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8df66987fe63ea280e6d4d6cdfaa5928dfbe8732 | subsection | 74 | 106 | Quality of Learned Policies and Meta-Algorithm Analysis. | After quantifying the estimation error of the value function returned by , it remains to translate that into a bound on the suboptimality of the returned policy:Assume we are in event . Then the policy \hat{\pi } = \hat{\pi }_{1:H}
returned by in Algorithm REF satisfiesV^{\hat{\pi }} \ge V^\star - p_{ul}^{\hat{\pi }} -... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
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faf39b4b74feaeb1d0393487cb9153af6fea1077 | subsection | 75 | 106 | Quality of Learned Policies and Meta-Algorithm Analysis. | Proposition REF states that for every learned state s \in ^{\textrm {learned}}_hV^\star (s) - Q^\star (s, \hat{\pi }_h)
\le 2 M\tau _V + 2 (H-h)(2M\tau _V + \sqrt{4M^2\tau _V + 2T_{\max }\tau _L} + 8\tau ).Using Lemma REF , we can show that \hat{\pi } yields expected return that is optimal up toV^\star - V^{\hat{\pi }}... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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08d56365e05ae6521268d0ce948e1c11505f4ed1 | subsection | 76 | 106 | Quality of Learned Policies and Meta-Algorithm Analysis. | More specifically, we bound the guaranteed gap as& 2 HM\tau _V + H^2(2M\tau _V + \sqrt{4M^2\tau _V + 2T_{\max }\tau _L} + 8\tau )
\\ \le & 2 MH\tau _V + 2MH^2\tau _V + 2MH^2 \sqrt{\tau _V} + H^2\sqrt{2T_{\max }\tau _L} + 8H^2\tau \\ \le & 6 MH^2\sqrt{\tau _V} + H^2\sqrt{2T_{\max }\tau _L} + 8H^2\tauand then set \tau , ... | {
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} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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f8b507937e5885e4ff588f2572da4ab1a220cb7f | subsection | 77 | 106 | Proof of Theorem | We now have all parts to complete the proof of
Theorem REF .For the calculation, we instantiate all the parameters asn_{\textrm {exp}}= & \frac{8}{\epsilon }\ln \left(\frac{4MH}{\delta }\right),
\quad n_{\textrm {eval}}= \frac{32}{\epsilon ^2}\ln \left( \frac{8MH}{\delta }\right),\quad n_{\textrm {train}}= 16K \left(\f... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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9da9ca0d56bb30c1187969049d8f73f5e56dfb46 | subsection | 78 | 106 | Proof of Theorem | For the
sample complexity, since T_{\max } is an upper bound on the number of calls to and at most M states are learned per level h \in [H], we collect a total of at most the following number of episodes:& (1+M)T_{\max }n_{\textrm {test}}+ M^2Hn_{\textrm {train}}\\
& = \tilde{O}\left(\frac{T_{\max }MH^2}{\epsilon ^2}\l... | {
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"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
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2e1aff767ddfd80291be0c9947c364ad6301dcf7 | subsection | 79 | 106 | Proof of Theorem | For the ease of presentation, we show the statement for = (\times \rightarrow [-1,1]) and all values scaled to be in [-1, 1]. By
linearly transforming all rewards accordingly, one obtains a proof for the
statement with all values in [0,1].We demonstrate a reduction from 3-SAT. Recall that an instance of
3-SAT is a Bool... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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3f9820a3b1a954f7dc91be8b3b32148ec8afe8ef | subsection | 80 | 106 | Proof of Theorem | To prove that Olive can encounter NP-hard problems, it therefore remains to show that running Olive on any MDP in
can generate the exact set of constraints in Equations (REF )-(REF ).The specification of Olive by only
prescribes that a constraint for one time step h among all that
have sufficiently large average Bellm... | {
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"Akshay Krishnamurthy",
"Alekh Agarwal",
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46c4fd47bd4e3dc5eeab6cc297be525a3a5c1047 | subsection | 81 | 106 | Proof of Theorem | Now there is positive average Bellman error in the initial state s_0 and with h_t = 1 the following constraints are added
f(s_0, \texttt {[try c_j]}) &= \max _b f(C_1, b) - 1/m
& \textrm {if } \pi _{f}(s_0) =& \texttt {[try c_j]}\\
f(s_0, \texttt {[solve]}) &= \frac{1}{m} \sum _{i=1}^m \max _{b} f(C_j,b)
& \textrm {if... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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6b5a5ab3262242041762d987186f632543694459 | subsection | 82 | 106 | Global Policy Algorithm | See Algorithm REF .
As the other algorithms, this method learns states using depth-first search.
The state identity test is similar to that of Valor at a high level: for any new path p, we derive an upper bound and a lower bound on V^\star (p), and prune the path if the gap is small. Unlike in Valor where both bounds a... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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491bb50504cd5552ee3e724749a4aefeba06b3e1 | subsection | 83 | 106 | Global Policy Algorithm | Once we change the global policy, however, all the pruned states need to be re-checked (Line REF ), as their optimal values are only guaranteed to be realized by the previous global policy and not necessarily by the new policy.[htbp]
InputaInput
OutputaOutput
myfunFunctiondfslearndfslearn
metaalgmain
testlearnedTestLea... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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a3f28e72b653b1706b8c6fc989541d1ad8d68369 | subsection | 84 | 106 | Computational efficiency | The algorithm contains three non-trivial computational components. In Eq.(REF ), a linear program is solved to determine the optimal value estimate of the current path given the value of one learned state (LP oracle).
