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We have also compared the obtained results with several feature selection algorithms like Particle Swarm Optimization (PSO) {{cite:596854276f3e16f6d17e4632dcb2589d92b99d0c}}, Grey Wolf Optimization (GWO) {{cite:26b2530d9979cfc65e36d65958bd6d09a427e7e8}}, Genetic Algorithm (GA) {{cite:d40392f7c6b7e29a35505e8c4d208ee19bbcbc8a}}, Harris Hawk Optimization (HHO) {{cite:52dcaba5386bfc34eba20be8fbbae4ab5e0ac6b0}}, Multi-Verse Optimization (MVO) {{cite:69876d472e767613d722904db72d8b87da6d3f51}}, Moth Flame Optimization {{cite:390ffba892898374f9d9c8944a1dc382aa2cb4bf}}, Whale Optimization Algorithm (WOA) {{cite:c8df49f047ad6a6ebc5866084d4dfa6be4534db9}} as shown in Table REF . The proposed method outperformed all the methods compared in this paper in 19 out of 25 cases in terms of fitness of selected features, which is reflected in classification accuracy. However, our model performed inferior in the case of {{formula:d4bf0798-bcd2-42f3-9e60-f4bc73d9ef97}} dataset with the best classification accuracy of 97.22% whereas most of the other methods were able to produce a superior feature subset, resulting in a classification accuracy of 100%. In the case of {{formula:be7a07e9-8ef8-4fc6-b04f-96f42a241394}} dataset, PSO, MVO, WOA and GWO performed the best with a classification accuracy of 76.62% as compared to 72.72% from the proposed RSO method. The MVO method performed the best in {{formula:c2f056bf-e490-41b0-8591-7134e52da23f}} dataset as compared to 78.67% from our proposed method. For {{formula:e2c7a869-ba9a-472b-8583-3af511c642ff}} dataset, all the methods produced a classification accuracy of 100% whereas the RSO method produced a result of 97.85%. MVO, MFO, WOA, and HHO produced similar results to the BSO method in {{formula:52751cee-b090-41dc-bc0f-8f71e3117663}} dataset with a classification accuracy of 95% as compared to 93.5% of our proposed method. In the case of {{formula:74338c83-7cbe-4c85-8082-c0ff41ab5575}} data, the GWO performs the best with a classification accuracy of 100% as compared to 98.31% from RSO. Our proposed method performs the best in all other datasets as shown in Table REF .
| m | 83a6246d85326902bf48faa87a20b57b |
The identification of K on MJD 57843 (March 31, 2017) is driven and supported by the following considerations. On MJD 57661 (September 30, 2016), when the source underwent another episode of enhanced radio emission activity (shaded gray area in Fig. REF ), the overall GMVA total intensity emission was dominated by component S1, found to be more than two times brighter than the core and S2 (right image in Fig.REF ). We interpret this finding as an indication that during that epoch the new component, K, is blended with and passing through S1 and that it emerges from the other side in the early months of 2017. Within this scenario, numerical simulations suggest that fluctuations in the position of a recollimation shock undergoing the crossing of a moving disturbance are expected {{cite:eaf8d6aacd1a4f929df709f7888fb5692f485a4a}}. We note that as K exits the S1 region and approaches S2, as predicted by simulations, both S1 and S2 show fluctuations in their positions with respect to the core (as is apparent in the left image of Fig.REF ). In more detail, from MJD 57661 (September 30, 2016) to MJD 57843 (March 31, 2017) component S2 moves away from the core from {{formula:99ca8dc6-8872-46bd-967d-4f327d8a92ac}} mas to {{formula:4906a016-10d3-40b2-9580-86dc5f5d8556}} mas, while S1 moves toward the core region from {{formula:8966230d-e169-4db9-989c-28fad8ba0fdb}} mas to {{formula:782f3640-a3ad-4e3e-8c2f-a24f44b74053}} mas.
| d | 8a67c2e3e08912d8ac67275e2c85e9d3 |
Using the isomoprhism between {{formula:1b823af4-43b5-46e4-9754-43e637980424}} and {{formula:3e4c5725-e4ca-4241-984b-1c755999f95f}} , we prove the theorem
in the context of {{formula:71cb0583-9035-48cf-8da3-00cd75676a64}} cocycles. Let us first introduce two auxiliary parameters
{{formula:735d421f-eef0-4efc-b8e6-77c57fd1c81a}} . Since {{formula:8fb25214-64bf-42d8-a408-80d662b68666}} , there exists {{formula:1690f5cc-1239-4fe7-b542-b4daca7e01ee}} such that
{{formula:383638ff-0f4d-41e5-b8dd-069bd5dd6266}}
and {{formula:dffd0bb6-d27f-4ae8-8b9b-6de8122ea96a}} satisfying:
{{formula:9df065c5-b594-4b24-ae87-7bf65e46607b}}
We will also use the auxiliary function
{{formula:066544ba-bb47-4d10-8973-033d1b632480}}
which satisfies {{formula:ed644e7f-b95c-49f9-b189-347d01dd1815}} , and
{{formula:96ae2151-2d27-4ba6-b87f-8c8ccce865cb}} and is monotonic increasing on {{formula:16c86dfe-d974-4501-9c0e-55b03aaab708}} .
We can now construct iteratively the sequence {{formula:c28fb621-b49c-4abf-9e09-7ad444eb022c}} . Let {{formula:0bdabe50-e917-4a0b-9b6c-f1e6808692e4}} ,
and choose {{formula:02c4852d-f457-49b3-903a-74c657bcdd4e}} satisfying:
{{formula:fc80282b-a84f-4ef3-aad1-3b9ad82a12f4}}
Since {{formula:de2289df-e342-4d87-8dc8-92817fb625c6}} , then there exist {{formula:457d3835-9cdb-4f17-8ab4-9ce007e2e082}} such that
{{formula:cb9e8892-2905-4f3e-bab2-b7206cc8e87a}}
with
{{formula:730053d7-5e66-4418-9d8e-7fd1a39d6a17}}
where {{formula:699b6e6a-a84d-47b7-ad82-b53468c014e1}} as in Proposition .
Now by Proposition , for any {{formula:39e11ef9-13db-48fc-9ff6-abc71e46202f}} , there exist
{{formula:cef451e9-667a-4fac-86f0-3bd7a1a49cb3}} such that
{{formula:e3e1c63b-a854-4b77-9bca-29146f0174be}}
satisfying
{{formula:f840d5a0-822c-4843-bd68-015ef710845e}}
Let {{formula:d820c9e0-698c-4473-99b9-85890e607195}} , then we have
{{formula:67b65f1e-4c74-4cc5-8300-bab10ca7ee15}}
We now separate three cases, following the regime to which {{formula:7d11bc1d-c278-42dd-9805-94807bb1c66a}} belongs.
{{formula:076c2be0-4f6e-4875-9252-0dfa6fbb1688}} : {{formula:0fc8df03-467e-400c-9a24-31c829772002}} is elliptic. Then there exists {{formula:a487ca91-6eed-4b28-aa02-8a72019be078}} such that
{{formula:4968ea28-1b5c-4fe6-9d5e-9e7ca51ba4f9}}
where {{formula:856904e6-6382-42bb-9892-0ca09c5c95c3}} . Since {{formula:64086d69-df5f-4eef-a52b-da1b8e4e9eca}} is rationally independent, there exists {{formula:0c48740f-fe04-4e60-b369-d4dca50aacc2}} such that
{{formula:11789663-d136-49b3-b2e2-12afb5862082}}
{{formula:71e29187-afe7-47d6-a1f6-6b1cc65f2a4f}} : {{formula:9f9a9f65-7e8a-416e-975c-2c4ca02a5b29}} is parabolic. In this case, without loss of generality, we assume the eigenvalues of {{formula:cfade701-552f-421c-b5f5-d35be0df0e89}} are {{formula:01a245be-ee24-4e2e-8fe2-564f3c8c255f}} . Then there exists {{formula:8e21c1ec-7631-4788-919f-ce4e0d6fed72}} such that
{{formula:f9b1e50c-11ca-4d9e-b69c-0a5bb4ea567d}}
Let
{{formula:d35a0f0f-cbe5-4097-b67a-b8500a964872}}
where {{formula:664a960b-347f-4934-976d-09fc6934b4cc}} , then {{formula:de8deed7-dc68-4ef7-bd40-5947722d6e73}} is
elliptic, and, moreover,
{{formula:5cb7cbe0-cc06-4312-bf6e-b5cdbf2ae73e}}
This situation has been transformed into {{formula:fa3c3b56-fc73-4ed9-ac56-4149bbe0a21d}} , which ends the proof for this case.
{{formula:a6573f5b-af1b-4d09-a46c-8138d50aa881}} : {{formula:5f764873-67d6-4f7c-b56c-8c0fb6a4cf46}} is hyperbolic. Let
the eigenvalues of {{formula:bdd18550-85ae-49e8-ae8a-2e9b0e8c61e0}} be {{formula:f2e82418-7ae3-414c-be6c-6e3031dfa0b1}} with {{formula:63e4a656-fd07-4d2a-84a4-b63def0eb4c4}} .
We first consider the case
{{formula:ddf202af-4b3e-43c9-9e90-807f1dc95aa8}}
In view of Proposition 18 of {{cite:961d76c6f6c25e2ee7d29515a393029432c8f889}}, there exists {{formula:e5131363-85f2-4af8-ab0a-fbf4c7a34a87}} , with
{{formula:15e22bcb-c744-466b-b6ea-84e7a2ff748c}}
such that
{{formula:3c55b6fc-b4f0-41e6-966d-55f09f462b71}}
Then
{{formula:1b4a505b-6f7c-4444-9a3f-ba8bc61785b9}}
with {{formula:c32b4f30-ee2d-4023-b1c8-eada549b300f}} , and
(REF ) implies that
{{formula:470f2ac9-ec5c-4c9f-8a34-1bf2a9dd1341}}
Consequently, {{formula:f5a04075-c396-472a-b6a3-34dcf2424195}} is uniformly hyperbolic by the usual cone criterion {{cite:bda52cf5ff872c7821c87033bb7fe12fe7944a56}}, which contradicts our assumptions.
Therefore,
{{formula:975fd92f-edd4-4c46-890a-94ac3857cfbd}} Consequently, there exists an elliptic matrix
{{formula:916aec9e-9d20-48f8-b6e0-94299aeb9d8c}} , such that
{{formula:013818a9-bf44-4c31-8b21-1fcf245f8748}}
Then by (REF ), we have
{{formula:85963724-c9e3-4f68-aa0c-7dccdcbaf9c0}}
which again transforms this case into
{{formula:aad6b504-35de-428a-a84d-d73c29a296bc}} , which concludes the proof.
Sub-linear growth of extended eigenfunction
Proposition 5.1
Let {{formula:9def9753-90ef-482f-9813-2cac2f534cda}} be rationally independent and fix {{formula:84f6dd8d-8d22-45b0-848c-13140b8801b7}} , {{formula:b58be615-ef3a-495b-91c1-bae760f8edda}} , and a
non-increasing sequence {{formula:ce225f5a-f2d1-4492-b6c4-4af1a18852bd}} satisfying {{formula:660e299f-c266-48c4-9e30-a250313a6cdb}} and
{{formula:ca3595fb-61de-443e-9df4-39e29cf18c29}} . Then there exists
{{formula:021ab238-1964-4351-b089-44084c96ee32}} which has
sub-linear growth with rate {{formula:3cf308b1-cf87-4477-848c-5048d9cbf363}} . Moreover, it satisfies
{{formula:106319f4-1991-4a02-9c73-4992bf5ec722}}
We construct {{formula:337347c7-cc54-466f-b4c8-2551815e183c}} iteratively. Firstly we construct the sequence {{formula:84836c3f-11bd-4ccd-95dd-de270b15e652}} . Let {{formula:2d4706ed-fe6f-4925-beb0-997faf2b81c9}} . Assuming we have constructed {{formula:0b75a123-7cdb-4508-9d35-59b59ab758b9}} . We choose {{formula:6d1c5dab-e0f1-4806-a530-5b1591399c78}} satisfying the following:
{{formula:ca01a241-6f65-46ea-9bc8-33e3965cbde8}}
Now we finish the proof that the cocycle {{formula:4a9507a7-e15b-4b9e-b2b1-b14f9dfc7179}} is not {{formula:48e03425-5748-4a80-8d75-a03cd8978e20}} -reducible,
proceeding by contradiction. Suppose that there exists {{formula:81dd0e23-709d-4f96-aacb-bc2f8416b4a7}} such that
{{formula:529d5ead-cd96-46cc-90b6-98d85191c32b}}
where {{formula:4911e07d-f362-4820-a375-0682836107d9}} .
Then, {{formula:44a8057b-c0d6-45a0-9381-8a8b30ab7722}} is diagonalizable. Otherwise {{formula:6e89bcf9-4b28-433d-a3e9-84e01fc5481f}} can be conjugate to a
Jordan block. Then the {{formula:a7b13736-d2df-42ea-ac3e-b0c97d65931f}} has linear growth on {{formula:d99d6cbc-2163-4703-b469-dad17b365fe8}} which contradicts equation
(). Combining equations () and (REF ) we have
{{formula:744029cf-136b-40d3-9739-93baa44ab08c}}
where {{formula:1e73d019-974d-4841-9e34-cc1e165e47cb}} . Define the linear operator on {{formula:3b88a499-d760-4350-9c3b-a8de10502870}} :
{{formula:c7d4e711-7a3f-44a4-bf2e-72f053ba706b}}
where {{formula:44e9b894-8fdd-4a1d-8b02-8f6dc2a1032c}} . Applying the Fourier transform to eq. (REF ),
we have that for every {{formula:2c460a3f-e2a0-4e85-8b46-dd079fad21a5}}
{{formula:9a8717cb-27ea-483d-81f2-f155fa74cb07}}
Since {{formula:075392d1-7788-4c18-bf19-7019a9873af9}} there exists {{formula:b1f59a4b-1823-49b0-875d-00d35c8eea17}} such that
{{formula:6f3eefeb-bfb3-4cfa-95fc-5636cac86241}} . Thus {{formula:5772713c-5a04-4f56-8a67-7221abefd905}} is an eigenvalue of {{formula:2e0ed26c-f66f-4184-a14f-60971ce3026b}} .
Therefore, the two eigenvalus of {{formula:d7a1cb3e-a539-4d66-9ae1-c1869c7cc05b}} are
{{formula:6036e9bb-3af6-4150-926c-d7ce454e2155}} ore
{{formula:6426ef28-ce0d-4b60-bb3f-857ecfbe00f2}} . Without loss of generality, we assume the eigenvalues to be the former. Since {{formula:1dfaa7db-2541-4fb5-b196-7993eac37732}} is diagonalizable there exists {{formula:ba62a980-7266-4d6c-821c-b4dbe4b7eb93}} such that
{{formula:756ae33d-4e39-438a-855f-b9e2a8b1f311}}
Let {{formula:ce3c1d60-e3ab-460f-8267-d8eed5723a76}} . We have
{{formula:fd135d15-f64e-478a-9c35-94512d18467c}}
Then there are two cases:
{{formula:24ed9c2d-b71d-4364-8c97-62fc5ce32920}} : For all {{formula:1c3b138c-b126-471b-901e-5cfd9f2a7401}} where {{formula:3d1400ef-17f0-4724-a3c9-888bd5d6ee06}} , we have {{formula:3f585a8e-f586-4736-857b-529dec1661d7}} .
In this case, combining () and {{formula:314a22f9-172c-49ea-a3dd-dbb2cd8909c1}} we have
{{formula:9d503bd6-9721-4b46-b5b9-0d18b5420388}}
where {{formula:3ef371ce-eb6e-4b04-8973-cc55b8111038}} . In the frequency domain, this implies that for every
{{formula:30e81b64-2206-428b-8521-64c6c8aea01d}} we have
{{formula:43168963-0bc9-400a-b7e5-5562e0685ce8}}
However, the eigenvalues of the operator {{formula:340222ec-e065-444a-837e-49ad1b7ad08a}} are
{{formula:66d0a62a-aa5d-43d7-a1ad-6e3c9923c6a2}} , and since{{formula:14bc6d9d-2f9a-4db1-953e-8cd86d9d7437}}
this implies that {{formula:fe4b38c9-04be-495f-9640-e21458bc6050}} . Thus we have
{{formula:8099e528-0781-4d8c-8ff7-09f4313c01c5}}
Due to {{formula:2bcb62a2-1d07-43c0-9fe4-ffe5c1952178}} and {{formula:a14ea321-8049-472e-ad06-06772dc58a54}} , we have {{formula:38623c40-fe2a-426a-ae9d-e353e1d51393}} which contradicts eq. ().
{{formula:a21b4042-b5cf-4ca6-b5de-59b9c5c5fdcf}} : There exists
{{formula:a8a4c6f9-fe5e-4410-96cc-40fd65c23db9}} , {{formula:200d647f-a167-41e0-b016-72ae70f66d92}} , such that {{formula:5276d24a-cfe1-421d-8e3f-cc79644a1cc8}} .
In this case, we just need to set {{formula:30a04b65-216e-4aaa-ae4c-af117735c8c3}} . Then we have
{{formula:534ef70b-f8cf-4a43-84b4-c0cc3bd06efe}}
This situation has been reduced
{{formula:d1a30764-c4ba-4601-97ae-c8daa81effbd}} , which concludes the proof.
{{formula:99a471c8-5853-4b44-bc19-38fa7473a80d}}
Proof of Theorem REF: The proof is as for Theorem REF , by
replacing Proposition REF by Proposition . {{formula:8ed54383-5e5e-4329-b40e-ba0e9b6c00bd}}
Proof of Theorem REF:
By {{cite:69382e6ecb8ddc4a97d7c55cbf4a01e242b39ebc}}, {{cite:dbf453389e147398f1b9887b20e057aa93cafa7d}}, for any {{formula:5261980b-9a9d-4e87-81d6-465d0b394a32}} , {{formula:ee19e86e-5197-487c-a117-557f7b7b95ae}} , there exist {{formula:23dff42c-d3d1-4f82-889c-d317b96775a1}} , such that if {{formula:fc749d16-9e75-4d75-b576-7ad7592f6dbf}} ,
{{formula:a25ee918-101e-40a8-a98b-ea2493cabac1}} , then {{formula:dd943837-b978-4ede-8aef-e6e168dfe76e}} . By Theorem REF , for any {{formula:44d905f8-20cc-40f4-b6c6-5825e7fd9e66}} , for any
{{formula:bc4aadf0-692c-4f6e-8744-2671d5f5e558}} , there exists {{formula:1fe04948-6358-4cae-b808-93981386e88f}} with {{formula:25034929-97af-4039-9b69-f6c9d7af0fc2}}
such that
{{formula:f28539ce-80a7-4bba-a949-8a998133fb94}}
Moreover, {{formula:e4585ee5-9b54-4132-8657-5198cfbc4758}} while {{formula:91c19f08-f4da-407f-bb46-4293f2d93ad1}} By Lemma , there exist {{formula:51ac9ac1-1468-42eb-98d6-de2bf048085d}} with {{formula:74322c44-8f6b-4169-853f-7bcde2c5c758}} and {{formula:b1bf6a52-6205-402c-81aa-f6df531f7e19}} such that
{{formula:bcdf58ed-aa56-48e5-97d7-9361e2ec6ba7}}
Let {{formula:e8963a9a-539d-4f7b-8577-ba44718ba471}} , then
{{formula:26e259c9-db2b-4b0f-a960-f6f4f96cb5eb}}
and write {{formula:dc65679a-e2a8-4b9e-8a01-40af7c88215c}} then
we have
{{formula:0beeff61-7a7a-4bc6-b46b-d6d7196474f9}}
Applying the Fourier transformatiomation to eq. {{formula:9bb2c16f-0757-4234-bfbb-ca6d3f8f7a28}} , we get
{{formula:ed0eb5a5-d563-4c60-88b9-27420d0edc8a}}
i.e., {{formula:63ff79ef-d284-4a86-b877-368d36236ca3}} , moreover, since {{formula:8637eca2-11b1-4f79-b3a9-98da82f49667}} while {{formula:b7de8fb8-fdb3-4254-87f1-95838c11b187}} then
the eigenfunction
{{formula:27cad1c8-1ba7-4b0f-83c8-7e35ca780be5}} .
Appendix: Proof of Proposition :
We follow the proof of the proposition as given in {{cite:dbf453389e147398f1b9887b20e057aa93cafa7d}}, {{cite:e6c2b4f1ac6e0918d19c8064e45ea95c0b401016}}. An alternative
proof can be obtained by use of the KAM normal form, following the method introduced in
{{cite:6d9e40db53a3e201de519628aba75a81644425b2}}. We need the following result, proved in {{cite:dbf453389e147398f1b9887b20e057aa93cafa7d}}, {{cite:e6c2b4f1ac6e0918d19c8064e45ea95c0b401016}}.
