A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.     

Oscillations and concentrations in weak solutions of the incompressible fluid equations

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.Partially supported by N.S.F. GrantPartially supported by N.S.F. Grant 84-0223 and 86-11110     

Strongly countable dimensional compacta form the Hurewicz set

The hyperspaces of strongly countable dimensional compacta of positive dimension and of strongly countable dimensional continua of dimension greater than 1 in the Hilbert cube are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. These facts hold true, in particular, for covering dimension dim and cohomological dimension dimG, where G is any Abelian group.     

N‐Acetyl‐l‐tyrosine methyl ester monohydrate at 293 and 123 K

The crystal structure of a protected l ‐tyrosine, namely N‐acetyl‐l ‐tyrosine methyl ester monohydrate, C₁₂H₁₅NO₄·H₂O, was determined at both 293 (2) and 123 (2) K. The structure exhibits a network of O—H...O and N—H...O hydrogen bonds, in which the water molecule plays a crucial role as an acceptor of one and a donor of two hydrogen bonds. Molecules of water and of the protected l ‐tyrosine form hydrogen‐bonded layers perpendicular to [001]. C—H...π interactions are observed in the hydrophobic regions of the structure. The structure is similar to that of N‐acetyl‐l ‐tyrosine ethyl ester monohydrate [Soriano‐García (1993). Acta Cryst. C 49 , 96–97].     

Cationic Heteroconjugation Constants in the OHN+ and NHO+ Systems in Propylene Carbonate: Dependence on Difference in Basicity

Cationic heteroconjugation equilibria of more than ninety systems consisting of substituted pyridines, their N-oxides, and trimethylamine N-oxide, i. e., in systems with mixed hydrogen bridges of type OHN⁺ (NHO⁺) were studied in propylene carbonate. Both experimental systems without proton transfer, BH⁺/B₁, and those with proton transfer, B₁H⁺/B, were explored. The stability of the mixed hydrogen bridges, OHN⁺ (NHO⁺), is compared with that of the OHC⁺-type bridges. The influence of the difference in basicity of the conjugate base of the proton donor and the proton acceptor on the presence of the proton transfer equilibria, and, consequently, the possibility of determination of the cationic heteroconjugation constant values is discussed.     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

Mechanistic studies on the binding of nitric oxide to a synthetic heme-thiolate complex relevant to cytochrome p450

The synthetic heme-thiolate complex (SR) in methanol binds nitric oxide (k(on) = (2.7 +/- 0.2) x10(6) M(-)(1) s(-)(1) at 25 degrees C) to form SR(NO). The binding of NO to the SR complex in a noncoordinating solvent, such as toluene, was found to be almost 3 orders of magnitude faster than that in methanol. The activation parameters DeltaH(), DeltaS(), and DeltaV() for the formation of SR(NO) in methanol are consistent with the operation of a limiting dissociative mechanism, dominated by dissociation of methanol in SR(MeOH). In the presence of an excess of NO, the formation of SR(NO) is followed by subsequent slower reactions. The substantially negative activation entropy and activation volume values found for the second observed reaction step support an associative mechanism which involves attack of a second NO molecule on the thiolate ligand in the initially formed SR(NO) complex. The following slower reactions are strongly accelerated by a large excess of NO or by the presence of NO(2)(-) in the SR/NO reaction mixture. They can be accounted for in terms of dynamic equilibria between higher nitrogen oxides (NO(x)()) and reactive SR species, which lead to the formation of a nitrosyl-nitrite complex of SR(Fe(II)) as the final product. This finding is clearly supported by laser flash photolysis studies on the SR/NO reaction mixture, which do not reveal simple NO photolabilization from SR(Fe(III))(NO), but rather involve the generation of at least three photoinduced intermediates decaying with different rate constants to the starting material. The species formed along the proposed reaction pathways were characterized by FTIR and EPR spectroscopy. The results are discussed in terms of their relevance for the biological function of cytochrome P450 enzymes and in context of results for the reaction of NO with imidazole- and thiolate-ligated iron(III) hemoproteins.     

Mechanistic information on the reversible binding of NO to selected iron(II) chelates from activation parameters

Mechanistic insight on the reversible binding of NO to Fe(II) chelate complexes as potential catalysts for the removal of NO from effluent gas streams has been obtained from the temperature and pressure parameters for the "on" and "off" reactions determined using a combination of flash photolysis and stopped-flow techniques. These parameters are correlated with those for water exchange reactions on the corresponding Fe(II) and Fe(III) chelate complexes, from which mechanistic conclusions are drawn. Small and positive Delta V(++) values are found for NO binding to and release from all the selected complexes, consistent with a dissociative interchange (I(d)) mechanism. The only exception in the series of studied complexes is the binding of NO to [Fe(II)(nta)(H(2)O)(2)](-). The negative volume of activation observed for this reaction supports the operation of an I(a) ligand substitution mechanism. The apparent mechanistic differences can be accounted for in terms of the electronic and structural features of the studied complexes. The results indicate that the aminocarboxylate chelates affect the rate and overall equilibrium constants, as well as the nature of the substitution mechanism by which NO coordinates to the selected complexes. There is, however, no simple correlation between the rate and activation parameters and the selected donor groups or overall charge on the iron(II) complexes.