Narrow Results

The first-order and the second-order directional derivatives of singular values are used to characterize the tangent cone, the normal cone and the second-order tangent set of the epigraph of the nuclear norm of matrices. Based on the variational geometry of the epigraph, the no gap second-order optimality conditions for the optimization problem, whose constraint is defined by the matrix cone induced by the nuclear norm, are established.

There has been a discrepancy between values of the pion-nucleon sigma term extracted by two different methods for many years. Analysis of recent high precision pion-nucleon data has widened the gap between the two determinations. It is argued that the two extractions correspond to different quantities and that the difference between them can be understood and calculated.

Dispersion relations along interior hyperbolas and a set of hyperbolas passing through the Cheng-Dashen point are used to calculate the pion-nucleon sigma term. The t-channel input is updated using the recent GWU partial wave solution and ππ phase shifts from calculations based on Roy equations. Obtained values for the sigma term are still within the error bars of the previous Karlsruhe result.