Proposition 6.1
Let {{formula:029068a5-c093-4cd1-9dd4-ce31b5dffc6b}} , {{formula:84e9f3e4-7886-4837-a005-721649773cf7}} . Suppose that
{{formula:1b83d2b1-4206-452e-bc45-c2bae1b737c3}} , {{formula:d37d24de-9fa8-475f-9587-9f22c6330b64}} . Then for any {{formula:df40e9fc-f84e-41c9-a2b3-b28560be20a6}} , there exists
constants {{formula:23c13e41-dc2d-427d-b400-ab3fb37ceada}} and {{formula:f8da3965-a4c9-4689-a62c-412372f154fe}} such that if
{{formula:b3a5ee08-6933-4b6d-8024-239c4c2f42ee}}
then there exists {{formula:75abd565-f550-4d3c-bb13-e2ded5232e07}} , {{formula:d7b04b0e-e792-4b02-85bc-b26bba66980c}} and {{formula:4d591077-7e00-4709-a664-dc51276565af}} such that
{{formula:fe40df94-fb37-4fa0-a943-15877ee8c18a}}
More precisely, letting {{formula:9995dc4b-d6be-43ba-bae2-f8fc989dfda4}} ,
{{formula:a5d91ee8-d2a9-4874-975b-24aab7765b7d}} , we can distinguish two cases:
(Non-resonant case) if for any {{formula:749ae8f5-f4e3-4c32-be21-5f6cd115dd4a}} with {{formula:88737e5f-2bf3-4de9-8dc0-15b73e16bc5d}} , we have
{{formula:ec4a9e11-e1e8-463f-90f3-ac875ece5110}}
then
{{formula:ab4594b6-45bc-4a67-b558-854d516594db}}
Moreover, {{formula:5786de2d-d951-49d0-bd8f-d73715c41309}} .
(Resonant case) if there exists {{formula:ef460f8d-ad0e-4b56-8bb9-3ff5708513b2}} with {{formula:cc56769c-589a-4696-95bd-00abe3a44479}} such that
{{formula:76af5bfa-822b-4f6f-a3c2-d344b1b4446c}}
then
{{formula:bc2b7db9-f260-42be-a797-a7f26c94acfc}}
Moreover, {{formula:9d9c1858-2f28-486e-b93a-5acb072de033}} can be written as
{{formula:c93ff4cb-1709-43a3-b64c-052e1f187dad}} with {{formula:50bf2fef-edad-4b95-ae71-7ef5deaba26e}} .
The proof of proposition follows by iteration of the proposition here above.
Consider the initial cocycle {{formula:3114f7f9-e364-475a-bd0f-0c26346f66cb}} , where {{formula:b76d7c52-9d86-43b7-a990-b768f43eba49}} ,
{{formula:aeafb52c-91be-4ba8-865f-93eb3d7b55ab}} . Without loss of generality, assume that {{formula:7628c722-612f-42ae-9e4b-5e7537d22dc1}} , as well as that
{{formula:de341fcd-b560-4583-909a-77a43343f065}}
where {{formula:ee0e6296-1226-4853-a553-a95a7ebba258}} is the constant defined in Proposition REF . Then we can define the sequence inductively. Let
{{formula:bb1f08bb-1267-40e0-a084-e471879b151c}} , {{formula:b8e2c71a-8431-429a-8e3e-b52e081cefc6}} , and assume that we are at the {{formula:6889fc7a-ecee-4dc2-87ff-8831f709794c}} KAM step,
i.e. we have already constructed {{formula:9e7c82c4-91c1-4e33-a3ca-2ef57732cc96}} such that
{{formula:90a3b96f-4c86-4f84-9efb-5b28068aa058}}
where {{formula:bf3e93f0-2e7d-4e02-bf13-94c2e000a051}} has eigenvalues {{formula:e6fd64f7-29e3-45dd-8e6d-fa11a8a0873f}} and
{{formula:83534505-07b5-4ef8-b93d-5a3bbe4f322c}}
for some {{formula:e72409e4-051b-4d2b-866f-5d448c2854bf}} , and define
{{formula:227a1e8e-f0f0-43d7-be68-a6729e168aa8}}
By our choice of {{formula:b715ac44-e3f5-416a-8a15-ce594db25e09}} , one can check that
{{formula:bbcfc5c1-4220-4cf5-b4b8-e47d4aa05c35}}
Indeed, {{formula:64f6c942-bd4f-4307-b5f7-5714fc2905e0}} on the left side of the inequality decays at least super-exponentially with {{formula:bf62b167-a3de-40e4-905b-5a6782fb8771}} , while {{formula:4deabedc-8c55-459b-9cfc-e5af18a04c9b}} on the right side decays exponentially with {{formula:03aa7723-d270-4855-951b-dcbd7ed1574a}} .
Note that {{formula:40d9db88-662a-43c2-a32c-4bf5cbd1155d}} implies that Proposition REF can be applied iteratively, consequently one can construct
{{formula:a40773f3-f9a6-4ef9-8135-789fbe15a9fe}}
such that
{{formula:82fe841c-7c8e-48c2-9b25-574a0f4d3401}}
More precisely, we can distinguish two cases:
Non-resonant case: If for any {{formula:56b2eff9-9944-463f-b0fb-5d63d50a5b9f}} with {{formula:3308fc61-01af-41ad-bd51-f80339f576ad}} , we have
{{formula:e370ee27-1d61-4bed-9523-3860de6f87dc}}
then
{{formula:1ed920e4-f48b-4ce4-9e00-ca74f1243ab0}}
Let {{formula:b34b6982-5bce-4294-b6ad-39614b430cf9}} , we have
{{formula:067b50c8-229a-4bf2-b48d-c6a686af7468}}
with estimate
{{formula:28778705-8821-46ce-afe6-050201f0775b}}
Resonant case: If there exists {{formula:25ff0c40-536d-435b-8f65-64b406713122}} with {{formula:ddf80c08-32b7-4a55-9b99-97430e2a4acf}} such that
{{formula:e9c6370d-15cf-4120-8407-284f52d799a5}}
with estimate
{{formula:de264711-e83d-4fef-89e6-cd9ae148ade2}}
Moreover, we can write {{formula:536cf093-f146-4c90-a2fb-f19cebe1fac7}} with estimate
{{formula:e65f0f38-9b1b-440b-af0d-aae71b0bdc29}}
Let {{formula:0de8605a-a7cc-460b-9839-69f18c9f1e64}} , then we have
{{formula:2fd324bc-0d2a-41fd-a07d-24a0a5865b74}}
with
{{formula:43d78915-e75c-42de-86b6-35e72342d239}}
The last inequality is possible since by our choise {{formula:3daa407b-825e-41c7-8bac-4722ed313d46}} .
Acknowledgements
N. Karaliolios was partially supported by LABEX CEMPI (ANR-11-LABX-0007-01) while a post-doc at
Université de Lille. He is grateful to his co-authors for their warm hospitality at
the Chern Institute. Q. Zhou was partially supported by support by NSFC grant (11671192,11771077), The Science Fund for Distinguished Young Scholars of Tianjin (No. 19JCJQJC61300) and Nankai Zhide Foundation.
| r | d1a7d824a685c72877b0ce9eb3c420b4 |
Computational imaging for geoscience: End-to-end learning strategies have become the state-of-the-art approaches for a variety of computational imaging problems, including among others denoising {{cite:d0e6a1b99d61c7bdcde9a6fc77a3390e2e06e7ef}}, {{cite:677e2e2962638d214f5bdeaa2ebcfe2a6fc249db}}, super-resolution {{cite:57565de9593d39dfb44f14e2397a8f90dfb2fc70}} and inpainting issues {{cite:2fe68cf783496a033f36fcbc0c4451b4b51bb3b3}}, {{cite:d5a8b3ecc35a39291ea483ee84d5f9c14d1587ea}}. While numerous applications to the observation and monitoring of geophysical processes exploit state-of-the-art deep learning schemes, the underlying physical laws naturally advocate for the design of physics-aware approaches. This is particularly true for space-time interpolation issues with very high missing data rates as targeted in our study to the reconstruction of sea surface dynamics from satellite data. We may also point out that video inpainting remain a challenge for deep learning {{cite:932e0c1f6c75b82d719c4df3656e9bc2e2385fff}}. Here, we exploit a classic inverse problem formulation to design our neural architecture. This allows us to make explicit the definition of the observation operators and the dynamical prior. As supported by our numerical experiments, the former is key to fully exploit SSH-SST synergies to improve the reconstruction of sea surface dynamics at finer scales. The proposed multimodal 4dVarnet scheme also relates to deep unfolding schemes {{cite:681c511738278870f2c42b6e2f45c3224c24984c}}. Our neural architecture implements an iterative gradient descent of the trainable variational cost. Rather than considering a reaction-diffusion formulation {{cite:ad35dd0e451bf449c15301d1d4897400d37ed3c5}} or an optimization scheme associated with proximal operators {{cite:d0470abdb4bb19702930bcfdcade520e01cc7e0b}}, our framework exploits the embedded automatic differentiation of the variational cost with a trainable gradient-based solver to speed up the optimization. This strategy also allows us to train jointly linear and non-linear observation operators with the dynamical prior and the solver. This is expected to contribute to reducing inversion biases {{cite:9492fba348aea5307e9e7a3a62ee338caf0a8709}}, {{cite:d840678d1b239ad42204f5f3d101703e8ae42428}}, {{cite:3b38d0898fc3e00fb614f845e6ecea1de0227bf4}} and improving the overall interpretability of the neural architecture.
| d | 29e76b6d50487a18e576f33e29bf02e3 |
DMFT for gradient dynamics:
In machine learning applications, the dynamics of RNNs at initialization is critical for training success.
Thus studying the autonomous dynamics of random RNNs, provides insights on training performance. Indeed, there is a deeper connection between the forward dynamics and the behaviour of gradients. The adjoint formalism introduced by Pontryagin et al. {{cite:f0aee456e12c229f4891527e6c5d07d65adb87e3}} explicitly shows that the (exponential) growth or decay of gradients is governed by the same state-to-state Jacobian which governs the local behavior in the forward dynamics. It is also known from empirical studies that one of the key difficulties in training RNNs is the problem of exploding or vanishing gradients {{cite:9530bb9ef0977a825a4d4eef38e1f9c002ea5493}}; this suggests that for efficient training we need gradients to be stable and {{formula:80a243a5-e7ab-4045-a307-f291d926deff}} . The theory suggests a way to achieve this – train at parameter regimes where the Jacobian eigenvalues are close to zero i.e. near critical lines. This is a more fine-grained perspective on why it might be easier to train at the “edge of chaos”. In the case of our gated RNN, in addition to critical lines, we also know that the update gate can make many parameter combinations marginally stable. The arguments above would then suggest that initializing in the marginally stable regions should be beneficial for taming the gradients during training. We have combined the MSRDJ field theory formalism to incorporate the adjoint framework for gradients, to to formulate a DMFT for gradients. The DMFT is typically, a tool typically used to study forward evolution, and our extension allows the powerful techniques of the DMFT to be used in the analysis of gradients. Fully exploring this connection will be undertaken in future work.
| d | 704df8b2007230a978c245aa442265cf |
In this section, we compare our method with {{cite:eae943f399d5bd0b42806819ca6731ea07da4652}}, {{cite:6af36245b36c5e9502c425b3dd7505be4bee5bfc}}, {{cite:3538b2cddd066cc4a7aaa38b0f51262085e16578}} on the synthetic face and CelebA dataset. Since the illumination information in unknown in CelebA dataset, image-based relighting is used to evaluate our proposed method and baseline methods in Fig. REF . More specifically, target lighting used for relighting is extracted from a reference image. The proposed method, SFSNet {{cite:eae943f399d5bd0b42806819ca6731ea07da4652}} and DPR {{cite:3538b2cddd066cc4a7aaa38b0f51262085e16578}} can extract the lighting of the reference image with its own lighting estimation method. Li et al. {{cite:6af36245b36c5e9502c425b3dd7505be4bee5bfc}} is a state-of-the-art portrait style transfer method, which takes the source image and reference image as input and relights the source image as the illumination of the reference image. Since Li et al. and DPR are applied on the L channel of the input images, the input images are transferred from RGB image to Lab image. Our method and
SFSNet are applied on the RGB channels of the input images. We notice that the edge of the relighted images of the proposed method is underexposed and some detail of face is lost. This is probably due to the gap between the training dataset and testing dataset. The results show that our proposed method can generate high-quality light estimations (1024 {{formula:49782b93-a021-4073-8c21-1af3e54393cf}} 1024). Compared with DPR and SFSNet, our proposed method successfully overcomes the over-exposed images and the over-lighting problem of the nose and eyes. Li et al. do not generate images under the correct lighting. Since Li et al. uses reference images as input, the high-resolution images would improve the relighting performance while the low-resolution images can not estimate the reference lighting accurately. Since the illumination information is known in synthetic faces, relighting based on the SH is applied to evaluate our proposed method and baseline methods by comparing with the ground truth images in Fig. REF . Our method can provide more face details and does not exhibit the over-lighting problem. In the second column, the skin colour of our relighted image is closer to the target image, since our proposed method input is a colour image, while the input to DPR is the L channel of the source image only. For further testing our proposed method, the RMSE-s between the target image and the relighted images of the different methods are calculated. The RMSE-s of ours (8.4 {{formula:2d615863-ab54-4960-8362-8449b3fb9c49}} ) is only 79{{formula:a37c8202-3d29-48fd-a88d-ec175c7d85a0}} of the RMSE-s of DPR (10.6{{formula:1757f7d0-16fa-47a2-85b0-0e52cbe56b22}} ), SFSNet (10.8{{formula:859b1edb-abea-4856-a0c5-bb7ff008205f}} ) and Li et al. (11.3{{formula:5f76ba2c-c7e5-4556-8037-e0d0b5d3ac38}} ). We also evaluate our proposed method on Multi-PIE shown in our Github page, due to the limitation of pages.
| m | 75bcf2eb5f48e0c89bb63cb77a2029ce |
For the comparison methods, we adopt the implementations from DomainBed {{cite:ed1c24dd84ebffbb7f982d9a6ddca96794f32c20}} while we change their network structures to be the same as ours for the comparison study.
We perform hyperparameter tuning for each comparison method to achieve its best performance on each task.
Specifically, we tune the following hyperparameters: learning rate is selected in {{formula:f657e044-fb74-43f0-b88d-c3c719cef543}} , batch size is set as 128.
We set the total training epochs as 500 and early stop patience to 30.
We run the experiments five times and report the average results.
F1 score is the main evaluation metric, and we also analyze the accuracy, precision, recall, and ROC curves in detailed analysis.
{{table:c00b3a06-0dd9-48ed-83fb-07f380dc54dd}}{{table:6808afd2-d995-45e7-83c6-e0299c90ce2d}}{{table:2746ec1d-2be7-435f-9581-460295374c50}} | m | 5edb72f0e375ec2fe34a502309491110 |
Even though our method leads to promising results, more work is required to investigate optimal ways to combine transformers and convolutions into strong GANs.
To the best of our knowledge, there are currently only two other GAN approaches that use both transformers and convolutions, GANsformer {{cite:0451bf5617cdd96619243a9b0e60b767daaa133a}} and VQGAN {{cite:95681aa4b36eae0e3d01343100c5e851f720bef0}}.
However, they have completely different setups.
While GANsformer and VQGAN integrate self-attention layers in-between the architecture in a sandwich like way, we keep them separated.
In particular, our approach consists of a purely transformer-based generator, and a fully CNN-based discriminator, thereby constraining the interaction between attention and convolutions.
Hence, our approach maintains relaxed inductive biases that characterize transformers in the generator, while leveraging the useful ones in the discriminator.
Last but not least, our frequency spectrum analysis has brought new insights regarding the impact of transformers on the generated images.
It shows, how a pure transformer based GAN framework, such as in TransGAN {{cite:fdc3298a940ebff90c061220237b168cd6a65ec2}}, seems to learn the frequency components in a more accurate manner.
Our hybrid model is able to maintain the well-matched spectrum, while achieving better or similar scores without requiring additional training constraints.
We think that these findings can lead to a new paradigm, where both transformers and convolutions are used to generate images.
| d | 0314ea9865b34db5a608994a5fda9ede |
During the pair pulsation, the progenitor of a PPISN may be a red supergiant
(RSG), a blue supergiant (BSG), a luminous blue variable, or Wolf–Rayet
star if the hydrogen envelope is completely shed away by continuous mass
loss {{cite:f36d0fe5854a80197f7bbe884b1fbdf31a74b019}}. The gravitational binding energy of the hydrogen
shell that will be blown away by the pulsation is
{{formula:d885fbb1-6dd8-40c6-9faa-124cdd9ec264}}
| d | 9b70a4b6425efad1581312910d02c653 |
In 1995, Dixon {{cite:f6d967d383a710438d9f0073690621b7f7f5da62}} showed via methods related to Bohr's inequality for power series {{cite:a813bbed799d55b95c4bdbfadb72bcad5f608bfe}}, that there are Banach algebras {{formula:3ee984d3-de62-483a-8ce1-7faa59a6bf67}} which are not isometrically operator algebras and yet satisfy a special case of von Neumann inequality: {{formula:a8c86f36-1a3d-4684-9111-a98ce447f55e}} for all {{formula:907e0699-4009-4eca-9728-8a00ab706b9d}} and polynomial {{formula:f96f5749-4461-42c6-be96-57acdbc621a5}} with {{formula:cbfab3ad-e624-4ed2-8e68-5765ce1a94a3}} . In the recent years, the generalization of Bohr's inequality for various classes of analytic functions becomes an active research area (see {{cite:fdccab3e7aadac4cd92ee4667a323b9313b77dd7}}, {{cite:2692b41f2e04ab75c7096a9ea4662f16f75f19f2}}, {{cite:a8c44091ad3792d899c8393c7d29c6b87107dfc3}}). The Bohr radius, for the holomorphic functions has been studied by Aizenberg et al. {{cite:72249b5aab4ab0e3a1be19d91f67cbf24e5e6d5d}}, Aytuna and Djakov {{cite:8fffd9c916ee15d7bf4f97c7997e932085b47d89}}. The Bohr phenomenon for the class of starlike log-harmonic mappings has been studied by Ali et al. {{cite:7939ae46b1e818da306ec20a5fc3d8f256ee81ea}}, and Ahamed and Allu {{cite:ee6591e222901709e748e88dbd3bc0e65655f22d}}. In 2018, Kayumov and Ponnusamy {{cite:fdccab3e7aadac4cd92ee4667a323b9313b77dd7}} introduced the notion of {{formula:aa3dc1f1-05d2-4ec4-9f5d-5fed3f67af13}} -Bohr radius for harmonic functions, and established result obtaining {{formula:c5f8c381-e0fc-42d3-8d7c-8425499656c7}} -Bohr radius for the class of odd analytic functions. The improved Bohr inequality for locally univalent harmonic mappings has been sudied by Evdoridis et al. {{cite:ba187cbd56f963b7ffe6497aaca6f1f927702190}}. In 2021, Huang et al. {{cite:b55b875207cadb702152099ae34fd0864a12ae38}} determined Bohr inequality for the class of harmonic mappings {{formula:64a2db44-dadc-4208-a033-b707a5fc2a36}} in the unit disk {{formula:62cb5c37-4509-4639-a9e9-7bc88832b941}} , where both {{formula:2853aa07-6754-40f9-afad-16c631c52312}} and {{formula:8a49f755-3720-4da3-a5d1-41a229e5e684}} are analytic and bounded in {{formula:5a17a053-5f79-4533-8c3c-c411c978f301}} and also investigated Bohr-type inequalities of harmonic mappings with a multiple zero at the origin. For more exciting aspects of Bohr phenomenon, we refer to {{cite:d2341fe8042bb2944f1c1ebf2f4ac83fff7d892d}}, {{cite:586e03490eb73d57f18390a89bb274211f2c619d}}, {{cite:bf7f5bc922fa804aed5b4d13ba82a94e1dda2cf2}}, {{cite:cc5ee74f8b8102cf75a992c70bf6911ef2dcd996}}, {{cite:6a9425560e5669c0d4439ffce801cba2049ca5c8}}, {{cite:ba187cbd56f963b7ffe6497aaca6f1f927702190}}, {{cite:910553ebb49bb211075d6b0b6dabe5504bb1f51b}}, {{cite:15fbf6623cd0267ecfa0dcc1084986cbb87d6146}}, {{cite:9571d3cf2c86b416cee253af3bf50cbb5fb5c7b6}}, {{cite:2692b41f2e04ab75c7096a9ea4662f16f75f19f2}}, {{cite:71830b4485bbc8baa03401597d6474fa3bfd0151}}, {{cite:87ce760cd89431c2fde3cff69ceb859b20df9681}} and the references therein.
| i | b89fc746d9c8ca72f8f36fcd4631a306 |
Our multistage classification method and potential scope for improvement. Our approach offers a general framework consisting of multiple algorithms in different stages: changepoint detection; the random forest algorithm in first stage classification; mean-shift hypothesis testing in second stage classification; post-correction algorithm.
