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578b8f7f5e377b0ed8eaac2a552666311c1019af | subsection | 363 | 399 | Local-global compatibility for | By Lemma REF and (REF ), we deduce that \Pi is also a subrepresentation of (\operatorname{\mathrm {I}nd}_{\overline{P}_1(\mathbb {Q}_p)}^{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)} \widetilde{\pi }^{\operatorname{\mathrm {a}n}})^{\operatorname{\mathrm {a}n}}/W. Hence (REF ) induces \operatorname{\mathrm {G}L}_3(\mat... | {
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396af60a7f4bb988e455216cee0c9091c3943b91 | subsection | 364 | 399 | Local-global compatibility for | The map f\mapsto \tilde{f} gives a section to (REF ), which concludes the proof.We refer to § REF , § REF for the definition of the subrepresentations \Pi ^r(\lambda , \psi _{\mathcal {L}_r})_0, \Pi ^r(\lambda , \psi _{\mathcal {L}_r}), \Pi ^r(\lambda , \psi _{\mathcal {L}_r})^+ of \widetilde{\Pi }^r(\lambda , \psi _{\... | {
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ce5530437373e233caf4527bc464ac73c5ff7268 | subsection | 365 | 399 | Local-global compatibility for | An injection \Pi ^r(\alpha ,\lambda , \psi _{\mathcal {L}_r})^+ \hookrightarrow \widehat{S}(U^{\wp }, W^{\wp })_{\overline{\rho }}[\mathfrak {m}_{\rho }] extends to:E\big (\Pi ^r(\alpha ,\lambda , \psi _{\mathcal {L}_r})^+, v_{\overline{P}_s}^{\infty }(\alpha ,\lambda ), Ev\big )\longrightarrow \widehat{S}(U^{\wp }, W^... | {
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161f78094dddb946916ec118a66edb8c70946aa1 | subsection | 366 | 399 | Local-global compatibility for | If \psi \notin E\psi _{\mathcal {L}_r}, we have E\psi + E\psi _{\mathcal {L}_r}=\operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, E) and an injection induced by \tilde{f}_0, \tilde{f}_1 (where S_{s,0} is defined as in § REF with \lambda =0) :\Pi ^r(\alpha ,\lambda , \psi _{\mathcal {L}_r})_0 \oplus _{S_{s,0}\otimes... | {
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4ec7ee7605434ad76ed5a76d4e63d00e859a90e2 | subsection | 367 | 399 | Local-global compatibility for | Let \rho : \operatorname{\mathrm {G}al}_F\rightarrow \operatorname{\mathrm {G}L}_3(E) be a continuous representation which is unramified at the places of D(U^p) and such that:\overline{\rho } is absolutely irreducible
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27438373e07f597ec7c9b9a682e194a80be2d6cb | subsection | 368 | 399 | Local-global compatibility for | By , we have:\operatorname{\mathrm {H}om}_{\operatorname{\mathrm {G}L}_3(\mathbb {Q}_p)}\big (\Pi (D),\widehat{S}(U^{\wp }, W^{\wp })[\mathfrak {m}_{\rho }]\big )
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efe1900fa33f4e9413d6f4361ffa6dabf1ffa941 | subsection | 369 | 399 | Appendix | The aim of this appendix is to give a complete proof of Proposition REF , for which we couldn't find precise references in the existing literature. | {
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59e72fc461945ae79069f298bf0a00dd0e7a5e88 | subsection | 370 | 399 | Deformations I | The main results of this section are Corollary REF and Corollary REF below.We keep the notation of § REF . We fix \overline{\rho }: \operatorname{\mathrm {G}al}_{\mathbb {Q}_p}\rightarrow \operatorname{\mathrm {G}L}_2(k_E) a continuous representation and let \pi (\overline{\rho }) be the smooth representation of \opera... | {
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bf5d07dec4d9c8e31f0ff76587b528f9b49fea3e | subsection | 371 | 399 | Deformations I | The following theorem follows from work of Kisin and Paškūnas (see ).Theorem H.1
The functor \textbf {V}_{\varepsilon ^{-1}} induces an isomorphism of groupoids:\operatorname{\mathrm {D}ef}_{\pi (\overline{\rho }), \operatorname{\mathrm {o}rtho}}^* {\sim } \operatorname{\mathrm {D}ef}_{\overline{\rho }}.Let \xi =(\rho... | {
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a96a682a50443890c45f37812da09c031fb3dba0 | subsection | 372 | 399 | Deformations I | The map \rho _{\xi }\mapsto \widehat{\pi }(\rho _{\xi }) is the p-adic local Langlands correspondence for \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p) (normalized as in ).Corollary H.2
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1ff4e556adcf35af6d8e85e4bea3c8c62ceb44e7 | subsection | 373 | 399 | Deformations I | By the proof of , one can find a finite \mathcal {O}_E-subalgebra A\subseteq E[\epsilon ]/\epsilon ^2 such that A[1/p]\cong E[\epsilon ]/\epsilon ^2 and a deformation \rho _{A,\xi } of \overline{\rho } over A such that \rho _{A,\xi }\otimes _{A} \mathcal {O}_E\cong \rho _{\xi }^0 (via the natural surjection A\twoheadri... | {
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07d4dbc24f50ce73d5ff937ae8d916d273b93886 | subsection | 374 | 399 | Deformations I | We know that \operatorname{\mathrm {D}ef}_{\overline{\rho }} is representable, hence so is \operatorname{\mathrm {D}ef}_{\pi (\overline{\rho }), \operatorname{\mathrm {o}rtho}}^* by Theorem REF .Let \overline{\zeta } :=\wedge ^2_{k_E} \overline{\rho } be the determinant of \overline{\rho } and recall that any element i... | {
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19afa2e31b38539f9db236801d628577a8cffb22 | subsection | 375 | 399 | Deformations I | Using the same arguments as in the proof of Lemma REF , we can then show that there exists \overline{\zeta }^{\prime }: \mathbb {Q}_p^{\times } \rightarrow (k_E[\epsilon ]/\epsilon ^2)^{\times } such that the center Z(\mathbb {Q}_p)\cong \mathbb {Q}_p^\times acts on \widetilde{\pi } by \overline{\zeta }^{\prime }\overl... | {
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114b4b864ba65582246be380c4ecdd3e196118fe | subsection | 376 | 399 | Deformations I | We thus deduce another k_E-linear morphism:\operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p)}(\pi (\overline{\rho }), \pi (\overline{\rho })) \longrightarrow \operatorname{\mathrm {H}om}(\mathbb {Q}_p^{\times }, k_E), \ \widetilde{\pi } \longmapsto \big (\overline{\zeta }^{\prime }\overline{\... | {
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edf9a9a31f81ebe0404f0806ee03993eac92e917 | subsection | 377 | 399 | Deformations I | \psi \mapsto \pi (\overline{\rho }) \otimes (1+\psi /2\epsilon )\circ \operatorname{\mathrm {d}et}) gives a section of (REF ) (resp. of (REF )).As in , we call \overline{\rho } generic if either \overline{\rho } is irreducible or \overline{\rho }\cong \begin{pmatrix}\delta _1 & * \\ 0 &\delta _2 \end{pmatrix} for \delt... | {
