Computation of Electromagnetic Properties of Molecular Ensembles

We outline a methodology for efficiently computing the electromagnetic response of molecular ensembles. The methodology is based on the link that we establish between quantum-chemical simulations and the transfer matrix (T-matrix) approach, a common tool in physics and engineering. We exemplify and analyze the accuracy of the methodology by using the time-dependent Hartree-Fock theory simulation data of a single chiral molecule to compute the T-matrix of a cross-like arrangement of four copies of the molecule, and then computing the circular dichroism of the cross. The results are in very good agreement with full quantum-mechanical calculations on the cross. Importantly, the choice of computing circular dichroism is arbitrary: Any kind of electromagnetic response of an object can be computed from its T-matrix. We also show, by means of another example, how the methodology can be used to predict experimental measurements on a molecular material of macroscopic dimensions. This is possible because, once the T-matrices of the individual components of an ensemble are known, the electromagnetic response of the ensemble can be efficiently computed. This holds for arbitrary arrangements of a large number of molecules, as well as for periodic or aperiodic molecular arrays. We identify areas of research for further improving the accuracy of the method, as well as new fundamental and technological research avenues based on the use of the T-matrices of molecules and molecular ensembles for quantifying their degrees of symmetry breaking. We provide T-matrix-based formulas for computing traditional chiro-optical properties like (oriented) circular dichroism, and also for quantifying electromagnetic duality and electromagnetic chirality. The formulas are valid for light-matter interactions of arbitrarily-high multipolar orders.     

Peptide synthesis in aqueous environments: the role of extreme conditions and pyrite mineral surfaces on formation and hydrolysis of peptides

A comprehensive study of free energy landscapes and mechanisms of COS-mediated polymerization of glycine via N-carboxy anhydrides (NCAs, "Leuchs anhydrides") and peptide hydrolysis at the water-pyrite interface at extreme thermodynamic conditions is presented. Particular emphasis is set on the catalytic effects of the mineral surface including the putative role of the ubiquitous sulfur vacancy defects. It is found that the mere presence of a surface is able to change the free energetics of the elementary reaction steps. This effect can be understood in terms of a reduction of entropic contributions to the reactant state by immobilizing the reactants and/or screening them from bulk water in a purely geometric ("steric") sense. Additionally, the pyrite directly participates chemically in some of the reaction steps, thus changing the reaction mechanism qualitatively compared to the situation in bulk water. First, the adsorption of reactants on the surface can preform a product-like structure due to immobilizing and scaffolding them appropriately. Second, pyrite can act as a proton acceptor, thus replacing water in this role. Third, sulfur vacancies are found to increase the reactivity of the surface. The finding that the presence of pyrite speeds up the rate-determining step in the formation of peptides with respect to the situation in bulk solvent while stabilizing the produced peptide against hydrolysis is of particular interest to the hypothesis of prebiotic peptide formation at hydrothermal aqueous conditions. Apart from these implications, the generality of the studied organic reactions are of immediate relevance to many fields such as (bio)geochemistry, biomineralization, and environmental chemistry.     

A Simpler Proof of the Excluded Minor Theorem for Higher Surfaces

We give a simple proof of the fact (which follows from the Robertson–Seymour theory) that a graph which is minimal of genusgcannot contain a subdivision of a large grid. Combining this with the tree-width theorem and the quasi-wellordering of graphs of bounded tree-width in the Robertson–Seymour theory, we obtain a simpler proof of the generalized Kuratowski theorem for each fixed surface. The proof requires no previous knowledge of graph embeddings.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

A network approach to the modelling of active magnetic bearings

The modelling of active magnetic bearings based on a network approach is considered. Unlike in the standard modelling approach, where a linearization of the current-force relation for the centred shaft position is used, network models permit to include the position dependence of the bearing force in the force model. This becomes necessary when model based controllers are used to stabilize a magnetically supported shaft in tracking applications.

The approach is based on the well known application of network models to magnetic circuits. Further simplifying assumptions are discussed which allow one to obtain a network with a limited number of lumped parameters describing the magnetic behaviour of a magnetic bearing. The modelling of a combined radial and axial bearing serves as an example for the application of the proposed approach. Furthermore, the fitting of the network based model to measured characteristic force curves is discussed. In this context, a method for including saturation effects in the model is sketched.     

A short proof of w1n (Hom(C2r+1, Kn+2)) = 0 for all n and A graph colouring theorem by Babson and Kozlov

We show that the n-th power of the first Stiefel-Whitney class of the ℤ₂-action on the graph complex Hom(C _2r+1, K _n+2) is zero, confirming a conjecture by Babson and Kozlov. This yields a considerably simplified proof of their graph colouring theorem, which is also known as the Lovsz conjecture. This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program “ Combinatorics, Geometry, and Computation” (No. GRK 588/2)     

Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics

Cauchy problems for a second order linear differential operator equation

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

The Erdős–Pósa Property for Odd Cycles in Graphs of Large Connectivity

Dedicated to the memory of Paul Erdős A graph G is k-linked if G has at least 2k vertices, and, for any vertices , , ..., , , , ..., , G contains k pairwise disjoint paths such that joins for i = 1, 2, ..., k. We say that G is k-parity-linked if G is k-linked and, in addition, the paths can be chosen such that the parities of their lengths are prescribed. We prove the existence of a function g(k) such that every g(k)-connected graph is k-parity-linked if the deletion of any set of less than 4k-3 vertices leaves a nonbipartite graph. As a consequence, we obtain a result of Erdős–Pósa type for odd cycles in graphs of large connectivity. Also, every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph. Received May 13, 1999/Revised June 19, 2000     

Off-equilibrium photon production during the chiral phase transition

In the early stage of ultrarelativistic heavy-ion collisions chiral symmetry is restored temporarily. During this so-called chiral phase transition, the quark masses change from their constituent to their bare values. This mass shift leads to the spontaneous non-perturbative creation of quark–antiquark pairs, which effectively contributes to the formation of the quark–gluon plasma. We investigate the photon production induced by this creation process. We provide an approach that eliminates possible unphysical contributions from the vacuum polarization and renders the resulting photon spectra integrable in the ultraviolet domain. The off-equilibrium photon numbers are of quadratic order in the perturbative coupling constants while a thermal production is only of quartic order. Quantitatively, we find, however, that for the most physical mass-shift scenarios and for photon momenta larger than 1 GeV the off-equilibrium processes contribute less photons than the thermal processes.