In Line REF , computing the value of each learned path can be reduced to multi-class cost-sensitive cl... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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dde5b021e63f219b65cc348d423d0de6df5839ca | subsection | 85 | 106 | Learning Values using Depth First Search. | We first show that if the current policy is close to optimal for all learned states, then the policy is also good on all states for which returns true.[Policy on Tested States]
Consider a call of at path p and level h and assume
the deviation bounds of Definition REF hold for all data sets collected during this and al... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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bf589b950be5940cc43ba96450d432aaeb78fd2d | subsection | 86 | 106 | Learning Values using Depth First Search. | Then the program in Line REF is always
feasible and after executing that line, we have \forall q \in \textsc {learned}(h),Q^{\hat{\pi }_{h+1:H}}(q, \hat{\pi }_h) \ge Q^{\hat{\pi }_{h+1:H}}(q,
\star ) - 3 \tau _{pol},where \star is a shorthand for \pi _{\hat{\pi }_{h+1:H}}^\star , the
policy defined in Assumption REF w.... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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14a0110c80662972dd1e77b72f1ad2a1d986bb35 | subsection | 87 | 106 | Learning Values using Depth First Search. | Now, using this inequality along with \hat{V}(q) = \max _{\pi \in \Pi } \mathbf {E}_{D_q} [K\lbrace a_h = \pi (x_h)\rbrace \bar{r}],
we can relate \hat{V}(q) and Q^{\hat{\pi }_{h+1:H}}(q, \star ):\hat{V}(q) \ge \hat{\mathbf {E}}_{D_q} [K \lbrace a_h = \pi _{\hat{\pi }_{h+1:H}}^\star (x_h)\rbrace \bar{r}] \ge Q^{\hat{\p... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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745a29f28e6cbfc7015d437b4b0936ce44b1efd9 | subsection | 88 | 106 | Learning Values using Depth First Search. | Then for all p \in \textsc {Learned}(h), the current policy satisfiesV^{\hat{\pi }_{h:H}}(p) \ge V^\star (p) - \phi _hat all times except between adding a new path and updating the policy.
Further, for all p \in \textsc {Pruned}(h) the currently policy satisfiesV^{\hat{\pi }_{h:H}}(p) \ge V^\star (p) - \phi _h - 8 \tau... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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168577f64ad990299ac81b7bdb6475c52aba4d8a | subsection | 89 | 106 | Learning Values using Depth First Search. | Hence, the second part of the statement also
holds for h which completes the proof.[Termination]
Assume the deviation bounds hold for all Data sets collected during the first T_{\max }= 3 M^2HK calls of and . The algorithm terminates during these calls and at all times for all h \in [H] it holds |\textsc {Learned}(h)|\... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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4688890fd36beec78cd2dbc1aec8084dcdc9486f | subsection | 90 | 106 | Learning Values using Depth First Search. | Furthermore, as long as all deviation
bounds hold, the number of learned paths per level is bounded by
|\textsc {Learned}(h)| \le M.We next show that the number of paths that have ever appeared in \textsc {Pruned}(h) is at most KM.
This is true since there are at most KM
recursive calls to at level h from level h-1 and... | {
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"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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79c9992f93f9bb749e3d0e4449d78b4a2fc2718f | subsection | 91 | 106 | Learning Values using Depth First Search. | As
such, Bernstein's inequality and a union bound over all \pi \in \Pi _h gives that with probability 1-\delta ^{\prime },|\hat{\mathbf {E}}_{D_q} [K\lbrace a = \pi (x)\rbrace \bar{r}] - \mathbf {E}_{q, \hat{\pi }_{h+1}:H}[K\lbrace a_h = \pi (x_h)\rbrace \bar{r}]| \le \sqrt{\frac{4 K\log (2
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} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
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0.04520409181714058... | |
05db4abffc160be6feb40097849cb2eab1562e7b | subsection | 92 | 106 | Learning Values using Depth First Search. | Using Lemma REF , this is sufficient to show that \mathbf {P}(\bar{}) \le \delta . | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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6621ea7af0777b260e7ba20328945d2bebe9a0bd | subsection | 93 | 106 | Oracle-Inefficiency of OLIVE | As explained in Section
Theorem REF follows directly from
Theorem REF
and Proposition REF by proof by contradiction with P \ne NP.
In the following two sections, we first prove Proposition REF and then Theorem REF . | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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b651fde349490c8d18e8dc3d91095390884fa31d | subsection | 94 | 106 | Proof for Polynomial Time of Oracles | [Proof of Proposition REF ]
We prove the claim for each oracle separatelyCSC-Oracle: For tabular functions, the objective can be decomposed as
n^{-1} \sum _{i=1}^n c^{(i)}(\pi (x^{(i)}))
= \sum _{x \in } n^{-1}
\sum _{i=1}^n \lbrace x = x^{(i)}\rbrace c^{(i)}(\pi (x)).
Each of the || terms only depend on \pi (x) but ... | {
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{
"arxiv_id": "",
"doi": "10.1016/0041-5553(80)90061-0",
"end": 1969,
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"raw": "Leonid G Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 1980.",
"... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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c0ced9e632c2d80458b1e3da6e2ce5ff24ebf333 | subsection | 95 | 106 | Proof for Polynomial Time of Oracles | Note that the initial ellipsoid can be set to any ellipsoid containing [0, 1]^{||} due to the normalization of rewards.
Further, the volume of the smallest ellipsoid can be upper bounded by a polynomial in \epsilon _{\textrm {feas}} using the fact that we only require a solution that is feasible up to \epsilon _{\textr... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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f9258a086818b97c9d6dcbb6a37f0f0a5b987d43 | subsection | 96 | 106 | MDP structure. | Let \psi be the 3-SAT instance with
variables x_{1:n} and clauses C_{1:m}. The state space for MDPs in consists of m+2n+1 states, two for each variable, one
for each clause, and one additional starting state. For each variable
x_i, there are two states x_i^0, x_i^1 corresponding to the
variable and its negation. Each c... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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038c3b18d852826e5199ef03302ddb91ab9f33a3 | subsection | 97 | 106 | MDP structure. | The
second set of actions are labeled [try C_j] (for
j \in [m]), which receives 1/m instantaneous reward and
transitions deterministically to c_j. Finally there is a
[solve] action that transitions uniformly to the
\lbrace C_j\rbrace _{j=1}^m states and receives zero instantaneous reward. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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3b0418335258a3e690b9032ba5f7edb7eeb9135c | subsection | 98 | 106 | The Optimal Value. | Consider the Olive optimization problem (REF ) on the
family of MDPs with constraints described above. Note that all MDPs in the
family generate identical constraints, so formulating the optimization
problem does not require determining whether \psi has a satisfying
assignment or not.Now, if \psi has a satisfying assig... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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427ada072b1b4ed0f1dfb3bcf7e20fd8be754e70 | subsection | 99 | 106 | Additional Barriers | In this section, we describe several further barriers that we must
resolve in order to obtain tractable algorithms in the stochastic
setting. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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07add96e7f2820b21bded6725c78819e3f2f4df5 | subsection | 100 | 106 | Challenges with Credit Assignment | We start with the learning step, ignoring the challenges with
exploration, and focus on a family of algorithms that we call
Bellman backup algorithms.
