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8821867646576398bdb290a27a4b6e4f8ff71f80 | subsection | 271 | 289 | Euclidean Conjugation | The "Euclidean Conjugation" acts on \mathbb {C}{P}^1 in an obvious way, \text{⌃}: \left[\pi ^{A^{\prime }}\right] \mapsto \left[\hat{\pi }^{A^{\prime }}\right] . This is now an involution but importantly it has no fix points. Its representation in inhomogeneous coordinates makes it clear that this is just antipodal map... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
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d6c9e30235f868c08f9f190e972b86aa5ff08e15 | subsection | 272 | 289 | Kahler structure | As one dimensional complex projective space, the Riemann sphere has a natural SU(2)-invariant Kähler structure.\hspace*{-35.56593pt}
J&= id\pi ^{A^{\prime }}\partial _{A^{\prime }} -i d\hat{\pi }^{A^{\prime }} \hat{\partial }_{A^{\prime }} = i d\zeta \partial _{\zeta } - i d\overline{\zeta }\partial _{\overline{\zeta }... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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416b4e3a0302316780fed078ebc4c9ca764bf185 | subsection | 273 | 289 | Holomorphic line bundles over | A holomorphic vector bundle bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection operator is holomorphic. Practically, this is equivalent to the fact that transition functions between different trivialisations have to be holomorphic with respect to... | {
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Three-Forms | [
"Yannick Herfray"
] | [
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206828e34d231563929cc06b96e2c5ffe19aec9d | subsection | 274 | 289 | Holomorphic line bundles over | The most condensed definition uses of homogeneous coordinates:\mathcal {O}(n)S^{\prime }\times \left\lbrace \left( \pi ^{A^{\prime }}, \chi \right) \sim \left( \lambda \pi ^{A^{\prime }}, \lambda ^n\chi \right) \right\rbrace .Then the projection clearly is\begin{array}{ll}
\Pi :& \left\lbrace
\begin{array}{lll}
\mathc... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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78a204424953da4e214b4f4a22b2e677c5e38215 | subsection | 275 | 289 | Holomorphic line bundles over | Then the associated section is,s_f: \left\lbrace \begin{array}{lll}
\mathbb {C}{P}^1&\rightarrow &\mathcal {O}(n) \\ \\
\left[\pi ^{A^{\prime }}\right] &\mapsto &
\left[\pi ^{A^{\prime }},f\left( \pi ^{A^{\prime }}\right) \right]=
\left[\lambda \pi ^{A^{\prime }},\lambda ^n f\left( p^{A^{\prime }}\right) \right]=
\left... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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705d2a41189fe899009813ee636a5e864bc4c9bb | subsection | 276 | 289 | Holomorphic line bundles over | On the other hand there are no holomorphic global section of \mathcal {O}(n) for n<0.Some of these holomorphic line bundles are of particular importance and deserve a name:The "tautological bundle", \mathcal {O}(-1), is the natural line bundle over \mathbb {C}{P}^1 such that its total space identifies with S^{\prime }:... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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589f392a58dcb1be84f91429c270439a65bb94b2 | subsection | 277 | 289 | Holomorphic line bundles over | This is better seen in charts:llllllllllll
: {{llc}
\mapsto & \Pi ^{-1}\left( U\right)\\
\left( \zeta , \chi \right) &\mapsto & \left( \zeta , \chi \partial _{\zeta } \right).
,
': {{llc}
\mapsto & \Pi ^{-1}\left( U^{\prime }\right)\subset S^{\prime }\\
\left( \zeta ^{\prime }, \chi ^{\prime }\right) &\mapsto & ... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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"hep-th",
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ea91dd9f8ce601bbee6035210ac5dd06b71d3efc | subsection | 278 | 289 | Hermitian metric on the | The total space of the \mathcal {O}(-1)line bundle \Pi :S^{\prime } \mapsto \mathbb {C}{P}^1 being equipped with an hermitian metric, g=\frac{1}{2} d\pi _{A^{\prime }}\odot d\hat{\pi }^{A^{\prime }} it induces a metric on the fibers. This is done by restricting g to the vertical tangent subspace \mathcal {V}:\mathcal {... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
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"math.MP"
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e1161ab3a0ed149c591257890892260d50546943 | subsection | 279 | 289 | Covariant derivative on | We now introduce the Chern connection associated with the Hermitian metric (REF ):a_{(n)} = -n \;\frac{\overline{\zeta }}{1+\zeta \overline{\zeta }}d\zeta .If \alpha ^{\prime }(\zeta ) is any \mathcal {O}(n,m)-valued k-form on \mathbb {C}{P}^1, then from (REF ), we can define its covariant derivative asd_a \alpha ^{\pr... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
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"math.MP"
] | 2,018 | en | Physics | [
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3a890d2da21e2a0e3e6ade695b8e93ae70a65bfd | subsection | 280 | 289 | Chern connection | The connection (REF ) has the following property: it is compatible with the complex structure, in the sense that a_{(n)}\Big |_{T^{(0,1)}}=0.