Moreover, the framework is very flexible as we can adopt more advanced algorithm choices at different stages of classification than what is used here.
We mention a few scenarios where further improvements can be made for each of the algorithms above.
High FDR for changepoint detection. As noted in section REF , results from both ecp and changepoint algorithms suffer from a high FDR, making the segmented series not identical to those produced by true changepoints, and therefore potentially affecting the performance of classification in later steps. The discrepancies arise partially because the idealistic assumptions of these methods are inconsistent with the noisy nature of the actual data. On the other hand, machine-learning based algorithms {{cite:f98f2e9b1eea600f06d8511d29e7f4d0c96bc3cc}} can be trained for detecting changepoints as well, and have the potential of being more robust against model misspecification, making them a better candidate for changepoint detection. Even so, our simple model-based assumptions which are visually verifiable, make our approach easier to interpret and plausible.
Deep learning based algorithm for first stage classification. Neural networks such as CNN can automatically select meaningful features from raw input data without feature engineering, which could potentially boost classification performance.
In the recent work of {{cite:bfd869d46382ca4c6aee88b3d6bd6ef2e5d61b0f}}, a CNN architecture is
proposed to classify 3-second window observations into sitting, standing and stepping.
They showed that CNN achieves higher accuracy than classical machine learning methods, such as logistic regression and random forest methods.
Their best result in terms of balanced accuracy is 83.3%, which is slightly higher than ours.
However, their result is not directly comparable with ours, as the granularity level in {{cite:bfd869d46382ca4c6aee88b3d6bd6ef2e5d61b0f}} is 3 seconds where the data considered must have consistent label in each window; on the other hand, our accuracy is calculated at the finest level, 0.1 second.
Mean-shift hypothesis testing using triaxial data. Our implementation in the second classification stage involves producing confidence intervals for the standing mean in one axis – one with the most significant "mean-shifting" phenomena. Even with this selection, we have found that there are some data whose degree of "mean-shifting" is not significant in any axis. In this scenario, it is natural to consider using information of all three axes. However, the choice of "confidence region" for the mean vector is more complicated as there isn't a natural ordering in three dimensions. As a result, different shapes of confidence regions need to be considered, and we leave this to future work with larger number of samples.
Validity of the proposed model.
The proposed model REF is based on our observation of the data and also the theoretical behavior of hip-worn accelerometer under different postures. However, we also observed a few data points which exhibit discrepancies between the proposed model and actual data.
Examples for the violation of model assumptions include (1) gradual shifting and rapid rotation of hip-worn device causing shifting of mean component within one daily file; (2) inseparable sit/stand mean components due to poor device positioning;
(3) significant noise during sitting epochs due to various reasons; etc. While we were successful in addressing these issues to some degree in the post-correction step, there could be more dedicated algorithms for resolving the above irregularities. For example, for (1) and (2) we have experimented with an algorithm utilizing a {{formula:c554e8fc-dc7d-457e-9688-426dffa961e8}} rotation matrix such that the rotated data have both stationary and maximum separability of different mean components. Again, we leave further consideration of these geometric transformations to future work with larger number of samples.
Generalization to other datasets. The dataset we use comprises of older women who are breast cancer survivors. We plan to investigate if our approach can be generalized for a different population, such as different age, biological sex or race and ethnic groups.
| d | 860c8c62a5bf4cf2f3a59a701e8e6dc6 |
Unlike solids and liquids, gases do not emit and absorb energy in a continuous manner but, they emit and absorb energies in discrete bands at resonant frequencies which give dark or bright lines on a spectrum. Further, the spectral lines get broadened and partially overlap each other due to internal collisions and the other effects. This makes spectral absorption coefficient calculation a challenging task which in turn makes the calculation of radiation heat transfer extremely difficult. The accurate calculation of absorption coefficient can be performed using some of the most popular available databases like high resolution transmission spectroscopic molecular absorption database (HITRAN) {{cite:5aa231d5f9529c343e0d9080a89a514949c861ee}}, carbon-dioxide spectroscopic database (CDSD) {{cite:d50b45e74971fccfa129560cdf79502dec0088e0}}, high temperature spectroscopic absorption parameter (HITEMP) {{cite:ecafa2bff04820f374b92a0b1f0b6671d6e9c57f}}, {{cite:0032f5e8cadd5ceed991e291bd4086fa93aed61b}} etc.
| i | e701dbbf44e6911ce120acde0478839f |
The limiting luminosities implied by our measurements can be
interpreted in a context of high-energy radiation possibly emitted
by either an expanding fireball/jet or by a remnant left over from
the NS–NS coalescence. The latter hypothesis is more interesting
in terms of constraints that can be obtained by the AGILE-GRID
observations. If the remnant is a magnetar-like system loaded with
a residual magnetic field and rapidly rotating {{cite:03fdcf51ba58f8866d60e798efa3dbe1cf57e3a5}}, {{cite:f1654fab0ae022bede44c8797c2ff286303fbdce}}, {{cite:b966788e8c4eab78bbc555862a460fb9c418eae4}}, {{cite:d9488b17ef424f08c39c01e1085087efa46d8c27}}, {{cite:99e4fe1ef1a6c49ea969edf443ac336220e9fb0d}}, we can
constrain its magnetic field assuming initial millisecond spin
periods. The electromagnetic emission (EM) by magnetic dipole
radiation of a star of radius {{formula:a573f5cc-ee53-4535-b740-a9f660b6b284}} with a poloidal magnetic field
{{formula:cc53609b-e295-41bf-b166-cefdab67a30c}} and angular frequency {{formula:795895be-f232-438c-91d6-adcc2647515f}} (with {{formula:83e382ca-1c06-46c1-a462-5913872be547}} as the
spin period) is {{formula:024e73ce-9c30-46dc-b231-97fa32fe58d4}}
(with {{formula:9100d3ab-1ba9-45c6-a277-f79c03a9c4b4}} as the speed of light). Neglecting GW radiation at late
times after coalescence, integration of the energy loss equation
leads to the dependence of {{formula:3502d535-a2eb-487f-8629-e97534513246}} as a function of time, {{formula:43e669cd-5ecc-4fcf-a8a9-9d37329299f3}} , with {{formula:6be83f41-4451-4017-a1a1-2196818dcb49}} as the initial
frequency and {{formula:e113ea60-bd28-4b5b-957e-f4dd58472c2a}} , where {{formula:a5c698ac-357b-409f-aa88-cbee9993969d}} is the compact object moment of
inertia in units of {{formula:db5b2693-6dcc-497c-811f-622460c7940b}} , {{formula:7eac08ef-6855-4e32-bad2-1843cc5180e3}} , {{formula:d2a2c35d-f7fc-4d8d-ae3f-83dce660eaae}} is the initial spin period in
units of {{formula:b2210023-d453-41a1-b836-6f5448315c0a}} , and {{formula:d60f04e0-4b44-424f-84a6-dacb0f284bc8}} is the radius of the
compact object in units of {{formula:33bdbc8f-5421-4998-8c8b-1aef7e60c06c}} cm.
For the EM-dominated regime of energy loss, we obtain the temporal
behavior of the spin-down luminosity {{cite:99e4fe1ef1a6c49ea969edf443ac336220e9fb0d}}
{{formula:9df614cc-c83f-4edb-8c85-33be5da5c31a}} , with {{formula:56975186-ece3-4e6d-a320-75aa36bf602b}} . It is interesting to
note that in the absence of absorption effects, the radiated
luminosity (in first approximation assumed here to be isotropic)
has the limiting behaviors {{formula:691889ca-aff4-4bbb-8508-dbaa1f2b4f4f}} for {{formula:8b34b7e8-ce17-4ba1-8fa3-4318880b3d97}} and
{{formula:bf5eb2a8-0606-42a5-97f2-6d8d81e222bc}} for {{formula:11339249-fd13-45b1-bced-e350e54b1920}} . In our case, we
can assume the remnant of radius {{formula:0407d2d9-3403-41d2-be09-ce28bef15b19}} and moment of inertia
{{formula:1e63b7b7-35ae-4ed1-a35d-c089094e9364}} to rotate with an initial millisecond spin period,
{{formula:5a91eb72-01c8-4c16-93fd-c6a77f9d3135}} . We then have the critical time {{formula:913af5e0-211b-4b82-9216-ff03eed475d0}} depending
only on the surface magnetic field {{formula:aff954d5-9dd0-4fbd-8d7b-fa416c138bbb}} .
| d | cbc24a50669dc32186ae04f7818d72ec |
Action Localization with Class-agnostic Attention. STPN {{cite:c65e6df9dffdcdb64edfe14bdaa6f8e56af26f1c}} proposed to learn attention through class-agnostic features but has a low performance as cross entropy loss alone does not train accurate attention signals. BG modeling {{cite:92c6471f07c8a262ef385ab676669af76c89f79f}} used a clustering loss to separate action from background. BG modeling {{cite:92c6471f07c8a262ef385ab676669af76c89f79f}} and BaSNet {{cite:12f3bf9ab42d110a278cfa5ed64847024722ba8a}} force all background frames to belong to one specific class which is not desirable as they do not share any common semantics. RPN {{cite:c6812e8ac344e67803d5182e034658b95fb53d07}}, and Huang et al. {{cite:2b8facd22dfbf785ba1007657835ed3ab022522c}} increase inter-class separateness by pushing action (or sub-action) features to their prototypes. Huang et al. {{cite:2b8facd22dfbf785ba1007657835ed3ab022522c}} outperforms RPN {{cite:c6812e8ac344e67803d5182e034658b95fb53d07}} by modeling the relations between sub-actions of each action. DGAM {{cite:34578affe87a9557ba9ab4a6ad1e70f1c6cefd02}} addressed the action-context confusion through imposing different attentions on different features with a generative model. EM-MIL {{cite:b5c43687b99c78eb3d440eb99bf1092a3c9e2f14}} employed Expectation-Maximization to capture complete action instances and outperformed DGAM {{cite:34578affe87a9557ba9ab4a6ad1e70f1c6cefd02}} on THUMOS14 dataset.
| m | 1a665cbfd6613fad65f3429eebbf80f9 |
Acknowledgements
We thank Ben Maybee, Donal O'Connell, and Alasdair Ross for many discussions on the KMOC formalism. We thank Ben Maybee and Donal O'Connell for useful comments
on the manuscript. We thank Fernando T. Brandt for useful correspondence and Luis A. Hernandez for informative discussions. This work was partially supported by the
STFC grant ST/P0000630/1. The author acknowledges financial support from the Open Physics Hub at the Physics and Astronomy Department in Bologna. Our figures were produced with the help of TikZ-Feynman {{cite:fc8e4c4ec850f2cc39a26be431154f3ae6d4b727}}. Some of the
calculations in this paper were done with Feyncalc {{cite:95f60866f27ff007771f46a37f5a12cbeabbab11}}, {{cite:c91779432f3d74b4521563acc2e2a528edafbc72}}, {{cite:a5aed6ace9d6319b9802baa2d79a4a725b5806a7}}.
| d | 99065fd0726f99eb90288906821c0bed |
Unfortunately, existing methods for irregularly sampled are not applicable to our problem. Approaches based on least-squares spectral analysis (e.g., the Lomb-Scargle (LS) periodogram {{cite:fb03cd3a133cfe66000b942560cefbfaefaa7593}}, {{cite:442e59ef3f0e0036219a6b4546ec966b8029ca9c}}) are oriented toward frequency-domain analysis and can't be applied directly to clustering, especially of sparsely sampled trajectories.
Model based estimators {{cite:d140dbe11abfaf7018ea35fd9f6da5d35586b4ef}} would require us to propose a model for pain dynamics before clustering.
| i | cd24b4f681adf519f1d545190977196d |
The most general statement of the Alexander method was formulated by Farb–Margalit {{cite:8c1de151f2b1d87d05bd4a6f98a1389956363c13}}, but the method can be traced back to the works of Dehn {{cite:fe6f848348cf8b9d6dd8440d4a922977519ab25f}}, {{cite:b369aecaa1aacf5d3f7e39771cefcbead6b47081}} and Thurston {{cite:acb8906bdee5fd04a5441140da0316b2c8c93bee}}, {{cite:7d301e64c0a210880746c9542f007355f4ddf322}}. All of these earlier results treated surfaces of finite type—that is, surfaces whose fundamental groups are finitely generated.
| i | 4abc0ade28654b5b53c3581ece174af3 |
To this end, by regarding rotation as an attack, we develop the ART-Point framework to improve the rotation robustness by training networks on inputs with Adversarial RoTations.
Like the general framework of adversarial training, ART-Point forms a classic min-max problem, where the max step finds the most aggressive rotations, on which the min step is performed to optimize the network parameters for rotation robustness.
For the max step, we propose an axis-wise rotation attack algorithm to find the most offensive rotating samples. Compared with the existing rotation attack algorithm {{cite:26d410454decbdc22bf452fc3cb49ebd027087bb}} that directly optimizes the transformation matrix, our method optimizes on the rotation angles which reduces the optimization parameters, while ensuring that the attack is pure rotation to serve for the adversarial training.
For the min step, we follow the training scheme of the original classifier to retrain the network on the adversarial samples. To overcome the problem of over-fitting on adversarial samples caused by label leaking {{cite:d0cf910d2806fac59ba0c96367ee296575806d7f}}, we construct a rotation pool that leverages the transferability of adversarial rotations among point cloud samples to increase the diversity of training data.
Finally, inspired by ensemble adversarial training {{cite:985eac2107a7fe45b209218f8969070fc82ce0bf}}, we contribute a fast one-step optimization method to solve the min-max problems. Instead of alternately optimizing the min-max problem until the model converges, the one-step method can quickly reach the final robust model with competitive performance.
| i | 633b009e831c2626189dc1bf07c2142c |
One must apply the Barrow entropy for analysing its associated phenomena because of the long-range interaction behaviour of gravity {{cite:9a31cb0ed7599a753f063a56c36523b1491c2e40}}. This research is motivated by the Barrow entropy {{cite:55fae27c84dc7171f2b5e5e56b3628ba1a4fab90}}, and based on the holographic theory, we introduced a new dark energy model with a time scale as an infrared cut-off. We have considered the IR cut-offs of the model as the age of the universe and the conformal time. During the cosmic evolution, we examine the nature of equation of state parameter {{formula:fa65d09e-bba6-4e83-8cd5-4124fda56024}} , deceleration parameter {{formula:7d95a4e4-7982-4593-97c7-fa77b1c07ffd}} , BADE energy density parameter {{formula:feaed311-d734-49e3-b224-9dce431d91f3}} and the squared sound speed {{formula:7cacf062-2475-4964-ade3-8ee818b833e8}} . We find that both of the presented models are classically unstable except the non-interacting NBADE for {{formula:b426a406-f6ed-4c86-9df5-e9cda4b88546}} and 0.4. Moreover, we discuss the importance of the presence of mutual interaction among the dark segments of the universe. We observed that the interacting models proposed here are classically unstable rather than the original TADE model based on the Tsallis entropy {{cite:e8c0202f4766ae23769d7c1fc865d420d4a7e24f}}. To resolve the late-time DE-dominated universe, both the BADE and NBADE models may be beneficial. Our research points out that for the cosmic evolution, the forecasts of the models are more sensitive to {{formula:f0844786-397a-47b5-baf2-59b928a2a0c8}} rather than {{formula:9308dcde-0376-4f81-bb66-aab2011fc2b6}} . We can achieve this by relating the identical curves, and this influences the primary conditions accepted for graphing the curves. In addition to the squared speed of sound, a performance challenges that the determination of ADE models is the holographic principle and also a complete review on their stability must reflect its non-local characteristics {{cite:862e3ff5c6b14ba0626aca0815583d5c1a653059}}. The latter approach of our aim is out with this research and also recognized as an earnest concern for future achievements.
| d | fb3751bd5f9b70c55df5833af2893ed9 |
Fig.REF displays {{formula:e53a17a4-6cd6-48ae-a1dc-7a61c8061eae}} as a function of R. Notice that for values of {{formula:bab9203c-58f3-4430-9bd4-8b893e54e150}} (UR region), {{formula:a3734c54-3fd4-4160-90af-f9bb25559749}} takes almost the same value for the three models ({{formula:02d1fe58-8f16-431d-97e8-4f34c626cf4c}} ). The reason lies to the fact that in the UR region where {{formula:fe9864d7-c6ea-4def-b8cb-48e179508952}} is larger and {{formula:287e634b-8648-4d81-957e-abe298888871}} , {{formula:c6f0cf12-14ce-42b3-9f1c-0186270f8e03}} (see eq.(REF )) and therefore in all models we have {{formula:3804ae62-c178-45d1-bfdb-9644f88355fc}} and almost the same differential equation {{formula:1323c69a-6d42-4f56-bd90-4158e2353205}} . In the NR region where {{formula:b487c1f0-67ab-4a21-9c55-f928e83984c5}} and {{formula:dbf24333-1899-4aff-979d-35b4596c9313}} , {{formula:91eb9389-d9b0-4e1c-9a32-b6ed8d29b997}} becomes constant. For example for {{formula:6fd35af6-39c0-4a4b-be47-65c5eaa6b6b1}} , {{formula:f5b16fc6-2a3b-4f30-b3be-f18af740d950}} , {{formula:e295f484-3241-4fe9-bc65-0fdd4111f25b}} and {{formula:28683273-d170-412a-8ef7-8237749cad98}} and {{formula:2d06aa99-ce1b-4894-96e9-994ed07a2450}} , {{formula:c46d67fe-aba1-4efb-bf1d-35c32645ca04}} , {{formula:4a3d62f7-eeeb-43a3-a71c-64898291a37c}} , {{formula:f955903f-4516-4e4b-9738-3890df09b678}} , (see table REF ). The reason lies to the fact that, in the NR region where {{formula:97ef3672-6b90-41ba-9182-0f1719afd285}} , {{formula:29e1bdea-a5f4-41f2-bcc7-6c511ce3ce11}} and {{formula:8fb601a0-c189-4d9f-98e2-16e0917ad4bb}} , one has {{formula:1f34423b-e726-45e8-9a9b-c36cc6331c5f}} for all models which leads to {{formula:46aa2a72-64d8-47cd-9c99-f467301797f6}} and therefore {{formula:caac74c0-61ab-4236-876d-06e2206a75c1}} becomes constant. The difference in the constant between the three models is due to the complicated expression of {{formula:07d0167e-180d-4e1a-a5d7-d4440fec1cce}} (if {{formula:ba309425-8d0b-4828-80e5-f2631f9156f1}} , where {{formula:4f1ee187-4eea-4958-865b-06abe7b2052e}} is a function of m or equivalently R because slightly different our model to an other. Any way, numerical results shows that in the NR case, {{formula:90c72a35-1c16-4868-b256-d7c50514fa73}} becomes slightly bigger than ours and almost twice that of Feng and al. In the intermediate region where {{formula:caaae29e-9e37-43cc-84f8-680b865f6e70}} ; {{formula:a4875f1b-043f-43c4-84c7-55dfa21e616c}} , {{formula:76d10f64-d193-46ef-be82-84bde3811c3d}} and {{formula:87de7783-59a8-4f86-a5cc-03d582a41f40}} . In fact, the difference between our model and Feng et al {{cite:196e511cae4f4e2bf582025c736f727a93f6902c}} (resp. Huang et al {{cite:4b4d8bb9f1ece3aa950f79ef6a915b284d1cdb3b}}) is due to the difference in the expression of the used differential equation {{formula:08e0c282-85ab-412f-9bdc-6b21174eab3d}} (see eqs. (REF ) and (REF )) (resp. to the difference in the experience of {{formula:6108723b-3cbe-4833-a96f-c30844ff18bf}} (see eq.(REF ))and U (see eq.(REF ) and {{formula:9053cf2e-f2d9-409b-bcad-da9b83e0576c}} ). Table REF , summarizes the behavior of {{formula:93e266db-ceac-428c-9ec7-a239ab4a133a}} in various regions.