A Bellman backup algorithm collects n samples from every
state and iterates the policy/value updates\hat{\pi }_h &= _{\pi \in \Pi _h} [lr]{\sum _{s \in _h}} \mathbf {E... | {
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"doi": "",
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"raw": "Damien Ernst, Pierre Geurts, and Louis Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 2005.",
"source_ref_id": "... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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674bb45a9aceb330f57e8a7612066219894c773e | subsection | 101 | 106 | Challenges with Credit Assignment | From the start state there are
two actions a,b where a transitions to x_{1,a} and b
transitions to x_{1,b}. From then on, there is just one action which
transitions from x_{h,z} to x_{h+1,z} z \in \lbrace a,b\rbrace .
The reward
from x_{H,a} is \textrm {Ber}(1/2+\epsilon ) and the reward
from x_{H,b} is \textrm {Ber}(1... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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8ea7d622d7b33018b847092aa6f8f316703a02f1 | subsection | 102 | 106 | A lower bound on the binomial tail. | The rewards from
x_{H,b} are drawn from \textrm {Ber}(1/2). Call this values
r_1,\ldots , r_n with average \bar{r}. We select g_{bad} if
\bar{r} \ge 1/2 + \epsilon /2^{H}. By Slud's lemma, the
probability is[ \bar{r} \ge 1/2 + \epsilon /2^{H}] \ge 1- \Phi \left( \frac{n \epsilon /2^{H} }{\sqrt{n/4}}\right)where \Phi is... | {
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{
"arxiv_id": "",
"doi": "",
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"raw": "Rémi Munos and Csaba Szepesvári. Finite-time bounds for fitted value iteration. Journal of Machine Learning Research, 2008.",
"source_ref_id": "6f51d77deaa045f... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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b7b0230a1892364034a6374646c39b27c4944121 | subsection | 103 | 106 | Challenges with Exploration | We now turn to challenges with exploration that arise when factoring
the Q-function class into the (g,\pi ) pairs, which works well in
the deterministic setting, as in Section . However, the
stochastic setting presents further challenges.
Our first construction shows that a decoupled approach using Olive's
average Bell... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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aa6c98a93c3baebf4ee32d2e8d5cb8215efc0024 | subsection | 104 | 106 | Challenges with Exploration | However, using this future-value function in
the optimization_{x_h \sim s_0} [r_h + \hat{g}_{h+1}(s^{\prime }) | a_h = \pi (x_h)],we see that all policies, including \pi ^\star have the same
objective value. When we choose any one of them but \pi ^\star , the “optimistic” value computed by maximizing Eq.(REF ) will be ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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14501d2f7407fb7c30986a631f3608b18b7bd5ca | subsection | 105 | 106 | Challenges with Exploration | These latter two function
haveg^\star (x_1) \triangleq 0, & \quad g^\star (x_2) \triangleq 1\\
g_{\textrm {bad}}(x_1) \triangleq \sqrt{\epsilon }, & \quad g_{\textrm {bad}}(x_2) \triangleq \sqrt{\epsilon }Now, let us calculate the square loss of these three value functions
to the roll-out achieved by \hat{\pi }.\textrm... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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7a7ae0b1747ae86355d521d0a7ea12903a11ad47 | abstract | 0 | 25 | Abstract | The aim of this article is to explore global and local properties of finite
groups whose integral group rings have only trivial central units, so-called
cut groups. For such a group we study actions of Galois groups on its character
table and show that the natural actions on the rows and columns are essentially
the sam... | {
"cite_spans": []
} | 10.1515/jgth-2020-0165 | 1808.03546 | Global and local properties of finite groups with only finitely many
central units in their integral group ring | [
"Andreas Bächle",
"Mauricio Caicedo",
"Eric Jespers",
"Sugandha Maheshwary"
] | [
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57df807f6823e1adcfbd51761102228721d48347 | subsection | 1 | 25 | Introduction | Let G be a finite group and let \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) denote the group of central units of the integral group ring \mathbb {Z}G. Clearly, \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) contains \pm \mathcal {Z}(G), where \mathcal {Z}(G) denotes the center of G. However, if \mathcal {Z}(\mathcal {U}(\mathbb... | {
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"raw": "G. K. Bakshi, S. Maheshwary, and I. B. S. Passi, Integral group rings with all central units trivial, J. Pure Appl. Algebra 221 (2017), no. ... | 10.1515/jgth-2020-0165 | 1808.03546 | Global and local properties of finite groups with only finitely many
central units in their integral group ring | [
"Andreas Bächle",
"Mauricio Caicedo",
"Eric Jespers",
"Sugandha Maheshwary"
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7ce003ac9d4e0a1be12a71fe82314f13a3ad7991 | subsection | 2 | 25 | Introduction | After setting up the necessary background in Section , we begin by observing the impact of the cut property of a group on its character table, see Section : We prove that for a cut group G, the number of rational valued irreducible characters of G equals the number of rational valued conjugacy classes of G by exhibitin... | {
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"Mauricio Caicedo",
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"Sugandha Maheshwary"
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b07ac0fc90782c9fd1688db51ae92cb63b5d2b4c | subsection | 3 | 25 | Preliminaries | Throughout the article, all groups considered are finite, unless otherwise stated explicitly. Let G be a group and x \in G be an element of G. The order of G is denoted by |G|. The order of x is denoted by o(x) and {G}(x) denotes the centralizer of x in G. Let y\in G. Then, by x \sim y we mean that “x is conjugate to y... | {
"cite_spans": []
} | 10.1515/jgth-2020-0165 | 1808.03546 | Global and local properties of finite groups with only finitely many
central units in their integral group ring | [
"Andreas Bächle",
"Mauricio Caicedo",
"Eric Jespers",
"Sugandha Maheshwary"
] | [
"math.GR",
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56609ca8c0033d6a4da5a9e2b78c2e526d1a0184 | subsection | 4 | 25 | Preliminaries | The group G is called (inverse) semi-rational if every element of G is (inverse) semi-rational in G.We begin by stating the following equivalent criteria for a cut group that are essential for this article.Proposition 2.1 (see , ) For a group G, the following statements are equivalent:G is a cut group.