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09cad92ce5bb7e0cd0a5f72802bd38830358d163 | subsection | 378 | 399 | Deformations I | By our assumptions on \overline{\rho }, we easily check that \dim _{k_E} \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}, \overline{\zeta }}(\overline{\rho }, \overline{\rho })=3. The result for \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {G}L}_2(\mathbb {Q}_p), \overline{\zeta }\... | {
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93976e3ed85eee31e3ed37e74ef2c34060cdca0e | subsection | 379 | 399 | Deformations I | Using Theorem REF , the third equality in Proposition REF and (and the representability of \operatorname{\mathrm {D}ef}_{\overline{\rho }}, \operatorname{\mathrm {D}ef}_{\pi (\overline{\rho }), \operatorname{\mathrm {o}rtho}}^*, \operatorname{\mathrm {D}ef}_{\pi (\overline{\rho }), \operatorname{\mathrm {o}rtho}}), we ... | {
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2a26f25ee4ad0e3fc7ade94b6bbda6f11ff396fe | subsection | 380 | 399 | Deformations I | As \operatorname{\mathrm {H}om}_{\mathcal {C}(\mathcal {O}_E)}(\pi (\overline{\rho })^{\vee }, \pi (\overline{\rho })^{\vee })=k_E and \dim _{k_E}\operatorname{\mathrm {E}xt}^1_{\mathcal {C}(\mathcal {O}_E)}(\pi (\overline{\rho })^{\vee }, \pi (\overline{\rho })^{\vee })<\infty , Schlessinger's criterion again implies ... | {
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local-global compatibility | [
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72847354aa4318a65bd9822b3d6be2e8bc0eef1a | subsection | 381 | 399 | Deformations I | Since \pi (\overline{\rho }) is admissible, we know M_A\otimes _{A[[H]]} k_E \cong \pi (\overline{\rho })^\vee \otimes _{k_E[[H]]} k_E is a finite dimensional k_E-vector space. By Nakayama's lemma (see e.g. ), we deduce M_A is finitely generated over A[[H]].(2) By and its proof, we have that M_A:=\operatorname{\mathrm ... | {
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8fbae819e94c5005ab63ebe86edac67b5d61e11d | subsection | 382 | 399 | Deformations I | Let \rho ^{\operatorname{\mathrm {u}niv}} be the universal deformation of \overline{\rho } over R_{\overline{\rho }} (for \operatorname{\mathrm {D}ef}_{\overline{\rho }}), \mathcal {N}\in \mathcal {C}(\mathcal {O}_E) the universal deformation of \pi (\overline{\rho })^{\vee } over R_{\overline{\rho }} (for \operatornam... | {
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2d82ccb71ba246c280517cb13e0b4425e8244a25 | subsection | 383 | 399 | Deformations I | For any \zeta :\mathbb {Q}_p^\times \rightarrow \mathcal {O}_E^\times such that \zeta \equiv \overline{\zeta } mod \varpi _E, we denote by \operatorname{\mathrm {D}ef}_{\overline{\rho }}^{\zeta } the subfunctor of \operatorname{\mathrm {D}ef}_{\overline{\rho }} of deformations with fixed determinant \zeta and by R_{\ov... | {
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local-global compatibility | [
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e36f13df7664772abf642707d29be88e83890021 | subsection | 384 | 399 | Deformations I | It is not difficult to see that the isomorphism in (REF ) induces a natural isomorphism (so that R_{\overline{\rho }}^{\zeta }\sim \over \rightarrow R_{\pi (\overline{\rho })^{\vee }}^{\zeta \varepsilon }):\operatorname{\mathrm {D}ef}_{\pi (\overline{\rho })^\vee , \mathcal {C}_{\zeta \varepsilon }(\mathcal {O}_E)} \si... | {
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11cc03fc32522d2fcd607a815d6207dca53d6ec8 | subsection | 385 | 399 | Deformations I | One easily sees 1^{\operatorname{\mathrm {u}niv}}\in \mathcal {C}(\mathcal {O}_E).Proposition H.10
We have \mathcal {N}\cong \mathcal {N}^{\zeta \varepsilon }\widehat{\otimes }_{\mathcal {O}_E} 1^{\operatorname{\mathrm {u}niv}}.We have that \mathcal {N}^{\zeta \varepsilon }\widehat{\otimes }_{\mathcal {O}_E} 1^{\opera... | {
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local-global compatibility | [
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984d06482fbd789d623690202c6b7dc1fd5568a0 | subsection | 386 | 399 | Deformations II | We prove here a key projectivity property of \mathcal {N}.We keep the previous notation and assumption (in particular \overline{\rho } satisfies (REF ) and is such that \operatorname{\mathrm {E}nd}_{\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}}(\overline{\rho })\cong k_E). We assume moreover p\ge 5 if \overline{\rho } ... | {
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aad33a4b162408d600017ca2c71fce771ebc2794 | subsection | 387 | 399 | Deformations II | Let \chi : Z_0\rightarrow \mathcal {O}_E^{\times } such that \chi ^2=\zeta (enlarging E if necessary and using Z_0\cong \mathbb {Z}_p), we deduce an isomorphism of \mathcal {O}_E[[K/Z_0]]-modules (using that \mathcal {O}_E[[K/Z_0]] is a local ring):(\mathcal {N}^{\zeta \varepsilon }\otimes \chi ^{-1}\circ \operatorname... | {
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local-global compatibility | [
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f129ee6715f424faf160797a50fa4eb41b07d2f2 | subsection | 388 | 399 | Deformations II | If \operatorname{\mathrm {T}or}^1_{S_1[[K/Z_0]]}(\mathcal {N}_1, k_E)=0, we get:0 \longrightarrow \mathcal {M}_1{\otimes }_{S_1[[K/Z_0]]} k_E \longrightarrow \mathcal {P}{\otimes }_{S_1[[K/Z_0]]} k_E \longrightarrow \mathcal {N}_1{\otimes }_{S_1[[K/Z_0]]} k_E \longrightarrow 0.Since \mathcal {P} is the projective envel... | {
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449b4b5b5878f36aa726ec28532f3c2ee3f32418 | subsection | 389 | 399 | Deformations II | Using the exact sequence:0 \longrightarrow S_1[[K/Z_0]]^{\oplus r} {x} S_1[[K/Z_0]]^{\oplus r} \longrightarrow \mathcal {O}_E[[K/Z_0]]^{\oplus r}(\cong \mathcal {N}_1/x)\longrightarrow 0,we easily deduce \operatorname{\mathrm {T}or}^1_{S_1[[K/Z_0]]}(\mathcal {N}_1/x, k_E) \sim \over \longrightarrow k_E^{\oplus r}, whic... | {
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} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
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bf25cb7c231d6b355559e9eb20cbadbc5bb533ed | subsection | 390 | 399 | Proof of Proposition | We finally prove Proposition REF .We keep the previous notation. We assume p\ge 5 and fix \rho :\operatorname{\mathrm {G}al}_{\mathbb {Q}_p}\rightarrow \operatorname{\mathrm {G}L}_2(E) as in Proposition REF , so that we have D_{\operatorname{\mathrm {r}ig}}(\rho )\cong D(\alpha , \lambda , \psi ) with D(\alpha , \lambd... | {
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local-global compatibility | [
