Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Riemannian Manifolds ◽  
Best Constants ◽  
Hyperbolic Spaces ◽  
High Dimensional ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Trudinger Inequality

AbstractThough there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ

HARDY INEQUALITIES ON RIEMANNIAN MANIFOLDS WITH NEGATIVE CURVATURE

2014 ◽  
Vol 16 (02) ◽  
pp. 1350043 ◽  
Author(s):  
QIAOHUA YANG ◽  
DAN SU ◽  
YINYING KONG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Distance ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Hardy Inequalities ◽  
Simply Connected

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the LpHardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Embedding Heterogeneous Information Network in Hyperbolic Spaces

2022 ◽  
Vol 16 (2) ◽  
pp. 1-23
Author(s):  
Yiding Zhang ◽  
Xiao Wang ◽  
Nian Liu ◽  
Chuan Shi
Keyword(s):  
Power Law ◽  
Hyperbolic Spaces ◽  
Hyperbolic Distance ◽  
Information Network ◽  
Extended Model ◽  
Optimization Strategy ◽  
Euclidean Spaces ◽  
Research Attention ◽  
Heterogeneous Information Network ◽  
Heterogeneous Information

Heterogeneous information network (HIN) embedding, aiming to project HIN into a low-dimensional space, has attracted considerable research attention. Most of the existing HIN embedding methods focus on preserving the inherent network structure and semantic correlations in Euclidean spaces. However, one fundamental problem is whether the Euclidean spaces are the intrinsic spaces of HIN? Recent researches find the complex network with hyperbolic geometry can naturally reflect some properties, e.g., hierarchical and power-law structure. In this article, we make an effort toward embedding HIN in hyperbolic spaces. We analyze the structures of three HINs and discover some properties, e.g., the power-law distribution, also exist in HINs. Therefore, we propose a novel HIN embedding model HHNE. Specifically, to capture the structure and semantic relations between nodes, HHNE employs the meta-path guided random walk to sample the sequences for each node. Then HHNE exploits the hyperbolic distance as the proximity measurement. We also derive an effective optimization strategy to update the hyperbolic embeddings iteratively. Since HHNE optimizes different relations in a single space, we further propose the extended model HHNE++. HHNE++ models different relations in different spaces, which enables it to learn complex interactions in HINs. The optimization strategy of HHNE++ is also derived to update the parameters of HHNE++ in a principle manner. The experimental results demonstrate the effectiveness of our proposed models.

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

scholarly journals Reflection-Like Maps in High-Dimensional Euclidean Space

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 872
Author(s):  
Zhiyong Huang ◽  
Baokui Li
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Orthogonal Transformation ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Domain ◽  
Euclidean Spaces ◽  
Line Cross ◽  
Klein Model

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

scholarly journals Towards sharp Bohnenblust–Hille constants

2018 ◽  
Vol 20 (03) ◽  
pp. 1750029 ◽  
Author(s):  
Daniel Pellegrino ◽  
Eduardo V. Teixeira
Keyword(s):  
Exponential Growth ◽  
Optimization Problems ◽  
Best Constants ◽  
Sharp Constants ◽  
Linear Forms ◽  
Entropy Formula ◽  
Optimal Constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

2010 ◽  
Vol 88 (1) ◽  
pp. 19-27 ◽  
Author(s):  
SORINA BARZA ◽  
JAVIER SORIA
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Best Constants ◽  
Sharp Constants ◽  
Dual Norm ◽  
Equivalent Norms ◽  
Weighted Lorentz Spaces ◽  
Weighted Lorentz Space

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

Publisher's Note: Packing hyperspheres in high-dimensional Euclidean spaces [Phys. Rev. E74, 041127 (2006)]

2007 ◽  
Vol 75 (2) ◽  
Author(s):  
Monica Skoge ◽  
Aleksandar Donev ◽  
Frank H. Stillinger ◽  
Salvatore Torquato
Keyword(s):  
High Dimensional ◽  
Euclidean Spaces
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