{{table:30c01d6b-6e93-4e2b-8e04-09f31400dfcc}}{{table:646339f2-80f1-4126-91c3-db021bcfabb7}} | r | 3ebfa758fdf824d8d7df887f6dbf2e01 |
Figure REF shows analogous results for CIFAR-10, using the PreAct Resnet-18 architecture as in {{cite:3ea3b9829291fef117bf5049510854215ea05140}}. Again {{formula:17d80fd1-6e2d-4b6a-aaa2-cf469affbda2}} -mixup succeeds in finding matches that lie significantly closer together as {{formula:0ed44e53-2110-4f5b-9eab-decf4892cda0}} increases. For each {{formula:5ff0f5da-2f20-43cc-84d8-b862ba81bfe7}} except the small {{formula:7437f5b2-1bc6-41c0-9360-ffcfde136f18}} , the best generalization performance is still for some {{formula:0f172124-99e5-4385-a1de-03bed00a8568}} , with {{formula:f33321fd-75ef-498d-9616-734bd75f8875}} yielding {{formula:436e526e-b294-4d0a-a3a6-f0c5556ac317}} accuracy for {{formula:5d496677-17c4-4561-b16f-0859e1ca3e8d}} compared to {{formula:0d07bf35-50c7-42b7-9681-32e343b57bcc}} accuracy for {{formula:e689f935-694b-4194-930f-66be5bd41038}} . While the best {{formula:a0382a4d-519d-4e25-a497-15cd1a2b8637}} -mixup performance exceeds that of the best 1-mixup by only 0.17%, recall that in this setting 1-mixup outperforms ERM by 1.4% {{cite:3ea3b9829291fef117bf5049510854215ea05140}}, so when combined with the low overall error rate, small gains are not surprising.
| r | 7e660b301140fa32a1669905e64a8507 |
Note that the trajectories selection method used in item 4 of our algorithm can be very simple - in our experiments we randomly select trajectories. However, our method is complementary to other work on selecting informative trajectories, so future work could select trajectories more systematically, for instance filtering out near-repeated trajectories as was done in {{cite:71d9f523068b0c46e45f101caf6bce0f5ccc2a11}} or selecting based on critical states as was done in {{cite:a7017361887f002d63acf7fc78196fa66320844a}}.
| m | 6e3a26f224f0edfeac8902b9365b50e3 |
We further evaluated proposed training pipeline along with ensemble of our two models, one pre-trained on CrowdHuman {{cite:a3685c28a21e5360469c3ce1a6895f623c518394}} and the other one on Wider Pedestrian {{cite:5cd444f54354d39a171519f88aaf987496f3f10c}}. Ensembling is performed by combining the detections followed by non-maxima suppression, using soft-nms {{cite:431704bef42ac04eeb9be7884d4449def0f3fccb}}.
Final results are evaluated on the dedicated serverhttps://github.com/cvgroup-njust/CityPersons
https://eurocity-dataset.tudelft.nl/eval/benchmarks/detection
{{formula:051d8534-6f0c-46fb-adba-314f918e0278}}: Correspond to our submissions and use of additional training data
(test set annotations are withheld) of CityPersons {{cite:2b523deb3a475223bbc95d5082beac46f943a9e2}} and ECP {{cite:41be4d6ba281d9c380ef39eb4da1332c85e53016}}, maintained by the benchmark publishers and frequency of submissions are constraint. Moreover, we have included results only for the published methods (detailed evaluations of all methods can be seen on the urls provided in the footnote REF ). Results are presented in Table REF and REF . Our submission (Cascade RCNN) achieves 1st and 2nd on both leaderboards respectively. These results serve as a reference for future methods. However, no other method to the best of our knowledge uses extra training data. Therefore, giving our submissions an unfair advantage. Finally, as stated above, fine-tuning on target set is not the goal of the paper and in many cases it is not practical. In this work, we argue in the favor of cross-dataset evaluation and its importance.
{{table:9591f62e-a2af-4f23-b8a3-0e0953237bf8}}{{table:31c5faf4-741a-4a67-81a5-223fe42e2b31}} | r | 872383e5326992672a23e331786217d5 |
Nonequilibrium Green-Kubo relations that relate macroscopic transport coefficients to local current observables have appeared in the literature in particular situations.
These studies can be categorized according to the method.
For a two-dimensional nonequilibrium viscous fluid {{cite:90b8544152799e60f445510f1a893f148ee2e24f}}, {{cite:d0006c1a4ac40f7c8ac7d00a593cf27feb100720}}, Green-Kubo relations were deduced by assuming that Onsager's regression hypothesis {{cite:cff1757403a18dedb2ae6d12910656823bb3d6d5}} remains valid around nonequilibrium steady states. An alternative approach utilizes the projection operator method {{cite:ff465595e4baf7d6cf785b3d6a3383c4a762ecd8}}, {{cite:02822ca8d3ca5debfa02188ec5e0957d16122c49}} adapted for non-Hamiltonian dynamics {{cite:f435b4a8a987e86ed9f602bdf47fbee0487857ea}}, {{cite:65f3a66f856ec03e60e67d4dabb3f8d04c0c48b8}}, {{cite:cb98a5689834c867d2d29028edcafa807507861d}}, {{cite:e6a30a54bb97c9d0f8238226435db323c8e1e72d}}.
The resulting Green-Kubo relations incorporate a time-reversed dynamics, apparently obscuring the interpretation of the resulting correlation functions.
This obstacle has been overcome for at least one specific model of a nonequilibrium active fluid {{cite:9447469454e607638fec9e50361fc3e6c13c7f15}}.
| i | 0f8fb7e5d7590915fae1a96a700f4b4c |
This work considers a nonlinear coupled fluid flow heat system.
Fluid flow is described by the
incompressible Navier-Stokes equations {{cite:73170660e2b56a95061d1a809a6fd5c5273f7c96}}, {{cite:4d6cf5a7003eb2c1e548c9e06a78d89b62bb83a1}} (for important
numerical developments, we refer to
{{cite:188f307cd868e7b821985ea62b43214620413b4c}}, {{cite:f30b8022a19633e3d1d8618eba6fdd2acc7a4250}}, {{cite:40045d6efffa06e2cb093f1ca2783863a422054a}}, {{cite:d29f6d7e3bc1c180af85b1062cd5aa2ae0715488}}, {{cite:fcf3175223bf0053c5a6f38f073ea1b5e837da86}}, {{cite:fd9a9df03ecba423503e0b9978c1c2b340a05247}}, {{cite:31d36fcd6a9ff7f36c7f00940334d9e0289e6ff0}}, {{cite:194cb7d4eafb3e66b2ef7d8286de1fe04bb0451c}}, {{cite:1fbfd44e928e8e833f7b119ad32b71e3a5d8c5d4}})
and the heat distribution
by an advection-diffusion equation. The resulting PDE (partial differential equation)
system is known as Boussinesq model {{cite:f34e39a570c56644c50e000c215bdeacc1af7c8d}}. This model has been
widely applied in various fields such as climate modeling {{cite:414fec8e3bd421497df60419ef17e964042ead6d}}
or earth mantle convection problems {{cite:6a9d13e41759b624d65fe30f87840642709ffe48}}.
Furthermore, the Boussinesq equation can serve as a sub-model within
laser material processing {{cite:eff49fdb3fc09b9aba0bb21fe0a835b22b082b13}} in wave guide modeling (e.g., {{cite:b84b23f405fcb47af3c41d371314145800ecc447}}, {{cite:e97b7e028c5f60d958b290dde0454fd5c788d88d}})
where heated material starts to flow
due to local heat sources. A mathematical analysis of the stationary
model that serves as our point of departure was done in {{cite:78f67ba7c4ccbff85475f2d741687e9e44c07cc1}}.
| i | d1226a51f7184af7368b46e95189578b |
For the first time Lubotzky and et. al {{cite:db1498b6943e2d25950a19cbf7704275204c7fa6}} and Margulis {{cite:428485e587750c44d05f84a5f542c2b10fd76405}} found the infinite families of Ramanajun graphs.
Adam and et. al proved the Conjecture REF for bipartite graphs. In addition, they showed that there are infinite families of {{formula:f5b1dbb6-262b-47c1-b3eb-7ac8af0c3300}} -biregular bipartite graphs such the second largest eigenvalue is less than
{{formula:5b4a9819-48b3-46ed-b8c9-cfebffb42636}} ({{cite:b49dbaf3011b99681e0f3940240f8bbfe3500018}}). For more information about the construction of Ramanajun graphs see {{cite:93f12d8a9fda8fbc644250c1a42623975e67c9e2}}, {{cite:0a395bd0ff898d0f2c90cd336f27fc35c8da2b99}}, {{cite:cfe990f17eb529c7ae36bc5af8264fe7362a094e}}, {{cite:8020b0e39ef26a2c406fc38d693b891bf863fe3b}}, {{cite:27bc44bf8c3ff8ffdd0fc2fb4990ea5438f68aad}}, {{cite:57351c2302c1ff26841550cdfef778dce2bb07db}}.
| i | 807c4e19ceec3afbd38eeaedd93b30a9 |
DFT analysis of interlayer charge transfer. The analysis of weak ferroelectricity in this work employs the VASP
package {{cite:d2a2b164f26406a990388a4530e1dd8552cba1b0}} with projector augmented
wave (PAW) pseudopotentials. We approximated the exchange correlation functional using the generalised gradient approximation (GGA) of Perdew, Burke and Ernzerhof {{cite:2d0e400ecd886ed4156128fc8a73d7e93b434839}}.
The cutoff energy for the plane-waves was set to 600 eV and the in-plane Brillouin zone sampled by a
{{formula:8cf8f9b5-d4d1-49ca-a3fb-c1d32f29740a}} grid. The in-plane lattice constants of the constituent monolayers are strained to the mean of their experimental values, taken from Refs. {{cite:c83942515af4767ab01455096767f3b932b79cd7}} and {{cite:df556bb9317e9cfd071b26eaaf56b159549d93b4}} for MoX{{formula:58d9ffca-e46b-46d0-be43-2635db116a50}} and WX{{formula:4e58d882-a82d-4994-9f23-28fed8806119}} , respectively, while the intralayer chalcogen-chalcogen distances are left unchanged for each constituent monolayer. In the inset in Fig. REF D we illustrate the DFT-computed behaviour of the difference between the plane-averaged electron potential energy for a WSe{{formula:995d2f14-bb16-49c0-a328-df0e1ba8c750}} /MoSe{{formula:20684e8a-9adb-41dc-a4c1-aa9d270bbb98}} heterobilayer (for 3R stacking and interlayer distance {{formula:c2ad742d-8995-4404-bdd2-71dddb9868ec}} Å), and that from the sum of the potential energies from isolated individual WSe{{formula:129d544c-6add-4ff9-abd7-fd9949eaebb9}} and MoSe{{formula:ec2bad49-e230-4cce-a1f4-89438a8b4807}} monolayers. To avoid having to artificially resolve the mismatch in potential energy which would arise at the boundary of a supercell containing a single heterobilayer, results for heterobilayers in this work were computed using a supercell containing two mirror-reflected images of the structure.
| m | 408bb6964b10834e19e1a0293190c8bf |
(Reinforcement) learning based methods for the TSP can be divided into two categories. The first category solves the problems in an end-to-end manner. Algorithms belonging to this category are usually based on deep neural networks. When receiving an input instance, they use the trained learning model to generate a solution directly. For example, Bello et al. {{cite:3d1069f8c6f4fa6639ffdd60fa263a14e16d864d}} address TSP by using the actor-critic method to train a pointer network. The S2V-DQN algorithm {{cite:88d5daae9a060177a446c50fb1d7f5e7a814758a}} applies reinforcement learning to train a graph neural network so as to solve several combinatorial optimization problems including the TSP. Goh et al. {{cite:92792a68f579192f2d97d6d0948dae9d85743bdd}} use an encoder based on a standard multi-headed transformer architecture and a Softmax or Sinkhorn {{cite:7b217ac02d873317500c96a030d55a72382b0977}} decoder to directly solve the TSP. There are also some algorithms based on deep reinforcement learning for variants of the TSPTW {{cite:da52abf365d38393f540751dbd61ea9c3f1d08e8}}, {{cite:41434355f562d4b58cf0705696309ee351ced197}}. These methods provide good innovations in the field of applying machine learning to solve combinatorial optimization problems. They can yield near-optimal or optimal solutions for small TSP instances with less than hundreds of cities. However, they are usually hard to scale to large instances (with more than thousands of cities) due to the complexity of deep neural networks.
| m | c9bc9e83ae23b83f16e9f5c4f6259e89 |
Section addresses the question of whether the remote state reduction of an entangled system by a local measurement on its entangled partner system constitutes a real disturbance. To this end, Section REF introduces a steering-like statistical inequality testable in QM or a GPT. Section REF develops a criterion to decide the reality of the remote disturbance of an entangled particle, a phenomenon we term “operationally real nonlocality”. We argue in Section REF that the experimental violation of the above inequality together with the satisfaction of a local signaling condition, entails fulfillment of the above criterion for the reality of the remote state reduction. An experimental test in QM for operationally real nonlocality is described in Section . The relationship of this type of nonlocality to Einstein-Podolsky-Rosen (EPR) steering and Bell nonlocality is pointed out in Section . We argue in Section that the reality of remote state reduction brings to the fore a fundamental distinction between relativistic signal locality and quantum no-signaling, and suggests that the latter should be understood as the consequence of a consistency condition in a class of GPTs, rather than as a basic principle inspired by relativistic signal locality. The conceptual implication of operationally real nonlocality for the well known EPR paradox {{cite:3df96046043eca0c3d4616ac6d0a62a15dd88252}} is discussed in Section , where it is pointed out that our result represents a different response to the paradox than the historical responses due to Bohr {{cite:a2dec81c68095045d01fed172fe378670d6c6061}} and Bell {{cite:1d1f5c279943834b0250b4b77f651fab4433e3a0}}. Finally, we present our conclusions and related discussions in Section .
| i | e2b03dc1e50e1d2e804c621292d7c234 |
If we return to the classification of the Galactic globular clusters made by {{cite:16790510f6c9f1fcddfc8c4d18505c1cc206a4a9}} on the basis of their kinematics (energies, and angular momenta), we can take a new look at the age-metallicity relation(s) of clusters which in their study are associated to different progenitors. For this, we make use of ages and metallicities from literature data, as reported by {{cite:ff8e32d32f40476f105a867ce9e500bf1d0511f4}} and {{cite:f2e063a72e9b33f63a4b5e38084c57e09c4a7d8c}}. We recall the reader some main differences and similarities among these studies. {{cite:ff8e32d32f40476f105a867ce9e500bf1d0511f4}} measured relative ages of a sample of 64 Galactic globular clusters, observed in the framework of the HST/ACS Survey of Galactic globular clusters. The corresponding ages, and errors, are reported in Table 4 of their paper, for a set of different theoretical isochrones, and metallicities for two abundance scales. We adopt in the following the ages corresponding to the theoretical isochrones of {{cite:e48403e607f8d844c0c0c25c687449a0e0eb6af9}} using the {{cite:a4e5ff470e28ab28749ed058a3bb479785776d9e}} abundance scale. {{cite:f2e063a72e9b33f63a4b5e38084c57e09c4a7d8c}} analyse 55 clusters – many of which are also in the {{cite:ff8e32d32f40476f105a867ce9e500bf1d0511f4}} – whose ages and metallicities are reported in Table 2 of their paper.
The resulting age-metallicity relations are shown in Fig. REF , where colours indicate the different galaxy progenitors to which {{cite:16790510f6c9f1fcddfc8c4d18505c1cc206a4a9}} associate the clusters: Gaia-Sausage Enceladus, Sagittarius, Helmi Streams, Sequoia, together with two additional groups, the Low-Energy clusters, and High-Energy clusters (see Table .).
Despite the differences in the shape and extension of the age-metallicity relations resulting from these two datasets, we notice that:
| d | 241019d815af778f8cdb2269d6750484 |
There are a lot of studies on the compressible MHD equations in the
literature. Here we mention some results on the multi-dimensional
case. For the two-dimensional case, Kawashima {{cite:5e09cd399183c850af6ead393a4d62bdd1ab9915}} obtained the
global existence of smooth solutions to the general
electromagnetofluid equations when the initial data are small
perturbations of some given constant state. For the
three-dimensional compressible MHD equations,
Umeda, Kawashima and Shizuta {{cite:1876a3613705ead2c2ddb7ebf01d6064c20dfb2a}} obtained the global
existence and the time decay of smooth solutions to the linearized
MHD equations. Li and Yu {{cite:506c1e73a3aba22a67eb5710e2fbc6f065c72217}} obtained the optimal decay rate
of classical solutions to the compressible MHD equations around a
constant equilibrium. The local strong solution to the
compressible MHD equations with general initial data was
obtained by Vol'pert and Khudiaev {{cite:d9b967d5eaf6fbb011749a1fa042209529ce93bc}}, and Fan and Yu {{cite:e1c42f7bd133917ef08c0e33d7ccfb01cefccf00}}.
Recently, Hu and Wang {{cite:a8bee4533f1c3495a41bdba8de15970923982c9f}}, {{cite:adbb0911bb27dd39421b56836bca48b53602fbd6}}, and Fan and Yu {{cite:d4a34d68c4335a66a8972a45883057970da41102}}
established the existence of global weak solutions to the compressible
MHD equations with general initial data; while in {{cite:fd30e39bec48e43d38add2e54f74ebddc6ee8851}} Zhang, Jiang and Xie
discussed a MHD model describing the screw pinch problem in plasma physics
and showed the global existence of weak solutions with symmetry.
| i | c47ba7bec16f43b3f037ed22e1391644 |
with parameters
({{formula:7311391f-f57e-42cc-a582-0b73fb57899d}} , {{formula:7de94aab-f4cc-458c-a187-27967ddf1081}} , {{formula:c28db3f1-7e2a-4526-8b9d-7227dffedc22}} ),
initial value
{{formula:bd31836a-265e-4333-9edd-10fac4df3832}} [0, 20]{{formula:7b6192af-a4bb-4b29-a4d1-645fc3f4e086}} In the first part of this experiment, the uncertainty calibration is evaluated across a large range of solver configurations and tolerances
({{formula:3b22d146-7c87-4609-a1b0-e79eefb0e340}} ,
{{formula:a10648d4-3587-4151-970c-dd76635cf655}} ).
To assess the quality of the uncertainty calibration, we use the {{formula:1ab7c4dc-8ea8-4b36-959f-7de8cb2aba4f}} -statistics
{{cite:42e8b442559196d052ebff9f93b785a5f6be1c1f}}
defined by
{{formula:d4636bdf-cd87-4537-97ea-2aed21a293c1}}
| m | 68346ef7a3114a0569c368fe1ceb8ac4 |
In summary, all the reference trends yielded {{formula:1813d825-c8f1-4899-b38c-3fe9c0d7f463}} measurements consistent
with the reverberation mapping values within their nominal uncertainties, with the decaying trends showing a slightly better agreement than the rising trends, which have a tendency to underestimate {{formula:8754adb3-19cd-47de-8233-4e5d35d3a500}} to a moderate degree. Unfortunately, the most reliable reference source – GRO J1655-40 during the 2005 decaying phase (hereafter GROD05) – has a fairly small range of {{formula:996c8604-8cb4-4c66-9c7a-8aa8881c911f}} during its spectral transition limiting its application to sources with relatively flat photon indices. Using the reverberation mapping values as calibration, it was determined that for AGN with steep spectra ({{formula:2c13b5de-068e-462f-8f07-d7e7f1a0655e}} ) the best estimate of {{formula:cfe5a65d-1d39-4249-a1f3-31caf763b4d2}} is obtained using the value derived from the rising phase of the 1998 outburst of XTE J1550-564 multiplied by a factor of 3 (hereafter 3*XTER98). Below, we summarize the general principles at the base of this technique; a more detailed explanation can be found in {{cite:d5ed163cb85a329b07e15abdd2f3fd0acb73394d}} and {{cite:8f64275bd69ccaff612c5047f4f6d29c2a00c43c}}. For completeness, in the Appendix we report the basic information on the reference sources, including the mathematical expression of their spectral trends, which is necessary to derive {{formula:31f1d4a3-58e1-48f8-b512-bc85994ec79a}} using the equation reported below.
| m | 08328a94fae52f052d025f407c6882ae |
{{cite:08ec69ea21c1f62e07be7e4f53d5822e51493b4f}} proposed a rigid-bubble model in the mesh-based simulations to overcome the issue of bubble integrity: in their model, bubbles were assumed to experience no deformation during their buoyant rise. Despite such a strong assumption, the model captured several key features that had been missed in previous AGN feedback studies, including the excitation of internal gravity waves and well-developed wakes of the buoyant bubbles. The model implied that long-lived intact bubbles could dramatically change our view of how AGN feedback works in galaxy clusters, which motivates this project.
| i | 730cde65d1db99db9f3bcd98eb0f5d11 |
An exciting direction that remains unexplored is to also train setter agents using the data produced by humans whose role it is to set tasks and questions. “Self-play”-like settings have proved powerful in developing agents in games, where agent roles are symmetric, and reward signals provide positive and negative feedback signals per experience {{cite:d0acec66e62cb39a8545895203e7dddfef0c32e9}}, {{cite:fee0811185ede4398bc13db2967bf0d1d15a8989}}. It is less clear, however, how one could leverage agent setter-agent solver dynamics in the Playhouse environment where roles are asymmetric and rewards do not exist, beyond using the extra data for generic representation learning.
| d | 88d16ecd81022a41676cde7dd5120dab |
We propose the use of a variant of the Xception architecture described by François Chollet in 2017 {{cite:11bcf77f9f84693cf5aacc11903a1179f3f63f2a}}, due to its low computational cost. This model recently achieved state of the art results in multiple computer vision tasks {{cite:806f3807cfcf8aa244ad3458169c5ec2a7dfd7f8}}, {{cite:ae7eff453a7a5a1efc6a40429a2a061923b32e33}}.
| m | 0fa81dc7e4d5da199afb018d03155402 |
Recently, physics-informed machine learning of dynamical (or Hamiltonian) systems has been receiving a significant interest owing to the wide variety of applications in the physical sciences. Greydanus et al. {{cite:77e546b6f4f0f2ef8813f3f44fcffbfeb5386412}} introduced Hamiltonian Neural Networks (HNNs) to learn Hamiltonian systems in an unsupervised fashion by virtue of a physics-informed loss function. HNNs satisfy important properties such as time reversibility and Hamiltonian conservation and were shown to be superior than data-driven neural networks. Along these lines, several studies proposed improved architectures for learning Hamiltonian systems such as physics-informed HNNs {{cite:ea73ffb20faa4aa1e8adb06f5f9669f81dc710c1}}, symplectic ODE-Net {{cite:6c0c940349828e37c2060b25e4af498168caab79}}, and symplectic neural nets {{cite:8708257c88bf36e4763092ca07ddb81de4974542}}. However, these efforts were all focused on learning Hamiltonian systems for classical mechanics problems like n-pendulum behavior and planetary motion, and not to perform Bayesian inference efficiently.
| i | 54f6561db5c5de5922cc9a2d699c69a1 |
Baselines.
