Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

A novel synthetic route to substituted pyranoanthocyanins with unique colour properties

Anthocyanins, isolated from natural sources by countercurrent chromatography, were reacted with cinnamic acids bearing at least one electron-donating substituent at the para-position. The resulting pyranoanthocyanins obtained by this simple one-step reaction were much less susceptible to pH shifts and retained their original colour over a wide pH-range. Through reaction with p-dimethylamino cinnamic acid, synthetic malvidin- and cyanidin-based anthocyanins with a unique violet hue were prepared.     

Molecular structure and pseudopolymorphism of squamtin A from Annona squamosa

The cyclopeptide, squamtin A (1, formula: C₃₉H₆₀O₁₁N₈S), was found to crystallized in two pseudopolymorphisms, i.e. 1·(H₂O)_3.5 and 1·(H₂O)_3.9. The composition of the amino acids and their linkage sequences are the same. The main differences between the two kinds of crystals lie in the positions and occupancies of the water molecules, the positions of the sulfur atoms and the conformation of the side chains. The absolute configuration of 1 is established by X-ray analysis in combination with the Marfey's analysis of its hydrolysates.     

A perturbation bound for the generalized polar decomposition

LetA be anm×n complex matrix. A decompositionA=QH is termed ageneralized polar decomposition ofA ifQ is anm×n subunitary matrix (sometimes also called a partial isometry) andH a positive semidefinite Hermitian matrix. It was proved that a nonzero matrixA ^m×n has a unique generalized polar decompositionA=QH with the property (Q ^H )=(H), whereQ ^H denotes the conjugate transpose ofQ and (H) the column space ofH. The main result of this note is a perturbation bound forQ whenA is perturbed.     

The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

苯胺气相N－烷基化反应研究Ⅱ．苯胺和乙醇常压催化合成N－乙基苯胺的工业应用

苯胺气相Ｎ－烷基化反应研究Ⅱ．苯胺和乙醇常压催化合成Ｎ－乙基苯胺的工业应用陈骏如，李瑞祥，韩银仙，李贤均（四川大学化学系成都６１００６４）Ｎ－乙基苯胺是制造染料、塑料、炸药和医药等许多化工产品的重要中间体，其合成方法过去多采用无机酸催化剂，如Ｈ＿２Ｓ...     

Concerted cycloaddition of chlorosulfonyl isocyanate to α-pinene stepwise rearrangement of the β-lactam cycloadduct to a γ-lactam

Chlorosulfonyl isocyanate adds to α-pinene in concerted fashion to give the unrearranged N-chlorosulfonyl-β-lactam 2 (75%). Thermolysis of cycloadduct 2 affords the rearranged N-chlorosulfonyl-γ-lactam 6 (60%) via a sequence of transient carbonium ions (45). Reductive hydrolysis of 2 and 6 gave, respectively, the NH-β-lactam 3 (70%) and the NH-γ-lactam 7 (70%). Proof of structure of the rearranged γ-lactams 6 and 7 are provided, respectively, by X-ray crystallographic analysis and the application of the Eu(dpm)₃ NMR shift reagent.     

Charge equilibria and pK values of 5-carboxypyranomalvidin-3-glucoside (vitisin A) by electrophoresis and absorption spectroscopy

Apparent pK values of the wine pigment, 5-carboxypyranomalvidin-3-glucoside (vitisin A), were determined using UV-vis spectroscopy, viz. pK_a1=0.98 (±0.10), pK_H1=4.51 (±0.03) and pK_H2=7.57 (±0.02). An additional ionisation constant at high pH (pK_a4=8.84±0.06) was established by high-voltage paper electrophoresis. These data in conjunction with previously published pK_a values determined by high-voltage electrophoresis suggest that in wine (pH 3.2-3.8), 5-carboxypyranomalvidin-3-glucoside exists as a complex mixture of hydrated and non-hydrated, partially ionised species with the predominant species being the quinonoidal base (λ_max 498 nm).     

Some positive cotes numbers for the chebyshev weight function

We show that, for the Chebyshev weight function (1–x ²)^–1/2, the Cotes numbers for the quadrature rule with nodes at the zeros of the ultraspherical polynomialP _n ^/() are nonnegative if and only if –1/2<1.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.