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037dd9676f090b21dd4753da6dfd0cfeaebbab18 | subsection | 22 | 72 | Probabilistic Strichartz estimates | We therefore view the radial randomization as a modest step towards probabilistic treatments of the geometric equations discussed in .Let f \in (\mathbb {R}^d) and A_{a,\delta } as in (REF ) with a \sim N . If \alpha ,p, and q satisfy (REF )-(), then\Vert |x|^\alpha A_{a,\delta } f \Vert _{L_t^qL_x^p} \lesssim N^{\frac... | {
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05f25b5970d440e6c872b9ab3076f5cb13f7fcb1 | subsection | 23 | 72 | Probabilistic Strichartz estimates | From the square-function estimate (Lemma REF ), Minkowski's integral inequality, Khintchine's inequality, and Corollary , it follows that&\Vert |x|^\alpha f^\omega \Vert _{{\sigma }{q}{p}}\\
&\Vert |x|^\alpha f^\omega _N\Vert _{L_\omega ^\sigma L_t^q L_x^p \ell _N^2}\\
&\le \Vert |x|^\alpha f^\omega _N \Vert _{\ell _N^... | {
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b7e903a1de13f082853e90587edc704a62193b8f | subsection | 24 | 72 | Probabilistic Strichartz estimates | Let 1\le p < \infty be a sufficiently large exponent. Using Proposition REF and Lemma , we have for all p \le \sigma < \infty that\Vert |x|^{\frac{3}{8}} f_N^\omega \Vert _{L_\omega ^\sigma L_t^{\frac{8}{3}} L_x^\infty }
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84acb5149c721a44461d9be4209be7b0b159bfc4 | subsection | 25 | 72 | Probabilistic Strichartz estimates | We fix t_0,t_1\in \mathbb {R} . By the fundamental theorem of calculus, it holds that\Vert \exp (it_1|\nabla |) f_N^\omega \Vert _{L_x^6} &\le \Vert \exp (it_0|\nabla |) f_N^\omega \Vert _{L_x^6} + \int _{[t_0,t_1]} \Vert \partial _t ( \exp (it|\nabla |) f_N^\omega ) \Vert _{L_x^6} \\
&\lesssim \Vert \exp (it_0|\nabla ... | {
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b78620383bd1d796b50ac3e034b71c83c0ccf22f | subsection | 26 | 72 | Probabilistic Strichartz estimates | REF ), Proposition REF , and the same argument as before, we obtain that\Vert |x|^{\frac{1}{2}} f^\omega _N \Vert _{{\sigma }{\infty }{\infty }} &\lesssim \Vert |x|^{\frac{1}{2}} |\nabla |^{\frac{3}{q}}f^\omega _N \Vert _{{\sigma }{\infty }{q}} \\
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16fc39ce5149e43e0fb8c675182526174d6a7a2f | subsection | 27 | 72 | An in/out decomposition | In this section, we describe a decomposition of solutions to the linear wave equation into incoming and outgoing components (see Figure REF ). This decomposition relies heavily on the spherical symmetry of the initial data. The in/out-decomposition can be derived in physical space by using spherical means, see e.g. . H... | {
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ed22d7f2cee2097bb5bec958c5006868949adab4 | subsection | 28 | 72 | An in/out decomposition | Then,\frac{\sin (t|\nabla |)}{|\nabla |} g(r) &= r^{-\frac{1}{2}} \int _0^\infty \sin (t\rho ) J_{\frac{1}{2}}(r\rho ) \widehat{g}(\rho ) \rho ^{\frac{1}{2}} \\
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e9741b8b6e7262ee6d73ded03a9699570b45071f | subsection | 29 | 72 | An in/out decomposition | This also allows us to savely leave out the arguments t+r and t-r in subsequent computations.From Plancherell's theorem, it follows that\Vert W_s[h](\tau )\Vert _{L_\tau ^2(\mathbb {R})} + \Vert W_c[h] \Vert _{L_\tau ^2(\mathbb {R})} \lesssim \Vert \rho h \Vert _{L_\rho ^2(\mathbb {R}_{>0})}~.As a consequence, we have ... | {
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1d0907ced6d04139b3e0b390c7e231e76cbb8656 | subsection | 30 | 72 | An in/out decomposition | To overcome this technical problem, we write&\partial _{x_j} F(t,x) \\
&= \frac{x_j}{r} \partial _r F(t,r) \\
&= - \frac{x_j}{r^3} \left( W_{\text{out}}[F](t-r) + W_{\text{in}}[F](t+r) \right) + \frac{x_j}{r^2} \left( -(\partial _\tau W_{\text{out}}[F])(t-r) + (\partial _\tau W_{\text{in}}[F])(t+r) \right)After a short... | {
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955161f10ad218d9833008aa170091fc51c333e0 | subsection | 31 | 72 | An in/out decomposition | Then, we have for all 2\le q \le \infty that\Vert W_s[f](\tau ) \Vert _{L^q_\tau (\mathbb {R})} + \Vert W_c[f](\tau ) \Vert _{L^q_\tau (\mathbb {R})} \lesssim (a\delta )^{\frac{1}{2}-\frac{1}{q}} \Vert f \Vert _{L_x^2(\mathbb {R}^3)} ~.Using Hölder's inequality, we have that|W_s[f](\tau )| + |W_c[f](\tau )| \lesssim \i... | {
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19175f65d650c002d19accf09da28a43ab1bab85 | subsection | 32 | 72 | An in/out decomposition | Using a combination of Khintchine's inequality, Minkowski's integral inequality, and Lemma , we have that&\Vert W_s[f_N^\omega ](\tau )\Vert _{L_\omega ^\sigma L_\tau ^q} \\
&\le \Vert W_s[f_N^\omega ](\tau )\Vert _{L_\tau ^q L_\omega ^\sigma } \\
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8acb8d7893b8e42009bf71d7233d383ee40ff17a | subsection | 33 | 72 | Local well-posedness and conditional scattering | Recall that the forced nonlinear wave equation is given by{\left\lbrace \begin{array}{ll}
-\partial _{tt} v + \Delta v = (v+F)^5~,\qquad (t,x) \in \mathbb {R}\times \mathbb {R}^3~.\\
v(t_0,x)= v_0 \in \dot{H}_x^1(), \qquad \partial _t v(t_0,x) = v_1\in L_x^2()~.
