Statistics of Persistent Events in the Binomial Random Walk: Will the Drunken Sailor Hit the Sober Man?

The statistics of persistent events, recently introduced in the context of phase ordering dynamics, is investigated in the case of the one-dimensional lattice random walk in discrete time. We determine the survival probability of the random walker in the presence of an obstacle moving ballistically with velocity v, i.e., the probability that the random walker remains up to time n on the left of the obstacle. Three regimes are to be considered for the long-time behavior of this probability, according to the sign of the difference between v and the drift velocity V¯ of the random walker. In one of these regimes (v>V¯), the survival probability has a nontrivial limit at long times which is discontinuous at all rational values of v. An algebraic approach allows us to compute these discontinuities as well as several related quantities. The mathematical structure underlying the solvability of this model combines elementary number theory, algebraic functions, and algebraic curves defined over the rationals.     

Light scattering from mesoscopic objects in diffusive media

No direct imaging is possible in turbid media, where light propagates diffusively over length scales larger than the mean free path .The diffuse intensity is, however, sensitive to the presence of any kind of object embedded in the medium, e.g. obstacles or defects. The long-ranged effects of isolated objects in an otherwise homogeneous, non-absorbing medium can be described by a stationary diffusion equation. In analogy with electrostatics, the influence of a single embedded object on the intensity field is parametrized in terms of a multipole expansion. An absorbing object is chiefly characterized by a negative charge, while the leading effect of a non-absorbing object is due to its dipole moment. The associated intrinsic characteristics of the object are its capacitance Q or its effective radius ,and its polarizability P. These quantities can be evaluated within the diffusion approximation for large enough objects. The situation of mesoscopic objects, with a size comparable to the mean free path, requires a more careful treatment, for which the appropriate framework is provided by radiative transfer theory. This formalism is worked out in detail, in the case of spherical and cylindrical objects of radius R, of the following kinds: (i) totally absorbing (black), (ii) transparent, (iii) totally reflecting. The capacitance, effective radius, and polarizability of these objects differ from the predictions of the diffusion approximation by a size factor, which only depends on the ratio .The analytic form of the size factors is derived for small and large objects, while accurate numerical results are obtained for objects of intermediate size .For cases (i) and (ii) the size factor is smaller than one and monotonically increasing with ,while for case (iii) it is larger than one and decreasing with . Received: 7 August 1998 / Accepted: 3 September 1998     

Synthesis and Crystal Structure of the Octametallic Ferrocene Containing Molybdenum(V) Cluster Mo4(μ3-O)4(μ2-O2P(CH2)2Fc)4O4 and Crystal Structure of 1,1′-Bis(Chloromethyl)Ferrocene

The reaction of MoO₂(acac)₂ with 1,1??-[phosphinicobis(methylene)]-ferrocene in ethanol at 120?°C and in the presence of triethylamine results in the octametallic cluster Mo₄(??₃-O)₄(??₂-O₂P(CH₂)₂Fc)₄O₄. If the reaction is carried out in the absence of triethylamine a mixture of products are obtained of which the identity of the compound produced was hypothesized but not definitively established. Mo₄(??₃-O)₄(??₂-O₂P(CH₂)₂Fc)₄O₄ is characterized by IR and ³¹P-NMR spectroscopy along with the single crystal X-ray determination of its structure. The crystal structure of 1,1??-bis(chloromethyl)ferrocene is also detailed.     

Syntheses, 95Mo NMR Spectroscopy and Structures of Distorted Cubic Mo4(��3-O)4(��2-O2P(CH2C6H5)2)4O4 and the Open Mixed-Valent Cluster, Mo4(��3-O)2(��2-O2P(CH2C6H5)2)6O6

The reaction of MoO₂(acac)₂ and dibenzylphosphinic acid in ethanol leads to a red distorted cubic tetrameric cluster, Mo₄(??³-O)₄(??²-O₂P(CH₂C₆H₅)₂)₄O₄, and a pink open mixed-valent cluster, Mo₄(??³-O)₂(??²-O₂P(CH₂C₆H₅)₂)₆O₆, when the reduction is carried out at 120 and 75 °C, respectively. ⁹⁵Mo NMR spectroscopy revealed a singlet for Mo₄(??³-O)₄(??²-O₂P(CH₂C₆H₅)₂)₄O₄ (1) at 584.9 ppm (????_1/2 = 4500 Hz) and two resonances for Mo₄(??³-O)₂(??²-O₂P(CH₂C₆H₅)₂)₆O₆ (2) at 238.8 ppm (????_1/2 = 1250 Hz) and 6.4 ppm (????_1/2 = 5999 Hz), which were assigned to the Mo(V) and Mo(VI) sites, respectively. DFT geometries and ⁹⁵Mo DFT-GIAO chemical shifts for Mo₄(??³-O)₄(??²-O₂P(CH₃)₂)₄O₄ (3) and Mo₄(??³-O)₂(??²-O₂P(CH₃)₂)₆O₆ (4) are consistent with X-ray crystallography and ⁹⁵Mo NMR of 1 and 2. The open complex, Mo₄(??³-O)₂(??²-O₂P(CH₂C₆H₅)₂)₆O₆·2(CH₂Cl₂), exhibits a central Mo(V)?CMo(V) single bond at 2.6217(5) Å with each Mo(V) atom bonded to one oxo (trans-disposed) terminal ligand.     

