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Rapid Mixing (rapid + mixing)

Distribution by Scientific Domains
Chemistry37%

Selected Abstracts

Rapid mixing of Gibbs sampling on graphs that are sparse on average

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2009
Elchanan Mossel
Abstract Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd,s-Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 - o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature ,, the mixing time of Gibbs sampling is at least n1+,(1/log log n). Recall that the Ising model with inverse temperature , defined on a graph G = (V,E) is the distribution over {±}Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of , or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < , and the Ising model defined on G (n, d/n), there exists a ,d > 0, such that for all , < ,d with probability going to 1 as n ,,, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd,s-Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub-graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and , < ,d, where d tanh(,d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source]

Redox properties of the couple compound I/native enzyme of myeloperoxidase and eosinophil peroxidase

FEBS JOURNAL, Issue 19 2001
Jürgen Arnhold
The standard reduction potential of the redox couple compound I/native enzyme has been determined for human myeloperoxidase (MPO) and eosinophil peroxidase (EPO) at pH 7.0 and 25 °C. This was achieved by rapid mixing of peroxidases with either hydrogen peroxide or hypochlorous acid and measuring spectrophotometrically concentrations of the reacting species and products at equilibrium. By using hydrogen peroxide, the standard reduction potential at pH 7.0 and 25 °C was 1.16 ± 0.01 V for MPO and 1.10 ± 0.01 V for EPO, independently of the concentration of hydrogen peroxide and peroxidases. In the case of hypochlorous acid, standard reduction potentials were dependent on the hypochlorous acid concentration used. They ranged from 1.16 V at low hypochlorous acid to 1.09 V at higher hypochlorous acid for MPO and from 1.10 V to 1.03 V for EPO. Thus, consistent results for the standard reduction potentials of redox couple compound I/native enzyme of both peroxidases were obtained with all hydrogen peroxide and at low hypochlorous acid concentrations: possible reasons for the deviation at higher concentrations of hypochlorous acid are discussed. They include instability of hypochlorous acid, reactions of hypochlorous acid with different amino-acid side chains in peroxidases as well as the appearance of a compound I,chloride complex. [source]

Investigation of impinging-jet crystallization with a calcium oxalate model system

AICHE JOURNAL, Issue 9 2003
Jean M. Hacherl
An impinging-jet crystallizer was investigated in this work to assess its operational sensitivity and reproducibility for the production of small, monodisperse crystals using calcium oxalate, a model system capable of forming multiple hydrates. The impinging-jet mixer provides rapid mixing of the reactant solutions through the impingement of two narrow reactant streams at high velocity. Impinging jet linear velocity and postjetting conditions were studied, with the jet operated in nonsubmerged mode. Hydrate form and crystal-size distribution (CSD) were determined using optical microscopy and image analysis techniques. The impinging jet consistently produced small, monodisperse crystals. However, at a high level of supersaturation, slight variations in the CSD were observed for apparently identical conditions, suggesting a degree of sensitivity in the system that could lead to difficulty in its application. An apparent trend between impinging-jet linear velocity and crystal size and number was observed, with more small crystals produced at higher linear velocity. [source]

Rapid mixing of Gibbs sampling on graphs that are sparse on average

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2009
Elchanan Mossel
Abstract Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd,s-Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 - o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature ,, the mixing time of Gibbs sampling is at least n1+,(1/log log n). Recall that the Ising model with inverse temperature , defined on a graph G = (V,E) is the distribution over {±}Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of , or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < , and the Ising model defined on G (n, d/n), there exists a ,d > 0, such that for all , < ,d with probability going to 1 as n ,,, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd,s-Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub-graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and , < ,d, where d tanh(,d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source]

Mixing in time and space for lattice spin systems: A combinatorial view

RANDOM STRUCTURES AND ALGORITHMS, Issue 4 2004
Martin Dyer
The paper considers spin systems on the d -dimensional integer lattice ,d with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we show that if the mixing time of the Glauber dynamics is O(n log n) then spin correlations decay exponentially fast with distance. We also prove the converse implication for monotone systems, and for general systems we prove that exponential decay of correlations implies O(n log n) mixing time of a dynamics that updates sufficiently large blocks (rather than single sites). While the above equivalence was already known to hold in various forms, we give proofs that are purely combinatorial and avoid the functional analysis machinery employed in previous proofs. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004 [source]

Random sampling of 3-colorings in ,2,

RANDOM STRUCTURES AND ALGORITHMS, Issue 3 2004
Leslie Ann Goldberg
Abstract We consider the problem of uniformly sampling proper 3-colorings of an m × n rectangular region of ,2. We show that the single-site "Glauber-dynamics" Markov chain is rapidly mixing. Our result complements an earlier result of Luby, Randall, and Sinclair, which demonstrates rapid mixing when there is a fixed boundary (whose color cannot be changed). © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004 [source]

On the swapping algorithm

RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2003
Neal Madras
Abstract The Metropolis-coupled Markov chain method (or "Swapping Algorithm") is an empirically successful hybrid Monte Carlo algorithm. It alternates between standard transitions on parallel versions of the system at different parameter values, and swapping two versions. We prove rapid mixing for two bimodal examples, including the mean-field Ising model. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 66,97, 2002 [source]

Enabling Continuous-Flow Chemistry in Microstructured Devices for Pharmaceutical and Fine-Chemical Production

CHEMISTRY - A EUROPEAN JOURNAL, Issue 25 2008
Norbert Kockmann Dr.
Abstract Microstructured devices offer unique transport capabilities for rapid mixing, enhanced heat and mass transfer and can handle small amounts of dangerous or unstable materials. The integration of reaction kinetics into fluid dynamics and transport phenomena is essential for successful application from process design in laboratory to chemical production. Strategies to implement production campaigns up to tons of pharmaceutical chemicals are discussed, based on Lonza projects. Mikrostrukturierte Bauteile bieten überragende Transporteigenschaften für schnelle Vermischung, erhöhte Wärme- und Stoffübertragung und können geringe Mengen gefährlicher oder instabiler Stoffe handhaben. Die Integration der Reaktionskinetik in die Transportvorgänge ist wichtig für erfolgreiche Anwendungen von der Prozesssynthese im Labor bis zur Produktion. Scale-up-Strategien für die Produktion von pharmazeutischen Chemikalien werden anhand von Lonza-Projekten gezeigt. [source]