For every x \i... | {
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e974fdd0c30fd15b962412e0e11b8e3e26b316e9 | subsection | 5 | 25 | Rationality | It is well-known that the number of real conjugacy classes of a group agrees with the number of its real irreducible characters, where the conjugacy class of x is called real if all its elements are real. Yet the analogous statement is not true if the field of reals is replaced by the field of rationals; there are grou... | {
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e2e5bd4265144eed69a62fc52094b9211665637a | subsection | 6 | 25 | Rationality | Since \mathbb {Q}(x) = \mathbb {Q}(\sqrt{-d}), d \geqslant 0, again by , we have that x is semi-rational in G, so that if j is a positive integer coprime to o(x), thenx^{j} \sim x ~\mathrm {or}~ x^{j} \sim x ^{m},~\mathrm { for~ some}~ m.Note that if x is real, then in view of the assumption \mathbb {Q}(x) = \mathbb {Q... | {
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24e6ab6f73ea7694b8be4c4f93ff0651e3e4db25 | subsection | 7 | 25 | Rationality | The group G of order 32 in Example REF below also has an irreducible character with field of character values \mathbb {Q}(\zeta _8) over the rationals.Example 3.4 In , J. Tent gave the following two groups:G = \langle \ a, b, c \ | \ a^2 = b^2 = c^8 = 1,\ b^c = b,\ b^a = bc^4,\ c^a = c^3\ \rangleandH = \langle \ a, b, ... | {
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ccdf6e649a455db235f62f59efbfabd57352dfcc | subsection | 8 | 25 | Nilpotent | As mentioned earlier, it is well known that an abelian group is cut, if and only if its exponent divides 4 or 6. As the cut property is quotient closed and the center of a cut group is again a cut group, we necessarily have for a cut group G:\text{for all } N \mathrel {G:\qquad \exp (\mathcal {Z}(G/N)) \mid 4\quad \tex... | {
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18b0194a7d786e5660e03b21e4f9ff126a517bd3 | subsection | 9 | 25 | Nilpotent | We first check that P_{2} is a \textsf {cut}\ group. By \cite [Corollary~3]{Mah18} this is equivalent with x^4 \in [x,P_2]=[x, G] for all x \in P_2. But this follows from the assumptions, since the image of x is contained in \mathcal {Z}(G/[x, G]). Similarly, P_{3} also is a \textsf {cut}\ group. Hence, it only remains... | {
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2975570fc57b8bdf6cd6314796d10d14bc8c7614 | subsection | 10 | 25 | Simple | So far, the properties of solvable cut groups have been explored. A complete classification of finite metacyclic cut groups is given in . A description of Frobenius cut groups can be found in (it turns out that those groups are always solvable). In this section, we give a complete classification of finite simple cut gr... | {
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b7b047a4f642f90558cd926db78a1b18815c2fbe | subsection | 11 | 25 | Simple | An inspection of the character tables of the remaining groups (for example in ATLAS or GAP) using Proposition REF reveals that of those exactly the groups listed above are cut groups.Note that in contrast to the case of finite simple groups, every infinite simple group I is a cut group i.e., has the property that \math... | {
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23aac02b6b4f936fc0dc59e3df167db8c5cc7dd3 | subsection | 12 | 25 | Local properties of | It was conjectured for a long time that being rational for 2-elements is governed by the Sylow 2-subgroup of a group. More precisely, already in Kletzing's book from 1984 it is referred to as a “long standing conjecture” that the Sylow 2-subgroup of a rational group is again rational (recall that every non-trivial rati... | {
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5e22e3118f81db635ec64dc2c4c7de87773cbfb7 | subsection | 13 | 25 | Local properties of | By Sylow's theorem, S is contained in a Sylow 3-subgroup P of G and x is inverse semi-rational in P. The other implication is clear.The above lemma is clearly false for p-elements, p \geqslant 5.It may be observed that if G is a cut group and P \in \operatorname{Syl}_3(G) then P is cut if and only if for all x \in P an... | {
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33b7f75b645471e633cd862f5a4175caa0dd8ce6 | subsection | 14 | 25 | Local properties of | If this series terminates in G, then G is called p-solvable. In this case, the p-length of G is defined to be the number of occurrences of the symbol p in the subscript of the first group in the upper p-series of G that equals G. For example, G has p-length at most 1 if and only if it has normal subgroups M and N, M \s... | {
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939f2adabb73cc734fd01cc3babbabe4afd2b909 | subsection | 15 | 25 | Local properties of | (Of course, we could also have used the description of Frobenius cut groups in .)
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489775506d19455936c0d13502b4ba31bb193ec0 | subsection | 17 | 25 | Local properties of | It is not hard to find cut groups that have Sylow 2-subgroups that are not cut (e.g. the groups with SmallGroupID [384, 18033] and [384, 18040] of order 384 = 2^7 \cdot 3). However, for the prime 3 things seem to behave differently. The following lemma shows that being inverse semi-rational for 3-elements is indeed a s... | {
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2ed166d97dcadcf5f8edc1ae72430eec3fdbea31 | subsection | 19 | 25 | Local properties of | Let x \in P. Then x is inverse semi-rational in G and hence also in some Sylow 3-subgroup Q of G by Lemma REF . Since N is a normal subgroup of 3^{\prime }-index, Q is contained in N and hence x is inverse semi-rational in N. Note that M is a normal complement for P in N, that is, N = MP is the semi-direct product of t... | {
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... | 10.1515/jgth-2020-0165 | 1808.03546 | Global and local properties of finite groups with only finitely many
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183c9dc39c2e2c006da8e38d1aa7314a81e69280 | subsection | 20 | 25 | Local properties of | By they have the following structure: |G| = 7\cdot 3^b and G contains a normal Frobenius subgroup of index 3 with \textup {O}_3(G) the Frobenius kernel. Moreover \textup {O}_3(G)T \in \operatorname{Syl}_3(G) for some subgroup T = \langle t \rangle of order 3 and G/\textup {O}_3(G) is the non-abelian group of order 21.
