3,396 research outputs found
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
Effective 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.Comment: Revision: added sections 6-8; minor correction
Fractional Calculus as a Macroscopic Manifestation of Randomness
We generalize the method of Van Hove so as to deal with the case of
non-ordinary statistical mechanics, that being phenomena with no time-scale
separation. We show that in the case of ordinary statistical mechanics, even if
the adoption of the Van Hove method imposes randomness upon Hamiltonian
dynamics, the resulting statistical process is described using normal calculus
techniques. On the other hand, in the case where there is no time-scale
separation, this generalized version of Van Hove's method not only imposes
randomness upon the microscopic dynamics, but it also transmits randomness to
the macroscopic level. As a result, the correct description of macroscopic
dynamics has to be expressed in terms of the fractional calculus.Comment: 20 pages, 1 figur
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