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210fd23c343d13f2287cf627b7c56c5b5f41a6e4 | subsection | 34 | 72 | Almost energy conservation and decay estimates | In this section, we prove new estimates for the solution to the forced NLW{\left\lbrace \begin{array}{ll}
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50218841fce06063543a5ef9ea4a5537627d0125 | subsection | 35 | 72 | Almost energy conservation and decay estimates | More precisely,
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1c712f74502a429bc79384a65b978c34e9f97ec5 | subsection | 36 | 72 | Almost energy conservation and decay estimates | Finally, the third summand in (REF ) only includes lower order error terms, and they are controlled in Section REF .The idea to integrate by parts in the energy increment has previously been used in , , .From the divergence formula (REF ), it follows that\frac{\mathrm {d}}{} E[v](t) &= \frac{\mathrm {d}}{} \int _{} T^{... | {
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18a24736360817a2eae8302966eb78710f62db05 | subsection | 37 | 72 | Almost energy conservation and decay estimates | Then, we have the Morawetz identity&\tfrac{2}{3} \int _{I} \int _{\mathbb {R}^3} \frac{v^6}{|x|} + \pi \int _I |v|^2(t,0) +
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6399c89de59fcbbcdfba54f90611de5f8aad7281 | subsection | 38 | 72 | Almost energy conservation and decay estimates | First, using Hardy's inequality, we have that&\left| \int _{\mathbb {R}^3} \partial _t v ~ \frac{x}{|x|}\cdot \nabla v - 4 \frac{v}{|x|} \partial _t v ~ {\Big |_{t=a}^{b}} \right| \\
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aa54b7ebf93e61a1dc35f56c890c1cbde798e24d | subsection | 39 | 72 | Almost energy conservation and decay estimates | Finally, we have that\left| \int _I \int _{\mathbb {R}^3} \frac{1}{|x|} \mathcal {N} v \right|
\lesssim \int _I \int _{} \frac{1}{|x|} |F| ( |F|+|v| )^4 |v| \lesssim \int _I \int _{} \frac{1}{|x|} |F| (|F|+|v|)^5 ~.In contrast to the case d=4 as in , , the energy and the Morawetz term are not strong enough to control t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1353/ajm.2020.0013",
"end": 342,
"openalex_id": "https://openalex.org/W3013597746",
"raw": "Benjamin Dodson, Jonas Lührmann, and Dana Mendelson. Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with ra... | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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3783139c7ec1b1b4ab105ac24f6807d2f714972a | subsection | 40 | 72 | Almost energy conservation and decay estimates | We will now use time-translation invariance to integrate it against a weight w\in L_\tau ^1(\mathbb {R}) .
[Figure: Interaction Flux Estimate][Forward Interaction Flux Estimate]
Let v be a solution to the forced NLW (REF ) on a compact time interval I=[a,b]\subseteq [0,\infty ) . Also, let w \in L_\tau ^1(\mathbb {R}) ... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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475139a6644edfd330743b001da3bc477b3dcde2 | subsection | 41 | 72 | Almost energy conservation and decay estimates | In contrast, the errors in () are of lower order, and they will be controlled in Section REF .To remember that the weight w in () should be integrated over {(-\infty , t-|x|]} , note that the contribution of the error \partial _t(F) v^5 should be weighted less as t\rightarrow -\infty and |x|\rightarrow \infty .By time-... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0673896fe12c1450304249d219b12e7d5e7e1863 | subsection | 42 | 72 | Almost energy conservation and decay estimates | Thus, we turn to the second summand in (REF ). Using integration by parts, we have that&5 \left| \int _I \int _{|x|\le t-\tau , t \in I} w(\tau ) F v^4 \partial _t v \right| \\
&= \left| \int _I \int _{} \left( \int _{-\infty }^{t-|x|} w(\tau ) \right) F \partial _t(v^5)\right| \\
&=\left| \int _{} \left( \int _{-\inft... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.01... |
ebbecb45618c1eb66a1c3fbdd9d7aa42a36b0d63 | subsection | 43 | 72 | Almost energy conservation and decay estimates | Then, we have that&\int _{I} \int _{} w(t+|x|) |v|^6(t,x) \\
&\lesssim \Vert w \Vert _{L_\tau ^1(\mathbb {R})} \sup _{t\in I} E[v](t) + \Vert w \Vert _{L_\tau ^1(\mathbb {R})} \Vert F \Vert _{L_t^\infty L_x^6()} \sup _{t\in I} E[v](t)^{\frac{5}{6}} \\
&~~+ \left| \int _I \int _{} \left( \int _{t+|x|}^{\infty } w(\tau )... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.020238108932971954,
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... |
d3a1f7c10e850cbca19aa2ca8db792ad94c69425 | subsection | 44 | 72 | Bootstrap argument | In this section, we introduce the quantities in the bootstrap argument to control the energy. For a given time interval I\subseteq \mathbb {R} , we define the energy\mathcal {E}_I := \sup _{t\in I} E[v](t) = \sup _{t\in I} \int _{\mathbb {R}^3} \frac{1}{2} (\partial _t v(t,x))^2 + \frac{1}{2} |\nabla v(t,x)|^2 + \frac{... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.007185841910541058,
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c6da2e878a2069a958b87adb5dcb76042e00dbc3 | subsection | 45 | 72 | Bootstrap argument | Instead, we define for each K \in 2^{\mathbb {N}} the operatorS_K w = K \langle K \tau \rangle ^{-2} * w~.[Interaction Flux Term]
Let (f_0,g_0)\in L_{}^2()\times \dot{H}_{}^{-1}() and assume that P_{\le 2^5}f_0= P_{\le 2^5} g_0 = 0 . Let F be the solution of the linear wave equation with data (f_0,g_0) , let \widetilde... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.009107505902647972,
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0.0020442306995391846,... |
ac87b73ac8572843e523908443e07cc8ec68b729 | subsection | 46 | 72 | Bootstrap argument | Then, we define\Vert F \Vert _{Y(I)} &:= \Vert N^{-\frac{3}{4} +\frac{1}{24\gamma } + \delta } |x|^{\frac{3}{8}} |\nabla |\Vert _{\ell _N^{\frac{8}{3}}L_t^{\frac{8}{3}}L_x^{\infty }(2^{\mathbb {N}}\times )} \\
&~+\Vert N^{-\frac{3}{4} + \frac{1}{24\gamma } + \frac{5\delta }{2}} |x|^{\frac{3+2\delta }{8}} |\nabla | F_N\... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.042364224791526794,
