The validity of nonlinear Langevin equations

In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Itô or Stratonovich concerns these higher orders, for which the equation is not valid anyway.     

Inclusive ϱ0,K *±(892) andf production in\bar p p Interactions at 12 GeV/c

Inclusive production of ?⁰,K ^*±(892), andf is studied in \(\bar p\) p interactions at 12 GeV/c. The inclusive cross sections for ?⁰,K ^*±(892), andf are found to be 6.7±0.3 mb, 1.0±0.2 mb, and 1.4±0.3 mb, respectively. The differential cross sections are presented as a function of c.m. rapidity, Feynmanx and square of the transverse momentump _T ² . Comparison with the correspondingpp data shows some interesting differences which can be attributed to the \(\bar p\) p annihilation. The results are compared with the predictions of the quark fusion model.     

Honeycomb nets with interpenetrating frameworks involving iminodiacetato-copper(II) blocks and bipyridine spacers: syntheses, characterization, and magnetic studies

Three coordination polymers of copper(II), viz. ([Cu(ida)(4,4'-bipyH)]ClO(4))( proportional, variant ) (1), ([Cu(2)(ida)(2)(micro-4,4'-bipy)].2H(2)O)( proportional, variant ) (2), and [Cu(2)(ida)(2)(bpa)]( proportional, variant ) (3) have been synthesized by the process of self-assembly using Cu(ida) [ida = iminodiacetate(2-)] as the building block and 4,4'-bipyridyl and 1,2-bis(4-pyridyl)ethane (bpa) as linkers. Crystals of 1 are orthorhombic, of space group Pna2(1), with a = 13.8956(12) A, b = 16.3362(16) A, c = 7.3340(12), and Z = 4. Both compounds 2 and 3 crystallize in monoclinic space group P2(1)/a with a = 10.1887(8) A (9.6779(10) A for 3), b = 8.0008(11) A (9.1718(10) A), c = 11.6684(9) A (12.9144(12) A), beta = 98.307(11) degrees (102.796(18) degrees ), and Z = 2 (2). Compound 1 has a zigzag chain structure with an extensive hydrogen-bonded network while compounds 2 and 3 are honeycomb (6,3) nets with interpenetrating structures. Variable temperature (2-300 K) magnetic study indicates the presence of weak antiferromagnetic interactions (J = 0.82 +/- 0.01 cm(-)(1)) in 1 and ferromagnetic in 2 (J = -0.45 +/- 0.05 cm(-)(1)) and 3 (J = -0.21 +/- 0.02 cm(-)(1)). The extent of planarity of the bridging "Cu-O-C-O-Cu" moiety, acting as the super-exchange pathway between the neighboring copper centers, probably controls the sign of the magnetic exchange coupling in these compounds.     

Modular bis(oxazoline) ligands for palladium catalyzed allylic alkylation: unprecedented conformational behaviour of a bis(oxazoline) palladium eta3-1,3-diphenylallyl complex

New families of enantiopure bis(oxazolines) with 4,5-trans (5 a-g) or 4,5-cis (6 c) stereochemistry at the individual rings have been prepared in high yield. Their eta(3)-allyl palladium complexes (8 a-g, 9 c and 10) have been used as catalytic precursors in allylic alkylation reactions with excellent enantioselectivities (up to 96 %) for the trans oxazoline derivatives, while Pd/6 c system was inactive. NMR studies on palladium eta(3)-1,3-diphenylallyl intermediates (11 a, c and e) showed the presence of syn/syn- and syn/anti-allyl isomers in solution; this resembles the first example of eta(3)-eta(1)-eta(3) isomerism in Pd allylic complexes containing bis(oxazolines) derived from malonic acid.     

