Rhodium-Catalyzed Enantioselective Synthesis,Structures, and Properties of Single and Double Azahelicene-Like Molecules

The enantioselective synthesis of aza[6] and [7]helicene-like molecules have been achieved by the cationic rhodium(I)/axially chiral biaryl bisphosphine complex-catalyzed intramolecular [2+2+2] cycloaddition of cyanodiynes. This protocol was successfully applied to the diastereo- and enantioselective synthesis of an S-shaped double aza[6]helicene-like molecule with a high ee value of 89 %. Although no epimerization and racemization were observed in the double carbo[6]helicene-like molecule at 80 °C, epimerization and racemization of the double aza[6]helicene-like molecule proceeded at 80 °C. This double aza[6]helicene-like molecule showed good fluorescent quantum yields and chiroptical responses under both neutral and acidic conditions.     

Organometallic Molecular Wires with Thioacetylene Backbones,trans-{RS-(C≡C)n}2Ru(phosphine)4: High Conductance through Non-Aromatic Bridging Linkers

In this work, the design, synthesis, and single-molecule conductance of ethynyl- and butadiynyl-ruthenium molecular wires with thioether anchor groups [RS=n-C₆H₁₃S, p-tert-Bu−C₆H₄S), trans-{RS−(C≡C)_n}₂Ru(dppe)₂ (n=1 ( 1^R ), 2 ( 2^R ); dppe: 1,2-bis(diphenylphosphino)ethane) and trans-(n-C₆H₁₃S−C≡C)₂Ru{P(OMe)₃}₄ 3^hex ] are reported. Scanning tunneling microscope break-junction study has revealed conductance of the organometallic molecular wires with the thioacetylene backbones higher than that of the related organometallic wires having arylethynylruthenium linkages with the sulfur anchor groups, trans-{p-MeS−C₆H₄-(C≡C)_n}₂Ru(phosphine)₄ 4 ⁿ (n=1, 2) and trans-(Th−C≡C)₂Ru(phosphine)₄ 5 (Th=3-thienyl). It should be noted that the molecular junctions constructed from the butadiynyl wire 2^R , trans-{ Au −RS−(C≡C)₂}₂Ru(dppe)₂ ( Au : gold metal electrode), show conductance comparable to that of the covalently linked polyynyl wire with the similar molecular length, trans-{ Au −(C≡C)₃}₂Ru(dppe)₂ 6³ . The DFT non-equilibrium Green's function (NEGF) study supports the highly conducting nature of the thioacetylene molecular wires through HOMO orbitals.     

Total Biosynthesis of Melleolides from Basidiomycota Fungi: Mechanistic Analysis of the Multifunctional GMC Oxidase Mld7

Mushroom terpenoids are biologically and chemically diverse fungal metabolites. Among them, melleolides are representative sesquiterpenoids with a characteristic protoilludane skeleton. In this study, we applied a recently established hot spot knock-in method to elucidate the biosynthetic pathway leading to 1α-hydroxymelleolide. The biosynthesis of the sesquiterpene core involves the cytochrome P450 catalyzing stepwise hydroxylation of the Δ⁶-protoilludene framework and a stereochemical inversion process at the C5 position catalyzed by short-chain dehydrogenase/reductase family proteins. The highlight of the biosynthesis is that the flavoprotein Mld7 catalyzes an oxidation-triggered double-bond shift accompanying dehydration and acyl-group-assisted substitution with two different nucleophiles at the C6 position to afford the Δ⁷-protoilludene derivatives, such as melleolide and armillarivin. The complex reaction mechanism was proposed by DFT calculations. Of particular importance is that product distribution is regulated by interaction with the cell membrane.     

DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods

In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. By using the analysis technique developed here, the theoretical convergence rate of the Sinc approximation with the DE transformation is also given for a class of functions for which the SE transformation is suitable. The result is quite similar to above; the convergence rate in the DE case is slightly lower than in the SE case. Numerical examples which support those two theoretical results are also given.     

On extensions of some Flugede–Putnam type theorems involving (p,k)‐quasihyponormal,spectral, and dominant operators

A Hilbert space operator S is called (p, k)‐quasihyponormal if S *^k ((S *S)^p – (SS *)^p )S^k ≥ 0 for an integer k ≥ 1 and 0 < p ≤ 1. In the present note, we consider (p, k)‐quasihyponormal operator S ∈ B (H) such that SX = XT for some X ∈ B (K,H) and prove the Fuglede–Putnam type theorems when the adjoint of T ∈ B (K) is either (p, k)‐quasihyponormal or dominant or a spectral operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic Estimates of the Distribution of Brownian Hitting Time of a Disc

The distribution of the first hitting time of a disc for the standard two-dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairly detailed asymptotic estimates of its density valid uniformly with respect to the point from which the Brownian motion starts.     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far.     

A converse of Loewner–Heinz inequality and applications to operator means

Let f(t)$f(t)$ be an operator monotone function. Then A?B$A?B$ implies f(A)?f(B)$f(A)?f(B)$, but the converse implication is not true. Let A?B$A?B$ be the geometric mean of A,B?0$A,B?0$. If A?B$A?B$, then B⁻¹?A?I$B^{-1}?A?I$; the converse implication is not true either. We will show that if f(λB+I)⁻¹?f(λA+I)?I$f{(\lambda B+I)}^{-1}?f(\lambda A+I)?I$ for all sufficiently small λ>0$\lambda >0$, then f(λA+I)?f(λB+I)$f(\lambda A+I)?f(\lambda B+I)$ and A?B$A?B$. Moreover, we extend it to multi-variable matrices means.     

Hydrodynamic limit for a spin system on a multidimensional lattice

Summary The hydrodynamic limit for a Markov process of [0, )-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation /t=2^–1P(), whereP is an increasing function which in a typical case equals const·².