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c9f77cd2b47cac8d6c6d92be7da11789711348bb | subsection | 21 | 25 | Local properties of | Consequently, every element of P is inverse semi-rational in P, i.e., P is cut.Note that the class of groups in (REF ) contains groups of 3-length 2. We do not know of any example of a cut group of odd order to which the above theorem does not apply. Theorem REF could also have been proved using the dual characterizati... | {
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593187ee273d77c080efb9d13c0cb3401d25308b | subsection | 22 | 25 | Existence of | In this section, we will give some indications that the class of cut groups is surprisingly large in all finite groups. Recall that a p-group can only be a cut group if p \in \lbrace 2, 3\rbrace . We show that in these cases, the ratio of cut groups tends to one in the logarithmic sense.Proposition 7.1 Let c(r) denote ... | {
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787c713d8804646a54d77437b1a8a0af5312c39a | subsection | 23 | 25 | Existence of | For instance, about 86.62\% of the groups of order at most 512 and 78.55% of groups of order at most 1023 are cut groups, whereas 0.57% of the groups of order at most 512 and 0.52% of groups of order at most 1023 are rational.
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abe0c9dff49538b4e987452342b838d3b45bbb92 | subsection | 24 | 25 | Existence of | The dashed graph gives the percentage of rational groups up to that order whereas the solid graph indicates the percentages of cut groups up to that order. Note that the upward bumps for the percentage of cut groups appear at 2-powers, whereas the (visible) bumps downwards for this percentage happen at orders of the fo... | {
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5f11998f0bb00358806d522d956852c1e733c6cf | abstract | 0 | 52 | Abstract | We study the existence of traveling wave solutions to a unidirectional
shallow water model which incorporates the full linear dispersion relation for
both gravitational and capillary restoring forces. Using functional analytic
techniques, we show that for small surface tension (corresponding to Bond
numbers between $0$... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
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a5777ffa77de750afa87ce06ec5ea709b6f6b8ec | subsection | 1 | 52 | “Full dispersion" models | It is well known that the Korteweg-de Vries (KdV) equationu_t+\sqrt{gd}\left(1+\frac{1}{6}d^2\partial _x^2\right)u_x+uu_x,approximates the full water wave problem in the small amplitude, long wavelength regime
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46265f1baa5a45cc93c0ac349b55db44ddb354b6 | subsection | 2 | 52 | “Full dispersion" models | In particular, in , , the authors conducted a detailed global bifurcation analysis of periodic traveling waves for (REF ) and concluded
that the branch of smooth periodic waves terminates in a non-trivial cusped solution – bounded solution with unbounded derivativeThe wave behaves like |x|^{1/2}
near the cusp.– that is... | {
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e64d62a50adf60b7d47649f161ee9e3cfd329c51 | subsection | 3 | 52 | Including surface tension | It is thus natural to consider the existence and behavior of solutions when additional physical effects are included. In this paper, we incorporate surface tension and consider
the following pseudodifferential equationu_t + ({\mathcal {M}}_{gd\tau } u + u^2)_x = 0.Here, u, x, and t are as in (REF ) above, and \mathcal ... | {
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40e2f1add24af09acde504341a921589517aa355 | subsection | 4 | 52 | Including surface tension | In Section below, we will apply these stability
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f0770fa48bc74cfa1f1543dee472783cdd4f3b0e | subsection | 5 | 52 | Formal computations and the main results | A routine nondimensionalization of (REF ) converts it tou_t + \left(\mathcal {M}_{\beta } u + u^2\right)_x = 0,where \mathcal {M}_\beta is the Fourier multiplier operator with symbolm_\beta (k) := \sqrt{\left(1+\beta k^2\right) {\tanh (k ) \over k}}.We will henceforth be working with this version of the system.
Substit... | {
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"Mathew A. Johnson",
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e1ec7839b2115a9e4e46907421932d5a275254f7 | subsection | 6 | 52 | Formal computations and the main results | In particular, (REF ) admits a unique non-trivial even solution in L^2({\mathbb {R}}) given by\sigma _\beta (X) := {3 \gamma _\beta \over 2 } \operatorname{sech}^2\left( {X \over 2} \right) = {1-3 \beta \over 4} \operatorname{sech}^2\left( {X \over 2} \right).Note that \sigma _\beta (X) is positive when \beta \in (0,1/... | {
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d2dfddf8502e5ec1c518aee1ab9951642d58892e | subsection | 7 | 52 | Formal computations and the main results | In Figure REF , note that when \beta \in (0,1/3) and c>1 (i.e. is supercritical)
there is a unique k_{\beta ,c}>0 at whichm_\beta (\pm k_{\beta ,c}) - c = 0.Thus {\mathcal {M}}_\beta - c cannot be inverted; the situation becomes more complicated. What occurs is that when \epsilon > 0 the main pulse \sigma _\beta , thro... | {
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2fabd338db97c2621ad930a5b5335cb8492803f6 | subsection | 8 | 52 | Formal computations and the main results | Specifically, there is a constant \delta >0 such that
the frequency of P_\epsilon lies in the interval [k_{\beta ,c_\epsilon } - \delta \epsilon ,k_{\beta ,c_\epsilon } +\delta \epsilon ] and for all r \ge 0 there is a constant C_{r}>0 for which
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d8e2ea1d30afa0e7b58828e54b8fa5a816efa631 | subsection | 9 | 52 | Conventions | Here we specify the notation for the function spaces we will be using along with some other conventions. | {
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3f1a82416e52cd4427c670f7672bf6b34e3d27d5 | subsection | 10 | 52 | Periodic functions | We let W^{r,p}_{\text{per}}:=W^{r,p}{(\mathbb {T}}) be the usual “r,p" Sobolev
space of 2 \pi -periodic functions. We denote L^p_{\text{per}}:=W^{0,p}_{\text{per}} and
H^r_{\text{per}}:=W^{r,2}_{\text{per}}.
