22,926 research outputs found
Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
Optimal linear prediction (also known as kriging) of a random field
indexed by a compact metric space
can be obtained if the mean value function
and the covariance function
of are known. We
consider the problem of predicting the value of at some location
based on observations at locations which
accumulate at as (or, more generally, predicting
based on for linear functionals ).
Our main result characterizes the asymptotic performance of linear predictors
(as increases) based on an incorrect second order structure
, without any restrictive assumptions on such as stationarity. We, for the first time, provide
necessary and sufficient conditions on for
asymptotic optimality of the corresponding linear predictor holding uniformly
with respect to . These general results are illustrated by an example on the
sphere for the case of two isotropic covariance functions.Comment: 36 page
Relativistic quantum clocks
The conflict between quantum theory and the theory of relativity is
exemplified in their treatment of time. We examine the ways in which their
conceptions differ, and describe a semiclassical clock model combining elements
of both theories. The results obtained with this clock model in flat spacetime
are reviewed, and the problem of generalizing the model to curved spacetime is
discussed, before briefly describing an experimental setup which could be used
to test of the model. Taking an operationalist view, where time is that which
is measured by a clock, we discuss the conclusions that can be drawn from these
results, and what clues they contain for a full quantum relativistic theory of
time.Comment: 12 pages, 4 figures. Invited contribution for the proceedings for
"Workshop on Time in Physics" Zurich 201
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