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c5482e5967824db8f0902c7d41109e277271abb3 | subsection | 7 | 41 | Our approach: Radiative neutrino scattering on electrons | In the present paper we consider a different realization of the idea of
employing the momentum transfer to the target rather than the recoil energy
of the target particle – radiative neutrino scattering on atomic electrons or
on free electrons in a conductor:\nu +e\rightarrow \nu +e+\gamma \,.In this case the emitted p... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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7d678320f5da3e351a1dc48683d4ee4d870f2b44 | subsection | 8 | 41 | Previous studies | The radiative neutrino scattering on electrons (REF ) was first
considered by Lee and Sirlin back in 1964 and since then has
been studied by many authors (see, e.g., , , , , , , ). To the best of our knowledge,
only two studies , concern the issue of
macroscopic coherence of the process.
In it was suggested to use rad... | {
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{
"arxiv_id": "",
"doi": "",
"end": 181,
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"raw": "T. D. Lee and A. Sirlin, “Possible method of determining the moment of charge of \\nu _e,” Rev. Mod. Phys. 36 (1964) 666.",
"source_ref_id": "c06f9b120419a554849a5a0c270412f9d9c04169",
"... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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... |
bcb1628e6a6ea5b309171e4573af8638fcfc47be | subsection | 9 | 41 | Previous studies | For neutrino scattering on a charged conductor the restoring force on
the electrons accelerated by the neutrino impact would still be
there, and would be due to both the pull from the positive ions and
push from the excess electrons. As a result, the \omega ^4
suppression would still be present. This is quite analogous... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1103/physreva.61.052714",
"end": 522,
"openalex_id": "https://openalex.org/W2000069680",
"raw": "J. P. J. Carney, R. H. Pratt, Lynn Kissel, S. C. Roy, and S. K. Sen Gupta, “Rayleigh scattering from excited states of atoms and ions,” Phys... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
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"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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... |
40a7e648ad0d9c9aa15050116eb99395227ac7ca | subsection | 10 | 41 | Radiative scattering with | In the present paper we shall consider neutrino detection through coherent
radiative neutrino scattering on atomic electrons or on free electrons in
a conductor. We will be assuming the energies of the incident neutrinos
to be higher than the corresponding
characteristic atomic frequencies \omega _{at} or plasma freque... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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0.0... |
8a2f793e942b53bc1b805bc35220739e661c5dc0 | subsection | 11 | 41 | The structure of this paper | The paper is organized as follows. In Section we consider
the radiative neutrino scattering on free non-relativistic electrons, both
without any additional kinematic constraints and assuming that the electron
recoil momentum is limited from above by a small value p_0, allowing for
macroscopic coherence of the process.... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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523ece1b4f87196f6d9036a8142a813b3460de8f | subsection | 12 | 41 | Radiative neutrino scattering on electrons | We shall consider the process\nu (k)+e(p)\rightarrow \nu (k^{\prime })+e(p^{\prime })+\gamma (k_\gamma )in the rest frame of the initial electron.
Herek=(\omega , \vec{k})\,,\quad p=(m, \vec{0})\,,\quad k^{\prime }=(\omega ^{\prime }, \vec{k}^{\prime })\,,\quad p^{\prime }=(E_{p^{\prime }}, \vec{p}\,^{\prime })\,,\quad... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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7424a007ed5649cce9b147567ac41d540ed9da43 | subsection | 13 | 41 | Weak interactions induced radiative
process | In calculating the cross section of
radiative neutrino scattering (REF ) on “spinless electron”
we take into account only the vector current part of the weak NC and CC
interactions of electrons since the axial-vector current does not
contribute to neutrino scattering on zero-spin targets.
[Figure: Leading order Feynman... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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e33ce4950242b432ed7de3576fe90857bfb563c9 | subsection | 14 | 41 | Weak interactions induced radiative
process | (REF ).To arrive at this result one has to make use of the Fierz
transformation and consider unpolarised
electrons in the limit when their recoil energy is non-relativistic in the
rest frame of the initial-state electrons. Note that, as we are interested
in coherent effects, the summation over the electron spin states ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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4d5d0121f4d26d7866ba26b04086ffbb444d8e01 | subsection | 15 | 41 | Weak interactions induced radiative
process | In our case a natural choice
of \omega _0 follows from the requirement that the photon energy exceed
the characteristic frequency of the target system, \omega _{at} for scattering
on atomic electrons or \omega _p\sim 10 eV for scattering on free electrons in
a conductor. As discussed in Section REF , this will allow on... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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... |
fe17664c957ebad197fbf44d39b38fa4f9410786 | subsection | 16 | 41 | Weak interactions induced radiative
process | (REF ).The cross section for the emission of photons with energies
\omega _\gamma \ge \omega _{\gamma \rm min}\equiv \omega _0 reads\sigma _w(\omega _\gamma >\omega _0)=\frac{G_F^2 g_V^2 e^2}{(2\pi )^3}\frac{p_0^4}{2m_e^2}
\Big \lbrace \ln (1/x)-\frac{3}{4}+x-\frac{1}{4}x^2 \Big \rbrace
\,.For \omega _0\ll \omega one ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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5c80fd6fa5672f4be647365f4971d535d208b0cf | subsection | 17 | 41 | Radiative scattering and the neutrino magnetic dipole moment | Let us now consider the radiative neutrino scattering process
(REF ) in the case when the neutrino-electron scattering is
mediated by the photon exchange due to neutrino magnetic or electric dipole
moments. In what follows we will for definiteness discuss the case of
neutrino magnetic dipole moment \mu _\nu . We will c... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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b5f2dbe8444318c81d36a41ed7029d44a548c44d | subsection | 18 | 41 | Radiative scattering and the neutrino magnetic dipole moment | Without constraining
|\vec{p}\,^{\prime }|, for the double and single differential cross sections we
find\frac{d^2\sigma _m}{d\omega _\gamma d\cos \theta _\gamma }=
\frac{\mu _\nu ^2 e^4}{(2\pi )^3}\frac{1}{4m_e^2}\cdot \frac{(\omega -\omega _\gamma )^2}{\omega _\gamma }\big (3-\cos ^2\theta _\gamma \big )
\,,\frac{d\s... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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d6684b6a7099422688ce031db0c264a6359b2104 | subsection | 19 | 41 | Radiative scattering and the neutrino magnetic dipole moment | (REF ), we find, to leading
order inp_0,\frac{d\sigma _m}{d\omega _\gamma }=
\frac{\mu _\nu ^2 e^4}{(2\pi )^3}\frac{1}{m_e^2}\cdot \frac{1}{6}
\frac{(\omega -\omega _\gamma )p_0^3}{\omega \omega _\gamma ^2}
\,,
\vspace*{2.84526pt}\sigma _m(\omega _\gamma >\omega _0)=
\frac{\mu _\nu ^2 e^4}{(2\pi )^3}\frac{1}{m_e^2}\cd... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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7c5e1e836101b17003e62963e41dc36e13b3523e | subsection | 20 | 41 | Radiative scattering and the neutrino magnetic dipole moment | Such an amplitude will then describe the transition of
\nu _\alpha to a neutrino \nu _\beta which may be
of the same or different flavour.