A silica-supported, switchable, and recyclable hydroformylation-hydrogenation catalyst

A homogeneous hydroformylation catalyst, designed to produce selectively linear aldehydes, was covalently tethered to a polysilicate support. The immobilized transition-metal complex [Rh(A)CO]+(1+)), in which A is N-(3-trimethoxysilane-n-propyl)-4,5-bis(diphenylphosphino)phenoxazine, was prepared both via the sol-gel process and by covalent anchoring to silica. 1+ was characterized by means of (31)P and (29)Si MAS NMR, FT-IR, and X-ray photoelectron spectroscopy. Polysilicate immobilized Rh(A) performed as a selective hydroformylation catalyst showing an overall selectivity for the linear aldehyde of 94.6% (linear to branched aldehyde ratio of 65). In addition 1-nonanol, obtained via the hydrogenation of the corresponding aldehyde, was formed as an unexpected secondary product (3.6% at 20% conversion). Under standard hydroformylation conditions, 1+ and HRh(A)(CO)(2)(1) coexist on the support. This dual catalyst system performed as a hydroformylation/hydrogenation sequence catalyst (Z), giving selectively 1-nonanol from 1-octene; ultimately, 98% of 1-octene was converted to mainly 1-nonanal and 97% of the nonanal was hydrogenated to 1-nonanol. The addition of 1-propanol completely changes Z in a hydroformylation catalyst (X), which produces 1-nonanal with an overall selectivity of 93%, and completely suppresses the reduction reaction. If the atmosphere is changed from CO/H(2) to H(2) the catalyst system is switched to the hydrogenation mode (Y), which shows a clean and complete hydrogenation of 1-octene and 1-nonanal within 24 h. The immobilized catalyst can be recycled and the system can be switched reversibly between the three "catalyst modes" X, Y, and Z, completely retaining the catalyst performance in each mode.     

Aminosulf(ox)ides as ligands for Iridium(I)-catalyzed asymmetric transfer hydrogenation

A new class of efficient catalysts was developed for the asymmetric transfer hydrogenation of unsymmetrical ketones. A series of chiral N,S-chelates (6-22) was synthesized to serve as ligands in the iridium(I)-catalyzed reduction of ketones. Both formic acid and 2-propanol proved to be suitable as hydrogen donors. Sulfoxidation of an (R)-cysteine-based aminosulfide provided a diastereomeric ligand family containing a chiral sulfur atom. The two chiral centers of these ligands showed a clear effect of chiral cooperativity. In addition, aminosulfides containing two asymmetric carbon atoms in the backbone were synthesized. Both the sulfoxide-containing beta-amino alcohols and the aminosulfides derived from 1,2-disubstituted amino alcohols gave rise to high reaction rates and moderate to excellent enantioselectivities in the reduction of various ketones. The enantioselective outcome of the reaction was favorably affected by selecting the most appropriate hydrogen donor. Enantioselectivities of up to 97% were reached in the reduction of aryl-alkyl ketones.     

Electrochemical activity of various types of aqueous In(III) species at a mercury electrode

An interpretation framework is presented which provides a straightforward means to characterise the electrochemical reactivity of aqueous ions together with their various hydrolysed counterparts. Our novel approach bypasses the more laborious strategy of solving rigorously, for all relevant species, the complete set of Butler-Volmer equations coupled to diffusion differential equations. Specifically, we consider the spatial variable via a Koutecký-Koryta type of differentiation between nonlabile and labile zones adjacent to the electrode. The theory is illustrated by an assessment of the electrochemical reactivity of aqueous In(III) species based upon proper comparison between relevant time scales of the involved interfacial processes, i.e. diffusion, (de)protonation of inner-sphere water, dissociation/release of H₂O and OH⁻, and electron transfer. The analysis reveals that whilst all In(III) species are labile on the experimental timescale with respect to (de)protonation and (de)hydration, there are large differences in the rates of electron transfer between \( \mathrm{In}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6^{3+} \) and the various hydroxy species. Specifically, in the case of \( \mathrm{In}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6^{3+} \), the rate of electron transfer is so slow that it replaces the traditional Eigen rate-limiting water release step in the overall passage from hydrated In³⁺ to its reduced metallic form; in contrast, the In(III) hydroxy species display electrochemically reversible behaviour.

