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a02b60ca7caeed89a8e59580d1b2926052aa287d | subsection | 264 | 279 | The group | Math.,
volume=20,
number=8,
pages=1850014, 43,
url=https://doi.org/10.1142/S0219199718500141,
review=3883405,Kobayashibook
author=Kobayashi, Shoshichi,
title=Differential geometry of complex vector bundles,
series=Publications of the Mathematical Society of Japan,
publisher=Princeton University Press, Princeton, NJ; Pr... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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708257d80117e81a6821444fa162be5f48276122 | subsection | 265 | 279 | The group | Topol.,
volume=13,
number=4,
pages=18771944,
url=http://dx.doi.org.umiss.idm.oclc.org/10.2140/gt.2009.13.1877,
review=2497314,McLeanMBSequencemisc
author=McLean, Mark,
title=Computing symplectic homology of affine varieties (using spectral
sequences),
date=2016,
url=http://www.math.stonybrook.edu/ markmclean/talks/spec... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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357be215d95325b3dd802681d0a8c2985722008a | subsection | 266 | 279 | The group | Symplectic Geom.,
volume=11,
number=1,
pages=109133,
url=http://projecteuclid.org.umiss.idm.oclc.org/euclid.jsg/1362146735,
review=3022923,Pascaleffunpublished
author=Pascaleff, James,
title=On the symplectic cohomology of log Calabi–Yau surfaces,
date=2013,
note=arXiv:1304.5298,Ritterarticle
author=Ritter, Alexander F... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
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] | 2,018 | en | Mathematics | [
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278a25c8838fab28f0691afef0bd97e4db0836ab | subsection | 267 | 279 | The group | Ann.,
volume=368,
number=1-2,
pages=367390,
url=https://doi.org/10.1007/s00208-016-1478-y,
review=3651577,SiebertGWarticle
author=Siebert, Bernd,
title=Algebraic and symplectic Gromov–Witten invariants coincide,
date=1999,
ISSN=0373-0956,
journal=Ann. Inst. Fourier (Grenoble),
volume=49,
number=6,
pages=17431795,
url=h... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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bc0a4593112d1de009142c5f86b03159fd09fee4 | subsection | 268 | 279 | The group | Given k\ge 2,\partial {m}_{k+1} &= 2 \, n_L(M,m) \, \widehat{M}_{k-1} \,+ \,\big ( n_{L_1} \, + \, n_{L_2} \big )\, \widehat{m}_k = \\
&= 2 \times 1 \, \widehat{M}_{k-1} \, + \, \big (1 + 1 \big )\, \widehat{m}_k = 2 \, \widehat{M}_{k-1} + 2 \, \widehat{m}_kand\partial {M}_{k} &= \big ( n_{L_1} \, + \, n_{L_2} \big )\,... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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65c5dd6fd29218cff0d56af08a98bfb505c8a507 | subsection | 269 | 279 | The group | We can conclude the following.Proposition 11.1SH_*(T^*S^2 ; \mathbb {Z}) = \mathbb {Z}\left\langle c, {m}_{1}, e, {M}_{k} - {m}_{k+1}, \widehat{M}_{k} \right\rangle \oplus \mathbb {Z}/ 2 \left\langle e + \widehat{m}_{1}, \widehat{M}_{k} + \widehat{m}_{k+1} \right\ranglewhere we take all k \ge 1.We can compare these res... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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5685ef16ddf2e7e75ec2e5a16afb72a6a3e985b0 | subsection | 270 | 279 | The group | Phys.,
volume=24,
publisher=Eur. Math. Soc., Zürich,
pages=271485,
review=3444367,AlbersFrauenfelderNegativeLineBundlesarticle
author=Albers, Peter,
author=Frauenfelder, Urs,
title=Floer homology for negative line bundles and Reeb chords in
prequantization spaces,
date=2009,
ISSN=1930-5311,
journal=J. Mod. Dyn.,
volume... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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cdedb36f5f8b35c6562ee9f630632bbff2cb7caa | subsection | 271 | 279 | The group | Topol.,
volume=7,
pages=799888,
url=https://doi.org/10.2140/gt.2003.7.799,
review=2026549,BiranBarriersarticle
author=Biran, Paul,
title=Lagrangian barriers and symplectic embeddings,
date=2001,
ISSN=1016-443X,
journal=Geom. Funct. Anal.,
volume=11,
number=3,
pages=407464,
url=http://dx.doi.org/10.1007/PL00001678,
revi... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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... |
7b90b052629b9c2791a31712dc22e2deea4e590b | subsection | 272 | 279 | The group | S.,
author=Yan, Jun,
title=The loop homology algebra of spheres and projective spaces,
date=2004,
booktitle=Categorical decomposition techniques in algebraic topology
(Isle of Skye, 2001),
series=Progr. Math.,
volume=215,
publisher=Birkhäuser, Basel,
pages=7792,
review=2039760,DiogoLisiSplitunpublished
author=Diogo, Lu... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
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71717dda6eb88ed4e076eb49594aea3877e25581 | subsection | 273 | 279 | The group | Math.,
volume=41,
number=6,
pages=775813,
url=http://dx.doi.org.umiss.idm.oclc.org/10.1002/cpa.3160410603,
review=948771,GanatraThesisbook
author=Ganatra, Sheel,
title=Symplectic Cohomology and Duality for the Wrapped Fukaya
Category,
publisher=ProQuest LLC, Ann Arbor, MI,
date=2012,
url=http://gateway.proquest.com.umi... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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98f1fd538b77fbf115155bf485311eb02327fb4d | subsection | 274 | 279 | The group | Math.,
volume=82,
number=2,
pages=307347,
url=https://doi.org/10.1007/BF01388806,
review=809718,HuRuanarticle
author=Hu, JianXun,
author=Ruan, YongBin,
title=Positive divisors in symplectic geometry,
date=2013,
ISSN=1674-7283,
journal=Sci. China Math.,
volume=56,
number=6,
pages=11291144,
url=http://dx.doi.org/10.1007/... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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bd1065e4e187875ebf5f7806a93aa59b2480f7a1 | subsection | 275 | 279 | The group | Math.,
volume=20,
number=8,
pages=1850014, 43,
url=https://doi.org/10.1142/S0219199718500141,
review=3883405,Kobayashibook
author=Kobayashi, Shoshichi,
title=Differential geometry of complex vector bundles,
series=Publications of the Mathematical Society of Japan,
publisher=Princeton University Press, Princeton, NJ; Pr... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