As the final-state neutrino is not detected, in calculating the cross
section of the process one has to sum over \beta . For the
ultra-relativistic neutrinos we confine ourselves to... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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7c977e1fb343e562cd1fbc20af7028f1a46fc221 | subsection | 21 | 41 | DM detection through radiative coherent scattering? | It would be interesting to extend the above considerations to detection of
other particles, such as DM.
Unfortunately, the mechanism of enhancement of the detection cross
section through macroscopic coherence considered here for neutrinos
would not work for non-relativistic projectiles.
The reason is actually
mostly ki... | {
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{
"arxiv_id": "",
"doi": "10.1103/physrevlett.118.031803",
"end": 1300,
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"raw": "C. Kouvaris and J. Pradler, “Probing sub-GeV Dark Matter with conventional detectors,” Phys. Rev. Lett. 118 (2017) no.3, 031803 [arXiv:... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
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8ede4058591cd47516d54c1617e3e2058af1d02d | subsection | 22 | 41 | Effects of atomic binding | In Section we considered radiative neutrino scattering
on free electrons. This is suitable for conduction electrons in metals;
however, for scattering on atomic electrons in dielectrics the effects of
atomic binding should in general be taken into account. We shall
show now that for the
kinematic regime of interest to... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
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ab624c7bef36ec7f15504914e7a04387faf8ae39 | subsection | 23 | 41 | Effects of atomic binding | \left\lbrace \frac{\langle i|e^{-i\vec{k}_f\vec{r}}
\vec{\rm p}
\vec{\epsilon }_{f}^{\hspace*{1.70717pt}*}
|n\rangle \langle n| e^{i\vec{k}_i\vec{r}}
\vec{\rm {p}}\vec{\epsilon }_i
|i\rangle }{E_n-E_i-\omega _i-i\varepsilon }+
\frac{\langle i|e^{i\vec{k}_i\vec{r}}
\vec{\rm {p}}\vec{\epsilon }_i
|n\rangle \langle n| e^{... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
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-0.0... |
5207554a7e89bde3d58301d6ca09644490cd4e4e | subsection | 24 | 41 | Effects of atomic binding | As we are interested in coherent scattering on a group of atoms,
by |i\rangle one should actually understand the ground state of such a
system. The cancellation then happens also in the case when this state is
spherically symmetric (i.e. has zero total angular momentum), even if the
ground states of the individual atom... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
0.012467892840504646,
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0.01... |
53c6765ccd1f64e5bf20208b6ee543225ceddad2 | subsection | 25 | 41 | Effects of atomic binding | The fact that the probability of the radiative
scattering without excitation or ionization of the target atoms is small is
already taken into account by
the suppression of the individual cross sections which we found upon
constraining the electron recoil momentum by
|\vec{p}\,^{\prime }|\lesssim p_0\sim 10^{-5} eV. | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.03244950994849205,
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-0... |
556989959044290352327f1981eae50d5821401d | subsection | 26 | 41 | Coherent effects and the cross sections | Let us now assess the effects of macroscopic coherence on the cross sections
of neutrino detection process (REF ).As discussed in Sections REF and REF , in order to
take possible macroscopic coherence effects into account one has to
multiply the elementary amplitude of the process
by the relevant structure factor (such... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.005078855436295271,
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0.01157719548791647,
-0.019872786477208138... |
ad3b21b62405966f493d82c5a517c93ebdd8c892 | subsection | 27 | 41 | Coherent effects and the cross sections | No such problems arise for radiative processes
discussed in the present paper.We shall now estimate the effects of possible macroscopic coherence on
radiative neutrino scattering on electrons. The corresponding cross
sections with the net electron recoil momentum constrained by
|\vec{p}\,^{\prime }|\le p_0 with a small... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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-0.... |
cbd53543d611d928654afe737aa72dd6241d9611 | subsection | 28 | 41 | Coherent effects and the cross sections | (REF ) and (REF ) we then find\frac{d\overline{\sigma }_m}{d\omega _\gamma }\simeq \frac{\mu _\nu ^2 e^4}{6}\,
\frac{(\omega -\omega _\gamma )}{\omega \omega _\gamma ^2}\,
\frac{n_e}{m_e^2}
\,,
\vspace*{2.84526pt}\overline{\sigma }_m(\omega _\gamma >\omega _0)\simeq \frac{1}{6}\,
\mu _\nu ^2 e^4
\frac{n_e}{m_e^2 \omeg... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
0.00366444094106555,
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0.013261422514915466,
-0.00017025056877173483,
-0.... |
bf73e785ea6ceb47ad08935fc8b72e480508ab20 | subsection | 29 | 41 | Coherent effects and the cross sections | Mutiplying the squared matrix element of the elementary
process by|F(\vec{k}-\vec{k}^{\prime }-\vec{k}_\gamma )|^2 \simeq N_e n_e (2\pi )^3
\delta ^3(\vec{k}-\vec{k}^{\prime }-\vec{k}_\gamma )\,,performing the integration over the momenta of the scattered neutrino
and the recoil electron as well as over the directions ... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
0.003171176416799426,
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0.042653560638427734,
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0.008420875295996666,
-0.015728119760751724,
... |
5da1f976d34f5bee2faf35861f20c5f64171876d | subsection | 30 | 41 | Coherent effects and the cross sections | (REF )n_e=N_A \rho (\rm {g/cm^3})Y_e\;{\rm cm}^{-3}\simeq (1.33\;{\rm keV})^3
\,\rho (\rm {g/cm^3})
\,,where N_A is the Avogadro constant, \rho is the density of the target
material, Y_e is the number of electrons per nucleon in the target,
and in the last (approximate) equality we have set Y_e=1/2. In the regime
\omeg... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.0255531407892704,
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0.0020... |
f7a7ee55abef8191ed68fceb3d0c8ddc731fd383 | subsection | 31 | 41 | Coherent effects and the cross sections | (REF ) in convenient units:\frac{d\overline{\sigma }_m}{d\omega _\gamma }\simeq 2.06\times 10^{-56}
\left(\frac{\mu _\nu }{10^{-12}\mu _B}\right)^2\,
\rho ({\rm g/cm^3})\,\left(\frac{100~{\rm eV}}{\omega _\gamma }\right)^2\,
~{\rm cm^2/eV}
\,.
