We investigate the depth overhead of stretch-amplification for pseudorandom generators in the black-box model. We show that any black box construction of a PRG $H: \{0,1\}^m \to \{0,1\}^M$ which makes $\mathrm{poly}(n)$ queries to a PRG $G: \{0,1\}^n \to \{0,1\}^N$ in $d$ adaptive phases during each evaluation (and with a black-box ... more >>>

We investigate the behavior of the non-negative rank of a matrix, $\hbox{rk}_+$, on Kronecker product, $\times$. Beasley et al. [2013] constructed matrices with $\hbox{rk}_+(M\times M)$ strictly smaller than $\hbox{rk}_+(M)^2$.
We will show that i) there exists an $M$ with $\hbox{rk}_+(M\times M)\leq O(\hbox{rk}_+(M)\log(\hbox{rk}_+(M))$.

Defining asymptotic non-negative rank as $\overline{\hbox{rk}}_+(M):= \lim_{k\rightarrow ... more >>>

We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant ... more >>>