16,171 research outputs found
Uniform test of algorithmic randomness over a general space
The algorithmic theory of randomness is well developed when the underlying
space is the set of finite or infinite sequences and the underlying probability
distribution is the uniform distribution or a computable distribution. These
restrictions seem artificial. Some progress has been made to extend the theory
to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary
distributions (by Levin). We recall the main ideas and problems of Levin's
theory, and report further progress in the same framework.
- We allow non-compact spaces (like the space of continuous functions,
underlying the Brownian motion).
- The uniform test (deficiency of randomness) d_P(x) (depending both on the
outcome x and the measure P should be defined in a general and natural way.
- We see which of the old results survive: existence of universal tests,
conservation of randomness, expression of tests in terms of description
complexity, existence of a universal measure, expression of mutual information
as "deficiency of independence.
- The negative of the new randomness test is shown to be a generalization of
complexity in continuous spaces; we show that the addition theorem survives.
The paper's main contribution is introducing an appropriate framework for
studying these questions and related ones (like statistics for a general family
of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of
Theorem 7 adde
Chaos and Complexity of quantum motion
The problem of characterizing complexity of quantum dynamics - in particular
of locally interacting chains of quantum particles - will be reviewed and
discussed from several different perspectives: (i) stability of motion against
external perturbations and decoherence, (ii) efficiency of quantum simulation
in terms of classical computation and entanglement production in operator
spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing,
and (iv) computation of quantum dynamical entropies. Discussions of all these
criteria will be confronted with the established criteria of integrability or
quantum chaos, and sometimes quite surprising conclusions are found. Some
conjectures and interesting open problems in ergodic theory of the quantum many
problem are suggested.Comment: 45 pages, 22 figures, final version, at press in J. Phys. A, special
issue on Quantum Informatio
Conditioning Gaussian measure on Hilbert space
For a Gaussian measure on a separable Hilbert space with covariance operator
, we show that the family of conditional measures associated with
conditioning on a closed subspace are Gaussian with covariance
operator the short of the operator to . We provide two
proofs. The first uses the theory of Gaussian Hilbert spaces and a
characterization of the shorted operator by Andersen and Trapp. The second uses
recent developments by Corach, Maestripieri and Stojanoff on the relationship
between the shorted operator and -symmetric oblique projections onto
. To obtain the assertion when such projections do not exist, we
develop an approximation result for the shorted operator by showing, for any
positive operator , how to construct a sequence of approximating operators
which possess -symmetric oblique projections onto
such that the sequence of shorted operators converges to
in the weak operator topology. This result combined with the
martingale convergence of random variables associated with the corresponding
approximations establishes the main assertion in general. Moreover, it
in turn strengthens the approximation theorem for shorted operator when the
operator is trace class; then the sequence of shorted operators
converges to in trace norm
The geometric complexity of special Lagrangian -cones
We prove a number of results relating various measures (volume, Legendrian
index, stability index, and spectral curve genus) of the geometric complexity
of special Lagrangian -cones. We explain how these results fit into a
program to understand the "most common" three-dimensional isolated
singularities of special Lagrangian submanifolds in almost Calabi-Yau
manifolds.Comment: Revised version accepted for publication in Inventiones Mathematicae.
46 pages, 2 tables. Reference added relating to Theorem B. Section 3.4.2,
section 4.2 and Appendix B streamlined. Typographical errors corrected and
references update
Classification of affine homogeneous spaces of complexity one
The complexity of an action of a reductive algebraic group G on an algebraic
variety X is the codimension of a generic B-orbit in X, where B is a Borel
subgroup of G. We classify affine homogeneous spaces G/H of complexity one.
These results are the natural continuation of the classification of spherical
affine homogeneous spaces, i.e., spaces of complexity zero.Comment: 22 pages, 7 table
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