\vspace*{2.84526pt}Here \mu _B=e/2m_e is the electron Bohr magneton. | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
0.010923654772341251,
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-0.006602250970900059,
... |
dea2957e8b9d45fd42ebc36c48733bf2403cc037 | subsection | 32 | 41 | Summary and discussion | We have considered the possibility of achieving macroscopic coherence
in neutrino detection experiments. For the elastic neutrino
scattering processes, coherence at macroscopic scales can only be
attained at the expense of unmeasurably small recoil energies of the
target particles, E_{rec}\sim 10^{-43} eV, and so is of... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.012047880329191685,
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0.0058217435143888,
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0.0019037023885175586,
-0.0233022328466177,
0.0... |
34e0241f26dac8f4cc9b08a2d79566932cafe158 | subsection | 33 | 41 | Summary and discussion | At the same
time, the usual increase of the radiative cross section at small photon
energies is limited by the requirement that \omega _\gamma exceed the
characteristic atomic frequencies \omega _{at} for neutrino scattering on
atomic electrons or plasma frequency \omega _p for scattering on free
electrons in a conduct... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1475-7516/2017/12/010",
"end": 1396,
"openalex_id": "https://openalex.org/W3104717429",
"raw": "E. Vitagliano, J. Redondo and G. Raffelt, “Solar neutrino flux at keV energies,” JCAP 1712 (2017) no.12, 010 [arXiv:1708.02248 [hep-ph]]... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.004901158157736063,
0.006419181823730469,
-0.044610071927309036,
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0.018719753250479698,
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-0.004664681851863861,
-0.04015516862273216,
... |
f70324e145601ebf6d15e9aa0b550556527b7bce | subsection | 34 | 41 | Summary and discussion | We have shown that for the conventionally
discussed non-relativistic dark matter particle
candidates this is not possible. The coherent enhancement mechanism
studied here may, however, work for the detection of relativistic
particles which usually exist in multi-component dark matter models,
an example being the “boost... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1088/1475-7516/2014/10/062",
"end": 337,
"openalex_id": "https://openalex.org/W2070577141",
"raw": "K. Agashe, Y. Cui, L. Necib and J. Thaler, “(In)direct Detection of Boosted Dark Matter,” JCAP 1410 (2014) no.10, 062 [arXiv:1405.7370 [h... | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.0067598046734929085,
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-0.044129468500614166,
0.... |
d61748a9bba80d37f6afd1b8e75156628c1e6fb2 | subsection | 35 | 41 | Appendix G: Kinematics of the process | Consider the kinematics of the processX(k)+e(p)\rightarrow X(k^{\prime })+e(p^{\prime })+\gamma (k_\gamma )\,,where X is
a projectile particle of mass M.
In the rest frame of the initial-state electron the 4-momenta of the incident
X-particle, initial-state electron, scattered
X-particle, final-state electron and emitt... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
0.013945686630904675,
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... |
d1766d72bc5cc3bf4ab50a7ae85a7c2507f3e037 | subsection | 36 | 41 | Appendix G: Kinematics of the process | \vspace*{2.84526pt}Here \theta _\gamma is the angle between \vec{k}_\gamma and \vec{k},
\theta _{\vec{p}^{\prime }(\vec{k}-\vec{k}^{\prime })} is the angle between
\vec{p}\,^{\prime } and \vec{k}-\vec{k}_\gamma , andR\equiv |\vec{k}-\vec{k}_\gamma |=\sqrt{\vec{k}^2+\omega _\gamma ^2
-2\omega _\gamma |\vec{k}|\cos \the... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.03762217238545418,
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0.0212521031498909,
-0.01335693895816803,
-0.016629... |
bd448e708ee21262cf878d456806d64fb10e6b9a | subsection | 37 | 41 | Appendix G: Kinematics of the process | This requires\omega _\gamma \simeq \frac{|\vec{k}|}{M}
|\vec{p}\,^{\prime }|\cos \theta _{\vec{p}^{\prime }(\vec{k}-\vec{k}^{\prime })}\,\ll \,|\vec{p}\,^{\prime }|\,.As discussed in Section REF , to achieve macroscopic coherence
one needs |\vec{p}\,^{\prime }|\lesssim 10^{-5} eV; condition (REF ) then
implies that the... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.04835619777441025,
-0.011280570179224014,
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0.010662770830094814,
0.004034767858684063,
-0.0... |
f0da05598bc8cdefa957f3d0742141f83150fec1 | subsection | 38 | 41 | Appendix H: 3-body phase space volume | We shall now consider the regime of relativistic projectiles assuming that
condition (REF ) is satisfied with a large margin. This will
allow us to treat the projectile as essentially massless.Consider the 3-body phase space volume integralR_3\,\equiv \int \frac{d^3p^{\prime }}{2E_{p^{\prime }}}\frac{d^3k_\gamma }{2\om... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.03876364976167679,
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0.03699333965778351,
-0.021350529044866562,
0.00167... |
abff8320dbb390fd80fcb7842dbd6f73283dedc3 | subsection | 39 | 41 | Appendix H: 3-body phase space volume | Using
\delta (\omega -\omega ^{\prime }-\omega _\gamma )=
2\omega ^{\prime }\delta [(\omega -\omega _\gamma )^2-{\omega ^{\prime }}^2] and
{\omega ^{\prime }}^{2}=\vec{k}^{\prime 2}=(\vec{k}-\vec{p}\,^{\prime }-\vec{k}_\gamma )^2,
we find\delta [(\omega -\omega _\gamma )^2-{\omega ^{\prime }}^2]\,=\,\delta \big (
2|\ve... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
-0.015852967277169228,
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0.029493536800146103,
-0.01675318367779255,
... |
a7acfeb17f9c147ac855116f36489b8a810528a3 | subsection | 40 | 41 | Appendix H: 3-body phase space volume | Therefore, the condition
|\vec{p}\,^{\prime }|\le p_0 puts a non-trivial constraint on x_\gamma =1-
\cos \theta _\gamma only when the expression on the right hand side of
eq. (REF ) is smaller than 2. This yields p_0/2<\omega _\gamma .
