random coin tossing a sequence of heads and tails is produced by repeatedly selecting a coin from two possible coins and tossing it the second coin is tossed at renewal times in a renewal process and the first coin is tossed at all other times the first coin is fair prob heads 1 2 and the second coin is known either to be fair or to have known bias 0 1 prob heads letting u k prob there is a renewal at time k we show that if k 0 u k 2 we can determine using only the sequence of heads and tails produced if the second coin had bias or 0 if we show that this is not possible

k homology asymptotic representations and unsuspended e theory connes and higson defined a bivariant homology theory e a b for separable c algebras the elements of e a b are taken to be homotopy classes of asymptotic momorphisms from sa k to sb k in symbols e a b sa sb k if a is k nuclear then e theory agrees with kasparov s bivariant k theory we show that in many cases one need not take suspensions to calculate the e theory group e a b for many a we show e a b a b k for all b among the a for which this is true are c0 x pt for x with the homotopy type of a finite connected cw complex this gives a concrete realization of k homology related to the brown douglas fillmore description for example k0 x arises as asymptotic representations k0 x c0 x k other a for which our isomorphic holds include the nonunital dimension drop intervals in this case there is no distinction between homomorphisms and asymptotic morphisms so we have succeeded in classifying all homomorphisms from a dimension drop interval to a stable c algebra this was subsequently used by elliott j reine angew math 443 1993 179 219 in the classification of certain c algebras the dimension drop interval may also be used to describe k x z n in terms of paths of asymptotic representations of c0 x

t t 1 transformation is not loosely bernoulli the t t problem open since 1971 is to demonstrate that the t t transformation a naturally arising k transformation is not bernoulli it is shown here not even to be loosely bernoulli

ergodic properties of color records abstract in a series of papers culminating in the dissertation of the second author under the direction of p w kasteleyn several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black white colored lattice and this under the sole assumption that the coloring is stationary ergodic and independent of the walking in this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk k 0 of successive times needed for the walker to go from the kth to the k 1 st black point as an example of the type of results obtained we show that if a certain induced color record process is strongly mixing in the ergodic theoretic sense then the average nk tends to the inverse of the density of black points as k tends to infinity and similarly that the distribution of nk tends to a limit that can be explicitly calculated in terms of the distribution of n0 for a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing our main goal however is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records

a strong law of large numbers for iterated functions of independent random variables we study sequences of random variables obtained by iterative procedures which can be thought of as nonlinear generalizations of the arithmetic mean we prove a strong law of large numbers for a class of such iterations this gives rise to the concept of generalized expected value of a random variable for which we prove an analog of the classical jensen inequality we give several applications to models arising in mathematical physics and other areas

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

strong invariance for local times let y 1 y 2 be a sequence of i i d random variables with distribution p y 1 k p k k 1 2 e y 1 0 e y 1 2 2 put t n y 1 y n and n x n k 0 k n t k x extending the result of revesz 1981 it is shown that for appropriate skorohod construction we have 1 mathop text sup limits x in mathbb z text l x n sigma 2 text sigma text 2 n x n o n 1 mathord left vphantom 1 4 varepsilon right kern nulldelimiterspace 4 varepsilon text a text s text provided all moments e y 1 m m 0 exists where l is the local time of a wiener process certain rate of convergence is given also under weaker conditions and for l x n 2 2 n x n too when x is fixed

nonlinear and nonparametric regression and instrumental variables we consider regression when the predictor is measured with error and an instrumental variable iv is available the regression function can be modeled linearly nonlinearly or nonparametrically our major new result shows that the regression function and all parameters in the measurement error model are identified under relatively weak conditions much weaker than previously known to imply identifiability in addition we exploit a characterization of the iv estimator as a classical correction for attenuation method based on a particular estimate of the variance of the measurement error this estimate of the measurement error variance allows us to construct functional nonparametric regression estimators making no assumptions about the distribution of the unobserved predictor and structural estimators that use parametric assumptions about this distribution the functional estimators uses simulation extrapolation or deconvolution kernels and the structural method uses bayesian markov chain monte carlo