Given our new multi-scan multi-body setting, we made adaptations to previous methods and compared to the following 4 baselines:
(1) PointNet++ {{cite:5d09f63c764001946727ddd6bce3c11eb0c7f984}}: We use the segmentation backbone to directly predict {{formula:ad6ee388-c82b-4736-b0fd-fed2dbe75535}} matrices. We aggregate the bottleneck features by taking the max before feeding it to the individual {{formula:3cccd10b-c19f-4a7d-9352-53422bff0233}} feature propagation modules.
(2) MeteorNet {{cite:1a046e34afbff6565c1202d79033d7a8d91b5d97}}: We use the MeteorNet-seg model proposed to directly predict the segmentations. Both PointNet++ and MeteorNet are supervised with the IoU loss (equ:loss:iou) which counts in the ambiguity of rigid body labeling.
(3) DeepPart {{cite:65f45301feafd1ac838df0acbd87cb95fe03248d}}: As this method only allows pairwise input, we associate multiple point clouds using sequential label propagation.
(4) NPP (Non-Parametric Part) {{cite:a487d9f567eca899b013db7c78ffa962bac6d70c}}: This algorithm does not need training and a grid search is conveyed for its many tunable parameters.
{{figure:0b273d05-dbb2-4274-add3-0dac27d4d2dd}} | r | ef0516e04f43b2fc04c6621e30add2c4 |
In this section we record a number of properties of certain boundary extensions of holomorphic maps between simply connected domains and the relationship of these boundary extensions to the harmonic measure of sets on the boundaries of these domains. In doing so, we use a number of classical results on the boundary behaviour of holomorphic maps, which can mostly be found in {{cite:d77d95b0524ab5e664c1c774c2b3a515d925db19}}.
| r | 9f462eb2a321f7c6a3429b4a3d11a741 |
Our focus is on the extension to {{formula:50f4df7c-60fd-4a1b-bd21-2eb06f81a95d}} -uniformly convex spaces of tools from the analysis
of fixed point iterations in linear spaces.
We are indebted to the works of Kuwae {{cite:d37db180defabb56b1ddd7eb2dbaed5fe78e5aa5}} and Ariza-Ruiz, Leuştean, López-Acedo, and
Nicolae {{cite:9c08d7488fa2d6611e6286a3f12099c7559f9ac5}}, {{cite:cef32f4b85af23088f9e988e4348bc276288d159}} who
studied firm nonexpansiveness in nonlinear spaces,
though the asymptotic behavior of averaged mappings in uniformly convex Banach spaces
was already studied by Baillon, Bruck and Reich in {{cite:f6e9ee7d25cc4a0d6e9ab262ea6be11ec7a8603c}}. Reich and Shafrir
established an approach to the study of convex combinations of nonexpansive mappings in
hyperbolic spaces {{cite:4b3d402ead6c259d4000a5de5b645d69342622ab}}, the foundations for which were developed in
{{cite:00d4132bd533417a3ccfbc44336ed61c732c229f}}.
Building on this, we follow the framework for nonconvex optimization established
in {{cite:80953e35af7461ce14d35cb4e89488ee971f708a}}
which is predicated on only two fundamental elements in a
Euclidean setting: pointwise almost {{formula:1ff2932b-d4dc-4c3f-9168-0c89bca7fd2d}} -averaging {{cite:80953e35af7461ce14d35cb4e89488ee971f708a}}
and metric subregularity {{cite:b97e3eb02b6b018a62e15ac493f3e09ba52b5403}}. Almost averaged mappings
are, in general, set-valued. In nonlinear metric spaces, there are several difficulties
that arise: first, there is no straight-forward generalization of the averaging property
since addition is not defined on general metric spaces; and second, multivaluedness,
which comes with allowing mappings to be expansive. The issue of multivaluedness
introduces technical overhead, but does not, at this early stage, seem to present any
conceptual difficulties. The issue of violations of averagedness and
nonexpansiveness is more fundamental. We show that such
violations are unavoidable if one wants to work with resolvents. The foundations for
working with these difficulties are established here, but we postpone until later
a direct study of resolvents on spaces with curvature bounded from above.
| i | 689e05207f4004d596348cf469401538 |
In our study, we have obtained a class of {{formula:475e1e83-38dc-4b51-83fb-df5e03595c09}} theories that exactly mimic {{formula:4640097b-ca00-4718-91aa-e3c4341ea7b6}} CDM expansion history, even if it is impossible to distinguish from the GR using measurements of the background cosmological parameters. Then, it is an exciting problem to see how the perturbations studies (such as growth factor, structure formations, or cosmological gravitational waves from GR) in these {{formula:51226af6-8aa1-43c6-8692-32b42f77d9b1}} theories can be verified experimentally {{cite:cda2dd445a9db2adceed4f8b1d97342bb3499c1c}}, {{cite:0c3975ac98e635c893706ccee4114e1449e2301d}}, {{cite:0cd88231984dfe63db1c93f3738b905f2d5d5b34}}. In fact, the {{formula:6aa18462-b7fa-4f8d-9e24-341d4ab16673}} theories presented in this manuscript can be tested through the solar system test, which rules out the Lagrangians to have a modified theory that works for both local and cosmological scales. Furthermore, we have restricted the reconstruction scheme to the flat FLRW cases, whereas in non-flat FLRW cases, one additional term for curvature will appear in the Hubble parameter, non-metricity scalar {{formula:c2c435df-03de-44f0-bc96-153adf571c17}} , and the Friedmann equations. As a result, a highly non-linear differential equation will arise for {{formula:4598ca53-5586-4d85-b9ec-b582463daf74}} , which is an open problem for readers. Hence, developing these types of modified theories adds a strong agreement in favor of inflation, dark matter, and dark energy in the context of a unified gravitational alternative theory.
| d | f4290cfd01d4841b2a7cba205df8e0fb |
The confinement of a finite thickness {{formula:0f250a2a-bbef-4650-8920-4f957762164a}} in the multilayer structure can quantize the momentum in the {{formula:e967a93a-dd58-4263-9e6a-f9b615ae7a01}} axis, leading to 2D subbands. If the multilayer structure center is set as the origin, we can choose {{formula:6acb4a03-2fff-4bf9-96a9-3b38032729f4}} to satisfy the open boundary conditions at {{formula:b84f5b92-eacc-4db7-b567-2c420171891d}} . The open boundary condition is also used in the study of the effect of finite size on the edge states in a quantum spin Hall system {{cite:576cffe69431ee5e5e6c78e67dec3907a60ea709}}. We then express the Hamiltonian in the {{formula:402cd977-283e-467b-9c9a-95de3963394b}} subband basis {{formula:442c5d89-1b42-46a5-a525-be54371a5870}} .
| m | e98f98bb7db6c4fcc0fe247a07eb18d5 |
The mid-latitudes, defined loosely as between {{formula:0e650d9a-c97e-42ac-97ec-b7f14fd9c9f2}} S{{formula:aeb4e4e1-e03b-4538-ad83-accc29215bb7}} S and northward of {{formula:ac1ca2cb-f20d-4444-afe7-14822f68ed01}} N, are where Neptune's brightest and strongest methane cloud activity is observed in the near-infrared {{cite:a17e0e8e6f1443a21572fef28d99e2b262827e8d}}, {{cite:557f3d80f3f13cebfe99578962ee7a9fcf86286d}}, {{cite:10f079d6774683ba480e84e9345006a163eaa357}}, {{cite:551a4bd63781cda0f3860d2fd9234904039190bc}}. In contrast, the equatorial region is nearly featureless. In addition, the mid-latitudes are colder than the equator and south pole in the mid-infrared {{cite:d1bced5a05f23b56969fa0d512e12a8e9083aab7}}, {{cite:18e4b75c0438b1383cc074e3c6e862b2a4f691c8}}. Combining these observations, a global circulation pattern is inferred at altitudes shallower than {{formula:ac7538a9-244c-44bb-9441-7e181ec9a72b}} bar (where the near- and mid-infrared probe): cold, enriched methane air rises at the mid-latitudes and travels to the equator and poles, where the methane-depleted air subsides and warms via adiabatic compression. However, this picture is complicated by the relative excess of gaseous CH{{formula:7a7b74e8-2f1b-404f-8e56-2b352df72875}} at the equator, which is more consistent with rising air {{cite:cf78f790fd54808e1f5633fdd19605d1b39d36ad}}, {{cite:551a4bd63781cda0f3860d2fd9234904039190bc}}, {{cite:77a3eb37f54f20563abf342f133c2b0ca94423b5}}. We show in this paper that H{{formula:e32d5380-ad4d-43d8-8641-a1f9c7e47115}} S is most abundant at the equator and southern mid-latitudes, in line with this meridional trend in CH{{formula:a001a14e-fad6-450c-9bfd-0dae3d80cb4a}} . At altitudes below methane condensation ({{formula:865cf612-b18d-422c-b3bb-058a4619f792}} bar), the aforementioned circulation scheme is, thus, at odds with the retrieved CH{{formula:05818340-75d9-49ec-a1f7-1ad69167e7ae}} and H{{formula:b26c942f-4b4a-4a50-92a7-4f2656938bec}} S abundances. Therefore, a more complicated picture of global circulation on Neptune is needed to explain each multi-wavelength observation. {{cite:e56f129bcc31ead0c89118a4cfe53ebddaedb4e6}} synthesize decades worth of analysis on the ice giants and argue that vertically-stacked circulation cells are necessary to bridge the observed patterns above and below 1 bar on both Uranus and Neptune.
{{figure:b87a072c-681c-4a32-adba-e1a9f6a4801e}} | d | 52bb584e203cd078b18ebc3857442bd0 |
Here we used a simple linear classifier to measure the quality of latent representations, which is an obvious simplification with regards to cortical processing.
Note however that also for more complex `readouts', organized latent representations allow for more efficient and faster learning {{cite:28e53c20f3114c95e578384abbec1725f634f6ae}}, {{cite:03eded50f0847d465a759862d816b9849b85e620}}, {{cite:b26661e89f9fa94160c40fd3df4b81793119a7da}}.
PAD{{formula:23a3adec-47c3-49a5-bc41-af5acfd91093}} assumed that training the linear readout does not lead to weight changes in the encoder network.
However, in cortical networks, downstream task demands likely shape the encoder, which could in our model be reflected in `fine-tuning' the encoder for specific tasks {{cite:ed90c9e6691922abe340e8301aba87ed5bef13a8}}.
| d | 25efc5b417c938abfb0bbe67c062f888 |
Topological properties of matter have always been discussed in relation with topological pumping.
In an enlighting gedankenexperiment, B. Laughlin related the topological nature of the
transverse conductivity of the dimension {{formula:f9a5ebbf-6f4a-4997-8fdf-785aa9741614}} quantum Hall state to a transfer of charge between
two edges in a Corbino geometry {{cite:f8bf35842b09fde104577e91d9000fe2e142ca50}}.
A modern interpretation views the Hall sample as effectively wrapped on a cylinder, realizing
a {{formula:caa0831e-fa25-4ac7-9aec-7513dc9e8a6c}} quantum hall charge pump as the enclosed magnetic flux is smoothly increased {{cite:5b5fe5bb32e7b0adf9bbd21c6e39383c0d7466e5}}, {{cite:aafc665cd2d3c79ffafe033f46d578f795738492}}.
This quantized adiabatic pumping of a time-dependent quantum system was soon generalized
beyond the Quantum Hall Effect to a driven one dimensional crystal by D. Thouless {{cite:176fad2b19aa2efbfd85961bdfaf711fb4bb76a8}}.
In a Thouless pump, exemplified by the Rice-Mele model {{cite:d9613b35502f75c9c9fefe9f8220216c9b42b45c}},
the time-varying flux of the quantum Hall effect is replaced
by a time-periodic phase {{formula:4dfd021b-0bfa-405f-acd5-b1eed5950d4f}} which accounts generically for the external drive.
The corresponding dynamics is periodic both in the linear spatial dimension of the crystal, but also in time.
In contrast with previous implementations of geometrical pumps, such topological pumps are characterized by a Chern number and were only recently realized
using cold atoms lattices {{cite:74789de89b49c4a3bcb762806674f2898f7eaae2}}, {{cite:b9cade4a38373c2d65969620e057930088970990}}, {{cite:8dd80a863d419e97333382cb2d8745dffc2138d2}},
optical waveguides {{cite:907c5a76a5f864b2ba5d7efd0a28f7269300752f}}, {{cite:61d2948489088ffaa01f8c4515ead14334b5b215}}, {{cite:20b4877f38535bb8d56fb90706cdb59c2f6ec573}}, magnetically coupled mechanical resonators {{cite:935aa25805098d3276e2bba2252888ebcae60619}} or stiffness-modulated elastic plates {{cite:89051b2eb9923995b32a995d297a4068c608831f}}.
| i | 5d8cdb5ef72edeccf3622ef53f2eac88 |
In this section, we compare the results of our algorithm with the state of art for every {{formula:23132522-0eea-4d0a-8493-a900d533533b}} and for every {{formula:49014570-000c-40e1-9983-ad97a22ca0a5}} . We averaged over 50 different realizations of {{formula:942c4472-12fa-4c79-8d19-4074b6f3bff8}} where {{formula:4373fd5b-9ca8-4ebc-97f4-e88b90c5927f}} is a tensor with random Gaussian components. We plot in the figure the correlation of the vector {{formula:34f4d392-0b72-4167-bd90-2f9e7416a747}} output by each algorithm with the signal vector {{formula:8cf747a2-c42b-4f8b-8a19-9a2e7fe71b61}} and plot the {{formula:ebac1284-2566-49cd-8085-d5e47be2f2c3}} confidence interval bars. The algorithms considered are SMPI (that we perform after symmetrizing the tensor {{formula:12a18e13-70ac-408a-9227-e3813659d201}} ), the Homotopy-based algorithm (Hom) {{cite:dffac76442daf50d23ce24ed173d28ccc1a06972}}, the Unfolding algorithm {{cite:78c54d839ce773b8b583414366c3b33e0d9d5d84}} which are considered as the two main successful algorithms for Tensor PCA, as well as the CP tensor decomposition algorithm of the Python package TensorLy {{cite:af8e62d8cc28b908ea79a9a9ab92ed5124bbb606}} used with a rank equal to one.
Similar results are obtained for {{formula:0d16a994-956c-42e5-a17e-aaa1dbd89810}} (only for SMPI and Homotopy) and on the non symmetric case and are provided in the Appendix.
| r | f1d600d98d874e4d014153f5b65c06e6 |
We consider three different photosynthetic exciton transfer complexes; the Fenna-Matthews-Olson (FMO) complex which appears in Green sulfur bacteria, the PC-645 protein, which is a sub-unit of the photosynthetic apparatus in cryptophyte algae, and LH2, part of the photosynthetic apparatus of the purple photosynthetic
bacterium Rhodopseudomonas acidophila (their schematic structures are plotted in the insets to Figs.2 (a), 2 (b) and 4, respectively). All three complexes were shown to exhibit coherent energy-transfer oscillations in non-linear 2D spectroscopy measurements {{cite:807c2fdb840d104f09dfa8f3b6dff1929e79aa76}}, {{cite:5dccfdb1fed0cbc56419fbe182ef8dc24781eafa}}, {{cite:92b594718f9a9311dc4020e66cf3062f60349aa4}}, {{cite:da11667167994c2080228de33cf3aa002a3f49f6}}, {{cite:fb8f3f72cacdb5595a9bed8d70f2e3525d371dad}}, {{cite:d06da48688feac2c53a9803c8a8f4918887edc22}}, {{cite:ea87e60d86e8a897a9821eab1429638c7d481347}}, {{cite:f80c379e10700851183b771e021e2e09da8a32de}}. The Hamiltonian parametrization of each complex was taken from previous literature {{cite:fa2775f59717b262048910f81c2ff42e5973a6b3}}, {{cite:38b9e46b883650ad8e3444992965e2766b813f3f}}, {{cite:ee0c78edc13672113919a3f3dbfcf29e7fa94017}}, {{cite:79bc9ee92624a5bb9c7342e05c4f98ea86de3df2}}, {{cite:9429c5591b5a05d48e426df8d4e3f064bd33cbcd}}, and are provided in the SM (section I). Some crystallographic measurements suggest an updated model of the FMO complex, containing eight bacteriochlorophylls (Bchls) instead of seven {{cite:e22c62dc63d5f0a9e5fde3da2232606dafcfa136}}, {{cite:076d2437926a41db53e5fbb440d20d0edda9f34a}}, a structure which has been parametrized and studied in the context of FMO energy transfer (e.g. {{cite:fb15a2a72fb88ab8e86a009ff4b295beea20f677}}, {{cite:200f1615e9eb69841125f785dfaf052385fa9a8e}}, {{cite:211cb8c3915c4ff61ec58022903e6abcabf3df43}}). Here we chose to focus on the seven Bchls model, as previous works demonstrated that the expected difference between the two models would be insignificant {{cite:211cb8c3915c4ff61ec58022903e6abcabf3df43}}, {{cite:200f1615e9eb69841125f785dfaf052385fa9a8e}}.
| m | e9792c40f0bfc54110864b9271fc3953 |
Finally, we want to discuss our motivation for considering {{formula:ebdddc8d-169b-4739-8daa-08c4b51fa4f0}} -adic
spatial rotations in the first place: it grows out of Volovich's
{{formula:e6934714-5252-4c1a-9155-7f5b7077542d}} -adic quantum theory, in which Euclidean space is replaced by
{{formula:7e3b46fb-f9ca-4ebd-95eb-9f6296b0d0a0}} -adic space to define the underlying phase space {{cite:e468ab5e08b2daca278d6b457b56a3416bc58901}},
see also {{cite:e6965effab4c1f85b3fe06b6375509eaa91a85ff}}, {{cite:27200c164931817f1afa222d72a9921cbd25c88f}}, {{cite:01adc3030604f26c6101b6d4f60e8c580077c243}}.
The idea is to realise quantum systems as unitary representations
of the symmetry group of the {{formula:9e891a92-fe99-4bfc-a855-8e63c916d7c2}} -adic space, according to Noether's
theorem {{cite:41f874750b4636418999eaef557c3a63ede4c94c}}, and this has been realised for the displacement operations
in position and momentum, resulting in a {{formula:72437f96-9704-4755-966b-672830523648}} -adic Heisenberg-Weyl
algebra of position and momentum operators.
What has not been done is to develop a quantum theory of p-adic
angular momentum. By the same philosophy, this is identical to the
classification of all the projective unitary irreducible representations
of {{formula:7957c6c1-05a3-4aed-b89a-d272a991fdfb}} . We get infinitely many such representations from
reducing the group modulo {{formula:7339b515-5c23-4a60-9e53-715bd4bee934}} , noting that {{formula:789e0148-8093-4f9b-b5fc-7d23e0f5abdf}} are
finite groups, for which the irreps can be found by standard tools {{cite:20b90287da2e1470c8288b09cfe14b9502ebcbd9}}.
Indeed, in {{cite:d9089227bef5ce4f8ce19fdc53bc09072a493153}}, {{cite:e5f4795c85e9ef041245fa899a6844214ddbbfa3}}, this has been done
for reduction modulo {{formula:0bf76313-9f3f-4957-a12f-79c366670874}} and {{formula:d1a4f67b-b4a2-4c90-93c8-c157036151fa}} , for certain odd primes.
An open question is whether all irreps of {{formula:2a4e14c5-5063-4d7b-b829-803761863969}} arise in this way.
