Complex Geometry and Dirac Equation

Complex geometry represents a fundamentalingredient in the formulation of the Dirac equation bythe Clifford algebra. The choice of appropriate complexgeometries is strictly related to the geometricinterpretation of the complex imaginary unit . We discuss two possibilities which appearin the multivector algebra approach: the₁₂₃ and ₂₁ complexgeometries. Our formalism provides a set of rules which allows an immediate translation between thecomplex standard Dirac theory and its version withingeometric algebra. The problem concerning a doublegeometric interpretation for the complex imaginary unit is also discussed.     

THE BIOGENETIC LAW AND ITS INFLUENCE ON THEORIES OF LEARNING MATHEMATICS

Reference to the history of mathematics has been widely used in discussions of the development of curricula, the problems students have in learning mathematics, and the development of concepts in the individual. This paper examines the background to the "Biogenetic Law" [1] which has influenced so much thinking in educational theory, and the use of the "principle of parallelism" where individual development is claimed to mirror the historical development of the subject matter. Interpretations of the history of mathematics which are used to justify these claims are examined, and the universality of the supposed concepts is challenged. Questions are raised about some of the fundamental tenets of Piagetian epistemology.     

Self‐Sufficient Formaldehyde‐to‐Methanol Conversion by Organometallic Formaldehyde Dismutase Mimic

The catalytic networks of methylotrophic organisms, featuring redox enzymes for the activation of one‐carbon moieties, can serve as great inspiration in the development of novel homogeneously catalyzed pathways for the interconversion of C₁ molecules at ambient conditions. An imidazolium‐tagged arene–ruthenium complex was identified as an effective functional mimic of the bacterial formaldehyde dismutase, which provides a new and highly selective route for the conversion of formaldehyde to methanol in absence of any external reducing agents. Moreover, secondary amines are reductively methylated by the organometallic dismutase mimic in a redox self‐sufficient manner with formaldehyde acting both as carbon source and reducing agent.     

Forced imbibition-a tool for separate determination of Laplace pressure and drag force in capillary filling experiments

When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces which are hard to quantify experimentally, particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure independently quantifies the Laplace pressure and Darcy's permeability as relevant physical parameters governing the imbibition process. From the latter one may then compute the effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing into the pore. In determining all these parameters independently, the consistency of our analysis of such forced imbibition processes is demonstrated.     

Speciation of butyltin compounds in marine sediments with headspace solid phase microextraction and gas chromatography – mass spectrometry

A method for the determination of organotin compounds (monobutyl = MBT, dibutyl = DBT, and tributyltin = TBT) in marine sediments by headspace Solid Phase Microextraction (SPME) has been developed. The analytical procedure involved 1) extraction of TBT, DBT and MBT from sediments with HCl and methanol mixture, 2) in situ derivatization with sodium tetraethylborate and 3) headspace SPME extraction using a fiber coated with poly(dimethylsiloxane). The derivatized organotin compounds were desorbed into the splitless injector and simultaneously analyzed by gas chromatography – mass spectrometry. The analytical method was optimized with respect to derivatization reaction and extraction conditions. The detection limits obtained for MBT, DBT and TBT ranged from 730 to 969 pg/g as Sn dry weight. Linear calibration curves were obtained for all analytes in the range of 30–1000 ng/L as Sn. Analysis of a standard reference sediment (CRM 462) demonstrates the suitability of this method for the determination of butyltin compounds in marine sediments. The application to the determination of TBT, DBT and MBT in a coastal marine sediment is shown.     

