Highly Modular P?OP Ligands for Asymmetric Hydrogenation: Synthesis,Catalytic Activity,and Mechanism

A library of enantiomerically pure P? OP ligands (phosphine–phosphite), straightforwardly available in two synthetic steps from enantiopure Sharpless epoxy ethers is reported. Both the alkyloxy and phosphite groups can be optimized for maximum enantioselectivity and catalytic activity. Their excellent performance in the Rh‐catalyzed asymmetric hydrogenation of a wide variety of functionalized alkenes (26 examples) and modular design makes them attractive for future applications. The lead catalyst incorporates an (S)‐BINOL‐derived (BINOL=1,1′‐bi‐2‐naphthol) phosphite group with computational studies revealing that this moiety has a dual effect on the behavior of our P? OP ligands. On one hand, the electronic properties of phosphite hinder the binding and reaction of the substrate in two out of the four possible manifolds. On the other hand, the steric effects of the BINOL allow for discrimination between the two remaining manifolds, thereby elucidating the high efficiency of these catalysts.     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

Geodesic Ham-Sandwich Cuts

Let P be a simple polygon with m vertices, k of which are reflex, and which contains r red points and b blue points in its interior. Let n = m + r + b. A ham-sandwich geodesic is a shortest path in P between two points on the boundary of P that simultaneously bisects the red points and the blue points. We present an O(n log k)-time algorithm for finding a ham-sandwich geodesic. We also show that this algorithm is optimal in the algebraic computation tree model when parameterizing the running time with respect to n and k.     

Enantiopure Narrow Bite‐Angle POP Ligands: Synthesis and Catalytic Performance in Asymmetric Hydroformylations and Hydrogenations

Herein is reported the preparation of a set of narrow bite‐angle P–OP ligands the backbone of which contains a stereogenic carbon atom. The synthesis was based on a Corey–Bakshi–Shibata (CBS)‐catalyzed asymmetric reduction of phosphomides. The structure of the resulting 1,1‐P–OP ligands, which was selectively tuned through adequate combination of the configuration of the stereogenic carbon atom, its substituent, and the phosphite fragment, proved crucial for providing a rigid environment around the metal center, as evidenced by X‐ray crystallography. These new ligands enabled very good catalytic properties in the Rh‐mediated enantioselective hydrogenation and hydroformylation of challenging and model substrates (up to 99 % ee). Whereas for asymmetric hydrogenation the optimal P–OP ligand depended on the substrate, for hydroformylation, a single ligand was the highest‐performing one for almost all studied substrates: it contains an R‐configured stereogenic carbon atom between the two phosphorus ligating groups, and an S‐configured 3,3′‐diphenyl‐substituted biaryl unit.     

Propanolysis of Esters Using Chlorotrimethylsilane

A variety of methyl esters are converted into the corresponding propyl esters upon treatment with l-propanol and chlorotrimethylsilane. Among them acyclic aliphatic esters have the best conversion rate.     

On triconnected and cubic plane graphs on given point sets

Let S be a set of n4 points in general position in the plane, and let h<n be the number of extreme points of S. We show how to construct a 3-connected plane graph with vertex set S, having max{3n/2,n+h−1} edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n4 is even and hn/2+1. The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.     

A Sufficient Condition for Lipschitz Stability of Controlled Invariant Subspaces

Given a pair of matrices (A, B) we study the Lipschitz stability of its controlled invariant subspaces. A sufficient condition is derived from the geometry of the set formed by the quadruples (A, B, F, S) where S is an (A, B)-invariant subspace and F a corresponding feedback.     

Retractability of set theoretic solutions of the Yang-Baxter equation

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.