2,521 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Local stability of ergodic averages
The mean ergodic theorem is equivalent to the assertion that for every
function K and every epsilon, there is an n with the property that the ergodic
averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show
that even though it is not generally possible to compute a bound on the rate of
convergence of a sequence of ergodic averages, one can give explicit bounds on
n in terms of K and || f || / epsilon. This tells us how far one has to search
to find an n so that the ergodic averages are "locally stable" on a large
interval. We use these bounds to obtain a similarly explicit version of the
pointwise ergodic theorem, and show that our bounds are qualitatively different
from ones that can be obtained using upcrossing inequalities due to Bishop and
Ivanov. Finally, we explain how our positive results can be viewed as an
application of a body of general proof-theoretic methods falling under the
heading of "proof mining."Comment: Minor errors corrected. To appear in Transactions of the AM
Universal Coding and Prediction on Martin-L\"of Random Points
We perform an effectivization of classical results concerning universal
coding and prediction for stationary ergodic processes over an arbitrary finite
alphabet. That is, we lift the well-known almost sure statements to statements
about Martin-L\"of random sequences. Most of this work is quite mechanical but,
by the way, we complete a result of Ryabko from 2008 by showing that each
universal probability measure in the sense of universal coding induces a
universal predictor in the prequential sense. Surprisingly, the effectivization
of this implication holds true provided the universal measure does not ascribe
too low conditional probabilities to individual symbols. As an example, we show
that the Prediction by Partial Matching (PPM) measure satisfies this
requirement. In the almost sure setting, the requirement is superfluous.Comment: 12 page
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear
operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n
x. We prove the following variational inequality in the case where T is power
bounded from above and below: for any increasing sequence (t_k)_{k in N} of
natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p,
where the constant C depends only on p and the modulus of uniform convexity.
For T a nonexpansive operator, we obtain a weaker bound on the number of
epsilon-fluctuations in the sequence. We clarify the relationship between
bounds on the number of epsilon-fluctuations in a sequence and bounds on the
rate of metastability, and provide lower bounds on the rate of metastability
that show that our main result is sharp
Physical Properties of Quantum Field Theory Measures
Well known methods of measure theory on infinite dimensional spaces are used
to study physical properties of measures relevant to quantum field theory. The
difference of typical configurations of free massive scalar field theories with
different masses is studied. We apply the same methods to study the
Ashtekar-Lewandowski (AL) measure on spaces of connections. We prove that the
diffeomorphism group acts ergodically, with respect to the AL measure, on the
Ashtekar-Isham space of quantum connections modulo gauge transformations. We
also prove that a typical, with respect to the AL measure, quantum connection
restricted to a (piecewise analytic) curve leads to a parallel transport
discontinuous at every point of the curve.Comment: 24 pages, LaTeX, added proof for section 4.2, added reference
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