A Darboux-type theorem for slowly varying functions

AbstractFor functionsg(z) satisfying a slowly varying condition in the complex plane, we find asymptotics for the Taylor coefficients of the function[formula]whenα>0. As applications we find asymptotics for the number of permutations with cycle lengths all lying in a given setS, and for the number having unique cycle lengths

Emergent Gravity from Noncommutative Spacetime

We showed before that self-dual electromagnetism in noncommutative (NC) spacetime is equivalent to self-dual Einstein gravity. This result implies a striking picture about gravity: Gravity can emerge from electromagnetism in NC spacetime. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy spacetime. We elucidate in some detail why electromagnetism in NC spacetime should be a theory of gravity. In particular, we show that NC electromagnetism is realized through the Darboux theorem as a diffeomorphism symmetry G which is spontaneously broken to symplectomorphism H due to a background symplectic two-form $B_{\mu\nu}=(1/\theta)_{\mu\nu}$, giving rise to NC spacetime. This leads to a natural speculation that the emergent gravity from NC electromagnetism corresponds to a nonlinear realization G/H of the diffeomorphism group, more generally its NC deformation. We also find some evidences that the emergent gravity contains the structure of generalized complex geometry and NC gravity. To illuminate the emergent gravity, we illustrate how self-dual NC electromagnetism nicely fits with the twistor space describing curved self-dual spacetime. We also discuss derivative corrections of Seiberg-Witten map which give rise to higher order gravity.Comment: 50 pages; Cosmetic revision and updated reference

Beurling slow and regular variation

We give a new theory of Beurling regular variation ( Part II). This includes the previously known theory of Beurling slow variation ( Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory