Computer-Assisted Bounds for the Rate of Decay of Correlations

1997 ◽  
Vol 189 (1) ◽  
pp. 237-257 ◽  
Author(s):  
Gary Froyland
Keyword(s):  
Decay Of Correlations ◽  
Computer Assisted ◽  
Rate Of Decay

scholarly journals Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

2014 ◽  
Vol 24 (10) ◽  
pp. 1430028 ◽  
Author(s):  
Vitor Araujo ◽  
Stefano Galatolo ◽  
Maria José Pacifico
Keyword(s):  
Exponential Decay ◽  
Chaotic Behavior ◽  
Decay Of Correlations ◽  
Statistical Properties ◽  
Computer Assisted ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Physical Measure ◽  
Exponential Decay Of Correlations ◽  
Statistical Behavior

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

scholarly journals A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

2012 ◽  
Vol 33 (2) ◽  
pp. 475-498 ◽  
Author(s):  
NICOLAI HAYDN ◽  
MATTHEW NICOL ◽  
TOMAS PERSSON ◽  
SANDRO VAIENTI
Keyword(s):  
Dynamical Systems ◽  
Return Time ◽  
Decay Of Correlations ◽  
Cantelli Lemma ◽  
Target Problem ◽  
Exponential Decay Of Correlations ◽  
Hyperbolic Dynamical Systems ◽  
Rate Of Decay ◽  
Dispersing Billiards ◽  
Lozi Maps

AbstractLet (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls Bi if μ(Bi)≥i−γ, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If μ(Bi)≥C1 /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that μ(Bi)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.

scholarly journals On mix-norms and the rate of decay of correlations

Nonlinearity ◽  
2021 ◽  
Vol 34 (6) ◽  
pp. 3762-3782
Author(s):  
Bryan W Oakley ◽  
Jean-Luc Thiffeault ◽  
Charles R Doering
Keyword(s):  
Decay Of Correlations ◽  
Rate Of Decay

3-D organization of fibrinogen receptors using computer reconstruction and whole-mount microscopy

1988 ◽  
Vol 46 ◽  
pp. 30-31
Author(s):  
E. T. O'Toole ◽  
R. R. Hantgan ◽  
J. C. Lewis
Keyword(s):  
Whole Blood ◽  
Colloidal Gold ◽  
Blood Platelets ◽  
Cell Contact ◽  
Computer Assisted ◽  
Adherent Cells ◽  
Differential Centrifugation ◽  
Computer Reconstruction ◽  
Sds Page ◽  
Whole Mount

Thrombocytes (TC), the avian equivalent of blood platelets, support hemostasis by aggregating at sites of injury. Studies in our lab suggested that fibrinogen (fib) is a requisite cofactor for TC aggregation but operates by an undefined mechanism. To study the interaction of fib with TC and to identify fib receptors on cells, fib was purified from pigeon plasma, conjugated to colloidal gold and used both to facilitate aggregation and as a receptor probe. Described is the application of computer assisted reconstruction and stereo whole mount microscopy to visualize the 3-D organization of fib receptors at sites of cell contact in TC aggregates and on adherent cells.Pigeon TC were obtained from citrated whole blood by differential centrifugation, washed with Ca++ free Hank's balanced salts containing 0.3% EDTA (pH 6.5) and resuspended in Ca++ free Hank's. Pigeon fib was isolated by precipitation with PEG-1000 and the purity assessed by SDS-PAGE. Fib was conjugated to 25nm colloidal gold by vortexing and the conjugates used as the ligand to identify fib receptors.

Computer Assisted Image Reconstruction of Oligomer Structure

1976 ◽  
Vol 34 ◽  
pp. 472-473
Author(s):  
A.M. Jones ◽  
A. Max Fiskin
Keyword(s):  
Three Dimensional ◽  
Depth Of Field ◽  
Computer Assisted ◽  
Projection Data ◽  
Two Dimensional ◽  
Single Particles ◽  
Dimensional Reconstruction ◽  
Computer Assisted Image ◽  
The Fourier Transform ◽  
Space Approach

If the tilt of a specimen can be varied either by the strategy of observing identical particles orientated randomly or by use of a eucentric goniometer stage, three dimensional reconstruction procedures are available (l). If the specimens, such as small protein aggregates, lack periodicity, direct space methods compete favorably in ease of implementation with reconstruction by the Fourier (transform) space approach (2). Regardless of method, reconstruction is possible because useful specimen thicknesses are always much less than the depth of field in an electron microscope. Thus electron images record the amount of stain in columns of the object normal to the recording plates. For single particles, practical considerations dictate that the specimen be tilted precisely about a single axis. In so doing a reconstructed image is achieved serially from two-dimensional sections which in turn are generated by a series of back-to-front lines of projection data.