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02c62be8edf4688805564220f6f56cafb5a76338 | subsection | 276 | 279 | The group | Topol.,
volume=13,
number=4,
pages=18771944,
url=http://dx.doi.org.umiss.idm.oclc.org/10.2140/gt.2009.13.1877,
review=2497314,McLeanMBSequencemisc
author=McLean, Mark,
title=Computing symplectic homology of affine varieties (using spectral
sequences),
date=2016,
url=http://www.math.stonybrook.edu/ markmclean/talks/spec... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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70df35567d88f2267e47cecdff165ed3b08ae413 | subsection | 277 | 279 | The group | Symplectic Geom.,
volume=11,
number=1,
pages=109133,
url=http://projecteuclid.org.umiss.idm.oclc.org/euclid.jsg/1362146735,
review=3022923,Pascaleffunpublished
author=Pascaleff, James,
title=On the symplectic cohomology of log Calabi–Yau surfaces,
date=2013,
note=arXiv:1304.5298,Ritterarticle
author=Ritter, Alexander F... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
"math.SG"
] | 2,018 | en | Mathematics | [
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6b8b5d0076acc770c23b9e6fb71606d3ae66e8dc | subsection | 278 | 279 | The group | Ann.,
volume=368,
number=1-2,
pages=367390,
url=https://doi.org/10.1007/s00208-016-1478-y,
review=3651577,SiebertGWarticle
author=Siebert, Bernd,
title=Algebraic and symplectic Gromov–Witten invariants coincide,
date=1999,
ISSN=0373-0956,
journal=Ann. Inst. Fourier (Grenoble),
volume=49,
number=6,
pages=17431795,
url=h... | {
"cite_spans": []
} | 10.1112/topo.12105 | 1804.08014 | Symplectic Homology of complements of smooth divisors | [
"Luís Diogo",
"Samuel T. Lisi"
] | [
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7844f721afef1f039ce3d1605a5cf82a43fc6895 | abstract | 0 | 47 | Abstract | We present Lampray: a multi-group long characteristics ray tracing method for
adaptive mesh radiation hydrodynamics in the Ramses code. It avoids diffusion,
captures shadows, and treats colliding beams correctly, and therefore
complements existing moment-based ray tracing in Ramses. Lampray includes
different options f... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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c88276a93fd0568d5f039fc12017aba8685d82d7 | subsection | 1 | 47 | Introduction | The interaction between radiation and matter is important for a majority of objects in the Universe, and often required for accurate,
quantitative numerical modelling in astrophysics. In the interstellar medium (ISM) the role of radiative stellar feedback versus feedback from
supernovae and galactic dynamics is debated... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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83544a01be67bea13f17503b47889a7e9f0bfd27 | subsection | 2 | 47 | Introduction | Both
flux-limited diffusion , , and the M1 method , have been
implemented on adaptive mesh for use in radiation-hydrodynamics. rayramses is a ray-tracing tool for post-processing of
cosmological simulations using curved rays to capture gravitational lensing, while provide an algorithm to estimate column densities
takin... | {
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} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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d7c7f6cd543c4c7e78c18ff3c15376fc6d04c9c3 | subsection | 3 | 47 | Numerical methods | Radiation hydrodynamics model the interaction and evolution of radiation and matter with a fluid description. Radiation can transport energy
and momentum through emission, absorption, and scattering. In addition, when high-energy photons are present radiation has important consequences
for the chemical composition of t... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
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32d92ede080effb9b0f6b509448aaec8cea0f3f0 | subsection | 4 | 47 | Numerical methods | The molecular weight of the gas is \mu = 1 / (\sum X_s / m_s).lampray does not compute the radiative pressure and momentum, and we have for clarity
left out other contributions from e.g. magnetic fields, self-gravity, and external forces, which are included in ramses.The dynamical time-scale of the fluid is dictated by... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
-0.004334619268774986,
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955f04ba5c434787618b2a00fb2a5a112c47b24e | subsection | 5 | 47 | Numerical methods | The mean intensity, which is the main input to compute the effect of radiation on matter, isJ_{\nu }\left({x}\right)=\frac{1}{4\pi }\int _{0}^{4\pi }I_{\nu }\left({x},\Omega \right)\mathrm {d}\Omega \,,while the intensity of the emitted radiation is given by the source functionS_{\nu }\left({x}\right)=\frac{j_{*,\nu }\... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0.0... | |
ca7c2d7625b4355c3b850628a346e31a72033527 | subsection | 6 | 47 | Numerical methods | In particular, for a photo-ionization or photo-dissociation reactionk_{j} = 4\pi \int ^\infty _{\nu _{\mathrm {th}}} \frac{\sigma _{\mathrm {j},\nu } J_\nu }{h\nu } \mathrm {d}\nu ,where \sigma _{j,\nu } is the frequency dependent photo cross section for reaction j and \nu _{\mathrm {th}} is the energy threshold of the... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
ccdfa6749df8dc208dd45d1aba3e43f5f18bb2f7 | subsection | 7 | 47 | Radiative transfer | The radiative transfer equation is solved explicitly using long characteristics ray tracing. We discretise the intensity in frequency into a set of bins
so I_{\textrm {bin}} = \int ^{\nu _{u\left(\textrm {bin}\right)}} _{\nu _{l\left(\textrm {bin}\right)}} I_\nu \textrm {d}\nu , where \nu _{u\left(\textrm {bin}\right)}... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0cb52685c68033efdbc5a4a3bd7cb19e58666679 | subsection | 8 | 47 | Radiative transfer | Finally, the chemical network is evolved in each cell ignoring any PdV work using the new radiation field in each cell, giving a net heating / cooling
term by which the total energy is changed. A new mean molecular weight and adiabatic index is also computed after the chemistry update.
