Thomas de Padova: Alles wird Zahl: Wie sich die Mathematik in der Renaissance neu erfand

Mathematische Semesterberichte -     

Editorial

Mathematische Semesterberichte -     

The Geodesics of Metric Connections with Vectorial Torsion

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for such connections from isometries of the conformally equivalent metric, and we explain in as much this result generalizes the Mercator projection which maps sphere loxodromes to straight lines in the plane. An example shows that Beltrami's theorem fails for this class of connections. We then study the system of differential equations describing geodesics in the plane for vector fields which are not gradients, and show among others that the Hopf–Rinow theorem does also not hold in general.     

Connections on Naturally Reductive Spaces,Their Dirac Operator and Homogeneous Models in String Theory

 Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections ∇ ^t joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D ^t corresponding to t=1/3 is the so-called ``cubic' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first eigenvalue of D <sup>1/3</sup>. This geometric situation can be used for constructing Riemannian manifolds which are Ricci flat and admit a parallel spinor with respect to some metric connection ∇ whose torsion T≠ 0 is a 3-form, the geometric model for the common sector of string theories. We present some results about solutions to the string equations and a detailed discussion of a 5-dimensional example. Received: 19 February 2002 / Accepted: 26 August 2002 Published online: 22 November 2002 RID="*" ID="*" This work was supported by the SFB 288 ``Differential geometry and quantum physics' of the Deutsche Forschungsgemeinschaft and the Max-Planck Society.     

Coverage-mediated suppression of blinking in solid state quantum dot conjugated organic composite nanostructures

Size-correlated single-molecule fluorescence measurements on CdSe quantum dots functionalized with oligo(phenylene vinylene) (OPV) ligands exhibit modified fluorescence intermittency (blinking) statistics that are highly sensitive to the degree of ligand coverage on the quantum dot surface. As evidenced by a distinct surface height signature, fully covered CdSe-OPV nanostructures (approximately 25 ligands) show complete suppression of blinking in the solid state on an integration time scale of 1 s. Some access to dark states is observed on finer time scales (100 ms) with average persistence times significantly shorter than those from ZnS-capped CdSe quantum dots. This effect is interpreted as resulting from charge transport from photoexcited OPV into vacant trap sites on the quantum dot surface. These results suggest exciting new applications of composite quantum dot/organic systems in optoelectronic systems.     

An enhanced host–guest electro‐optical polymer system using poly(norbornene‐dicarboximides) via ROMP

High glass transition temperature poly(N‐cyclohexyl‐5‐norbornene‐2,3‐dicarboximide)s (NDI)s prepared by ring opening metathesis polymerization yielded polymers with a narrow polydispersity and well‐controlled molecular weight materials when using the Grubbs first generation initiator. Polymers produced using the Grubbs second generation initiator could not be controlled easily. By initiator selection it was also possible to synthesize polymers with either 98 or 52% trans microstructures. These materials were employed as electro‐optic (EO) polymer hosts for high molecular hyperpolarizability (β) phenyl vinylene thiophene vinylene bridge chromophores. This chromophore was modified by the incorporation of a tert‐butyldiphenylsilane group. The addition was able to further increase its EO coefficient (r₃₃) to reach 93 pm/V in a trans rich poly(NDI) produced by the Grubbs first generation initiator, compared to a benchmark chromophore / polymer combination. We investigated in detail the relationship between polymer microstructure and their absolute molecular weight on forming the best host–guest with the high β chromophore. Our results indicate that by utilizing a very simple host–guest system a high r₃₃ can be realized. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Water‐proton relaxivity of hyperbranched polymers carrying TEMPO radicals

High water‐soluble hyperbranched poly(styrene) (HPS) polymers carrying stable 2, 2, 6, 6‐tetramethylpiperidine‐1‐oxyl (TEMPO) radicals, HPS‐N‐TEMPO, HPS‐Im‐TEMPO, and HPS‐Im‐(TEMPO)₂, were prepared in ca. 60% introducing yield. HPS‐N‐TEMPO and HPS‐Im‐TEMPO were determined to be nearly spherical shapes of the diameter of 2.4 ± 0.6 and 2.2 ± 0.6 nm, respectively, by transmission electron microscope (TEM) images. The values of water‐proton relaxivity, r₁, at 25 MHz, 0.59 T, and 25 °C were 6.0, 5.2, and 14 mM^?1 sec^?1 for HPS‐N‐TEMPO, HPS‐Im‐TEMPO, and HPS‐Im‐(TEMPO)₂, respectively. The spin‐lattice relaxation time (T₁)‐weighted images in phantom were also observed. Copyright © 2009 John Wiley & Sons, Ltd.     

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple (Mⁿ,g,) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second-order operator Ω acting on spinor fields. In case of a naturally reductive space and its canonical connection, our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated G₂-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of -parallel spinors.     

Observation of enhanced energy transfer in individual quantum dot-oligophenylene vinylene nanostructures

The temporal and spectral properties of luminescence from individual CdSe quantum dot-oligophenylene vinylene nanostructures (single quantum dots with conjugated organic ligands coordinated to the surface) are profoundly modified relative to blended films of the same components. These kinds of composite quantum dot-conjugated organic systems have attracted significant interest as a way to improve efficiency in photovoltaic device applications. By direct functionalization of the dot surface with the conjugated organic ligands, we realize a significant enhancement in energy transfer and luminescence stability.