Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.     

Tubular Structures of Calcium Carbonate: Formation,Characterization, and Implications in Natural Mineral Environments

Chemical gardens are self-assembled tubular precipitates formed by a combination of osmosis, buoyancy, and chemical reaction, and thought to be capable of catalyzing prebiotic condensation reactions. In many cases, the tube wall is a bilayer structure with the properties of a diaphragm and/or a membrane. The interest in silica gardens as microreactors for materials science has increased over the past decade because of their ability to create long-lasting electrochemical potential. In this study, we have grown single macroscopic tubes based on calcium carbonate and monitored their time-dependent behavior by in situ measurements of pH, ionic concentrations inside and outside the tubular membranes, and electrochemical potential differences. Furthermore, we have characterized the composition and structure of the tubular membranes by using ex situ X-ray diffraction, infrared and Raman spectroscopy, as well as scanning electron microscopy. Based on the collected data, we propose a physicochemical mechanism for the formation and ripening of these peculiar CaCO₃ structures and compare the results to those of other chemical garden systems. We find that the wall of the macroscopic calcium carbonate tubes is a bilayer of texturally distinct but compositionally similar calcite showing high crystallinity. The resulting high density of the material prevents macroscopic calcium carbonate gardens from developing significant electrochemical potential differences. In the light of these observations, possible implications in materials science and prebiotic (geo)chemistry are discussed.     

Effect of the External Mirror Feedback Strength in the Dynamics and Spectrum of the Injected Semiconductor Lasers

We analytically investigate the effect of an external mirror on the stability of an injected semiconductor laser, the latter treated as injected damped oscillators. In the studied configuration, the injected semiconductor laser with an external mirror is under the influence of a chaotic oscillating optical signal that is generated by a similar semiconductor master laser. We derive our results in terms of the damping rate and resonance frequency. We show that the external mirror can eliminate the unstable modes of the injected laser at low frequencies. Furthermore, the mirror can enhance the damping of the oscillation modes of the injected semiconductor laser; consequently, the driven response of the injected laser may have a broad spectrum, even wider than that of the chaotic driving signal. We also show results for the bandwidths of the injection amplitude and phase increment as functions of the injection rate and feedback strength of the external mirror. In addition, we use bifurcation and time-series curves to describe the dynamical behavior of the injected laser. We identify the feedback strength of injected laser, relative to that of the master laser, which induces synchronization between the injected-laser oscillation modes and the master laser.     

ON THE REARRANGEMENT OF N-ARYL SULFIMIDES

Abstract

N-Aryl-S,S-dialkylsulfimides, 1, with R¹ = alkyl other than CH₃, have been rearranged by heating in ethanol yielding o-alkylthiomethyl-anilines, 2, as main products. Isomeric o-methylthioalkyl-anilines, 14, are formed in minor amounts only. Reactions of sulfimides, 1, with R¹ = CH₃, with certain alkylating or acylating agents yielded o-methylthiomethylated, N-alkylated or -acylated products 9. Mechanistic considerations are discussed. The rearrangement of sulfimides 1 has been assumed to occur via [2,3]-sigmatropic reactions of intermediate azasulfonium ylids 3. Attempts to resolve (+)-camphor-10-sulfonates of N-aryl sulfimides failed, but optically active N-aryl sulfimides could be obtained by reaction of anilines with optically active sulfoxides and P₄O₁₀. Optically active 2,6-disubstituted sulfimides, 1, could be rearranged in ethanolic KOH to yield optically active cyclohexadienimines 12, indicating a transfer of asymmetry from sulfur to carbon and supporting the assumption of a sigmatropic rearrangement.     

The Inclusion Properties of a New Watersoluble Sulfonated Calix[4]resorcinarene towards Alkylammonium and N-Methylpyridinium Cations

The inclusion behaviour of a new water-soluble sulfonated calix[4]resorcinarene towards alkylammonium and N-methylpyridinium cations has been investigated on the basis of ¹H NMR spectroscopy and pH-potentiometry data. The inclusion of the N-methylpyridinium cation has been found to be dependent on pH with the preferable inclusion of the methyl substituent in alkaline and the aromatic ring in neutral aqueous media.     

Incomplete data problems in X-ray computerized tomography

Summary In the present paper we study truncated projections for the fanbeam geometry in computerized tomography. First we derive consistency conditions for the divergent beam transform. Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography. We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition.The work of the authors was supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4     

Native ethylene glycol in wine: Application of a dead volume free,very fast “deans heart-cut” system on-line with multi-chromatography

Data about the existence of native HO? CH₂? CH₂? OH (MEG) in natural wines and champagne has led to delicate questions because of legal decisions already made to declare wine or champagne illegal for sale if the MEG concentrations found are above 10 mg/liter. Action has been taken because of the DEG (diethylene glycol) disaster in European wines of 1985…86. An incorrect legal decision was made due to the belief that MEG cannot be produced biochemically by grape vines. A further reason may be lack of correct data on native MEG trace concentrations, as a result of the special behavior of this diol. As first member of a homologous series whose higher members (C₄) are normally found in all wines at quite high concentration levels, MEG shows extremely adsorptive behavior. The solution of the chromatographic problems is summarized in this paper. MEG concentrations in Riesling as example are in the range of 2 to 6 mg/liter and can easily be increased by biotechnological steps to a level of around 10 to 60 mg/liter. This is again due to the specific adsorptive behavior of MEG, which can be enriched on filter surfaces and displaced when the wine acidity changes with changing types. In order to control and guarantee the qualitative and quantitative results of MEG analyses we used a combination of Deans heart cutting on-line with Multi-Chromatography. It was easy to produce false data by many otherwise useful single column or two-dimensional separation processes, etc.     

Optimal control of Markovian jump processes with partial information and applications to a parallel queueing model

We consider a stochastic control problem over an infinite horizon where the state process is influenced by an unobservable environment process. In particular, the Hidden-Markov-model and the Bayesian model are included. This model under partial information is transformed into an equivalent one with complete information by using the well-known filter technique. In particular, the optimal controls and the value functions of the original and the transformed problem are the same. An explicit representation of the filter process which is a piecewise-deterministic process, is also given. Then we propose two solution techniques for the transformed model. First, a generalized verification technique (with a generalized Hamilton–Jacobi–Bellman equation) is formulated where the strict differentiability of the value function is weaken to local Lipschitz continuity. Second, we present a discrete-time Markovian decision model by which we are able to compute an optimal control of our given problem. In this context we are also able to state a general existence result for optimal controls. The power of both solution techniques is finally demonstrated for a parallel queueing model with unknown service rates. In particular, the filter process is discussed in detail, the value function is explicitly computed and the optimal control is completely characterized in the symmetric case.     

Alkene–Tetrazine Ligation for Imaging Cellular DNA

5‐Vinyl‐2′‐deoxyuridine (VdU) is the first reported metabolic probe for cellular DNA synthesis that can be visualized by using an inverse electron demand Diels–Alder reaction with a fluorescent tetrazine. VdU is incorporated by endogenous enzymes into the genomes of replicating cells, where it exhibits reduced genotoxicity compared to 5‐ethynyl‐2′‐deoxyuridine (EdU). The VdU–tetrazine ligation reaction is rapid (k≈0.02 M ^?1 s^?1) and chemically orthogonal to the alkyne–azide “click” reaction of EdU‐modified DNA. Alkene–tetrazine ligation reactions provide the first alternative to azide–alkyne click reactions for the bioorthogonal chemical labeling of nucleic acids in cells and facilitate time‐resolved, multicolor labeling of DNA synthesis.