The timestep can be adaptively su... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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43dc06bee64b098a46fe1fcc67aa4e5beee8372f | subsection | 9 | 47 | Coupling to chemistry | We have implemented two methods that solve Eqn.(REF ), Eqn.(REF ) and Eqn.(REF ). The first method
(Sect. REF ) solves it for any set of photo-reactions and in connection with an arbitrary chemical network using the krome
package. It however requires a fairly restrictive time step if a correct propagation speed of fast... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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a770fa4db702bf2f806d3b532dd169fe74d9989c | subsection | 10 | 47 | Photo-chemistry with KROME | krome is a framework that generates code from a description of a chemical network to efficiently solve its time-dependent evolution. It is able to compute photo-chemical rate coefficients based on frequency-binned mean intensities and on frequency-dependent photo cross-sections where available, and include the latter t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1101,
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"raw": "Abel T., Norman M. L., Madau P., 1999, , 523, 66",
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"start": 977
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{
"arxiv_id": "",
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"e... | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
05193fe5d66544cb482a2e717826d3cb9ffc3dfc | subsection | 11 | 47 | Photo-chemistry with KROME | In this case, the optical depth element is\Delta \tau _{\nu }=\sum _{\mathrm {r}\in \mathcal {P}}\sigma _{\nu ,\mathrm {r}}n_{s(\mathrm {r})}\Delta l\,,where the sum is over all photo-reactions \mathcal {P}: s(\mathrm {r}) + \gamma \rightarrow p(\mathrm {r}). s(\mathrm {r}) is the reactant and
p(r) is the product in re... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0.... | |
ca7d120b5980ecf8c37f111ed867a4ba618495c3 | subsection | 12 | 47 | Photo-chemistry with KROME | In our example calculation of the structure of a photo-ionized region irradiated with the spectrum from (Sect. REF ), we find that the most accurate method is to use the unweighted average, combined with careful placement of bin limits. To describe the ionization transitions of H and He as accurately as possible with f... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0.... | |
80efa1e1a71616ce08755cf182beae7c2be1a1b1 | subsection | 13 | 47 | Photo-chemistry with KROME | The optical depth can be expressed in terms of the
dust column density, as well as the visual extinction as\tau _{\mathrm {bin}} = \kappa _{\mathrm {bin,d}}f_{\mathrm {d}}\frac{N_{\mathrm {H}}\,m_\mathrm {p}}{ x_{\mathrm {H}}} = \gamma _\mathrm {bin}A_\mathrm {v}\, .Here, f_{\mathrm {d}} is the dust mass fraction, N_{\... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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2918e0736a4e040de84efa1715b2e05eecccc527 | subsection | 14 | 47 | Hydrogen ionization with | In an explicit method, the evolution of radiatively driven ionization fronts can require a very small time step and cell size to resolve
correctly the ionization front, and obtain the right propagation speed. c2ray is an explicitly photon-conserving
and causally implicit photo-ionization method that takes the ionizatio... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
a6e2767c07db11f9452670d29001dbd8e9345786 | subsection | 15 | 47 | Interstellar radiation field as diffuse emission | In earlier 3D dynamical models of PDRs, the external radiation field has been simulated as an ISRF, attenuated through
a column density measured from the edge of the simulation box , . While this works well when
modelling an isolated cloud that stays more or less centred in the simulated box, when simulating turbulent ... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0.036... | |
03f2d27e08c1a4f7e88ae03822c08f94bb468967 | subsection | 16 | 47 | Self-shielding of H2 and CO | The self-shielding of \textrm {H}_2 and CO needs to be treated carefully to get qualitatively correct results for the two most abundant
molecules in the ISM. Since this requires that one follows in detail their excited states and the associated frequencies ,
this is not currently achievable in 3D radiation hydrodynamic... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
875ddc3916158616289c33f44e0eafa1b5864efc | subsection | 17 | 47 | Self-shielding of H2 and CO | 23, slightly relabelled here)I_{\mathrm {bin}}\left(n\right)=\sum _{i=0}^{n-1}\Delta l_{i}j_{*,\mathrm {bin},i}
\mathcal {S}_{\mathrm {bin}}\left(\sum _{j=i}^{n}\mathbf {n}_{spec,j}\Delta l_{j}\right)\,,where I_{\mathrm {bin}}\left(n\right)
is the intensity in the frequency range “bin” in cell n, \Delta l_{i}
is the pa... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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6c46f3d4573abfeb10d051458ff3b7aa4b868bc5 | subsection | 18 | 47 | Thermal balance of dust grains | Grain surface chemistry depends on the temperature of the dust grains, and is used in our chemical network for calculating
the {H2} formation rate, and the freeze-out and evaporation of {H2O} and {CO} ices. The kinetic cooling or
heating of the gas also depends on the temperature differential between dust and gas. In w... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
713235558295efce415b4bd7176b1f9151eb1b2c | subsection | 19 | 47 | Thermal balance of dust grains | Then the radiation absorbed by the dust grains can be calculated as\Gamma _\mathrm {abs} = \sum _{i \in \mathrm {bins}} J_i \tilde{\Gamma }_\mathrm {abs,i}\,,where\tilde{\Gamma }_\mathrm {abs,i} = \int _{E_\mathrm {min, i}}^{E_\mathrm {max, i}} {\rm d}E\, \int {\rm d}a\, \pi a^2 \varphi (a) \frac{Q_\mathrm {abs}(a,E)}{... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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0.00... | |
0cd6160913009e6a227d8c52d0e507b52367f461 | subsection | 20 | 47 | Adsorption and desorption of CO and H | For the adsorption reaction rate coefficients per gas molecule for species s, k_\mathrm {ads,s}, we follow , .k_\mathrm {ads,s} = \sigma _s v_\mathrm {gas,s} n_\mathrm {dust}\,,where \sigma _s is the cross section, v_\mathrm {gas,s} is the thermal velocity of species s, and n_\mathrm {dust} is the number