Biodegradable polyester neuroprostheses promote the reconnection of separated peripheral nerve stubs

1992 ◽  
Vol 50 (2) ◽  
pp. 1094-1095
Author(s):  
Beverly L. Giammara ◽  
Jennifer S. Stevenson ◽  
Peggy E. Yates ◽  
Robert H. Gunderson ◽  
Jacob S. Hanker
Keyword(s):  
Image Analysis ◽  
Basement Membrane ◽  
Periodic Acid ◽  
Light And Electron Microscopy ◽  
Computer Assisted ◽  
Type Iii Collagen ◽  
Image Analysis System ◽  
Biodegradable Polyester ◽  
Adult Rats ◽  
Type Iv

An 11mm length of sciatic nerve was removed from 10 anesthetized adult rats and replaced by a biodegradable polyester Vicryl™ mesh sleeve which was then injected with the basement membrane gel, Matrigel™. It was noted that leg sensation and movement were much improved after 30 to 45 days and upon sacrifice nerve reconnection was noted in all animals. Epoxy sections of the repaired nerves were compared with those of the excised segments by the use of a variation of the PAS reaction, the PATS reaction, developed in our laboratories for light and electron microscopy. This microwave-accelerated technique employs periodic acid, thiocarbohydrazide and silver methenamine. It stains basement membrane or Type IV collagen brown and type III collagen (reticulin), axons, Schwann cells, endoneurium and perineurium black. Epoxy sections of repaired and excised nerves were also compared by toluidine blue (tb) staining. Comparison of the sections of control and repaired nerves was done by computer-assisted microscopic image analysis using an Olympus CUE-2 Image Analysis System.

High-resolution measurement of birefringent fine structure in living cells using a new polarized light microscope

1995 ◽  
Vol 53 ◽  
pp. 54-55
Author(s):  
Rudolf Oldenbourg
Keyword(s):  
Light Microscope ◽  
Fluorescent Dyes ◽  
Polarized Light ◽  
Processing System ◽  
Video Camera ◽  
Molecular Organization ◽  
Computer Assisted ◽  
Image Recording ◽  
Birefringent Crystal ◽  
Technological Advances

The recent renaissance of the light microsope is fueled in part by technological advances in components on the periphery of the microscope, such as the laser as illumination source, electronic image recording (video), computer assisted image analysis and the biochemistry of fluorescent dyes for labeling specimens. After great progress in these peripheral parts, it seems timely to examine the optics itself and ask how progress in the periphery facilitates the use of new optical components and of new optical designs inside the microscope. Some results of this fruitful reflection are presented in this symposium.We have considered the polarized light microscope, and developed a design that replaces the traditional compensator, typically a birefringent crystal plate, with a precision universal compensator made of two liquid crystal variable retarders. A video camera and digital image processing system provide fast measurements of specimen anisotropy (retardance magnitude and azimuth) at ALL POINTS of the image forming the field of view. The images document fine structural and molecular organization within a thin optical section of the specimen.

Computer-Assisted Analysis of Multi-Color Confocal Microscopic Datasets: Automated Light Microscopic Discrimination of Synaptic Relationships

1996 ◽  
Vol 54 ◽  
pp. 284-285
Author(s):  
M Wessendorf ◽  
A Beuning ◽  
D Cameron ◽  
J Williams ◽  
C Knox
Keyword(s):  
Nerve Fibers ◽  
Confocal Scanning Laser Microscopy ◽  
Computer Assisted ◽  
Nerve Terminals ◽  
Neuronal Somata ◽  
Scanning Laser Microscopy ◽  
Confocal Scanning ◽  
Entire Cell ◽  
Confocal Scanning Laser ◽  
Assisted Analysis

Multi-color confocal scanning-laser microscopy (CSLM) allows examination of the relationships between neuronal somata and the nerve fibers surrounding them at sub-micron resolution in x,y, and z. Given these properties, it should be possible to use multi-color CSLM to identify relationships that might be synapses and eliminate those that are clearly too distant to be synapses. In previous studies of this type, pairs of images (e.g., red and green images for tissue stained with rhodamine and fluorescein) have been merged and examined for nerve terminals that appose a stained cell (see, for instance, Mason et al.). The above method suffers from two disadvantages, though. First, although it is possible to recognize appositions in which the varicosity abuts the cell in the x or y axes, it is more difficult to recognize them if the apposition is oriented at all in the z-axis—e.g., if the varicosity lies above or below the neuron rather than next to it. Second, using this method to identify potential appositions over an entire cell is time-consuming and tedious.

Serial thin sectioning for pre- and post-embedding immunohistochemistry: Achieving consistent and reliable results for a 3-D reconstruction

1993 ◽  
Vol 51 ◽  
pp. 328-329
Author(s):  
Antonia M. Milroy
Keyword(s):  
Nervous System ◽  
Computer Assisted ◽  
Serial Sectioning ◽  
Electron Microscopic ◽  
High Quality ◽  
New Techniques ◽  
3 Dimensional ◽  
Multiple User ◽  
Thin Sectioning ◽  
Subcellular Level

In recent years many new techniques and instruments for 3-Dimensional visualization of electron microscopic images have become available. Higher accelerating voltage through thicker sections, photographed at a tilt for stereo viewing, or the use of confocal microscopy, help to analyze biological material without the necessity of serial sectioning. However, when determining the presence of neurotransmitter receptors or biochemical substances present within the nervous system, the need for good serial sectioning (Fig. 1+2) remains. The advent of computer assisted reconstruction and the possibility of feeding information from the specimen viewing chamber directly into a computer via a camera mounted on the electron microscope column, facilitates serial analysis. Detailed information observed at the subcellular level is more precise and extensive and the complexities of interactions within the nervous system can be further elucidated.We emphasize that serial ultra thin sectioning can be performed routinely and consistently in multiple user electron microscopy laboratories. Initial tissue fixation and embedding must be of high quality.

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