SHIFTLIKE APPROXIMATIONS AND TOPOLOGICAL MIXING

Abstract

We show that every map in the group G of self-homeomorphisms of the bisequence space can be approximated by homeomorphisms which “look like” the shift map and are expansive. By removing a certain open set of maps from G, we obtain a closed subspace M which contains all mixing maps. If φ · M then any shiftlike approximation to φ is topologically strong mixing. Thus the strong mixing expansive maps are dense in M. Further the weak mixing maps form a dense Gδ sets in M.     

Method for enhancing the mixing of a compressible mixing layer

Numerical simulations have been done for a compressible mixing layer, in which the inflow speed on the low speed side was made to have periodic undulations, so as to see if this method could enhance the mixing effect of the layer. Systematic computations for a 2-D compressible mixing layer with Mach numberM _e = 0.6 have been done, and the results showed that the proposed method was indeed effective in enhancing the mixing.     

Ergodicity and mixing of non-commuting epimorphisms

We study mixing properties of epimorphisms of a compact connectedfinite-dimensional abelian group X. In particular, we show thata set F, with |F| > dim X, of epimorphisms of X is mixingif and only if every subset of F of cardinality (dim X) + 1is mixing. We also construct examples of free non-abelian groupsof automorphisms of tori which are mixing, but not mixing oforder 3, and show that, under some irreducibility assumptions,ergodic groups of automorphisms contain mixing subgroups andfree non-abelian mixing subsemigroups.     

2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension

V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a stationary process (ξ_i )_i2ℤ, and the question remains open today. In 1978, F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold mixing problem, the socalled 3-dot system, but in the context of stationary random fields indexed by ℤ². In this work, we first present an attempt to adapt Ledrappier's construction to the onedimensional case, which finally leads to a stationary process which is 2-fold but not 3-fold mixing conditionally to the σ-algebra generated by some factor process. Then, using arguments coming from the theory of joinings, we will give some strong obstacles proving that Ledrappier's counterexample can not be fully adapted to one-dimensional stationary processes.     

Weakly coupled analysis of a blade in multiphase mixing vessel

Two or more physical systems frequently interact with each other, where the independent solution of one system is impossible without a simultaneous solution of the others. An obvious coupled system is that of a dynamic fluid-structure interaction. [8] In this paper a computational analysis of the fluid-structure interaction in a mixing vessel is presented. In mixing vessels the fluid can have a significant influence on the deformation of blades during mixing, depending on speed of mixing blades and fluid viscosity. For this purpose a computational weakly coupled analysis has been performed to determine the multiphase fluid influences on the mixing vessel structure. The multiphase fluid field in the mixing vessel was first analyzed with the computational fluid dynamics (CFD) code CFX. The results in the form of pressure were then applied to the blade model, which was the analysed with the structural code MSC.visualNastran forWindows, which is based on the finite element method (FEM). (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological weak mixing and quasi-Bohr systems

A minimal dynamical system (X, T) is called quasi-Bohr if it is a non-trivial equicontinuous extension of a proximal system. We show that if (X, T) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is:AG′=G implies weak mixing.) The converse does not hold. In fact there are nontrivial quasi-Bohr systems which are weakly mixing of all orders. Our main tool in the proof is a theorem, of independent interest, which enhances the general structure theorem for minimal systems.     

Fast mixing for independent sets,colorings, and other models on trees

We study the mixing time of the Glauber dynamics for general spin systems on the regular tree, including the Ising model, the hard‐core model (independent sets), and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper (Martinelli, Sinclair, and Weitz, Tech. Report UCB//CSD‐03‐1256, Dept. of EECS, UC Berkeley, July 2003) in the context of the Ising model, for establishing mixing time O(nlog n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition. We also discuss applications of our framework to reconstruction problems on trees. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

On extensions of Hoeffding’s inequality for panel data

Hoeffding’s inequality provides a probability bound for the deviation between the average of n independent bounded random variables and its mean. This paper introduces two inequalities that extend Hoeffding’s inequality to panel data, which consists of several mutually independent sequences of dependent data with strong mixing or with a dependence structure being even more general than strong mixing. One is denoted as the Bosq’s Extension which is an extension of Bosq’s inequality (Bosq, 1993) to panel data and the other one is called the Triplex Extension, which extends the Triplex inequality (Jiang, 2009) to panel data. The Bosq’s Extension provides a tighter upper probability bound, while the Triplex Extension is more relaxed in assumption allowing unboundedness and more general dependence than strong mixing. We also apply these two inequalities to establish the convergence rate of empirical risk minimization for high dimensional panel data with variable selection.     

