Synthesis of Alkylcyclophosphanes from Red Phosphorus – molecular structure of i-Pr4P4 by gas electron diffraction

A polyphosphide of the type (NaP)_X 1 was obtained from red phosphorus and sodium in presence of small amounts of naphthalene in 1,2-dimethoxyethane as solvent. 1 reacts with alkyl halides in dependence on the kind of alkyl group with formation of different alkylcyclophosphanes. As definite compounds Me₅P₅ 2 , Et₅P₅ 3 (with not definite constitution), i-Pr₄P₄ 4 and t-Bu₄P₄ 5 could be isolated. 1 was identified by X-ray diffraction, 2 – 5 by means of their ¹H, ¹³C, ³¹P n.m.r. and mass spectra. The molecular structure of 4 was identified by electron diffraction.     

Continuous functions with impermeable graphs

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.     

(Non-)Distributivity of the product for $$\sigma $$ σ -algebras with respect to the intersection

Archiv der Mathematik - We study the validity of the distributivity equation $$\begin{aligned} ({\mathcal {A}}\otimes {\mathcal {F}})\cap ({\mathcal {A}}\otimes {\mathcal {G}}) ={\mathcal...     

$${{\varvec{L}}}^{{\varvec{p}}}$$ L p -Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic,General Growth Setting

Journal of Theoretical Probability - We present a unified approach to $$L^p$$ -solutions ( $$p > 1$$ ) of multidimensional backward stochastic differential equations (BSDEs) driven by...     

Darstellung und Reaktionen von Natriumtellurid,Na2Te – Kristallstruktur von [Na(CH3OH)3]2Te2

Preparation and Reactions of Sodium Telluride, Na₂Te – Crystal Structure of [Na(CH₃OH)₃]₂Te₂ Sodium telluride, Na₂Te 1 , is easily obtained by reaction of sodium with hexagonal tellurium in 1,2-dimethoxy-ethane in the presence of catalytic amounts of naphthalene. Upon exposure to oxygen 1 reacts with methanol forming [Na(CH₃OH)₃]₂Te₂ 2 , which crystal structure was determined. Compound 1 , prepared in situ, can be conveniently converted into dialkyl tellurides, R₂Te (R = C₂H₅, i-C₃H₇, CH₂ = CH? CH₂ 3 ).     

Direct formation of beta-glycosides of N-acetyl glycosamines mediated by rare earth metal triflates

A direct, mild and efficient protocol for the preparation of beta-glycosides of N-acetyl glucosamine (GlcNAc) and N-acetyl galactosamine (GalNAc) has been developed using peracetylated beta-GlcNAc and beta-GalNAc as donors. All rare Earth metal triflate promoters screened were found to promote glycosylation with Sc(OTf)(3) being superior in terms of reaction rate. Simple alcohol glycosylation was found to proceed smoothly in refluxing dichloromethane, whereas higher temperatures under microwave conditions were needed to attain acceptable yields with less reactive, carbohydrate based glycosyl acceptors. The protocol developed was applied to provide the first example of direct chemical formation of a disaccharide using both GlcNAc as a glycosyl donor and acceptor. The alpha-acetate donor was found to be significantly less reactive than the corresponding beta-anomer necessitating higher reaction temperatures under which glycoside anomerisation was found to occur. It was established, that the anomerisation only took place in the presence of both Sc(OTf)(3) and acetic acid.     

Existence,uniqueness and regularity of the projection onto differentiable manifolds

Annals of Global Analysis and Geometry - We investigate the maximal open domain $${\mathscr {E}}(M)$$ on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ can be...