The parallel complexity class NC^1 has many equivalent models such as
polynomial size formulae and bounded width branching
programs. Caussinus et al. \cite{CMTV} considered arithmetizations of
two of these classes, #NC^1 and #BWBP. We further this study to
include arithmetization of other classes. In particular, we show that
counting paths ... more >>>

We give an explicit (in particular, deterministic polynomial time)
construction of subspaces $X
\subseteq \R^N$ of dimension $(1-o(1))N$ such that for every $x \in X$,
$$(\log N)^{-O(\log\log\log N)} \sqrt{N}\, \|x\|_2 \leq \|x\|_1 \leq \sqrt{N}\, \|x\|_2.$$
If we are allowed to use $N^{1/\log\log N}\leq N^{o(1)}$ random bits
and ... more >>>

We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

1. Noise-resistant. A syntactically multilinear ... more >>>