Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userColumn randomization and almost-isometric embeddings
69
0
0.0
(
0
)
Authors
Shahar Mendelson
Ask ChatGPT about the research
No Arabic abstract
The matrix $A:mathbb{R}^n to mathbb{R}^m$ is $(delta,k)$-regular if for any $k$-sparse vector $x$, $$ left| |Ax|_2^2-|x|_2^2right| leq delta sqrt{k} |x|_2^2. $$ We show that if $A$ is $(delta,k)$-regular for $1 leq k leq 1/delta^2$, then by multiplying the columns of $A$ by independent random signs, the resulting random ensemble $A_epsilon$ acts on an arbitrary subset $T subset mathbb{R}^n$ (almost) as if it were gaussian, and with the optimal probability estimate: if $ell_*(T)$ is the gaussian mean-width of $T$ and $d_T=sup_{t in T} |t|_2$, then with probability at least $1-2exp(-c(ell_*(T)/d_T)^2)$, $$ sup_{t in T} left| |A_epsilon t|_2^2-|t|_2^2 right| leq Cleft(Lambda d_T deltaell_*(T)+(delta ell_*(T))^2 right), $$ where $Lambda=max{1,delta^2log(ndelta^2)}$. This estimate is optimal for $0<delta leq 1/sqrt{log n}$.
rate research
Read More
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penroses theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metric agrees with the Unruh temperature measured by an observer moving along hyperbolae in $R^{4, 1}$.
We consider the class $Psi_d$ of continuous functions $psi colon [0,pi] to mathbb{R}$, with $psi(0)=1$ such that the associated isotropic kernel $C(xi,eta)= psi(theta(xi,eta))$ ---with $xi,eta in mathbb{S}^d$ and $theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $Psi_d$ as combinations of $1$-Schoenberg coefficients. We also give expressions for the computation of Schoenberg coefficients of the exponential and Askey families for all even dimensions through recurrence formula. Problem 3 regards the curvature at the origin of members of $Psi_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along pi_1-injective boundary components with amenable fundamental group.
We construct isometric and conformally isometric embeddings of some gravitational instantons in $mathbb{R}^8$ and $mathbb{R}^7$. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to $3$. For $mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
