Mild and efficient Cs2CO3-promoted synthesis of phosphonates

A mild and convenient synthesis for phosphonates using cesium carbonate (Cs₂CO₃), tetrabutylammonium iodide (TBAI) and DMF was developed at room temperature. Numerous dialkyl phosphites were screened using a diverse array of alkyl halides and these reaction conditions were found to be highly efficient producing various phosphonates exclusively in moderate to high yields.     

Ligand and base additive effects on the reversibility of nucleophilic addition in palladium-catalyzed allylic aminations monitored by nucleophile crossover

A nucleophile crossover experiment was used to monitor the reversibility of nucleophilic addition of benzylamine to π-allylpalladium complexes. Dppe, dppp, dppb, and PHOX showed more crossover than PPh₃ and dppm in both DMF and dichloromethane. Crossover was inhibited by the addition of DBU or Cs₂CO₃, but much less elimination to diene side products was observed with Cs₂CO₃. Analysis of percent crossover vs. percent reaction completion using the PHOX ligand revealed that with added DBU or Cs₂CO₃ crossover only began occurring after 100% completion had been reached.     

The anionic alkylation of easily reducible arenes. A photochemical route to nucleophilic aromatic substitution of anthracene by organolithiums

High yield nucleophilic addition ensues upon mixing tetrahydrofuran solutions of benzyllithium or cyclooctadienyllithium and anthracene at low temperatures At least part of this addition proceeds by a single electron transfer pathway. Photolysis of the resulting adducts leads to the elimination of lithium hydride, giving net nucleophilic substitution in approximately 50% yield. The analogous reaction fails with naphthalene, where photolysis of the organolithium/ arene mixture leads to dimeric products derived from the organolithium.     

The 6Li,6Li-INADEQUATE Experiment: A New Tool for 1D- and 2D-NMR Studies of Organolithium Compounds

The 1D- and 2D-⁶Li, ⁶Li-INADEQUATE experiments are described as new tools for the detection of scalar coupled nonequivalent ⁶Li nuclei in organolithium clusters. Practical applications of these sequences are demonstrated for the ⁶Li-NMR spectra of (E)-1-lithio-2-(2-lithiophenyl)-1-phenylhex-1-ene ( 1 ) and (E)-2-lithio-1-(2-lithiophenyl)-1-phenylpent-1-ene ( 2 ), where signals due to dimers and monomers can be distinguished. The performance of the 2D-⁶Li, ⁶Li-INADEQUATE and the ⁶Li, ⁶Li-COSY-45-LR experiment are compared. The ⁶Li chemical shifts of 1 and 2 are discussed.     

On Rado's Boundedness Conjecture

We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a₁,a₂, and a₃ are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a₁x₁+a₂x₂+a₃x₃=0, then for every finite coloring of the positive integers there is a monochromatic solution to a₁x₁+a₂x₂+a₃x₃=0.     

Learning to Make Sense of Division of Fractions: One K–8 Preservice Teacher's Perspective

Current reform documents in mathematics education recommend that teachers help students develop both conceptual and procedural understandings. However, teachers often do not possess the in-depth mathematical reasoning necessary to accomplish this goal. The purpose of this article is to provide one way in which preservice teachers can come to better understand the mathematics they will teach.     

MO Study of flavin–protein interactions in flavodoxin catalysis

MINDO /3 calculations have been performed on the Clostridium MP flavodoxin active site (a complex of the redox active coenzyme flavin mononucleotide sandwiched between the side chains of methionine and tryptophan) at various redox levels using coordinates derived from x-ray diffraction studies of the holoenzyme. Frontier orbital indices were calculated and indicate that reduction of the flavin is accompanied by induced polar states in the amino acid side chains. This stabilization of charge by the amino acid side chains could account for the reaction rate enhancement of flavin reduction catalyzed by flavodoxin. Frontier orbitals for free flavin, for the flavodoxin bound flavin without the amino acid side chains, and for the oxidized Desulfovibrio vulgaris flavodoxin active site were computed for comparison.     

A bipartite analogue of Dilworth’s theorem for multiple partial orders

Let r be a fixed positive integer. It is shown that, given any partial orders <₁, …, <_r on the same n-element set P, there exist disjoint subsets A,BP, each with at least n^1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an such that every element of A is larger than every element of B in the partial order <_i, or (2) no element of A is comparable with any element of B in any of the partial orders <₁, …, <_r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,BC, each with at least n^1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B.     

Perturbed Dolbeault operators and the homology Todd class

This paper discusses the role played by perturbed Dolbeault operators in relating the coherent sheaf and elliptic operator perspectives on the homology of projective varieties. Among the consequences are index formulas for perturbed Dolbeault operators.