87 research outputs found
Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions
This paper is aimed to show the essential role played by the theory of
quasi-analytic functions in the study of the determinacy of the moment problem
on finite and infinite-dimensional spaces. In particular, the quasi-analytic
criterion of self-adjointness of operators and their commutativity are crucial
to establish whether or not a measure is uniquely determined by its moments.
Our main goal is to point out that this is a common feature of the determinacy
question in both the finite and the infinite-dimensional moment problem, by
reviewing some of the most known determinacy results from this perspective. We
also collect some properties of independent interest concerning the
characterization of quasi-analytic classes associated to log-convex sequences.Comment: 28 pages, Stochastic and Infinite Dimensional Analysis, Chapter 9,
Trends in Mathematics, Birkh\"auser Basel, 201
Reconstruction of the early Universe as a convex optimization problem
We show that the deterministic past history of the Universe can be uniquely
reconstructed from the knowledge of the present mass density field, the latter
being inferred from the 3D distribution of luminous matter, assumed to be
tracing the distribution of dark matter up to a known bias. Reconstruction
ceases to be unique below those scales -- a few Mpc -- where multi-streaming
becomes significant. Above 6 Mpc/h we propose and implement an effective
Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the
Zel'dovich approximation is well satisfied and reconstruction becomes an
instance of optimal mass transportation, a problem which goes back to Monge
(1781). After discretization into N point masses one obtains an assignment
problem that can be handled by effective algorithms with not more than cubic
time complexity in N and reasonable CPU time requirements. Testing against
N-body cosmological simulations gives over 60% of exactly reconstructed points.
We apply several interrelated tools from optimization theory that were not
used in cosmological reconstruction before, such as the Monge-Ampere equation,
its relation to the mass transportation problem, the Kantorovich duality and
the auction algorithm for optimal assignment. Self-contained discussion of
relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour
clarifications in the text, additional material on the history of the
Monge-Ampere equation, improved description of the auction algorithm, updated
bibliography. Version 3: several misprints correcte
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