scholarly journals Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

2018 ◽  
Vol 20 (05) ◽  
pp. 1750050 ◽  
Author(s):  
Alvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Dirichlet Problem At Infinity ◽  
Gromov Boundary

In this paper, we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.

scholarly journals A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Hyperbolic Manifolds ◽  
Necessary Condition ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Gromov Boundary

AbstractWe study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) isoperimetric inequality, in terms of their Gromov boundary, improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition to have isoperimetric inequality and, therefore, it can be removed as hypothesis.

CHEEGER ISOPERIMETRIC CONSTANTS OF GROMOV-HYPERBOLIC SPACES WITH QUASI-POLES

2000 ◽  
Vol 02 (04) ◽  
pp. 511-533 ◽  
Author(s):  
JIANGUO CAO
Keyword(s):  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Spaces ◽  
Connected Components ◽  
Local Geometry ◽  
Complete Manifold ◽  
Cheeger Constant ◽  
Gromov Hyperbolic ◽  
Geometric Boundary ◽  
Dirichlet Problem At Infinity

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger constant h(X) may be zero if any of the above assumption on X is removed. Applications of this isoperimetric estimate include the solvability of the Dirichlet problem at infinity for non-compact Gromov-hyperbolic manifolds X above. In addition, we show that the Martin boundary ∂ΔX of such a space X is homeomorphic to the geometric boundary X(∞) of X at infinity.

scholarly journals Cross ratios and cubulations of hyperbolic groups

2021 ◽  
Author(s):  
Jonas Beyrer ◽  
Elia Fioravanti
Keyword(s):  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Optimal Length ◽  
Rigidity Result ◽  
Cube Complex ◽  
Geodesic Currents ◽  
Gromov Hyperbolic ◽  
Cube Complexes ◽  
The Relationship ◽  
Gromov Boundary

AbstractMany geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

Studies on Leukemogenic and Sarcomagenic Viruses

1970 ◽  
Vol 28 ◽  
pp. 156-157
Author(s):  
Leon Dmochowski
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Morphological Properties ◽  
Mode Of Development ◽  
Relationship Of ◽  
The Relationship

Electron microscopy has proved to be an invaluable discipline in studies on the relationship of viruses to the origin of leukemia, sarcoma, and other types of tumors in animals and man. The successful cell-free transmission of leukemia and sarcoma in mice, rats, hamsters, and cats, interpreted as due to a virus or viruses, was proved to be due to a virus on the basis of electron microscope studies. These studies demonstrated that all the types of neoplasia in animals of the species examined are produced by a virus of certain characteristic morphological properties similar, if not identical, in the mode of development in all types of neoplasia in animals, as shown in Fig. 1.

The relationship of IGE receptor topography to signaling activity in RBL-2H3 mast cells

1991 ◽  
Vol 49 ◽  
pp. 236-237
Author(s):  
J.R. Pfeiffer ◽  
J.C. Seagrave ◽  
C. Wofsy ◽  
J.M. Oliver
Keyword(s):  
Mast Cells ◽  
Protein A ◽  
Scattered Electron ◽  
Ige Receptor ◽  
Receptor Complexes ◽  
Ige Binding ◽  
Electron Imaging ◽  
Small Clusters ◽  
Relationship Of ◽  
The Relationship

In RBL-2H3 rat leukemic mast cells, crosslinking IgE-receptor complexes with anti-IgE antibody leads to degranulation. Receptor crosslinking also stimulates the redistribution of receptors on the cell surface, a process that can be observed by labeling the anti-IgE with 15 nm protein A-gold particles as described in Stump et al. (1989), followed by back-scattered electron imaging (BEI) in the scanning electron microscope. We report that anti-IgE binding stimulates the redistribution of IgE-receptor complexes at 37“C from a dispersed topography (singlets and doublets; S/D) to distributions dominated sequentially by short chains, small clusters and large aggregates of crosslinked receptors. These patterns can be observed (Figure 1), quantified (Figure 2) and analyzed statistically. Cells incubated with 1 μg/ml anti-IgE, a concentration that stimulates maximum net secretion, redistribute receptors as far as chains and small clusters during a 15 min incubation period. At 3 and 10 μg/ml anti-IgE, net secretion is reduced and the majority of receptors redistribute rapidly into clusters and large aggregates.

Severe Paradoxical Dysphonia in Two Young Women

1993 ◽  
Vol 2 (3) ◽  
pp. 52-55 ◽  
Author(s):  
Michael Collins ◽  
Robert McDonald ◽  
Robert Stanley ◽  
Timothy Donovan ◽  
C. Frank Bonebrake
Keyword(s):  
Young Women ◽  
Theoretical Basis ◽  
Therapeutic Regimen ◽  
Instrumental Data ◽  
Relationship Of ◽  
The Relationship

This report describes an unusual and persistent dysphonia in two young women who had taken a therapeutic regimen of isotretinoin for intractable acne. We report perceptual and instrumental data for their dysphonia, and pose a theoretical basis for the relationship of dysphonia to this drug. We also provide recommendations for reducing the risk of acquiring a dysphonia during the course of treatment with isotretinoin.

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