density of dus... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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b043ed6a788d6cd459929beda00b335ceaa2a8ab | subsection | 21 | 47 | Adsorption and desorption of CO and H | The cosmic-ray photodesorption rate , isk_\mathrm {cr-des,s} = f(70\,{\rm K})\,k_\mathrm {th-des,s}(70\,{\rm K})\,,where the fraction of time spent by the grain at 70 K is f(70\,{\rm K})=3.16\times 10^{-19}\zeta /\zeta _0,
\zeta _0=1.3\times 10^{-17} s^{-1}, and \zeta is the cosmic ray ionization rate. Although for the... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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... | |
ccc556fbb9eafd7bc37e8780b2f1a45fbf8eff57 | subsection | 22 | 47 | Implementation | The hybrid-characteristics method described for point-sources in and diffuse radiation in works well on patch-based hierarchical adaptive meshes. ramses, however, has a cell-based octree, based on the fully threaded tree by , and the “patches” that would be connected by short-characteristics rays, would consists of onl... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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ef23179346eb1c9bca7d5238a69aa43388c2cd56 | subsection | 23 | 47 | Ray Tracing | The radiation field is split into two components: A diffuse field for extended sources of emission, and a point source field. The diffuse field is discretised in solid angle into a set of equal-solid-angle directions using the healpix scheme , and for each direction into a set of parallel rays that cover the computatio... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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5c0d2c123e15524546834febd07e2c85405cb81e | subsection | 24 | 47 | Ray Tracing | Going from level \ell to \ell +1 we can proceed as just described. Lets label by d the distance from the ray plane cell centre to the projected cell centre in units of the cell size at level \ell , and the vector of indices at level \ell +1 by {i}_{\ell +1}=(2i, 2i+1)+q_{\ell +1}, whereq_{l+1}=
{\left\lbrace \begin{arr... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
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376630b7d08f30ae839f22ffd8dd86fe62c81f1d | subsection | 25 | 47 | Ray Tracing | We use the healpix nested pixel index as unique ray id for matching up ray points and load balancing rays. | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
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14cb39897cd895f105480457346fe3958a4beb45 | subsection | 26 | 47 | Interpolation | When interpolating from cells to rays, we are interpolating in three dimensions from a uniform grid so we can get a second-order accurate, conservative interpolation with triangular-shaped cloud (TSC) interpolation. Where support points are missing, they are reconstructed from the eight surrounding parent cells using c... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
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] | 2,018 | en | Physics | [
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857dadbf8b7d3eb9ed3887610304d60c6677102e | subsection | 27 | 47 | Interpolation | Selected results are summarised in Tab. REF . Ray discretisation and error for tests I3 and I8 are shown in Fig. REF .Based on these tests, we choose to use the unstructured variant of TSC interpolation in the plane perpendicular to the main ray direction and the R1 coverage strategy.
[Table: Summary of tests of interp... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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] | 2,018 | en | Physics | [
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ae9c7c5cb601d3b72283a6f4e23cf797e780dc71 | subsection | 28 | 47 | Parallelisation | The ramses grid is distributed over different MPI ranks, and when constructing the hash table of rays in Sect. REF , we need to merge them before constructing ray points. This is done by extracting an array of the keys and values on each rank, gathering them on a single master rank, inserting them into the master rank'... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
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bf9d21ac88531e36cde917cd3b92be4095e9f77d | subsection | 29 | 47 | Scaling considerations | As described, each rank must have access to a global hash table of all rays. Inevitably, for a large enough number of ranks, merging of tables will dominate computational time, and holding the table will dominate memory usage. Here we describe a way to solve this problem. The method is however not currently implemented... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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764991fcdf73275507193b26629d67a26789e0f5 | subsection | 30 | 47 | Boundary conditions | We have implemented the following boundary conditions for the radiation. A prescribed incoming intensity (I(0)=I_{0}, I(L)=I_{L}), which is useful in test problems, radiative equilibrium (I(0)=S(0), I(L)=S(L)), and periodic boundaries for the radiation. Periodicity is useful in situations with periodic boundaries for t... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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7bebbdc3d1169cbbae0b39799c2334ebe015d4be | subsection | 31 | 47 | Tests | lampray is applied to a set of test problems where the result can be compared to a well-known analytical solution or to other established codes. The tests include the expansion of an HII region around a new star (Test 1), trapping of an ionization front by a dense clump of gas (Test 2), photo-evaporation of a dense clu... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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924951065c045d9fd0c41463bfda7ce4dd804981 | subsection | 32 | 47 | TEST 1: HII region expansion | The cosmological code comparison papers and include a series of four tests with the expansion of an HII region around a point source. We reproduce three of these here, starting with a monochromatic source in an isothermal, static, uniform density medium, then replacing the source with a T=10^5 K black body and includin... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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416c84458d6197f83e7783cb809aac3188819c43 | subsection | 33 | 47 | TEST 1.1: Isothermal gas | The first test reproduces Test 1, which has monochromatic radiation with photon energy 13.6 eV, is isothermal at T=10^4 K, has box length L=6.6 kpc, and is resolved by 128^3 cells.