On the other hand, the constraint |\vec{p}\,^{\prime }|\le p_0 is only non-trivial
w... | {
"cite_spans": []
} | 10.1007/JHEP10(2018)045 | 1806.10962 | Coherent scattering and macroscopic coherence: Implications for
neutrino, dark matter and axion detection | [
"Evgeny Akhmedov",
"Giorgio Arcadi",
"Manfred Lindner",
"Stefan Vogl"
] | [
"hep-ph",
"astro-ph.CO",
"hep-ex",
"nucl-ex"
] | 2,018 | en | Physics | [
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0.008843991905450821,
-0... |
226f456371eccb12fec3eca18c95ad2fbc1709f6 | abstract | 0 | 30 | Abstract | Long-period EUV pulsations, recently discovered to be common in active
regions, are understood to be the coronal manifestation of thermal
non-equilibrium (TNE). The active regions previously studied with EIT/SOHO and
AIA/SDO indicated that long-period intensity pulsations are localized in only
one or two loop bundles. ... | {
"cite_spans": []
} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
"F. Auchère",
"Z. Mikić",
"G. Aulanier",
"K. Bocchialini",
"E. Buchlin",
"J. Solomon",
"E. Soubrié"
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c5831cc5f36b2fcd8daeeb21b4ff2456e61d2bd5 | subsection | 1 | 30 | Introduction | Solving the coronal heating problem remains one of the biggest challenge in astrophysics. How can the tenuous plasma that constitutes the highest layer of the solar atmosphere be maintained at temperatures two orders of magnitude higher than that of the solar surface? One of the fundamental facets of this problem is to... | {
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93daf84c4738a8eb2b09569e7844771aba824b99 | subsection | 2 | 30 | Introduction | In particular, these pulsations are very common in coronal loops. They have also been observed with the coronal channels of the Atmospheric Imaging Assembly , on board the Solar Dynamics Observatory , during the six first years of the AIA archive . studied three examples of such events in detail, with respectively peri... | {
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72f59ad106392803b28bbfbaa0c96a5dc67b2e65 | subsection | 3 | 30 | Parameter-space scan | For this parameter-space study, we use the same 1D hydrodynamic code as in and in Paper I.
The 1D description is particularly suited for this kind of study as multiple configurations of loops can be easily tested. The loop geometries used in these simulations, except for one loop (loop A, see Section REF ), are from th... | {
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f4e137764142370c194b27704efb3b468b38224e | subsection | 4 | 30 | Method and parameters explored | We choose to focus on three different loop geometries and to scan various heating configurations for these loops. In addition of the loop geometry that matches the pulsating loop bundle already used in Paper I (noted here as loop B), we use a semi-circular geometry as a control sample (loop A) and we picked another loo... | {
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9c13b3c0d7204d7391e36a42b7f5577c8ce4009a | subsection | 5 | 30 | Method and parameters explored | The scan cube is then (H_1, \lambda _1, \lambda _2).For each simulation, we can define the heat flux, i.e. the total heat that the loop receives over its length, normalized to the first loop footpoint cross-section area :Q_0= \frac{1}{A_0} \int _{0}^{L} H(s) \times A(s)\, ds \hspace{28.45274pt}[\mathrm {W} \, \mathrm {... | {
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b63351a25ff85d9a0a6434f5a84a31025a10ea50 | subsection | 6 | 30 | Loop geometries | In Figure REF , we present the loop geometry of loop A, the test case semi-circular geometry we use for the first set of simulations.
In Figure REF , we present the loop geometry of loops B and C which are from two field lines extracted from the LFFF extrapolations of the active regions presented in Figure REF .
These ... | {
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1d67c428129a4e607ab13d56fd1a6f490e3b52e4 | subsection | 7 | 30 | Loop A | This loop is a semi-circular loop with the same length as loop B, i.e L=367 \, \mathrm {Mm}. The magnetic field along the loop is given by:B(s) = B_0 + B_1(e^{-s/l}+e^{-(L-s)/l})with B_0 = 1 \, \mathrm {G}, B_1 =10 \, \mathrm {G} et l=14 \, \mathrm {Mm} . The loop expansion factor reaches a value of 11 at the loop apex... | {
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bf3007be0f4b6966fb7b5f3d9860bfb025556736 | subsection | 8 | 30 | Loop B | This is the same loop that we studied in Paper I. It corresponds to the pulsating loop bundle detected in AIA data. It is quite a large and asymmetric loop with L=367 \, \mathrm {Mm}. As in Paper I, s=0 corresponds to the eastern footpoint, while s=L corresponds to the western footpoint. The apex is at an altitude of 8... | {
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8f448626ea234336eea2587155b78eb9459d44c9 | subsection | 9 | 30 | Loop C | This field-line corresponds to a non-pulsating loop bundle in the AIA data. It is located in the small active region at the West of NOAA AR 11499. We did not find any intensity pulsations in any of the loops of this region. The loop chosen is shorter than the previous ones, with L=139 \, \mathrm {Mm}. As for loop B, s=... | {
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} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
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4916d0804a55dc6257693444b020ffccf004bdd0 | subsection | 10 | 30 | Exploration of heating parameter space | For each loop geometry, we present hundreds of simulations, which is still only a fraction of the parameter space we explored. We choose here to focus only on the area of the parameter space surrounding the simulations showing TNE cycles.The results of the exploration for each loop geometry are displayed using grid plo... | {
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0225f012eb14595535e8f1696b62bb531b2b73f3 | subsection | 11 | 30 | Physical limitations on the domains explored | As mentioned earlier, we explored a very large range of heating for each loop geometry, in particular in terms of scale heights: \lambda _{1} and \lambda _{2}. We did not limit the study to specific ranges that would be appropriate to the magnetic field configuration of each loop system. As a consequence, it is likely ... | {
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e47bb9e6f8a1521c2e4014413e60a967884f456d | subsection | 12 | 30 | Criteria to distinguish between the different behaviors | Within the parameter-space, we detect the TNE cases and determine the nature of the condensations, using only the temperature profiles. | {
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314960a3a7aac0c5555225fcd97d8651099474a4 | subsection | 13 | 30 | TNE events | They are detected within the parameter-space using Fourier analysis. We look at periods between 2 and 16 hours as we did for the AIA observations. For each simulation, we look at the evolution of the temperature averaged around the loop apex. We do not consider the beginning of these temperatures curves, i.e. the first... | {
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bacc6d1df489f5e6d18a48b40236731980686eeb | subsection | 14 | 30 | Distinction between ICs and CCs | We look at the nature of the condensations not only around the apex but all along the coronal part of the loops. In fact if some CCs occur low enough in one of the loop legs, the evolution of temperature around the loop apex is not very different from an IC case (see e.g. the first, noted as IC and the third, noted as ... | {
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77252ff5219a44eac1b38c5606a321ac54e19d30 | subsection | 15 | 30 | Loop A | For this group of simulations H_0=1 \times 10^{-7} \, \mathrm {W} \, \mathrm {m}^{-3}. We scan three values of H_1: 640 H_0, 1280 H_0, and 2560 H_0.