We suspect that this problem cannot be solved using standard
{{formula:ae2ed501-18a7-43a3-814f-76c08c922113}} -adic techniques based on Hensel's Lemma, but may require non-standard
{{formula:15b044ef-c756-4aff-aca2-5b1e9048a9be}} -adic analysis and Gretel's Lemma {{cite:edb3769a800792984f29c8a49897cdd397ca4738}}.
| d | de761f0b0f0411ff34f16985553f2efc |
For the experiments, the network weights and biases were initialized with unsupervised greedy layerwise learning of SDSAE. The weights, biases were then fine-tuned with each of the proposed fine-tuning procedures. Finally the FRC mentioned in Section III.B was trained on data obtained at the last hidden layer. The results for all experiments have been tabulated in Table REF . For each dataset in the table, the first row presents the average classification accuracy values (Accuracy), the second row presents the average number of rules generated by the FRC (Rule Size) and the sparsity parameter value {{formula:9e33a120-bf9f-4030-8b82-1c73973b5993}} that gave these optimal results. The third row mentions the network architecture (Net. Arch.) by mentioning the number of units present in the input and hidden layer(s).
The network architecture was intuitively selected, and multiple architectures were not tested and compared. Bengio {{cite:d6503a37a1ddc1f059794d23f65ef11b151e6fe8}} mentions that as long as the network has sufficient flexibility to model the non-linear relationships between features, the performance of networks does not depend very strongly on network architecture. Deciding the optimal network architecture for the proposed FRC, is beyond the scope of this paper.
In general, an appropriate specification of
{{formula:03e9bab5-ddaa-4957-89c5-f6ed244e5954}} without prior knowledge is difficult. Moreover, as networks become deeper, the resultant features and classification performance can be significantly affected by the value chosen for {{formula:10845501-f128-4b42-afb7-ac894906e06b}} . To find a reasonable value, {{formula:ac74ad5f-a5b2-4d55-ba9e-cb2871122b6b}} was varied across {0.1,0.2,.....,0.9} and the value which gave the best average classification performance was chosen. It may be noted that during the layerwise training of AEs {{cite:30183dbd9d296b641474b65495c9351f211ff74d}}, {{cite:7b70d23ab3031ae25a78e6363ff37a2f09d02fd0}}, {{cite:20fc83f10d0a28138f00d9280f70dd2150c3bcf0}}, white noise of 10dB was added to the input of AEs for incorporating the denoising constraint {{cite:4bf126acf3e39467b16ba21e7cae0938010e4d8a}}.
{{table:24b2be62-50f4-4fa8-947f-85773edccead}} | r | f796fa8ec51776294abda2fc462cbfb0 |
The CNN architecture consists of 5 convolutional layers of 64 feature
maps and 3-by-3 convolution kernels, max-pooling with
size and stride of (2,2), and two fully connected layers as
illustrated in the figure REF . Dropout(0.5) is
applied to the all convolutional and fully-connected layers to
increases generalisation {{cite:35aebfd7f0c73ad36aab6a1f60f435ab906b9238}}.
This system showed 75% of accuracy at the end of training.
{{figure:9a99ffe0-4931-4c30-94b4-901aa546b83f}}{{table:8b977095-93db-4ae0-8a41-3284caeffd27}} | d | 3f5d0a691d0c74391f014e6d6cf334d6 |
Low-ionization emission lines (LILs - H{{formula:c003ac30-21bc-4de0-a794-35329ca0fdb3}} best studied feature) in quasars frequently show asymmetric profiles. The H{{formula:26df4a13-3697-429e-8770-cd45077649ed}} broad component (H{{formula:b73f5454-bb20-47e8-8a82-36426040332d}} ) in Population B sources is usually well described by a double Gaussian profile with one of the components centered on the rest-frame and the second one redshifted by {{formula:bd50d8ae-8a40-40de-8be7-e85cbeb26abe}} 1000 – 3000 km s{{formula:f7cd2e13-780f-4c3f-b526-a144b57eeab1}} {{cite:bec397575acafbc46cd2fcb730ca5cb87f27e218}}, the very broad component (VBC). Population A sources rarely show the redshifted component. High-ionization line (Civ{{formula:cb1984c3-f28d-4bb0-b1ef-466c54885ac6}} 1549 best studied prototype) profiles usually show blueward shifts/asymmetries in Population A sources. Pop. B objects also show weak or moderate strength LILs like Feii and the Caii IR triplet {{cite:7ef42f9ba42119f559d910d875bd7adac317bd6a}}, {{cite:10a5b7fdcbfa9f6dfc9452524741abbc1248383f}}. Usually Population B quasars do not show any strong soft X–ray excess {{cite:91e5114706857290e7f376c940c13261ff9187c3}}, {{cite:0e45eb4dcf5161f6a101c4536827518f6e003bbe}}. Largely radio-quiet (RQ) population A sources usually show symmetric profiles well–modeled with a Lorentz function {{cite:2a42aa4cccacc3ddd069eb505de53c8927c47cc6}}, {{cite:bec397575acafbc46cd2fcb730ca5cb87f27e218}}, {{cite:bc69386847701439f581906e32abc69e373748a1}}. Population A sources with the narrowest H{{formula:1e761298-49f7-4f5d-b72f-dc5a1491ffe4}} profiles ({{formula:5c450316-e500-4ef4-b29d-47a313c5945f}} 2000 km/s) are often called narrow line Seyfert 1 sources (NLSy1), but in no sense represent a distinct class of quasars {{cite:aa1f43a02829518711b57d368087ff7ad1644b5c}}, {{cite:2d9667bc389d5abfe7cffd7e13f0163ec246963f}}.
| i | ff6495a43b089bfd78ca1bd937bb6c0b |
We have included three DNN models for denoising: QRNN {{cite:448e2689d98697b842f80a3b9a61b7b43b92989b}}, MemNet {{cite:2d732dc49e664fa823828135d0485885fa6e8bc1}}, and HSID-CNN {{cite:f2cbe6dd3e4502242bc9e0997f2fdb218a2cfec2}}, {{cite:448e2689d98697b842f80a3b9a61b7b43b92989b}}.
The base implementation of QRNN has 2D and 3D versions, where 2D and 3D refer to the type of convolution used.
MemNet and HSID-CNN cannot use 3D convolutions with the default structure, as they will exhaust the available GPU memory during training.
Furthermore, we implemented a version of MemNet with a trainable HyRes step before the standard network layers, named MenNetRes.
Pre-trained versions of these methods are available within HyDe.
While these methods do not encompass all of the state-of-the-art methods, they show how DNN methods can be implemented in HyDe.
| m | 80e7310667b85f381345232c68f5a993 |
The well-established DFT + DMFT method combines realistic band structure calculation by DFT with the non-perturbative many-body treatment of local interaction effects in DMFT {{cite:d4998b50b694df069c33b6e9e7ceee4445722c76}}, {{cite:9ba5b397f9bb305bf7adc58254144073872596f5}}.
The DFT + DMFT method has achieved great success in abundant correlated materials. It is really appropriate for describing electronic structures of strongly correlated materials. To account for the strongly correlated 5{{formula:436e76db-6934-49b3-ab55-ea924954894f}} electrons, we perform charge fully self-consistent calculations to explore the detailed electronic structure of PuB{{formula:00d3187c-31bd-44f2-93fd-bb26ae63e507}} ({{formula:b9129ba9-8f0b-4b42-8fab-0fbff0106eca}} =1, 2, 6, 12) using DFT + DMFT method. The implementation of this method is divided into two separate parts. The DFT part is solved by using the WIEN2K code {{cite:f7239f41a1213c2fe9e603424f773c42a3b9682e}} which implements a full-potential linear augmented plane-wave (FP-LAPW) formalism. The DMFT part is solved by employing the EDMFTF package {{cite:6ad0c0bd02b181cac0a4a2ea85551e0836f7e817}} which preserves stationarity of the DFT + DMFT functional, and is able to obtain high precision total energy and force.
| m | b1ebf293cd2d6aacbe8ac15f1ecc8a95 |
For context, we first sketch the derivation of the electronic contribution electron-hole kernel matrix elements, as detailed by Rohlfing and Louie {{cite:bc71d6607f2d24ad80f1817c88096f37a7bb2e0c}}, {{cite:4956096e6251787eb31bf9fed0806634b0845ce0}}. Adopting a plasmon-pole model, we express the electronic contribution to the screened Coulomb interaction in real-space as {{cite:bc71d6607f2d24ad80f1817c88096f37a7bb2e0c}}:
{{formula:76062b7f-22fa-40a8-af82-f9e7603304d9}}
| m | 44ee2c5e4a9319041266f4ad9f895807 |
Ytn = tTgn(s, Xsn, Zsn; (-n, h-n)(s, Xsn)) ds - tT(Zsn)T dWs,
in the sense of (cf. {{cite:869efe4ae363b2f54cf2e563f796b5454bb51ca8}}, {{cite:a1a98d4f69ae62323000f213239260cfbbad5dcd}}, {{cite:032a6ba183ec7b507c94d8ca0dee9a91a59fa768}})
{{formula:ee1f20d1-d636-4a5f-bfe3-d9b4ef84b9e3}}
| m | e019d2859cd60ded1032c9ca74a5da14 |
and use a line search algorithm {{cite:a75d15d4d68d275033416969505c4aa716c402ce}} to find {{formula:97cb88b2-8baa-4730-8fb4-2df9fdd7cd32}} , and update {{formula:2aa61f6f-7f5e-4714-b5d6-573a79e0657c}} .
| m | fa41fbe32d8e58ad4912fab040ce2e6a |
We prove in this paper that {{formula:f7f73772-a460-43aa-bae9-3a50a760cbe0}} is a {{formula:cc6a8c16-545f-4e80-9b76-9197f830cd5d}} -critical scheme in the sense of {{cite:f6f233e07b26474409df35681a8f06b5f2ce8928}}. Of course {{formula:9f1bfe6c-ec04-45fd-a126-6a31464fb21d}} does not always come from a {{formula:fec74ae9-8b1e-4462-9406-1c13c2e66f25}} -shifted symplectic derived scheme.
Also {{formula:fe3ddc87-5811-4b54-a070-69c088ac44c1}} does not always have a symmetric obstruction theory in {{cite:a1cfd16945c24f906187e0a485e17ed66154354c}}.
Let {{formula:6ee82a5b-733c-489b-bd27-88f4886bb5bb}} be the canonical line bundle for the {{formula:04ae5d5a-58b5-4337-a732-fe58fb606a0e}} -critical scheme {{formula:58284dee-94b3-428e-aad1-2666cdd9bad5}} defined in
{{cite:f6f233e07b26474409df35681a8f06b5f2ce8928}}. The {{formula:4288eeff-3be8-4bde-811c-82196b4fdf69}} -critical scheme {{formula:1882c619-b8a2-4ac4-a0f2-4eb34bf24cd6}} has an orientation, i.e., a square root {{formula:a4143f18-e837-48f5-9d17-998e9ada6e9e}} exists, which is proved in {{cite:df1201758c8b6797a5a158b2a68bc36f53f2c2e6}}, or {{cite:e1bd65e5e2a52d339e49deab85dba326b203ba44}}. So from {{cite:e77dec3cc1257d8f8e0ecc991c5a92350af364b0}}, there is a unique global motive {{formula:05515f36-62d4-475d-b75f-3fb1c57fa17a}} , where {{formula:2f139387-1e81-4bd4-8124-3d18254c682e}} and {{formula:5879f345-7650-4af1-8050-29ae6b625236}} is the
equivariant Grothendieck ring of varieties.
On each {{formula:d0e722e8-5467-409e-b8f9-faadcb8cac88}} -critical chart {{formula:1f4df0a0-dfb1-449c-8e49-5d7a37853be6}} of the {{formula:6965c5e3-1b0c-4473-a906-4e798624e505}} -critical scheme {{formula:45922268-7918-4912-899d-95ca851b4d32}} , the motive
{{formula:e5396364-a456-48ab-a273-70d24d4d8775}}
| i | 7486e4bf0300f4d29e831dfbe70cd14c |
Through the visualization results in Figure REF , we find that the position encoded by the CNN model is the main reason to cause the shortcut solution. According to the previous work {{cite:bd37aa090aaea62b2eb08f17d20301f38cc18dcf}}, the position information is mainly introduced by the zero paddings. So a straightforward way to avoid the shortcut solution is to use different padding methods (e.g., replicate padding and reflect padding). Another strategy is to give up padding operation and crop the feature boundary as the network goes in deeper {{cite:da05f1c224ae42febfb231071d98d40315ce10f8}}.
We first conduct experiments on these methods and present the results in Table REF .
We observe that these strategies can improve the baseline accuracy and alleviate the shortcut effect, but the performance is still not promising.
This is because changing padding methods do not eliminate the source of the position information, although different padding methods make it harder to learn to encode position information.
In comparison, our proposed STFC{{formula:82a2fa70-0d3a-42fe-b57e-606f65b8e807}} method achieves the improvements of 29.6 [email protected] on J-HMDB for pose tracking, -37.4 RMSE({{formula:c310165e-458a-4d79-9a8a-a33df4a1e87d}} ) on 300VW for face landmark tracking and 42.5 {{formula:d261207b-f883-49ef-8881-bb9cc4033ad4}} &{{formula:fad5c52d-1bae-448b-acad-88458e9e2363}} on DAVIS-2017 for video object segmentation.
This strongly demonstrates the effectiveness of our method to address the shortcut issue.
| m | f652b8334eaa9157a61344d3b8aa80ec |
After sampling, we had for each image 51 points, in ascending order, that represented the luminance CDF curve. Based on {{cite:e4c9057fcc23c3b5035263eb163b48d65780d85a}}, we implemented a machine learning framework with, as input, the dataset of 51-sampled raw images and then each artist's 51-sampled edited images. We used a Gaussian Process Regressor (GPR) with 5-fold cross-validating.
| m | 8a5910f1298f7d2eec578e0a5b5c67c2 |
The multiple testing problem is fundamental to high throughput data, and the necessary statistical stringency
makes detection of features of interest difficult unless their effect size is large. We have shown here for
representative data sets that the power to detect most true effects is very low. Unless this problem can be
overcome, then the promise of high throughput data may never be realized. Weighted p-values provide a
framework for using external information to prioritize the data features that are most likely to be true
effects. Many studies
{{cite:81fc18d1a7a5e2c955e1a77c677456b4f5083b10}}, {{cite:4adb804d3ab18866b46e7325c10afa6cb1e33c5a}}, {{cite:f641411cf8d0d3bbdd0e752d97e453c2ff598208}}, {{cite:4ed49449607c0a34fa8f4c71f2bd7dc6e0cf4e14}}, {{cite:8f29e7678c8436bd4716fb106184382849bc10c5}}, {{cite:8a6f76578ae9efd5dd5794b6b8b583daf63ae1c4}}, {{cite:6a2e278c045dc64b6a209872b59d3bffc58cf75d}}, {{cite:f548e188d09337b355bcd1e5076ed3959a69773b}}, {{cite:d038e85c886f4cbbb16a2f7be6bc4770562acfab}}, {{cite:7b393ed6bb768173ed4dd6e2992442bf2d83a543}}, {{cite:4feb878a3c18827222b896b1885889a871c2f0d5}}, {{cite:d6ab796642f3dc25f487334d8153077ef0e14a28}}, {{cite:5231ae541692b633ddf3a10f394c5810bb1bf5e3}}, {{cite:a070ddb3b423dfe4aea4d71390e1cf34ffb76b2d}}
have proposed methods of p-value weighting and techniques have steadily advanced. However, all existing
methods require either 1) difficult to attain knowledge of effect sizes or effect size distributions or 2)
grouping tests by the covariate and then estimating properties of the groups. While group-based methods are
powerful in many scenarios, their effectiveness may suffer when true features of interest are rare and/or
have effect sizes such that power will be low.
| d | 83b490704b67a00236e47dbc6afe3f54 |
On the other hand, an additional coherence known as SGC can be produced between two degenerate lower levels with non-orthogonal dipole moments in a {{formula:e4ccf0c7-264d-4d35-aa41-0820ac914cd4}} -type atomic system {{cite:02aa228b704ce35966f10af68e9abe4be100b2c6}}. Some interesting results due to SGC have been observed such as spectral line elimination {{cite:e7e473f19c4ca7c677fc13f4a40604c5105ba9aa}} enhancement in quantum interference {{cite:13215bca8986fb0bad2172501e7c23b39c406822}} and gain without inversion {{cite:bfcc1f3e77f3e2721d66f3c2f8ef049c0149cfb0}}. In this paper, we investigate an improved version of the nontraditional QHE by utilizing SGC between two lower levels. The emission and absorption cross sections related to the output field strongly depend on the SGC co-efficient. A coupling field and two thermal reservoirs interact with a three-level EIT medium in our proposed system. SGC compensates the losses in the EIT medium and as a result, the spectral brightness and emission cross-section of the QHE can be improved.
| i | 45b00bcae00cd8e2dc9c17b7eaa6a444 |
The alternative evaluation methods we find more promising are qualitative rather than quantitative.
These aim only to check whether {{formula:ebf6ad1a-aca9-446d-8f50-7fde73c22d19}} is very wrong.
Perhaps the simplest approach is to plot {{formula:9cccdf2b-00ee-404e-8b4f-b7c76b4be99f}} together with {{formula:2090d849-a37c-4b80-a848-da2e01846b71}} for an example where the true posterior is known.
This is ubiquitous in the literature {{cite:aa8ebe20641a15f7a0b73d3046df8daa5e9b7e06}}, {{cite:bfc791d1d2b9b35d682378236b6160de203771d8}}, {{cite:c9e2ed80c91068a3a5c792ba0987ea552fb7009f}}, {{cite:f0f0b756f2dab35da306612a4aa87c502d8472a1}}, {{cite:374f859dadd8081410114ba3c8391ce9c9ad8fb4}}, {{cite:3da2c045623a836477878b46ca0f22c696162837}}, {{cite:c78df80226b148b54a6ad2072309d139f9495bd3}}, for good reason.
Plotting is easy to implement and makes the most important differences between {{formula:4ca9dd3c-154f-437f-8950-2edace6d6860}} and {{formula:fcb8222d-1411-4413-a059-3caef1b1263e}} visually obvious, without requiring the user to decide beforehand what features they care about.
| m | 421bdff493108eeffec3a9eaa1692d30 |
Finally, consider the setting where the quantum state {{formula:bdcf3e8d-1b16-4c05-8388-2ebb4ada7a94}} is obtained from an exponentially long classical vector {{formula:eae9edc4-cae0-4997-bc7d-6b9c99f618ed}} stored in classical RAM (random access memory).
In this setting, constructing an SQ access is less demanding than constructing quantum state inputs (although both require an exponential amount of time). Hence, it is reasonable to compare classical algorithms with SQ access to quantum algorithms with quantum state inputs when the quantum states are obtained from classical data.
We emphasize that in this setting, quantum algorithms that have been dequantized {{cite:f733d97d0d4511ac396f2a04a7199da12e5c728c}}, {{cite:57a79debb771ff85ff505a044aeb31e6b28f9e0c}}, {{cite:8822f82241847e7dd4ef9c2bf4f6e01641de848d}}, {{cite:34fccee3d63f26819803059baf3dc82d620e3db1}} do not yield an exponential speedup.