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0... |
5a311fe5e10d21b54344337cd61be622a37cb312 | subsection | 47 | 72 | Bootstrap argument | More precisely, let \eta > 0 be given and assume that \Vert F \Vert _{Y(\mathbb {R}) } < \infty . Then, there exists a finite number J=J(\eta ,\Vert F\Vert _{Y(\mathbb {R})}) and a partition of \mathbb {R} into finitely many intervals I_1,\hdots , I_J such that \Vert F \Vert _{I_j} < \eta for all j=1,\hdots ,J .[Almost... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04308348149061203,
0.031336214393377304,
0.017270008102059364,
0.027491655200719833,
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0.021999426186084747,
-0.02552360668778419... |
58473323432e7b326961a6f06c5b9cff1700fbd2 | subsection | 48 | 72 | Bootstrap argument | For \sigma \ge 8/3 , it follows from Minkowski's integral inequality and Lemma that&~~~~\Vert N^{-\frac{3}{4}+ \frac{1}{24\gamma }+ \delta } |x|^{\frac{3}{8}} |\nabla |^\omega \Vert _{L_\omega ^\sigma \ell _{N}^{\frac{8}{3}} L_t^{\frac{8}{3}} L_x^\infty } \\
&\le \Vert N^{-\frac{3}{4}+ \frac{1}{24\gamma }+ \delta } |x|... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03559156507253647,
-0.009622847661376,
-0.01805523969233036,
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-0.... |
17fbff6ba3626fde2e9bec6266183cbd0e827e3c | subsection | 49 | 72 | Bootstrap argument | Using Corollary , the terms involving
\Vert W_*[|\nabla | ] \Vert _{L_\tau ^p} lead to the restrictions > \max \left( (1-\tfrac{1}{\gamma }) (\tfrac{1}{2}-\tfrac{1}{24}), 0 \right) + \max \left( 1-\tfrac{1}{12\gamma },0 \right)~.Since 0 < \gamma \le 1 , this leads to s > \max (1-\frac{1}{12\gamma }, 0 ) . Using Lemma ... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.028032682836055756,
0.02171502821147442,
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57bad8ff56ec9712a94b999dfbb32428013b75f7 | subsection | 50 | 72 | Control of error terms | In this section, we estimate the error terms in Proposition , Lemma , and Proposition . Before we begin with our main estimates we prove an auxiliary lemma.Let w\in L^1_\tau (\mathbb {R}) be nonnegative. Let K \in 2^\mathbb {N} be arbitrary, and let S_K be defined byS_K w = K ~ \langle K \rho \rangle ^{-2} * w ~.Then, ... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.015544399619102478,
0.061079271137714386,
0.013279097154736519,
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0.0209902785718441,
0.0... |
d3768ec1823a1c41ac86f9f8d9d2e9c903f96ba3 | subsection | 51 | 72 | Control of error terms | Using an integral formula from , we have that&K^3 \int _{} |{\Psi }(K(y-x))| w(t-|x|) \\
&= K^3 \int _{} |{\Psi }(Kx)| w(t-|y-x|) \\
&\lesssim K^3 \int _{0}^\infty |{\Psi }(Kr)| \left( \int _{|x|=r} w(t-|y-x|) \right) \\
&= K^3 \int _{0}^\infty |{\Psi }(Kr)| \left( \int _{|y-x|=r} w(t-|x|) \right) \\
&= K^3 \int _{0}^\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 647,
"openalex_id": "https://openalex.org/W646012940",
"raw": "Christopher D. Sogge. Lectures on nonlinear wave equations. Monographs in Analysis, II. International Press, Boston, MA, 1995.",
"source_ref_id": "db8ff283a29a90... | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.00420415448024869,
0.07416403293609619,
0.0036853114143013954,
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0.02194400690495968,
-0.00... |
6729af3f552edccf0a303e1e3882bdb949fdf9ce | subsection | 52 | 72 | Control of error terms | We have that&\frac{K^3}{|y|} \int _{4|y|}^\infty \int _{r-|y|}^{r+|y|} r |{\Psi }(Kr)| w(t-\rho )\rho \\
&\lesssim \frac{K^3}{|y|} \int _{4|y|}^\infty \int _{r-|y|}^{r+|y|} r^2 |{\Psi }(Kr)|w(t-\rho ) \\
&\le \frac{K^3}{|y|}\Vert w \Vert _{L_\tau ^1(\mathbb {R})} \int _{0}^\infty |{\Psi }(Kr)| r^2 \\
&\le \frac{1}{|y|}... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02631465159356594,
0.052476752549409866,
0.0013061983045190573,
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0.0066701918840408325,
0.00986227393150329... |
3dc5ebab88a8265b22ed0a26f71c5ed306d2b7ad | subsection | 53 | 72 | Control of error terms | Using the in/out-decomposition and Lemma , it follows that&\quad ~\Vert |x|^{\frac{1}{3}} \big ( |\nabla | \widetilde{F}_N \big )^{\frac{1}{3}} P_K v \Vert _{L_{t,x}^6()}^6 \\
&\lesssim \Vert ()^{\frac{1}{3}} P_K v \Vert _{L_{t,x}^6()}^6 + \Vert ()^{\frac{1}{3}} P_K v \Vert _{L_{t,x}^6()}^6\\
&\lesssim \Vert S_K( ||^2)... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.00455749174579978,
0.05449158325791359,
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-0.018611347302794456,
0.... |
37b3d77e4a2dc7d8a5b5c38953ea61fc4fd845c9 | subsection | 54 | 72 | Energy increment | In this section, we control the main error term in the energy increment.
[Main error term in energy increment]
Let F be as in Definition and let v\colon I \times \rightarrow \mathbb {R} be a solution of (REF ). Then, it holds that\left| \int _{I} \int _{\mathbb {R}^3} (|\nabla |) v^5 \right| \lesssim ( \mathcal {F}_{I}... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.018627965822815895,
0.041222576051950455,
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-0.009489431045949459,
-0.02259460836648941,
... |
21013e664fb228e8ae76f3cc7bb6ea8cdcb8f4aa | subsection | 55 | 72 | Energy increment | Using Proposition REF and Corollary , it follows that&\left| \int _{I} \int _{\mathbb {R}^3} (|\nabla |) \prod _{j=1}^5 P_{K_j} v\right| \\
&\le \Vert |x|^{\frac{3}{8}} |\nabla | \Vert _{{\frac{8}{3}}{\infty }()}^{\frac{2}{3}} \Vert |x|^{\frac{1}{3}} (|\nabla |)^{\frac{1}{3}} P_{K_5} v \Vert _{L_{t,x}^6()}~ \prod _{j=2... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03314272314310074,
0.037323713302612305,
-0.009536925703287125,
-0.0038662697188556194,
-0.022079890593886375,
0.022522402927279472,
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0.008293310180306435,
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0.011039945296943188,
-0.022201962769031525... |
e47c95158d5ac8e86182a60b56c4896f26dc3da7 | subsection | 56 | 72 | Morawetz estimate | In this section, we control the main error term in the Morawetz estimate. The main new difficulty is the weight x/|x| .