the connes higson construction is an isomorphism let a be a separable c algebra and b a stable c algebra containing a strictly positive element we show that the group ext 1 2 sa b of unitary equivalence classes of extensions of sa by b modulo the extensions which are asymptotically split coincides with the group of homotopy classes of such extensions this is done by proving that the connes higson construction gives rise to an isomorphism between ext 1 2 sa b and the e theory group e a b of homotopy classes of asymptotic homomorphisms from s2a to b

concurrent games with tail objectives we study infinite stochastic games played by two players over a finite state space with objectives specified by sets of infinite traces the games are concurrent players make moves simultaneously and independently stochastic the next state is determined by a probability distribution that depends on the current state and chosen moves of the players and infinite proceed for an infinite number of rounds the analysis of concurrent stochastic games can be classified into quantitative analysis analyzing the optimum value of the game and e optimal strategies that ensure values within e of the optimum value and qualitative analysis analyzing the set of states with optimum value 1 and e optimal strategies for the states with optimum value 1 we consider concurrent games with tail objectives i e objectives that are independent of the finite prefix of traces and show that the class of tail objectives is strictly richer than that of the w regular objectives we develop new proof techniques to extend several properties of concurrent games with w regular objectives to concurrent games with tail objectives we prove the positive limit one property for tail objectives the positive limit one property states that for all concurrent games if the optimum value for a player is positive for a tail objective f at some state then there is a state where the optimum value is 1 for the player for the objective f we also show that the optimum values of zero sum strictly conflicting objectives games with tail objectives can be related to equilibrium values of nonzero sum not strictly conflicting objectives games with simpler reachability objectives a consequence of our analysis presents a polynomial time reduction of the quantitative analysis of tail objectives to the qualitative analysis for the subclass of one player stochastic games markov decision processes

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

mean field criticality for percolation on planar non amenable graphs the critical exponents and are proved to exist and to take their mean field values for independent percolation on the following classes of infinite locally finite connected transitive graphs 1 non amenable planar with one end 2 unimodular with infinitely many ends

variations on a theme by mark kac mark kac s theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations most of which are formulated in terms of 0 1 processes whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0 this condition is dropped in part of the variations two others refer to the variance of the recurrence time and two to the poincare cycle of a dynamical system most variations consist in inequalities and formal identities for the mean first arrival time and subsequent recurrence times for the given state

time series analysis and monte carlo methods for eigenvalue separation in neutron multiplication problems a time series approach has been applied to the nuclear fission source distribution generated by monte carlo mc particle transport in order to calculate the non fundamental mode eigenvalues of the system the novel aspect is the combination of the general technical principle of projection pursuit for multivariate data with the neutron multiplication eigenvalue problem in the nuclear engineering discipline proof is thoroughly provided that the stationary mc process is linear to first order approximation and that it transforms into one dimensional autoregressive processes of order one ar 1 via the automated choice of projection vectors the autocorrelation coefficient of the resulting ar 1 process corresponds to the ratio of the desired mode eigenvalue to the fundamental mode eigenvalue all modern mc codes for nuclear criticality calculate the fundamental mode eigenvalue so the desired mode eigenvalue can be easily determined this time series approach was tested for a variety of problems including multi dimensional ones numerical results show that the time series approach has strong potential for three dimensional whole reactor core the eigenvalue ratio can be updated in an on the fly manner without storing the nuclear fission source distributions at all previous iteration cycles for the mean subtraction lastly the effects of degenerate eigenvalues are investigated and solutions are provided

mixture models with a prior on the number of components abstracta natural bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with symmetric dirichlet weights and put a prior on the number of components that is to use a mixture of finite mixtures mfm the most commonly used method of inference for mfms is reversible jump markov chain monte carlo but it can be nontrivial to design good reversible jump moves especially in high dimensional spaces meanwhile there are samplers for dirichlet process mixture dpm models that are relatively simple and are easily adapted to new applications it turns out that in fact many of the essential properties of dpms are also exhibited by mfms an exchangeable partition distribution restaurant process random measure representation and stick breaking representation and crucially the mfm analogues are simple enough that they can be used much like the corresponding dpm properties consequently many of the powerful methods developed for inference in dpms