It is also compatible with the Hermitian metric (REF ) in the sense thatd_{a} h_{n}= d\left(\left(1+\zeta \overline{\zeta }\right)^{-n} \right) + \left( a_{(n)}+\overline{a}_{(n)... | {
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Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
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290fb14ddb3988e4944f30f713b95d9c294cee6a | subsection | 281 | 289 | The Twistor Space of Complexified Anti-Self-Dual Space-Times | We here review how to construct the twistor space M) of an anti-self-dual complexified space-time (cf , for the original references, , for pedagogical presentations). We especially emphasise how the self-dual connection D= d + A on space-time gives a \mathcal {O}(2)-valued one-form \tau on the associated twistor space.... | {
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Three-Forms | [
"Yannick Herfray"
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79ddee881175a3c2c7100204ca04facef44dbf69 | subsection | 282 | 289 | The Twistor Space of Complexified Anti-Self-Dual Space-Times | We note \Pi \colon \mathbb {PF}(M) \rightarrow \mathbb {PT}(M) the projection operator.We then have the classical double fibration picture:x1) \mathbb {PF}(M);
x2) [left = 2cm of x1] ;
x3) [right = 2cm of x1];y2) [below = 1.5cm of x2] \mathbb {PT}(M);
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"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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"math.MP"
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200fd0fdb9054df6a9d6858083f6d5f39227824d | subsection | 283 | 289 | The Twistor Space of Complexified Anti-Self-Dual Space-Times | A tangent vector to p then corresponds to a certain vector field on \hat{p} that “connects" \hat{p} to an other infinitesimally close integral surface :X(x) = V(x){}^{AA^{\prime }} D_{AA^{\prime }} + \beta (x){}^{B^{\prime }} \frac{\partial }{\partial \pi ^{B^{\prime }}}Being a “connecting vector field", it is defined ... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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86cfb8d1c1a47d70a0e16dde331413f4905baf08 | subsection | 284 | 289 | The Twistor Space of Complexified Anti-Self-Dual Space-Times | Contracting this form with the connecting vector field (REF ) we get a scalar field on \hat{p},\tau (X) = \beta _{A^{\prime }}\pi ^{A^{\prime }}.Now \tau defines a one-form on M) if and only if this scalar field is constant along \hat{p} i.e\pi ^{A^{\prime }}\nabla _{AA^{\prime }} \left( \tau (X) \right) =0.Making use ... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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0044572a0137974253424a6f38a7962862c7e981 | subsection | 285 | 289 | 3d Gravity Conventions | In this section, we review some basic facts about 3D gravity. Our notations are standard for the gravity literature. | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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1065d878162f9d9c4855ade662fcd26bd42b0749 | subsection | 286 | 289 | Einstein-Cartan frame formalism in 3D | Let \left(e^{i}\right)_{i\in \lbrace 1,2,3\rbrace } be orthogonal frame field so that the 3D metric isds^2 =e^i \otimes e^j \eta _{ij},where \eta _{ij}={\rm diag}(1,1,1). We raise and lower indices with the metric \delta _{ij}, and the spin-connection is the set of one-forms w^{ij}=w^{[ij]}. The anti-symmetry is the st... | {
"cite_spans": []
} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
] | 2,018 | en | Physics | [
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3a015a61e38466c8c1d1abc25ba1099907f5126c | subsection | 287 | 289 | Matrix notations | It is very convenient to get rid of the internal i,j,\ldots indices at the expense of making all objects \mathfrak {su}(2)-valued. The Lie algebra generators \left(\sigma ^{i}\right)_{i \in \lbrace 1,2,3\rbrace } are taken such that they satisfy\textrm {Tr}(\sigma _i \sigma _j) = - \frac{1}{2} \delta _{ij}, \qquad [\si... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
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a8f1b4c0f1e8533548c6509074ff1bbbb03a160d | subsection | 288 | 289 | Chern-Simons formulation | The two sets of equations \nabla {{e}}=0, {{f}}={{e}}\wedge {{e}} can be combined as the real and imaginary parts of a single complex-valued equation by introducing \mathfrak {sl}(2,-valued fields{{a}}_{\pm } {{w}}\pm \sqrt{\Lambda } {{e}}.Here and in what follows \sqrt{\Lambda } will stand for i\sqrt{|\Lambda |} when ... | {
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} | 1807.11376 | New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's
Three-Forms | [
"Yannick Herfray"
] | [
"gr-qc",
"hep-th",
"math-ph",
"math.MP"
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bffb50c1e5d3ba29af695cfa91940f6dc530ee76 | abstract | 0 | 20 | Abstract | Both humans and the sensors on an autonomous vehicle have limited sensing
capabilities. When these limitations coincide with scenarios involving
vulnerable road users, it becomes important to account for these limitations in
the motion planner. For the scenario of an occluded pedestrian crosswalk, the
speed of the appr... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
"cs.SY"
] | 2,018 | en | Computer Science | [
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186e31c8f5b9761945ed623c4abc991765637ef5 | subsection | 1 | 20 | Introduction | Autonomous vehicles rely on sensors to provide information about the world to decision-making algorithms. Just like humans have limited sensing capabilities, the sensors on an autonomous vehicle are also susceptible to limitations. GPS requires open skies. Cameras require certain weather and lighting conditions. Radar,... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
"cs.SY"
] | 2,018 | en | Computer Science | [
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67cbb8392524dc9b5956af9c0a3d02748a6c1176 | subsection | 2 | 20 | Related Work | Motion planning under uncertainty is a large topic of study in the robotics community. For an autonomous vehicle, motion planning encompasses both lateral and longitudinal motion which are controlled through a combination of steering and acceleration. A sampled-based motion planning technique known as Rapidly-Exploring... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
"cs.SY"
] | 2,018 | en | Computer Science | [
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126e0b1bf0c498bb34972e6ea9a4bf1a2fd1a46d | subsection | 3 | 20 | Infrastructure | For this work, the simulation environment is two-fold. The first part is a nonlinear vehicle dynamic model that simulates vehicle pose information and how the vehicle maneuvers in space. The other aspect is the simulation of the occupancy grid. | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
"cs.SY"
] | 2,018 | en | Computer Science | [
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203f077a290a7a6beef024e743c48272c0428d41 | subsection | 4 | 20 | Vehicle Motion | A vehicle is controlled by commanding a steer angle (\delta ) and longitudinal acceleration (a_{\text{x}}). The vehicle motion is simulated using a lumped axle vehicle model, where the two front tires are lumped as one front tire and the two rear tires are lumped as one rear tire (also known as a bicycle or single-trac... | {
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} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
"cs.SY"
] | 2,018 | en | Computer Science | [
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6875bf55c7fb2db7e63acfc8a9d5fa2a4003f4ca | subsection | 5 | 20 | Occupancy Grid | The occupancy grid is a discretized top-down view of the world around the vehicle. To emulate the Velodyne HDL-32E lidar, the longitudinal range of the occupancy grid is limited to 70 in front of the vehicle. Because the width of the roadway in the scenario is much less than the range of the lidar, the lateral range of... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
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] | 2,018 | en | Computer Science | [
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e64db6f7e3b2aba5da29f37c12555e2525ccc452 | subsection | 6 | 20 | POMDP | A POMDP makes decisions based on the history of observations o_{1:t}. To reduce the information stored, the history is summarized in a belief state b, which is a distribution over states. The optimal policy is represented as a set of alpha vectors, which convert the belief state to an action. A POMDP model takes a simi... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
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"cs.AI",
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] | 2,018 | en | Computer Science | [
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99e285f89c81c8f5d78c8c47a066f496c86a02ee | subsection | 7 | 20 | State Space | The state space is represented in a low dimensional subspace that captures pose and motion of the vehicle as well as perception information. The components of the state considered in this work are:V: current velocity of the vehicle
D: distance along the path
C: event of pedestrian crossingSpeed and distance along the... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
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"cs.AI",
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f0e2257aef9834c5c6ed32e390992b46613a6599 | subsection | 8 | 20 | Action Space | The vehicle actuation considered here is longitudinal acceleration. Commanded longitudinal acceleration is determined by proportional speed control. Thus, the POMDP action space is a speed scaling factor applied to the desired speed in the longitudinal control. After discretization of the action space, the actions are ... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
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96f51e86623044b186d56323a27b20e9f09f312f | subsection | 9 | 20 | Observation Space | The observation space captures information the agent observes after taking an action. In this work, the observations are provided just from the lidar sensors. Two types of observations are considered:N: number of unobservable tiles
C: detection of pedestrian crossingTo simplify the problem size, the number of unobserv... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
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b80aafde2933be22b1cd8a2e9c9d6f563e635e2e | subsection | 10 | 20 | Reward Model | The reward function in this POMDP formulation is designed with the following objectives in mind:Encourage the vehicle to drive to the end of the path.
