Mixing (physics)

Abstract: Density functional approximations for the exchange‐correlation energy EDFAxc of an electronic system are often improved by admixing some exact exchange Ex: Exc≊EDFAxc+(1/n)(Ex−EDFAx). This procedure is justified when the error in EDFAxc arises from the λ=0 or exchange end of the coupling‐constant integral ∫10 dλ EDFAxc,λ. We argue that the optimum integer n is approximately the lowest order of Gorling–Levy perturbation theory which provides a realistic description of the coupling‐constant dependence Exc,λ in the range 0≤λ≤1, whence n≊4 for atomization energies of typical molecules. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second‐order perturbation theory with the generalized gradient approximation.

Abstract: SUMMARY Thispaperconsidersmixed,orrandomcoeAcients,multinomiallogit (MMNL)modelsfordiscreteresponse, andestablishesthefollowingresults.Undermildregularityconditions,anydiscretechoicemodelderivedfrom random utility maximization has choice probabilities that can be approximated as closely as one pleases by a MMNLmodel.PracticalestimationofaparametricmixingfamilycanbecarriedoutbyMaximumSimulated LikelihoodEstimationorMethodofSimulatedMoments,andeasilycomputedinstrumentsareprovidedthat make the latter procedure fairly eAcient. The adequacy of a mixing specification can be tested simply as an omittedvariabletestwithappropriatelydefinedartificialvariables.Anapplicationtoaproblemofdemandfor alternativevehiclesshowsthatMMNL provides aflexible and computationally practical approach todiscrete response analysis. Copyright # 2000 John Wiley & Sons, Ltd.

Abstract: This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.

Abstract: Concepts and Definitions. Fickian Diffusion. Turbulent Diffusion. Shear Flow Dispersion. Mixing in Rivers. Mixing in Reservoirs. Mixing in Estuaries. River and Estuary Models. Turbulent Jets and Plumes. Design of Ocean Wastewater Discharge System. An Estimate for the Density of Seawater. Fluid Properties. References

Abstract: Plane turbulent mixing between two streams of different gases (especially nitrogen and helium) was studied in a novel apparatus Spark shadow pictures showed that, for all ratios of densities in the two streams, the mixing layer is dominated by large coherent structures High-speed movies showed that these convect at nearly constant speed, and increase their size and spacing discontinuously by amalgamation with neighbouring ones The pictures and measurements of density fluctuations suggest that turbulent mixing and entrainment is a process of entanglement on the scale of the large structures; some statistical properties of the latter are used to obtain an estimate of entrainment rates Large changes of the density ratio across the mixing layer were found to have a relatively small effect on the spreading angle; it is concluded that the strong effects, which are observed when one stream is supersonic, are due to compressibility effects, not density effects, as has been generally supposed