Automorphisms that admit an approximation by periodic transformations

Summary We are concerned with a class of automorphisms which can be approximated by periodic transformations. We show that such an automorphism T is ergodic and that a relationship exists between the spectrum of T, the measure of the sets which are invariant under T, and the ergodicity of the powers of T. Finally, we exhibit conditions under which T is weakly mixing and not strongly mixing.This research was supported by NSF Grant No. GP-7490.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

Optimization of chemical reactor networks with respect to flow configuration

In Ref. 1, Jackson treated for the first time an optimization problem in which changes in chemical reactor performance were related to small changes in the mixing pattern or flow configuration. His algebraic structure could handle arbitrary kinetic schemes, but had the severe limitation in that onlyglobal mixing in the reactor network was considered. In the present paper, it is shown how the approach can be further generalized to take into accountlocal mixing in the network, by introducing continuous-stirred tank reactors of given volume at a newly defined type of nodal point. Relatively simple algebraic equations are derived, which can be used as a basis for optimizing performance by successive adjustments in the mixing pattern. Finally, the new equations are applied to a simple example, and the results are shown to agree with those obtained by direct calculation.The author is deeply indebted to Professor F. H. Shair for his constant encouragement and for many helpful discussions. This work was supported by the U.S. Atomic Energy Commission under Project Agreement No. 1, Grant No. AT (04-3)-767.     

Limit laws for the maximum and minimum of stationary sequences

Summary The class of non-degenerate joint limiting distributions for the maximum and minimum of stationary mixing sequences is determined. These limit distributions are of the form, H(x, ) –H(x, –y), where H(x,y) is a bivariate extreme value distribution. Unlike the classical result for i.i.d. sequences, the maximum and minimum of stationary mixing sequences may be asymptotically dependent. We also give a sufficient condition for the asymptotic independence of the maximum and minimum. Finally, an example of a stationary sequence satisfying the mixing condition D in Leadbetter but which is not weakly mixing is constructed.This research was supported in part by the National Science Foundation grant MCS 80-05483     

Interactions between compressibility and heat release in turbulent mixing layers

Reduced turbulence activity and growth rate of a compressible mixing layer when the convective Mach number increases, are well-known phenomena. Heat release due to combustion has similar effects and important implications for applications such as scramjet engines the efficiency of which depends on a proper mixing of oxidizer and fuel. To understand the underlying mechanisms, highly resolved DNS of inert and reacting, temporally evolving mixing layers at different convective Mach numbers and Reynolds numbers ensuring a self-similar state are performed and analyzed with a particular focus on the pressure-strain correlations in the Reynolds-stress transport equations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

The family of pairwise independently determined(PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages 1/N N-1∑n=0 f_1(T~nx)···f_d(T~(dnx)),N→∞,almost surely converge.     

Rank one lightly mixing

A rank one transformationT was constructed by Chacón that is weakly mixing but not mixing. We will show thatT is lightly mixing, not partially mixing, and not lightly 2-mixing. Partially supported by an NSF Postdoctoral Research Fellowship.     

On the mixing coefficients of piecewise monotonic maps

We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse ϕ-mixing. If its invariant density is uniformly bounded away from 0, it is ϕ-mixing iff all images of all orders are big, in which case it is ψ-mixing. Among β-transformations, non-ϕ-mixing is generic. In this sense, the asymmetry of ϕ-mixing is natural.     

Rapid decay of correlations for nonuniformly hyperbolic flows

We show that superpolynomial decay of correlations (rapid mixing) is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which Young proved exponential decay of correlations. The proof combines techniques of Dolgopyat and operator renewal theory.

It follows from our results that planar periodic Lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing.

     

Generic mixing transformations are rank 1

In 2007, S. V. Tikhonov introduced a complete metric on the space of mixing transformations. This metric generates a topology called the leash topology. Tikhonov posed the following problem: what conditions should be satisfied by a mixing transformation T for its conjugacy class to be dense in the space of mixing transformations equipped with the leash topology. We show the conjugacy class to be dense for every mixing transformation T. As a corollary, we find that a generic mixing transformation is rank 1.     

强混合序列部分和的强逼近

具有 φ（ｘ）一致可积的混合序列的强逼近问题（其中 φ（ｘ）／ｘ２＋ δ↑∞,δ＞ ０）是本文所要论述的主题．文章给出的结论弥补了［１］中强混合序列的强逼近与独立序列之间的空隙,同时推广了［１］中的结论     

Invariance principles under a two-part mixing assumption

Weak and strong invariance principles are established for strictly stationary sequences satisfying a mixing assumption which has two “parts”, one based on the strong mixing condition with a polynomial mixing rate and the other based on the ϱ-mixing condition.