Functional equations on abelian groups with involution

Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ _I ² =₁ g _l (x)h _l (y),x, y∈G, where the functionsf,g ₁,h ₁ to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.     

Students’ Reasoning About One-Object Stochastic Phenomena in an ICT-Environment

This paper focuses on the different ways in which students in lower secondary school (14–16 year olds) experience compound random events, presented to them in the form of combined junctions. A carefully designed ICT environment was developed enabling the students to interact with different representations of such structures. Data for the analysis was gathered from two interview sessions. The analysis of the interaction is based on constructivist principles on learning; i.e. we adopted a student-oriented perspective, taking into consideration the different ways students try to make sense of chance encounters. Our results show how some students give priority to geometrical and physical concerns, and we discuss how seeking causal explanations of random phenomena may have encouraged this. With respect to numerically oriented models a division strategy appears to stand out as the preferred one.     

Acousto-optic mode coupling of partially coherent light in two-mode optical fibers

An acoustic pulse propagating on a two-mode fiber can act as a beam splitter in a scanning interferometer. When this device is employed in white-light interferometry, the effects of distributed coupling and dispersive interferometer arms need to be considered. A theory suitable for treating acousto-optic interaction of partially coherent light in a moving interaction region was developed. It was found that differential optical dispersion should be negligible and the acoustic pulse length short. Also the coherence time should be short but long compared to the intermodal group delay difference over a pulse length. Experiments with long acoustic pulses were performed, and fairly good agreement with theory was obtained.     

TEM investigations on PECVD-titanium carbide layers

TiC_x-PECVD-layers were characterized by TEM. EDX analysis and electron diffraction. TiC_x-layers deposited using benzene showed a columnar structure, at which the column size decreases with rising excess carbon content. TiC_x-layers deposited using n-heptane presented a lamellar structure, at which the lamellar thickness diminishes with an increasing excess carbon content. In dependence on the layer thickness a periodic progress of the element contents was observed, at which a maximum for Ti and Cl correlates with a minimum for C. It was found that the incorporated chlorine is bonded to titanium. The lattice parameter depends on the chlorine content. Using TiCl₄/H₂/Ar-gas mixtures without any hydrocarbon, layers containing TiH₂ are formed.     

Structure and Magnetic Properties of Ce3Ge0.66In4.34 and Ce11Ge4.74In5.26

New indides Ce₃Ge_0.66In_4.34 and Ce₁₁Ge_4.74In_5.26 were synthesized from the elements by arc‐melting and subsequent annealing at 870 K. Single crystals were grown through special annealing procedures in sealed tantalum tubes in a high‐frequency furnace. Both compounds were investigated on the basis of X‐ray powder and single crystal data: I4/mcm, La₃GeIn₄ type, a = 848.8(1), c = 1192.0(2) pm, Z = 4, wR2 = 0.0453, 499 F² values, 17 variables for Ce₃Ge_0.66In_4.34 and I4/mmm, Sm₁₁Ge₄In₆ type (ordered version of the Ho₁₁Ge₁₀ type), a = 1199.3(2), c = 1662.0(3) pm, wR2 = 0.0507, 1217 F² values, 41 variables for Ce₁₁Ge_4.74In_5.26. The Ce₃Ge_0.66In_4.34 structure shows a mixed Ge/In occupancy on the 4c Wyckoff position. This site is octahedrally coordinated by cerium atoms. These octahedra share all edges, leading to a three‐dimensional network. The latter is penetrated by a two‐dimensional indium substructure which consists of flattened tetrahedra at In–In distances of 291 and 300 pm. The Ce₁₁Ge_4.74In_5.26 structure contains three crystallographically independent germanium sites. The latter are coordinated by eight or nine cerium neighbors. These CN8 and CN9 polyhedra are condensed to a complex network which is penetrated by a three‐dimensional indium network with In–In distances of 301–314 pm. The 16m site shows a mixed In/Ge occupancy. Chemical bonding in both compounds is dominated by the p elements. Both ternaries studied exhibit localized magnetism due to the presence of Ce³⁺ ions. The compound Ce₃GeIn₄ remains paramagnetic down to 1.72 K, whereas Ce₁₁Ge₄In₆ orders ferromagnetically at T_C = 7.5 K.     

