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A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved.
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This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $mathbb{H}^n$, for $ngeq 2$. For that purpose, the kissing number is replaced by the kissing function $kappa(n, r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $kappa(n, r)$, we study their asymptotic behaviour and show, in particular, that $lim_{rto infty} frac{log kappa(n,r)}{r} = n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.
We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without $2$-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.
In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is called the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential $chi^2$ testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.
We establish sharp Hardy-Adams inequalities on hyperbolic space $mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $alpha>0$ there exists a constant $C_{alpha}>0$ such that [ int_{mathbb{B}^{4}}(e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2})dV=16int_{mathbb{B}^{4}}frac{e^{32pi^{2} u^{2}}-1-32pi^{2} u^{2}}{(1-|x|^{2})^{4}}dxleq C_{alpha}. ] for any $uin C^{infty}_{0}(mathbb{B}^{4})$ with [ int_{mathbb{B}^{4}}left(-Delta_{mathbb{H}}-frac{9}{4}right)(-Delta_{mathbb{H}}+alpha)ucdot udVleq1. ] As applications, we obtain a sharpened Adams inequality on hyperbolic space $mathbb{B}^{4}$ and an inequality which improves the classical Adams inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.
We make a brief historical review to the moment model reduction to the kinetic equations, particularly the Grads moment method for Boltzmann equation. The focus is on the hyperbolicity of the reduced model, which is essential to the existence of its classical solution as a Cauchy problem. The theory of the framework we developed in last years is then introduced, which may preserve the hyperbolic nature of the kinetic equations with high universality. Some lastest progress on the comparison between models with/without hyperbolicity is presented to validate the hyperbolic moment models for rarefied gases.
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