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
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19eb72a46f4fcab8e74000baa9de20c74a0e79dd | subsection | 11 | 52 | Functions on | We let W^{r,p}:=W^{r,p}({\mathbb {R}}) be the usual “r,p" Sobolev
space of functions defined on {\mathbb {R}}. For q \in {\mathbb {R}} putW^{r,p}_{q} := \left\lbrace u \in L^2({\mathbb {R}}) : \cosh ^q(x) u(x) \in W^{r,p}({\mathbb {R}}) \right\rbrace .These are Banach spaces with the naturally defined norm.
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ace011afbe596357eb65cb4570a632ea673f3737 | subsection | 12 | 52 | Spaces of operators | For Banach spaces X and Y we let B(X,Y) be the space
of bounded linear operators from X to Y equipped with the usual induced topology. | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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62a57b66fc3f5340b4c7b66af14a836354ae9def | subsection | 13 | 52 | Big | Suppose that
Q_1 and Q_2 are positive quantities (like norms) which depend upon
the smallness parameter \epsilon , the regularity index r, the decay rate q and some collection of elements \eta which live in a Banach space X.When we write “Q_1 \le C Q_2" we mean “there exists C>0,
\epsilon _0 > 0, q_0>0, \delta >0 such ... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
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4b824bcfad8a5548641eb42dc4a2122890ff3495 | subsection | 14 | 52 | Fourier analysis: | We use following normalizations and notations for the Fourier transform and its inverse:\widehat{f}(k):={\mathcal {F}}[f](k):={1 \over 2 \pi } \int _{\mathbb {R}}f(x)e^{-ikx} dx \quad \text{and}\quad {(x):={\mathcal {F}}^{-1}[g](x) := \int _{\mathbb {R}}g(k)e^{ikx} dk.
} | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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b75aea26b77985122471e21f4badb94ba1b14211 | subsection | 15 | 52 | Solitary waves of depression when | The following theorem on the existence of solitary waves in a certain class of pseudodifferential equations was proved in :Theorem 3
Suppose that there exists \delta _*>0 such that
n: (-\delta _*,\delta _*) \rightarrow {\mathbb {R}} is C^{2,1} (that is, its second derivative exists and is uniformly Lipschitz continuou... | {
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"raw": "J. D. Wright and A. Stefanov. Small amplitude traveling waves in the full-dispersion whitham equation. Preprint, 2018. arXiv:1802.10040.",
... | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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8525be9f9cad08a567f63bc3626952b823c347d9 | subsection | 16 | 52 | Solitary waves of depression when | In this problem, however, we have the additional information that m_\beta (k) grows like |k|^{1/2} for large |k| and hence {\mathcal {M}}_\beta is “like" \partial _x^{1/2}. With this, a straightforward bootstrapping argument demonstrates that the solutions are smooth. We omit these details. | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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b91b790cbbe9df2a569d164a85ffd0e597234433 | subsection | 17 | 52 | Generalized solitary waves when | In this section we prove Theorem REF . Throughout we fix \beta \in (0,1/3) and we will, for the most part, not track how quantities depend on this quantity.Our results hold for any such choice of \beta but we make no claims upon how they depend on \beta and in particular we make no claims about what happens at \beta \r... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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d24d5864497c18ba4ecabf090a4b46a75817bd69 | subsection | 18 | 52 | A necessary solvability condition | We begin our proof of Theorem REF by doing something that is doomed to fail. Nevertheless, we believe that understanding
the mechanism behind this failure is an important step in the journey to the proof of Theorem REF . Throughout, r\ge 1 (a regularity index) is fixed but arbitrary and
q>0 (a decay rate) is taken to b... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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65ea0e371f9a1e369369af77cc2b74a879db9c3e | subsection | 19 | 52 | A necessary solvability condition | \end{split}Then the formal expansion of {\mathcal {M}}_\beta ^\epsilon in (REF ) indicates that
J_0 \sim \epsilon ^2 \partial _X^4 \sigma _\beta . This argument can be made rigorous by way of Fourier analysis:Lemma 4
There exists q_0>0 so that for any q \in [0,q_0] and r \ge 0 we have\Vert J_0 \Vert _{r,q} \le C_r \ep... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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0fb953d75d774dcbfd4418b3de94899d3eb52ac0 | subsection | 20 | 52 | Beale's method | Beale encountered nearly the same obstacle encountered in Section REF in his work on the full gravity-capillary water wave problem .
In his investigation, he made the remarkable observation that
just as the special frequency K_\epsilon causes difficulties at the linear level, it also points to a way out.
Indeed, observ... | {
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"arxiv_id": "",
"doi": "10.1002/cpa.3160440204",
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"raw": "J. T. Beale. Exact solitary water waves with capillary ripples at infinity. Comm. Pure Appl. Math., 44:211?257, 1991.",
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"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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842b8395ebc166413fcdb0ff8f3d94446b5b1fab | subsection | 21 | 52 | Beale's method | Specifically,
introducing the notation{\Phi _\epsilon ^a(X)}:= \phi _\epsilon ^a(K_\epsilon ^a X)we attempt to construct solutions of the profile equation (REF ) for 0<\epsilon \ll 1 of the formW_\epsilon (X) = \sigma _\beta (X) + a {\Phi _\epsilon ^a(X)} + R(X)where now both R \in E^r_q and a \in {\mathbb {R}} are unk... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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7b2712c8f5a0d4ecbd10f8303fd07d7842c4c036 | subsection | 22 | 52 | Beale's method | Specifically:Lemma 7 There exists \epsilon _0>0 and q_0>0 such that, for all r \ge 0, q \in [0,q_0] and \epsilon \in (0,\epsilon _0]
we have\left\lbrace \begin{aligned}\Vert J_3\Vert _{r,q} &\le C_r \epsilon ^{-r} |a|\Vert R\Vert _{r,q}\\
&\hspace{-20.0pt}\quad \text{and}\quad \Vert J_3-\widetilde{J}_3\Vert _{r,0} \le ... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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9866b4cf2c39e8cfda6e96ee88b8b26bc4da0f65 | subsection | 23 | 52 | The linear problem | The left hand side of (REF ) is linear in R and a.