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3e7a64451c6227d9b72bee465876ae4d6163ede9 | subsection | 391 | 399 | Proof of Proposition | It follows from and Proposition REF that the Banach space \Pi (equipped with the supremum norm) is an R_{\overline{\rho }}-admissible continuous representation of \operatorname{\mathrm {G}L}_2(\mathbb {Q}_p) in the sense of .Lemma H.12
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da290a1516e088c08c5fca864faa69fe5ffb9b0d | subsection | 392 | 399 | Proof of Proposition | But:[18] {\operatorname{\mathrm {H}om}_{\mathcal {O}_E}\big (\operatorname{\mathrm {H}om}_{\mathcal {O}_E}^{\operatorname{\mathrm {c}ts}}(R_{\overline{\rho }},\mathcal {O}_E)\widehat{\otimes }_{R_{\overline{\rho }}} \pi ^{\operatorname{\mathrm {u}niv}}(\overline{\rho }),\mathcal {O}_E\big )}& \\
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9d9a82f8b78fc1cb7127f23f0a3a98e53f08a247 | subsection | 393 | 399 | Proof of Proposition | The result follows then from Remark REF (2) and the fact \operatorname{\mathrm {H}om}_{\mathcal {O}_E}^{\operatorname{\mathrm {c}ts}}(R_{\overline{\rho }}/\mathcal {I}_{\widetilde{\rho }},\mathcal {O}_E) is free of rank one over R_{\overline{\rho }}/\mathcal {I}_{\widetilde{\rho }}\cong \mathcal {O}_E[\epsilon ]/\epsil... | {
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local-global compatibility | [
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a7bed7b33f45470e7b84df76b1b6e9b3a75474c9 | subsection | 394 | 399 | Proof of Proposition | A point x=(\rho _x, \delta _x)\in (\operatorname{\mathrm {S}pf}R_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}}\times \mathcal {T} lies in X if and only if there is a T(\mathbb {Q}_p)-embedding \delta _x \hookrightarrow J_B(\Pi ^{R_{\overline{\rho }}-\operatorname{\mathrm {a}n}}[\mathfrak {p}_{\rho _x}])=J_B(\wideh... | {
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4d527c35f4340531493497d2e16acb3ae962341c | subsection | 395 | 399 | Proof of Proposition | The pull-back \mathcal {M}_1:=j^* \mathcal {M} is thus a coherent sheaf on X_{\rm tri}(\overline{\rho }).It follows from Proposition REF that \mathcal {N} is finitely generated and projective as S[[\operatorname{\mathrm {G}L}_2(\mathbb {Z}_p)]]-module where S=\mathcal {O}_E[[x,y]]\hookrightarrow R_{\overline{\rho }}. I... | {
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40dea3adbf04b6f79bd4fd42b2002eb76768dae8 | subsection | 396 | 399 | Proof of Proposition | By the global triangulation theory (, ) and using similar arguments as in , we have the following facts:the morphism X_{\rm tri}(\overline{\rho }) \longrightarrow (\operatorname{\mathrm {S}pf}R_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}} induces an isomorphism:
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"raw": "Kedlaya, K., Pottharst, J., and Xiao, L. Cohomology of arithmetic families of (\\varphi , \\Gamma )-modules. J. Amer. Math. Soc. 27 (2014), 1043–1115.",
"source_ref_id": "39edc5a035e574bed649a... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
"math.RT"
] | 2,018 | en | Mathematics | [
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7c9e1a65b700c8c3a91c8d9d384ae8b5672e1b04 | subsection | 397 | 399 | Proof of Proposition | Since the rigid space (\operatorname{\mathrm {S}pf}R_{\overline{\rho }})^{\operatorname{\mathrm {r}ig}}\times \mathcal {T} is nested (), so are its closed subspaces X and X_{\rm tri}(\overline{\rho }), and it follows that the composition:v: \operatorname{\mathrm {S}pec}E[\epsilon ]/\epsilon ^2 \longrightarrow U \longri... | {
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"doi": "10.24033/ast.782",
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"raw": "Bellaïche, J., and Chenevier, G. Families of Galois representations and Selmer groups. Astérisque 324 (2009).",
"source_ref_id": "dd425b4ffeec179... | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
"math.NT",
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a6bd3e6fbe76dc7d4edc493d031ec930b584da99 | subsection | 398 | 399 | Proof of Proposition | Fix an extension in \operatorname{\mathrm {E}xt}^1_{\operatorname{\mathrm {t}ri}}(\rho , \rho ), i.e. a trianguline deformation \widetilde{\rho } of \rho over E[\epsilon ]/\epsilon ^2, by (REF ) and what is below (REF ), we have that (x^{k_1}|\cdot |(1+\psi _{v,1}\epsilon )\operatorname{\mathrm {u}nr}(\alpha ), x^{k_2}... | {
"cite_spans": []
} | 1803.10498 | Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ and
local-global compatibility | [
"Christophe Breuil",
"Yiwen Ding"
] | [
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e5820727e525195d029a26d045af5ab6748327e5 | abstract | 0 | 21 | Abstract | Blockchains have a storage scalability issue. Their size is not bounded and
they grow indefinitely as time passes. As of August 2017, the Bitcoin
blockchain is about 120 GiB big while it was only 75 GiB in August 2016. To
benefit from Bitcoin full security model, a bootstrapping node has to download
and verify the enti... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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396c0dea792479995e3fafe7c75c0faa707ed63d | subsection | 1 | 21 | Trustless Bitcoin | Within a decade, blockchains have become extremely popular, and have
been used to implement several widely-used
crytocurrencies , and smart-contract
services . A blockchain implements a tamper-proof
distributed ledger in which public transactions can be recorded in
a close-to-irrevocable manner. Recorded transactions a... | {
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{
"arxiv_id": "",
"doi": "10.2139/ssrn.3440802",
"end": 159,
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"raw": "S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008. [Online]. Available: http://www.cryptovest.co.uk/resources/Bitcoin%20paper%20... | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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dcd49414a36736a99c66a06064d9b190853798b9 | subsection | 2 | 21 | Trustless Bitcoin | Diet nodes must be able to detect any tampering of the UTXO set itself, at a cost that remains affordable for low-resource devices, both in terms of communication and computing overhead.The rest of this report is structured as follows. We first present the Bitcoin protocol in more detail (Section ), and explain the wor... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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5d5ba9bfd451ea29fbcfe109ec2119c0f38d82f3 | subsection | 3 | 21 | The Bitcoin system | A blockchain is a decentralized ledger composed of blocks containing transactions.