algebraic ℤ actions of entropy rank one we investigate algebraic z d actions of entropy rank one namely those for which each element has finite entropy such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery this leads to numerous alternative characterizations of entropy rank one both geometric and algebraic we then compute the measure entropy of a class of skew products where the fiber maps are elements from an algebraic z d action of entropy rank one this leads via the relative variational principle to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures we use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms

winning quick and dirty the greedy random walk as a strategy to complete games quickly we investigate one dimensional random walks where the step length increases deterministically upon each return to the origin when the step length after the kth return equals k the displacement of the walk x grows linearly in time asymptotically the probability distribution of displacements is a purely exponentially decaying function of x t the probability e t l for the walk to escape a bounded domain of size l at time t decays algebraically in the long time limit e t l l t2 consequently the mean escape time t lln l while tn l2n 1 for n 1 corresponding results are derived when the step length after the kth return scales as k for 0

comparison of various orthogonal polynomials in hp version time finite element method four types of orthogonal polynomials legendre chebyshev hermite and integrated legendre are evaluated as basis functions in the time finite element method to solve initial value problems governed by second order differential equations condition numbers of the augmented stiffness matrix for the selected problems are estimated by increasing the number of polynomial terms in the expansion results for the cpu time using an ibm 3090 300e vf computer and the estimated condition numbers using imsl subroutine dlfcrg for each of four basis functions are presented the augmented stiffness matrix for the case of integrated legendre polynomials is found to have the best condition number behavior and that for the hermite polynomial the worst the integrated legendre polynomials require the most cpu time whereas the chebyshev polynomials require the least

strong control conservative point estimation and simultaneous conservative consistency of false discovery rates a unified approach summary the false discovery rate fdr is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses benjamini and hochberg introduced this quantity and proved that a particular step up p value method controls the fdr storey introduced a point estimate of the fdr for fixed significance regions the former approach conservatively controls the fdr at a fixed predetermined level and the latter provides a conservatively biased estimate of the fdr for a fixed predetermined significance region in this work we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent in particular the fdr point estimates can be used to define valid fdr controlling procedures in the asymptotic setting we also show that the point estimates can be used to estimate the fdr conservatively over all significance regions simultaneously which is equivalent to controlling the fdr at all levels simultaneously the main tool that we use is to translate existing fdr methods into procedures involving empirical processes this simplifies finite sample proofs provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence

a mathematical framework for critical transitions bifurcations fast slow systems and stochastic dynamics abstract bifurcations can cause dynamical systems with slowly varying parameters to transition to far away attractors the terms critical transition or tipping point have been used to describe this situation critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance the main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions we shall focus on early warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions we suggest that the mathematical theory of fast slow systems provides a natural definition of critical transitions since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast slow systems the interplay between sample path techniques partial differential equations and random dynamical systems is highlighted each viewpoint provides potential early warning signs for critical transitions since increasing variance has been suggested as an early warning sign we examine it in the context of normal forms analytically numerically and geometrically we also consider autocorrelation numerically hence we demonstrate the applicability of early warning signs for generic models we end with suggestions for future directions of the theory

multi component random model of diffusion in chaotic systems we extend our recent study robnik et al 1997 j phys a math gen 30 l803 of diffusion in strongly chaotic systems the random model to systems composed of several weakly coupled ergodic components by this we mean that the system as a whole is ergodic but the typical time for the transition from one to another component is very long much longer than the ergodic time inside each individual component thus for short times the system behaves like a single component ergodic system and the random model applies neglecting the coupling to other components at times much longer than the transition time the system behaves like an ergodic system without internal structure without decomposition into several components and the random model applies again with different parameters at intermediate times there is the crossover regime which we describe in detail analytically for a two component system and test it numerically in a double billiard system butterfly billiard

ergodic and bernoulli properties of analytic maps of complex projective space we examine the measurable ergodic theory of analytic maps f of complex projective space we focus on two different classes of maps ueda maps of p n and rational maps of the sphere with parabolic orbifold and julia set equal to the entire sphere we construct measures which are invariant ergodic weak or strong mixing exact or automorphically bernoulli with respect to these maps we discuss topological pressure and measures of maximal entropy h f h top f log deg f we find analytic maps of p 1 and p 2 which are one sided bernoulli of maximal entropy including examples where the maximal entropy measure lies in the smooth measure class further we prove that for any integer d 1 there exists a rational map of the sphere which is one sided bernoulli of entropy log d with respect to a smooth measure

