A facile and practical one-pot ‘catch and release’ synthesis of substituted guanidines

A ‘catch and release’ two-step one-pot protocol has been developed for the facile and practical synthesis of substituted guanidines from thioureas and various amines utilizing readily available brominated polystyrene resin.     

Wavelets for multichannel signals

In this paper, we introduce and investigate multichannel wavelets, which are wavelets for vector fields, based on the concept of full rank subdivision operators. We prove that, like in the scalar and multiwavelet case, the existence of a scaling function with orthogonal integer translates guarantees the existence of a wavelet function, also with orthonormal integer translates. In this context, however, scaling functions as well as wavelets turn out to be matrix-valued functions.     

Julia directions of meromorphic functions and their derivatives

We prove that if a transcendental meromorphic function has no Julia direction and is bounded on a path to ￥ \infty then there is a common Julia direction for all derivatives. Related statements are obtained under the assumption that f is o(?{ | z | }) o(\sqrt{\mid z \mid}) or O(?{ | z | }) O(\sqrt{\mid z \mid}) on a path to ￥ \infty . Further we disprove a conjecture of Frank and Wang by means of a counterexample.     

Integrated optical components on atom chips

We report on the integration of small-scale optical components into silicon wafers for use in atom chips. We present an on-chip fibre-optic atom detection scheme that can probe clouds with small atom numbers. The fibres can also be used to generate microscopic dipole traps. We describe our most recent results with optical microcavities and show that a sufficiently high finesse can be achieved to enable single-atom detection on an atom chip. The key components have been fabricated by etching directly into the atom chip silicon substrate.     

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

C1-continuous time integration based on cubic Hermite interpolation

We discuss a C¹-continuous time integration method based on piecewise cubic Hermite approximation. This method, denoted as p2-scheme, belongs to a class of one-step integration methods derived recently [1]. It exhibits a convergence rate of order four and shows properties similar to variational integrators, such as an excellent long-term energy preservation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational chemistry and molecular simulations of phosphoric acid

Quantum mechanics (QM) calculations, molecular dynamics (MD) simulations using the condensed‐phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field, and the atom‐centered density matrix propagation (ADMP) approach have been used to investigate properties of phosphoric acid (PA). QM using B3LYP/6‐31++G(d,p) density functional theory were used to calculate gas‐phase proton affinities and interaction energies of PA and its derivatives. Detailed single coordinate driving, followed by quadratic synchronous transit optimization was used to determine energy barriers for different proton transfer (PT) pathways. Determined energy barrier heights in ascending order are (unit: kJ/mol): H₃O⁺→H₃PO₄ (0); H₄P₂O₇→H₃PO₄ (2.61); H₃PO₄→H₂PO (5.31); H₄PO→H₃PO₄ (～7.33); H₃PO₄→H₄P₂O₇/H₃PO₄→H₃PO₄ (15.99); H₄P₂O₇→H₂O (28.61); H₃PO₄→H₂O (47.14). The COMPASS force field was used to study condensed‐phase properties of PA. Good agreement between experimental data and MD results including density, radial distribution functions, and self‐diffusion coefficient at different temperatures provides validation of the COMPASS force field for PA. Finally, preliminary ADMP studies on a cluster of three PA molecules shows that the ADMP approach can reasonably describe the PT and self‐dissociation processes in PA. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FPR) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L²-Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.