The transactions, the blocks, and the resulting chain obey a few core rules that ensure the system remains tamper-proof.
Great care is required when modifying these rules, as even minor changes might break the blockchain's properties and... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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e573a7d3626cea45d67a985c7e6072f1720e0d75 | subsection | 4 | 21 | Overview | In a blockchain system such as Bitcoin, the blockchain proper
((B_k)_{k \in \mathbb {Z}_{\ge 0}}, label [baseline=(char.base)]
shape=circle,draw,inner sep=0.5pt,fill,text=white] (char) 1; in Figure REF ) is maintained by a peer-to-peer
network of miners. Each block B_k
links to the previous block B_{k-1}
by including i... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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... | |
b520bf69bf13efd2d233ff69906795d33c5e9506 | subsection | 5 | 21 | Recording a new transaction | To transfer 8 bitcoins from herself to Bob, the user Alice must first create a valid transaction (label [baseline=(char.base)]
shape=circle,draw,inner sep=0.5pt,fill,text=white] (char) 2;) that contains information proving she actually owns the 8 bitcoins (with a cryptographic signature using asymmetric keys), and enco... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
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1c2bd2529a21def82df9c08e29974da319b6aa07 | subsection | 6 | 21 | Irrevocability of deep blocks | Because blocks are produced at a limited rate such that all the miners receive block B_k before they can successfully mine a concurrent block B_{k^{\prime }}, honest miners are highly likely to extend the chain when producing a new block, ensuring a consistent system state with high probability.
The views of individual... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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a569f992a791d8271e79f2f5d47d3700b97669e6 | subsection | 7 | 21 | Transactions, Blocks, and UTXO set | To benefit from the full security of Bitcoin, Bob should verify the
validity of the new block that contains Alice's payment to him
(label [baseline=(char.base)]
shape=circle,draw,inner sep=0.5pt,fill,text=white] (char) 9; in Figure REF ) in
addition to verifying the validity of Alice's transaction. This is
because Alic... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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5d308a643e120edb20e471f4a561434d63ec5808 | subsection | 8 | 21 | Checking block validity | A block is valid if and only if it meets the following two conditions.(BV1) Its header respects the blockchain's Proof-of-Work predicate.
(BV2) It only contains valid transactions (which we discuss further below).BV1: The Proof-of-Work predicate makes it very difficult for malicious actors to alter the blockchain in a... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
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66ac61f376ce5cc4a0122e6748e3c3ba9573b590 | subsection | 9 | 21 | Checking block validity | Figure REF shows the validity mechanisms included in a typical Bitcoin transaction. In this example, Alice uses 3 coins she owns (the transaction's inputs
[baseline=(char.base)]
shape=circle,draw,inner sep=0.5pt,fill,text=white] (char) 1;) to pay 7 Bitcoins to Bob, and 4 to Tux (the transaction's outputs [baseline=(cha... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
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7b511990b08ecf619f35e9931f169f1654a38776 | subsection | 10 | 21 | The set of Unspent TransaCTion Outputs (UTXO set) | While the validity of a block's header (BV1) only requires
access to the current block B_k, and to the header of its
predecessor B_{k-1}, verifying transactions (BV2) requires a
lot more information. Verifying the ownership challenges of input
coins (TV1), and their amount (TV2) requires access to the
transactions reco... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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ddd2bbfbc5c10161192ab41a30a242a7abeb06e5 | subsection | 11 | 21 | The limitations of SPV nodes | In spite of its benefits, constructing a local UTXO set is costly: in order to obtain the set, a node must first download the entire chain (120 GiB as of August 2017, see Figure REF and validate it (a lengthy process that can take hours on high-end machines), even if only the latest block is relevant to its interest.Be... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
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3f156c9e90d466f6909f3c195a2fc32ae629e518 | subsection | 12 | 21 | The Dietcoin system | To address the vulnerabilities of
SPV nodes and to improve the confidence mobile users can have in
recent transactions, we propose Dietcoin, an extension to Bitcoin-like
blockchains.
Although our proposal can be applied to most existing
Proof-of-Work blockchains using the UTXO model for coins, we describe Dietcoin in t... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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... | |
b67309a5e31d96f31a17377e85f6dfc14e5b7df8 | subsection | 13 | 21 | Sharding the UTXO set | To enable the operation of diet nodes, Dietcoin-enabled full nodes
need (i) to provide diet nodes with shards of the
UTXO set, while (ii) enabling them to verify that these shards are
authentic.To satisfy (i), Dietcoin-enabled full nodes store the UTXO set resulting
from the application of the transactions in each bloc... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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6555b6bfd4f1d7e15b9e2d45644eb967022ae39e | subsection | 14 | 21 | Linking blocks with the UTXO set | With reference to Figure REF , consider a Dietcoin-enabled miner that
is mining block B_k, and let \textsc {UTXO}_k be the state of the UTXO set
after applying all the transactions in B_k. The miner stores the
root of the UTXO Merkle tree associated with \textsc {UTXO}_k as an
unspendable output in the first transactio... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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9e4ffaa3a5b391ce05493d786339666d8cf7ebab | subsection | 15 | 21 | Extended verification | Diet nodes have the ability to extend their confidence in a block by iterating the verification process towards its previous blocks.
By doing so, diet nodes ensure the correctness of the UTXO Merkle root present in block B_{k-1} used to verify the correctness of block B_k.
The extended verification can be performed on ... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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b63d81919959a184add3ab64ee336ada3d420f3e | subsection | 16 | 21 | Detailed Operation | Equipped with the knowledge of Dietcoin's basic mechanisms, we can now
detail the verification process carried out by diet
nodes. Algorithm REF depicts the actions taken by a diet
node when its user starts the application using Dietcoin and compares
it with those taken by legacy SPV clients. Black dotted
lines [\bullet... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
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] | 2,018 | en | Computer Science | [
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... | |
6722465f5c7172347839857641137535b13c84a4 | subsection | 17 | 21 | Detailed Operation | Once the verification process has terminated, the diet node returns
the transactions associated with the local user to the application (line 14). | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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... | |
7e00bad488c732c17d9ea5f1675c767f93363617 | subsection | 18 | 21 | Related work | Making the UTXO set available for queries between nodes has been discussed several times in the Bitcoin community over the past few years.
