scholarly journals Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds

2021 ◽  
Vol 9 (1) ◽  
pp. 219-253
Author(s):  
Hiroshi Tsuji
Keyword(s):  
Ricci Curvature ◽  
Convex Sets ◽  
Convex Geometry ◽  
Type Inequality ◽  
Model Space ◽  
Functional Inequalities ◽  
Curvature Bounds ◽  
Strongly Convex ◽  
Weighted Ricci Curvature ◽  
The One

Abstract In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

scholarly journals Parametric study of hyperfine structure of Zr II even-parity levels

2016 ◽  
Vol 94 (12) ◽  
pp. 1310-1313 ◽  
Author(s):  
Safa Bouazza
Keyword(s):  
Hyperfine Structure ◽  
Ion Beam ◽  
Model Space ◽  
Laser Fluorescence ◽  
Beam Laser ◽  
Extended Space ◽  
Experimental Values ◽  
The One ◽  
Fast Ion ◽  
Laser Fluorescence Spectroscopy

Until now experimental hyperfine structure (hfs) data of 12 even-parity Zr II levels were given in the literature. Recently new hyperfine splitting measurements of 11 other Zr II levels, of the same parity are achieved, applying fast-ion-beam laser-fluorescence spectroscopy. The hfs of these 23 gathered levels has been analysed by simultaneous parametrisation of the one- and two-body interactions, first in model space (4d + 5s)3 and secondly in extended space. For the three lowest configurations, radial parameters of the magnetic dipole A and quadrupole electric B factors are deduced in their entirety for 91Zr II, compared and discussed with calculated values, available in the literature, and also with ours, computed by means of the ab initio method. For instance we give the main experimental values of the extracted single-electron hfs parameters of 4d25s: [Formula: see text] = –2701 MHz, [Formula: see text] = –122.4 MHz, and [Formula: see text] = –113.5 MHz.

scholarly journals Smoothing Riemannian metrics with Ricci curvature bounds

1996 ◽  
Vol 90 (1) ◽  
pp. 49-61 ◽  
Author(s):  
Xianzhe Dai ◽  
Guofang Wei ◽  
Rugang Ye
Keyword(s):  
Ricci Curvature ◽  
Riemannian Metrics ◽  
Curvature Bounds

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

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