on the bernoulli property for certain partially hyperbolic diffeomorphisms abstract we address the classical problem of equivalence between kolmogorov and bernoulli property of smooth dynamical systems in a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to anosov derived from anosov on 3 torus we prove that kolmogorov and bernoulli properties are equivalent in our approach we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms as an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from anosov diffeomorphism of 3 torus and its linearization our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi isometric

spectral triples and wavelets for higher rank graphs abstract in this paper we present a new way to associate a finitely summable spectral triple to a higher rank graph via the infinite path space of moreover we prove that this spectral triple has a close connection to the wavelet decomposition of which was introduced by farsi gillaspy kang and packer in 2015 we first introduce the concept of stationary k bratteli diagrams in order to associate a family of ultrametric cantor sets and their associated pearson bellissard spectral triples to a finite strongly connected higher rank graph we then study the zeta function abscissa of convergence and dixmier trace associated to the pearson bellissard spectral triples of these cantor sets and show these spectral triples are regular in the sense of pearson and bellissard we obtain an integral formula for the dixmier trace given by integration against a measure and show that is a rescaled version of the measure m on which was introduced by an huef laca raeburn and sims finally we investigate the eigenspaces of a family of laplace beltrami operators associated to the dirichlet forms of the spectral triples we show that these eigenspaces refine the wavelet decomposition of l 2 m which was constructed by farsi et al

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

pointwise redundancy in lossy data compression and universal lossy data compression we characterize the achievable pointwise redundancy rates for lossy data compression at a fixed distortion level pointwise redundancy refers to the difference between the description length achieved by an nth order block code and the optimal nr d bits for memoryless sources we show that the best achievable redundancy rate is of order o spl radic n in probability this follows from a second order refinement to the classical source coding theorem in the form of a one sided central limit theorem moreover we show that along almost any source realization the description lengths of any sequence of block codes operating at distortion level d exceed nr d by at least as much as c spl radic nloglogn infinitely often corresponding direct coding theorems are also given showing that these rates are essentially achievable the above rates are in sharp contrast with the expected redundancy rates of order o log n reported by various authors our approach is based on showing that the compression performance of an arbitrary sequence of codes is essentially bounded below by the performance of shannon s random code we obtain partial generalizations of the above results for arbitrary sources with memory and we prove lossy analogs of barron s lemma barron 1985

the dual cheeger constant and spectra of infinite graphs abstract in this article we study the top of the spectrum of the normalized laplace operator on infinite graphs we introduce the dual cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the cheeger constant controls the bottom of the spectrum moreover we show that the dual cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized laplace operator shrinks to one point

large deviations estimates for self intersection local times for simple random walk in mathbb z 3 we obtain large deviations estimates for the self intersection local times for a simple random walk in dimension 3 also we show that the main contribution to making the self intersection large in a time period of length n comes from sites visited less than some power of log n this is opposite to the situation in dimensions larger or equal to 5 finally we present an application of our estimates to moderate deviations for random walk in random sceneries

optimal geostatistical model selection in many fields of science predicting variables of interest over a study region based on noisy data observed at some locations is an important problem two popular methods for the problem are kriging and smoothing splines the former assumes that the underlying process is stochastic whereas the latter assumes it is purely deterministic kriging performs better than smoothing splines in some situations but is outperformed by smoothing splines in others however little is known regarding selecting between kriging and smoothing splines in addition how to perform variable selection in a geostatistical model has not been well studied in this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on approximately unbiased estimation of mean squared prediction errors using a data perturbation technique the proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors and is shown to be optimal in terms of two mean sq

