12,749 research outputs found
A Gravitational Theory of the Quantum
The synthesis of quantum and gravitational physics is sought through a
finite, realistic, locally causal theory where gravity plays a vital role not
only during decoherent measurement but also during non-decoherent unitary
evolution. Invariant set theory is built on geometric properties of a compact
fractal-like subset of cosmological state space on which the universe is
assumed to evolve and from which the laws of physics are assumed to derive.
Consistent with the primacy of , a non-Euclidean (and hence non-classical)
state-space metric is defined, related to the -adic metric of number
theory where is a large but finite Pythagorean prime. Uncertain states on
are described using complex Hilbert states, but only if their squared
amplitudes are rational and corresponding complex phase angles are rational
multiples of . Such Hilbert states are necessarily -distant from
states with either irrational squared amplitudes or irrational phase angles.
The gappy fractal nature of accounts for quantum complementarity and is
characterised numerically by a generic number-theoretic incommensurateness
between rational angles and rational cosines of angles. The Bell inequality,
whose violation would be inconsistent with local realism, is shown to be
-distant from all forms of the inequality that are violated in any
finite-precision experiment. The delayed-choice paradox is resolved through the
computational irreducibility of . The Schr\"odinger and Dirac equations
describe evolution on in the singular limit at . By contrast,
an extension of the Einstein field equations on is proposed which reduces
smoothly to general relativity as . Novel proposals for
the dark universe and the elimination of classical space-time singularities are
given and experimental implications outlined
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
Euclidean Distance Matrices: Essential Theory, Algorithms and Applications
Euclidean distance matrices (EDM) are matrices of squared distances between
points. The definition is deceivingly simple: thanks to their many useful
properties they have found applications in psychometrics, crystallography,
machine learning, wireless sensor networks, acoustics, and more. Despite the
usefulness of EDMs, they seem to be insufficiently known in the signal
processing community. Our goal is to rectify this mishap in a concise tutorial.
We review the fundamental properties of EDMs, such as rank or
(non)definiteness. We show how various EDM properties can be used to design
algorithms for completing and denoising distance data. Along the way, we
demonstrate applications to microphone position calibration, ultrasound
tomography, room reconstruction from echoes and phase retrieval. By spelling
out the essential algorithms, we hope to fast-track the readers in applying
EDMs to their own problems. Matlab code for all the described algorithms, and
to generate the figures in the paper, is available online. Finally, we suggest
directions for further research.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine
- change of title in the last revisio
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