The positive semidefinite rank of a nonnegative (m×n)-matrix S is the minimum number q such that there exist positive semidefinite (q × q)-matrices A1, . . . , Am, B1, . . . , Bn such that S(k, l) = trA∗kBl. The most important lower bound technique on nonnegative rank only uses the zero/nonzero pattern of the matrix. We characterize the power of lower bounds on positive semidefinite rank based ...

The problem of finding mr(F,G) and describing Gk(F ) has recently attracted considerable attention, particularly for the case in which F = R (see [29, 17, 26, 25, 27, 13, 33, 5, 9, 22, 2, 11, 6, 7, 10, 18, 4]). The minimum rank problem over R is a sub-problem of a much more general problem, the inverse eigenvalue problem for symmetric matrices: given a family of real numbers, find every symmetr...

Abstract It is shown that the relative distance in Frobenius norm of a real symmetric order- d tensor rank-two to its best rank-one approximation upper bounded by $$\sqrt{1-(1-1/d)^{d-1}}$$ 1 - ( / d ) <...

A bstract We study the Higgs branches of five-dimensional $$ \mathcal{N} N = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid’s pagodas, and Laufer’s examples. Our approach consists in reducing to IIA with D6-branes O6-planes, computing open-string s...