Farkas’ identities with quartic characters

The Ramanujan Journal - Farkas in (On an Arithmetical Function II. Complex Analysis and Dynamical Systems II. Contemporary Mathematics, American Mathematical Society, Providence, 2005) introduced...     

Phosphonium-Ring-Fused Bicyclic Metallafuran Complexes of Ruthenium and Osmium

Metallafuran complexes with a fused five-membered phosphonium ring were synthesized from reactions between terminal ynones HC≡C(C=O)R and cis-[Ru/Os(dppm)₂Cl₂] (dppm=1,1-bis(diphenylphosphino)methane). A metal–vinylidene-involving pathway was found to be an energetically feasible formation mechanism for these complexes. These phosphonium-containing metallafurans, like many phosphonium-functionalized drugs, have the ability to induce mitochondrial dysfunction. They also exhibit stronger cytotoxicity against several human cancer cell lines in comparison with their metal precursors and the classic anticancer drug cisplatin. Overall, this work provides structural and mechanistic insights for the rational design of functional metallacycles via activation of alkynes by Ru^II and Os^II centers.     

Results and Estimates on Multiple Solutions of Lidstone Boundary Value Problems

We consider the following boundary value problem (-1)ⁿy(²ⁿ)=F(t,y), n≥ 1, t ∈ (0,1), y⁽²ⁱ⁾(0)=y⁽²ⁱ⁾(1)=0, 0≧i≧n-1. Criteria are developed for the existence of two and three positive solutions of the boundary value problem. In addition, for special cases we establish upper and lower bounds for these positive solutions. Several examples are also included to dwell upon the importance of the results obtained.     

Convenient one-pot synthesis of trans-1,2-diaryloxiranes from the direct coupling of benzyl halides

trans-1,2-Diaryloxiranes were conveniently prepared in an one-pot reaction by the direct coupling of benzyl halides in the presence of silver oxide and DMSO under mild conditions.