We claim it is a bijection in an appropriate sense. Specifically we have the following linear solvability result.Proposition 8
There exists \epsilon _0>0 and q_0>0 for which the following hold when
\epsilon \in (0,\epsilon _0], q \in (0,q_0] and r \ge 0. There are lin... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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5bd6e7a98017875b472b64dabe692513215d5b61 | subsection | 24 | 52 | The linear problem | Moreover, the analyticity of \sigma _\beta and the fact that K_\epsilon = {\mathcal {O}}(1/\epsilon ) implies |\widehat{\sigma _\beta }(2K_\epsilon )| is exponentially small in \epsilon .
Consequently \chi _\epsilon and \chi _\epsilon ^{-1} are bounded uniformly in \epsilon for 0<\epsilon \ll 1.
It follows that we can ... | {
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"raw": "Timothy E. Faver and J. Douglas Wright. Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity. SIAM J. Math. Anal., 50(1):182–250, 2018.",
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5acd5521412f5d092432a38cc26aadfb89cc7cbf | subsection | 25 | 52 | The linear problem | Specifically\begin{split}
\Vert {\mathcal {P}}_\epsilon F\Vert _{r,q} &\le \Vert F\Vert _{r,q} + 2\chi _\epsilon ^{-1} \left|\widehat{F}(K_\epsilon ) \right|\Vert \sigma _\beta \Phi _\epsilon ^0\Vert _{r,q}\\
&\le \left(1 +C_q K_\epsilon ^{-r} \Vert \sigma _\beta \Vert _{r,q} \Vert \Phi _\epsilon ^0\Vert _{W^{r,\infty ... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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2f97d8ca8d8f792369a75368a9959106fdfe6eb5 | subsection | 26 | 52 | The linear problem | To make this step requires a critical feature of {\mathcal {L}}_\epsilon ^{-1} {\mathcal {P}}_\epsilon : it is small perturbation of -\gamma _\beta ^{-1} (1-\partial _X^2)^{-1}.
Specifically, if we put{\mathcal {G}}_\epsilon := \epsilon ^{-1} \left({\mathcal {L}}_\epsilon ^{-1} {\mathcal {P}}_\epsilon +\gamma _\beta ^{... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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2f61d99b39166b5cbdc66e933d1ea23278e253ba | subsection | 27 | 52 | The linear problem | Following (for instance) Appendix D.10 of , this observation can be extended to the weighted
space E^r_q via operator conjugation. Specifically:Lemma 11
There exists q_0 >0 such that for all r \ge 0 and q \in [0,q_0] the operator {\mathcal {S}}_0
is a bounded and invertible map from E^r_q \rightarrow E^r_q. In particu... | {
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"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
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5766b47cad3876aece0366a26f927a8cd073fb3f | subsection | 28 | 52 | Nonlinear solvability | We now return to constructing a solution of the form (REF ) to the nonlinear equation (REF ). Thanks to Proposition REF
we see that solving (REF ) is equivalent to solving the fixed point problem\begin{split}
R = {\mathcal {R}}_\epsilon \left(J_0 + J_1 + J_2 + J_3 \right)=:{\bf {N}}^1_\epsilon (R,a)\\
a = {\mathcal {A... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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06864887b887bdbbf32bb756369e21af8594a741 | subsection | 29 | 52 | Nonlinear solvability | For all r \ge 1 there exist \kappa _r>0 such that \epsilon \in (0,\epsilon _0] implies\Vert {\bf {N}}_\epsilon ^1(R,a)\Vert _{r,q_*} \le \kappa _r\left(\epsilon ^2 +
\epsilon ^{-r+1/2} a^2 + \epsilon ^{-r+1/2} |a| \Vert R\Vert _{r-1/2,q_*} + \Vert R\Vert ^2_{r-1/2,q_*} \right),|{\bf {N}}_\epsilon ^2(R,a)| \le \kappa _r... | {
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} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
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5d771afd3dfc65c4797ebe63fa780412218ba8fe | subsection | 30 | 52 | Nonlinear solvability | To begin, let R_0:=0, a_0 :=0 and for n\ge 0 defineR_{n+1}:={\bf {N}}_\epsilon ^1(R_{n},a_{n}) \quad \text{and}\quad a_{n+1}:={\bf {N}}_\epsilon ^2(R_{n},a_{n}).We first claim that for each fixed r\ge 1 there exists \epsilon _{r} > 0 such that\Vert R_{n}\Vert _{r,q_*} \le 2 \kappa _r \epsilon ^2 \quad \text{and}\quad |... | {
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"Mathew A. Johnson",
"J. Douglas Wright"
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c49117d6b76f6c04721d0445cf3e33b289b87865 | subsection | 31 | 52 | Nonlinear solvability | For the inductive step, if we assume (REF ) then (REF ) implies by way of the estimates (REF ) and (REF ) that\Vert R_{n+1}\Vert _{r,q_*} \le \kappa _{r,q_*}\left(\epsilon ^2 +
\epsilon ^{-r+1/2} (2 \kappa _{r,q_*} \epsilon ^{r+2})^2 + \epsilon ^{-r+1/2} |2 \kappa _{r,q_*} \epsilon ^{r+2}|2 \kappa _{r,q_*} \epsilon ^2 ... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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4cdb73b558c38c48c912c85aa81732ccd6377033 | subsection | 32 | 52 | Nonlinear solvability | Using (REF )
together with the estimates (REF ) and (REF ) we find directly that\begin{split}
\Vert R_{n+1} - R_n\Vert _{r,0}=&\Vert {\bf {N}}_\epsilon ^1(R_n,a_n)-{\bf {N}}_\epsilon ^1(R_{n-1},a_{n-1})\Vert _{r,0}\\ \le &\kappa _r \epsilon ^{-r} (\Vert R_n\Vert _{r,q_*}+\Vert R_{n-1}\Vert _{r,q_*}+|a_n|+|a_{n-1}|)|a_n... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
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dbdacf49fc828d937018fbc114f9218d10ce65a3 | subsection | 33 | 52 | Nonlinear solvability | Taken together, it follows that\Vert R_{n+1} - R_n\Vert _{r,0} + \epsilon ^{-r}|{a}_{n+1} - a_n| \le 8 \kappa ^2_{r}\epsilon ^2 \left(\Vert R_n - R_{n-1}\Vert _{r,0}+ \epsilon ^{-r} |{a}_n-{a}_{n-1}|\right),which, in turn, gives\Vert R_{n+1} - R_n\Vert _{r,0} + \epsilon ^{-r}|{a}_{n+1} - a_n| \le \left(8 \kappa ^2_{r}\... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
