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We prove asymptotic completeness for operators of the form on , , where is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials , (if , then we require ), as well as real-valued potentials satisfying a global Kato condition. The class of admissible perturbations also contains first-order differential operators of the form for suitable vector potentials . Our main technical statement is a new limiting absorption principle, which we prove using techniques from harmonic analysis related to the Stein-Tomas restriction theorem
If is a finitely generated group with generators , then an infinite-order element is a distortion element of provided that where is the word length of in the generators. Let be a closed orientable surface, and let denote the identity component of the group of -diffeomorphisms of . Our main result shows that if has genus at least two and that if is a distortion element in some finitely generated subgroup of , then for every -invariant Borel probability measure . Related results are proved for or . For a Borel probability measure on , denote the group of -diffeomorphisms that preserve by . We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to with infinite image
We construct a lifting from Siegel cusp forms of degree to Siegel cusp forms of degree . For , our result is a partial solution of a conjecture made by Miyawaki [27, page 307] in 1992. In particular, we can calculate the standard -function of a cusp form of degree 3 and weight 12, which is in accordance with Miyawaki's conjecture. We give a conjecture on the Petersson inner product of the lifting in terms of certain -values
In this article we study the connection between dimers and Harnack curves discovered in . We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions
We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative -spaces with . One of our results states that given a map , where (, being a von Neumann algebra), is completely bounded if and only if factors through a direct sum of a -column space and a -row space. We also obtain several operator-space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative -space () with values in a -column space for every ( being the index conjugate to ). These results are the -space analogues of the recent works on the operator-space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine-type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup-type tensor norm that turns out to be particularly fruitful when applied to subspaces of noncommutative -spaces (). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative -spaces, is equal to the factorization norm through a -row space
It has been known for a long time that a nonsingular real algebraic curve of degree in the projective plane cannot have more than even ovals. We show here that this upper bound is asymptotically sharp; that is to say, we construct a family of curves of degree such that , where is the number of even ovals of the curves. We also show that the same kind of result is valid when dealing with odd ovals