If a pedestrian is detected, then the vehicle should yield to the pedestrian. Thus, non-zero scale factors are penalized when a pedestrian crossing event is true.
Addi... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
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0108d6eeccb05ab1428ec2ba098e1857a26dbedf | subsection | 11 | 20 | State-Transition Model | The dynamics of the system are not actually stochastic, but rather uncertainty is introduced from the crude discretization of the state space. Also, the event of a pedestrian crossing is modeled as a random process. The following parameters characterize the state-transition model:Speed scaling and changes in speed are ... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
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a6027392e04c26865ea598b48a3a966a29856faa | subsection | 12 | 20 | Observation Model | In a typical POMDP problem, the observation model is defined as the conditional probability of observing each observation o given the current state s and the action a taken to get there: Pr(o|s,a). For this work, it is assumed the action does not contribute to the observation o given s. Thus, the dependence on a is dro... | {
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} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
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25dad6a5a362494b87c1fd08ba28b9750d67f390 | subsection | 13 | 20 | QMDP | QMDP is an offline method to approximate an optimal POMDP solution, and assumes at the next time step the state will be fully observable. QMDP is well suited for this problem because the actions are not information gathering. The algorithm is akin to value iteration for MDP, except it iterates over the belief state. Th... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
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2b1407c368252adb40c0d762bde900fa8d6b9372 | subsection | 14 | 20 | Policy Execution | The execution of the POMDP policy follows Algorithm REF . The belief state is initialized to a uniform distribution. Using the current belief state, an optimal action is calculated using the set of alpha vectors and is executed in the vehicle longitudinal control. In the next control loop (not represented in Algorithm ... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
"cs.RO",
"cs.AI",
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a8dee112db0ffda533fc3c791fcd5e96570ef006 | subsection | 15 | 20 | Evaluation Framework | To evaluate the POMDP approach for speed control in an uncertain environment, an oracle approach and baseline approach are also implemented. | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
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42cad407c79b95b212b72b0cffde5445ac32595a | subsection | 16 | 20 | Oracle | The oracle approach does not use unobservable information from the occupancy grid. Instead, perfect sensing is assumed and exact pose information about the scenario is used to allow the autonomous vehicle to maneuver around the parked vehicle at the speed limit since it will also know with perfect knowledge whether a p... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
] | [
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dcdebfb225b2ea442dfbe42b2d833ab5b6a3968a | subsection | 17 | 20 | Baseline | The baseline speed scale factor is determined by a simple function inversely proportional to the number of unobservable tiles. The speed scale factor is discretized into 10 bins, where a scale factor of 1.0 means the number of unobservable spaces is negligible and the vehicle can maneuver at the desired speed and a sca... | {
"cite_spans": []
} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
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f00bda3ba6d8317b1acb1eabe306deb54478b4cc | subsection | 18 | 20 | Simulation Results | The examples for comparison are a pedestrian hidden behind the occluded vehicle in the pedestrian crosswalk (Fig. REF ), and a pedestrian in the crosswalk not behind the occluding vehicle (Fig. REF ). Since the pedestrian is in the crosswalk, all vehicles must yield. However, if an approach is unable to account for a p... | {
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} | 1802.06314 | Autonomous Vehicle Speed Control for Safe Navigation of Occluded
Pedestrian Crosswalk | [
"Sarah Thornton"
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54adffcac9e024849168f30b1ac252a5cf229869 | subsection | 19 | 20 | Conclusions | The preliminary results presented above demonstrate how well the baseline and oracle approaches work. Next steps are to keep improving the POMDP problem formulation. In particular, the observation model needs to be improved. Running many simulations with the pedestrian randomly placed throughout the crosswalk should al... | {
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Pedestrian Crosswalk | [
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7cb32234e1ed54000cbc6fcdc61e2398ee1d68a6 | abstract | 0 | 13 | Abstract | Observational tests of stellar and Galactic chemical evolution call for the
joint knowledge of a star's physical parameters, detailed element abundances,
and precise age. For cool main-sequence (MS) stars the abundances of many
elements can be measured from spectroscopy, but ages are very hard to
determine. The situati... | {
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} | 10.3847/1538-4357/aaee74 | 1802.06663 | Precise Ages of Field Stars from White Dwarf Companions | [
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192e016f4473cc32d393eec7bfd1775f9e55342f | subsection | 1 | 13 | Introduction | The two members of a binary star systems are stars born at nearly the same time
from the material of the same element composition, but usually with different
masses. Binary stars are not only interesting in themselves but offer a wide
range of avenues to measure stellar properties and learn about stellar physics.
