scholarly journals The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Tim Austin
Keyword(s):  
Metric Spaces ◽  
Metric Geometry ◽  
Bernoulli Shifts ◽  
Additional Information ◽  
Covering Numbers ◽  
Model Spaces ◽  
Connectedness Property ◽  
Sofic Entropy ◽  
Sofic Groups

AbstractBowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

scholarly journals ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

2016 ◽  
Vol 4 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lower Bound ◽  
Probability Distributions ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Entropy Theory ◽  
Reverse Inequality ◽  
Cartesian Products ◽  
Model Spaces ◽  
Sofic Entropy ◽  
Sofic Groups

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

scholarly journals COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

2018 ◽  
Vol 98 (3) ◽  
pp. 422-433
Author(s):  
BORIS GOLDFARB ◽  
JONATHAN L. GROSSMAN
Keyword(s):  
Metric Spaces ◽  
General Context ◽  
Metric Geometry ◽  
Finitely Generated ◽  
Coherence Property ◽  
Finitely Generated Groups ◽  
Permanence Property ◽  
Permanence Properties ◽  
New Framework

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Quasi-metric geometry

2014 ◽  
Author(s):  
◽  
Dan Brigham
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Linear Structure ◽  
Abstract Objects ◽  
Metric Geometry ◽  
Natural Structure ◽  
University Of Missouri ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Every time one sees |x-y|, one is looking at a specific metric acting on x and y, whatever they may happen to be, usually numbers or vectors. The notion of the distance between two objects is one of the most fundamental and ubiquitous in many branches of mathematics. A quasi-metric is a generalization of the familiar notion of metric. This dissertation examines what happens in this new setting of quasi-metrics. In particular, in the first chapter we introduce quasi-metrics, provide examples of them, then, given an arbitrary quasi-metric, develop a procedure which allows us to construct a better quasi-metric. Then we look at topological matters, such as openness and continuity. After that, we look at functions on abstract objects called groupoids, which is yet another step toward generality, since the objects we consider here contain the class of quasi-metrics. Dealing with groupoids is useful because it provides a natural structure into which quasi-metrics and quasi-norms fit. After these preliminary chapters, we then introduce linear structure, meaning the quasi-metrics studied are defined on sets in which one can add two points together, and multiply points by numbers, as this is not possible in an abstract set. Next we quantify smoothness of quasi-metric spaces, and throw in measures. For the first six chapters, we worked within a given quasi-metric space, assigning to points the distance between them. The seventh and final chapter deals with the "distance'' between two distinct quasi-metrics.

scholarly journals METRIC GRAPH RECONSTRUCTION FROM NOISY DATA

2012 ◽  
Vol 22 (04) ◽  
pp. 305-325 ◽  
Author(s):  
MRIDUL AANJANEYA ◽  
FREDERIC CHAZAL ◽  
DANIEL CHEN ◽  
MARC GLISSE ◽  
LEONIDAS GUIBAS ◽  
...  
Keyword(s):  
Real World ◽  
Metric Spaces ◽  
Data Sets ◽  
Metric Geometry ◽  
Metric Graphs ◽  
Real World Data ◽  
Data Set ◽  
World Data ◽  
Metric Graph ◽  
Graph Reconstruction

Many real-world data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs.19 Building on the notions of correspondence and Gromov-Hausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs a metric graph that is homeomorphic to the underlying metric graph and has bounded distortion of distances. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.

Production of New and Revised A.A.V.S.O. Charts

1979 ◽  
Vol 46 ◽  
pp. 368
Author(s):  
Clinton B. Ford
Keyword(s):  
Variable Star ◽  
Standard Form ◽  
Original Data ◽  
University Of Pennsylvania ◽  
Standard Format ◽  
Additional Information ◽  
The Gift ◽  
The University

A “new charts program” for the Americal Association of Variable Star Observers was instigated in 1966 via the gift to the Association of the complete variable star observing records, charts, photographs, etc. of the late Prof. Charles P. Olivier of the University of Pennsylvania (USA). Adequate material covering about 60 variables, not previously charted by the AAVSO, was included in this original data, and was suitably charted in reproducible standard format.Since 1966, much additional information has been assembled from other sources, three Catalogs have been issued which list the new or revised charts produced, and which specify how copies of same may be obtained. The latest such Catalog is dated June 1978, and lists 670 different charts covering a total of 611 variables none of which was charted in reproducible standard form previous to 1966.