slew maneuver of flexible space structures using time finite element analysis second orderinitial value problems recently kim and cho 1 developed a penalized weighted residual method that conforms well to the parallel computing scheme park and kapania 2 studied several orthogonal polynomials to e nd the good basis function in the time e nite element method suk and kim 3 proposed a method to analyze the dynamic systems with specie ed initial and e nal conditions in this note the slew maneuver of a e exible space structure is analyzed using the time e nite element method fem the slew dynamics of a e exible space structure is governed by coupled equations in slewand e exible modes using time discretization the slew modeequation canbe converted to an algebraic matrix equation that incorporates the effect of structural e exibility and the control input torque also the structural mode is reduced to a fourth order partial differential equation that is a function of the solution of the slew mode the novel feature of this study is that the original coupled system is split into two coupled sets of equations in which one set corresponding to the slew mode can be solved explicitly because the slew and structural modes of the e exible structure satisfy the characteristics of a self adjoint system it is also possible to reduce the orderof themodel by solving a generalized eigenvalue problem to demonstrate and verify the proposed formulation slew dynamics by open loop control input is simulated and is compared with conventional fem techniques

asymptotic ergodicity of the eigenvalues of random operators in the localized phase we prove that for a general class of random operators the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of minami spectra of random operators and related topics 2011 minami conjectured this to be the case for discrete anderson model in the localized regime we also provide a local analogue of this result from the asymptotics ergodicity one can recover the statistics of the level spacings as well as a number of other spectral statistics our proofs rely on the analysis developed in germinet and klopp spectral statistics for random schrodinger operators in the localized regime 2010

tail triviality for sums of stationary random variables we study tail sigma fields and loss of memory associated with sums of stationary integer valued random variables an application concerns convergence in distribution of interarrival times in zero one sequences

rational maps are d adic bernoulli freire lopes and mafine proved that for any rational map f there exists a natural invariant measure f 5 maiine showed there exists an n 0 such that fn af is measurably conjugate to the one sided dn shift with bernoulli measure d 15 in this paper we show that f uf is conjugate to the one sided bernoulli d shift this verifies a conjecture of freire lopes and maine 5 and lyubich 11

mixing properties for random walk in random scenery consider the lattice z d d geq 1 together with a stochastic black white coloring of its points and on it a random walk that is independent of the coloring a local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position under weak assumptions on the probability distributions governing walk and coloring we prove asymptotic independence of local sceneries perceived at times 0 and n in the limit as n rightarrow infty and at times 0 and t k in the limit as k rightarrow infty where t k is the random k th hitting time of a black point an immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits i e of t k 1 t k as k rightarrow infty the limit distribution is expressed in terms of the distribution of the first hitting time t 1 the proof uses coupling arguments and ergodic theory

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

semi invertible extensions of c algebras abstract we prolong the list of c algebras which have the property that all extensions by a stable c algebra are semi invertible in particular it is shown to include group c algebras both reduced and full of certain amalgamated free products of amenable groups as well as all free products of nuclear c algebras with amalgamation over a common nuclear c subalgebra

reconstructing a random scenery observed with random errors along a random walk path we show that an i i d uniformly colored scenery on observed along a random walk path with bounded jumps can still be reconstructed if there are some errors in the observations we assume the random walk is recurrent and can reach every point with positive probability at time k the random walker observes the color at her present location with probability 1 and an error y k with probability the errors y k k 0 are assumed to be stationary and ergodic and independent of scenery and random walk if the number of colors is strictly larger than the number of possible jumps for the random walk and is sufficiently small then almost all sceneries can be almost surely reconstructed up to translations and reflections

entropy and dyadic equivalence of random walks on a random scenery abstract for any 1 1 measure preserving map t of a probability space consider the t t 1 endomorphism and the corresponding decreasing sequence of algebras we demonstrate that if the decreasing sequence of algebras generated by t t 1 and s s 1 are isomorphic then t and s must have equal entropies as a consequence if the t t 1 endomorphism is isomorphic to the s s 1 endomorphism then the entropy of t is equal to the entropy of s central to this is a relationship between feldman s f metric 1976 israel j math 24 16 38 and vershik s v metric 1970 dokl akad nauk sssr193 748 751

on the number of infinite geodesics and ground states in disordered systems we study first passage percolation models and their higher dimensional analogs models of surfaces with random weights we prove that under very general conditions the number of lines or in the second case hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to as corollaries we show that in any dimensiond 2 the number of ground states of an ising ferromagnet with random coupling constants equals with probability one 2 or proofs employ simple large deviation estimates and ergodic arguments