For each value of H_1, \lambda _1 and \lambda _2 are scanned between 2\% and 11\% of L, i.e. we test 10 values between 7.3 Mm and 44.0 Mm. Every combination is tested, so ... | {
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764d7ac09c2f85c8e756b7c4860d77558957452a | subsection | 16 | 30 | Loop A | The velocities are lower close to TNE cases.We use the velocity mapsWe do not show the velocity maps in this paper for conciseness. However, Figure REF allows us to identify the SE cases, without looking at the velocity maps. to analyze the loops without cycles. For each H_1, in the region where \lambda _1 > \lambda _2... | {
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8fa45b938c90838fb1f36a7aa25f72fd148030e0 | subsection | 17 | 30 | Loop B | We use the same H_0, i.e. 1 \times 10^{-7} \, \mathrm {W} \, \mathrm {m}^{-3}, for the simulations with this loop geometry. We scan two valuesIn our analysis, we scanned a third value of H_1: 320 \, H_0, however we detected no TNE cases for this scan. of H_1: 640 \, H_0, as presented in Figure REF , and 1280\, H_0, as ... | {
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} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
"F. Auchère",
"Z. Mikić",
"G. Aulanier",
"K. Bocchialini",
"E. Buchlin",
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d4ea5172b01f8c144786ed9a297e0625a3208087 | subsection | 18 | 30 | Loop B | But we observe the same pattern of velocity evolution within the parameter space, i.e. higher velocities when the heating is highly stratified, \lambda _1 + \lambda _2 < 60 \, \mathrm {Mm} in that case. Moreover, as for loop A, velocities at the apex are lower in the TNE domain.At each side of the TNE domain, the simul... | {
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} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
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e90c52b9ef7954df3307482c1e0b82df5c0318a8 | subsection | 19 | 30 | Loop C | For this last loop geometry, we parameterize the heating function with H_0=2 \times 10^{-6} \, \mathrm {W} \, \mathrm {m}^{-3}. We scan three values of H_1: 20 H_0, as presented in Figure REF , 40 H_0, as presented in Figure REF , and 80 H_0, as presented in Figure REF .
For each value of H_1, \lambda _1 and \lambda _2... | {
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} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
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"K. Bocchialini",
"E. Buchlin",
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592f7cae0728f6ae49a0edc09449314a49892613 | subsection | 20 | 30 | Conditions that favor TNE and constraints on the heating | Scanning the parameter space of heating configurations for different loop geometries, we have noticed that the distribution of the occurrence of TNE depends on the loop geometry.
However, from this study, it seems that we are able to produce TNE-favorable conditions for any loop geometry. TNE will occur if the heating ... | {
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... | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
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65f459610583ab21e293023bbc5ee465a58b8a17 | subsection | 21 | 30 | Exploration de-correlated from the magnetic field strength | For each of the geometries tested, not all the stratified heating configurations lead to TNE. The area where TNE occurs is limited to some range in the heating parameter space. This leads to the question as to whether the area explored within the parameter space is realistic.
In particular, the heating is somewhat corr... | {
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f39b349b7a4370cbab817f505b537d605855b601 | subsection | 22 | 30 | Common characteristics of TNE events | From the analysis of the flows, using in particular the averaged apex velocity (see Section REF ), we noticed that the siphon flows are more intense for short heating scale heights. Moreover, they tend to be weaker close to the TNE conditions (see Figures REF to REF ).We examine also some characteristics of the cycles ... | {
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} | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
"C. Froment",
"F. Auchère",
"Z. Mikić",
"G. Aulanier",
"K. Bocchialini",
"E. Buchlin",
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4ca04bc13cc1048842411e84d93d06ee6f23a827 | subsection | 23 | 30 | Period of the TNE cycles | We can notice an evident dependence on the loop length. The periods are from 2.5 hr to 15.5 hr for loop A, and from 5.5 hr to 15.5 hr loop B, which are both 367 Mm long, and from 2.4 hr to 5.9 hr for loop C, which is 139 Mm long. This dependence has already been seen in the EIT event statistics of , the AIA event stati... | {
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"K. Bocchialini",
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48f91e7d37ee2bca9cd7f61cf975cfef48e29c63 | subsection | 24 | 30 | Time lag between | We compute the time lag between the temperature and density evolution. This delay is also a characteristic of TNE cycles; it is a signature of TNE when combined with the periodicity. It also explains the systematic cooling pattern observed between EUV channels, the intensity peaking first in the hotter channels and the... | {
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"K. Bocchialini",
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b4c1a60f080c2415e22a194bd2abc56efc6cf449 | subsection | 25 | 30 | EUV pulsations and coronal rain | Figure REF shows three TNE cases for loop B. We display the temperature, density and velocity evolution along the loop, for three days of simulation, i.e. about 8 evaporation/condensation cycles .
These simulations are extracted from Figure REF and thus correspond to H_1 = 12.8 \times 10^{-5} \, \mathrm {W} \, \mathrm ... | {
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"raw": "Froment, C., Auchère, F., Bocchialini, K., et al. 2015, The Astrophysical Journal, 807, 158. http://iopscience.iop.org/0004-637X/807/2/158",
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cdec5fe1ddadb89fc4941653e337438f3272aaa2 | subsection | 26 | 30 | EUV pulsations and coronal rain | For comparison, we add the same plot from the AIA observations. The intensity is given along the loop defined by a smoothed version of the orange contourNote that the length of the observed loop is then a bit shorter that the length of the simulated loops (derived form LFFF extrapolations). It does not affect our analy... | {
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0d47f06e3108f27c150e2fd4aa25c96624feb408 | subsection | 27 | 30 | Are all the results of these simulations realistic? | We have previously seen that some of the TNE cases can reproduce very well the average behavior observed with AIA, in case of long-period intensity pulsations (see also the results of Paper I).
