_J = 71,492$ km) from Jupiter impacts of micrometer-sized particles have been recorded which could be particles on bound orbits about Jupiter.

H. Krüger E. Grün A. Graps A. Heck S. Lammers