stable limit laws for randomly biased walks on supercritical trees we consider a random walk on a supercritical galton watson tree with leaves where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree the biases are chosen independently on distinct edges each one according to a given law that satisfies a logarithmic nonlattice condition we determine the condition under which the walk is sub ballistic and in the sub ballistic regime we find a formula for the exponent 0 1 such that the distance x n moved by the walk in time n is of the order of n we prove a stable limiting law for walker distance at late time proving that the rescaled walk n x n converges in distribution to an explicitly identified function of the stable law of index this paper is a counterpart to ben arous et al ann probab 40 2012 280 338 in which it is proved that in the model where the biases on edges are taken to be a given constant there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement it is randomization of edge biases that is responsible for the emergence of the stable limit in the present article while also introducing further correlations into the model in comparison with the constant bias case the derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone the paper may be considered as a sequel to ben arous and hammond comm pure appl math 65 2012 1481 1527 since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical galton watson tree

retrieving random media benjamini asked whether the scenery reconstruction methods of matzinger see e g 21 22 20 can be done in polynomial time in this article we give the following answer for a 2 color scenery and simple random walk with holding we prove that a piece of the scenery of length of the order 3 n around the origin can be reconstructed up to a reflection and a small translation with high probability from the first 2 310 n observations with a constant 0 independent of n thus the number of observations needed is polynomial in the length of the piece of scenery which we reconstruct the probability that the reconstruction fails tends to 0 as n

growth of uniform infinite causal triangulations we introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added a relation to a random walk on the integer half line is shown this relation is used to estimate the geodesic distance of a given triangle to the rooted boundary in terms of the time of the growth process and to determine from this the fractal dimension furthermore convergence of the boundary process to a diffusion process is shown leading to an interesting duality relation between the growth process and a corresponding branching process

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

a phase transition in random coin tossing suppose that a coin with bias theta is tossed at renewal times of a renewal process and a fair coin is tossed at all other times let mu theta be the distribution ofthe observed sequence of coin tosses and let u n denote the chance of a renewal at time n harris and keane showed that if sum n 1 infty u n 2 infty then mu theta and mu 0 are singular while if sum n 1 infty u n 2 infty and theta is small enough then mu theta is absolutely continuous with respect to mu 0 they conjectured that absolute continuity should not depend on theta but only on the square summability of u n we show that in fact the power law governing the decay of u n is crucial and for some renewal sequences u n there is a phase transition at a critical parameter theta c in 0 1 for theta theta c the measures mu theta and mu 0 are mutually absolutely continuous but for theta theta c they are singular we also prove that when u n o n 1 the measures mu theta for theta in 1 1 are all mutually absolutely continuous

the galton board limit theorems and recurrence we study a particle moving in r2 under a constant external force and bouncing off a periodic array of convex domains scatterers the latter must satisfy a standard finite horizon condition to prevent ballistic collision free motion this model is known to physicists as galton board it is also identical to a periodic lorentz gas previous heuristic and experimental studies have suggested that the particle s speed v t should grow as t1 3 and its coordinate x t as t2 3 we prove these conjectures rigorously we also find limit distributions for the rescaled velocity t 1 3v t and position t 2 3x t in addition quite unexpectedly we discover that the particle s motion is recurrent that means that a ball dropped on an idealized galton board will roll down but from time to time it should bounce all the way back up with probability one

one more pathology of c algebraic tensor products we define a collection of tensor product norms for c algebras and show that a symmetric tensor product functor on the category of separable c algebras need not be associative

penalized weighted residual method for the initial value problems to satisfy the need for network parallel computing in the whole time domain a penalized weighted residual formulation for the second order initial value problems is developed and its time finite element approximation is presented to impose properly both the initial displacement and velocity the initial velocity is imposed to the weighted residual equation as a penalized form that is constructed such that the approximated initial velocity can satisfy the initial condition in the average sense by this procedure the penalized weighted residual formulation makes it possible to handle the whole time domain of investigation and to use the conventional displacement based finite element technique without any other artificial routine through several numerical tests it is confirmed that the present method gives very accurate solutions for various systems arising in the second order initial value problems and presents promising characteristics for network parallel computation in the whole time domain of investigation