At the final time, the ionization front is spherical (Fig. REF ) with ionization structure in good agreement with the benchmark codes. The ... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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2a428421875c30ef07fc804fed5244c92b372cbd | subsection | 34 | 47 | TEST 1.2: Non-isothermal gas | The second test reproduces Test 2, which has initial temperature T=100 K and includes heating and cooling. The box length is still L=6.6 kpc. Again, the ionization structure agrees well, especially with the ray tracing codes (Fig. REF ), and the temperature structure is similar to that of especially rsph, with signific... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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bf977a15b274fa4ecf9cf8e0a0f986d335599528 | subsection | 35 | 47 | TEST 1.3: Hydrodynamics | The third test reproduces Test 5. It has the same parameters as Test 1.2, except that the box length is L=15 kpc and hydrodynamics is included. The position of the ionization front (Fig. REF ) follows the agreement of all the benchmarked codes except Enzo, which is monochromatic. In density and Mach number, the double ... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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8e0a8aa046629eb37d57047882c41626ad985078 | subsection | 36 | 47 | TEST 2: ionization front trapping by a dense clump | This test is a replication of Test 3 from the code comparison paper . It tests hydrogen photo-ionization chemistry and related heating and cooling processes, as well as the diffusivity of radiation transport and ability to form a shadow. The setup is briefly repeated here. The simulated volume is resolved by 128^3 cell... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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09a77ba5a73a8fbe55da6e0670e2f915d8a65510 | subsection | 37 | 47 | TEST 3: Photo-evaporation of a dense clump | This test corresponds to Test 7 from the code comparison paper . It is identical to the above test, except hydrodynamics is included, allowing the clump to evaporate. At 1 Myr, the gas is still almost static, and the structure resembles the previous test, with virtually no flaring from the ionization front (Fig. REF ).... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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d83e1ec4a3c79cf89945166436f20001c162eaab | subsection | 38 | 47 | TEST 4: Plane-parallel PDR | A motivation for developing the present method is to study the effect of the interstellar radiation field on molecular clouds. Therefore we
wish to test it on the case of a PDR. The problem is so rich in micro-physical processes that no consensus solution exists, even in the
simplified case of a uniform density, plane ... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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46926d96e300335ed7a5430c4d165a2f42424513 | subsection | 39 | 47 | TEST 4: Plane-parallel PDR | REF ) as is used in . The ionization transition to H+ and He+, and the dissociation transition for H2 agrees well with Richings' result (Fig. REF ). There is roughly a 50% deviation in the optically thin ionization degree of He+, which is due to the coarseness of frequency bins. The locations of the molecular transitio... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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51ce581efc38287c33f221764e9ffc4d398fadb3 | subsection | 40 | 47 | TEST 4: Plane-parallel PDR | In the top panel are shownH2: n_{{H2}} / n_{{H}_\mathrm {tot}}, CO: n_{{CO}} / n_{{C}_\mathrm {tot}}, OH: n_{{OH}} / n_{{O}_\mathrm {tot}} and H2O: n_{{H2O}} / n_{{O}_\mathrm {tot}}.In the lower panel are shown E: n_{{e-}} / n_{{H}_\mathrm {tot}}, H: n_{{H}} / n_{{H}_\mathrm {tot}},H+: n_{{H+}} / n_{{H}_\mathrm {tot}},... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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489ad82a6cf308bd2c528e9dde9c8433070d7bc9 | subsection | 41 | 47 | TEST 4: Plane-parallel PDR | They are similar to the previous test in that they use constant pressure, with n T=10^3\, \mathrm {K}\, \mathrm {cm}^{-3} as a model for an ISM in pressure equilibrium, except that at high densities a temperature floor is imposed at T=10\, \textrm {K} for the gas temperature, to emulate an isothermal molecular cloud. T... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
"astro-ph.GA"
] | 2,018 | en | Physics | [
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cc3bc65132f14fc7d30c1bcce17590a7d9c61248 | subsection | 42 | 47 | TEST 5: Spherical PDR | The final PDR test uses the same density and temperature profile as above, but with r being the radius in a spherically symmetric cloud. 192 healpix directions are used (N_\mathrm {side}=4), with outflow boundaries for the radiation on all faces. The emission is set such that we get one Black flux at a distance of r_0+... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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4749559ea6caeda2eec094c5de3ff6173c6399f5 | subsection | 43 | 47 | TEST 5: Spherical PDR | In the 1D solution, Eqn.() is used to account for the clouds spherical shape.][Figure: Test 5: Same as Fig. , but with two extra adaptive mesh levels. Refinement is applied where the relative gradient in H, H+ or H2 exceeds 0.2, and the abundance of that species is above 1% by mass.]The test is repeated with two additi... | {
"cite_spans": []
} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
"Troels Haugbølle",
"Tommaso Grassi"
] | [
"astro-ph.SR",
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9db80d41e5efcc485150207085d3ad1ed2c7e236 | subsection | 44 | 47 | Performance and scaling | The weak-scaling properties of the photo-chemistry solver with krome are measured on the spherical cloud setup of Test 5, except using 48 healpix directions instead of 192. The test is performed on the local HPC cluster at University of Copenhagen using nodes with 2x 10-core Xeon E5-2680v2 running at 2.8 GHz with 64 GB... | {
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} | 1809.05541 | Lampray: Multi-group long characteristics ray tracing for adaptive mesh
radiation hydrodynamics | [
"Troels Frostholm",
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"Tommaso Grassi"
] | [
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870e453eb39d3a8c250183ad6a78def53b7190d8 | subsection | 45 | 47 | Summary and future work | We have presented a new method we call lampray, and its implementation into the ramses code, for solving the time-independent radiative transfer problem on an adaptive octree mesh by means of tracing rays, that each cover the entire length of the domain. The diffuse radiation field is represented by rays that uniformly... | {
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8edb1195b1ff6d666fae4cdbff85c6deec7401cc | subsection | 46 | 47 | Summary and future work | Electron and OH density in the optically thick part differs due to the omission of metals that would donate electrons if included.A new method for including an interstellar radiation field in periodic boundary gas simulations has been presented, in which the interstellar field is modelled as an emissivity in the diffus... | {
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adfff87b06630d728c81ed85c0833f7ed08fb101 | abstract | 0 | 56 | Abstract | Long before we learn to construct the field of rational numbers (out of the
ring of integers) at university, we learn how to calculate with fractions at
school. When it comes to "numbers", we are used to a commutative
multiplication, for example 2*3=6=3*2. On the other hand --even before we can
write-- we learn to talk... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
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0868beff054e4161b796599de7f7856f90a6eeb5 | subsection | 1 | 56 | Introduction | Since most of the literature on free fields is almost
inaccessible without a degree in mathematics and difficult
without a specialization in algebra we want to provide
an introduction with focus on the application.
