Synthesis of a (thienylethenyl)benzimidazole platinum acetylide complex

A new platinum acetylide complex based on 6-dialkylaminobenzimidazol-2-yl-vinyl-2-thiophene-5-ylethyne was synthesized in seven steps and 2% overall yield.     

Utilization of the hydroxyalkylation reaction to prepare bis(benzocrown ethers)

The hydroxyalkylation reaction has been used to condense benzocrown ethers with various aldehydes and ketones. The condensation reactions are catalyzed by triflic or sulfuric acid. The products from the reactions are bis(benzocrown ethers) and they are formed in good yields (42-98%, 13 examples).     

Enantioselective synthesis of C2-functionalized, N-protected morpholines and orthogonally N,N'-protected piperazines via organocatalysis

In this Letter, we describe a novel three-step, one-pot procedure for the enantioselective synthesis of N-benzyl protected morpholines and orthogonally N,N'-protected piperazines with chiral alkyl groups installed at the C2 position of each heterocyclic core via organocatalysis. This methodology allows for the rapid preparation of functionalized morpholines and piperazines that are not readily accessible through any other chemistry in good to excellent % ee (55-98% ee).     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

Tunneling from a 3-dimensional quantum well in an electric field: an analytic solution

We derive the stationary tunneling solution for charged particles moving in a spherical, 3-dimensional zero-range potential plus a constant electric field. From the analytic expression for the wave function we calculate the distribution of the current inside and outside the vacuum barrier. At low field strengths there is a constant spreading of the tunnel current orthogonal to the direction of the applied field. At intermediate field strengths the exact results for the current distribution behave different from the semiclassical predictions.     

Contractive projections and operator spaces

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces , , generalizing the row and column Hilbert spaces and , and we show that an atomic subspace that is the range of a contractive projection on is isometrically completely contractive to an -sum of the and Cartan factors of types 1 to 4. In particular, for finite-dimensional , this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w-closed -triples without an infinite-dimensional rank 1 w-closed ideal.