Bistable Retinal Schiff Base Photodynamics of Histidine Kinase Rhodopsin HKR1 from Chlamydomonas reinhardtii

The photodynamics of the recombinant rhodopsin fragment of the histidine kinase rhodopsin HKR1 from Chlamydomonas reinhardtii was studied by absorption and fluorescence spectroscopy. The retinal cofactor of HKR1 exists in two Schiff base forms RetA and RetB. RetA is the deprotonated 13‐cis‐retinal Schiff base (RSB) absorbing in the UVA spectral region. RetB is the protonated all‐trans RSB absorbing in the blue spectral region. Blue light exposure converts RetB fully to RetA. UVA light exposure converts RetA to RetB and RetB to RetA giving a mixture determined by their absorption cross sections and their conversion efficiencies. The quantum efficiencies of conversion of RetA to RetB and RetB to RetA were determined to be 0.096 ± 0.005 and 0.405 ± 0.01 respectively. In the dark thermal equilibration between RetA and RetB with dominant RetA content occurred with a time constant of about 3 days at room temperature. The fluorescence emission behavior of RetA and RetB was studied, and fluorescence quantum yields of ?_F(RetA) = 0.00117 and ?_F(RetB) = 9.4 × 10^?5 were determined. Reaction coordinate schemes of the photodynamics are developed.     

Two hydrogen-bond-cross-linked molybdenum (VI) network polymers: synthesis, crystal structures and cyclooctene epoxidation with H2O2

Two molybdenum (VI) hydrogen-bonded network polymers [MoO₂F₄]·(4,4′-H₂bpd)(H₂O)₂ (1) and [MoO₂Cl₃(H₂O)]·(4,4′-H₂bpd)Cl (2) (bpd = bipiperidine) have been synthesized and examined as catalysts for epoxidation of cyclooctene. Complexes of the Mo compounds containing the bpd ligand are prepared and characterized by infrared spectroscopy, thermogravimetric and elemental analyses. They have been structurally characterized by single crystal X-ray diffraction analysis. The structures of both the complexes are shown to be comprised of molybdenum and two protonated N-ligand cations that have resulted in a cross-linked hydrogen-bonded network structure. These complexes are applicable as catalysts for the cis-cyclooctene epoxidation reactions with hydrogen peroxide as a source of oxygen and NaHCO₃ as a cocatalyst. It has been observed that the formation of the oxidant peroxymonocarbonate ion, HCO₄ ⁻ by hydrogen peroxide and bicarbonate enhances the epoxidation reaction. Both the complexes have exhibited a good activity and a very high selectivity for the formation of cyclooctene oxide. An erratum to this article can be found at     

Dynamic Frequency Shifts of Complexed Ligands: An NMR Study of -[1-C,1-H]Glucose Complexed to theEscherichia coliPeriplasmic Glucose/Galactose Receptor

The¹³C multiplet structure of -[1-¹³C,1-²H]glucose complexed to theEscherichia coliperiplasmic glucose/galactose receptor has been studied as a function of temperature. Asymmetric multiplet patterns observed are shown to arise from dynamic frequency shifts. Multiplet asymmetry contributions resulting from shift anisotropy–dipolar cross correlations were found to be small, with optimal fits of the data corresponding to small, negative values of the correlation factor, χ^CD-CSA. Additional broadening at higher temperatures most probably results from ligand exchange between free and complexed states. Effects of internal motion are also considered theoretically, and indicate that the order parameter for the bound glucose is ≥0.9.     

Temperature and velocity flow fields measurements using ultrasonic computer tomography

This paper presents a measurement method to determine the velocity flow field and the temperature in a cross-section of an aerosol chamber by means of the ultrasonic computer tomography. The required high measurement resolution of the propagation time of the ultrasonic impulse through the medium is obtained by a special signal processing technique. Since the propagation direction of a sonic wave in a non-stationary medium is not straight-lined, a ray linking procedure was developed that traces the rays. The combination of the precise propagation time measurement, the ray linking method and a vector algebraic reconstruction technique leads to a␣computer tomographic measurement system for the determination of the temperature distribution and the velocity flow field in a cross-section. Received on 9 June 1997     

Beitr?ge zur Braunsteinanalyse nach dem Fresenius-Will'schen Verfahren

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