The recently developed theory of wavelets has a remarkable property to \zoom in" on very short lived frequency phenomena, such as transients in signals and singularities in functions, and hence provides an ideal tool to study localized changes. This paper proposes a wavelet method to estimate jump and sharp cusp curves of a function in the plane. The method is rst to compute wavelet transformat...

We provide a uniform vanishing result for the graded components of finite length Koszul module associated to subspace K⊆∧2V as well sharp upper bound its Hilbert function. This purely algebraic statement has interesting applications study number invariants finitely generated groups, such Alexander invariants, Chen ranks, and degree growth virtual nilpotency class. For instance, we explicitly af...