Surface modification of cetyltrimethylammonium bromide-capped gold nanorods to make molecular probes

A chemical procedure to replace the cetyltrimethylammonium bromide (CTAB) cap on gold nanorods (GNRs) fabricated through seed-mediated growth with organothiol compounds [3-animo-5-mercapto-1,2,4-triazole (AMTAZ) and 11-mercaptoundecaonic acid (MUDA)] was developed to reduce the cytotoxity of GNRs and facilitate further biofunctionalization. Compared to phosphatidylcholine (PC) modification, our procedure yields stable GNRs that are biocompatible and suitable for whole-cell studies. The PC-, AMTAZ-, and MUDA-activated GNRs all showed low cytotoxicity. By choosing different organothiols, net positive or negative charges could be created on the nanorod surface, for different applications. Gold nanorod molecular probes (GNrMPs) were fabricated by subsequent attachment of antibodies to the activated GNRs and were used to visualize and detect cell surface biomarkers in normal and transformed human breast epithelial cells, demonstrating the potential of developing novel biosensors using gold nanorods. The sensitivity of GNrMPs made from organothiol-activated GNRs is considerably higher than that of CTAB/PC-activated GNRs, demonstrating that the protocol reported here is favored in developing molecular probes using GNRs.     

Pre-organization of the core structure of E-selectin antagonists

A new class of N-acetyl-D-glucosamine (GlcNAc) mimics for E-selectin antagonists was designed and synthesized. The mimic consists of a cyclohexane ring substituted with alkyl substituents adjacent to the linking position of the fucose moiety. Incorporation into E-selectin antagonists led to the test compounds 8 and the 2'-benzoylated analogues 21, which exhibit affinities in the low micromolar range. By using saturation transfer difference (STD)-NMR it could be shown that the increase in affinity does not result from an additional hydrophobic contact of the alkyl substituent with the target protein E-selectin, but rather from a steric effect stabilizing the antagonist in its bioactive conformation. The loss of affinity found for antagonists 10 and 35 containing a methyl substituent in a remote position (and therefore unable to support to the stabilization of the core) further supports this hypothesis. Finally, when a GlcNAc mimetic containing two methyl substituents (52 and 53) was used, in which one methyl was positioned adjacent to the fucose linking position and the other was in a remote position, the affinity was regained.     

One‐ and Two‐Component Bottle‐Brush Polymers: Simulations Compared to Theoretical Predictions

Scaling predictions for bottle‐brush polymers with a rigid backbone and flexible side chains under good solvent conditions are discussed and their validity is assessed by a comparison with Monte Carlo simulations of a simple lattice model. It is shown that typically only a rather weak stretching of the side chains is realized, and then the scaling predictions are not applicable. Also two‐component bottle brush polymers are considered, where two types (A,B) of side chains are grafted, assuming that monomers of different kind repel each other. In this case, variable solvent quality is allowed. Theories predict “Janus cylinder”‐type phase separation along the backbone in this case. The Monte Carlo simulations, using the pruned‐enriched Rosenbluth method (PERM) give evidence that the phase separation between an A‐rich part of the cylindrical molecule and a B‐rich part can only occur locally. The correlation length of this microphase separation can be controlled by the solvent quality. This lack of a phase transition is interpreted by an analogy with models for ferromagnets in one space dimension.

     

Functionalization of bolalipid nanofibers by silicification and subsequent one-dimensional fixation of gold nanoparticles

In the present work, we describe the successful stabilization of bolalipid nanofibers by sol-gel condensation (silicification) of tetraethoxysilane (TEOS) or 3-mercaptopropyltriethoxysilane (MP-TEOS), respectively, onto the nanofibers. The conditions for an effective and reproducible silicification reaction were determined, and the silicification process was pursued by transmission electron microscopy (TEM). The resulting bolalipid-silica composite nanofibers were characterized by means of differential scanning calorimetry (DSC), TEM, (13)C, and (31)P NMR spectroscopy. Finally, the novel silicified bolalipid nanofibers were used as templates for the fixation of 5 and 2 nm AuNPs, respectively, resulting in one of the rare examples of one-dimensional AuNP arrangements in aqueous suspension.     

Über die Synthese von 5-Phenyl-1H-thieno[3,4—e]1,4-diazepin-2(3H)-on

The synthesis of 5-phenyl-1H-thieno[3.4—e]1.4-diazepin-2(3H)-one (16), its alkylation in position 1 and the preparation of the intermediate 3-(4-aminothienyl)-phenylketone (8) are described. In addition thieno[3.4—b]pyridin-2(1H)-one (7) is described.