We show that, given an n-dimensional normed space X, a sequence of $N={(8/\epsilon )}^{2n}$ independent random vectors ${\left(X_i\right)}_{i=1}^N$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $\Gamma :ℝ\to {ℝ}^N$ defined by $\Gamma x={(⟨x,X_i⟩)}_{i=1}^N$ embeds X in ${\ell }_{\infty }^N$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ${\ell }_{\infty }^N$ with asymptotically best possible relation between N, n, and ε.

We consider a variation of the standard Hastings–Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations...

Microscopic prolate spheroids in a given volume of an opaque material are considered. The extremes of the shape factor of the spheroids are studied. The profiles of the spheroids are observed on a random planar section and based on these observations we want to estimate the distribution of the extremal shape factor of the spheroids. We show that under a tail uniformity condition the Maximum domain of attraction is stable. We discuss the normalising constants (n.c.) for the extremes of the spheroid...

Acute triangles are defined by having all angles less than $\pi /2$, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n\ge 3$, acuteness is defined by demanding that all dihedral angles between $(n-1)$-dimensional faces are smaller than $\pi /2$. However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness...

In this paper, we study the size of the giant component $C_G$ in the random geometric graph $G=G(n,r_n,f)$ of $n$ nodes independently distributed each according to a certain density $f(·)$ in ${[0,1]}^2$ satisfying ${inf}_{x\in {[0,1]}^2}f\left(x\right)gt;$$0$. If $\frac{c_1}{n}\le r_n^2\le c_2\frac{logn}{n}$ for some positive constants $c_1$, $c_2$ and $n{r}_n^2⟶\infty$ as $n\to \infty$, we show that the giant component of $G$ contains at least $n-\mathrm{o}\left(n\right)$ nodes with probability at least $1-{\mathrm{e}}^{-\beta n{r}_n^2}$ for all $n$ and for some positive constant $\beta$....