These... | {
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... | 10.3847/1538-4357/aaee74 | 1802.06663 | Precise Ages of Field Stars from White Dwarf Companions | [
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3d5f99ca6b43ba9297f7853c10655a93a4ed1ecd | subsection | 2 | 13 | Introduction | In this regime, WD-MS wide binaries may be the best way forward to reach \sim 10\% age precision.This paper is organized as follows: in Section we describe the identification of likely WD-MS binary systems that have TGAS information on the MS component; in Section we then exploit the resulting precise luminosity info... | {
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8f6cc5bb0b89dd4ba118dfd86f6a78d451a41da9 | subsection | 3 | 13 | Identification of Candidate WD - MS Wide Binaries | We aim to identify WD-MS wide binary candidates without using the actual luminosity (or apparent magnitude) or detailed color of the possible WD component, as these quantities should subsequently serve as constraints on the WD's age. We cannot also rely on only spectroscopically confirmed WDs, as this would severely li... | {
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3bddcbfedfe67dc6a9278e71b7f9cf253809db4e | subsection | 4 | 13 | Identification of Candidate WD - MS Wide Binaries | The specifics are detailed in Appendix .This above selection left us with a wide binary sample of about 150 objects, where we expect
the companions to the TGAS MS stars to be either fainter MS stars or WDs.
Adopting the parallax-distance to the primary MS, we can construct a color –
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913b17594fdafc01b58f9678c07446984c69c48a | subsection | 5 | 13 | Age Constraints on the Wide Binary Systems | We are now left with a set of 91 candidate WDs (cWD), whose distances are precisely constrained by the parallaxes to their
companions. Of those, 15 are brighter (Figure REF ,
red circles) than the predictions from the 0.5\,{\rm M}_\odot
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6d20cbb601a3aff404db1ca49bb0af4785933c38 | subsection | 6 | 13 | Age Constraints on the Wide Binary Systems | Without spectroscopy, we do not know which objects are
H-atmosphere (DA) WDs and which are DBs. Fortunately for our analysis,
nature makes predominantly DA WDs (\sim 75%; ), and it is therefore a good
initial assumption that those cWDs that have posterior distance
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067d1ce4e5f7781bab99dc5618a91c1eb8c62c2b | subsection | 7 | 13 | Age Constraints on the Wide Binary Systems | For WDs with {T}_{\rm eff}\le 5000 K, issues arise both
in our present understanding of their atmospheres and possibly with additional
sources of energy release during crystallization .
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477254ce47479a1c7d519248167071b37b05e9e0 | subsection | 8 | 13 | Discussion and Outlook | In this paper we carried out a pilot study for one of the many applications of
using Gaia data to constrain stellar properties. We identified systems where
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2bc9cc8fed85b157f2544c7dbfd81cc9740f75a3 | subsection | 9 | 13 | GPS1 | In this section, we detail the selection query we performed on TGAS and GPS1 catalogs.Matching GPS1 against TGAS will report all the stars from GPS1 within some radius that could potentially be associated with a TGAS bright star.
If we also filter on parallax and motion similarity this will only give co-moving pairs.
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c22e11767443c8f25955518d917b6c6fc2e14ea3 | subsection | 10 | 13 | GPS1 | Finally, we also added color terms that avoid having main-sequence objects and we also select good photometry for their SED analysis. Based on empirical definitions we added the following selections:\begin{split}
& \left|(g - i) -1.6 \times (g - r) + 0.1 \right| < 0.15\, \mathrm {mag},\\
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2ce3f20bc8fa343269534ebc00739b76b6019e40 | subsection | 11 | 13 | GPS1 | As GAVO is currently the only service providing the GPS1 catalog, the field names correspond to their definition, and may vary when using other sources (e.g., VizieR, Gaia Archive).SELECTdb.obj_id, db.ra, db.dec, db.e_ra, db.e_dec, db.pmra, db.e_pmra,db.pmde, db.e_pmde, db.magg, db.e_magg, db.magr, db.e_magr,db.magi,... | {
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9ee80fbbdff16fe28826059f7f8a52bbe4b41c78 | subsection | 12 | 13 | Catalogs | In this section we describe the content of the catalog generated during this study.The catalog contains the photometric and astrometric data for all of the WD candidates of this study. For each star, we also provide the mean, median and standard deviation of the posterior PDF of the WD properties, esp. age and ZAMS mas... | {
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0ecceae7904b9e3832efdc3eb9374df9861de6ab | abstract | 0 | 13 | Abstract | Understanding the mechanisms of neural communication in large-scale brain
networks remains a major goal in neuroscience. We investigated whether
navigation is a parsimonious routing model for connectomics. Navigating a
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149247add2faa1ec3267695ed5e73c409f6a0261 | subsection | 1 | 13 | Overview | This supporting document provides details on the acquisition and preprocessing of the analyzed connectivity data. We also provide descriptions of the network measures and methods applied throughout this work. Finally, we include supplementary and replication analyses that indicate the robustness and universality of the... | {
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a6cde848730f7b256a266795b25d79fae7165868 | subsection | 2 | 13 | Mouse. | The Allen Institute for Brain Science mapped the mesoscale topology of the mouse nervous system by means of anterograde axonal injections of a viral tracer . Using two-photon tomography, they identified axonal projections from the 469 injections sites to 295 target regions. Building on these efforts, Rubinov and collea... | {
