A counterexample in a unique continuation problem

Unique continuation for the elasticity system and a counterexample for second order elliptic systems

In this paper we study the global unique continuation property for the elasticity system and the general second order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand, for the anisotropic system, if the coefficients...

Consistency of a counterexample to Naimark's problem.

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assumin...

The Minimum Rank Problem: a counterexample

We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rat...

Unique Continuation and an Inverse Problem for Hyperbolic Equations across a General Hypersurface

For a hyperbolic equation p(x, t)∂ t u(x, t) = ∆u(x, t)+ ∑n j=1 qj(x, t)∂ju+ qn+1(x, t)∂tu+r(x, t)u in R n×R with p ∈ C and q1, ...., qn+1, r ∈ L∞, we consider the unique continuation and an inverse problem across a non-convex hypersurface Γ. Let Γ be a part of the boundary of a domain and let ν(x) be the inward unit normal vector to Γ at x. Then we prove the unique continuation near a point x0...

Strong Unique Continuation for Solutions of a p(x)-Laplacian Problem