A Σ3 Boolean circuit has 3 levels of gates. The input level is comprised of OR gates each taking as inputs 2, not necessarily distinct, literals. Each of these OR’s feeds one or more AND gates at the second level. Their outputs form the inputs to a single OR gate at the output level. Using the projection technique of Paturi, Saks, and Zane, it is shown that the smallest Σ3 Boolean circuit testi...

A U(2)L×U(2)R chiral theory of pseudoscalar, vector, and axial-vector mesons has been proposed. VMD has been revealed from this theory. The physical processes of normal parity and abnormal parity have been studied by using the same lagrangian and the universality of coupling has been revealed. Two new mass relations between vector and axial-vector mesons have been found. Weinberg’s first sum ru...

For a subset A of a finite abelian group G we define Σ(A) = {∑a∈B a : B ⊂ A}. In the case that Σ(A) has trivial stabiliser, one may deduce that the size of Σ(A) is at least quadratic in |A|; the bound |Σ(A)| ≥ |A|2/64 has recently been obtained by De Vos, Goddyn, Mohar and Šámal [2]. We improve this bound to the asymptotically best possible result |Σ(A)| ≥ (1/4− o(1))|A|2. We also study a relat...