Bryan Bishop published a comprehensive list of such proposals that share some of the following goals: (i) enabling faster node bootstrap, (ii) strengthening the security guarantees... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 401,
"openalex_id": "",
"raw": "B. Bryan, “[bitcoin-dev] Protocol-Level Pruning,” Nov. 2017. [Online]. Available: https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2017-November/015297.html",
"source_ref_id": "ff0d4fd2... | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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7115f6cc44564f1b7931c2bd634d6ddd4821cdd4 | subsection | 19 | 21 | Related work | Long distance links are particularly well adapted to navigate a well-identified subset of a blockchain (such as a package's releases). They were not however directly designed to handle the kind of dependencies captured by the UTXO model.An entirely different approach to scaling blockchains for lightweight nodes is the ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-030-51280-4_27",
"end": 413,
"openalex_id": "https://openalex.org/W3042319774",
"raw": "A. Kiayias, A. Miller, and D. Zindros, “Non-interactive proofs of proof-of-work,” Tech. Rep. 963, 2017. [Online]. Available: https://eprin... | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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0.0022467849776148796... | |
6684d421fc460ef4e9baf846930488cd2dc0eb98 | subsection | 20 | 21 | Conclusion | In this report, we have presented the design of Dietcoin, that proposes a new form of Bitcoin nodes that strengthens the security guarantees of lightweight SPV nodes by bringing them closer to those of full Bitcoin nodes. The Dietcoin protocol enables low-resource nodes to verify the transactions contained in blocks wi... | {
"cite_spans": []
} | 1803.10494 | Dietcoin: shortcutting the Bitcoin verification process for your
smartphone | [
"Davide Frey",
"Marc X. Makkes",
"Pierre-Louis Roman",
"François Taïani",
"Spyros Voulgaris"
] | [
"cs.DC",
"cs.CR"
] | 2,018 | en | Computer Science | [
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... | |
2515558e68abbe29ddba9326211bff7ed981f00c | abstract | 0 | 32 | Abstract | We present the linearized metrizability problem in the context of parabolic
geometries and subriemannian geometry, generalizing the metrizability problem
in projective geometry studied by R. Liouville in 1889. We give a general
method for linearizability and a classification of all cases with irreducible
defining distr... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03954697027802467,
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... | |
cc6252e92fe43a1ced8be106b1b81fd7f4937467 | subsection | 1 | 32 | Introduction | Many areas of geometric analysis and control theory deal with distributions on
smooth manifolds, i.e., smooth subbundles of the tangent bundle. Let {\mathcal {H}}\leqslant TM be such a distribution of rank n on a smooth m-dimensional manifold
M. A smooth curve c\colon [a,b] \rightarrow M (a\leqslant b\in {\mathbb {R}})... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.022324534133076668,
-0.0006852434016764164,
-0.03848426416516304,
-0.012527985498309135,
0.021988827735185623,
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0.002439600182697177,
0.03668365254998207,
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0.0047647543251514435,
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0.006786628160625696,
0.0011816514888778329,
... | |
1578c852d98fa7827ec726b63587579982675d79 | subsection | 2 | 32 | Introduction | Among the horizontal curves joining two points, it may be important to
find those which are optimal in some sense, for example those of shortest
length with respect to a horizontal metric. Horizontal metrics also allow for
the definition of a hypo-elliptic sublaplacian , allowing methods of
harmonic analysis to be appl... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.015063751488924026,
0.013262816704809666,
-0.005654630251228809,
0.007501351181417704,
0.016147363930940628,
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0.02499941736459732,
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0.015124799683690071,
0.02380896918475628,
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0.007470827084034681,
0.04398554190993309,
0.0... | |
23d50191428381dc03b1b501e566f22e80ad30e2 | subsection | 3 | 32 | Introduction | In section , we
describe the linearization principle and prove Theorem REF . We give
examples, and in particular show how explicit formulae can be obtained not
only for the homogeneous model, but also for so-called normal
solutions. Section is devoted to the main classification
result. We conclude by giving examples (... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.054449114948511124,
0.015107798390090466,
0.01053730770945549,
-0.014505012892186642,
-0.0017482698895037174,
-0.032413095235824585,
0.0160234235227108,
0.00676417350769043,
0.03387809544801712,
0.011025641113519669,
-0.008271138183772564,
0.002878493396565318,
0.002048708964139223,
0.0... | |
cb94fa175ed2948251ae7fd916dac39e0593a193 | subsection | 4 | 32 | Background and motivating examples | We work throughout with real smooth manifolds M, real Lie groups P and
real Lie algebras {\mathfrak {p}} (e.g., we view \mathop {{GL}\hbox{}}\nolimits (n, as a real Lie group and
\mathop {\mathfrak {gl}\hbox{}}\nolimits (n, as a real Lie algebra).A (real or complex) P-module W is a finite dimensional (real or
complex) ... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.027389684692025185,
-0.0104904780164361,
-0.020080681890249252,
-0.03402729704976082,
-0.007118266075849533,
-0.05941806733608246,
0.03823874518275261,
0.0077515095472335815,
0.03677389398217201,
0.0432741753757,
-0.01736460253596306,
-0.007099192589521408,
-0.013008193112909794,
0.0068... | |
4c5c863ceece39874130e619b24a45e75756e798 | subsection | 5 | 32 | Parabolic geometries and Weyl structures | Let P\leqslant G be a closed Lie subgroup of a Lie group G, whose Lie algebra
{\mathfrak {p}}\leqslant {\mathfrak {g}} has nilpotent radical {\mathfrak {n}}\trianglelefteq {\mathfrak {p}}.Definition 2.1 A Cartan geometry of type G/P on a smooth manifold M
is a principal P-bundle {\mathcal {G}}\rightarrow M equipped wit... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.042239315807819366,
0.01779300533235073,
-0.01779300533235073,
-0.00925510935485363,
0.010140182450413704,
-0.0446198545396328,
0.06592262536287308,
0.024858323857188225,
0.025941776111721992,
0.048434820026159286,
-0.017335209995508194,
-0.025148261338472366,
0.02873433195054531,
0.008... | |
9aec9b97423aa7c01131c03928a210b0e38e5291 | subsection | 6 | 32 | Parabolic geometries and Weyl structures | Then Cartan geometries of type G/P are
called parabolic geometries and have several distinctive features which
we briefly explain and illustrate in the examples below (see for
further details).First, the Killing form of {\mathfrak {g}} induces a duality between {\mathfrak {p}}^\perp and
{\mathfrak {g}}/{\mathfrak {p}}... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03482941538095474,
0.05390775576233864,
-0.007436736952513456,
-0.01427822932600975,
-0.007772515527904034,
-0.02304663509130478,
0.0628822073340416,
0.0333336740732193,
0.019551482051610947,
0.058028679341077805,
-0.0180862657725811,
-0.021169325336813927,
0.007745806127786636,
0.01430... | |
c096e7a55a6d873b763f4a8ecbab6e94fedfa72f | subsection | 7 | 32 | Summary | A manifold M with a parabolic geometry of type
G/P comes equipped with: a filtration of the tangent bundle TM, a G_0
structure on \mathop {{gr}\hbox{}}\nolimits (TM), and a distinguished class of G_0-connections (the
Weyl connections).There are general results , stating that these data
are often sufficient to determine... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.030166180804371834,
0.0243526678532362,
0.003330181585624814,
-0.01461007446050644,
-0.011070087552070618,
-0.0646963119506836,
0.0328364297747612,
0.028457220643758774,
0.017074331641197205,
0.0326533243060112,
-0.005226057954132557,
-0.036803655326366425,
0.015311968512833118,
0.01382... | |