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18ef9ec0ea46b6f4bdd6fdcfcb4702220f3ddddb | subsection | 3 | 13 | Macaque. | Markov and colleagues applied 1615 retrograde tracer injections to 29 of the 91 areas of the macaque cerebral cortex, spanning occipital, temporal, parietal, frontal, prefrontal and limbic regions , . This resulted in a 29 \times 29 directed interregional sub-network of the macaque coritco-cortical connections. Connect... | {
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4ff99061efba8bf9d4e13373782a33306145ba88 | subsection | 7 | 13 | Connectome thresholding. | Density-based thresholding of connection weights was applied to the group-averaged connectome in order to filter out potentially noisy and spurious connections . A density threshold of 15% consists of keeping the top 15% highest weighted connections and deleting the remaining ones. Since the true connection density of ... | {
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"Caio Seguin",
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07f9726d55bc19086d4f9da098b21627ca84675d | subsection | 8 | 13 | Null network models. | Four null network models were used in this work: (M1) topologically randomized (rewired) networks, (M2) spatially randomized (repositioned) networks, (M3) progressively randomized networks and (M4) progressively clusterized networks. Null models (M1) and (M2) were employed in the sections Navigability of the human conn... | {
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"Caio Seguin",
"Martijn P. van den Heuvel",
"Andrew Zalesky"
] | [
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cd87e0c51f2643c66a2f6e47279bcaf4079d8e4f | subsection | 9 | 13 | Navigation centrality. | Binary and weighted betweenness centralities were computed using the Brain Connectivity Toolbox, representing the absolute number of shortest paths that pass through each node of the network. Similarly, the navigation centrality of node i defined as c(i) = \sum _{s \ne i \ne t} \pi _{st}(i), where \pi _{st}=1 if the su... | {
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} | 10.1073/pnas.1801351115 | 1801.07938 | Navigation of brain networks | [
"Caio Seguin",
"Martijn P. van den Heuvel",
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] | [
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3dbf8c035a66b5107bef97ead2cb7f7179791f09 | subsection | 10 | 13 | Navigation and functional connectivity. | Our implementation of navigation identifies paths based on the Euclidean distance between network nodes. Thus, a natural definition of navigation path length is to consider the physical distance traversed from one node to another. For our FC analysis, we defined navigation path length from i to j as \Lambda ^{nav}_{ij}... | {
"cite_spans": []
} | 10.1073/pnas.1801351115 | 1801.07938 | Navigation of brain networks | [
"Caio Seguin",
"Martijn P. van den Heuvel",
"Andrew Zalesky"
] | [
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] | 2,018 | en | Quantitative Biology | [
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8572da1b444b256399028cfc49f1f60353bdc163 | subsection | 11 | 13 | Distance scaling of connection weights. | A common practice in connectomics studies based on white matter tractography is to scale streamline counts by average fiber lengths . This is motivated by the notion that certain tractography algorithms are biased towards overestimating the streamline count of long fiber bundles. We chose not to perform distance-based ... | {
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6384cca74e1a93c0ee926a0718844900c53a7157 | subsection | 12 | 13 | Distance scaling of connection weights. | (A) Whole brain correlations between FC and Euclidean distance (blue), weighted shortest paths (orange), binary shortest paths (yellow) and navigation distance (green). (B) Scatter plot between navigation distances and FC for all brain regions. Green line indicate the linear fit between the two measures. (C-D) Right he... | {
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42f183df939a3018a657de2b0922b949b7a98128 | abstract | 0 | 59 | Abstract | We investigate the existence of weak expanding solutions of the harmonic map
flow for maps with values into a smooth closed Riemannian manifold. We prove
the existence of such solutions in case the target manifold is isometrically
embedded as a hypersurface of some Euclidean space and the initial condition is
a Lipschi... | {
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} | 1801.08012 | Existence of expanders of the harmonic map flow | [
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42ba5f0b6b8d15fa864bae3b634117c04ff8bfbe | subsection | 1 | 59 | Introduction | In this paper, we consider the Cauchy problem for the harmonic map flow of maps (u(t))_{t\ge 0} from \mathop {\rm \mathbb {R}}\nolimits ^n, n\ge 3 to a closed smooth Riemannian manifold (N,g) isometrically embedded in some Euclidean space \mathop {\rm \mathbb {R}}\nolimits ^m, m\ge 2. More precisely, we study the parab... | {
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"Alix Deruelle",
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c0ab49361102e298689beabc8d9fc4b1f56f030b | subsection | 2 | 59 | Introduction | Indeed, if u is an expanding solution in the previous sense then the map U(x):=u(x,1) for x\in \mathbb {R}^n, satisfies the elliptic system\left\lbrace \begin{aligned}&\Delta _f U+A(U)(\nabla U,\nabla U)=0,\quad \mbox{on $\mathbb {R}^n$},&\\
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\end{aligned}
\right.where, ... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
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9e3e1b120b1197b32463b36392162eb2d777dd1a | subsection | 3 | 59 | Introduction | Our main result is the followingTheorem 1.1
Let n\ge 3 and m\ge 2 be two integers and let u_0:\mathbb {R}^n\rightarrow (N,g)\subset \mathop {\rm \mathbb {R}}\nolimits ^m be a Lipschitz 0-homogeneous map which is homotopic to a constant.Then there exists a weak expander u(\cdot ,1)=:U(\cdot ) of the harmonic map flow c... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
