Clustering Dynamic Class Coupling Data to Measure Class Reusability Pattern

10.1017/etds.2020.101 ◽

2020 ◽

pp. 1-47

Author(s):

RYOKICHI TANAKA

Keyword(s):

Hausdorff Dimension ◽

Radon Measure ◽

Hyperbolic Group ◽

Hyperbolic Metric ◽

Hyperbolic Groups ◽

Group Invariant ◽

Hyperbolic Metrics ◽

Abstract Weshow that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

Statistical behaviour of the leaves of Riccati foliations

10.1017/s0143385708001028 ◽

2009 ◽

Vol 30 (1) ◽

pp. 67-96 ◽

Cited By ~ 6

Author(s):

CH. BONATTI ◽

X. GÓMEZ-MONT ◽

R. VILA-FREYER

Keyword(s):

Lebesgue Measure ◽

Geodesic Flow ◽

Conformal Structure ◽

Holomorphic Foliation ◽

Horocycle Flow ◽

Linear Ordinary Differential Equations ◽

Positive Time ◽

Measure Class ◽

Dimension One

AbstractWe introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

Some spherical functions on hyperbolic groups

Journal of Topology and Analysis ◽

10.1142/s1793525320500429 ◽

2020 ◽

pp. 1-50

Author(s):

Adrien Boyer

Keyword(s):

Slow Growth ◽

Point Of View ◽

Hyperbolic Groups ◽

Spherical Functions ◽

Decay Estimates ◽

The Family ◽

Boundary Representations ◽

Measure Class ◽

Gromov Boundary ◽

Parameter Deformation

We investigate properties of some spherical functions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson–Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (also called Haagerup’s inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. We also prove that the family of boundary representations studied in this paper, which can be regarded as a one parameter deformation of the boundary unitary representation, are slow growth representations acting on a Hilbert space admitting a proper 1-cocycle.

Regularity at infinity of compact negatively curved manifolds

10.1017/s0143385700007999 ◽

1994 ◽

Vol 14 (3) ◽

pp. 493-514

Author(s):

Ursula Hamenstädt

Keyword(s):

Tangent Bundle ◽

Compact Manifold ◽

Negative Curvature ◽

Periodic Points ◽

Geodesic Flows ◽

Stable Foliation ◽

Curved Manifolds ◽

Negatively Curved ◽

Unit Tangent Bundle ◽

AbstractIt is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

A Cauchy problem for the Tartar equation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001700 ◽

2002 ◽

Vol 132 (2) ◽

pp. 395-418 ◽

Cited By ~ 2

Author(s):

Sergey Sazhenkov

Keyword(s):

Cauchy Problem ◽

Velocity Field ◽

Velocity Fields ◽

Well Posedness ◽

Oscillatory Solutions ◽

Linear Transport Equation ◽

Weak Limits ◽

Bounded Open Subset ◽

Measure Class ◽

Image Position

In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H-measures defined on the space of positions and frequencies. The present paper is devoted to the investigation of the Tartar equation which describes the evolution of the H-measure μt associated with a sequence of oscillatory solutions of the linear transport equation in cases when a given solenoidal velocity field v(x, t) is sufficiently smooth. Here, (t, x, y) ∈ (0, T) × Ω × S1, 0 < T < +∞, Ω is a bounded open subset of R2 and S1 is the unit circle in R2, given coefficients Yij = Yij(y) are infinitely smooth.Assuming that v belongs to , we establish the well posedness of Cauchy problem for the Tartar equation in the same measure class as the H-measures are in. For this purpose, we develop and use an extension of the theory of Lagrange coordinates for a case of non-smooth solenoidal velocity fields.

An example of an amenable action from geometry

10.1017/s0143385700004016 ◽

1987 ◽

Vol 7 (2) ◽

pp. 289-293 ◽

Cited By ~ 8

Author(s):

R. J. Spatzier

Keyword(s):

Compact Manifold ◽

Negative Curvature ◽

Universal Cover ◽

Constant Negative Curvature ◽

Type Iii ◽

Natural Measure ◽

Amenable Action ◽

AbstractLet M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.

Mindfulness-Based Stress Reduction Intervention in Chronic Stroke: a Randomized, Controlled Pilot Study

Mindfulness ◽

10.1007/s12671-021-01751-0 ◽

2021 ◽

Author(s):

Juliana V. Baldo ◽

Krista Schendel ◽

Sandy J. Lwi ◽

Timothy J. Herron ◽

Denise G. Dempsey ◽

...

Keyword(s):

Pilot Study ◽

Active Control ◽

Stress Reduction ◽

Chronic Stroke ◽

Retention Rates ◽

Brain Health ◽

Stroke Patients ◽

Post Intervention ◽

Mindfulness Based Stress Reduction ◽

Abstract Objectives Mindfulness-Based Stress Reduction (MBSR) involves training in mindful meditation and has been shown to improve functioning across a range of different disorders. However, little research has focused on the use of MBSR in stroke patients, and previous MBSR studies typically have not included an active control condition to account for non-specific factors that could contribute to the observed benefits. Methods We conducted a pilot study of MBSR in chronic stroke patients, comparing MBSR to an active control condition. Half of participants were randomly assigned to a standard 8-week MBSR class, and the other half of participants were assigned to an 8-week Brain Health class matched for schedule, instructor, and format. Participants were assessed pre- and post-intervention by blinded examiners on a neuropsychological battery that included primary outcome measures of psychological and cognitive functioning. Participants were also given an anonymous questionnaire following the post-intervention testing session to measure class satisfaction. Results Both the MBSR and Brain Health classes were rated favorably by participants. Recruitment and retention rates were high, and methods for participant randomization and examiner blinding were successful. Class implementation in terms of execution was also successful, as rated by outside experts. Conclusions This study established the feasibility of conducting MBSR and Brain Health classes in a chronic stroke population. Trial Registration https://ClinicalTrials.gov, NCT #: 02600637

Amenability of groupoids and asymptotic invariance of convolution powers

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/772/15482 ◽

2021 ◽

pp. 69-92

Author(s):

Theo Bühler ◽

Vadim Kaimanovich

Keyword(s):

Random Walk ◽

Locally Compact Group ◽

Probability Measures ◽

Locally Compact ◽

Von Neumann ◽

Original Definition ◽

Convolution Powers ◽

Measure Class ◽

Definition Of

The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich–Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implies, in particular, that the amenability of a measure class preserving group action is equivalent to the existence of a random environment on the group parameterized by the action space, and such that the tail of the random walk in almost every environment is trivial.