Looking beyond the TNE cases, we can ask ourselves if the non TNE cases produced in the parameter space are realistic. Only a ... | {
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d43738c2f7a0ab7b4a52402b917e31b065d6a27a | subsection | 28 | 30 | Summary | In this paper, we explored a large range of dynamics, scanning different regimes of thermal non-equilibrium (TNE) and other behaviors in coronal loops.
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... | 10.3847/1538-4357/aaaf1d | 1802.04010 | On the occurrence of thermal non-equilibrium in coronal loops | [
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5eafa48afde8b2751f7e55e10b85a8219f5011a6 | subsection | 29 | 30 | Summary | We found that the period (from 2.4 to 15.5 hrs) is increasing with the length of the loop and with the maximum temperature reached. These periods tend also to be longer for CC compared to IC for the same loop geometry. The time delay between the temperature and density evolution, characteristic of TNE events when combi... | {
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0a3b71c6312cb9588624311d81082198ffc37a38 | abstract | 0 | 223 | Abstract | An introduction and overview of constructive reverse mathematics. | {
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1d12b5b3f519e48d8b5a4ccfb8dd9fec38d36645 | subsection | 1 | 223 | Constructive Mathematics | Almost all proofs in traditional mathematics invoke the law of excluded middle ([PR:LEM]LEM) at some point. Sometimes this use is as obvious as starting a proof of \varphi by “let us assume \varphi is false” followed by a derivation of a contradiction, and sometimes this use is subtly hidden in details, as it is in the... | {
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5276c79cb9de007027e8b0b3cfc36955fd3da24d | subsection | 2 | 223 | Constructive Reverse Mathematics | The focus of reverse mathematics—as opposed to normal, everyday mathematics—is not to find what theorems we can prove from certain axioms, but to ask which axioms are also necessary to prove certain theorems. As such the idea is neither new nor revolutionary. However, applying this approach systematically to some fragm... | {
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0bd371fc2529e8f3928e750de48cc087e7af4038 | subsection | 3 | 223 | Constructive Reverse Mathematics | As such they provide details of why certain statements cannot be proved constructively. Often, this gives us the exact information needed to “constructivise” a theorem. For example if a theorem is shown to be equivalent to the uniform continuity theorem (see Section ) it is most likely to be true constructively if we a... | {
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2f2cc873c75bed4d4721e89ba62b7035750e9140 | subsection | 4 | 223 | A very Short History of Constructive Reverse Mathematics | It is generally accepted that the following 1984 result by Julian and Richman is the first result in constructive reverse mathematics *Theorem 2.4.A new, alternative, proof can be found in .Proposition The fan theorem for decidable bars ([PR:FAN]FAN_{\Delta }) is equivalent to the statement that every uniformly continu... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e5d0a132f47e07528a1d7be4e8eb386baaa140d9 | subsection | 5 | 223 | A very Short History of Constructive Reverse Mathematics | Since, under the BHK interpretation *Chapter 5 of the logical connectives this implies that we have either proved Fermat's last theorem or found a counterexample, Brouwer rejected such theorems T.Sequences like \alpha _n and \beta _n might become obsolete for this purpose, since the underlying problem might get solved;... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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96e91fae1aa650b8f2c41d60901ff05f403fcc04 | subsection | 6 | 223 | A very Short History of Constructive Reverse Mathematics | We would also like to add that if the point of an Brouwerian counterexample were only to show that some statement T implied an unacceptable statement and is therefore unacceptable itself, then there would be no point in distinguishing between, say, [PR:LPO]LPO and [PR:LLPO]LLPO and simply work with the weakest one. The... | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
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11154188b412cf81a1c90a23fcfca8c043998e08 | subsection | 7 | 223 | Foundational Aspects | In the tradition of Bishop-style constructive mathematics eB67,dB85,dBlV06 we will be working informally, in the same sense that most mathematicians work informally. That is not to say that we will be vague or imprecise, but that we are happy to skip details for the sake of readability and clarity of ideas. However, we... | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4378f92f848e34937721a88c9d42da68ce1d13ac | subsection | 8 | 223 | Foundational Aspects | These are the following principles.[Sec:Choice]ACC
If S is a subset of \mathbb {N}\times B, and for each n \in \mathbb {N} there exists b \in B such that (n, b) \in S, then there is a function f: \mathbb {N}\rightarrow B such that (n,f(n)) \in S for each n \in \mathbb {N}.
[Sec:Choice]ADC
If a \in A and S \subset A \ti... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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f1de1ca2e4fd2f68d97d6a9eb59bb17e4c234cb6 | subsection | 9 | 223 | Foundational Aspects | Large parts of traditional Bishop style constructive mathematics could be rendered choice-free by switching to represented reals and making similar definitions and arguments for other objects such as point-wise continuous functions and so on.
However, there are also uses of [Sec:Choice]ACC and [Sec:Choice]ADC of a mor... | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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26b44f22c39b0e2a46abf1a62e0bfd8869e6acdd | subsection | 10 | 223 | Overview and Plan | Contrary to Simpson style reverse mathematics, in which most theorems fall into one of the “big five” categories,Although recent work has shown that there are more than the big five. there is a plethora of principles that have been considered in constructive reverse mathematics, with a quick count totalling about 17 ma... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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11ce56854240b6100402434abef8e4dd46a07e8d | subsection | 11 | 223 | Overview and Plan | Actually, [PR:WMP]WMP is true everywhere, but fits better into this chapter than into the chapter about [PR:BDN]BD-N.
[font=,baseline=(current bounding box.north)]
[mylightblue!200](90:0.4cm) circle (0.5cm);
[mylightblue!200] (210:0.4cm) circle (0.5cm);
[mylightblue!80] (330:0.4cm) circle (0.5cm);
(90:0.4cm) circle ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0cde854193ae435e1ac2fe2158bcbb83b2aa1756 | subsection | 12 | 223 | Version History | The numbering of theorems and equations is kept consistent between minor version increases, for example from 1.0 to 1.1, but not between major version increases, for example from 1.9 to 2.0. Minor minor version changes, such as 1.1 to 1.1.2, are reserved for correction of typos and changes in presentation, but not math... | {
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} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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e43d768c6ecb7b46d8f3fe433366cc347300a184 | subsection | 13 | 223 | [PR:LEM]LEM and [PR:WLEM]WLEM | The possibly strongest of all omniscient principles is the law of excluded middleOften, such as in , it is called the principle of excluded middle—PEM. Of course it also well known under its Latin name tertium non datur. itself.