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770007819e7fcc31a4c2c7b93bd8a59828110a37 | subsection | 34 | 52 | Nonlinear solvability | As E^r_0\times {\mathbb {R}} is clearly a Hilbert space, it follows that exists (R_\epsilon ,a_\epsilon ) \in E^r_0 \times {\mathbb {R}} such that(R_n,a_n) \underset{E^r_0 \times {\mathbb {R}}}{\longrightarrow }(R_\epsilon ,a_\epsilon ) \quad \text{as $n \rightarrow \infty $}.Our next goal is to show that, in fact, the... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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f83d28b9301f0f9b6bea10130b4e5f681a227aaa | subsection | 35 | 52 | Nonlinear solvability | Furthermore, since norms on Hilbert spaces are lower semi-continuous with respect to weak limits, we know from (REF ) that\Vert R_\epsilon \Vert _{r,q_*} \le 2 \kappa _r \epsilon ^2 \quad \text{and}\quad |a_\epsilon | \le 2 \kappa _r \epsilon ^{r+2}.The next step is to show that the pair (R_\epsilon ,a_\epsilon )\in E^... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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4869a45923c7935d997cecd72333bd14878ded57 | subsection | 36 | 52 | Nonlinear solvability | Indeed, if there were another pair (\widetilde{R}_\epsilon ,{\widetilde{a}}_\epsilon )\in E^r_q\times {\mathbb {R}} that satisfies both (REF ) and (REF ), then it is apparent that\Vert R_{\epsilon } - \widetilde{R}_\epsilon \Vert _{r,0} + \epsilon ^{-r}|{a}_{\epsilon } - {\widetilde{a}}_\epsilon |
= \Vert {\bf {N}}_\ep... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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8643b6469ab91a95f336b287194ca05fbb97fba9 | subsection | 37 | 52 | Nonlinear solvability | By putting P_\epsilon (x) = a_\epsilon \Phi _\epsilon ^{a_\epsilon }(x/\epsilon ) we have proven Theorem REF . | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
5b3354a1493d3377b8c31ac523aac1fe54f213c5 | subsection | 38 | 52 | Discussion on Stability | In this final section, we briefly consider the spectral stability of the generalized solitary waves constructed in Theorem REF . Specifically, we are interested
in the ability of these generalized solitary waves to persist when subject to small perturbations. As we will see, a necessary condition for our small generali... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.na.2015.08.019",
"end": 650,
"openalex_id": "https://openalex.org/W2962822343",
"raw": "Vera Mikyoung Hur and Mathew A. Johnson. Modulational instability in the Whitham equation with surface tension and vorticity. Nonlinear Anal.,... | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.... | |
f07c385eb1814ecf290fd7e0ea4d1288b7d1d2cc | subsection | 39 | 52 | Discussion on Stability | Together, this implies the
following:\lambda \in \sigma (L_\epsilon )\quad \Rightarrow \quad \pm \lambda ,~\pm \bar{\lambda }\in \sigma (L_\epsilon ).It follows that the pattern w_\epsilon is spectrally stable if and only if \sigma (L_\epsilon )\subset {\mathbb {R}}i.To study the spectrum of L, note that from Theorem R... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02534223161637783,
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3ec012ce328d4e394fdbc355b10f9a39c32af6e4 | subsection | 40 | 52 | Discussion on Stability | We now study the spectral stability of these oscillations.As the operator \widetilde{L}_\epsilon has periodic coefficients its spectrum can be studied via Floquet-Bloch theory, from which it can be easily shown that
non-trivial solutions of \widetilde{L}_{\epsilon }v=\lambda v
can not be integrable over {\mathbb {R}}: ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4614-6995-7",
"end": 621,
"openalex_id": "https://openalex.org/W360456325",
"raw": "Kapitula T. and K. Promislow. Spectral and dynamical stability of nonlinear waves, volume 185 of Applied Mathematical Sciences. Springer, New ... | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.03653222694993019,
-0.0327172... | |
aae4d9df8dd6079b0fe79b89376cd5fe1c0104c7 | subsection | 41 | 52 | Discussion on Stability | (km_\beta (k))^{\prime }=m(0);
(3)
the phase velocities of the fundamental mode and the second harmonic coincide, i.e. m(k)=m(2k);
(4)
\Delta _{BF}(k)=0.It is interesting to note that possibilities (1)-(3) are purely linear, not depending on any nonlinear effects.
Note since the waves P_\epsilon are necessarily supercr... | {
"cite_spans": []
} | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.018708186224102974,
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-0.... | |
ad86f4b40046ef34b956b1a24125b867ab4e9cf7 | subsection | 42 | 52 | Proofs of Technical Estimates | In this Appendix, we prove a number of technical lemmas used throughout the paper. To prove Lemma REF , we need the following general result:Lemma 15
Suppose that h(Z) is a complex valued function with the following properties:h(Z) is analytic on the closed strip \overline{\Sigma }_q = \left\lbrace |\Im Z | \le q\righ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1215/s0012-7094-80-04719-5",
"end": 913,
"openalex_id": "https://openalex.org/W1676487141",
"raw": "J. Thomas Beale. Water waves generated by a pressure disturbance on a steady stream. Duke Math. J., 47(2):297–323, 1980.",
"source_... | 1807.11469 | Generalized Solitary Waves in the Gravity-Capillary Whitham Equation | [
"Mathew A. Johnson",
"J. Douglas Wright"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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