One of the main hurdles is the huge number of concepts
and definitions (for precise formulations), needin... | {
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553ffa28213f1b070dac27709a24e77b40156f33 | subsection | 2 | 56 | Representing Elements | First of all, we need a suitable representation for the elements
in the free field
\mathbb {F} = \mathbb {K}(\!\langle X\rangle \!) of the
free associative algebra \mathbb {K}\langle X\rangle over the
commutative field \mathbb {K} (for example the rational numbers
\mathbb {Q} or the real numbers \mathbb {R}) and the (f... | {
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b49e9f772b3443f42989c07dd46247b9f026ea7e | subsection | 3 | 56 | Representing Elements | So here we could write \pi _r = (1, a, v).)For the polynomial f = xy+yx-yz \in \mathbb {K}\langle X\rangle
an ALS \mathcal {A}_f = (u,A,v) of dimension n=4 is (the
zeros are replaced by lower dots to emphasize the structure)\begin{bmatrix}
1 & -x & -y & . \\
. & 1 & . & -y \\
. & . & 1 & z-x \\
. & . & . & 1
\end{bmat... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
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f6550376f1973b72b34519a0b2c73f06519d55b3 | subsection | 4 | 56 | Representing Elements | A (square) matrix
A \in R^{n \times n} is called full, if
A = PQ with P \in R^{n \times m} and Q \in R^{m \times n}
implies m \ge n
. To show that the full matrices over the free associative algebra
are those which are invertible over the free field (and vice versa)
is very difficult. For details we refer to
. Importan... | {
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a313c102fd9d9f4ecedc6f8a4243e2ce43e80087 | subsection | 5 | 56 | Free Fractions | The main idea (of free fractions) is as simple as in the usage of “classical”
fractions (for elements in \mathbb {Q}):
calculating, factorizing and minimizing
(or cancelling), for example&\frac{2}{3} \cdot \frac{3}{4}
= \frac{6}{12} = \frac{2\cdot 3}{2\cdot 2 \cdot 3}
= \frac{1}{2}\quad \text{or}\\
&\frac{1}{2} + \frac... | {
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fc9066905dce9e3bfa0ea83f382eee6efb043447 | subsection | 6 | 56 | Free Fractions | If we describe (elementary)
transformations in the following, they always refer to
the system matrix A.\underbrace{\begin{bmatrix}
\underline{1} & \underline{0} & \underline{0} & \underline{0}
\end{bmatrix}}_{\mathbox{2cm}{u, left hand side}}
=
\underbrace{\begin{bmatrix}
t_1 & t_2 & t_3 & t_4
\end{bmatrix}}_{\mathbox{... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
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2e376fee529f699c7ec80dc18b601e6bed338e2e | subsection | 7 | 56 | Free Fractions | In this case we say “f is of type (*,1)” or 1 \in L(f) if
1 \in L(\mathcal {A}) respectively
“f is of type (1,*)” or 1 \in R(f) if 1 \in R(\mathcal {A}).Now we will transform this representation step by step such that
the solution of both systems of equations, that is, the computation
of s and t, becomes easier. Those ... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
"Konrad Schrempf"
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c98b9943351b0e56c5e7cac9e0840a7fe0c43445 | subsection | 8 | 56 | Free Fractions | Either by factorization (Section )
or by “abstract” refinement (Section ).
But we should not worry here. The examples in the beginning
are such that we can easily minimize them by “hand”
respectively check their minimality.A last note concerning the system matrix A.
We always write it in the compact form with (at most)... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
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62d5a01d5f44063cf47c053f9fc3e08e355bc8e9 | subsection | 9 | 56 | Left and Right Minimization Steps | For practical computations we repeatedly have to make
admissible linear systems smaller. In concrete situations
it is possible to minimize them. Later, in Section
we will see that there are some subtle details behind the
rather simple looking (left and right) “minimization steps”.
Let us take a closer look on the exam... | {
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ae48e4746e8341c664e5963a31aa3ce97f488e76 | subsection | 10 | 56 | Left and Right Minimization Steps | For the inverse we use the minimal inverse
(Theorem REF ).
And finally, using the multiplication
(Proposition REF )
we obtain an ALS of dimension 5 for f f^{-1}.
for f f^{-1}= 1
with f = xy - z,\mathcal {A} = \left(
\begin{bmatrix}
1 & . & . & . & .
\end{bmatrix},
\begin{bmatrix}
1 & -x & z & . & . \\
. & 1 & -y & . & ... | {
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bad9f6884b292cb88fc1d64ba908c6e8814434a6 | subsection | 11 | 56 | Left and Right Minimization Steps | Given an ALS \mathcal {A} = (u,A,v) with v = [0,\ldots ,0,\lambda ]^{\!\top }
we write also write \mathcal {A} = (1,A,\lambda ).Remark.