GnRH binding RNA and DNA Spiegelmers: a novel approach toward GnRH antagonism

Mirror-image oligonucleotide ligands (Spiegelmers) that bind to the pharmacologically relevant target gonadotropin-releasing hormone I (GnRH) with high affinity and high specificity have been identified using the Spiegelmer technology. GnRH is a decapeptide that plays an important role in mammalian reproduction and sexual maturation and is associated with several benign and malignant diseases. First, aptamers that bind to D-GnRH with dissociation constants of 50-100 nM were isolated out of RNA and DNA libraries. The respective enantiomers of the DNA and RNA aptamers were synthesized, and their binding to L-GnRH was shown. These Spiegelmers bind to L-GnRH with similar affinity to that of the corresponding aptamers that bind to D-GnRH. We further demonstrated dose-dependent inhibition of GnRH-induced Ca(2+) release in Chinese hamster ovary cells that were stably transfected with the human GnRH receptor.     

ansa-Metallocene derivatives XXXIII. 2-Dimethylamino-substituted bis-indenyl zirconium dichloride complexes with and without a dimethylsilyl bridge: syntheses, crystal structures and properties in propene polymerization catalysis

Bis(2-N,N-dimethylamino-indenyl) zirconium dichloride, (2-(CH₃)₂N-C₉H₆)₂ZrCl₂, and dimethylsilyl-bridged bis(2-N,N-dimethylamino-indenyl) zirconium dichloride, (CH₃)₂Si(2-(CH₃)₂N-C₉H₅)₂ZrCl₂, were prepared by reaction of the corresponding ligand lithium salts with ZrCl₄ in toluene. Diffractometric structure determinations reveal C₂-symmetric complex geometries for both complexes. An increased electron density at the Zr center of the dimethylamino-substituted complexes is indicated by reduction potentials which are 0.3–0.4 V more negative than those of their unsubstituted analogs. When activated with methyl aluminoxane in toluene solution, (CH₃)₂Si(2-(CH₃)₂N-C₉H₅)₂ZrCl₂ catalyzes the polymerization of propene to polymers with a microstructure comparable with that of polymers produced with other Me₂Si-bridged bis(indenyl)ZrCl₂ complexes, but with a substantially increased fraction of i-propyl end groups derived from alkyl exchange between Zr-polymer and Al---Me species.     

PHYTOCHROME EVOLUTION: A PHYLOGENETIC TREE WITH THE FIRST COMPLETE SEQUENCE OF PHYTOCHROME FROM A CRYPTOGAMIC PLANT: (Selaginella martensii SPRING)

We have sequenced cDNA and genomic clones coding for phytochrome of the fern Selaginella. On the amino acid level, this phytochrome shares sequence homologies with phytochromes of higher plants which range between 62 (phytochrome B of Arabidopsis) and 55 (56)% [phytochrome C of Arabidopsis (Avena)]. Introns in the Selaginella gene are short and occupy positions known from phytochrome sequences of higher plants. A rooted phylogenetic tree based on mutation distances puts Selaginella phytochrome closest to the hypothetical ancestor. A similar tree arises if the tree is constructed with partial sequences (about 200 amino acids) around the chromophore attachment site. An extension of this tree by sequences of other cryptogamic plants (Mougeotia, Ceratodon, Psilotum) shows all these sequences including those of the phytochromes B and C of Arabidopsis on a branch, well separated from the branch formed by phytochromes known to accumulate in etiolated plants. The rooted phytochrome phylogenetic tree, however, is difficult to reconcile with the fossil record.