[Main error term in Morawetz estimate]
Let F be as in Definition and let v be a solution of (REF ). Then,&\left| \int _I \int _{\mathbb {R}^3} \frac{x}{|x|}\cdot \nabla _x(F)~ v^5 \right| \\
&\lesssim... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.014164215885102749,
0.03584912046790123,
-0.024575715884566307,
0.01586514338850975,
0.0013662709388881922,
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0.016277026385068893,
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-0.021646765992045403,
0.02997596561908722,
0.0409... |
d7abcf48417563a4da2caffceaf9625cb16a570f | subsection | 57 | 72 | Morawetz estimate | Using (REF ), we estimate&\left| \int _I \int _{\mathbb {R}^3} P_L\Big (\frac{x}{|x|} \Big )\cdot \nabla _x(F_N)~ \prod _{j=1}^5 P_{K_j} v ~ \right| \\
&\le \int _{I} \int _{\mathbb {R}^3} \left|P_L\Big (\frac{x}{|x|} \Big )\right| ~\frac{1}{|x|^{\frac{1}{3}}} \left( ||+ || \right)^{\frac{1}{3}} |\nabla _xF_N|^{\frac{2... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.046456046402454376,
0.05729172006249428,
-0.02875269204378128,
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-0.028462722897529602,
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0.028584815561771393,
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0.018970061093568802,
-0.02156451903283596,
0.004... |
1884e0973d3ef2c7b6c2e8c995b153bf32facc01 | subsection | 58 | 72 | Morawetz estimate | We have that& \int _{I} \int _{\mathbb {R}^3} \left|P_L\left(\frac{x}{|x|} \right)\right|~ \frac{1}{|x|^{\frac{1}{3}}} |F_N|^{\frac{1}{3}} |\nabla _xF_N|^{\frac{2}{3}} \prod _{j=1}^5 |P_{K_j} v|\\
&\lesssim \Vert P_L( \frac{x}{|x|}) \Vert _{L_{t,x}^\infty ()} ~ \prod _{j=2}^5 \Vert |x|^{-\frac{1}{6}} P_{K_j} v \Vert _{... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.... |
3dab41289786aba4a315be4dba57f62cf4ac45e6 | subsection | 59 | 72 | Morawetz estimate | Consequently, we have that L \ge 2^{-4} N > 1 . Using Lemma REF , it follows that
|P_L( \frac{x}{|x|})| \lesssim (L|x|)^{-1} . This yields&\left| \int _I \int _{\mathbb {R}^3} P_L\left(\frac{x}{|x|} \right)\cdot \nabla _x(F_N)~ \prod _{j=1}^5 P_{K_j} v ~ \right| \\
&\lesssim L^{-1} \int _I \int _{\mathbb {R}^3} \frac{... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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4f7c7d5662ca97b2d795d6260004ef4ec0b88131 | subsection | 60 | 72 | Interaction flux estimate | In this section, we control the main error terms in the interaction flux estimate. The main difficulty is the weight \int _{-\infty }^{t-|x|} w(\tau ). First, we recall a radial Sobolev embedding.For any v \in L_t^\infty \dot{H}_{}^1() , we have\sup _{K\in 2^{N}}\Vert |x|^{\frac{1}{2}} P_K v \Vert _{L_{t,x}^\infty ()} ... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0fec1b8b06bbf3ff5b27aff6b1490bb5466f36bc | subsection | 61 | 72 | Interaction flux estimate | Then, it holds that&\left| \int _{I} \int _{\mathbb {R}^3} \left( \int _{-\infty }^{t-|x|} w \right) (|\nabla | \widetilde{F}) v^5 \right| \\
&\lesssim \Vert w \Vert _{L_\tau ^1(\mathbb {R})} \Vert F \Vert _{Y_I}^{\frac{2}{3}} ( \mathcal {F}_{I} + \Vert F\Vert _Z^2 \mathcal {A}_I )^{\frac{1}{6}} \mathcal {A}_I^\frac{7}... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e99d8e0f128d0fab28ef463e79ae55c05de12a17 | subsection | 62 | 72 | Interaction flux estimate | We have that&\left| \int _{I} \int _{\mathbb {R}^3} P_L\left( \int _{-\infty }^{t-|x|} w \right) (|\nabla | ) \prod _{j=1}^5 P_{K_j} v ~ \right|\\
&\lesssim \Vert P_L\big ( \int _{-\infty }^{t-|x|} w \big ) \Vert _{L^{\infty }_{t,x}()}~ \Vert |x|^{\frac{3}{8}} |\nabla | \Vert _{{\frac{8}{3}}{\infty }()}^{\frac{2}{3}} ~... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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e3ed2b5116bcee7a9b5af7db6ec50493e1ccda68 | subsection | 63 | 72 | Interaction flux estimate | Using Proposition REF , Corollary and Lemma REF , we obtain that&\left| \int _{I} \int _{|x|\ge 1} P_L\left( \int _{-\infty }^{t-|x|} w \right) (|\nabla | ) \prod _{j=1}^5 P_{K_j} v ~ \right| \\
&\le \Vert \langle x \rangle ^{-2} P_L \left( \int _{-\infty }^{t-|x|} w \right) \Vert _{L_{t,x}^2()} \prod _{j=3}^5 \left( \... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... |
c63fad19d0ed8f0bd438b2c482a7292f5a83417b | subsection | 64 | 72 | Interaction flux estimate | Using Lemma
REF and the boundedness of the Hardy-Littlewood maximal function M , we obtain that& \Vert \langle x \rangle ^{-2} P_L \left( \int _{-\infty }^{t-|x|} w \right) \Vert _{L_{t,x}^2()}\\
&= \Vert \langle x \rangle ^{-2} P_L \left( \int _{t-|x|}^{t} w \right) \Vert _{L_{t,x}^2()}\\
&\lesssim L^{-1} \Vert \langl... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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5072f15bc5a7b2487183d358961372babf0ec87e | subsection | 65 | 72 | Interaction flux estimate | Thus, we write&\left| \int _{I} \int _{|x|\le 1} P_L \left( \int _{-\infty }^{t-|x|} w \right) (|\nabla | ) \prod _{j=1}^5 P_{K_j} v ~ \right|\\
&\lesssim \Vert |x|^{\frac{1}{6}} P_L\Big ( \int _{-\infty }^{t-|x|} w \Big ) \Vert _{L_{t,x}^{12}(I\times \lbrace |x|\le 1 \rbrace )} \Vert |x|^\frac{2}{3} |\nabla | \Vert _{... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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cc99600cbbdc779292abcdf3f96ccef7420e1326 | subsection | 66 | 72 | Interaction flux estimate | Let F be as in Definition and let v\colon I \times \rightarrow \mathbb {R} be a solution of (REF ). Then, it holds that\left| \int _I \int _{\mathbb {R}^3} w(t-|x|) F v^5 \right| \lesssim \Vert w \Vert _{L_\tau ^2(\mathbb {R})} \mathcal {F}_I^{\frac{1}{2}} \mathcal {E}_I^{\frac{1}{2}} + \Vert w \Vert _{L_\tau ^{12}(\ma... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0... |
