Approximate Real Symmetric Tensor Rank

We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.Comment: Fixed few typos and error in writing of Algorithm 1. To appear in Arnold Mathematical Journa

ψ α-Estimates for Marginals of Log-Concave Probability Measures

We show that a random marginal π F (μ) of an isotropic log-concave probability measure μ on R n exhibits better ψ α-behavior. For a natural variant ψ' α of the standard ψ α-norm we show the following: (i) If k ≤√n, then for a random F ε G n,k we have that π F (μ) is a ψ' 2- measure. We complement this result by showing that a random π F (μ) is, at the same time, super-Gaussian. (ii) If k = n δ, 1/2 < δ < 1, then for a random F ε G n,k we have that π F (μ) is a ψ' α(δ)-measure, where α(δ) = 2δ/3δ-1. © 2011 American Mathematical Society

On the distribution of the ψ2-norm of linear functionals on isotropic convex bodies

It is known that every isotropic convex body K in Rn has a subgaussian direction with constant This follows from the upper bound for the volume of the body Ψ 2(K) with support function . The approach in all the related works does not provide estimates on the measure of directions satisfying a ψ2-estimate with a given constant r. We introduce the function and we discuss lower bounds for ψ K (t), Information on the distribution of the ψ2-norm of linear functionals is closely related to the problem of bounding from above the mean width of isotropic convex bodies. © 2012 Springer-Verlag Berlin Heidelberg

Procedural volume and outcomes with radial or femoral access for coronary angiography and intervention

Abstract not availableSanjit S. Jolly, John Cairns, Salim Yusuf, Kari Niemela, Philippe Gabriel Steg, Matthew Worthley, Emile Ferrari, Warren J. Cantor, Anthony Fung, Nicholas Valettas, Michael Rokoss, Goran K. Olivecrona, Petr Widimsky, Asim N. Cheema, Peggy Gao, Shamir R. Mehta for the RIVAL Investigator