47497cd58f436cd9895cb474a7ca864d9a36caa1 | subsection | 8 | 32 | Projective parabolic geometries | We begin with some examples in which {\mathfrak {p}}^\perp is abelian, hence the
filtration of {\mathfrak {g}}/{\mathfrak {p}} is trivial and (REF ) is a {\mathbb {Z}}-grading of
{\mathfrak {g}} as a Lie algebra, with {\mathfrak {p}}_0 in degree 0 and {\mathfrak {m}},{\mathfrak {p}}^\perp in degree
\pm 1 (also called a... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.05497182160615921,
0.0020831909496337175,
-0.0018571303226053715,
-0.005093516316264868,
0.019122624769806862,
-0.013040927238762379,
0.053659334778785706,
0.01903105527162552,
0.027501171454787254,
0.03916093707084656,
-0.04242689162492752,
0.008416702039539814,
-0.014292367734014988,
... | |
c4532dbac74d3a927d8778f399b6b848f9a65dca | subsection | 9 | 32 | Projective geometry | in dimension m may be viewed
as a parabolic geometry of type G/P where G=\mathop {{PGL}\hbox{}}\nolimits (m+1,{\mathbb {R}}) and P is the
parabolic subgroup of block lower triangular matrices with blocks of sizes m
and 1. Here {\mathfrak {m}}={\mathbb {R}}^m, {\mathfrak {p}}_0 = \mathop {\mathfrak {gl}\hbox{}}\nolimits... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03314894810318947,
0.011133590713143349,
0.006509220693260431,
0.004422149155288935,
0.014529374428093433,
-0.002169104292988777,
0.0076958369463682175,
0.04828880354762077,
0.011736437678337097,
0.019642125815153122,
-0.04636579751968384,
0.0009247786365449429,
0.02353392355144024,
-0.... | |
7f8fb345bd26d02616ccaf642924c57e99440b87 | subsection | 10 | 32 | (Almost) c-projective geometry | is a
complex analogue of projective geometry , , , with
G=\mathop {{PGL}\hbox{}}\nolimits (m+1, and P\leqslant G block lower triangular as in the projective
case, so the homogeneous model G/P is complex projective space ^{m}
viewed as a real homogeneous space. A parabolic geometry of this type on a
2m-manifold M is giv... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.04088471829891205,
-0.00817694328725338,
-0.0019565168768167496,
-0.011990816332399845,
0.003098771907389164,
-0.05119743198156357,
-0.0033543012104928493,
0.03853537142276764,
0.000673625327181071,
0.0013482040958479047,
-0.03633858263492584,
0.014355418272316456,
0.00796336680650711,
... | |
64033e2738afaf4aa90dda9eca7acbd06a53d70b | subsection | 11 | 32 | (Almost) grassmannian geometries | are
generalizations of real projective geometry with G=\mathop {{PGL}\hbox{}}\nolimits (m+k,{\mathbb {R}}) and P
block lower triangular with blocks of size m and k. The homogeneous model
G/P is the grassmannian of k-planes in {\mathbb {R}}^{m+k}. On a parabolic geometry
of this type, the G_0-structure is given by an id... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.05436026304960251,
0.008035002276301384,
0.0009676531190052629,
-0.009774775244295597,
-0.008965933695435524,
-0.04792005196213722,
0.036199480295181274,
0.05045340582728386,
0.023303795605897903,
0.040533650666475296,
-0.044898342341184616,
0.0061044651083648205,
0.012628612108528614,
... | |
cea58e789a35d4b3980bd7bb33900d5f08dc8f70 | subsection | 12 | 32 | Parabolic geometries on filtered manifolds | We now turn to the examples of greater interest to us, in which {\mathcal {H}} is a
proper subbundle of TM. In fact, in these examples, the geometry is often
entirely determined by the distribution {\mathcal {H}}, as we now discuss.Given a smooth manifold M of dimension m, equipped with a distribution
{\mathcal {H}}={\... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.051501818001270294,
-0.006624604109674692,
-0.012783922255039215,
-0.019404713064432144,
0.007387368008494377,
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0.04567430168390274,
0.011235511861741543,
0.031090255826711655,
0.03221914544701576,
-0.04808463528752327,
-0.00004511033330345526,
-0.014172152616083622,
... | |
2ebe90ff63d3613f7209d311fa0cba666510832e | subsection | 13 | 32 | Parabolic geometries on filtered manifolds | However, if
we work only with horizontal (or partial) connections,
i.e., restrict the Weyl connections to covariant derivatives in {\mathcal {H}}
directions only, then the theory is as simple as in the |1|-graded case: the
Lie bracket between {\mathfrak {m}} and {\mathfrak {p}}^\perp in {\mathfrak {g}} induces a Lie br... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.09499455988407135,
0.0034836933482438326,
0.005727299023419619,
0.02507648803293705,
0.011584330350160599,
0.0009195729508064687,
0.01441554632037878,
0.05094663053750992,
0.013660046271979809,
0.01659047044813633,
-0.03711869567632675,
-0.012141415849328041,
0.010424370877444744,
-0.00... | |
e5c6920dd80bc93eeea11d8c6d8dd1907d8e2e94 | subsection | 14 | 32 | Free distributions | are parabolic geometries
with G=\mathop {{SO}\hbox{}}\nolimits (n+1,n) and P block lower triangular with blocks of sizes n,
1, n, where the inner product is defined on the standard basis
e_0,e_1\ldots e_{2n} by \langle e_i,e_{n+1+i}\rangle =\langle e_n,e_n\rangle =\langle e_{n+1+i},e_i\rangle
=1 for 0\leqslant i\leqsl... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03337875381112099,
-0.00495036831125617,
-0.0077306595630943775,
0.016704631969332695,
0.019831983372569084,
0.003966396674513817,
0.05025119334459305,
0.04259299859404564,
-0.0061097764410078526,
0.039755500853061676,
-0.0583670549094677,
-0.009137967601418495,
0.0075971754267811775,
0... | |
11d322b8b444bd73a073068142d17f058cc6b881 | subsection | 15 | 32 | Free CR or quaternionic CR distributions | are obtained by replacing \mathop {\mathfrak {so}\hbox{}}\nolimits (n+1,n) with {\mathfrak {g}}=\mathop {\mathfrak {su}\hbox{}}\nolimits (n+1,n) or \mathop {\mathfrak {sp}\hbox{}}\nolimits (n+1,n),
again with (complex or quaternionic) blocks of sizes n, 1, n, and {\mathfrak {p}}
being block lower triangular . Elements ... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
0.0019263623980805278,
0.0017232360551133752,
-0.031554196029901505,
-0.013358655385673046,
-0.010703708976507187,
-0.05181724205613136,
0.04971159249544144,
0.02487105503678322,
-0.004863588139414787,
0.020568210631608963,
-0.011100425384938717,
0.015380382537841797,
-0.04348620027303696,
... | |
72a1252a8df7b8c724093724b88d45db02192a9a | subsection | 16 | 32 | First BGG operators, local metrizability of the homogeneous model,
and normal solutions | Let {\mathcal {G}}\rightarrow M,\,\theta be a Cartan geometry of type G/P. The extension of
{\mathcal {G}} by the left action of P on G is a principal G-bundle \tilde{{\mathcal {G}}}=
{\mathcal {G}}\times _P G with G-connection \tilde{\theta }\colon \tilde{{\mathcal {G}}}\rightarrow {\mathfrak {g}}, and
(by constructio... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.05759754404425621,
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-0.00... | |
b0a963600323f1464a2f555c322afb2d0052ce6f | subsection | 17 | 32 | First BGG operators, local metrizability of the homogeneous model,
and normal solutions | By , such explicit formulae also
apply on general curved geometries to the so called normal solutions,
which are those induced by parallel sections of the corresponding tractor
bundle. We discuss this further in §REF . | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.0... | |
fdbb4bbfe95d5ecf1e98678906381b1626567835 | subsection | 18 | 32 | First order operators | In , the second and third authors developed a theory of invariant
first order linear operators for parabolic geometries, generalizing work of
Fegan in the conformal case (cf. ).We first fix some notation. The Killing form of {\mathfrak {g}} induces a nondegenerate
invariant scalar product on {\mathfrak {p}}_0={\mathfr... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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-0.02... | |
f04f457fd1863702a1d55d7b67aa00fcfe41ef7d | subsection | 19 | 32 | First order operators | Thus\begin{split}
c_{\lambda , \mu , \alpha } &= c_{\lambda ^{\prime },\mu ^{\prime },\alpha ^{\prime }}+\tfrac{1}{2}\bigl (
(\alpha _0, 2\lambda ^0+\alpha ^0)-(\alpha ^0,\alpha ^0)\bigr )\\
&= c_{\lambda ^{\prime },\mu ^{\prime },\alpha ^{\prime }} + (\lambda ^0, \alpha ^0).
