Photoinduced Luminescence Blinking and Bleaching in Individual Single‐Walled Carbon Nanotubes

The temporal evolution of photoluminescence in individual single‐walled carbon nanotubes (SWNT) under strong laser irradiation is studied and pronounced blinking and bleaching is observed, caused by photoinduced oxidation that subsequently quenches mobile excitons. The nanotubes are isolated with sodium cholate and spun onto either a glass or mica surface. Their bleaching behavior is investigated for variable laser intensities in air and argon atmosphere. The decay rate for luminescence bleaching generally increases with higher laser intensity, however saturating on mica substrates, which is attributed to limited availability of oxygen in the vicinity of the nanotubes. Step‐like events in the luminescence time traces corresponding to single oxidation events are analyzed regarding relative step height and suggest an exciton diffusion range of about 105 nm.     

Rhodium(I)‐Catalyzed Enantioselective Activation of Cyclobutanols: Formation of Cyclohexane Derivatives with Quaternary Stereogenic Centers

The activation of carbon–carbon σ bonds is a complementary method to access uncommon and difficult‐to‐prepare organometallic species. Herein, we describe the activation of tert‐cyclobutanols through an enantioselective insertion of a chiral rhodium(I) complex into the C? C σ bond of the cyclobutane, forming a quaternary stereogenic center and an alkyl‐rhodium functionality that initiates ring‐closure reactions. This technology provides access to a variety of substituted cyclohexane derivatives with quaternary stereogenic centers. The formation of different product families can be controlled by the employed set of reaction conditions and additives. In general, high yields and excellent enantioselectivities of up to 99 % ee are obtained.     

Sugawara-Type Constraints in Hyperbolic Coset Models

In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E ₁₀/K(E ₁₀) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E ₁₀ Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in Damour et al. (Class. Quant. Grav. 24:6097, 2007) that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E ₁₀ Weyl group and of the fact that the E ₁₀ model contains both D = 11 supergravity and D = 10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of ‘open constraint algebras’ in traditional canonical approaches to gravity.     

Ueber die Convolvulaceen-Glykoside

Ohne Zusammenfassung     

Area-preserving diffeomorphisms and supermembrane Lorentz invariance

The Lie algebra of area-preserving diffeomorphisms on closed membranes of arbitrary topology is investigated. On the basis of a harmonic decomposition we define the structure constants as well as two other tensors which appear in the supermembrane Lorentz generators. We derive certain identities between these tensors and analyze their validity when the areapreserving diffeomorphisms are approximated bySU(N). One of the additional tensors can then be identified with the invariant symmetric three-index tensor ofSU(N), while the second has no obvious analog. We prove that the Lorentz generators are classically conserved in the light-cone gauge for arbitrary membrane topology, as a consequence of these tensor identities. This formulation allows a systematic study of the violations of Lorentz invariance in theSU(N) approximation.     

On Topological Lower Bounds for Algebraic Computation Trees

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set \(\Sigma \subset {\mathbb R}^n\) is bounded from below by

$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\Sigma ))}{m+1} -c_2n, \end{aligned}$$

where \(\mathrm{b}_m(\Sigma )\) is the mth Betti number of \(\Sigma \) with respect to “ordinary” (singular) homology and \(c_1,\ c_2\) are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36–43, 1997) for locally closed semialgebraic sets in terms of the total Borel–Moore Betti number. We also prove that if \(\rho :\> {\mathbb R}^n \rightarrow {\mathbb R}^{n-r}\) is the projection map, then the height of any tree deciding membership in \(\Sigma \) is bounded from below by

$$\begin{aligned} \frac{c_1\log (\mathrm{b}_m(\rho (\Sigma )))}{(m+1)^2} -\frac{c_2n}{m+1} \end{aligned}$$

for some positive constants \(c_1,\ c_2\). We illustrate these general results by examples of lower complexity bounds for some specific computational problems.