An unusually robust triple bond: synthesis, structure and reactivity of 3-alkynylcyclopropenes

Several 1,2-diphenyl- and 1,2,3-triphenyl-3-alkynylcyclopropenes have been prepared in moderate to very good yields by the reaction of acetylenic nucleophiles with the appropriate cyclopropenylium salt. Single crystal X-ray structures of four of the cyclopropenes were obtained. Stereoselective reduction of the triple bond failed in all cases, whereas model compounds lacking the cyclopropene moiety were reduced successfully. A rational for this lack of reactivity is proposed. The solution-phase thermochemistry of the 3-alkynyl-1,2,3-triphenylcyclopropenes was explored, affording 3-alkynyl-1H-indenes in moderate to good yields.     

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

Preparation and Ir, 31P,and 93Nb NMR spectroscopic investigation of phosphine derivatives of η5-C5H5Nb(CO)4

UV irradiation of η⁵-C₅H₅Nb(CO)₄ in the presence of the phosphine ligands L (L = 2 PEt₃, Ph₂P(CH₂)₂PPh₂ (p₂(n), n = 1–5), cis-Ph₂PCH=CHPPh₂ (c-dpe)), and the mixed arsine-phosphine ligands Ph₂AsCH₂CH₂PPh₂ (arphos) and o-C₆H₄(AsPh₂)(PPh₂) (pab) yields the well defined complexes cis-[η⁵-C₅H₅Nb(CO)₂L]. The monosubstituted species η⁵-C₅H₅Nb(CO)₃L have been characterized spectroscopically. P₂Ph₄ forms mono- and dinuclear, mono- and biligate carbonylniobium complexes.Shielding of the ⁹³Nb nucleus increases in the sequences (i) Ph₂As- < Ph₂P-, (ii) chelate 4-ring < chelate 5-ring and (iii) η⁵-C₅H₅Nb(CO)₂L < η⁵-C₅H₅Nb(CO)₃L < η⁵-C₅H₅Nb(CO)₄, and ³¹P coordination shifts decrease in the order c-dpe > pab > arphos > p₂(2) > p₂(5) > p₂(4) ～ PEt₃ > p₂(3) > p₂(1). The trends generally parallel those for the corresponding NMR parameters of the vanadium complexes. Paramagnetic contributions to the overall shielding are smaller for the ⁹³Nb than for the ⁵¹V nucleus, and this is explained in terms of increased covalency and decreased π-interaction in the niobium complexes.     

Monotonicity properties of the quantum mechanical particle density: an elementary proof

The negative quantum mechanical particle density in its dependence on the potential in the Hamiltonian is a nonlinear operator. We give an explicit proof of the monotonicity of this operator for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.     

Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions

We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti (Ann Scuola Norm Sup Pisa Cl Sci 3(4): 697–718, 1976) and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for Hardy-Rellich inequalities.     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.     

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.