The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

Synthesis, characterization, and electrochemical studies on [1.1]ferrocenophanes containing aluminum, gallium, and indium

The synthesis, characterization, structure, and electrochemistry of [1.1]ferrocenophanes, bridged by the heavier group 13 elements aluminum (1a), gallium (1b), and indium (1c), are described and discussed. Compounds 1a-c have been synthesized from dilithioferrocene and intramolecularly coordinated group 13 element dihalides Ar'EX(2) (Ar' = 2-(Me(2)NCH(2))C(6)H(4); EX(2) = AlCl(2), GaCl(2), InI(2)). Although the synthesis and characterization of 1a by single-crystal X-ray analysis has been described recently (Braunschweig, H.; Burschka, C.; Clentsmith, G. K. B.; Kupfer, T.; Radacki, K. Inorg. Chem. 2005, 44, 4906), compounds 1b and 1c are described for the first time. The galla (1b) and the inda (1c) [1.1]ferrocenophane have been characterized by single-crystal X-ray determination [1b: C(38)H(40)Fe(2)Ga(2)N(2), monoclinic, P2(1)/c, a = 10.3467(5) Angstroms, b = 11.6311(4) Angstroms, c = 14.0747(7) Angstroms, beta = 105.931(2) degrees, Z = 2; 1c: C(38)H(40)Fe(2)In(2)N(2), monoclinic, P2(1)/c, a = 10.5522(7) Angstroms, b = 11.8476(8) Angstroms, c = 13.9855(9) Angstroms, beta = 104.990(3) degrees, Z = 2]. All three compounds 1a-c are anti conformers with trans orientations of the two donating NMe(2) groups. For the [1.1]ferrocenophane 1a, an unprecedented fully reversible two-electron redox process was observed by cyclic voltammetry, whereas the corresponding Ga and In species exhibit a more conventional stepwise redox chemistry. According to the Robin-Day classification, 1a is a class I and 1b and 1c are class II species. In addition to the reversible processes, compound 1a shows an irreversible oxidation at higher voltages accompanied by adsorption processes. The irreversible adsorption process was investigated with an electrochemical quartz crystal microbalance (EQCM).     

Hybrid approach combining chemometrics and likelihood ratio framework for reporting the evidential value of spectra

Many chemometric tools are invaluable and have proven effective in data mining and substantial dimensionality reduction of highly multivariate data. This becomes vital for interpreting various physicochemical data due to rapid development of advanced analytical techniques, delivering much information in a single measurement run. This concerns especially spectra, which are frequently used as the subject of comparative analysis in e.g. forensic sciences. In the presented study the microtraces collected from the scenarios of hit-and-run accidents were analysed. Plastic containers and automotive plastics (e.g. bumpers, headlamp lenses) were subjected to Fourier transform infrared spectrometry and car paints were analysed using Raman spectroscopy. In the forensic context analytical results must be interpreted and reported according to the standards of the interpretation schemes acknowledged in forensic sciences using the likelihood ratio approach. However, for proper construction of LR models for highly multivariate data, such as spectra, chemometric tools must be employed for substantial data compression. Conversion from classical feature representation to distance representation was proposed for revealing hidden data peculiarities and linear discriminant analysis was further applied for minimising the within-sample variability while maximising the between-sample variability. Both techniques enabled substantial reduction of data dimensionality. Univariate and multivariate likelihood ratio models were proposed for such data. It was shown that the combination of chemometric tools and the likelihood ratio approach is capable of solving the comparison problem of highly multivariate and correlated data after proper extraction of the most relevant features and variance information hidden in the data structure.     

Bisignate resonance Raman optical activity: a pseudo breakdown of the single electronic state model of RROA?

Raman optical activity (ROA) spectra were calculated for a set of conformers of astaxanthin, which is a non‐rigid molecule exhibiting strong resonance enhancement in the visible range. Single electronic state theory of the Resonance ROA (RROA) predicts the spectrum to be monosigned. For astaxanthin, it appeared that some of the conformers exhibit different sign of the bands than the other conformers. As a result, the conformer population averaged spectrum of astaxanthin can exhibit both signs of the bands, or be monosigned depending on which conformers are dominating, that reflects a departure from the single electronic state approximation. Moreover, use of different basis sets and/or density functional theory (DFT) functionals results in different conformer populations, thus yielding again either monosigned ROA spectrum or with bands of both signs. Consequences of these two findings for the astaxanthin RROA spectrum are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of nanopore arrangement and structural features of anodic alumina layers formed by two-step anodizing in oxalic acid using the dedicated executable software