a cell dynamical system model of chemical turbulence a cellular automaton like caricature of chemical turbulence on an infinite one dimensional lattice is studied the model exhibits apparently turbulent space time patterns to make this statement precise the following problems or points are discussed 1 the infinite system size limit of such cell dynamical systems and its observability is defined 2 it is proved that the invariant state in the large system size limit of the turbulent phase exhibits spatial patterns governed by a gibbs random field 3 potential characteristics of turbulent space time patterns are critically surveyed and a working definition of weak turbulence is proposed 4 it is proved that the invariant state of the turbulent phase is actually weak turbulent furthermore we conjecture that the turbulent phase of our model is an example of a k system that is not bernoulli

all extensions of c _r left mathbb f_n right are semi invertible it is shown that an extension of the reduced group c algebra of a free group by the compact operators can be made asymptotically split by addition of another extension which admits a completely positive lifting

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

simulating sensitivities of conditional value at risk conditional value at risk cvar is both a coherent risk measure and a natural risk statistic it is often used to measure the risk associated with large losses in this paper we study how to estimate the sensitivities of cvar using monte carlo simulation we first prove that the cvar sensitivity can be written as a conditional expectation for general loss distributions we then propose an estimator of the cvar sensitivity and analyze its asymptotic properties the numerical results show that the estimator works well furthermore we demonstrate how to use the estimator to solve optimization problems with cvar objective and or constraints and compare it to a popular linear programming based algorithm

stochastic wave propagation in maxwell s equations we use the spectral representation to describe the random electromagnetic fields which are coupled by maxwell s equations with a random source term the covariance matrix functions of the random electromagnetic fields are expressed in terms of the spectral matrix function of the random source term and the green function of maxwell s equations when the random source term is gaussian the electromagnetic fields exhibit diffusive scaling limits in the sense of distributions

a level 1 large deviation principle for the autocovariances of uniquely ergodic transformations with additive noise a large deviation principle ldp at level 1 for random means of the type m n equiv frac 1 n sum limits j 0 n 1 z j z j 1 text n 1 2 is established the random process zn n 0 is given by zn xn n n 0 1 2 where xn n 0 and n n 0 are independent random sequences the former is a stationary process defined by xn tn x0 x0 is uniformly distributed on the circle s1 t s1 s1 is a continuous uniquely ergodic transformation preserving the lebesgue measure on s1 and n n 0 is a random sequence of independent and identically distributed random variables on s1 is a continuous real function the ldp at level 1 for the means mn is obtained by using the level 2 ldp for the markov process vn xn n n 1 n 0 and the contraction principle for establishing this level 2 ldp one can consider a more general setting t 0 1 0 1 is a measure preserving lebesgue measure phi left 0 left 1 right right to mathbb r is a real measurable function and n are independent and identically distributed random variables on mathbb r for instance they could have a gaussian distribution with mean zero and variance 2 the analogous result for the case of autocovariance of order k is also true

on a class of quantum channels open random walks and recurrence we study a particular class of trace preserving completely positive maps called pq channels for which classical and quantum evolutions are isolated in a certain sense by combining open quantum random walks with a notion of recurrence we are able to describe criteria for recurrence of the walk related to this class of channels positive recurrence for open walks is also discussed in this context

local limit theorem for randomly deforming billiards we study limit theorems in the context of random perturbations of dispersing billiards in finite and infinite measure in the context of a planar periodic lorentz gas with finite horizon we consider random perturbations in the form of movements and deformations of scatterers we prove a central limit theorem for the cell index of planar motion as well as a mixing local limit theorem for piecewise holder continuous observables in the context of the infinite measure random system we prove limit theorems regarding visits to new obstacles and self intersections as well as decorrelation estimates the main tool we use is the adaptation of anisotropic banach spaces to the random setting

