Some Numerical Implications of the Hardy and Littlewood Analysis of the 3-Primes Problem

We explore some of the numerical implications of the famous 1922 paper Some Problems of Partitio Numerorum III: On the Expression of a Number as a Sum of Primes of Hardy and Littlewood. In particular, we prove that if the Generalized Riemann Hypothesis holds, then Hardy and Littlewood's analysis yields that every odd number greater than 1.24 × 10⁵⁰ is a sum of three primes.     

Metal Coordinated Phosphinite Derivatives

Abstract

The following phosphinites as derivatives of the unknown diphenylphosphinous acid with phosphorus(III) of the type (C₆H₅)₂POX (X=H, NH₄, S(O)R, P(C₆H₅)₂) are normally unstable in the free state. However, they can be stabilized in the protecting coordination sphere of transition metal complexes by building them up by direct or indirect methods. The simple starting molecule is, in principle, diphenylphosphane, (C₆H₅)₂PH, which can be functionalized by oxidation or coordination and subsequent substitution and reduction reactions as it is shown in the following scheme:     

Cell‐Derived Vesicles for Single‐Molecule Imaging of Membrane Proteins

A new approach is presented for the application of single‐molecule imaging to membrane receptors through the use of vesicles derived from cells expressing fluorescently labeled receptors. During the isolation of vesicles, receptors remain embedded in the membrane of the resultant vesicles, thus allowing these vesicles to serve as nanocontainers for single‐molecule measurements. Cell‐derived vesicles maintain the structural integrity of transmembrane receptors by keeping them in their physiological membrane. It was demonstrated that receptors isolated in these vesicles can be studied with solution‐based fluorescence correlation spectroscopy (FCS) and can be isolated on a solid substrate for single‐molecule studies. This technique was applied to determine the stoichiometry of α3β4 nicotinic receptors. The method provides the capability to extend single‐molecule studies to previously inaccessible classes of receptors.     

Static structure function of turbulent flow from the Navier-Stokes equations

The velocity field of a fully developed isotropic turbulent flow is decomposed into a smoothed fieldu _i ^(r) and a strongly fluctuating part _i ^(r) depending on a lengthr which is varied through all scales. The eddies described byu _i ^(r) loose their energy either by direct dissipation or by energy transfer to the _i ^(r) -eddies. Both contributions can be traced back to the second order static structure functionD(r) and the Lagrangian time correlation function. The latter can also be evaluated in terms ofD(r). The energy balance then gives an integro-differential equation forD(r) which determinesD(r) uniquely. The solution is not only in agreement with the scaling behaviour in the viscous and inertial subranges as predicted by dimensional arguments, but it also gives the correct transition between the two regimes. A comparison with experimental data is offered.     

Complex Chemical Synthesis of Unstable Sulfinyl- and Sulfonyl Derivatives of Phosphorus(III)

Abstract

The phosphorus(III) homologues of sulfonamides, R′₂P-S(O)₂R, are hitherto unknown. This is also in the case of phosphorus(V) derivatives: sulfinylphosphinates undergo symmetrization reactions to the homogeneous anhydrides. Sulfinylphosphinites, the O-isomers of sulfonylphosphanes, can only be built up as unstable ligands in the coordination sphere of transition metals, where the Br(CO)₄Mn-fragment seems to be more stable than the (CO)₅M-fragment (M=Cr, Mo, W). Thus, we now synthesized the iodo complexes I(CO)₄Mn[P(C₆H₅)₂X] (X=H, SiMe₃) as suitable starting material for the synthesis of SO-, SO₂-, RSO- and RSO₂-derivatives of coordinated phosphorus(III). Because of the surprising cis-position of the iodo and silyl group (x-ray structure analysis) the silyl-phosphane complex 2 is also of high interest to 1,2-elimination reactions to give metal-phosphorus double bonds.