On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

2004 ◽  
Vol 108 (1) ◽  
pp. 111-124 ◽  
Author(s):  
François Guéritaud
2018 ◽  
Vol 22 (09n10) ◽  
pp. 814-820
Author(s):  
Yingying Jia ◽  
Ling Xu ◽  
Bangshao Yin ◽  
Mingbo Zhou ◽  
Jianxin Song

Beginning with 5,10,15-triarylporphyrin-nickel complex, five meso-to-meso directly linked porphyrin-diazaporphyrin triads were successfully prepared for the first time through a series of reactions including formylation via Vilsmeier–Haack reaction, condensation with pyrrole, bromination with [Formula: see text]-Bromosuccinimide (NBS), oxidation with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ), metal-templated cyclization of dibromodipyrrin-metal complexes with NaN[Formula: see text] and demetalization. All these triads were comprehensively characterized by [Formula: see text]H NMR, high-resolution mass spectrometry and UV-vis absorption. In addition, the structure of compound 6Ni was unambiguously determined by X-ray diffraction analysis, which showed that the two dihedral angles are both 86.65 (4)[Formula: see text] between each mean plane of porphyrin and that of central diazaporphyrin The UV-vis absorption spectra disclosed that the longest wavelengths of Soret bands and Q bands for these triads were observed at 429 and 642 nm, respectively. In contrast to diazaporphyrin-porphyrin dyads, diazaporphyrin dimers and diazaporphyrin monomers reported previously the molar extinction coefficients, particularly for triad 8Ni are much higher.


1985 ◽  
Vol 48 ◽  
Author(s):  
Howard T. Sawhill ◽  
Linn W. Hobbs

ABSTRACTNi/NiO interface structures were investigated using TEM, and the observed structures were compared with current heterophase interface models. Relative magnitudes of Ni/NiO interfacial energies were obtained from measurements of dihedral angles at triple grain junctions between Ni and NiO grains. Extra reflections in diffraction patterns from oxide grains adjacent to the Ni/NiO interface were compared with kinematical structure factor calculations for several proposed structures.


1992 ◽  
Vol 57 (3) ◽  
pp. 954-969 ◽  
Author(s):  
Michael Rathjen

AbstractLet KP− be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V ≔ universe of sets) be a Δ0-definable set function, i.e. there is a Δ0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and V ⊨ ∀x∃!yφ(x, y). In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection of those functions which are Σ1-definable in KP− + Σ1-Foundation + ∀x∃!yφ(x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of Δ0-Dependent Choices to the latter theory.


2019 ◽  
Vol 23 (11n12) ◽  
pp. 1592-1602 ◽  
Author(s):  
Yumeela Ganga-Sah ◽  
Elahe Tajbakhsh ◽  
Rachel H. Platel ◽  
Wen Zhou ◽  
Daniel B. Leznoff

The preparation and structural characterization of a pair of scandium(III) phthalocyanine hydroxide complexes were achieved by reaction of PcScCl with alkali metal alkoxides, likely via hydrolysis of soluble PcSc-alkoxide intermediates. A Sc[Formula: see text]Li[Formula: see text]-OH)[Formula: see text] cubane supported by two distorted Pc rings of the form (PcSc)[Formula: see text]-OH)[Formula: see text]Li[Formula: see text](THF)(DME) was isolated from the reaction of PcScCl with LiO[Formula: see text]Pr, while a simpler alkali-metal-free [Pc[Formula: see text]Sc[Formula: see text]-OH)[Formula: see text](THF)] was obtained from addition of NaO[Formula: see text]Bu; both structures are reminiscent of bent metallocenes, with dihedral angles between the two Pc rings of 50.8 and 37.7[Formula: see text]respectively. A soluble PcScOH material can also be obtained directly via hydrolysis of insoluble PcScCl in approximately 95:5 THF:water. Reduction of the Pc ring of PcScCl using KC[Formula: see text] is reversible and generates Pc[Formula: see text] and Pc[Formula: see text]-containing materials that were characterized via UV-vis spectroscopy and, where appropriate EPR and [Formula: see text]H NMR spectroscopy; analogous reductions of the PcScOH-based species were irreversible. Exposure of the air-sensitive, reduced PcScCl-based species to ambient atmosphere generated PcScOH materials analogous to the direct hydrolysis route.


1971 ◽  
Vol 4 (2) ◽  
pp. 193-200 ◽  
Author(s):  
Brailey Sims

It is known that in a B*-algebra every self-adjoint element is hermitian. We give an elementary proof that this condition characterizes B*-algetras among Banach*-algebras.


2011 ◽  
Vol 84 (1) ◽  
pp. 98-102 ◽  
Author(s):  
WARREN B. MOORS

AbstractIn this paper we provide an elementary proof of James’ characterization of weak compactness in separable Banach spaces. The proof of the theorem does not rely upon either Simons’ inequality or any integral representation theorems. In fact the proof only requires the Krein–Milman theorem, Milman’s theorem and the Bishop–Phelps theorem.


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