extensions of the reduced group c algebra of a free product of amenable groups abstract we prove that the unitary equivalence classes of extensions of c r g by any unital stable c algebra taken modulo extensions which split via an asymptotic homomorphism form a group which can be calculated from the universal coefficient theorem of kk theory when g is a free product of a countable collection of countable amenable groups

the capacity of finite state markov channels with feedback we consider a class of finite state markov channels with feedback we first introduce a simplified equivalent channel model and then construct the optimal stationary and nonstationary input processes that maximize the long term directed mutual information furthermore we give a sufficient condition under which the channel s shannon capacity can be achieved by a stationary input process the corresponding converse coding theorem and direct coding theorem are proved

time domain finite element analysis of dynamic systems in this study dynamic analyses of systems are conducted using the time domain finite element analysis when initial and final conditions are specified dynamics of systems can be analyzed using the time marching procedure based on hamilton s weak principle a new method is developed to describe the motion of a system and is extended to accommodate the initial displacement as a state variable system matrices are constructed to evaluate time histories of the system from initial time to a specified time without the information on the velocity and momentum at each time step galerkin s weak principle is used to construct element matrices in the time domain and assemble the whole matrices the neumann boundary condition is modified to reflect the effect of initial displacement and velocity that are inherent to dynamic problems through the momentum conservation of the system the matrices obtained thereafter conserve the symmetricity and enable us to reduce the model order by the systematic approach spatial propagation equation is built up and two point boundary conditions are used to estimate the unknown initial conditions at one end of the beam modal domain analysis is introduced to reduce the size of matrices it can be seen that the matrices constructed using the time domain finite element analysis are applicable to the model reduction that is analogous to the space domain model reduction methodology several numerical examples show that a consequence of the suggested method can be applied to describe the motion of various dynamic systems successfully and to validate the effect of the model reduction in the time domain

on stability ofc algebras letabe a unitalc algebra i e aadmits a countable approximate unit it is proved thatais stable i e ais isomorphic toa k where k is the algebra of compact operators on a separable hilbert space if and only if for each positive elementa aand eache 0 there exists a positive elementb asuch that ab eandx x a xx bfor somexina using this characterization it is proved among other things that the inductive limit of any sequence of unital stablec algebras is stable and that the crossed product of a unital stablec algebra by a discrete group is again stable

a long time tail for random walk in random scenery consider a simple random walk on d whose sites are colored black or white independently with probabilityq resp 1 q walk and coloring are independent letn k be the number of steps by the walk between itskth and k 1 th visits to a black site i e the length of itskth white run and let k e n k q 1 our main result is a proof that lim k k d 2 k 1 q q d 2 2 d 2 d 2 since it is known thatq 1 k e n 1 n k 1 b e n 1 b e n k 1 b withb the event that the origin is black exhibits a long time tail in the run length autocorrelation function numerical calculations of k 1 k 100 ind 1 2 and 3 show that there is an oscillatory behavior of k for smallk this damps exponentially fast following which the power law sets in fairly rapidly we prove that if the coloring is not independent but is convex in the sense of fkg then the decay of k cannot be faster than

belief propagation for continuous state spaces stochastic message passing with quantitative guarantees the sum product or belief propagation bp algorithm is a widely used message passing technique for computing approximate marginals in graphical models we introduce a new technique called stochastic orthogonal series message passing sosmp for computing the bp fixed point in models with continuous random variables it is based on a deterministic approximation of the messages via orthogonal series basis expansion and a stochastic estimation of the basis coefficients via monte carlo techniques and damped updates we prove that the sosmp iterates converge to a neighborhood of the unique bp fixed point for any tree structured graph and for any graphs with cycles in which the bp updates satisfy a contractivity condition in addition we demonstrate how to choose the number of basis coefficients as a function of the desired approximation accuracy and smoothness of the compatibility functions we illustrate our theory with both simulated examples and in application to optical flow estimation

asymptotically brownian skew products give non loosely bernoulli k automorphisms a quite natural exponential mixing condition on a real valued cocyclef implies that the cocycle is asymptotically approximable by brownian motion to a degree better thant 1 2 this in turn implies that the skew extension by a positive entropy ergodic flow using the cocycle is not loosely bernoulli in complete analogy to kalikow st t 1 argument kal