Prädikatives und funktionales Denken in der Wahrscheinlichkeitsrechnung

In order to emphasize functional thinking in mathematical education, arguments are brought forward for an early and frequent use of tree diagrams in teaching stochastics. Since set and tree diagrams can be regarded as two sides of the same matter, Bayes’ theorem, which is strongly associated with predicative set diagrams, need not be dealt with explicitly any longer. Set diagrams are replaced by tree diagrams that are labelled in detail and by tree inversion as a functional instrument of dealing with conditioned probabilities. This technique is demonstrated by three typical problems in the context of conditioned probability. Finally, two letters by Pascal to Fermat are analysed in order to illustrate that predicative and functional approaches were already pursued in the early stages of probability calculus in the 17^th century.     

Erratum: Preparation of size‐controlled In2O3 nanoparticles

The article referenced above was first published online on 7 August 2007 with incorrect pagination; the pagination has now been corrected online and in print. Copyright © 2007 John Wiley & Sons, Ltd.     

Synthesis of optically active poly(N‐propargylsulfamides) with helical conformation

A novel chiral N‐propargylsulfamide monomer ( 1a ) and its enantiomer ( 1b ) were synthesized and polymerized with (nbd)Rh⁺B^?(C₆H₅)₄ as a catalyst providing poly(1) (poly( 1a ) and poly( 1b )) in high yields (≥99%). Poly(1) could take stable helices in less polar solvents (chloroform and THF), demonstrated by strong circular dichroism signals and UV–vis absorption peaks at about 415 nm and the large specific rotations; but in more polar solvents including DMF and DMSO, poly(1) failed to form helix. Quantitative evaluation with anisotropy factor showed that the helical screw sense had a relatively high thermal stability. These results together with the IR spectra measured in solvents showed that hydrogen bonding between the neighboring sulfamide groups is one of the main driving forces for poly(1) to adopt stable helices. In addition, copolymerization of monomer 1a and monomer 2 was conducted, the solubility of poly(1) was improved drastically. However, the copolymerization had adverse effects on the formation of stable helices in the copolymers. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 500–508, 2007     

WAXS studies on block copolymers of ethylene and propylene

Wide-angle X-ray scattering from presumed block copolymers of polypropylene (PP) and ethylene-propylene copolymer (EPR), i.e., PP-EPR and PP-EPR-PP, synthesized by sequential polymerization with δ-TiCl₃? Et₂AlCl, was examined and compared with WAXS of mechanical blends and chain-transfer mixtures of PP and EPR with comparable compositions. The peak at 2θ = 20° for both the copolymers and the mixtures was attributed to the γ modification of PP in EPR. A strong variation in the ratio of diffraction intensities I₀₄₀/I₁₁₀ of PP in block copolymers and mixtures was explained in terms of crystallite growth in different directions. Analysis of the patterns and calculation of crystallinity, crystallite size, and lattice parameters led to the conclusion that block structure existed in the prepared copolymers.     

Pd (II)‐catalyzed vinyl addition polymerization of novel functionalized norbornene bearing dimethyl carboxylate groups

Three novel functionalized polynorbornenes (PNB) with pendant dimethyl carboxylate group (carboxylates—acetate, propionate, and butyrate) are synthesized as a vinyl‐type with a palladium (II) catalyst in high yield. The effects of size of substitutents, molar ratio of monomer to catalyst, solvent polarity, reaction time, and temperature on the polymerization of exo‐norbornene dimethyl propionate were systematically investigated. The low molar ratio and temperature, as well as high polarity of solvent, and long reaction time, are favorable for the enhancement of the monomer conversion, especially, the solvent have an obvious effect on the catalyst activity. The resulting poly(cis‐norbornene‐exo‐2,3‐dimethyl carboxylates) (PNB‐dimethyl carboxylates) show good solubility in common organic solvent and high thermal stability up to 360 °C. The glass transition temperature was detected by DMA at 331, 324, and 318 °C for acetate, propionate, and butyrate, respectively. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3391–3399, 2007     

Synthesis and light emitting properties of polyacetylenes having pendent fluorene groups

Five novel fluorene‐containing polymers, poly[(9,9‐dimethylfluoren‐2‐yl)acetylene] ( PFA1 ), poly[(1‐pentyl‐2‐(9,9‐dimethylfluoren‐2‐yl)acetylene) ( PFA2 ), poly[1‐decyl‐2‐(9,9‐dimethylfluoren‐2‐yl)acetylene] ( PFA3 ), poly[1‐phenyl‐2‐(9,9‐dimethylfluoren‐2‐yl)acetylene] ( PFA4 ), and poly[1‐(3,4‐difluorophenyl)‐2‐(9,9‐dimethylfluoren‐2‐yl)acetylene] ( PFA5 ) were synthesized by the polymerization of the corresponding fluorene‐substituted acetylenic monomers ( M1–M5), using WCl₆, MoCl_5, and TaCl₅ as catalysts and n‐Bu₄Sn as a cocatalyst. The synthesized polymers were thermally stable and readily soluble in common organic solvents. The degradation temperatures for a 5% weight loss of the polymers were ∼352–503 °C under nitrogen. PFA1–PFA5 show emission peaks from 402 to 590 nm. Besides, their electroluminescent properties were studied in heterostructure light‐emitting diodes (LEDs), using PFA2–PFA5 as an emitting layer. The PFA5 device revealed an orange‐red emission peak at 602 nm with a maximum luminescence of 923 cd/m² at 8 V. A device with the ITO/PEDOT/ a mixture of PFA2 (98 wt %) and PFA5 (2 wt %)/Ca/Al showed near white emission. Its maximum luminance and current efficiency are 450 cd/m² at 15 V and 1.3 cd/A, respectively. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 519–531, 2006     

