منابع مشابه
Randomness and Differentiability
We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) We show that a real number z ∈ [0, 1] is computably random if and only if every nondecreasing computable function [0, 1] → R is differentiable at z. (2) A real number z ∈ [0, 1] is weakly 2-random if and only if every almost everywhere differentiable computable function [0, 1] → R is diff...
متن کاملRandomness and Differentiability (long Version)
We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a real number z ∈ [0, 1] is weakly 2-random if and only if each almost everywhere differentiable computa...
متن کاملFréchet directional differentiability and Fréchet differentiability
Zaj́ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvalued function on a normed linear space, the se...
متن کاملEffective genericity and differentiability
We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x . This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions. 2010 Mathematics Subject Classification 03E15; 03F60 (p...
متن کاملEfficiency and Hadamard Differentiability
Suppose an estimator sequence Tn. is asymptotically efficient for a. • smooth functional 1'1;. Then a Hadamard differentia.ble functional rf> applied to Tn is efficient for i f> 0 11':. More generally, Hadamard dif-ferentiability ma.y be used to esta.blish asymptotic efficiency. Three non-trivial applications are discussed: random right censoring, random truncation and estima.ting a. point symm...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2015
ISSN: 0002-9947,1088-6850
DOI: 10.1090/tran/6484