On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets

An analysis of crystal cleavage in the passage from atomistic models to continuum theory

We study the behavior of atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.Comment: The final publication is available at springerlink.co

Supersymmetry and eigensurface topology of the planar quantum pendulum

We make use of supersymmetric quantum mechanics (SUSY QM) to find three sets of conditions under which the problem of a planar quantum pendulum becomes analytically solvable. The analytic forms of the pendulum's eigenfuntions make it possible to find analytic expressions for observables of interest, such as the expectation values of the angular momentum squared and of the orientation and alignment cosines as well as of the eigenenergy. Furthermore, we find that the topology of the intersections of the pendulum's eigenenergy surfaces can be characterized by a single integer index whose values correspond to the sets of conditions under which the analytic solutions to the quantum pendulum problem exist

Supersymmetry and eigensurface topology of the spherical quantum pendulum

We undertook a mutually complementary analytic and computational study of the full-fledged spherical (3D) quantum rotor subject to combined orienting and aligning interactions characterized, respectively, by dimensionless parameters $\eta$ and $\zeta$. By making use of supersymmetric quantum mechanics (SUSY QM), we found two sets of conditions under which the problem of a spherical quantum pendulum becomes analytically solvable. These conditions coincide with the loci $\zeta=\frac{\eta^2}{4k^2}$ of the intersections of the eigenenergy surfaces spanned by the $\eta$ and $\zeta$ parameters. The integer topological index $k$ is independent of the eigenstate and thus of the projection quantum number $m$. These findings have repercussions for rotational spectra and dynamics of molecules subject to combined permanent and induced dipole interactions.Comment: arXiv admin note: text overlap with arXiv:1404.224

Topology of surfaces for molecular Stark energy, alignment and orientation generated by combined permanent and induced electric dipole interactions

We show that combined permanent and induced electric dipole interactions of polar and polarizable molecules with collinear electric fields lead to a sui generis topology of the corresponding Stark energy surfaces and of other observables - such as alignment and orientation cosines - in the plane spanned by the permanent and induced dipole interaction parameters. We find that the loci of the intersections of the surfaces can be traced analytically and that the eigenstates as well as the number of their intersections can be characterized by a single integer index. The value of the index, distinctive for a particular ratio of the interaction parameters, brings out a close kinship with the eigenproperties obtained previously for a class of Stark states via the apparatus of supersymmetric quantum mechanics.Comment: 22 pages, including 2 tables and 8 figure

"weit vor dem Ende abgebrochen [...]" : Kafkas fragmentarisches Schreiben

Der vorliegende Beitrag - die stark überarbeitete Fassung des Vortrages "Lektüren des Unausdeutbaren ", den der Verfasser im Januar 2004 im Rahmen der kulturwissenschaftlichen Vortragsreihe GrenzBereiche des Lesens hielt - ist auch erschienen in: literatur für leser 27 (2004), Heft 4, S. 181-199. Ein Beispiel für die Schwierigkeit der literarischen Lektüre gibt Friedrich Schmidt. Er untersucht Lektüremöglichkeiten für das formale wie semantische "Abgebrochensein" des literarischen Kunstwerks der Moderne, wie es in den Fragmenten Kafkas seinen exponierten Ausdruck findet. In diesen Texten tritt zur äußeren Unabgeschlossenheit des Textkorpus ein brüchiges Sinngefüge: die Endlosigkeit der Reflexionen und Handlungszüge, die Heterogenität der Erzählfiguren, die Inkonsistenz jeder Bedeutungskonstruktion von Seiten des Lesers. Indem überdies der Abbruch, als verlorene Schrift oder verschwiegene Botschaft, in Kafkas Fragmenten explizit zum Thema wird, erhebt sich der Text gleichzeitig zum metareflexiven Kommentar: er vollzieht selbst, wovon er spricht. Insofern handelt er – sprachskeptisch – vom Defizit seines eigenen Ausdrucks, das letztlich auch von Lektüren nie vollständig eingeholt werden kann