The Intersection of Two Infinite Matroids

Conjecture: Let M and N be two matroids (possibly of infiniteranks) on the same set S. Then there exists a set I independentin both M and N, which can be partitioned as I=HK, where sp_M(H)sp_N(K)=S.This conjecture is an extension of Edmonds' matroid intersectiontheorem to the infinite case. We prove the conjecture when oneof the matroids (say M) is the sum of countably many matroidsof finite rank (the other matroid being general). For the proofwe have also to answer the following question: when does thereexist a subset of S which is spanning for M and independentin N?     

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of \(K_{n,n}\) is partitioned into n sets, each of size n, then there is a partial rainbow matching of size \(n-1\). He proved that there is a partial rainbow matching of size \(n(1-\frac{D_n}{n!})\), where \(D_n\) is the number of derangements of [n]. This means that there is a partial rainbow matching of size about \((1- \frac{1}{e})n\). Using a topological version of Hall’s theorem we improve this bound to \(\frac{2}{3}n\).     

A Type of Auxiliary for Native Chemical Peptide Ligation beyond Cysteine and Glycine Junctions

Native chemical ligation enables the chemical synthesis of proteins. Previously, thiol‐containing auxiliary groups have been used to extend the reaction scope beyond N‐terminal cysteine residues. However, the N‐benzyl‐type auxiliaries used so far result in rather low reaction rates. Herein, a new N^α‐auxiliary is presented. Consideration of a radical fragmentation for cleavage led to the design of a new auxiliary group which is selectively removed under mildly basic conditions (pH 8.5) in the presence of TCEP and morpholine. Most importantly and in contrast to previously described auxiliaries, the 2‐mercapto‐2‐phenethyl auxiliary is not limited to Gly‐containing sites and ligations succeed at sterically demanding junctions. The auxiliary is introduced in high yield by on‐resin reductive amination with commercially available amino acid building blocks. The synthetic utility of the method is demonstrated by the synthesis of two antimicrobial proteins, DCD‐1L and opistoporin‐2.     

On the Lifetime of the Transients (NP)?(CH3)n (NP=Ag0, Au0, TiO2 Nanoparticles) Formed in the Reactions Between Methyl Radicals and Nanoparticles Suspended in Aqueous Solutions

Methyl radicals react in fast reactions, with rate constants k>1×10⁸ M ^?1 s^?1, with Au⁰, Ag⁰ and TiO₂ nanoparticles (NPs) dispersed in aqueous solutions to form intermediates, (NP)? (CH₃)_n, in which the methyl groups are covalently bound to the NPs. These intermediates decompose to form ethane. As n≥2 is required for the formation of C₂H₆, the minimal lifetime (τ) of the methyls bound to the NPs, (NP)? CH₃, can be estimated from the rate of production of the CH₃^. radicals and the NPs concentration. The results obtained in this study, using a very low dose rate γ‐source for NP=Ag⁰, Au⁰, and TiO₂ point out that τ of these intermediates is surprisingly long, for example, ≥8 and ≥188 sec for silver and gold, respectively. These data point out that the NP? C bond dissociation energies are ≥70 kJ mol^?1. Under low rates of production of CH₃^., that is, when the rate of formation of ethane is very low, other reactions may occur, consequently the mechanism proposed is “broken”. This is observed in the present study only for TiO₂ NPs. These results have to be considered whenever alkyl radicals are formed near surfaces. Furthermore, the results point out that the rate of reaction of methyl radicals with (NP)? (CH₃)_n depends on n, that is, the number of methyl radicals bound to the NPs affect the properties of the NPs.     

Low-degree rational curves on hypersurfaces in projective spaces and their fan degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.     

An observation concerning Ryser's conjecture

If in anr-partiter-graphH everyk edges intersect (wherek>2), then (H)[(r–1)/(k–1)].     

A Tree Version of Kőnig's Theorem

Kőnig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed. Received January 27, 2000/Revised November 2, 2000 RID=" " ID=" " The research of the first author was supported by grants from the Israel Science Foundation, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.