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e132f75e8980f790b9ef7def661e35a774b345a3 | subsection | 4 | 59 | Introduction | Since the initial condition u_0 is allowed to have large local-in-space energy, it is likely that uniqueness will fail. In particular, the authors do not know if the solution produced by Theorem REF coming out of a 0-homogeneous harmonic map will stay harmonic.Now a few words about the proof of Theorem REF . A direct p... | {
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... | 1801.08012 | Existence of expanders of the harmonic map flow | [
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ee97c659476e4666de4cfb5d2bb3ca64b07a836c | subsection | 5 | 59 | Introduction | Then for any K>0, there exists a smooth Homogeneous Ginzburg-Landau expander u_K coming out of u_0:\left\lbrace \begin{aligned}&\Delta _fU_K+K(1-|U_K|^2)U_K=0,&\\
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\end{aligned}
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"Alix Deruelle",
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8b7598b43647a74a587c39d41b44220bdf4fccbf | subsection | 6 | 59 | Introduction | In this case, the homogeneity is of degree -1. To prove Theorem REF , we proceed similarly to their work by using the Leray-Schauder degree theory. For this, one needs a path of initial conditions (u_0^{\sigma })_{0\le \sigma \le 1)}:\mathbb {S}^{n-1}\rightarrow N connecting the restriction u_0^0 of u_0 to \mathbb {S}^... | {
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"Alix Deruelle",
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aae1e45a35bf8e389ebb39e920cbc185bc6f532b | subsection | 7 | 59 | Deformation theory of expanders (with small energy) | From now on, let u:\mathbb {R}^n\rightarrow N be an expanding solution of the harmonic map flow, fixed once and for all. We consider the linearisation of equation (REF ) around u. | {
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} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
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4cd16dd7e7c13dcb578804b9f62a964a97c1a5e2 | subsection | 8 | 59 | Deformation theory of expanders (with small energy) | It takes the following time-dependent form: if u+h is an expander then,\partial _t(u+h)&=&\Delta (u+h)+A(u+h)(\nabla (u+h),\nabla (u+h))|,is equivalent to\partial _th&=&\Delta h+ A(u+h)(\nabla (u+h),\nabla (u+h))-A(u)(\nabla u,\nabla u)\\
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"Alix Deruelle",
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40f859987f90813f2c37602917ec42338eaae9f2 | subsection | 9 | 59 | Deformation theory of expanders (with small energy) | The number \sbox
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}\rule [0.5/2]{}{0.5pt}h\Vert ^p_{L^p(P(x,r))} is the normalized p-norm of h on the parabolic neighborhood P(x,r).Finally, as in , we introduce a somewhat intermediate function space Y:Y:=\left\lbrace R:\mathbb {R}^n\rightarrow \mathbb {R}^m\quad |\quad \sup _{t\in \ma... | {
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5526f918a8613d25dc2e55218822eaa218b6f98d | subsection | 10 | 59 | Deformation theory of expanders (with small energy) | Besides, uniqueness holds in a neighbourhood of u for the topology of X.The proof follows closely the one in the paper which in turn is motivated by the paper .First of all, let us fix some map v_0\in BMO(\mathbb {R}^n,N) and define the map T: X\rightarrow X as follows:T(h):=K_t\ast (v_0-u_0)+\int _0^tK_{t-s}\ast R(u,\... | {
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9f66803a2c92fb3ce0cbe34dab1ad5b6864066c3 | subsection | 11 | 59 | Existence of smooth expanders with arbitrary energy | Let (N,g) be a closed Riemannian manifold which is isometrically embedded in some Euclidean space \mathop {\rm \mathbb {R}}\nolimits ^m. Then there exists a tubular neighbourhood T_{2\delta _N}(N) of N such that the projection map \Pi _N:T_{2\delta _N}(N)\rightarrow N is well-defined and smooth.
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5120898030eb343d030a4f6a6054ba8c4f08f106 | subsection | 12 | 59 | Existence of smooth expanders with arbitrary energy | Then for any K>0, there exists a regular Chen-Struwe expanding solution u_K coming out of u_0\left\lbrace \begin{aligned}&-\Delta _fU_K+K\chi ^{\prime }\left( d^2_{N}(U_K)\right)\nabla \left( \frac{d^2_{N}}{2}\right)(U_K)=0,&\\
&\lim _{t\rightarrow 0^+}u_K(\cdot ,t)= u_0\quad \mbox{in the weak sense. }
\end{aligned}
\r... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00222-013-0468-x",
"end": 1508,
"openalex_id": "https://openalex.org/W1966573859",
"raw": "Hao Jia and Vladimír Šverák. Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-s... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.02... | |
e6c00d114cd777f40f70b10d801a9863c406a28e | subsection | 13 | 59 | Existence of smooth expanders with arbitrary energy | Notice that the time-dependent map V_K(\cdot ,t):=V_K(\cdot /\sqrt{t}) will satisfy the bounds(\sqrt{t}+|x|)^{2}|V_K(x,t)|+(\sqrt{t}+|x|)^{3}|\nabla V_K(x,t)|\le Ct,\quad \forall x\in \mathop {\rm \mathbb {R}}\nolimits ^n,\quad \forall t>0,\quad k\in \lbrace 0,1\rbrace .Denote by U_0(t) the caloric extension of the map... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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-0... | |
3761b14bcfcd0d73999600d36ed9131ab6bfebdb | subsection | 14 | 59 | Existence of smooth expanders with arbitrary energy | Then the caloric extension U_0 of u_0 satisfies&&\Vert U_0\Vert _{L^{\infty }}\le \Vert u_0\Vert _{L^{\infty }}, \\
&&\sup _{x\in \mathop {\rm \mathbb {R}}\nolimits ^n}(1+|x|)|\nabla ^kU_0|(x)\le C(k,u_0),\quad \forall k\ge 1,\\
&&(1+|x|)d_N(U_0(x))\le (1+|x|)|U_0(x)-u_0(x/|x|)|\le C(u_0).Moreover, if u_0 is in C^3_{lo... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
b0b16c5d80f5d169972a7178d71a12b67ca50bfe | subsection | 15 | 59 | Existence of smooth expanders with arbitrary energy | In particular, this implies thatU_0(x)=u_0(x/|x|)+\textit {O}((1+|x|)^{-1}),as x tends to +\infty .Therefore, d_N(U_0(x))\le |U_0(x)-u_0(x/|x|)|=\textit {O}((1+|x|)^{-1}).Now, if u_0 is in C^3_{loc}(\mathop {\rm \mathbb {R}}\nolimits ^n\setminus \lbrace 0\rbrace ), the decay on the first and the second derivatives of U... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.... | |
