Multivariate Tight Wavelet Frames with Few Generators and High Vanishing Moments.
Tight wavelet frames are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with tight wavelet frame construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to tight wavelet frames with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline tight wavelet frame constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity.
We study in this paper the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the 'entangling' part of the interaction. We then specialize to a spin chain with nearest-neighbour interactions and numerically find a new type of quantum-phase transition related to the strength of a random external field, i.e. the time-reversal-breaking one-body interaction term.