S-Structures for k-Linear Categories and the Definition of a Modular Functor

Ideas from string theory and quantum field theory have beenthe motivation for new invariants of knots and 3-dimensionalmanifolds which have been constructed from complex algebraicstructures such as Hopf algebras [17, 22], monoidal categorieswith additional structure [24], and modular functors [14, 23].These constructions are closely related. Here we take a unifyingcategorical approach based on a natural 2-dimensional generalisationof a topological field theory in the sense of Atiyah [1], andshow that the axioms defining these complex algebraic structuresare a consequence of the underlying geometry of surfaces.     

From stochastic pulse optimization to a stereoselective laser pulse sequence: simulation of a chiroptical molecular switch mounted on adamantane

Quantum dynamical simulations for the laser-controlled isomerization of 1-(2-cis-fluoroethenyl)-2-fluorobenzene mounted on adamantane are reported based on a one-dimensional electronic ground-state potential and dipole moment calculated by density functional theory. The model system 1-(2-cis-fluoroethenyl)-2-fluorobenzene supports two chiral and one achiral atropisomers upon torsion around the C-C single bond connecting the phenyl ring and ethylene group. The molecule itself is bound to an adamantyl frame which serves as a model for a linker or a surface. Due to the C3 symmetry of the adamantane molecule, the molecular switch can have three equivalent orientations. An infrared picosecond pulse is used to excite the internal rotation around the chiral axis, thereby controlling the chirality of the molecule. In order to selectively switch the molecules--independent of their orientations-- from their achiral to either their left- or right-handed form, a stochastic pulse optimization algorithm is applied. A subsequent detailed analysis of the optimal pulse allows for the design of a stereoselective laser pulse sequence of analytical form. The developed control scheme of elliptically polarized laser pulses is enantioselective and orientation-selective.     

Braids,mapping class groups,and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. The first author was supported by Inha University research grant.     

Dramatically Enhanced Fluorescence of Heteroaromatic Chromophores upon Insertion as Spacers into Oligo(triacetylene)s

In continuation of a previous study on the modulation of π‐electron conjugation of oligo(triacetylene)s by insertion of central hetero‐spacer fragments between two (E)‐hex‐3‐ene‐1,5‐diyne ((E)‐1,2‐diethynylethene, DEE) moieties (Fig. 1), a new series of trimeric hybrid oligomers ( 14 – 18 and 22 – 24 , Fig. 2) were prepared (Schemes 1–3). Spacers used were both electron‐deficient (quinoxaline‐based heterocycles, pyridazine) and electron‐rich (2,2′‐bithiophene, 9,9‐dioctyl‐9H‐fluorene) chromophores. With 19–21 (Scheme 4), a series of transition metal complexes was synthesized as potential precursors for nanoscale scaffolding based on both covalent acetylenic coupling and supramolecular assembly. The UV/VIS spectra (Fig. 3) revealed that the majority of spacers provided hetero‐trimers featuring extended π‐electron delocalization. The new hybrid chromophores show a dramatically enhanced fluorescence compared with the DEE dimer 13 and homo‐trimer 12 (Fig. 5). This increase in emission intensity appears as a general feature of these systems: even if the spacer molecule is non‐fluorescent, the corresponding hetero‐trimer may show a strong emission (Table 2). The redox properties of the new hybrid chromophores were determined by cyclic voltammetry (CV) and rotating‐disk voltammetry (RDV) (Table 3 and Fig. 5). In each case, the first one‐electron reduction step in the hetero‐trimers appeared anodically shifted compared with DEE dimer 13 and homo‐trimer 12 . With larger spacer chromophore extending into two dimensions (as in 14 – 18 , Fig. 2), the anodic shift (by 240–490 mV, Table 3) seems to originate from inductive effects of the two strongly electron‐accepting DEE substituents rather than from extended π‐electron conjugation along the oligomeric backbone, as had previously been observed for DEE‐substituted porphyrins.     

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let ${\mathcal{T}}$ be an ideal triangulation of M. The deformation variety ${\mathfrak{D}(\mathcal{T})}$ , a subset of which parameterises (incomplete) hyperbolic structures obtained on M using ${\mathcal{T}}$ , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.     

Extrinsic homogeneity of parallel submanifolds

We consider parallel submanifolds M of a Riemannian symmetric space N and study the question whether M is extrinsically homogeneous in N, i.e. whether there exists a subgroup of the isometry group of N which acts transitively on M. Provided that N is of compact or non-compact type, we establish the extrinsic homogeneity of every complete irreducible parallel submanifold of N whose dimension is at least three and which is not contained in any flat of N.     

Spherical aberration correction in tandem with exit-plane wave function reconstruction: interlocking tools for the atomic scale imaging of lattice defects in GaAs.

With the availability of resolution boosting and delocalization minimizing techniques, for example, spherical aberration correction and exit-plane wave function reconstruction, high-resolution transmission electron microscopy is drawing to a breakthrough with respect to the atomic-scale imaging of common semiconductor materials. In the present study, we apply a combination of these two state-of-the-art techniques to investigate lattice defects in GaAs-based heterostructures at atomic resolution. Focusing on the direct imaging of stacking faults as well as the core structure of edge and partial dislocations, the practical capabilities of both techniques are illustrated. For the first time, we apply the technique of bright-atom contrast imaging at negative spherical aberration together with an appropriate overfocus setting for the investigation of lattice defects in a semiconductor material. For these purposes, the elastic displacements associated with lattice defects in GaAs viewed along the 110 zone axis are measured from experimental images using reciprocal space strain map algorithms. Moreover, we demonstrate the benefits of the retrieval of the exit-plane wave function not only for the elimination of residual imaging artefacts but also for the proper on-line alignment of specimens during operation of the electron microscope--a basic prerequisite to obtain a fair agreement between simulated images and experimental micrographs.     

Vergleichende Untersuchungen zur Struktur von 4‐Dimethylaminopyridin‐Addukten

Comparative Structural Studies on 4‐Dimethylaminopyridine‐Adducts Lewis acid‐base adducts of the type dmap—MMe₃ (M = Al 1 , Ga 2 , In 3 , Tl 4 ) as well as dmap—AlCl₃ ( 6 ) and dmap—Al(t‐Bu)₃ ( 7 ) were synthesized by reaction of MR₃ with 4‐dimethylamino‐pyridine (dmap) whereas dmap—AlH₃ ( 5 ) was obtained from AlH₃·Et₂O. 1 — 7 were characterized by means of NMR (¹H, ¹³C{¹H}) and mass spectrometry and elemental analysis. In addition, their solid state structures were determined by single crystal X‐ray diffraction studies. A comparison of the structural parameters reveales the influence of both electronic (Lewis acidity of the group 13 atom) and steric interactions on the structure and stability of as prepared Lewis acid‐base adducts.