\end{split}If we fix \lambda ^{\prime },\al... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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c9fe137b0d9b4059eba3c1e5deff6f59bdebc5a0 | subsection | 20 | 32 | The algebraic linearization condition | Let ({\mathcal {G}}\rightarrow M,\theta ) be a parabolic geometry of type (G,P) and let
be the socle of the p-module g/p, whose central weights form a basis
of z(p0)*. As we have seen, GPGP g/pTM defines a (bracket generating) ``horizontal^{\prime \prime } distribution HTM.
Our aim is to construct compatible subrieman... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.02... | |
8b6396065c1a012a8d0704b766c0bbf4df7d3b1f | subsection | 21 | 32 | The linearization principle | Suppose first for simplicity that B\leqslant S^2 is absolutely irreducible and
satisfies the ALC (so has at most two irreducible components) and let
\pi ={{id}}_{*\otimes B}-\zeta \circ b be the projection onto \ker (b\colon *\otimes B\rightarrow . The linearization method constructs a
(pseudo-riemannian) metric on {\m... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.028... | |
543a655257f5c97f7565472058325a106df46bb1 | subsection | 22 | 32 | The linearization principle | Hence by Schur^{\prime }s lemma and §\ref {ss:firstorder}
(i.e.,~\cite {SS}):
\begin{equation*}
[\hspace{-1.66656pt}[\cdot ,\Upsilon ]\hspace{-1.66656pt}]\mathinner {\centerdot }\end{equation*}\eta =
(\zeta \circ b)([\hspace{-1.66656pt}[\cdot ,\Upsilon ]\hspace{-1.66656pt}]\mathinner {\centerdot })
=(b)(0
)
for nonze... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
d30e75e54bef677be8a24a8256b0c7701ac5583e | subsection | 23 | 32 | The linearization principle | Then for all
i{1,...r} there are induced line bundles Li and invariant
first order linear operators Di acting on sections of Bi Li such that there is a bijection between nondegenerate solutions
i:i{1,...r} of the equations Di (i) = 0, and
nondegenerate sections of B* with |Hy = 0 for
some Weyl connection .
Define b_i... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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6b012b7c31c57757ae7c599e374146a4756c4eda | subsection | 24 | 32 | The linearization principle | As before,
we define \sharp _\eta (\Upsilon )=\sum _{\alpha \in {\Sigma }_0}\ell _\alpha b(\Upsilon _\alpha \otimes \eta ), so that if
\hat{\nabla }|_{{\mathcal {H}}}^{\vphantom{y}}=\nabla |_{{\mathcal {H}}}^{\vphantom{y}}+\Upsilon then\hat{\nabla }|_{{\mathcal {H}}}^{\vphantom{y}}\eta =\nabla |_{{\mathcal {H}}}^{\vpha... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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523df22b96f64d13fdef08948df160e4083935a0 | subsection | 25 | 32 | Example: projective geometry | Let us illustrate the metrizability procedure by showing how the well-known
example of projective geometry , , , fits into the
general method. Here {\mathfrak {g}}=\mathop {\mathfrak {sl}\hbox{}}\nolimits (n+1,{\mathbb {R}})= \mathop {\mathfrak {gl}\hbox{}}\nolimits (\oplus * and S^2
is irreducible. Since *S2(*0 S2,
wh... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
9feadcb10b926d1f6e721fe9337939c5071538c2 | subsection | 26 | 32 | The metric tractor bundle | As we have seen
in §REF , the homogeneous model G/P is always locally metrizable,
and solutions in the kernel of a given first BGG operator are induced by
parallel sections of a corresponding metric tractor bundle. In general,
if M has nontrivial curvature, not all solutions to a linearized
metrizability problem will c... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.0... | |
ab349a22bf736f24df59cb090e1e677f64ab263a | subsection | 27 | 32 | The metric tractor bundle | Our convention is such that =vv has components
\begin{gather*}
\nu ^{ab}= \lambda ^{a}\tilde{\lambda }^{b} + \tilde{\lambda }^{a}\lambda ^{b};\quad \sigma ^b=\lambda ^b\tilde{\tau }+\tau \tilde{\lambda }^b ;\quad \kappa =\tau \tilde{\tau };\quad \psi _b^c=\ell _{b}\tilde{\lambda }^c+
\lambda ^{c}\tilde{\ell }_{b} ;\\
\... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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ac94e89dc02011ff87d50ca2ff58562fd6d9a5a3 | subsection | 28 | 32 | The metric tractor bundle | The normal solution is the projection onto
S^2{\mathcal {H}} of \exp ({x})\cdot \Phi , which is given by\eta ^{ab}(x,y)=
\nu ^{ab} +x^{(a}\sigma ^{b)}-y^{c(a}\psi ^{b)}_c
+\tfrac{1}{2} {x}^c x^{(a} \psi _c^{b)}+x^{(a} y^{b)c}\xi _c
+\tfrac{1}{2} x^{a}{x}^{b}\psi ^c_c\\
+\tfrac{2}{3} x^{a} x^{b}x^c\xi _c
-\tfrac{1}{3} x... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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3357d569c73613b27ef0b34d555494a46b251730 | subsection | 29 | 32 | Classification of metric parabolic geometries with irreducible | We have seen that the linearizability problem of the existence of compatible
subriemannian metrics on parabolic geometries reduces to a purely algebraic
question related to the number of components in certain tensor products of the
{\mathfrak {p}}_0-modules and its dual *. In fact, we are only interested in
the actions... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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5948cd3d1ece3cf33dd242f2080ba1b0a0348699 | subsection | 30 | 32 | Classification of metric parabolic geometries with irreducible | \end{}
\begin{}[!ht]
\begin{}{|l|l|l|l|l|}
\hline Case& Diagram \Delta _\ell for {\mathfrak {p}},B & Real simple {\mathfrak {g}} & Growth \\
\hline A_\ell ^{h}&\mbox{{}\scriptsize \hspace{5.0pt}$\diagup $}\diagdown \end{}\hrulefill \end{}\hrulefill 1\strut \bulletWe have seen that the linearizability problem of the ex... | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
661dc59806732e5ce1e403982939579666181565 | subsection | 31 | 32 | Classification of metric parabolic geometries with irreducible | \end{}
\begin{}[!ht]
\begin{}{|l|l|l|l|l|}
\hline Case& Diagram \Delta _\ell for {\mathfrak {p}},B & Real simple {\mathfrak {g}} & Growth \\
\hline A_\ell ^{h}&\mbox{{}\scriptsize \hspace{5.0pt}$\diagup $}\diagdown \end{}\hrulefill \end{}\hrulefill 1\strut \bullet | {
