Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

Protection Measures against Product Piracy and Application by the Use of AM

Presently the implications Additive Manufacturing (AM) on intellectual properties are discussed in public. Here AM is often mentioned as a driver for product piracy as it allows to produce and to copy objects with any geometries. Imitators need a lot of information to copy an object accurately. As reverse engineering has been identified as the most important information source for product imitators, AM can also help to reduce the threat of product piracy when correctly applied in the product development. Due to the layer wise production process that allows the manufacturing of very complex shapes and geometries, the reverse-engineering process can be complicated by far. By this, quite contrary to the public opinion, AM can increase the needed effort of imitators and strongly reduce the economic efficiency of product piracy. This paper will show different protection measures and a methodological approach of how to apply these measures to a product. Beside the protective effect some measures allow a traceability of parts over the product’s lifecycle and thus support the quality management of AM processes and additively produced parts.Mechanical Engineerin

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the H$_3^+=\{$p$^+,$p$^+,$p$^+,$e$^-,$e$^-\}$ molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result for this molecular ion with rovibrational results including non-adiabatic corrections.Comment: 29 pages, 3 figures, 4 table

[(CH3)3Si]14Si7O21 - ein neuer cyclischer Kieselsäuretrimethylsilylester

Durch Trimethylsilylierung des Bariumchloridsilicates 2BaO - 3BaCl2 - 2SiO2 wurde ein bisher unbekannter Kieselsäuretrimethylsilylester hergestellt, der mit Hilfe der Kapillargaschromatographie, Massenspektroskopie29Si-NMR-Spektroskopie und Dünnschichtchromatographie charakterisiert wurde. Aus den Untersuchungen folgt für den Ester die Konstitution eines Cycloheptakieselsäuretrimethylsilylesters (Tetradecakistrimethylsiloxicycloheptasiloxan) der Formel [(CH3)3Si]14Si7O21. Die Verwendung des Esters als Standardsubstanz wird diskutiert

Synthese, Aufbau und Eigenschaften käfigartiger vinyl- und allylsilylierter Kieselsäuren

Durch Umsatz des Tetramethylammoniumsilicats [N(CH3)4]8Si8O20 · 69 H2O mit Vinyldimethylchlorsilan (I) bzw. Divinyltetramethyldisiloxan oder Allydimethylchlorsilan wurden der Vinyldimethylsilylkieselsäureester [CH2CHCH2(CH3)2Si]8Si8O20 und Allyldimethylsilyl-kieselsäureester [CH2CHCH2(CH3)2Si]8Si8O20 erhalten. Mit Hilfe der Gaschromatographie, Massenspektroskopie sowie 1H-und 29Si-NMR wurden die Verbindungen charakterisiert und eine käfigartig aufgebaute Doppelvierring (D4R)-Struktur des Kieselsäuregerüsts nachgewiesen, an deren 8 terminalen O-Atomen des Kieselsäuregerüsts nachgewiesen, an deren 8 terminalen O-Atomen des Kieselsäuregerüsts Vinyldimethylsiyl- bzw. Allydimethylsilylgruppen gebunden sind. Durch Silylierung mit I und Trimethylchlorsilan sind in Abhängigkeit vom Mischungsverhältnis Vinyldimethylsilytrimethysily1-D4R-Kieselsäureester mit einer mittleren Anzahl ungesättigter Gruppen < 8 herzustellen

Ein neues Verfahren für namensbasierte Zufallsstichproben von Migranten

The set of best methods for sampling mi- grant populations includes name-based sampling. So far this is done using either ad-hoc lists or onomastic dictionaries for the classi cation of names. This paper pro- poses a new name-based procedure, which uses a Bayes-classi er for the n-grams of the name. The new procedure is fault-tol- erant of alternate spellings, and also allows the classi cation of names that are not found in dictionaries. It was tested using the names of about 1.600 foreigners in the PASS panel. Finally, a CATI survey based on the new method in Hesse is described

Sequential decoupling of negative-energy states in Douglas-Kroll-Hess theory

Here, we review the historical development, current status, and prospects of Douglas--Kroll--Hess theory as a quantum chemical relativistic electrons-only theory.Comment: 15 page

Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules

We present a new class of tensor network states that are specifically designed to capture the electron correlation of a molecule of arbitrary structure. In this ansatz, the electronic wave function is represented by a Complete-Graph Tensor Network (CGTN) ansatz which implements an efficient reduction of the number of variational parameters by breaking down the complexity of the high-dimensional coefficient tensor of a full-configuration-interaction (FCI) wave function. We demonstrate that CGTN states approximate ground states of molecules accurately by comparison of the CGTN and FCI expansion coefficients. The CGTN parametrization is not biased towards any reference configuration in contrast to many standard quantum chemical methods. This feature allows one to obtain accurate relative energies between CGTN states which is central to molecular physics and chemistry. We discuss the implications for quantum chemistry and focus on the spin-state problem. Our CGTN approach is applied to the energy splitting of states of different spin for methylene and the strongly correlated ozone molecule at a transition state structure. The parameters of the tensor network ansatz are variationally optimized by means of a parallel-tempering Monte Carlo algorithm

Reliable estimation of prediction uncertainty for physico-chemical property models

The predictions of parameteric property models and their uncertainties are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational methods. Here, we discuss the calibration of property models in the light of bootstrapping, a sampling method akin to Bayesian inference that can be employed for identifying systematic errors and for reliable estimation of the prediction uncertainty. We apply bootstrapping to assess a linear property model linking the 57Fe Moessbauer isomer shift to the contact electron density at the iron nucleus for a diverse set of 44 molecular iron compounds. The contact electron density is calculated with twelve density functionals across Jacob's ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer-shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06-0.08 mm/s and 0.04-0.05 mm/s, respectively, the latter being close to the average experimental uncertainty of 0.02 mm/s. Furthermore, we show that both model parameters and prediction uncertainty depend significantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the coefficient of correlation, r2, or the root-mean-square error, RMSE) should not be inferred from a single data set. This study presents the first statistically rigorous calibration analysis for theoretical Moessbauer spectroscopy, which is of general applicability for physico-chemical property models and not restricted to isomer-shift predictions. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.Comment: 49 pages, 9 figures, 7 table

Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators

We present new results on the block-diagonalization of Dirac operators on three-dimensional Euclidean space with unbounded potentials. Classes of admissible potentials include electromagnetic potentials with strong Coulomb singularities and more general matrix-valued potentials, even non-self-adjoint ones. For the Coulomb potential, we achieve an exact diagonalization up to nuclear charge Z=124 and prove the convergence of the Douglas-Kroll-He\ss\ approximation up to Z=62, thus improving the upper bounds Z=93 and Z=51, respectively, by H.\ Siedentop and E.\ Stockmeyer considerably. These results follow from abstract theorems on perturbations of spectral subspaces of operators with gaps, which are based on a method of H.\ Langer and C.\ Tretter and are also of independent interest