b693d9d7c583198530350af14e0add988aa679b8 | subsection | 67 | 72 | Lower order error terms | [Control of lower order error terms]
Let F be as in Definition and let v be a solution of (REF ). | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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4e4da8d3a9055a54b684e7d2234e97a57c1dc591 | subsection | 68 | 72 | Lower order error terms | Then, it holds that&\int _{I} \int _{} |F|^5 \left( |\partial _t v| + \frac{|v|}{|x|} + |\nabla v | \right) &&\lesssim \Vert F \Vert _{Y_I}^5 \mathcal {E}_I^{\frac{1}{2}} ~,\\
&\int _{I} \int _{} |F|^2 |v|^3 \left( |\partial _t v| + \frac{|v|}{|x|} + |\nabla v | \right) &&\lesssim \Vert F \Vert _{Y_I}^2 \mathcal {A}_I^... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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6cd727c63aac9f91e9b19f4b50375d9bcbbc971a | subsection | 69 | 72 | Proof of the main theorem | In this section, we collect all previous estimates to prove the a priori energy bound (Theorem ). Using the conditional scattering result of , we finish the proof of Theorem .[Proof of Theorem ]By time-reversal symmetry, it suffices to prove that \sup _{t\in [0,\infty )} E[v](t) < \infty . Let \frac{1}{2}\ge \eta _0> 0... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1353/ajm.2020.0013",
"end": 175,
"openalex_id": "https://openalex.org/W3013597746",
"raw": "Benjamin Dodson, Jonas Lührmann, and Dana Mendelson. Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with ra... | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
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92ae055302dbd9a5a517ffe823593a990cf53922 | subsection | 70 | 72 | Proof of the main theorem | First, recall that from the definition of \Vert F \Vert _Z and the embedding \ell _1 \hookrightarrow \ell _2 , we have that&\sum _{*\in \lbrace \text{out},\text{in} \rbrace } \sum _{p\in \lbrace 2,4,24 \rbrace } \left( \sum _{N\ge 2^5 } (N^{-\frac{1}{6\gamma }+2\delta } + N^{-2+2\delta }) \left( \Vert W_{*}[|\nabla |F_... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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8904941e580dbb5947db050bb352a0400e34c3a4 | subsection | 71 | 72 | Proof of the main theorem | Combining (REF ), (REF ), and (REF ), we arrive at&\le C(\mathcal {E}_j +1 ) + \eta _0 + \eta _0 (+)~, \\
+ &\le C( +1 ) + \tfrac{1}{2} (+ )~.Finally, choosing \eta _0 > 0 sufficiently small depending on C= C(\Vert F \Vert _Z ) , we obtain that+1 \le \tilde{C}~ ( \mathcal {E}_j+1) ~.By iterating this inequality finitel... | {
"cite_spans": []
} | 10.2140/apde.2020.13.1011 | 1804.09268 | Almost sure scattering for the radial energy critical nonlinear wave
equation in three dimensions | [
"Bjoern Bringmann"
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ea86ad79d2bc41705252ba834d6462d6615f735f | abstract | 0 | 15 | Abstract | The arcp URI scheme is introduced for location-independent identifiers to
consume or reference hypermedia and linked data resources bundled inside a file
archive, as well as to resolve archived resources within programmatic
frameworks for Research Objects. The Research Object for this article is
available at http://s11... | {
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} | 10.1109/eScience.2018.00018 | 1809.06935 | The Archive and Package (arcp) URI scheme | [
"Stian Soiland-Reyes",
"Marcos Cáceres"
] | [
"cs.DL",
"cs.SI"
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6779290612c8a4b3f6c4c24d2fba5c1de4bdd540 | subsection | 1 | 15 | Background | Archive formats like BagIt have been recognized as important for preservation and transferring of datasets and other digital resources . More specific examples include COMBINE archives for systems biology, CDF for astronomy data, as well as the more general HDF5 which is also used for meteorological data. For the purpo... | {
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"raw": "J.A. Kunze, J. Littman, L. Madden, J. Scancella, C. Adams (2018): The BagIt File Packaging Format (V1.0). Internet Engineering Task Force. https://datatracker.ietf.org/doc/html/draft-kunze-bagit-16"... | 10.1109/eScience.2018.00018 | 1809.06935 | The Archive and Package (arcp) URI scheme | [
"Stian Soiland-Reyes",
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72ca393a778bb924bcbeba5535837f4f91679b17 | subsection | 2 | 15 | The Archive and Package (arcp) URI scheme | Inspired by the app URL scheme we defined the Archive and Package (arcp) URI scheme , an IETF Internet-Draft which specifies how to mint URIs to reference resources within any archive or package, independent of archive format or location.The primary use case for arcp is for consuming applications, which may receive an ... | {
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"arxiv_id": "",
"doi": "10.5281/zenodo.1320264",
"end": 238,
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"Stian Soiland-Reyes",
"Marcos Cáceres"
] | [
"cs.DL",
"cs.SI"
] | 2,018 | en | Computer Science | [
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1bcd87e0b2174b2c9d9015148ed5cdbdbb3f773c | subsection | 3 | 15 | Identifier structure | By definition an arcp identifier is an URI with three parts, as shown in figure REF .