Convergent Beam Electron Diffraction

1978 ◽  
Vol 36 (3) ◽  
pp. 304-315
Author(s):  
G. Lehmpfuhl
Keyword(s):  
Electron Diffraction ◽  
Diffraction Pattern ◽  
Crystal Surface ◽  
Diffraction Spot ◽  
Crystal Thickness ◽  
Large Area ◽  
Electron Microscopic ◽  
Additional Information ◽  
Microscopic Investigations

Introduction In electron microscopic investigations of crystalline specimens the direct observation of the electron diffraction pattern gives additional information about the specimen. The quality of this information depends on the quality of the crystals or the crystal area contributing to the diffraction pattern. By selected area diffraction in a conventional electron microscope, specimen areas as small as 1 µ in diameter can be investigated. It is well known that crystal areas of that size which must be thin enough (in the order of 1000 Å) for electron microscopic investigations are normally somewhat distorted by bending, or they are not homogeneous. Furthermore, the crystal surface is not well defined over such a large area. These are facts which cause reduction of information in the diffraction pattern. The intensity of a diffraction spot, for example, depends on the crystal thickness. If the thickness is not uniform over the investigated area, one observes an averaged intensity, so that the intensity distribution in the diffraction pattern cannot be used for an analysis unless additional information is available.

Microtubule nucleation sequence studied by time-resolved x-ray scattering and electron microscopy

1983 ◽  
Vol 41 ◽  
pp. 766-767
Author(s):  
Eva-Maria Mandelkow ◽  
Eckhard Mandelkow ◽  
Joan Bordas
Keyword(s):  
Electron Microscopy ◽  
Dynamic Equilibrium ◽  
Time Resolved ◽  
X Ray ◽  
Additional Information ◽  
X Ray Scattering ◽  
Low Concentrations ◽  
Microtubule Growth ◽  
Ray Patterns ◽  
Ray Scattering

When a solution of microtubule protein is changed from non-polymerising to polymerising conditions (e.g. by temperature jump or mixing with GTP) there is a series of structural transitions preceding microtubule growth. These have been detected by time-resolved X-ray scattering using synchrotron radiation, and they may be classified into pre-nucleation and nucleation events. X-ray patterns are good indicators for the average behavior of the particles in solution, but they are difficult to interpret unless additional information on their structure is available. We therefore studied the assembly process by electron microscopy under conditions approaching those of the X-ray experiment. There are two difficulties in the EM approach: One is that the particles important for assembly are usually small and not very regular and therefore tend to be overlooked. Secondly EM specimens require low concentrations which favor disassembly of the particles one wants to observe since there is a dynamic equilibrium between polymers and subunits.

Edge Penetration Effects in the Low-Loss Electron Image in the SEM

1988 ◽  
Vol 46 ◽  
pp. 202-203
Author(s):  
Oliver C. Wells
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
Beam Energy ◽  
Secondary Electron ◽  
Sharp Edge ◽  
Beam Diameter ◽  
Low Loss ◽  
Additional Information ◽  
Electron Image ◽  
Scanning Electron

The low-loss electron (LLE) image in the scanning electron microscope (SEM) is useful for the study of uncoated photoresist and some other poorly conducting specimens because it is less sensitive to specimen charging than is the secondary electron (SE) image. A second advantage can arise from a significant reduction in the width of the “penetration fringe” close to a sharp edge. Although both of these problems can also be solved by operating with a beam energy of about 1 keV, the LLE image has the advantage that it permits the use of a higher beam energy and therefore (for a given SEM) a smaller beam diameter. It is an additional attraction of the LLE image that it can be obtained simultaneously with the SE image, and this gives additional information in many cases. This paper shows the reduction in penetration effects given by the use of the LLE image.

Questions and Answers

2000 ◽  
Vol 5 (2) ◽  
pp. 3-3
Author(s):  
Christopher R. Brigham ◽  
James B. Talmage
Keyword(s):  
Nervous System ◽  
Peripheral Nervous System ◽  
Spinal Nerve ◽  
Sensory Loss ◽  
Residual Symptoms ◽  
Additional Information ◽  
Questions And Answers ◽  
Motor Nerves ◽  
Symptoms And Signs ◽  
Limited Motion

Abstract Lesions of the peripheral nervous system (PNS), whether due to injury or illness, commonly result in residual symptoms and signs and, hence, permanent impairment. The AMA Guides to the Evaluation of Permanent Impairment (AMA Guides) describes procedures for rating upper extremity neural deficits in Chapter 3, The Musculoskeletal System, section 3.1k; Chapter 4, The Nervous System, section 4.4 provides additional information and an example. The AMA Guides also divides PNS deficits into sensory and motor and includes pain within the former. The impairment estimates take into account typical manifestations such as limited motion, atrophy, and reflex, trophic, and vasomotor deficits. Lesions of the peripheral nervous system may result in diminished sensation (anesthesia or hypesthesia), abnormal sensation (dysesthesia or paresthesia), or increased sensation (hyperesthesia). Lesions of motor nerves can result in weakness or paralysis of the muscles innervated. Spinal nerve deficits are identified by sensory loss or pain in the dermatome or weakness in the myotome supplied. The steps in estimating brachial plexus impairment are similar to those for spinal and peripheral nerves. Evaluators should take care not to rate the same impairment twice, eg, rating weakness resulting from a peripheral nerve injury and the joss of joint motion due to that weakness.

Sign in / Sign up

Export Citation Format

Share Document