e088e0fa0e3f7bb0385910f163c8052b027d7e3a | subsection | 16 | 59 | Existence of smooth expanders with arbitrary energy | If n=3, we consider a barycentric approximation of u_0 as follows: if \eta :\mathop {\rm \mathbb {R}}\nolimits ^n\rightarrow [0,1] denotes any smooth function such that \eta \equiv 1 if |x|\ge 2 and \eta \equiv 0 if |x|\le 1, we defineU_0^b:=(1-\eta )P+\eta u_0,where P\in N is fixed.The properties of U^b_0 can be summa... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.002518... | |
380bf8cf70e97baddda25352be1dca5cda37a200 | subsection | 17 | 59 | Existence of smooth expanders with arbitrary energy | Note that this path is chosen inside the homotopy class of [u_0]\in \pi _{n-1}(N).Thus, solving (REF ) amounts to solving the static Chen-Struwe equation&&\Delta _fV_K-K\chi ^{\prime }\left( d^2_{N}(U_0^{\sigma }+V_K)\right)\nabla \left( \frac{d^2_N}{2}\right)(U_0^{\sigma }+V_K)=0,\quad V_K\in X.If V\in X, K>0 and \sig... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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df5d7eb12772aa63bb9ceb280e20344e91277dea | subsection | 18 | 59 | Body | In this section, we prove that the map F_K^{\sigma }:X\rightarrow X is compact and continuous.Note that the map F_K^{\sigma } can also be formally interpreted asF_K^{\sigma }(V)(x,t)=-K\int _0^t\int _{\mathop {\rm \mathbb {R}}\nolimits ^n}\mathcal {K}^{\sigma }_{t-s}(x,y)\frac{1}{s}\chi ^{\prime }(d^2_{N}(U_0^{\sigma }... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
0.0038206973113119602,
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... | |
110a0858885735507b019b9d0e9cf5a7e33a49e7 | subsection | 19 | 59 | Body | To do so, given V\in X, we first solve the following Dirichlet problem&&\Delta _fW_R-K\chi ^{\prime }(d^2_{N}(U_0^{\sigma }))d_{U_0^{\sigma }}d_N(W_R)\nabla d_N(U_0^{\sigma })=Q(U_0^{\sigma },V),\quad \text{ on } B(0,R)\subset \mathop {\rm \mathbb {R}}\nolimits ^n,\\
&&W_R=0,\quad \text{ on } \partial B(0,R),\\
&&\text... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.... | |
2b8dde87eaa1f007adc480dd473bde1e6e6bf3a5 | subsection | 20 | 59 | Body | Therefore,Q(U_0^{\sigma },V)=d_N(U_0^{\sigma })\nabla d_N(U_0^{\sigma })+d_N(U_0^{\sigma })\textit {O}(V)+V\ast V,where, if A and B are two tensors, A\ast B denotes any contraction of linear combinations of the tensor product A\otimes B. By using Lemma REF together with (REF ), one gets:\sup _{x\in \mathop {\rm \mathbb... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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... | |
944a8ab867966f98f9d79f9810fb02eca7d82642 | subsection | 21 | 59 | Body | Now observe that the function f^{-1} is a good barrier function since\Delta _ff^{-1}&=&-f^{-2}\Delta _ff+2|\nabla f|^2f^{-3}\\
&=&-f^{-1}\left(1-2f^{-2}\frac{|x|^2}{4}\right)\\
&\le &-Cf^{-1},for some positive constant C. Indeed, one can check that for n\ge 2 we have\inf _{x\in \mathop {\rm \mathbb {R}}\nolimits ^n} \l... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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-... | |
7edcc26ec6e07fddc6a6ac5de2a608600689ecfa | subsection | 22 | 59 | Body | This is the content of the next proposition.Proposition 3.8
The solution F_K^{\sigma }(V) satisfies the weighted a priori C^1 bound\sup _{\mathop {\rm \mathbb {R}}\nolimits ^n}f^{3/2}|\nabla F_K^{\sigma }(V)|\le C(n,m,K)\Vert Q(U_0^{\sigma },V)\Vert _{X}.Strictly speaking, the solution F_K^{\sigma }(V)=:F(V) is in C^{... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.008089627139270306,
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... | |
2adb523fcdda76ee38119e47d9b37b2eb5d1e571 | subsection | 23 | 59 | Body | Moreover, by interior parabolic Schauder estimates applied to the corresponding time-dependent solution F(V)(x,t):=F(V)(x/\sqrt{t}) defined on \mathop {\rm \mathbb {R}}\nolimits ^n\times \mathop {\rm \mathbb {R}}\nolimits _+, one gets the bounds\sup _{x\in \mathop {\rm \mathbb {R}}\nolimits ^n}f(x)\Vert F(V)\Vert _{C^{... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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0.0... | |
9331f3d883f813a1b972957b35b11811e9a84024 | subsection | 24 | 59 | Body | Now, as we know that the function f^{2}|\nabla F(V)|^2 is bounded, the function f^{2}|\nabla F(V)|^2-k^{-1}\ln f-Af^{-1} goes to -\infty as x goes to +\infty . Therefore, it attains its maximum. The maximum principle applied to the previous differential inequality impliesf^{2}|\nabla F(V)|^2-k^{-1}\ln f-Af^{-1}\le 2Ck^... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.06652097404003143,
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0.... | |
9d4029684c91c73bc79d99e2e119b8a65c09adf2 | subsection | 25 | 59 | Body | In particular, by Taylor's theorem,\sup _{\mathop {\rm \mathbb {R}}\nolimits ^n}f^{2+i/2}|\nabla ^i(Q(U_0,V)-Q(U_0,0))|\le C(K,n,m,\Vert V\Vert _{X}),\quad i=0,1.As in the proof of Proposition REF , one can use a barrier function of the form f^{-2} in order to prove that G(V) decays like f^{-2} uniformly with respect t... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.022153440862894058,
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0.... | |
fb21c54fd964e6556640a033c98e06e070e581d1 | subsection | 26 | 59 | Body | Moreover, this last fact also yields the desired estimate if the maximum is attained on the boundary.By interior parabolic Schauder estimates, one has the following corollary.Corollary 3.11
For any k\ge 0, there is a positive constant M(K,k) uniform in \sigma \in [0,1] and K>0 such that if V\in X is a fixed point of t... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03266105800867081,
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0.... | |
