6,591 research outputs found
Every countable model of set theory embeds into its own constructible universe
The main theorem of this article is that every countable model of set theory
M, including every well-founded model, is isomorphic to a submodel of its own
constructible universe. In other words, there is an embedding that
is elementary for quantifier-free assertions. The proof uses universal digraph
combinatorics, including an acyclic version of the countable random digraph,
which I call the countable random Q-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a
set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph,
closely connected with the surreal numbers. The proof shows that contains
a submodel that is a universal acyclic digraph of rank . The method of
proof also establishes that the countable models of set theory are linearly
pre-ordered by embeddability: for any two countable models of set theory, one
of them is isomorphic to a submodel of the other. Indeed, they are
pre-well-ordered by embedability in order-type exactly .
Specifically, the countable well-founded models are ordered by embeddability in
accordance with the heights of their ordinals; every shorter model embeds into
every taller model; every model of set theory is universal for all
countable well-founded binary relations of rank at most ; and every
ill-founded model of set theory is universal for all countable acyclic binary
relations. Finally, strengthening a classical theorem of Ressayre, the same
proof method shows that if is any nonstandard model of PA, then every
countable model of set theory---in particular, every model of ZFC---is
isomorphic to a submodel of the hereditarily finite sets of . Indeed,
is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at
http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe.
(v2 adds a reference and makes minor corrections) (v3 includes further
changes, and removes the previous theorem 15, which was incorrect.
Generation of monochromatic electrostatic waves of large amplitude in a bounded beam-plasma system
Monochromatic electrostatic waves of large amplitude were excited by the interaction of an electron beam with a bounded plasma. These waves were identified as resonant beam modes, which are amplified by multiple reflexion in a cavity. Nonlinear effects, such as the generation of harmonies and sidebands, were observe
The Counterpart Principle of Analogical Support by Structural Similarity
We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures
Two quantum Simpson's paradoxes
The so-called Simpson's "paradox", or Yule-Simpson (YS) effect, occurs in
classical statistics when the correlations that are present among different
sets of samples are reversed if the sets are combined together, thus ignoring
one or more lurking variables. Here we illustrate the occurrence of two
analogue effects in quantum measurements. The first, which we term
quantum-classical YS effect, may occur with quantum limited measurements and
with lurking variables coming from the mixing of states, whereas the second,
here referred to as quantum-quantum YS effect, may take place when coherent
superpositions of quantum states are allowed. By analyzing quantum measurements
on low dimensional systems (qubits and qutrits), we show that the two effects
may occur independently, and that the quantum-quantum YS effect is more likely
to occur than the corresponding quantum-classical one. We also found that there
exist classes of superposition states for which the quantum-classical YS effect
cannot occur for any measurement and, at the same time, the quantum-quantum YS
effect takes place in a consistent fraction of the possible measurement
settings. The occurrence of the effect in the presence of partial coherence is
discussed as well as its possible implications for quantum hypothesis testing.Comment: published versio
Sub-shot-noise photon-number correlation in mesoscopic twin-beam of light
We demonstrate sub-shot-noise photon-number correlations in a (temporal)
multimode mesoscopic ( detected photons) twin-beam produced by
ps-pulsed spontaneous non-degenerate parametric downconversion. We have
separately detected the signal and idler distributions of photons collected in
twin coherence areas and found that the variance of the photon-count difference
goes below the shot-noise limit by 3.25 dB. The number of temporal modes
contained in the twin-beam, as well as the size of the twin coherence areas,
depends on the pump intensity. Our scheme is based on spontaneous
downconversion and thus does not suffer from limitations due to the finite gain
of the parametric process. Twin-beams are also used to demonstrate the
conditional preparation of a nonclassical (sub-Poissonian) state.Comment: 5 pages, 5 (low-res) figures, to appear on PR
Robust generation of entanglement in Bose-Einstein condensates by collective atomic recoil
We address the dynamics induced by collective atomic recoil in a
Bose-Einstein condensate in presence of radiation losses and atomic
decoherence. In particular, we focus on the linear regime of the lasing
mechanism, and analyze the effects of losses and decoherence on the generation
of entanglement. The dynamics is that of three bosons, two atomic modes
interacting with a single-mode radiation field, coupled with a bath of
oscillators. The resulting three-mode dissipative Master equation is solved
analytically in terms of the Wigner function. We examine in details the two
complementary limits of {\em high-Q cavity} and {\em bad-cavity}, the latter
corresponding to the so-called superradiant regime, both in the quasi-classical
and quantum regimes. We found that three-mode entanglement as well as two-mode
atom-atom and atom-radiation entanglement is generally robust against losses
and decoherence,thus making the present system a good candidate for the
experimental observation of entanglement in condensate systems. In particular,
steady-state entanglement may be obtained both between atoms with opposite
momenta and between atoms and photons
De l'intermédiation à la prescription : le cas de la télévision
NAaudiovisuel;prescription;marché;distribution
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