scholarly journals New Riemannian manifolds with Lp-unbounded Riesz transform for p > 2

2020 ◽  
Author(s):  
Alex Amenta
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Fractal Structure ◽  
Arbitrary Dimension ◽  
Riesz Transform

Abstract We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $$L^p(M)$$ L p ( M ) for all $$p > 2$$ p > 2 . This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.

scholarly journals Alternative Approach to Infinity

1974 ◽  
Vol 64 ◽  
pp. 99-99
Author(s):  
Peter G. Bergmann
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Quotient Space ◽  
Three Dimensional ◽  
Cauchy Data ◽  
Space Time ◽  
Equivalence Classes ◽  
Null Infinity ◽  
Alternative Approach ◽  
Dimensional Domain

Following Penrose's construction of space-time infinity by means of a conformal construction, in which null-infinity is a three-dimensional domain, whereas time- and space-infinities are points, Geroch has recently endowed space-infinity with a somewhat richer structure. An approach that might work with a large class of pseudo-Riemannian manifolds is to induce a topology on the set of all geodesics (whether complete or incomplete) by subjecting their Cauchy data to (small) displacements in space-time and Lorentz rotations, and to group the geodesics all of whose neighborhoods intersect into equivalence classes. The quotient space of geodesics over equivalence classes is to represent infinity. In the case of Minkowski, null-infinity has the usual structure, but I0, I+, and I- each become three-dimensional as well.

Percolation on Penrose Tilings

1998 ◽  
Vol 41 (2) ◽  
pp. 166-177 ◽  
Author(s):  
A. Hof
Keyword(s):  
Large Class ◽  
Arbitrary Dimension ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Aperiodic Tilings ◽  
Infinite Clusters ◽  
Almost All ◽  
Percolation Processes ◽  
Penrose Tilings

AbstractIn Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ℤd, and to other percolation processes, including Bernoulli bond percolation.

Riesz transform and related inequalities on non-compact Riemannian manifolds

2003 ◽  
Vol 56 (12) ◽  
pp. 1728-1751 ◽  
Author(s):  
Thierry Coulhon ◽  
Xuan Thinh Duong
Keyword(s):  
Riemannian Manifolds ◽  
Riesz Transform

QUASI-REGULAR DIRICHLET FORMS ON FREE RIEMANNIAN PATH SPACES

2009 ◽  
Vol 12 (02) ◽  
pp. 251-267 ◽  
Author(s):  
FENG-YU WANG ◽  
BO WU
Keyword(s):  
Large Class ◽  
Free Path ◽  
Riemannian Manifolds ◽  
Diffusion Coefficients ◽  
Dirichlet Forms ◽  
Integration By Parts ◽  
Constant Diffusion ◽  
Integration By Parts Formula ◽  
Noncompact Riemannian Manifolds ◽  
Initial Distributions

The integration by parts formula on free path spaces over noncompact Riemannian manifolds is established for initial distributions with densities in [Formula: see text]. As an application, a large class of Dirichlet forms with (unbounded and non-constant) diffusion coefficients are constructed on free Riemannian path spaces, which are quasi-regular under mild curvature conditions.

scholarly journals Stable Configurations of Repelling Points on Flat Tori

2019 ◽  
Vol 14 (2) ◽  
pp. 87-102
Author(s):  
Marina Nechayeva ◽  
Burton Randol
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Final Section ◽  
Hyperbolic Manifolds ◽  
Analytic Approach ◽  
Flat Tori

AbstractFlat tori are analyzed in the context of an intrinsic Fourier-analytic approach to electrostatics on Riemannian manifolds, introduced by one of the authors in 1984 and previously developed for compact hyperbolic manifolds. The approach covers a large class of repelling laws, but does not naturally include laws with singularities at the origin, for which possible accommodations are discussed in the final section of the paper.

scholarly journals A relation between biharmonic Green’s functions of simply supported and clamped bodies

1976 ◽  
Vol 61 ◽  
pp. 59-71 ◽  
Author(s):  
James Ralston ◽  
Leo Sario
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elastic Plate ◽  
Boundary Data ◽  
Arbitrary Dimension ◽  
Point Load ◽  
Classification Theory ◽  
Thin Elastic Plate ◽  
Sufficient Condition ◽  
Simply Supported

The deflection, under a point load, of a thin elastic plate clamped at the edges is the biharmonic Green’s function β with the boundary data β = ∂β/∂n = 0. If the boundary of the region is reasonably smooth, the construction of β offers no difficulty. In contrast, nothing is known about the existence of β in the general case. The purpose of our study is to give a sufficient condition for the existence of β on a given Riemannian manifold of arbitrary dimension and to construct β. Our results will have, apart from their physical meaning in elasticity, consequences in the biharmonic classification theory of Riemannian manifolds.

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