Hamiltonian-connected tournaments

We characterize weakly Hamiltonian-connected tournaments and weakly panconnected tournaments completely and we apply these results to cycles and bypasses in tournaments with given irregularity, in particular, in regular and almost regular tournaments. We give a sufficient condition in terms of local and global connectivity for a Hamiltonian path with prescribed initial and terminal vertex. From this result we deduce that every 4-connected tournament is strongly Hamiltonian-connected and that every edge of a 3-connected tournament is contained in a Hamiltonian cycle of the tournament and we describe infinite families of tournaments demonstrating that these results are best possible.     

3-connected planar spaces uniquely embed in the sphere

We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in has a unique accumulation point in , then has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.

     

The Theorem of Fáry and Milnor for Hadamard Manifolds

Suppose M is a complete noncompact surface C² immersed in E³ with K 0. It is proved that M is a cylinder if H 2 for some > 0. Moreover, the pinching constant 2 is optimal.     

On planarity of compact, locally connected, metric spaces

Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅ or K _3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅, K _3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.     

PTIME Parametric Verification of Safety Properties for Reasonable Linear Hybrid Automata

This paper identifies an industrially relevant class of linear hybrid automata (LHA) called reasonable LHA for which parametric verification of convex safety properties with exhaustive entry states can be verified in polynomial time and time-bounded reachability can be decided in nondeterministic polynomial time for non-parametric verification and in exponential time for parametric verification. Properties with exhaustive entry states are restricted to runs originating in a (specified) inner envelope of some mode-invariant. Deciding whether an LHA is reasonable is shown to be decidable in polynomial time.     

Determining conformal transformations in from minimal correspondence data

In this paper, we derive a method to determine a conformal transformation in n‐dimensional Euclidean space in closed form given exact correspondences between data. We show that a minimal data set needed for correspondence is a localized vector frame and an additional point. In order to determine the conformal transformation, we use the representation of the conformal model of geometric algebra by extended Vahlen matrices— 2 ×2 matrices with entries from Euclidean geometric algebra (the Clifford algebra of ). This reduces the problem on the determination of a Euclidean orthogonal transformation from given vector correspondences, for which solutions are known. We give a closed form solution for the general case of conformal (in contrast, anti‐conformal) transformations, which preserve (in contrast, reverse) angles locally, as well as for the important special case when it is known that the conformal transformation is a rigid body motion—also known as a Euclidean transformation—which additionally preserves Euclidean distances. Copyright © 2011 John Wiley & Sons, Ltd.     

Planar and infinite hypohamiltonian and hypotraceable graphs

Chvátal raised the question whether or not planar hypohamiltonian graphs exist and Grünbaum conjectured the nonexistence of such graphs. We shall describe an infinite class of planar hypohamiltonian graphs and infinite classes of planar hypotraceable graphs of connectivity two (resp. three). Infinite hypohamiltonian (resp. htpotraceable) graphs are also described. It is shown how the study of infinite hypotraceable graphs leads to a new infinite family of finite hypotraceable graphs.     

Saturation of the all-optical Kerr effect in solids

We discuss the influence of the higher-order Kerr effect (HOKE) in wide bandgap solids at extreme intensities below the onset of optically induced damage. Using different theoretical models, we employ multiphoton absorption rates to compute the nonlinear refractive index by a Kramers-Kronig transform. Within this theoretical framework we provide an estimate for the appearance of significant deviations from the standard optical Kerr effect predicting a linear index change with intensity. We discuss the role of the observed saturation behavior in practically relevant situations, including Kerr lens mode-locking and supercontinuum generation in photonic crystal fibers. Furthermore, we present experimental data from a multiwave mixing experiment in BaF2, which can be explained by the appearance of the HOKE.     

Few-cycle-driven XUV laser harmonics: generation and focusing

We have investigated the use of sub-10-fs near-infrared laser pulses to generate high-order harmonic radiation efficiently in the wavelength region between 30 to 10 nm. The ultrashort rise time of the driver pulses allows harmonics to be produced at low ionization levels and hence to grow coherently over propagation lengths becoming comparable to the XUV absorption lengths in the noble gas medium. As a result, absorption-limited harmonic generation has been extended to the 10-nm range for the first time. Harmonic conversion efficiencies of (3-4)᎒^ъ in the range of 10-13 nm in neon and some two orders of magnitude higher at around 30 nm in argon have been obtained in simple gas tube targets under these conditions. Preliminary focusing tests with 13-nm harmonic radiation have been carried out with a specially designed zoneplate and a spherical Mo/Si multilayer mirror and have resulted in spot sizes of about 2 microns. Our experiments aim at paving the way to nonlinear optics in the soft-X-ray regime.