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrigh... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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be0079cfbff9f29df41f5df9b0ac293ab176156a | subsection | 14 | 223 | [PR:LEM]LEM and [PR:WLEM]WLEM | Hence the first alternative holds and we are done.Conversely let \varphi be any syntactically correct closed statement and consider the setS = \mbox{$\left\lbrace \,x \, | \, x = 0 \wedge (\varphi \vee \lnot \varphi ) \,\right\rbrace $} \ .Then the assumption that S=\emptyset leads to a contradiction and thus, S \ne \e... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0e2f8faaf67e9b4f8b4776aedc967976f38efd56 | subsection | 15 | 223 | [PR:LEM]LEM and [PR:WLEM]WLEM | To this end let S \subset \mathbb {R} and B\in \mathbb {N} be such that\forall {s \in S} : {-B \leqslant s \leqslant B} \ .By *Proposition 4.3, Chapter 1 it suffices to show that \lnot \lnot S is order located; that is for all a,b \in \mathbb {R} with a<b\left( \lnot \forall {s \in S} : {s \leqslant a} \right) \ \vee \... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d465d1a391b9ae68414fea4ede4e30cc87101d26 | subsection | 16 | 223 | [PR:LEM]LEM and [PR:WLEM]WLEM | Since \lnot (\lnot \varphi \wedge \varphi ) is provable in intuitionistic logic we have \lnot \lnot \varphi \vee \lnot \varphi ; that is [PR:WLEM]WLEM holds.Conversely assume that \lnot (\varphi \wedge \psi ). By [PR:WLEM]WLEM either \lnot \varphi or \lnot \lnot \varphi . It is easy to see that in the second case the a... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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c982a0dc0678b40f4f7b002faedfa7272cba5a7a | subsection | 17 | 223 | [PR:LEM]LEM and [PR:WLEM]WLEM | To see that REF implies [PR:LEM]LEM we will show that it implies REF . Clearly \lnot \lnot \varphi implies (\lnot \varphi \Rightarrow \varphi ), and hence \varphi . To see that REF implies [PR:LEM]LEM, we simply take \psi \equiv \top and \vartheta \equiv \bot . Similarly REF implies [PR:LEM]LEM by choosing \varphi \equ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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bc6a3023ec14043375feeb34215766d91dcb9abc | subsection | 18 | 223 | [PR:LPO]LPO | The ubiquitousness of the limited principle of omniscience ([PR:LPO]LPO) in analysis might only be rivalled by [PR:LLPO]LLPO's. This is mainly due to the fact, that real numbers and sequences feature prominently in analysis, and [PR:LPO]LPO tells us everything we want to know about both.
[leftmargin=2em,rightmargin=3em... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3438f26f7ca054b41c5be090fd1992b2a6bc49cb | subsection | 19 | 223 | Basic equivalencies of [PR:LPO]LPO | Mostly taken directly from are the following equivalences:Proposition 2.1 The following are equivalent to [PR:LPO]LPO\forall {x \in \mathbb {R}} : {x < 0 \, \vee \, x = 0 \, \vee \, 0 < x}
For every binary sequence (a_n)_{n \geqslant 1} we can decide whether
\exists {n \in \mathbb {N}} : {\forall {n \geqslant N} : {... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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2b01ae78678319dd108ee1b70960757d4af3b60f | subsection | 20 | 223 | Basic equivalencies of [PR:LPO]LPO | By [PR:LPO]LPO either a_n=0 for all n \in \mathbb {N} or there exists n\in \mathbb {N} such that a_n=1. That is we can decide whether x \ne r_n for all n\in \mathbb {N} or whether there exists n \in \mathbb {N} such that x=r_n.Conversely let (a_n)_{n \geqslant 1} be a decreasing binary sequence. Definea = \sum _{n \geq... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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b4b3b6899a5d6fc5621b7ae7200e9dcf8a938b37 | subsection | 21 | 223 | Basic equivalencies of [PR:LPO]LPO | Using countable choice we can assume that there is a binary sequence such thata_n = {\left\lbrace \begin{array}{ll} 0 & \Rightarrow f(n) \in A \\ 1 & \Rightarrow f(n) \in B
\end{array}\right.}Using [PR:LPO]LPO and countable choice iteratively we can fix another binary sequence (b_n)_{n \geqslant 1} such thatb_n = {\lef... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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8ef70ff8be34f553d09192d3578f69ba92c0325a | subsection | 22 | 223 | Basic equivalencies of [PR:LPO]LPO | If this set is decidable, then either 1 \in A or 1 \notin A. In the first case there exists n\in \mathbb {N} with a_n=1. In the second case there cannot be such an n. Hence, in that case, \forall {n \in \mathbb {N}} : {a_n=0}. | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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07c1afdff99229ac99c80c4cd201b746dae49d85 | subsection | 23 | 223 | Metastability | In a program suggested by Tao , it is proposed to recover the “finite” (constructive) content of theorems by replacing them with logically (using classical logic) equivalent ones that can be proved by “finite methods.” For example, since there often is no way to attain the Cauchy condition it is replaced with the follo... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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d6222c94c85d8185e56d713d23c74f2975a55c13 | subsection | 24 | 223 | Metastability | Hence a_{i}=0 for all i > N, that is we only need to check finitely many entries to see if (a_n)_{n \geqslant 1} consists of 0s or whether there is a term equalling 1.One might thus hope that there is maybe a chance that every bounded, metastable sequence converges. However, also this statement is equivalent to [PR:LPO... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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aba85f47de313f16ea2477fc0c59371831b13a84 | subsection | 25 | 223 | Metastability | In the first case, since (a_n)_{n \geqslant 1} has at most one 1, for all i \in [f(1)+1,f(f(1)+1)] we have a_{i}=0. In both cases there exists m such that, regardless of \varepsilon >0, we have\forall {i,j \in [m,f(m)]} : {*{a_{i} - a_{j}} = 0 < \varepsilon } \ ;that is (a_n)_{n \geqslant 1} is metastable. Now if this ... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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6f9b295c0596a7ec2fa1bf581217f14f8114544b | subsection | 26 | 223 | Hillam's theorem | Hillam's theorem states that if f:[0,1] \rightarrow [0,1] is a continuous map and one defines a sequence by choosing an arbitrary x_{0} \in [0,1] and then taking x_{n+1} = f(x_n) for all n > 0, then*{x_n - x_{n+1}} \rightarrow 0 \iff (x_n)_{n \geqslant 1} \text{ is Cauchy} \ .The interesting direction here is the one f... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.0495295375585556,
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0.010238545946776867,
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-0.038207631558179855,
0.026138046756386757,... | |
db755298dd9b8f9779656527b1f449f0bdfec826 | subsection | 27 | 223 | Hillam's theorem | Then, by Bernoulli's inequalityx_{\infty }/2 < x_n = 1 - (1-a)^n \leqslant 1 - 1 - N(-a) = Na \ .Hence a > x_{\infty }/(2N), after dividing by N, and therefore a >0. | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.06196674704551697,
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9cfd39a18ab9cbb60a60fd77a6b969ece27d938d | subsection | 28 | 223 | Cardinality | It is a well known exercise in first year mathematics to prove that there is a bijection between [0,1), and (0,1) and that therefore both sets have the same cardinality.Proposition 2.9
[PR:LPO]LPO is equivalent to the existence of a strongly extensional bijectionTo be precise: by bijection we obviously mean surjective... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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-0.02441473864018917,
-0.004... | |
4ebf0ec189b848f502f397956fdab0139945ac74 | subsection | 29 | 223 | [PR:WLPO]WLPO | Slightly weaker than [PR:LPO]LPO is the weak limited principle of omniscience.
