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We give an explicit construction of an ??-biased set over k bits of size O(k/??2 log(1/??))5/4This improves upon previous explicit constructions when e is roughly (ignoring logarithmic factors) in the range [k-1.5,k-0.5]. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus. Studying...
We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most O(k3 log d). The previous best rank bound known was 2O(k2)(log d)k-2 by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Omega(k log d)). Our rank bound significantly improves (dependence...
Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks. These models are a special form of compartmental systems, which involve mass- and energy-balance relations. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent...
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