2a1d32bbfb139587b1707d64de6308e01f336553 | subsection | 27 | 59 | Well-posedness of the homogeneous Chen-Struwe equation for small initial data | In this subsection, we assume that \sigma is close to 1 and hence U_0^\sigma is close to a constant map in the sense that\Vert U_0^{\sigma }-c\Vert _{L^\infty }+\sup _{x\in \mathop {\rm \mathbb {R}}\nolimits ^n}(1+|x|)|\nabla U_0^{\sigma }(x)|\le C(u_0^\sigma ),where c\in \mathbb {S}^{m-1} and where \lim _{\sigma \righ... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
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2748f8b1c6fd62772326e6cc84b308fbf5213945 | subsection | 28 | 59 | A Bochner formula | It is a straightforward adaptation from to get the following crucial Bochner formula:Proposition 3.13 (Bochner formula)
Let u:\mathop {\rm \mathbb {R}}\nolimits ^n\times (0,T)\rightarrow \mathop {\rm \mathbb {R}}\nolimits ^m be a smooth solution to the Homogeneous Chen-Struwe flow:\partial _tu-\Delta u+\frac{K}{t}\chi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01161997",
"end": 400,
"openalex_id": "https://openalex.org/W2034075141",
"raw": "Yun Mei Chen and Michael Struwe. Existence and partial regularity results for the heat flow for harmonic maps. Math. Z., 201(1):83–103, 1989.",
... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.00583490589633584,
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-0... | |
bd3085df602f595d735b0b3008485ee465ef1ca8 | subsection | 29 | 59 | A Bochner formula | If d_{N}(u)\le 2\cdot \delta _N, by using the fact that \chi ^{\prime } is nonnegative we obtain(\partial _t-\Delta )e_K(u)+|\nabla ^2u|^2+\frac{K^2}{t^2}\chi ^{\prime 2}d^2_{N}(u)&\le &\frac{1}{2t^2}K^2d^2_{N}(u)+c|\nabla u|^4,for some uniform positive constant c independent of K>0.Therefore, in all cases, this gives ... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.04833829402923584,
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... | |
42a08b4684667bd7e1e4eecce40c43fe6b293a7f | subsection | 30 | 59 | An energy inequality and a monotonicity formula | We define the L^2_{loc} norm at scale R>0 of a map u:\mathop {\rm \mathbb {R}}\nolimits ^n\rightarrow \mathop {\rm \mathbb {R}}\nolimits ^m in H^1_{loc}(\mathop {\rm \mathbb {R}}\nolimits ^n,\mathop {\rm \mathbb {R}}\nolimits ^m) as follows:\Vert \nabla u\Vert ^2_{L^2_{loc,R}}:=\sup _{x_0\in \mathbb {R}^n}_{B(x_0,R)}|\... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03798634186387062,
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... | |
1b47b68b24ec296e30b8348f4aa7299409f83b4c | subsection | 31 | 59 | An energy inequality and a monotonicity formula | Then,E_{K,x_0}(u(t))&\le & \left(1+C\left(n,m,\Vert \nabla u_0\Vert _{L^2_{loc}},t\right)\right)\Vert \nabla u_0\Vert ^2_{L^2(B(x_0,1))},\quad \forall x_0\in \mathop {\rm \mathbb {R}}\nolimits ^n,\\
E_{K,loc}(u(t))&\le & \left(1+c_n \left(e^{c_nt}-1\right)\right)\Vert \nabla u_0\Vert _{L^2_{loc}}^2, \quad t>0,where \li... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.005733677186071873,
0.0516984649002552,
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0.005874825641512871,
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0.01913514733314514,
-0.010254241526126862,
-0.004306933842599392,
-0... | |
ffe34aefe91d3bcef9a3ab3dcbb1a6ba62176cfe | subsection | 32 | 59 | An energy inequality and a monotonicity formula | We multiply the Homogeneous Chen-Struwe flow equation by \phi _{x_0}^2\partial _tu where \phi _{x_0}:\mathbb {R}^n\rightarrow \mathbb {R}_+ is a smooth function with compact support in B(x_0,2) which equals 1 on B(x_0,1) and whose gradient is less than c, and then we integrate by parts to get\int _{\mathbb {R}^n}|\part... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.028212817385792732,
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-0.03500281646847725,
0.0018071690574288368,
... | |
5f06a13494c4025e3bb97aabd9ebc43a6e50e01d | subsection | 33 | 59 | An energy inequality and a monotonicity formula | Therefore, by remark (REF ), one gets in particular:_{B(x_0,1)}|\nabla u(t)|^2dx\le \Vert \nabla u_0\Vert ^2_{L^2_{loc,1}}+c_n\int _0^t\Vert \nabla u(s)\Vert ^2_{L^2_{loc,1}}dt,which implies:\Vert \nabla u(t)\Vert ^2_{L^2_{loc}}\le \Vert \nabla u_0\Vert ^2_{L^2_{loc}}+c_n\int _0^t\Vert \nabla u(s)\Vert ^2_{L^2_{loc}}ds... | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.05067332461476326,
0.052871204912662506,
-0.020269328728318214,
-0.022619839757680893,
0.010653608478605747,
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0.02524508349597454,
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-0.007566656917333603,
0.00819625798612833,
0... | |
790ce1b8779e0276a7b1863ef8f5ca9c555e47ca | subsection | 34 | 59 | An energy inequality and a monotonicity formula | We get\Vert (\nabla u(t))\chi _R\Vert _{L^2_{loc}}^2&\le & \Vert (\nabla u_0)\chi _R\Vert _{L^2_{loc}}^2+c_n\int _0^t\Vert (\nabla u(s))\chi _R\Vert _{L^2_{loc}}^2ds+\frac{c_n}{R^2}\int _0^t\Vert \nabla u(s)\Vert _{L^2_{loc}}^2ds\\
&\le &\Vert (\nabla u_0)\chi _R\Vert _{L^2_{loc}}^2+c_n\int _0^t\Vert (\nabla u(s))\chi ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01161997",
"end": 674,
"openalex_id": "https://openalex.org/W2034075141",
"raw": "Yun Mei Chen and Michael Struwe. Existence and partial regularity results for the heat flow for harmonic maps. Math. Z., 201(1):83–103, 1989.",
... | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
-0.03509979322552681,
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-0.008118734695017338,
-0.019961100071668625,
-0.004074627999216318,
... | |
fd25dba86eca09901bf44ad509e3942ca9b3dfa0 | subsection | 35 | 59 | An energy inequality and a monotonicity formula | We start with a Pohozaev identity like:Proposition 3.16 (Pohozaev identity)
Let u:\mathop {\rm \mathbb {R}}\nolimits ^n\times (0,T)\rightarrow \mathop {\rm \mathbb {R}}\nolimits ^m be a smooth solution to the Homogeneous Chen-Struwe flow (with parameter K>0). | {
"cite_spans": []
} | 1801.08012 | Existence of expanders of the harmonic map flow | [
"Alix Deruelle",
"Tobias Lamm"
] | [
"math.DG"
] | 2,018 | en | Mathematics | [
0.02166757918894291,
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0.033722080290317535,
-0.00655749486759305,
-0.003246322274208069,
-0.0... |
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