Furthermore, our results show that classical algorithms could be significantly more powerful than quantum algorithms with quantum states that encode classical data in the amplitudes.
| d | 5b4e84325a452643ab0476066b73393a |
Regarding future work, there are a few ways one might consider building upon what has been proposed here. Firstly, a natural extension of our models is to consider a mixture model, with our SIS or SIM models functioning as mixture components, which would allow one to capture heterogeneity in the observations, opening the door to answering question (c) of sec:intro. Secondly, on a more pragmatic note, one could also take steps to scale-up our approach computationally. For example, one might be able to circumvent the need to use the exchange algorithm if the normalising constant for a particular distance metric was derived, as was the case for the CER model in {{cite:25f2f1e25694a6ced8d9f2b92c778bd9b081fe24}}. Finally, if one is able to make an exchangeability assumption for each observation, that is, the order in which paths arrive is not of interest, then a slightly modified model structure could be considered, reminiscent of the latent Dirichlet allocation (LDA) model {{cite:455db397c468f7376e9e9f831ca554f1efd9ac8a}}. Namely, one could assume each observation was drawn from some mixture distribution over paths, with mixture components being shared between observations but mixture proportions differing. This would also have a natural non-parametric extension via the hierarchical Dirichlet process (HDP) {{cite:a431ced460b27db30bc36e46c1d02930528288e8}}. It would be interesting to see how the inferences from such an approach compare with ours, at least qualitatively, and whether any computational benefit would be achieved.
| d | 18feb0cb108ad3ffd662b2a81009b494 |
Prior Work: The existing work for RIS reflection pattern design normally requires explicit channel knowledge {{cite:142fe5ca2281113201d9b114dc70d69ffe1a8aff}}, and does not focus on designing a codebook {{cite:9afa3c452f55c57c64d8a1b1a3edc34fb601d9c5}}. Furthermore, the reflecting elements are normally assumed to have continuous phases for the ease of optimization {{cite:3464fce6c6c8821af6f156ee7c0fef2f740416ca}}. Besides, the prior work on RIS beamforming design {{cite:142fe5ca2281113201d9b114dc70d69ffe1a8aff}}, {{cite:9afa3c452f55c57c64d8a1b1a3edc34fb601d9c5}}, {{cite:3464fce6c6c8821af6f156ee7c0fef2f740416ca}} generally ignored the possible non-stationarity of the RIS channels {{cite:ec21bc00dc5761d3935607131d461ce86ed0a09f}}.
| i | 08af9b5223e13a5ba7725354197c8524 |
where {{formula:9409378e-8f37-4ca9-b1a2-4db587e68088}} represents the nuclear norm {{cite:6a62ae7290e1e78d26bf8fc448640232d9dcaa0d}} ({{formula:32201673-7001-4ee9-9abb-9aafa9090aab}} , with {{formula:ac6a8934-6990-423c-89f1-abcb97b6eeb2}} denoting the {{formula:6cc935ba-5609-4460-ad09-ee60c0e941d6}} -th singular value of the matrix {{formula:ba3b34bc-cbf5-4c67-9b1e-5fa896da8084}} ) and {{formula:6fcd53d2-b66b-4f8a-83bd-cde1ac37d272}} is a positive constant. While singular value thresholding (SVT) {{cite:6a62ae7290e1e78d26bf8fc448640232d9dcaa0d}} provides good theoretical guarantees for NNM, all singular values are shrunk equally by the NNM algorithm, which neglects the different significance of different singular values and thus leads to an unsatisfactory matrix rank estimation. Meanwhile, the NNM algorithm is prone to over-shrink the matrix rank components (i.e., deviated from the original rank components), thus limiting its capability in various image reconstruction applications. To overcome the limitations of the NNM algorithm, most recently, many excellent rank minimization algorithms, such as WNNM {{cite:3438f3f340703cdf95e078ffe2300cd316f75f56}} and rank residual constraint (RRC) {{cite:5163fd94724b346bd51f707e1e58fe8ba261a6e4}} have been developed
for computational imaging settings.
| m | 72ac0f288c2efb12b23a1d36ac54082f |
Let {{formula:301603cc-d28b-4d4b-82f4-b5362b4a71d5}} be a partition of {{formula:b65f5c17-88e8-4ac2-b2ea-38936ab6fe55}} , {{formula:7453f859-3c35-4471-9fe2-82b7c9d5a7d7}} and {{formula:c667b35b-8f5c-48e4-82f8-19a7e3ac2f59}} . To construct a numerical scheme, given a sequence of values {{formula:f2c74aa1-4e13-4228-b51e-5a97656e45df}} we need to define an approximation of the fractional derivative {{formula:76956cde-a352-4e67-a6b8-eee0c4f97d12}} on this grid. A standard approach is to apply the fractional derivative to a linear interpolant of this data. This is a standard approach to constructing numerical methods including Fredholm and Volterra integal equations {{cite:2be5ee216e1f9a9f533f515c636da81b6032c2c9}}, {{cite:c603428693a8a4c523605cf8c7e7f90a6e92cadf}}. In the case of fractional derivatives this approach is often called the L1 scheme {{cite:abac149bf7d3ecc14d8360b25a52f8dae9a35638}}, {{cite:e7b66541c63bbb31218b5d669575965c45fc7c49}}. We give the details next.
| m | 9d56b1de275647944608f767816dcc71 |
with {{formula:db54f2fa-f767-4dc7-8e5f-e780cc34cfae}} denoting the mutual information {{cite:3c2cc2ad709184118ffcc6aa0a8d4f91f013d4d4}} between X and Y, and {{formula:5c7ecaaf-ff2d-4c64-ba72-781e693ca284}} a non-negative real number. It is well known that the constraint set is convex and compact with respect to the topology of weak convergence {{cite:891b5efe65f528bee56a7c7c76639e72ec7a8422}}, {{cite:565387a9436cb7f32922a8b406dbdb4a206e8a45}}. Using the lower semi-continuity of {{formula:bd8cb004-20da-4b41-9151-f20d13a64580}} and the compactness of the constraint set, then, from the extreme value theorem, the minimum in (REF ) is attained.
| r | 88b45007e61d19b29eb87873bfa4677a |
Generative modelling describes a class of models in which we want to learn a distribution {{formula:2bbcbfb4-69a5-4918-beae-864f65e78670}} to approximate the true distribution of data {{formula:bb6ee921-b4f6-490d-8938-b40bea582f52}} in a dataset. Understanding the generative process behind some data is often very useful, as it helps us build useful abstractions and causal relationships are typically naturally embedded {{cite:c79af4071222bff2c5c5a69d384b42dd82f9f18f}}. We can sample directly from this distribution to generate new example data that captures some statistical properties and features of the data. A subset of this are latent variable models, which condition on unobserved variables {{formula:b9fd97f9-002d-48da-a502-332b30dcc111}} , with a family of deterministic functions {{formula:9fcb5676-6feb-40b2-9891-86ec9defb5ed}} mapping {{formula:059ff2c5-b973-4753-968f-fb1a80b8fea2}} to the input data space via a decoder network, which is parameterised by {{formula:5d667f69-aebd-40e0-902e-1acb6f08ab54}} . The marginal {{formula:ede3f49f-7f45-41a0-9681-491923f5de8f}} is also referred to as the model evidence; the probability of seeing the entire data under the generative model with parameters {{formula:2d18dd4c-85a3-4946-a6eb-058e1e2a3eb7}} . The aim is maximum likelihood estimation of {{formula:3c8343ef-6c00-45e3-97f3-b44df9ddeed0}} with respect to {{formula:3203c396-f0d5-4c58-8a32-efce0e66703f}} , but this is often intractable. Variational autoencoders (VAEs) {{cite:2d1c6b51f743c11a4d2078f56310143a3dff8c15}} resolve this problem using variational Bayesian inference together with probabilistic graphical models, reproducing this `bottleneck' architecture and borrowing the name `autoencoder' due to the presence of distinct but coupled encoder and decoder modules that give resemblance to the high-level structure of a classical autoencoder (Figure REF b). The encoder describes a variational distribution {{formula:a12d314c-d146-45de-a8f5-f33e7f24e72b}} that approximates the true posterior {{formula:1f145cd4-4389-4359-b05c-e3604f17de4d}} , which itself would provide an ideal encoding of {{formula:5a5ccd6e-91f1-43e7-8e02-bd1d40a9e840}} into {{formula:19eef3ca-b4d7-4f5a-a5fc-0b6dcf8d82a6}} . During training, it learns both to encode some information about each data point through the latent representation {{formula:0334b08a-e031-4647-b8a9-6f88e98873ee}} and stochastic sampling from this distribution allows the model to abstract and generate new candidate data {{formula:251f54c8-f891-4902-b3d5-62a64c0c0995}} that accurately reconstructs inputs from the assumed underlying random process. This process enables it to discriminate artefacts despite never being explicitly exposed to any, since we assume different latent mechanisms govern the waveform behaviour of any artefacts, and so model prediction over an input containing such an artefact will have low reconstruction probability and high reconstruction error {{cite:4b0c109f046088f32284690e2bc552ca4495e448}}. For further detail, comprehensive summaries of the VAE can be found in the literature e.g. {{cite:c79af4071222bff2c5c5a69d384b42dd82f9f18f}}, {{cite:e3b57ed3b13808370cca579ed1ac195dbc74c2e4}}, {{cite:5a1180b111824aa7a80e8ba8be7155bea2d410d7}}.
| i | 36da41273e7f4701d1210b956b81b47e |
It should moreover be possible to remove the condition on {{formula:e702c40e-9f61-41fd-8c35-241655e87187}} in the non-acyclic case.
It is currently used to produce sections of determinantal line bundles, which we use to check local reductivity of the stack of semistable representations in order to apply the existence criteria {{cite:f21632fe4668486660fdb11135f01d3ced009eae}} in arbitrary characteristic.
If {{formula:983b06cd-05b9-4496-a2dd-073187e2daa4}} has characteristic 0, we can use good moduli spaces instead of adequate moduli spaces,
and the required local reductivity follows from {{cite:b787dd0fe098bac38d25df54c5bb399a4641daaf}}, {{cite:f21632fe4668486660fdb11135f01d3ced009eae}}. We also use determinantal sections in the proof of semiampleness.
| r | 8e02c35a0c29841081eecbc967edff0a |
The theory of one-parameter {{formula:5f893370-4481-4cc8-ab3a-bf752a89834b}} -semigroups (strong continuous) of linear transformations {{formula:335eeb62-15d7-40ff-a01b-8b8e2df949d5}} on the Banach space {{formula:d65ade9e-1e5d-4914-a187-859536bfe9ce}} introduced in the pioneering papers {{cite:a3851a8aa68297e714b31eeca77535c9de956678}}, {{cite:1a2ba40ffb6ce3109cf2897faf7f9c0c43f52fd6}} states the conditions for the closed linear operator {{formula:54f81b3e-071a-4539-a011-046755af8184}} with a dense domains {{formula:8b91ff34-9eaf-43bc-b0fe-1a486a98d8c7}} to be a generator of {{formula:70e26561-3a94-4c2e-8f82-39c2fe647e06}} such that {{formula:08c56260-cd30-47d7-be41-d43249fcf08e}} . If {{formula:3cc21306-162c-49bd-b8dc-a875eb6ac679}} is the Banach space of nuclear operators in a Hilbert space {{formula:53899952-b8d3-4e6b-ab95-a21ff4cbbed4}} the claim of strong continuity for orbits of {{formula:7fd38991-4d00-443f-98dc-02d4fd5a97df}} possessing the property of non-increasing a trace is equivalent to weak continuity that is {{formula:7392624e-e45a-4e03-b9b0-7f8aa931028a}} if {{formula:a3fe7977-0f32-4964-829e-f4e611fc1a48}} for all {{formula:a7ed662d-11b1-4468-8fc5-d742d698b270}} and {{formula:07081f92-fd7f-451a-882e-49bc495f95f5}} (the algebra of all bounded operators in {{formula:2d4d8703-a28c-4810-9ddf-b6e113de1aa9}} ) {{cite:92793676a8b4b5d56c12909d7a91834d58bd34f6}}. In {{cite:1a81ee8c3073721f5b190df3f6e9d68f62a939a2}} it was shown that the perturbation of the generator {{formula:b541af06-fad3-4c44-8610-4d725e976b77}} by a linear map {{formula:9617de8e-6aec-4cb5-8066-daf2143f7794}} satisfying some additional conditions can be represented in the form of integral equation including the operator-valued measure generated by {{formula:905c7231-0754-4da4-89df-bd1ed35c7b93}} . Together with a perturbation of {{formula:221626cf-6786-4d18-a932-b3af65f6f89b}} it is naturally to consider the corresponding perturbation of the adjoint semigroup {{formula:47520801-70eb-4415-b75b-9b3102649469}} on the algebra {{formula:b4986e8b-0dc7-4a06-b737-ec00dab754c8}} . We realize this construction starting directly with the measure. As an example we construct a perturbation of
the semigroup of non-unital *-endomorphisms on the algebra of canonical anticommutation relations (CAR). As a result of this perturbation we obtain the flow of shifts on the CAR algebra {{cite:057ff0bc3f569eeeac38af49c472648bf8bee83f}}. Earlier we have announce our result for the CAR algebra in {{cite:ffcff1337b45bd124df55301af9116f3bb552eed}}. Note that the perturbations of a semigroup on {{formula:715dbde0-85b3-49aa-824d-2eaa98fcbc2d}} generated by the perturbation of the generator {{formula:40a472e6-99ae-4344-92d7-1d26023f3a40}} of the corresponding preadjoint semigroup on {{formula:58ceb2f0-0655-4248-bbaa-2fdb58f3b616}} by a linear map with the domain containing {{formula:e848239f-2140-4180-892a-6bb8a15a079e}} form the basis for the construction of non-standard quantum dynamical semigroups {{cite:6919a50b928828330d0631bf137bd65e5252384c}}, {{cite:91ea5f7cb9c88b5e5f35c28b3a4410dc36ea88d1}}.
| i | bdc1c6d2b977e38f0cccff3ef3942e1f |
While Hamiltonian systems are important in the mathematical sciences, there are some important restrictions that limit the general applicability of the Hamiltonian framework. Amongst the most important restrictions are systems exhibiting dissipation and systems that cannot be brought easily into a canonical Hamiltonian form, such as dynamical systems with an odd number of degrees of freedom. While for the latter, the more general Poisson geometry and associated Poisson integrators are available {{cite:2af0ccd182f2cd02ae363668afdd6f8e65d64059}}, these numerical schemes are not as universally applicable as the symplectic schemes for canonical Hamiltonian systems.
| i | 241cafa3a57cdc83ce8894945390d6f5 |
The artificial potential field method and its variants are a family of
planners whose instances offer simple and fast computation for mobile
robot obstacle avoidance
{{cite:d7d840502fab11bad0e87c2140ea0106cf05dc90}}, {{cite:eb0c15f1e47c544308a926e5a96dd8ee6dadb438}}, {{cite:9ac1eea509ecd77927eff195d548dae249f69492}}, {{cite:bfcd7f084d3ec1cba4156a1e6f9d8b4324438595}}, {{cite:5c5525822c43df1ad3ea159a93e40d22dd05357f}}.
While the potential field is particularly attractive due to its elegance
and simplicity, there are substantial shortcomings inherent to this
method such as a lack of consideration for robot kinematics, dynamics,
as well as local minimum problems with regard to world geometry
{{cite:51a62d8bfd9e862d80febda40becdb193935d5a7}}, {{cite:7263062850f3be2460cc8a34d608eaca41169b05}}.
Significant efforts were made to alleviate those
problems {{cite:67af61b124ece358bef5652312b1facebc29279d}}, {{cite:5453c40592ae3021d9df051301306cce644b6fcb}}.
When implemented as reactive planners, the family of APFs nevertheless
directly map robot state and sensor observation to available actions,
offering better computational performance than deliberative planners.
They share valuable traits with perception space methods,
such as minimal sensor processing and planning complexity in the
egocentric robot frame. Integrating reactive methods and perception
space methods can leverage their fast compute properties and the
limited deliberation associated with local planning modules informed by
a global planner.
| m | e8c99e0d987e21c1ce34bf06f20608fe |
It has long been suggested that this generalized interacting stellar winds model of PN formation {{cite:a819972414dbf05b3c4699a550ca50a563ea50b5}} is too simple to explain the wild variety of PN morphologies {{cite:33d1524637c9ff4cff781762d45b785a18c9d802}}, {{cite:af19b4f54908de28750fcb4399325ad5871c92bd}}.
Several mechanisms were soon invoked to explain this conundrum {{cite:8bbaa16909f7e04f2f76761324fc37e16bb2cab0}}, involving binary star interactions, including jet-like ejections during the late AGB or early post-AGB phases {{cite:9e30983441c0815020b60ddba4ac9500d8c45bdf}}.
| i | 3713008eb38765b8c2e78db775a903e4 |
A traditional metric to calculate the distance between two lists of ordered elements is the Kendall's {{formula:f6ff2a52-9950-42ce-a90b-09b3abcce777}} distance {{cite:1829ff4596a3034e242443e1dd62858f200eef78}}, which considers the number of pairwise swaps of adjacent elements necessary to make the lists similarly ordered. However, Kendall's {{formula:6a8c224c-00ba-4198-99c1-f237d831aaa3}} distance assumes that both lists are composed of the same elements. Since we are interested in the evolution of tag vocabularies over time, this assumption is not valid in our case: tag vocabularies are likely to contain different tags at different times due to the constant inclusion of new tags.
| m | c0e2788a3080a903bdef002947b18f21 |
The numerical solution of FP equations has been widely studied. There are several methods (see e.g. {{cite:11a22fa3fc0bc0b3f86a0c645e55bbb1f5a62025}}, {{cite:93c20612d073a557bf1d48d5dc14ed6d56d81950}}, {{cite:26c280879e5b539f60fe937d2b9610824990d4df}}, {{cite:c96407b399269cfc53b9d98c58c9c5c6618cb717}}, {{cite:491d1c8fbf0c7eb5bff5e9e7f931d1170c185318}}, {{cite:d62d2993698ab80139d2e7b731c69b1d890da231}}, {{cite:1721df4e1848861688b7ca9e797d8e3c0b252e65}} and the references therein) based on the popular finite difference scheme proposed by Chang and Cooper in {{cite:ee2259666e58bdb7945472e6c4915337b6eb73cd}}, which, in order to be explicit and stable, requires a parabolic CFL condition on the discretization steps.
In the framework of MFGs, in {{cite:5ce0a89879a7f7eed06ffb6c3d071785c9d6f650}} and {{cite:68b8a8350ab4f6d1a9df461b33106a68bfc66dbf}} the authors propose a semi-implicit finite difference scheme and a Semi-Lagrangian (SL) type scheme, respectively, to approximate the solutions to FP equations. The scheme proposed in {{cite:68b8a8350ab4f6d1a9df461b33106a68bfc66dbf}}, which does not impose a CFL condition and hence allows for large time steps compared to space steps, has been extended in {{cite:c9737f3871e7729cb16723b522aa5b6d0e5abcd9}} to deal with nonlinear FP equations and in {{cite:35d44de77f664eda8ec952d579a29c775f9feabf}} to approximate FP equations with non-local diffusions terms.
| i | cb88d60cb55a41ddf2f531fd8e1f0d3e |
Before estimating the freeze out temperature, it is first useful to expand the annihilation cross section as a power series in velocity. Because DM is non-relativistic at freeze out, the lower velocity modes will be the most significant annihilation channels. This expansion essentially encodes the temperature dependence of the annihilation mode given that {{formula:b052d4ae-9060-4499-a314-c358b02edecd}} for non-relativistic particles. In many models, a single annihilation mode dominates. In this case, we can parametrize the thermally averaged annihilation cross section as {{formula:fbcc837c-6850-4aff-9420-3d50bae2bd43}} where we employ the dimensional parameter {{formula:6ad0e8ce-1a9c-455b-b946-0b2d82c7b804}} and {{formula:6b54c920-1c5c-4394-a5ff-433efea64f43}} has no explicit temperature dependence {{cite:ea5307a43ac93d428aba19f44c5a46affea7bd78}}. The {{formula:f248786e-13b5-4b90-98cd-2d620b318dc6}} -wave annihilation mode corresponds to {{formula:98d04ad4-26de-4967-b34a-89761091dc8f}} , {{formula:8479c1c3-34b0-49d2-a814-a29b6ade02d5}} -wave to {{formula:05481e29-f471-45fe-91c1-661edaa04d8b}} , and so on. Assuming a dominant annihilation mode, one may estimate that freeze out occurs when
{{formula:90082809-5e75-4e98-adb4-66e6a1c6ab9a}}
| d | 5d58d970a97282f12bc0f08b8a4db66f |
In summary, we have constructed an infinite class of modular eigenmodes ({{formula:8e09eaa1-66fb-4208-b292-71cc72c9dce7}} ) for the single interval in the vacuum of CFT{{formula:e0bb4466-5050-4932-9591-07c68629a55f}} . These are expressed as smeared integrals of the stress tensor components and thus exist in any CFT{{formula:8d94ca60-240a-4c0b-b39d-bb48e79e4ccd}} Such smeared intergrals of the stress tensor has appeared in several different contexts recently, for instance in the study of the light ray operators {{cite:5ecbaa2b92878eb274f7180d50c724c0e85cd9ea}}-{{cite:a6ad44279de6ed9bbcee144e332966b9c84a2278}} as well in the context of the so-called dipolar quantization of CFT{{formula:b1dd9a0a-7b14-4ff5-9422-d072c8165b2f}} as discussed in {{cite:da9efabb4f74d2a485f6fae797a2160910f567d7}},{{cite:e342121033e8887b9b4038b5c5ae11262a541872}}. We thank Bartek Czeck for bringing these works on the dipolar quantization to our attention..
Our construction of these eigenmodes are intimately tied to the causal diamond of {{formula:38fbd271-018d-4d0a-9764-a32c5a9e0dcd}} . This fact manifests itself in many of its interesting features. For instance, one way in which this connection to the causal diamond manifests itself is in the way {{formula:04dfb6a3-2d2b-41bd-9296-9d9672a77f9c}} acts on OPE blocks. We showed that this action is identical to the action of conformal generators on local primary fields in CFT{{formula:c0f60883-0b3a-47ab-949d-03d1973f4a44}} . Coupled with the fact that the OPE blocks have a local description as fields living on the k-space, which is the space of causal diamonds of the CFT{{formula:27c723f1-04aa-4ed0-bd25-e888f644f2d3}} , this hints at the possibility of finding an equivalent effective description of the CFT on k-space. We argued that on this k-space, the {{formula:9ce5c1c6-474b-44c7-ad04-3accc5ee9b33}} seem to generate 1d diffeomorphisms along two independent directions. Unfortunately our discussions are only at a kinematic level, and it would be nice if these ideas can be made more concrete.
| d | 9c98ab374c721fcf54f20e78124e3e77 |
I will devote an extensive fifth chapter to modelling the modes of the bent waveguide, combining analytical considerations and numerical modelling. In this chapter, I find solutions to the Helmholtz equation {{cite:0fcc883bcf92b557bf666c0cbeb8a68786b24b03}} defining the modes of an optical waveguide, going beyond the approximations commonly used in the literature. Thanks to this, I will be able to correctly predict the number of modes in a bent waveguide and their spatial distribution. The correctness of my results is verified with numerical simulations obtained in COMSOL.
