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Register a new userRemarks on Hilbert identities, isometric embeddings, and invariant cubature
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Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoirs method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small indices with few points are provided, which are used to update Shatalovs table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.
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In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the Moller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral.
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 $R^{4, 1}$.
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 $0<delta leq 1/sqrt{log n}$.
J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
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.
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