[leftmargin=2em,rightmargin=3em,skipabove=1em,skipbelow=1em, innerleftmargin=-1em,innerrightmargin=0em, nobreak=true, linecolor=blueish, linewidth=5pt, bottomline=false, topline=false, rightline=false, backgroundcolor=blueish!10]([PR:WLPO]WL... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.06629416346549988,
0.014429358765482903,
0.00033645928488112986,
-0.007664881180971861,
-0.01214782614260912,
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0.04129650443792343,
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0.0013792207464575768,
-0.0007005793158896267,
... | |
aa2b2580177e13e2596ef85ada116a60c1b95fc0 | subsection | 30 | 223 | [PR:WLPO]WLPO | The “weak support”
S^w_f = \mbox{$\left\lbrace \,x \in [0,1] \, | \,\lnot \left(f(x) = 0 \right) \,\right\rbrace $}
of a point-wise continuous function f:[0,1] \rightarrow \mathbb {R} is located whenever it is inhabited.
(Strong Intermediate Value Theorem) For any point-wise continuous function f: [0,1] \rightarrow... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.045746099203825,
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-0.007000007666647434,
-0.0023670324590057135,
0.0... | |
e98edb08392968c4e360e92152b20c4de96f30e8 | subsection | 31 | 223 | [PR:WLPO]WLPO | Construct the piecewise linear function f byf(x) = {\left\lbrace \begin{array}{ll}
(3-3{a}) x - 1& x \in *{0,\frac{1}{3}} \ , \\
-{a} & x \in *{\frac{1}{3},\frac{2}{3}} \ , \\
3(1+{a}) x -2 - 3{a} & x \in *{\frac{2}{3},1} \ .
\end{array}\right.}
[Figure: We think of a \geqslant 0 as being very small. The function f is... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03462886065244675,
0.03496447205543518,
-0.02608605846762657,
-0.021143436431884766,
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0.0005587145569734275,
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-0.0022005343344062567,
0.025247031822800636,
... | |
636a0a0bbdd954e578614aab605409919bd2883c | subsection | 32 | 223 | [PR:WLPO]WLPO | Now consider f:[0,1] \rightarrow \mathbb {R} defined byf(x) = {\left\lbrace \begin{array}{ll}
a & \text{for } x \in *{0,\frac{1}{2}} \\
(1-a)2x -1 +2a & \text{for } x \in *{\frac{1}{2},1}
\end{array}\right.}Then f(1)=1, so 1 \in S_{f}. If the support is located that means that \delta =d(0,S_{f}) exists. Now either \del... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.06895726919174194,
0.03737728297710419,
-0.0268353633582592,
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-0.006777492817491293,
-0.016446003690361977,
0.012845580466091633,
0.014386439695954323,
0.002... | |
2cf07d1a8309f3b8117961b4ca203285abb1762c | subsection | 33 | 223 | [PR:WLPO]WLPO | One technical point worth pointing out is that one needs to use a variation of Ishihara's tricks .As a direct consequence of this we also have that the Cantor-Bernstein-Schräder theorem implies [PR:WLPO]WLPO, even when restricted to functions between subsets of \mathbb {R}, since x \mapsto \frac{1+x}{2} and the identit... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.06343269348144531,
0.04874976724386215,
-0.013126111589372158,
0.044781409204006195,
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-0.026099594309926033,
-0.02307753451168537,
0... | |
2bb578cd5be66228226ab13874d69afe392bf528 | subsection | 34 | 223 | The Rising Sun Lemma, and [PR:LPO]LPO and [PR:WLPO]WLPO | We can use Proposition REF to show two more equivalences of [PR:LPO]LPO and [PR:WLPO]WLPO; namely (versions of) the Rising Sun Lemma, which can be used to prove the Hardy-Littlewood maximal inequality *pp. 143.Proposition 3.8
[PR:LPO]LPO is equivalent to the following statement.Consider a continuous function f:[0,1] \... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.034459151327610016,
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0.010179031640291214,
-0.023364465683698654,
0... | |
ae93815c047f2b24ede39be749ba028b67e17f0e | subsection | 35 | 223 | The Rising Sun Lemma, and [PR:LPO]LPO and [PR:WLPO]WLPO | To see this consider the piece-wise linear function f:[0,1] \rightarrow [0,1] defined by
f(x) = {\left\lbrace \begin{array}{ll}
-2x +1 & x \in *{0,\frac{1}{3}} \\
x & x \in *{\frac{1}{3},\frac{2}{3}} \\
-2x + 2& x \in *{\frac{2}{3},1} \ .
\end{array}\right.}
For this function the set E from the Rising Sun Lemma is *{\... | {
"cite_spans": []
} | 1804.05495 | Constructive Reverse Mathematics | [
"Hannes Diener"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04632522910833359,
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-0.029189778491854668,
-0.02267433889210224,
-0.030502021312713623,
0.0... |
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