Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Applications of computability theory to prime models and differential geometry
Applications of computability theory to prime models and differential geometry
Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space
Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space
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To each computable enumerable (c.e.) set Awith aparticular enumeration {As}sεω,there is associated a settling function mA(x), where mA(x) is the last stage when a numberless than or equal to xwas enumerated into A. In[7], R.W. Robinson classified the complexity of c.e. sets into twogroups of complexity based on whether or not the settling functionwas dominant. An extension of this idea to a more refined orderingof c.e. sets was first introduced by Nabutovsky and Weinberger in[6] and Soare [9], for application to differential geometry. Therethey defined one c.e. set Ato settling time dominateanother c.e. set B(B stA) if for every computable function f, for allbut finitely many x, mB(x) f(mA(x)). In [4] Csima and Soareintroduced a stronger ordering, where BsstAif for all computable fandg, for almost all x, mB(x) f(mA(g(x))). We give a survey ofthe known results about these orderings, make some observations,and outline the open questions.