In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation o...

Based on some identities for the determinant of completely positive maps of rank two, concurrences are calculated or estimated from below.

Let V be a vector space over K. For each nonnegative integer k let Pk(V ) be the set of k-subspaces of V . For positive d (≤ dimV ), let Ld(V ) be the set of pairs (U,W ) with U ∈ Pd−1(V ), W ∈ Pd+1(V ), and U ≤ W . We also set B− d (V ) = Pd−1(V ), B + d (V ) = Pd+1(V ), and Bd(V ) = B − d (V ) ∪ B + d (V ). In the case V = K and d = 2, we can view P2(V ) and L2(V ) as the points and lines of ...

We study the geometry underlying the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebrai...

It is known that a geometry with rank r and no minor isomorphic to the (q + 2)-point line has at most (qr − 1)/(q − 1) points, with strictly fewer points if r > 3 and q is not a prime power. For q not a prime power and r > 3, we show that qr−1 − 1 is an upper bound. For q a prime power and r > 3, we show that any rank-r geometry with at least qr−1 points and no (q + 2)-point-line minor is repre...

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank target is known in advance and this information can be useful process. paper, first, we revisit recently proposed rank-based heuristic for “known-rank” establish condition under which generated sequence quasi-Fejér convergent solution set. Then, by including accelerati...

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 ) <...