High asymmetric induction in diels-alder additions of cyclopentadiene to acrylates derived from isoborneol

Starting from (R)-(+)- and from (S)-(?)-camphor the chiral alcohols $\text{1}$, $\text{2}$ and $\text{3}$ have been prepared; their acrylates II underwent TiCl₂(OR)₂-promoted Diels-Alder additions to cyclopentadiene giving efficiently in a predictable manner either the (2R)- or the (2S)-adducts III with up to virtually quantitative asymmetric induction.     

The separation of palladium and platinum by ion flotation

Differences in the ion flotation properties of palladium(II) and platinum(IV) chloro complexes in aqueous solutions are used to achieve separations of these metals. The anionic chloro complex PtCl^2-₆ is floated selectively with cationic surfactants of the type, RNR'₃Br, from solutions of PdCl^2-₄ and various concentrations of hydrochloric acid. The palladium(II) does not float from solutions of ? 3.0 M HCl and the platinum(IV) floated from these solutions can be recovered free of palladium. However, the separation is incomplete as much of the platinum(IV) is also unfloated from these solutions. Quantitative separations are obtained by conversion of the palladium(II) to the cationic ammine, Pd(NH₃)₄²⁺ with aqueous ammonia prior to flotation. The anionic chloro complex of platinum(IV) is unaffected by the presence of ammonia and is floated quantitatively with the surfactant n-hexadecyltri-n-propylammonium bromide from 0.01 M ammonia solutions.     

Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

Foundations of Computational Mathematics - This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian...     

Avoidance of split overlaps

We generalize Axel Thue’s familiar definition of overlaps in words, and observe that there are no infinite words avoiding split occurrences of these generalized overlaps. We give estimates for the length of the longest finite word that avoids split overlaps. Along the way we prove a useful theorem about repeated disjoint occurrences in words — an interesting natural variation on the classical de Bruijn sequences.     

Eigenvalues Estimate for the Neumann Problem of a Bounded Domain

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Ω in a given complete (not compact a priori) Riemannian manifold (M,g). For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As applications, we prove that if the Ricci curvature of (M,g) is bounded below Ric  ^g ≥−(n−1)a ², a≥0, then there exist constants A _n >0,B _n >0 only depending on the dimension, such that

Which distance-hereditary graphs are cover–incomparability graphs?

In this paper we deal with cover–incomparability graphs of posets, or briefly C–I graphs. These are graphs derived from posets as the edge-union of their cover graph and their incomparability graph. We answer two recently posed open questions. Which distance-hereditary graphs are C–I graphs? Which Ptolemaic (i.e. chordal distance-hereditary) graphs are C–I graphs? It follows that C–I graphs can be recognized efficiently in the class of all distance-hereditary graph whereas recognizing C–I graphs in general is known to be NP-complete.     

Sensitivity analysis of composite piecewise smooth equations

This paper is a contribution to the sensitivity analysis of piecewise smooth equations. A piecewise smooth function is a Lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible B-derivative at this point. We emphasise the role of functions of the typef=g °h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf=g ° is a Lipschitzian homeomorphism near a given point for every sufficiently close toh with respect to theC ¹-topology. Revised version of part of the paper “Sensitivity analysis and Newton’s method for composite piecewise smooth equations”.     

Two characterisations of the minimal triangulations of permutation graphs

A minimal triangulation of a graph is a chordal supergraph with an inclusion-minimal edge set. Minimal triangulations are obtained from adding edges only to minimal separators, completing minimal separators into cliques. Permutation graphs are the comparability graphs whose complements are also comparability graphs. Permutation graphs can be characterised as the intersection graphs of specially arranged line segments in the plane, which is called a permutation diagram. The minimal triangulations of permutation graphs are known to be interval graphs, and they can be obtained from permutation diagrams by applying a geometric operation, that corresponds to the completion of separators into cliques. We precisely specify this geometric completion process to obtain minimal triangulations, and we completely characterise those interval graphs that are minimal triangulations of permutation graphs.     

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d  -space R^d${\mathbb{R}}^d$, with d?2$d?2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z^d${\mathbb{Z}}^d$. In the particular case when the Delone set X   in R^d${\mathbb{R}}^d$ comes from a primitive substitution tiling of R^d${\mathbb{R}}^d$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X   to the lattice βZ^d$\beta {\mathbb{Z}}^d$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.