The other case f^{-1}f = 1 is somewhat more difficult
because we must not change the first component in the left family.
The trick here is, to work with an “extended” ALS for
1 \cdot f... | {
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6c27c4d77f73692fb693f87dce4efd162016968a | subsection | 12 | 56 | Calculating | One of the main parts of this section is the
construction of a minimal admissible linear system
for the inverse (of an element in the free field)
in Section REF .
The following (simple) construction (of an ALS for the
inverse) is from Proposition REF .
We assume that we have given the inverse of a monomial
f = xyz by t... | {
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8133a29fe6e3980e9548ace15e4dd7df3b8ac797 | subsection | 13 | 56 | Calculating | We denote by I_n the identity matrix
and \Sigma _n the permutation matrix that reverses the order
of rows/columns (of size n)
respectively I and \Sigma if the size is clear from the context. | {
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f083ef7be5fd65faa084efac9c3b522b70946229 | subsection | 14 | 56 | Preliminaries | Let \mathbb {K} be a commutative field,
\overline{\mathbb {K}} its algebraic closure and
X = \lbrace x_1, x_2, \ldots , x_d\rbrace be a finite (non-empty) alphabet.
\mathbb {K}\langle X\rangle denotes the free associative
algebra (or free \mathbb {K}-algebra)
and \mathbb {F} = \mathbb {K}(\!\langle X\rangle \!) its uni... | {
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e7ee69d704fdbfa5defc7b47e7518d8eb9262a6b | subsection | 15 | 56 | Preliminaries | A linear representation of f is a triple \pi _f = (u,A,v) with
u \in \mathbb {K}^{1 \times n}, full
A = A_0 \otimes 1 + A_1 \otimes x_1 + \ldots + A_d \otimes x_d, that is, A is invertible over \mathbb {F},
A_\ell \in \mathbb {K}^{n\times n},
v \in \mathbb {K}^{n\times 1}
and f = u A^{-1}v.
The dimension of \pi _f is \... | {
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a051478ce3996bd6d44820d7190ea04ddc3edea2 | subsection | 16 | 56 | Preliminaries | The following definition is a special case
of the more general admissible systems
and the slightly more general linear representations
.Definition 2.6 (Admissible Linear Systems, Admissible Transformations
)
A linear representation \mathcal {A} = (u,A,v) of f \in \mathbb {F}
is called admissible linear system (ALS) f... | {
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20b927f66c6c6b6a4a21a67104025e1711e6c333 | subsection | 17 | 56 | Preliminaries | An admissible transformation (P,Q) for an ALS \mathcal {A}
is called polynomial if it has the form(P,Q) = \left(
\begin{bmatrix}
1 & \alpha _{1,2} & \ldots & \alpha _{1,n-1} & \alpha _{1,n} \\
& \ddots & \ddots & \vdots & \vdots \\
& & 1 & \alpha _{n-2,n-1} & \alpha _{n-2,n} \\
& & & 1 & \alpha _{n-1,n} \\
& & & & 1
\e... | {
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245012236c5a04c28f1ca83ab14774bb69ea3f4a | subsection | 18 | 56 | Minimal Systems | The main idea is to start with minimal admissible linear
systems and construct minimal ones for the rational operations
(scalar multiplication, sum, product, inverse).
We already have seen the
minimal monomial:Proposition 2.9 (Minimal Monomial
)
Let k \in \mathbb {N} and f= x_{i_1} x_{i_2} \cdots x_{i_k} be a monomial... | {
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39453bc2a4b3bd032725d25ffb0a89fb82406b93 | subsection | 19 | 56 | Rational Operations | “Basic” rational operations (on the level of admissible linear systems)
are easy to formulate. For the multiplication we can provide alternative
constructions yielding minimal admissible linear systems immediately
in special cases, for example the minimal polynomial multiplication
(Proposition REF ).Proposition 2.10 (R... | {
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5b034ee895d3e560807b23631f13b9c9c831ad21 | subsection | 20 | 56 | Rational Operations | Then there
exists an admissible transformation (P,Q) such that
the first column of PAQ is [1,0,\ldots ,0]^{\!\top } and
Pv = [0,\ldots ,0,\lambda ]^{\!\top } for some \lambda \in \mathbb {K}.Proposition 2.12 (Multiplication Type (*,1)
)
Let f,g\in \mathbb {F} \setminus \mathbb {K} be given by the
admissible linear sys... | {
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1817920fd3e6fe082dc0f732927bc5dcd6cd8103 | subsection | 21 | 56 | Disjoint Addition | For disjoint elements f,g \in \mathbb {F}
, that is, \operatorname{rank}(f+g) = \operatorname{rank}(f) + \operatorname{rank}(g),
the addition from
Proposition REF is minimal.
For further details we refer to the remarks after
.
An important result of Cohn and Reutenauer
is the primary decomposition (of elements in the
f... | {
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f43d099a11696fb89ea6e9bbb0c99047055c66ac | subsection | 22 | 56 | Minimal Inverse | The derivation of the minimal inverse in
consists of two major steps (motivated in the beginning
of this section): keeping the form for f = (f^{-1})^{-1}
and distinguishing different cases to ensure minimality.
Notice especially the remark before
how to transfer admissible linear systems into the appropriate
form.The... | {
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66948dff221090c6bcc711cc3805c1dc8d009a16 | subsection | 23 | 56 | Minimal Inverse | \\
-\lambda b & -b^{\prime }\Sigma & -a \\
. & . & 1
\end{bmatrix},
1 \right)
\quad \text{for}\quad \mathcal {A} = \left(1,
\begin{bmatrix}
a & b^{\prime } & b \\
a^{\prime } & B & b^{\prime \prime } \\
. & . & 1
\end{bmatrix},
\lambda \right).f of type (0,0) yields f^{-1} of type (1,1) with \dim (\mathcal {A}^{\prime ... | {
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4a96aa8e8d021191b60991758531463c94deb6a9 | subsection | 24 | 56 | Rational Identities | Using the minimal inverse (Theorem REF )
and the rational operations (Proposition REF )
one can already show non-trivial rational identities very systematically by “hand”.