"cite_spans": []
} | 1803.10482 | Subriemannian metrics and the metrizability of parabolic geometries | [
"David M. J. Calderbank",
"Jan Slovak",
"Vladimir Soucek"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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7f58967de131156ac8dd99f709895d85a2e494d8 | abstract | 0 | 73 | Abstract | A recently-introduced class of probabilistic (uncertainty-aware) solvers for
ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to
initial value problems. These methods model the true solution $x$ and its first
$q$ derivatives \emph{a priori} as a Gauss--Markov process $\boldsymbol{X}$,
which is... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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... | |
0e377d8ac823a2ba57a258febaed93f8d6e78254 | subsection | 1 | 73 | Introduction | A solver of an initial value problem (IVP) outputs an approximate solution \hat{x} \colon [0,T] \rightarrow \mathbb {R}^d of an ordinary differential equation (ODE) with initial condition:x^{(1)}(t)
\frac{\mathrm {d}x}{\mathrm {d}t} (t)
=
f \left( x(t) \right),\; \forall t\in [0,T], \qquad x(0)=x_0 \in \mathbb {R}^d.(W... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1395,
"openalex_id": "https://openalex.org/W3175461480",
"raw": "H. Poincaré, Calcul des probabilités, Gauthier-Villars, Paris, 1896.",
"source_ref_id": "9360f48cc28b37c724de9fb0c42ba47422b4f41c",
"start": 1141
},
... | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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0.... | |
8068e73de7e58e2fab3fc666921959f972295693 | subsection | 2 | 73 | Introduction | These equivalences (i.e. the equality of the filtering posterior mean and the classical method) are only known to hold in the case of the integrated Brownian motion (IBM) prior and noiseless evaluations of f (in terms of our later notation, the case R\equiv 0), as well as under the following restrictions:Firstly, for q... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 676,
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"raw": "M. Schober, D. Duvenaud, and P. Hennig, Probabilistic ODE solvers with Runge–Kutta means, in Advances in Neural Information Processing Systems (NIPS), 2014, https://papers.nips.cc/paper/5451-probabi... | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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2039a962ec6e4cb5df59c2a9571688c2777adde9 | subsection | 3 | 73 | Contribution | Our main results—lemma:Psiwithdelta,theorem:GlobalTruncation—provide local and global convergence rates of the ODE filter when the step size h goes to zero.
lemma:Psiwithdelta shows local convergence rates of h^{q+1} without the above-mentioned previous restrictions—i.e. for a generic Gaussian ODE filter for all q \in ... | {
"cite_spans": []
} | 1807.09737 | Convergence Rates of Gaussian ODE Filters | [
"Hans Kersting",
"T. J. Sullivan",
"Philipp Hennig"
] | [
"math.NA",
"cs.LG",
"cs.NA",
"math.ST",
"stat.CO",
"stat.ML",
"stat.TH"
] | 2,018 | en | Mathematics | [
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85be09e8acb159324632d2e97e96608d74d0b4ad | subsection | 4 | 73 | Related work on probabilistic ODE solvers | The Gaussian ODE filter can be thought of as a self-consistent Bayesian decision agent that iteratively updates its prior belief {X} over x \colon [0,T] \rightarrow \mathbb {R}^d (and its first q derivatives) with information on \dot{x} from evaluating f.Here, the word `Bayesian' describes the algorithm in the sense th... | {
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1dd67f8fb6dd344cf8585442b96d193ebd472f7c | subsection | 5 | 73 | Related work on probabilistic ODE solvers | Therefore, it is in a way more similar to classical non-stochastic solvers than to sampling-based stochastic solvers.Accordingly, the convergence results in this paper concern the convergence rate of the posterior mean to the true solution, while the theoretical results in , , , , , provide convergence rates of the var... | {
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fba228215e0c840f88f65ea92a67af1f62fcfe08 | subsection | 6 | 73 | Relation to Filtering Theory | While Gaussian filtering was first applied to the solution of ODEs in , it has previously been analysed in the filtering, data assimilation as well as linear system theory community.
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21eb063e97dd1c5796a9b94b61f81034f9ff397b | subsection | 7 | 73 | Outline | The paper begins with a brief introduction to Gaussian ODE filtering in sec:introductionofalgorithm.
Next, sec:regularityofflow,sec:auxiliaryboundsonintermediatequantities provide auxiliary bounds on the flow map of the ODE and on intermediate quantities of the filter respectively.
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f50d0f65abf9d5e39cedce1c0ba2ca921946cc65 | subsection | 8 | 73 | Notation | We will use the notation [n] = \lbrace 0,\dots ,n-1\rbrace .
For vectors and matrices, we will use zero-based numbering, e.g. x=(x_0,\dots ,x_{d-1}) \in \mathbb {R}^d .
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75d4c2214961d1b7c6439fde5e3cb56de393b9fb | subsection | 9 | 73 | Gaussian ODE filtering | This section defines how a Gaussian filter can solve the IVP IVP.
In the various subsections, we first explain the choice of prior on x, then describe how the algorithm computes a posterior output from this prior (by defining a numerical integrator {\Psi }), and add explanations on the measurement noise of the derivati... | {
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32459e01cc031c14c9728a3868b96e70b4c09294 | subsection | 10 | 73 | Prior on | In PN, it is common to put a prior measure on the unknown solution x.
Often, for fast Bayesian inference by linear algebra , this prior is Gaussian.
To enable GP inference in linear time by Kalman filtering , we restrict the prior to Markov processes.
As discussed in , a wide class of such Gauss–Markov processes can be... | {
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