Bernoulli Shift Research Articles

Group amenability and actions on [formula omitted]-stable C*-algebras

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if G contains a copy of F2, then there exist uncountably many, non-cocycle conjugate strongly outer actions of G on any tracial, unital, separable C*-algebra that absorbs tensorially the Jiang-Su algebra. Similar conclusions hold for outer actions on McDuff II1 factors. We moreover show that G is amenable if and only if the Bernoulli shift on any finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods are inspired by Jones' work [27], and consist in a careful study of weak containment for the Koopman representations of certain generalized Bernoulli actions.

Asymptotic Behavior of Density in the Boundary-Driven Exclusion Process on the Sierpinski Gasket

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

CLINICAL EFFICACY OF TOCILIZUMAB IN THE TREATMENT OF SEVERE COVID-19 IN THE VOLYN REGION OF UKRAINE

Treatment of coronavirus disease COVID-19 is currently unresolved. Whether it is advisable to use Tocilizumab (TM), whether it is effective and safe for patients with Covid-19 - this question remains open to physicians.The aim of the study: To evaluate the safety and therapeutic efficacy of TM on the course and prognosis of severe COVID-19 during treatment with standard treatment according to the National Protocol without antiviral therapy.Materials and methods: Retrospective cohort study of adult patients (≥18 years) with severe COVID-19, who were admitted to the infectious department №2 KP «Volyn Regional Clinical Hospital» from September 2020 to December 2020.The statistical analysis included a group of patients who received TM and recovered (n=42), a group of patients who recovered without TM (n=59), and a group of patients who received or did not receive TM and died (n=58). Statistical methods of interval estimation, null hypothesis according to U Mann-Whitney criteria, binomial criteria, and T-test for two independent dichotomous samples according to the Bernoulli scheme using SPSS Statistics 26 stepwise direct method LR were used for the analysis.Results: 77.7% of patients with severe COVID-19 who received TM recovered, however, based on the analysis of data from a cohort of patients with severe COVID-19 who did not receive TM, it was found that recovery did not depend on TM. Adverse events associated with TM were noted in 3.7%. It has been found that TM reduces recovery time in the hospital and reduces the risk of being admitted to the intensive care unit. The average value of C-reactive protein (СRP) in the group of recovering those who received TM - 114.1 mg/l, and elevated levels of Procalcitonin (PC) above normal by 59.52% with an average value of 0.35 ng / ml.Conclusions: The high statistical significance of the obtained results in terms of therapeutic efficacy and safety of TM makes it possible to implement the obtained results in wide practice for the use of this drug in the treatment of severe COVID-19.

A new proof of the dimension gap for the Gauss map

AbstractIn [4], Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map T(x)=1/x mod 1 satisfy a ‘dimension gap’ meaning that for some c > 0, supp dim μp < 1– c, where μp denotes the (pushforward) Bernoulli measure for the countable probability vector p. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.

Boundary and Rigidity of Nonsingular Bernoulli Actions

Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},\mu_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{\mu_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigidity result in the non measure preserving case. For the proof, we use anti-symmetric Fock spaces and left creation operators to construct the boundary and therefore the assumption of having two base points is crucial.

Regularization of $p$-Adic Distributions Associated to Functions on $p$-Adic Fields With Moderate Variation

The $p$-adic distributions attached to ordinary functions defined on $p$-adic fields with moderate variation are studied. We first give a sufficient growth condition on ordinary functions to construct $p$-adic distributions. Then a moderate variation condition on functions for regularization of these $p$-adic distributions is imposed which provides a general method to construct $p$-adic measures. The $p$-adic integrals against these measures are also explicitly transformed to integrals against Bernoulli measures.

Probing bacterial cell wall growth by tracing wall-anchored protein complexes

The dynamic assembly of the cell wall is key to the maintenance of cell shape during bacterial growth. Here, we present a method for the analysis of Escherichia coli cell wall growth at high spatial and temporal resolution, which is achieved by tracing the movement of fluorescently labeled cell wall-anchored flagellar motors. Using this method, we clearly identify the active and inert zones of cell wall growth during bacterial elongation. Within the active zone, the insertion of newly synthesized peptidoglycan occurs homogeneously in the axial direction without twisting of the cell body. Based on the measured parameters, we formulate a Bernoulli shift map model to predict the partitioning of cell wall-anchored proteins following cell division.

Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence

This paper is devoted to the estimation of partial derivatives of multivariate density functions. In this regard, nonparametric linear wavelet-based estimators are introduced, showing their attractive properties from the theoretical point of view. In particular, we prove the strong uniform consistency properties of these estimators, over compact subsets of , with the determination of the corresponding convergence rates. Then, we establish the asymptotic normality of these estimators. As a main contribution, we relax some standard dependence conditions; our results hold under a weak dependence condition allowing the consideration of mixing, association, Gaussian sequences and Bernoulli shifts.

The influence of flow model selection on finite element model parameter estimation using Bayesian inference.