[Figure: Structure of arcp identifier]The arcp Internet-Draft specifies three initial redprefix values: uuid, ni and name, each which defines how to identify a particular archive by a corresponding greennamespace. These namespaces are... | {
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{
"arxiv_id": "",
"doi": "",
"end": 86,
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"raw": "T. Berners-Lee, R. Fielding, L. Masinter (2005): Uniform Resource Identifier (URI): Generic Syntax. RFC Editor. RFC 3986 https://doi.org/10.17487/rfc3986",
"source_ref_id": "9fd23e5bb542a12ae16... | 10.1109/eScience.2018.00018 | 1809.06935 | The Archive and Package (arcp) URI scheme | [
"Stian Soiland-Reyes",
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] | 2,018 | en | Computer Science | [
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0.... |
2619f8b6c204d6ab974f0a54407c45cc17e6b820 | subsection | 4 | 15 | UUID-based identifiers | The simplest case for temporary sandbox processing of an archive with arcp is to generate a new random UUIDv4 , e.g.:c6179148-3cde-4435-8e66-304453f89d59From this the corresponding arcp URI is:<arcp://uuid,c6179148-3cde-4435-8e66-304453f89d59/>This base URI can be used when resolving relative URI references, e.g. if
<m... | {
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45623818fc9aa39aee31227fae93f672494572f6 | subsection | 5 | 15 | Hash-based identifiers | For this arcp defines a hash-based method, where the bytes of the archive file is used to find a checksum-based identifier based on the
Naming Things With Hashes (ni) URI scheme . For instance if the sha-256 checksum of a Zip file is in hexadecimal:7f83b1657ff1fc53b92dc18148a1d65dfc2d4b1fa3d677284addd200126d9069After b... | {
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f394c86bcb1951c5cb14295b17f01f072f11008f | subsection | 6 | 15 | Name-based identifiers | Finally, paying homage to its origin in app URLs, arcp can use a system-based app name. This is a suggested mechanism for resolving resources of an application package installed in a runtime system like Android applicationId or Java package name, where an application identifier can be directly reused in arcp for URIs w... | {
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8e6a040557596fbf7c424d565eabbb1579f4db34 | subsection | 7 | 15 | Archive fragments | Without using arcp one could in theory still reference files within archives at an URL with # fragments:<http://example.com/download/archive13.zip#data/survey.csv>Unlike formats like text/html
or application/pdf, most archive media formats like application/zip unfortunately do not define a fragment syntax, and some maj... | {
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7271a3e71b47b1879e5c305dfbd86fef95d36811 | subsection | 8 | 15 | File URIs | As argued above, file URLs that represent local directories are fragile and not globally unique. It is perhaps less known that file URLs
can specify a host name:<file://host.example.com/home/alice/extracted/archive13/>The above references a file path on the machine with the fully qualified domain name (FQDN)
host.examp... | {
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2e81f95c1b9a8f79bc7acc4702e2756c132a8df9 | subsection | 9 | 15 | JAR URLs | If we restrict usage to ZIP files at a known URL, then they are in theory also valid JAR files, and we can address files with the
jar URL scheme:<jar:http://example.com/download/archive13.zip!/data/survey.csv>Here relative URIs may not parse well, as it is easy to accidentally climb out of !/, and technically the JAR U... | {
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336d186595e079ae8d4e68b9146fa5d782c6dac6 | subsection | 10 | 15 | Object Reuse and Exchange proxies | OAI-ORE defines proxies to represent a resource as aggregated in a collection; these can be used to model archives , but ORE proxies face two problems: How to represent the file path, and how to identify the proxy so it can be used as a reference in Linked Data. The resource must be identified using two triples of ore:... | {
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a5b5d8f6aeee7ab447d5856a0b6e1b5813b5bedd | subsection | 11 | 15 | Publishing file systems as Linked Data | F2R , using the Nepomuk File Ontology , defines a way to publish file systems as Linked Data, where a server endpoint exposes the files and their file system metadata.F2R URIs are localized to an endpoint and an free-text named file system, e.g. mysource, and files are identified with UUIDs:<http://f2r.example.com/myso... | {
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e426a7c40eae3c03e1e94ddbb49614e0286323c7 | subsection | 12 | 15 | EPUB canonical fragment identifiers | EPUB is a standard for hypermedia eBooks.
RO Bundle is based on the
EPUB Open Container Format .
EPUB Canonical Fragment Identifiers can link to nested XML elements of an publication using a variation of
XPath with
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fd3f69916dc503e6428c85fcb83cac2d91e76842 | subsection | 13 | 15 | arcp implementations | The
arcp Python library was developed to help creating, parsing and validating arcp URIs. In particular it can
generate arcp based on random UUIDs, URL locations, names and hashing archive bytes. The arcp parser recognize the arcp prefix and can extract UUIDs or hashes, and can generate the corresponding .well_known/ni... | {
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8c53c6d0bc95562f5a19344ea4cd6890b8a5c407 | subsection | 14 | 15 | Conclusion | This article propose the arcp identifier scheme for resources within archives using formats like ZIP, tar and BagIt, and suggest arcp is useful for identifying standalone Research Objects and for processing Linked Data embedded in archives. The Internet-Draft draft-soilandreyes-arcp is under consideration by IETF’s App... | {
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0ac0e4726929e4b868ef6a3ac1ccbd36494afd2f | abstract | 0 | 21 | Abstract | Classical epidemiology has focused on the control of confounding but it is
only recently that epidemiologists have started to focus on the bias produced
by colliders. A collider for a certain pair of variables (e.g., an outcome Y
and an exposure A) is a third variable (C) that is caused by both. In a
directed acyclic g... | {
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} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
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696c7f108a2a14cec3f075ec3a6b6a87075846d1 | subsection | 1 | 21 | Abstract | Classical epidemiology has focused on the control of confounding but it is only recently that epidemiologists have started to focus on the bias produced by colliders. A collider for a certain pair of variables (e.g., an outcome Y and an exposure A) is a third variable (C) that is caused by both. In a directed acyclic g... | {
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Non-Communicable Disease Epidemiological Data: a reproducible illustration
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4248b5232c3799352bb1bbe28bd1ca93ac3a624d | subsection | 2 | 21 | Keywords | epidemiological methods, causality, noncommunicable disease epidemiology. | {
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7e578594a01c69950ce0f77a7e22bc2e6bfc3d92 | subsection | 3 | 21 | Key messages box | [roundcorner=10pt, backgroundcolor=black!10, linecolor=black!5]Paradoxical associations between an outcome and exposure are common in epidemiological studies using observational data.
