Polynomials on Banach Spaces: Zeros and Maximal Points

AbstractA maximal point of a polynomial on a Banach space is a point in the unit ball at which the polynomial attains its norm. Lower bounds are given for the distances between zeros and maximal points

Unbounded violation of tripartite Bell inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized GHZ states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not lead to extremal quantum correlations. The results are based on tools from the theories of operator spaces and tensor norms which we exploit to prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras.Comment: Substantial changes in the presentation to make the paper more accessible for a non-specialized reade

The generalized Rademacher functions

In [A & G], the authors introuced the so-called generalized Rademacher functions and used the to prove that every continuous multilinear form A : $c_0$ x ... x$c_0$ $\longrightarrow$ $C$

Array CGH identifies reciprocal 16p13.1 duplications and deletions that predispose to autism and/or mental retardation

Autism and mental retardation (MR) are often associated, suggesting that these conditions are etiologically related. Recently, array-based comparative genomic hybridization (array CGH) has identified submicroscopic deletions and duplications as a common cause of MR, prompting us to search for such genomic imbalances in autism. Here we describe a 1.5-Mb duplication on chromosome 16p13.1 that was found by high-resolution array CGH in four severe autistic male patients from three unrelated families. The same duplication was identified in several variably affected and unaffected relatives. A deletion of the same interval was detected in three unrelated patients with MR and other clinical abnormalities. In one patient we revealed a further rearrangement of the 16p13 imbalance that was not present in his unaffected mother. Duplications and deletions of this 1.5-Mb interval have not been described as copy number variants in the Database of Genomic Variants and have not been identified in >600 individuals from other cohorts examined by high-resolution array CGH in our laboratory. Thus we conclude that these aberrations represent recurrent genomic imbalances which predispose to autism and/or MR