Recently, Bayesian estimation coupled with finite element modeling has been demonstrated as a viable tool for estimating vocal fold material properties from kinematic information obtained via high-speed video recordings. In this article, the sensitivity of the parameter estimations to the employed fluid model is explored by considering Bernoulli and one-dimensional viscous fluid flow models. Simulation results indicate that prescribing an ad hoc separation location for the Bernoulli flow model can lead to large estimate biases, whereas including the separation location as an estimated parameter leads to results comparable to that of the viscous fluid flow model.

Bernoulli disjointness

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $\beta G$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable $G$-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if $G$ is a countable icc group, then it admits a free, minimal, proximal flow.

Max-min theorems for weak containment, square summable homoclinic points, and completely positive entropy

We prove a max-min theorem for weak containment the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained a Bernoulli shift is $G\curvearrowright X/Y$-ergodic in the presence of $G\curvearrowright X$. We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal algebraic actions whose associated convolution operator is injective. We also deduce from our techniques that algebraic actions whose square summable homoclinic group is dense have completely positive entropy when the acting group is sofic.

Ergodicity and Totality of Partitions Associated with the RSK Correspondence

The asymptotic properties of sequences of partitions ( $$\sigma$$ -algebras) associated with the Robinson–Schensted–Knuth correspondence in spaces with Bernoulli measures are studied.

A Novel Image Encryption Algorithm Based on Bernoulli Shift Map

This paper puts forward a chaotic encryption algorithm based on Bernoulli Shift Map. This algorithm adopts dynamic encryption key and uses plaintext image parameters to control generation of chaotic key, and enhance connection between chaotic key and plaintext data; the iterative operation precision of chaotic map during encryption is high, avoiding cycle of chaotic system iteration process. The NPCR and UACI analysis, adjacent pixels correlation analysis and image information entropy analysis on encrypted image show this encryption algorithm has favorable encryption characteristics such as larger space for key, sensitivity to plaintext and random distribution of ciphertext.

Analysis of Instantaneous Feedback Queue with Heterogeneous Servers

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments are presented.

-spectra of measures on planar non-conformal attractors

AbstractWe study the $L^{q}$ -spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha }$ and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the $L^{q}$ -spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

Invariant means and the structure of inner amenable groups

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G . Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first ℓ 2 -Betti number of G with that of the stabilizer subgroups. In addition, for any marked finitely generated nonamenable group G we establish a uniform isoperimetric threshold for Schreier graphs G / H of G , beyond which the group H is necessarily weakly normal in G . Even more can be said in the particular case of an atomless mean for the conjugation action—that is, when G is inner amenable. We show that inner amenable groups have fixed price 1 , and we establish cocycle superrigidity for the Bernoulli shift of any nonamenable inner amenable group. In addition, we provide a concrete structure theorem for inner amenable linear groups over an arbitrary field. As a special case of inner amenability, we consider groups which are stable in the sense of Jones and Schmidt, obtaining a complete characterization of linear groups which are stable. Our analysis of stability leads to many new examples of stable groups; notably, all nontrivial countable subgroups of the group H ( R ) , of piecewise-projective homeomorphisms of the line, are stable.

Exact dimensional for Bernoulli measures and the Gauss map

Exact dimensional for Bernoulli measures and the Gauss map

Methods for Computing a System with Instantaneous Feedback and Variable Input Stream Intensity

We study the Markov model of a servicing system with one server and instantaneous feedback. After servicing is complete, a part of the calls, according to the Bernoulli scheme, either leave the system or immediately return to receive re-servicing, which requires a positive (random) server switching time. The intensity of the incoming stream depends on the state of the server, which can be in operating mode or in switching mode. We find the ergodicity condition for the corresponding two-dimensional Markov chain and propose three methods for studying it: the method of generating functions, the method of spectral expansion, and the method of space merging. We present the results of numerical experiments.

A short proof of Bernoulli disjointness via the local lemma

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $\Gamma$ is an infinite discrete group, then every minimal $\Gamma$-flow is disjoint from the Bernoulli shift $2^\Gamma$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovasz Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems.

Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets

We study multifractal properties of Bernoulli measure μp supported on homogeneous Cantor set determined by ([0,1],(2)k≥1,(ck)k≥1), where 0<p<1/2 and ck is either d1 or d2 with 0<d1<d2<1/2. Let θ_ and θ‾ be the lower and upper limits of the occurrence frequency of d1 in (ck)k≥1, respectively. It is proved by Wu (2005) [14] that, if θ_=θ‾ then the multifractal formalism holds. After given some conditions on (ck)k≥1, we get multifractal spectra for any 0≤θ_<θ‾≤1. We show that if θ_=0 and θ‾=1, the refined formalism does not hold, while Olsen's concave multifractal formalism holds in some cases. This answers a question asked by Olsen (1995) [10]. We also show that if 0<θ_<θ‾<1, Olsen's concave multifractal formalism does not hold.