48,474 research outputs found
A strong form of almost differentiability
We present a uniformization of Reeken's macroscopic differentiability (see [5]), discuss its relations to uniform differentiability (see [6]) and classical continuous differentiability, prove the corresponding chain rule, Taylor's theorem, mean value theorem, and inverse mapping theorem. An attempt to compare it with the observability (see [1, 4]) is made too. © 2009 Springer Science+Business Media, Inc.CEOCFCTFEDER/POCT
Randomness and differentiability in higher dimensions
We present two theorems concerned with algorithmic randomness and
differentiability of functions of several variables. Firstly, we prove an
effective form of the Rademacher's Theorem: we show that computable randomness
implies differentiability of computable Lipschitz functions of several
variables. Secondly, we show that weak 2-randomness is equivalent to
differentiability of computable a.e. differentiable functions of several
variables.Comment: 19 page
A note on Hadamard differentiability and differentiability in quadratic mean
We proof that Hadamard differentiability in addition with usual assumptions on the loss function for M estimates implies differentiability in quadratic mean. Thus both concepts are exchangeable. --Hadamard differential,Differentiability in quadratic mean
The Lip-lip equality is stable under blow-up
We show that at generic points blow-ups/tangents of differentiability spaces
are still differentiability spaces; this implies that an analytic condition
introduced by Keith as an inequality (and later proved to actually be an
equality) passes to tangents. As an application, we characterize the -weak
gradient on iterated blow-ups of differentiability spaces.Comment: minor corrections: change of normalization of the measures; The final
version will appear in Calc. Var. PD
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