Numerical homogenization of H(curl)-problems

If an elliptic differential operator associated with an $\mathbf{H}(\mathrm{curl})$-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding $\mathbf{H}(\mathrm{curl})$-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest order N\'ed\'elec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, $\mathbf{H}(\mathrm{curl})$-stable, quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh-size) in the $\mathbf{H}(\mathrm{curl})$ norm are obtained provided the right-hand side belongs to $\mathbf{H}(\mathrm{div})$. With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first order corrector, including corresponding quantitative error estimates without the requirement of scale separation

Coalgebraic Behavioral Metrics

We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra $\alpha\colon X \to HX$ for a functor $H \colon \mathrm{Set}\to \mathrm{Set}$, we define a framework for deriving pseudometrics on $X$ which measure the behavioral distance of states. A crucial step is the lifting of the functor $H$ on $\mathrm{Set}$ to a functor $\overline{H}$ on the category $\mathrm{PMet}$ of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If $H$ has a final coalgebra, every lifting $\overline{H}$ yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction

Towards Trace Metrics via Functor Lifting

We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, showing under which conditions also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set. More precisely, for a coalgebra on Set we determinize it, thus obtaining a coalgebra in the Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we can equip the final coalgebra with a behavioral distance. The trace distance between two states of the original coalgebra is the distance between their images in the determinized coalgebra through the unit of the monad. We show how our framework applies to nondeterministic automata and probabilistic automata

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures

Transformation of Ammonium Dicyanamide into Dicyandiamide in the Solid

Ammonium dicyanamide NH4[N(CN)2] was synthesized through aqueous ion exchange. The crystal structure was investigated by single-crystal X-ray diffraction (P21/c, a = 378.67(6) pm, b = 1240.9(3) pm, c = 911.84(14) pm, β = 91.488(18)°, Z = 4). It derives from the CsCl structure type. Medium strong hydrogen bonds between NH4+ and [N(CN)2]- ions are indicative of the observed formation of dicyandiamide H4C2N4 during heating. According to DSC and temperature-dependent X-ray powder diffractometry, this isomerization is exothermic and occurs between 102 and 106°C in the solid. The reaction represents the isolobal analogue to the classical synthesis of urea by heating NH4OCN. While other alkali and alkaline earth dicyanamides undergo trimerization or polymerization of their anions during heating, ammonium dicyanamide thus shows a different reactivity

Höhenatmung, Phosphat-Protein-Wechselwirkung: Die Sequenz der Hämoglobine des Goldhamsters (Mesocricetus aureatus) und des zweihöckerigen Kamels (Trampeltier, Camelus ferus, Camelidae)

The sequence of the main component of the hemoglobin of the hamster and that of the camel (Camelus ferus) is given. The sequence is obtained automatically by the sequenator using the quadrol and the propyne programme. The sequence of the α-and β-chains of the hamster is compared with the human hemoglobin; the sequence of the hemoglobin of the camel (Camelus ferus, Cameiidae), in comparison to the llama, there are to be found in the α-chains five amino acid exchanges, in the β-chains are two exchanges - in β2 and β76 - only. The sequence β2 in camel - the P2-glycerate contact - is histidine: This sequence sustains the interpretation of the high altitude respiration of the llama as mutation β2His to Asn. The amino acid sequence of the hemoglobin of Camelus ferus and Camelus dromedarius is identical

In design we trust: dealing with the innovation imperative

This research was funded by the Arts and Humanities Research Council (Grant number: AH/J005126/1).Preprin