The following proof is from
.Example 2.14 (Hua's Identity
)
We have:x - \bigl (x^{-1}+ (y^{-1}- x)^{-1}\bigr )^{-1}= xyx.Minimal admissible linear ... | {
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285a53b65b8c71ec786166018f9dd3356ba03f2a | subsection | 25 | 56 | Rational Identities | \\ -1
\end{bmatrix}, \quad s =
\begin{bmatrix}
x^{-1}+ (y^{-1}-x)^{-1}\\
(y^{-1}-x)^{-1}\\
-(1-xy)^{-1}\end{bmatrix}.This system is minimal and —after adding row 3 to row 1
(to eliminate the non-zero entry in the right hand side)—
we apply the (minimal) inverse of type (0,0):\begin{bmatrix}
-1 & -1 & -x & . \\
. & -y &... | {
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d3c172e01b394c53b660985ef01d00f40ede49bb | subsection | 26 | 56 | Factorizing | Since the whole factorization theory originated from a “small”
problem of the minimization of linear representations,
it should lead as a thread through this section.
Somehow this theory has become independent and is interesting
now from a purely algebraic point of view since it
enables to view the free field as a “rin... | {
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d371ba96543e05eb12aaeac011f3e644d796d0fe | subsection | 27 | 56 | Factorizing | And this upper right block of zeros is that one
coming from multiplication (1,*),
see also
.
This yields a “natural” correspondence between
factorizations and upper right zero block structure in the
system matrix (assuming zero entries in the corresponding
components of the right hand side).In other words: One can find... | {
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fd0aa8e20ed0ec88646a4af159661abe9bcc3dec | subsection | 28 | 56 | Minimal Polynomial Multiplication | As an introduction one could take
the multiplication of x and 1-yx
using Proposition REF ,
see also
. The following lemma is needed in Section REF
and (the proof of) the following proposition.Lemma 3.1 ()
Let \mathcal {A} = (u,A,v) = (1,A,\lambda ) be an ALS of dimension n\ge 2
and \mathbb {K}-linearly dependent left... | {
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5414aed6e72c713bef406f87039225403b0a9008 | subsection | 29 | 56 | Polynomial Factorization | The polynomial factorization theory depends on minimal
(polynomial) admissible linear systems. How to obtain such
systems directly is discussed in Section REF .
How to construct them in general is discussed in
Section REF .Remark.
Notice that, although we use (general) admissible linear systems
here to represent polyno... | {
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f85629adc6f17ae549fd89798a7b12a875342120 | subsection | 30 | 56 | Factorization Theory | The general factorization theory is somewhat difficult.
Although it seems to be clear from
the polynomials how it should be, the path to
the divisibility equivalence
(Theorem REF ) is long and stony.
One needs a notion of left (respectively right) divisibility
on the level of minimal admissible linear systems.
This is ... | {
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236b9761b4bcee55fe39b4982a74a1370a5b3017 | subsection | 31 | 56 | Minimal Factor Multiplication | Given two minimal admissible systems,
under which conditions are the multiplications
from Proposition REF and REF
minimal? A special case is the minimal polynomial
multiplication (Proposition REF ).
The general answer is given in
within the (framework of the) general factorization theory. | {
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11080e6cdb606c72941b3d53a716a5b85ba37e61 | subsection | 32 | 56 | General Factorization | Like in the general (minimal) multiplication in the previous subsection
we have to distinguish several cases for the
factorization
. Looking for zero (lower left and upper right) blocks
(of appropriate size)
in the system matrix of a minimal ALS
(similar to the polynomial factorization) is rather
natural when we want t... | {
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178e7bfd601c4cb92c3c06d5300f99991482119c | subsection | 33 | 56 | Examples Factorization | Polynomial factorization is illustrated in detail (step by step) in
.
The general factorization (of a regular element)
is discussed briefly in
. | {
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"raw": "K. Schrempf. On the factorization of non-commutative polynomials (in free associative algebras). Journal of Symbolic Computation, 2018.",
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8599ff8e6148c858cd71ef4617d0d2dad4a85405 | subsection | 34 | 56 | Minimizing | The basic idea of the minimization (of a linear representation)
with left and right minimization steps is surprisingly simple.
If the block structure becomes coarser and a “look” is not
sufficient any more, row and column transformations can be
found by solving a linear system of equations.
That is the essential conten... | {
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3fb89e462d8afe248bc94753c414288278274b28 | subsection | 35 | 56 | Preliminaries and a Standard Form | To be able to formulate statements —in particular
for the minimization— in a convenient way, we need
some notation which formalizes what we have already
used, namely to describe an ALS (and admissible
transformations) in terms of block rows and
columns instead of (single) rows and columns.
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bc14e99be274f4b304b3d996437098b728f4517f | subsection | 36 | 56 | Preliminaries and a Standard Form | The admissible linear system \mathcal {A} is called refined
if all pivot blocks are refined.Definition 4.3 (Standard Admissible Linear System
)
A minimal and refined
ALS \mathcal {A} = (u,A,v) = (1,A,\lambda ),
that is, v = [0,\ldots , 0, \lambda ], is called
standard.Remark.
For a polynomial p given by a standard ALS... | {
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d17803801fa558b0c536ede2057a9b12854159f0 | subsection | 37 | 56 | The Word Problem | One of the difficulties in free fields is (that of) the word problem,
that is, to check whether two admissible linear systems represent
the same element. A solution to the word problem is
. Unfortunately it is hard to apply practically already
for systems of dimension 3.
If those systems are given by minimal admissible... | {
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