Serendipitous synthesis of a ditwistane: a one-step access!

4-tert-Butyl-2-cyclohexen-1-one dimerizes in THF solution via its kinetic enolate, leading to di-tert-butylditwistane 8 in up to 36% yield (−78 °C → room temp., protonolysis, flash chromatography). X-ray crystallography shows that 8 incorporates one R and one S enantiomer of the starting ketone; none of the diastereomeric ditwistanes epi-8, epi’-8 or iso-8 was isolated. This means that the formation of 8 proceeds with mutual kinetic resolution and 100% induced diastereoselectivity.     

Non-projectability of polytope skeleta

We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal and Ziegler (2010) and the wedge product surfaces of Rörig and Ziegler (2011) and complement their results.     

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.     

Fermionic atoms in a three dimensional optical lattice: observing Fermi surfaces, dynamics, and interactions

We have studied interacting and noninteracting quantum degenerate Fermi gases in a three-dimensional optical lattice. We directly image the Fermi surface of the atoms in the lattice by turning off the optical lattice adiabatically. Because of the confining potential, gradual filling of the lattice transforms the system from a normal state into a band insulator. The dynamics of the transition from a band insulator to a normal state is studied, and the time scale is measured to be an order of magnitude larger than the tunneling time in the lattice. Using a Feshbach resonance, we increase the interaction between atoms in two different spin states and dynamically induce a coupling between the lowest energy bands. We observe a shift of this coupling with respect to the Feshbach resonance in free space which is anticipated for strongly confined atoms.     

Explicit Solution of a Non-Linear Filtering Problem for Lévy Processes with Application to Finance

In this paper we explicitly solve a non-linear filtering problem with mixed observations, modelled by a Brownian motion and a generalized Cox process, whose jump intensity is given in terms of a Lévy measure. Motivated by empirical observations of R. Cont and P. Tankov we propose a model for financial assets, which captures the phenomenon of time inhomogeneity of the jump size density. We apply the explicit formula to obtain the optimal filter for the corresponding filtering problem.     

Binary boron-rich borides of magnesium: single-crystal investigations and properties of MgB(7) and the new boride Mg(∼5)B(44)

Single crystals of dark-red MgB(7) were grown from the elements in a Cu-melt. The crystal structure (Pearson symbol oI64; space group Imma; a = 10.478(2) ?, b = 5.977(1) ?, c = 8.125(2) ?, 2842 reflns, 48 params, R(1)(F) = 0.018, R(2)(I) = 0.034) consists of a hexagonal-primitive packing of B(12)-icosahedra and B(2)-units in trigonal-prismatic voids. According to the UV-vis spectra and band structure calculations MgB(7) is semiconducting with an optical gap of 1.9 eV. The long B-B distance of 2.278 ? within the B(2)-unit can be seen as a weak bonding interaction. The new Mg(～5)B(44) occurs beside the well-known MgB(12) as a byproduct. Small fragments of the black crystals are dark-yellow and transparent. The crystal structure (Pearson symbol tP196, space group P4(1)2(1)2, a = 10.380(2) ?, c = 14.391(3) ?, 4080 reflns, 251 params, R(1)(F) = 0.025, R(2)(I) = 0.037) is closely related to tetragonal boron-II (t-B(192)). It consists of B(12)-icosahedra and B(19+1)-units. With a charge of -6 for the B(19+1)-units and a Mg-content of ～20 Mg-atoms per unit cell the observed Mg content in Mg(～5)B(44) is quite close to the expected value derived from simple electron counting rules. All compositions were confirmed by EDXS. The microhardness was measured on single crystals for MgB(7) (H(V) = 2125, H(K) = 2004) and MgB(12) (H(V) = 2360, H(K) = 2459).     

An invariant of nonpositively curved contact manifolds

We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can be approximated by isotopies of a given contact structure.