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f8965c2d873dcce8864d1fcba4467a29bb5b9b8d | subsection | 4 | 21 | Introduction | During the last 30 years, classical epidemiology has focused on the control of confounding . It is only recently that epidemiologists have started to focus on the bias produced by colliders in addition to confounders , . Directed acyclic graphs (DAGs) can help to visualize the assumed structural relationships between t... | {
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824259329600e0c46b4bd0e5d3f50348ff041094 | subsection | 5 | 21 | Review of confounding | Confounding arises from common causes of the exposure (A) and the outcome (Y). Note that in Figure 1A, both the outcome (Y) and the exposure (A) share a common “parent” (direct cause). Y and A are both called “descendants” of W as they are both caused by W. The confounder wholly or partially accounts for the observed a... | {
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fec860fa3fbbbb4baa1d6a4b2066d9d98284e6e2 | subsection | 6 | 21 | Demonstration of confounding and regression adjustment | We now demonstrate adjustment for confounding via linear regression models. In Box 1 we show how to generate data consistent with the DAG from Figure 1A after which we run two different regression models. The confounder W is generated as a standard normal random variable i.e. with mean 0 (\mu =0) and variance 1 (\sigma... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
] | [
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5ca049796074c1d82314cfcbf7f5f7052da30dcc | subsection | 7 | 21 | Collider structure | Unlike in Figure 1A, where the causal arrows start from W, in Figure 1B they now point towards C from A and Y. If we condition on C (e.g. using regression or stratification), we will create collider bias. The common effect C is referred to as a collider on the path A \rightarrow C \leftarrow Y because two arrow heads c... | {
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Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
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b90e0e8ee8a8236d624379a0bfec4cbae8477313 | subsection | 8 | 21 | Collider structure | The true causal coefficient of the exposure A is 0.3, and the coefficients for the association of the collider C with the exposure A and the outcome Y are 1.2 and 0.9, respectively (Box 2).Box 2
[roundcorner=10pt, backgroundcolor=black!10, linecolor=black!5]library(visreg) # load package to visualize regression output... | {
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"doi": "10.1093/aje/155.2.176",
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"raw": "Hernan MA. Causal Knowledge as a Prerequisite for Confounding Evaluation: An Application to Birth Defects Epidemiology. American Journal of Epid... | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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3f8d7c8d4a7bee839c182276f875c2698bc1607a | subsection | 9 | 21 | Data generation | Based on a motivating example in non-communicable disease epidemiology, we generated a dataset with 1,000 observations to contextualize the effect of conditioning on a collider. Nearly 1 in 3 Americans suffer from hypertension and more than half do not have it under control . Increased levels of systolic blood pressure... | {
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Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
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cb3e769ef3bceb282e3c240d03d3078a532f2c1b | subsection | 10 | 21 | Data generation | The true causal effect of sodium intake on SBP is 1.05 (i.e., \text{Systolic blood pressure}\, = \,\beta _{1}\,\times \,\text{sodium}\,+\,\beta _{2}\,\times \,\text{age}\,+\,\varepsilon ; where \beta _1= 1.05, \beta _2 = 2.00 and \varepsilon is a standard normally distributed error). The coefficients for the associatio... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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28ba4272ebb416ef58633fa86bccb2361257abf6 | subsection | 11 | 21 | Data generation | The model specifications are shown here below; in Box 4 we show how to fit and visualize the corresponding models in R.Models specification:\text{Systolic Blood Pressure in mmHg} = \beta _{0}\,+\,\beta _{1}\,\times \,\text{Sodium in gr} \,+\, \varepsilon\text{Systolic Blood Pressure in mmHg} = \beta _{0}\,+\,\beta _{1}... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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a479eeee4bcee4ddab5409dd5da29f16d1e62efc | subsection | 12 | 21 | Data generation | The model specifications are the same as described above but now with a binary outcome (hypertension); in Box 5 we show how to fit and visualize the corresponding models in R using a forest plot function.Box 5
[roundcorner=10pt, backgroundcolor=black!10, linecolor=black!5]## Models fit on multiplicative scalelibrary(dp... | {
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} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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999c9ad95a8990e808c8a1634605fa7c1991890e | subsection | 13 | 21 | Data generation | The odds ratio for the effect of sodium on hypertension similarly suggests that it is protective (i.e., for one unit increase in sodium intake the risk of hypertension decreases by 98\%) (Figure 5).\begin{figure}[H]\begin{center}\includegraphics[scale=0.47]{Figure4.png}\caption{Collider effect for the illustration: Uni... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
] | [
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e8cb8415a7d79b2037eaf54b045e3982ab22da9c | subsection | 14 | 21 | Data generation | \newline\textbf{Box 6}\begin{mdframed}[roundcorner=10pt, backgroundcolor=black!10, linecolor=black!5]\small\begin{verbatim}# Monte Carlo SimulationsR<-1000true <- rep(NA, R)collider <- rep(NA,R)se <- rep(NA,R)set.seed(050472)for(r in 1:R) {if (r# Function to generate datagenerateData <- function(n){Age_years <- rnorm(n... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
] | [
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5c9b54b5c59db549e72ea38dabd15278d284eaae | subsection | 15 | 21 | Data generation | Table 3 shows different values for the true causal effect of sodium intake on SBP and the estimated causal effect for different values of the association between PRO (i.e., the collider) with sodium intake (\alpha _1) and SBP (\alpha _2) in the collider model, and assuming \alpha _1 = \alpha _2 (i.e., the same magnitud... | {
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"doi": "10.1097/01.ede.0000042804.12056.6c",
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"raw": "Greenland S. Quantifying Biases in Causal Models: Classical Confounding vs Collider-Stratification Bias. Quantifying Biases in Caus... | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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5e508c94110428b11e0ffc4e9de56bc2d11e2e3c | subsection | 16 | 21 | Conclusion | We investigated a situation where adding a certain type of variable to a linear regression model, called a “collider”, led to bias with respect to the regression coefficient estimates while still improving the model fit. DAGs are based on subject matter knowledge and are vital for identifying colliders. Determining if ... | {
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"raw": "Pearce N, Richiardi L. Commentary: three worlds collide: Berkson's bias, selection bias and collider bias. International journal of epidemiology. 2014;43(2):521–524.",
"source_ref_id": "c8d1e1... | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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24c3561f17363455c80f4d22138e6f5099063cdc | subsection | 17 | 21 | Competing Interests | The authors declare that they do not have any conflict of interest associated with this research and the content is solely the responsibility of the authors. | {
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} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
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"Mireille E. Schnitzer"
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d2f8eb65d7b0edd690489efb4bfaa3b5fe7ef99a | subsection | 18 | 21 | Funding | Miguel Angel Luque Fernandez is supported by the Spanish National Institute of Health, Carlos III Miguel Servet I Investigator Award (CP17/00206). Maria Jose Sanchez Perez is supported by the Andalusian Department of Health. Research, Development and Innovation Office project grant PI-0152/2017. Anand Vaidya was suppor... | {
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} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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3b8656a722ae70a01150c4e49d09b95ce02460a9 | subsection | 19 | 21 | Authors’ contributions | The article and Shiny application arise from the motivation to disseminate the principles of modern epidemiology among clinicians and applied researchers. MALF developed the concept, designed the study, carried out the simulation, analysed the data, and wrote the article. DRS and MALF developed the shiny application. A... | {
"cite_spans": []
} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
"Daniel Redondo-Sanchez",
"Maria Jose Sanchez Perez",
"Anand Vaidya",
"Mireille E. Schnitzer"
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dfbe8d065dd9f62398deb8172412e990e3669a48 | subsection | 20 | 21 | Tables | Table 1. Coefficients and standard errors of the linear association between Y (outcome) and A (exposure) illustrating confounding and collider effects, n = 1,000
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} | 10.1093/ije/dyy275 | 1809.07111 | Educational Note: Paradoxical Collider Effect in the Analysis of
Non-Communicable Disease Epidemiological Data: a reproducible illustration
and web application | [
"Miguel Angel Luque-Fernandez",
"Michael Schomaker",
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edfa6054c65a84fa18ab192d4cc1c9fc63a64540 | abstract | 0 | 144 | Abstract | In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and... | {
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} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
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1b37c8aece787a4b2ee1b2c47a85ab7f0add9258 | abstract | 1 | 144 | Abstract | It parametrizes rescaling classes of
weak $F$-matroid structures on $M$, and its unit group is coincides with the
inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if
its foundation is the regular partial field, and a non-regular matroid $M$ is
binary if and only if its foundation is the field... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
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3fbe24677e435f5a5e929f3281cd13a37c496ece | subsection | 2 | 144 | Body | The moduli space of matroids
Matthew Baker and Oliver Lorscheid>*!/-5pt/@(The moduli space of matroidsMatthew Baker
mbaker@math.gatech.edu
School of Mathematics, Georgia Institute of Technology, Atlanta, USAOliver Lorscheid
oliver@impa.br
Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, BrazilIn , Nath... | {
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... | |
56e69b4325b714e7006597c7bbd1698e30f54cb3 | subsection | 3 | 144 | Body | We show that the unit group of k_M^w can be canonically identified with the Tutte group of M, originally introduced by Dress and Wenzel.