Anodic porous alumina layers were fabricated by a two-step self-organized anodization in 0.3 M oxalic acid under various anodizing potentials ranging from 30 to 60 V at two different temperatures (10 and 17 ^°C). The effect of anodizing conditions on structural features and pore arrangement of AAO was investigated in detail by using the dedicated executable publication combined with ImageJ software. With increasing anodizing potential, a linear increase of the average pore diameter, interpore distance, wall thickness and barrier layer thickness, as well as a decrease of the pore density, were observed. In addition, the higher pore diameter and porosity values were obtained for samples anodized at the elevated temperature, independently of the anodizing potential. A degree of pore order was investigated on the basis of Delaunay triangulations (defect maps) and calculation of pair distribution or angle distribution functions (PDF or ADF), respectively. All methods confirmed that in order to obtain nanoporous alumina with the best, hexagonal pore arrangement, the potential of 40 V should be applied during anodization. It was confirmed that the dedicated executable publication can be used to a fast and complex analysis of nanopore arrangement and structural features of nanoporous oxide layers.     

A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions

In this paper we discuss a topological treatment for the planar system 0.1 $$\begin{aligned} z'=f(t,z)+g(t,z) \end{aligned}$$ where $f:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ and $g:\mathbb {R}\times \mathbb {R}^{2}\longrightarrow \mathbb {R}^{2}$ are $T$ -periodic in time and $g(t,z)$ is bounded. Namely, we study the effect of $g(t,z)$ in two different frameworks: isochronous centers and time periodic systems having subharmonics. The main tool employed in the proofs consists of a topological strategy to locate fixed points in the class of orientation preserving embedding under the condition of some recurrence properties. Generally speaking, our topological result can be considered as an extension of the main result in Brown (Pac J Math 143:37–41, 1990) (concerning two cycles) to any recurrent point.     

The oxidative degradation of dibenzoazepine derivatives by cerium(IV) complexes in acidic sulfate media

The kinetics of the oxidation of imipramine and desipramine using cerium(IV) complexes were studied in the presence of a large excess of azepine derivative (TCA) in acidic sulfate media using UV-Vis spectroscopy. The reaction proceeds via dibenzoazepine radical formation, identified by EPR measurements. The kinetics of the first degradation step were studied independently of the further slower degradation reactions. Linear dependences, with zero intercept, of the pseudo-first-order rate constants (k(obs)) on [TCA] were established for both dibenzoazepine radical formation processes. Rates of reactions decreased with increasing concentration of the H(+) ion indicating that cerium(IV) as well as both reductants exist in an equilibrium with their protolytic forms. The activation parameters for the degradation of dibenzoazepine derivatives in the first oxidation stage were as follows: ΔH(≠) = 39 ± 2 kJ mol(-1), ΔS(≠) = -28 ± 8 J K(-1) mol(-1) for imipramine and ΔH(≠) = 39 ± 2 kJ mol(-1), ΔS(≠) = -28 ± 6 J K(-1) mol(-1) for desipramine, respectively. Imipramine and desipramine radicals dimerized leading to an intermediate radical dimer, which decayed in a first-order consecutive decay process. These two further reactions proceed with rates which are characterized by non-linear dependences of the pseudo-first-order rate constants (k(obs)) on [TCA]. The degradation reaction of the intermediate radical dimer leads to an uncharged dimer as a final product. Mechanistic consequences of all the results are discussed.     

Structure of the stapled p53 peptide bound to Mdm2

Mdm2 is a major negative regulator of the tumor suppressor p53 protein, a protein that plays a crucial role in maintaining genome integrity. Inactivation of p53 is the most prevalent defect in human cancers. Inhibitors of the Mdm2-p53 interaction that restore the functional p53 constitute potential nongenotoxic anticancer agents with a novel mode of action. We present here a 2.0 ? resolution structure of the Mdm2 protein with a bound stapled p53 peptide. Such peptides, which are conformationally and proteolytically stabilized with all-hydrocarbon staples, are an emerging class of biologics that are capable of disrupting protein-protein interactions and thus have broad therapeutic potential. The structure represents the first crystal structure of an i, i + 7 stapled peptide bound to its target and reveals that rather than acting solely as a passive conformational brace, a staple can intimately interact with the surface of a protein and augment the binding interface.