Polymerization of cyclopentadiene initiated by methylaluminoxane

The polymerization of cyclopentadiene (CPD) was effectively initiated by methylaluminoxane (MAO) to generate poly(cyclopentadiene) (polyCPD). The effects on the polymerization of some reaction parameters such as the monomer concentration, the initiator concentration, and solvents were investigated. The conversion of CPD was monitored with gas chromatography to investigate the reaction kinetics. The polymerization rate was proportional to the concentrations of MAO in the first order and of the CPD monomer in the second order, and a reasonable cationic polymerization mechanism was suggested on the basis of the kinetic study. PolyCPD obtained at a low temperature could be dissolved in toluene or chloroform, and this indicated lower cross‐coupling during the polymerization reaction. ¹H NMR and IR analysis of the polymer indicated that there were almost equal amounts of 1,2‐enchainment and 1,4‐enchainment in the polymer chain. The measurement of polyCPD showed its unique properties as a potential candidate for stable wrappings or electronic packaging materials. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 264–272, 2006     

Synthesis and micellization of amphiphilic poly(sebacic anhydride)–poly(ethylene glycol)–poly(sebacic anhydride) block copolymers

Amphiphilic biodegradable block copolymers [poly(sebacic anhydride)–poly(ethylene glycol)–poly(sebacic anhydride)] were synthesized by the melt polycondensation of poly(ethylene glycol) and sebacic anhydride prepolymers. The chemical structure, crystalline nature, and phase behavior of the resulting copolymers were characterized with ¹H NMR, Fourier transform infrared, gel permeation chromatography, and differential scanning calorimetry. Microphase separation of the copolymers occurred, and the crystallinity of the poly(sebacic anhydride) (PSA) blocks diminished when the sebacic anhydride unit content in the copolymer was only 21.6%. ¹H NMR spectra carried out in CDCl₃ and D₂O were used to demonstrate the existence of hydrophobic PSA domains as the core of the micelle. In aqueous media, the copolymers formed micelles after precipitation from water‐miscible solvents. The effects on the micelle sizes due to the micelle preparation conditions, such as the organic phase, dropping rate of the polymer organic solution into the aqueous phase, and copolymer concentrations in the organic phase, were studied. There was an increase in the micelle size as the molecular weight of the PSA block was increased. The diameters of the copolymer micelles were also found to increase as the concentration of the copolymer dissolved in the organic phase was increased, and the dependence of the micelle diameters on the concentration of the copolymer varied with the copolymer composition. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1271–1278, 2006     

Synthesis and characterization of polyimides derived from (3‐amino‐2,4,6‐trimethylphenyl)‐(3′‐aminophenyl)methanone and aromatic dianhydrides

A series of new polyimides were prepared via the polycondensation of (3‐amino‐2,4,6‐trimethylphenyl)‐(3′‐aminophenyl)methanone and aromatic dianhydrides, that is, 3,3′,4,4′‐biphenyltetracarboxylic dianhydride (BPDA), 4,4′‐oxydiphthalic anhydride, 3,3′,4,4′‐benzophenone tetracarboxylic dianhydride, and 2,2′‐bis(3,4‐dicarboxyphenyl) hexafluoropropane dianhydride. The structures of the polyimides were characterized by Fourier transform infrared and NMR measurements. The properties were evaluated by solubility tests, ultraviolet–visible analysis, differential scanning calorimetry, and thermogravimetric analysis. The two different meta‐position‐located amino groups with respect to the carbonyl bridge in the diamine monomer provided it with an unsymmetrical structure. This led to a restriction on the close packing of the resulting polymer chains and reduced interchain interactions, which contributed to the solubility increase. All the polyimides except that derived from BPDA had good solubility in strong aprotic solvents, such as N‐methyl‐2‐pyrrolidinone, N,N′‐dimethylacetamide, N,N‐dimethylformamide, and dimethyl sulfone, and in common organic solvents, such as cyclohexanone and chloroform. In addition, these polyimides exhibited high glass‐transition values and excellent thermal properties, with an initial thermal decomposition temperature above 470 °C and glass‐transition temperatures in the range of 280–320 °C. The polyimide films also exhibited good transparency in the visible‐light region, with transmittance higher than 80% at 450 nm and a cutoff wavelength lower than 370 nm. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1291–1298, 2006