We also show that the sub-pasture k_M^f of k_M^w generated by “cross-ratios”, which we call the foundation of M, parametrizes rescaling classes of weak F-matroid structures on M, and ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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6a8f8d557ca0fe66e20f27143f7558d87a4d71cb | subsection | 4 | 144 | Pullbacks of matroid bundles | The pullback ^\ast ({\mathcal {M}}) of a matroid bundle {\mathcal {M}} on Y along a morphism :X\rightarrow Y of pasteurized ordered blue schemes is defined by the following lemma.Lemma 1.1
Let :X\rightarrow Y be a morphism in \operatorname{OBSch}^\pm and {\mathcal {L}} an invertible sheaf on Y. Let E be a non-empty fi... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
5a85e22c419f4ee17b3494150e91e54ef40a96da | subsection | 5 | 144 | Compatibility with matroids over ordered blueprints | In the following, we verify that matroid bundles over \operatorname{Spec}B correspond bijectively to B-matroids in a functorial way.Proposition 1.2
Let B be a pasteurized ordered blueprint, E a non-empty finite ordered set, r a natural number and X=\operatorname{Spec}B. Then the map\begin{array}{cccc}
\Phi _B: & \big ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
c71ed8469910845a70b17fb1e9142f1a3ebd89f0 | subsection | 6 | 144 | Compatibility with matroids over ordered blueprints | The inclusion _B:B\rightarrow \Gamma (X,{\mathcal {O}}_X) as constant sections is an isomorphism of ordered blueprints, which implies that any two Grassmann-Plücker functions \Delta ,\Delta ^{\prime }:\binom{E}{r}\rightarrow B are different if _B\circ \Delta and _B\circ \Delta ^{\prime } are different. Moreover, this i... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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de795f3c86a11f176d5a3f619533030ab0e2aabb | subsection | 7 | 144 | Compatibility with matroids over ordered blueprints | By Lemma REF , the pullback ^\ast (\widetilde{M}) is represented by the Grassmann-Plücker function _{{\mathcal {O}}_X}^\#\circ _B\circ \Delta :\binom{E}{r}\rightarrow \Gamma (Y,^\ast ({\mathcal {O}}_Y)) where Y=\operatorname{Spec}C.The result now follows from the commutativity of the diagram{tikz/fig8}
\begin{}[row sep... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.0... | |
f47d1ea7b7d04ff9c9255e369aa8f8d3c4e42972 | subsection | 8 | 144 | Example of a matroid bundle over the projective line over | In this example, we investigate matroid bundles of rank 2 on E=\lbrace 1,2,3,4\rbrace over the projective line {\mathbb {P}}^1_{\mathbb {K}}=\operatorname{Proj}\big ({\mathbb {K}}[T_0,T_1]\big ). We review some general facts that we will use below.Since {\mathbb {K}}^\bullet =\lbrace 0,1\rbrace , the underlying monoid ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-... | |
a8bd46d9034a8020d801572b3466338ff4385967 | subsection | 9 | 144 | Example of a matroid bundle over the projective line over | Since T_0 is contained in the maximal ideal of {\mathbb {K}}[T_0,T_1]_{(T_0)} and T_1 is contained in the maximal ideal of {\mathbb {K}}[T_0,T_1]_{(T_1)}, there is a unique minimal set of global sections that generates {\mathcal {O}}(d): for d=0, this set is \lbrace 1\rbrace and for d>0, this set is \lbrace T_0^d,T_1^d... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
4be18466d645028bd80c84bf442dfb03dc24dd63 | subsection | 10 | 144 | Example of a matroid bundle over the projective line over | Let \Delta :\binom{E}{2}\rightarrow \lbrace 0,T_0,T_1\rbrace be a function. Since \lbrace \Delta _{i,j}\rbrace _{\lbrace i,j\rbrace \in \binom{E}{2}} has to generate {\mathcal {O}}(1) in order for \Delta to be a Grassmann-Plücker function, we must have \Delta _{i,j}=T_0 and \Delta _{k,l}=T_1 for some 2-subsets \lbrace ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
b6227c36cf2ef814c77686272f58480ed04276f0 | subsection | 11 | 144 | The moduli functor of matroids | Let E be a non-empty finite ordered set and r a natural number. We extend the functor \operatorname{{Mat}}(r,E):\operatorname{{OBlpr}}^\pm \rightarrow \operatorname{Sets} to the functor\begin{array}{cccc}
\operatorname{{Mat}}(r,E): & \operatorname{OBSch}^\pm & \longrightarrow & \operatorname{Sets}\\[5pt]
& X & \longmap... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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... | |
532cc2d1cabd3d042fd129ed8b5c7478da090bdb | subsection | 12 | 144 | The moduli space of matroids | We define the matroid space of rank r on E as the ordered blue scheme\textstyle \operatorname{Mat}(r,E) \quad = \quad \operatorname{Proj}\Big ( \, {\mathbb {F}}_1^\pm \big [ \, x_I \, \big | \, I\in \binom{E}{r} \, \big ]\!\sslash \!\operatorname{{Pl}}(r,E) \, \Big ),where \operatorname{{Pl}}(r,E) is generated by the P... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-... | |
420d38540fa7a08e1c131c8e64fff26602ed1765 | subsection | 13 | 144 | The moduli space of matroids | The universal matroid bundle is the class {\mathcal {M}}_\textup {univ} of \Delta _\textup {univ}, which is a matroid bundle of rank r on E over \operatorname{Mat}(r,E).The following theorem shows that the pair (\operatorname{Mat}(r,E),{\mathcal {M}}_\textup {univ}) represents the moduli functor \operatorname{{Mat}}(r,... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.018642988055944443,
... |
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