     

Novel Mo(V)-dithiolene compounds: characterization of nonsymmetric dithiolene complexes by electrospray ionization mass spectrometry

Three new Mo(V) dithiolene compounds have been synthesized by addition of alkynes ((Me(3)Si)(2)C(2) (TMSA), (Me(3)Si)(2)C(4), and (Ph)(2)C(4) to MoO(2)S(2)(2-) in a MeOH/NH(3) mixture: [Mo(2)(O)(2)(mu-S)(2)(eta(2)-S(2))(eta(2)-S(2)C(2)H(2))](2)(-) 1, [Mo(2)(O)(X)(mu-S)(2)(eta(2)-S(2))(eta(2)-S(2)C(2)Ph(C(2)Ph))](2-) 2 (X = O or S), and [Mo(2)(O)(2)(mu-S)(2)(eta(2)-S(2))(eta(2)-S(2)C(2)H(C(2)H))](2-) 3. The structure of 1 as determined by single-crystal X-ray diffraction study (space group Pbca, a = 13.3148(1) A, b = 15.7467(4) A, c = 28.4108(7) A, V = 5956.7(2) A(3)) is discussed. 2 and 3 have been identified by ESMS (electrospray mass spectrometry), (1)H NMR, (13)C NMR, and infrared spectroscopies. This investigation completes our previous study devoted to the addition of DPA (C(2)Ph(2)) to MoO(2)S(2)(2-) which led to [Mo(2)(O)(X)(mu-S)(2)(eta(2)-S(2))(eta(2)-S(2)C(2)Ph(2))](2-) 4 (X = O or S). A reaction scheme is proposed to explain the formation of the different species present in solution. The reactivity of the remaining nucleophilic site of these complexes (eta(2)-S(2)) toward dicarbomethoxyacetylene (DMA) is also discussed.     

On the asymptotic approximations of the solutions of a system of two non-linearly coupled harmonic oscillators

A procedure, closely related to the averaging method, is given for the construction of asymptotic approximations for the solutions of the Lagrangian system

$\text{d}\text{x}{}_1\text{d}\text{t}\text{=x}{}_2\text{,}\text{d}\text{x}{}_2\text{d}\text{t}\text{=?x}{}_1\text{+ε}\text{∑}\text{i, j=1}\text{4}\text{α}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{,}\text{d}\text{x}{}_3\text{d}\text{t}\text{=x}{}_4\text{,}\text{d}\text{x}{}_4\text{d}\text{t}\text{=?4(1+δ)x}{}_3\text{+ε}\text{∑}\text{i, j=1}\text{4}\text{β}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{, x}{}_1\text{(0)=a}{}_0\text{, x}{}_2\text{(0)=0, x}{}_3\text{(0)=b}{}_0\text{, x}{}_4\text{(0)=0,}$

where ε > 0 and δ > 0 are small parameters, a_ij = a_ji and β_ij = β_ji are real constants. The solutions can be represented as follows:

$\text{x}{}_1\text{=a}\text{cos}\text{(t+ξ), x}{}_2\text{=a}\text{sin}\text{(t+ξ), x}{}_3\text{=b}\text{cos}\text{(2t+ψ), x}{}_4\text{=2b}\text{sin}\text{(2t+ψ).}$

The asymptotic approximations (in a certain well-defined sense) of a, b, ξ and ψ are given. It is shown that a rapid variation of the phase, ψ(t), will take place if b → 0 for the case that δ ≈ ε or δ ? ε. In addition for the case δ ≈ ε the phenomenon of bifurcation will be discussed.     

Measurements of the spin rotation parameterR in high energy elastic scattering and helicity amplitudes at Serpukhov energies

The spin rotation parameterR in pp and ⁺ p elastic scattering at 45 GeV/c has been measured at the Serpukhov accelerator, for ¦t¦ ranging from 0·2 to 0·5 (GeV/c)². The results are presented, together with previousR measurements at lower energies, and are compared with the predictions of Regge pole models. The equality of the values forR in proton-proton and pion-proton scattering, within the experimental errors, is a test of factorization of the residues.Ans-channel helicity amplitude analysis for pion-nucleon scattering at 40 GeV/c is made using all available data, Significant results are obtained for the isoscalar exchange amplitudes. The helicity flip in isoscalar exchange is non negligible. The energy dependence of this amplitude, at 6, 16 and 40 GeV/c, is compared with predictions of Reggs pole models.Presented by L.van Rossum at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.