Reflection Positivity and Monotonicity

ar X iv : 0 70 5 . 07 12 v 2 [ m at h - ph ] 9 M ay 2 00 7 REFLECTION POSITIVITY AND MONOTONICITY

We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

ar X iv : 0 70 5 . 07 12 v 1 [ m at h - ph ] 4 M ay 2 00 7 REFLECTION POSITIVITY AND MONOTONICITY

We prove the most general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

On Testing Positivity/Monotonicity/Convexity of nonparametric Signals

Reflection Positivity for Majoranas

We establish reflection positivity for Gibbs trace states defined by a certain class of Hamiltonians that describe the interaction of Majoranas on a lattice. These Hamiltonians may include many-body interactions, as long as the signs of the associated coupling constants satisfy certain restrictions. We show that reflection positivity holds on an even subalgebra of Majoranas.

Tensor-product monotonicity preservation

The preservation of surface shape is important in geometric modelling and approximation theory. In geometric modelling one would like to manipulate the shape of the surface being modelled by the simpler task of manipulating the control points. In scattered data approximation one might wish to approximate bivariate data sampled from a function with a shape property by a spline function sharing t...