Indachlorins: Nonplanar Indanone‐Annulated Chlorin Analogues with Panchromatic Absorption Spectra between 300 and 900 nm

Indaphyrins, pyrrole‐modified porphyrins containing a cleaved pyrrole β,β′‐bond and two annulated indanone moieties, possess unusually broadened and redshifted UV/Vis spectra because of their π‐expanded chromophores. The parent free base indaphyrin has been crystallographically characterized, highlighting its strongly ruffled conformation incorporating a helimeric twist. It was shown to be susceptible to regiospecific derivatizations at the opposite side of the ring‐cleaved pyrrole (dihydroxylation, followed by functional group transformations of the resulting diol functionality), generating indaphyrin‐based chlorin analogues, indachlorins, that incorporate a dihydroxypyrroline, pyrrolindione, oxazolone, or a morpholine moiety. Structural modifications resulted in further broadening and hyper‐ and bathochromic shifts of the optical spectra, some of which possess a nearly panchromatic absorption between 300 to well above 900 nm. The extents to which these modifications affect their solid‐state conformations were analyzed.     

A characteristic property of the Euclidean disc

In this paper the following is proved: Let K ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a smooth strictly convex body, and let L ⊂ $ \mathbb{E}^2 $ \mathbb{E}^2 be a line. Assume that for every point x ∈ L/K the two tangent segments from x to K have the same length, and the line joining the two contact points passes through a fixed point in the plane. Then K is an Euclidean disc.     

On a Helly-type question for central symmetry

Periodica Mathematica Hungarica - We study a certain Helly-type question by Konrad Swanepoel. Assume that X is a set of points such that every k-subset of X is in centrally symmetric convex...     

Measure partitions using hyperplanes with fixed directions

We study nested partitions of R^d obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among k sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any t measures there is a path formed only by horizontal and vertical segments using at most t - 1 turns that splits them by half simultaneously, and optimal masspartitioning results for chessboard colourings of R^d using hyperplanes with fixed directions.     

Unfolding physics from the algebraic classification of spinor fields

After reviewing the Lounesto spinor field classification, according to the bilinear covariants associated to a spinor field, we call attention and unravel some prominent features involving unexpected properties about spinor fields under such classification. In particular, we pithily focus on the new aspects — as well as current concrete possibilities. They mainly arise when we deal with some non-standard spinor fields concerning, in particular, their applications in physics.     

IRK Methods for Index 2 and 3 DAEs: Starting Algorithms

When semi-explicit differential-algebraic equations are solved with implicit Runge-Kutta methods, the computational effort is dominated by the cost of solving the non-linear systems. That is why it is important to have good starting values to begin the iterations. In this paper we study a type of starting algorithms, without additional computational cost, in the case of index-2 and index-3 DAEs. The order of the starting values is defined, and by using DA-series and rooted trees we obtain their general order conditions. If the RK method satisfies some simplifying assumptions, then the maximum order can be obtained.