In this chapter I emphasize the context of basic research: I show that the bent waveguide is described by equations analogous to the equations of the dynamics of a quantum particle in a space with axial symmetry.
The motion of such a particle is described using an effective image based on a fictitious quantum potential related to quantum centrifugal force {{cite:a5cdf544dc809d3ada18217d8abdc725a226c6ba}}, {{cite:30bc9a0fdd3c0408018d477459dbb8da8768aa28}}, {{cite:6e2bb77eabcf5489261b9f8fb38c96d203ffd263}}, {{cite:ffb034d19ad0aa7b770a97fb4e1a575bbd992bd5}}, {{cite:318da5c064d4e547ebd790c7eb87b25e64d877b2}}, {{cite:e1cece1351446fd30f1239d42bb434f58223f3fc}}. According to Feynmann's famous remark that "the same equations have the same solutions" {{cite:39bc36a37885cd848a94a208d91b70cbf7ddabc2}}, due to the analogical equations, a bent waveguide can provide an experimental platform for studying fictitious quantum forces. The theoretical considerations described above use the mathematical apparatus described in detail in the introductory Chapter 2. There, I will introduce basic information about the statistical methods used in my work and about the properties of non-classical light. In addition, the appendices devoted to the form of the differential operator in cylindrical coordinates, the analysis of the influence of photon propagation in the optical fibre on the photon wave function and the Schrödinger equation for a particle in two-dimensional space with axial symmetry will be helpful to understand the content.
| i | a9e964562f7dbc59ddaf284ae079fa5d |
We have extended our original work on cluster false positive rates {{cite:02b34c8df201f2f483ef3b35b43945ad89262902}}, {{cite:c9bf224532b6af1c653d1ff9d1103bd479968f5d}} to two-sided tests, showing that parametric methods perform worse for two-sided tests. RFT p-values depend on a number of approximations:
| d | 2f98b65c329e972a5a170226842bd3d3 |
In the traditional teacher-student knowledge distillation framework, the existing methods all use a certain type of teacher knowledge to supervise the same type of student knowledge, such as soft target{{cite:926a410cdd034e6368f5ff43fd8c012278fa15eb}}, hints{{cite:fe295e634714cebc6ca5f0cfe9eb7f74d57ebbb7}}, attention map{{cite:ada0d21e010ff45d11667952a891a76974a76ebb}}, relationship between samples{{cite:df93bf876b2e619067b50b5d1982f065351e8b1f}} or layers{{cite:81bf8b288eb69b7115c0e9c3cf9873d17152b4ea}}. Inspired by CAM-loss, we propose a different idea to match different types of knowledge between teacher and student, that is, to directly supervise the CAAMs generated by the student network with the CAMs generated by the teacher network. We call it CAAM-CAM matching (CCM) knowledge distillation. The experimental results show that CCM can effectively improve the accuracy and convergence rate of the student network.
| i | 7d1320014e4d4fde0028c528d4d652a8 |
For the strange-quark contribution, the results published in {{cite:96611885d58360bb15555d0d908104194308f331}}, {{cite:2d469cc488a0c823d73fbbb211b1b6fe8626cb0d}}, {{cite:e8f019e75657e80df24f3e575accb011a1cd8c0a}} are in good agreement. It suggests that the scale-setting determination is not the cause of the discrepancy observed in the light-quark contribution.
Concerning the charm quark contribution, a small tension is observed between BMW {{cite:2d469cc488a0c823d73fbbb211b1b6fe8626cb0d}} and the RBC/UKQCD {{cite:e8f019e75657e80df24f3e575accb011a1cd8c0a}} collaboration who found {{formula:05033726-6f4f-40a6-83a6-bdf4661ae419}} and {{formula:6808adb6-ce38-4249-bfa1-e5b88437ebf7}} , respectively. In addition to the light-quark contribution, results for the quark disconnected contribution might be valuable to understand the tension in the quark-disconnected contribution discussed in Section REF .
| m | d0c9bd6317e1b4cefe278d85af3f4205 |
It is interesting to look for more general solutions in the {{formula:dfa6acc3-e33f-478d-899f-996f8d23769f}} case in particular solutions carrying both electric and magnetic charges of the same gauge fields. In this paper, we have given only some representative examples of the possible solutions which carry either electric or magnetic charges of a given gauge field. Another direction is to find an embedding of the solutions given here in string/M-theory. Solutions in pure {{formula:6a218af8-a04d-49c1-970c-6a58eeb8c354}} and {{formula:175bc65d-9811-4177-953c-8e50646028d2}} gauged supergravities can be embedded in ten and eleven dimensions using consistent truncations given respectively in {{cite:14c349e95208ff34fde58496ef774a891a8d6819}}, {{cite:3407f638e33aff413259e8a3ac2915b23d079de1}} and {{cite:bc55af85c192b3a6537a7767ab26be773e8f82fe}}. It would be useful to find similar embedding for the solutions in matter-coupled gauged supergravities. It could also be of particular interest to study the dual three-dimensional {{formula:e6a4e327-3600-4311-a4e3-80b71b03ad79}} SCFTs with topological twists and compute microscopic entropy of the black holes. Finally, it would be interesting to study similar solutions in other gauged supergravities such as {{formula:ddd35124-8a8c-498f-9a87-d7bae0c2a514}} -deformed {{formula:1854b5d6-e1a3-46bb-b5d4-351a2e2f68ed}} gauged supergravity and {{formula:d5734134-fd47-4550-a134-255070d20978}} truncation of massive type IIA on {{formula:6dbee6d4-dc76-4914-aaf7-332ba2504a55}} given in {{cite:1eca0b29f1a0839e781468301263dad58432a499}} and {{cite:43b4983a7458f03ea25a43697e4f895d07bc8430}}, respectively.
| d | 48cb67423d0750bb30d49c31876f2af2 |
Multiwavelength observation provides a systematical study on the GRB radiation mechanism. In particular, Fermi-LAT
takes an important role on the GRB observations in the GeV energy band. Here, it is helpful to discuss the GRB emission in the GeV energy band for
GRB 1901114C and GRB 180720B.
GRB 190114C detected by Fermi-LAT has a short-time flaring feature within 10 s after the trigger.
{{cite:170abc1ff887d9a72dec5b9afca12585a7237b0b}} suggested that the feature can be explained by the SSC mechanism in the reverse shock regime.
The general GRB radiation process in the reverse shock regime has been investigated {{cite:073bfae15cbeec08e1262eee790b6d657e4f4f1c}}, {{cite:4dae68683dfda7ce677b40fa3a03580aff581356}}. It is found that this short-time brightening feature has been also shown in some other GRBs in the Fermi-LAT catalog {{cite:25e7426081b2046fedcac4728dfd2d1dea0dddd6}}.
GRB 090510 and GRB 130427A are two examples {{cite:80b6ea23798ff8442d85075d1b0f0be592bf509f}}, {{cite:8297b82d198174a690f7a53ec5038fcff362b17c}}. Recently, {{cite:a4e6f8a2983194ec36afb682c3a69ef34d32de11}} comprehensively studied
the GRB flaring feature detected by Fermi-LAT, and the SSC mechanism in the reverse shock regime
can be successfully applied to explain the feature.
In this paper, we utilize the jitter radiation process in the kinetic turbulence framework to explain the GRB emission in the
TeV energy band.
The jitter photons can be scattered by the relativistic electrons that produce the jitter photons themselves. The process is very similar to the SSC mechanism, and it is called jitter self-Compton (JSC) mechanism {{cite:adccfb574628395793bd875ad80f657ffa1a34c9}}. The SSC mechanism uses the photons produced by the synchrotron radiation to be the seed photons, and the JSC mechanism uses the photons produced by the jitter radiation to be the seed photons. If both the jitter radiation and the synchrotron radiation can successfully produce the same seed photons, the JSC and SSC mechanisms can produce the same photons of the bright peaks in the GeV energy band.
Here, we further note that the successful JSC mechanism is dependent on the maximum electron Lorentz factor. The maximum number that the relativistic electrons can be accelerated is dependent on the particle acceleration. We propose that the kinetic turbulence takes an important effect on the particle acceleration. In this paper, the power-law index of the electron energy distribution that we adopted to calculate the fluxes of GRB 190114C and GRB 180720B is originated from the results of the kinetic turbulence acceleration.
Some electrons should reach the Lorentz factor of {{formula:8af2bf61-7879-4e83-b1fd-d1759304fed7}} {{cite:adccfb574628395793bd875ad80f657ffa1a34c9}}. We realize in Equation (6) that the relativistic electrons can be effectively accelerated.
Thus, the JSC mechanism can be applied to explain the brightening feature in the GeV energy band for some GRBs.
| d | 9ba6a7caa2f59fbe8e928a0c265a4c70 |
A major limitation of our study is that we have used synthetic experimental data. The use of synthetic experimental data provides an ideal setting to validate and quantify the accuracy of our model. Ideally, one would select low-resolution in-vivo data as the experimental data, perform DA with CFD, and assess the accuracy of the results with in-vitro measurements such as PIV, which is considered the gold-standard method in experimental fluid mechanics {{cite:247602add4cdd50747f767045aa00899a48d4615}} and is commonly used in hemodynamics research {{cite:70e7bbdca7c7c7ed38e2f2ee10678b4bba347d4c}}, {{cite:13ab7518fb79173031511817c37ab912350dffd3}}, {{cite:b288c19fed795e208dd894667a0756ca1922867d}}, {{cite:3f820d63a33070b9105e8abc6ede5c86c064c09b}}. However, unfortunately, this approach does not perform a real validation. In our DA method, we are performing ultra-high-resolution CFD simulation (quadratic high order elements), and therefore the numerical error is sufficiently small. The major error in such CFD simulations comes from uncertainty in patient-specific parameters. In-vitro PIV has the exact same issue as CFD. The information that we use in setting up CFD and PIV experiments are the same (boundary conditions and constitutive properties), questioning the utility of experimental validation in this case. Therefore, we generated synthetic experimental data where we have an exact ground truth simulation to validate our model. Of course, the success of our method with real-world data remains to be investigated and is expected to be challenging. The difficulty here is due to the lack of a clear validation benchmark. Also, assessing the covariance matrix in real-world experimental data is not trivial. We should point out that under the scenario where we exactly know the CFD simulation parameters, but we have a low-resolution and numerically dissipative CFD solver where numerical discretization error is prominent, we can assess our DA method by leveraging PIV data as benchmark for validation. Our work has additional limitations. We did not consider non-Newtonian rheology models, but our ROM-KF model is data-driven and could be readily applied to CFD data obtained from more complex models. The use of ROM within our model will generate error in reconstructing the fine-scale flow structures depending on the complexity of the flow.
| d | 07143fa173af630854755d0dc43d25e4 |
The diffusion map finds new variables that reflect nonlinear combinations of metabolic capabilities and returns them in the order of their importance {{cite:51cf635c70b22cfed9565419bff7d7fb063ca21c}}, {{cite:50335a63e6176d01923989eedf6146fcf6a4eaa6}}. Each variable assigns coordinate entries to the genomes that can then be used to order genera, from the most negative to the most positive entries, along different niche dimensions. Dimensions can then be interpreted by analyzing the strategies of taxa near the extrema of the orderings {{cite:a8145bdab5e4b0e6dbfd37083d72b358f97a063d}}, corresponding to large positive or negative (i.e. far from zero) variable entries.
{{figure:d3aedc30-c434-473f-b4e6-11a58967b391}} | r | a0dd20bae557d12b3c2fe5c85f9d95f1 |
The reweighting procedure, necessitated by the difference between the transformed (reference) and the target distribution, was performed using the importance sampling or the free energy perturbation method. This is the simplest possible reweighting which still provides accurate results. Since we have access to the normalized probability density of the transformed distribution, more elaborate procedures could be used for the reweighting, for example adiabatic switching {{cite:6bfcee7da0dd82b1dd6c07876cc6e35e867f1347}} and annealed importance sampling {{cite:927c78a117a60cba64cff25b97656aef08b8e5f6}}.
| d | f3fff33c93b98e058c5b857d79335150 |
On the other hand, the EOS in a lower density region is also gradually constrained through terrestrial nuclear experiments, but still, there are large uncertainties in EOS parameters (or in neutron star properties) constrained from terrestrial experiments. For instance, the fiducial value of the density-dependent nuclear symmetry energy {{formula:86c1d5da-3aaa-48c6-a0df-effb022cf81b}} is {{formula:1acbee93-60e4-4ae3-a676-01463f0df4ba}} MeV {{cite:5c7557b6f592e6741ba9197b4a956721cc4cf07a}}, {{cite:8c18ee1a4eba727730350f2eb9ee06f879465cee}}, while the constraints of {{formula:01caa8c5-342f-400e-b9fb-d65c10689968}} obtained recently seem to be significantly larger than the fiducial value {{cite:457e885d456f65e7581828d00cbcc2222f0febcd}}, {{cite:7a0f7de6756579c6e47696a4211bd84d0ad7395a}}. This is because one has to usually transform the experimental constraint to the EOS parameters, even if the information determined via experiments is associated with some aspects of nuclear EOS. Then, one can eventually discuss the neutron star mass and radius as a solution of the Tolman-Oppenheimer-Volkoff (TOV) equation. Anyway, the terrestrial experiments are definitely crucial for understanding the neutron star EOS as well as the astronomical observation of neutron stars.
| i | be09cfdd57c086a45344659c0dee48d3 |
We use the KITTI dataset to evaluate the performance of our method under real road scenes. KITTI2012 dataset contains static scenes only, while the KITTI2015 dataset has more challenging dynamic scenes. Following {{cite:8ed17d02ca66938451d11744e226a2bad701d91c}}, {{cite:b1bb7a17b35a3a4527439bb4e7971c4fd295290b}}, {{cite:e0ff9defa43cc6b0f9ad5d91411f13506e3ee922}}, we use the training split, which has ground truth of camera intrinsics, poses, and depth maps collected by LIDAR. All methods above-mentioned were trained on other datasets and evaluated on this training split. In line with previous works[DGC, GLU, GOC, COTR], We employ the Average End-point Error (AEPE) and percentage of optical flow outliers (Fl) as evaluation metrics. Here, inliers are defined as AEPE{{formula:77bfb912-4616-4af7-92e3-db88c256a1fe}} 3 pixels or {{formula:ba704aac-d8ab-4909-82bc-ddd42a1f7ef2}} . Same with COTR, We sample {{formula:633d5094-c9e7-4164-9480-e23eda47cf8d}} points for a fair comparison.
| r | 8a2f6b928f77e8d4c7a485e807e74a79 |
In addition to comparing with baselines such as Majority, normal fine tuning and prompt-based method LM-BFF, we conduct more experiments to verify the effectiveness of our proposed method PromptDA as a plug-in module. Because different template choices can result in a large variance of performance {{cite:a66e9c0732c1be0d0c3eed48ba8aeceb18a9a78f}}, we design two groups of experiments, namely template-free and template-augmented, to show that our method can improve over standard prompt-based tuning method regardless of template design. For the template-augmented group of experiments, we manually choose “It is [MASK]" as the template, following {{cite:e6864db5145f9a3fcf3b5cf0fbd83e9060800f08}}. For the template-free group of experiments, we only append “[MASK]" in the input. We report the results of PromptDA in Table REF when the size of data augmentation is {{formula:e70dbfbe-092d-499a-91db-daf12fa8ffe5}} (i.e., {{formula:cd376c2b-7bd1-4b6c-8ca2-ddadbf1cce9e}} ). We also consider two scenarios that the label words are derived manually or with our automatic label augmentation mechanism.
We choose 8 samples ({{formula:9a6819b6-bf16-4214-93e2-7dbb74ba6756}} ) per class as the few-shot setting of our main experiments. For fair comparison, we choose the same random seed of training set sampling as LM-BFF. We train for 10 epochs for each dataset following {{cite:e6864db5145f9a3fcf3b5cf0fbd83e9060800f08}}. We report the average performance and standard variance of our result over five runs of sampling for each dataset. The main results can be seen in Table REF .
{{table:6554f7fa-5afe-4181-8a0c-935dec74af35}} | r | 94cb1e1fed2b959e5934e6cb3bee523d |
From Fig. REF -left, while the values of {{formula:5a33c577-ed0a-4881-9af4-c44dd317e9d6}}
are constrained by the current LHC data {{cite:4a9e53ffd7ed821ae9aae336dfb022ab1580e513}},
the branching ratio {{formula:6a89be80-a5ca-4cb7-ba98-dba167ea0d5a}} is modified drastically
with respect to the SM, it could be almost null as 138% enhanced
with respect to the SM. From the right panel, one learns that the
Higgs decays into gauge bosons and fermions can be reduced/enhanced
by {{formula:3676913b-7c72-4fc3-9e37-ddfe371b0f73}} and {{formula:e7170ef2-06fb-48cc-a990-c5cf93f9e384}} respectively. Therefore,
more precise Higgs measurements will tighten these ranges and put
more constraints on the parameter space. For the considered parameter
space, the oblique parameter given in (REF ) takes the values
{{formula:e4b76ace-a799-42c1-ada6-4e9438e2f235}} .
| d | f86c6e30a1b0e1afbb85f62f0f3f8c82 |
The following result can be found in {{cite:1538f30d7c5521ed241324576908bdeb92ba9ab4}}.
| r | bc245178733b04c3bfbf8015c579d169 |
The realization of our scheme is within reach of current state-of-the-art experimental technology. The
ultrastrong and deep-strong coupling regimes have been realized in several platforms, such as
superconducting circuits {{cite:844be856c4f85cffe4f5f827b15a81fc58a60583}}, {{cite:0f017f76d4d3a73eb43ee1cabee1cc82177e41e3}}, {{cite:1dc63960eb43fac248a8f4eaa65a935979453c8f}}, {{cite:71b635d2bdaae21cd7656a86087f332656c84942}}, intersubband polaritons
{{cite:987230202bd22381101f794653485a578925adaf}}, Landau polaritons {{cite:2ad3a09df5f1137fc08696ecb5943eb17b5c607f}}, {{cite:638b5f6514c6eb91ca5637cd0251f7900c7115a3}}, organic molecules {{cite:40984b8ee82983948aa2716856cb02fd8c41568c}}, {{cite:93f5757963e2fa0b9a5e73bb53f04b66a11d9601}}, and optomechanics {{cite:d02e5f5b0fa682dea8a65b3de4114b621a2e90f2}}. Hence our method is
a practical approach that can lead to the construction of on-demand multi-photon sources.
| d | 55d21893c67fea3e16c26e0903834c42 |
Thus we conclude that the simplest approach to gravity bootstrap, discussed in Section , fails. However, it shall be noted that the whole bootstrap paradigm must not be reduced to mechanism is spirit on the naive bootstrap. Namely, it is established that the cubic interaction described by general relativity (REF ) is the only interaction consistent with causality {{cite:58d374e2c95824da62a2e7ad7ca1ba78132ccd76}}, {{cite:8acb5027cdb58334b5fec61502431cfe8e8f3184}}, {{cite:075a246cfe91f729373f100d40d80ae45beebae8}}, {{cite:9f1fbb44362413d354b1aa367c89c02e931fcc67}}. And the spin principle also provides a way to reconstruct the complete nonlinear gravity theory with a certain assumption on the structure of its interaction {{cite:c12b7f344a3ebb9846032b69194912fa3aa518a1}}, {{cite:998b85d8934369eeafdec7d44d16909cb24782c0}}.
| d | d4de38f4b310d7718b5ee18f57424774 |
From Section REF , finding an optimal policy is equivalent to solving {{formula:fc033617-3d46-48f3-b90b-9fd24ae44bbd}} . It has been noted since as far back as {{cite:407f80f0cefd9b6dc59fad0bebb627d3e85c89f9}} that the operator {{formula:c8f05516-fc41-4c9f-b68a-0caa42718478}} can be treated as the gradient of an unknown function. We review here the results from {{cite:d949b6cb105a7da9de2de34e127003f8f79d0736}}, where the authors build upon this analogy to define first-order methods for MDPs, extend Nesterov's acceleration and Polyak's momentum to MDPs and present novel lower bounds on the performances of value iteration algorithms. We also review novel connections between Mirror Descent {{cite:5c1cff288b8fa5fb7567775af057e6f81ba0f1f3}}, Primal-Dual Algorithm {{cite:900725212ec6bb31a5e12f0f9107efd7df123c5d}}, and Value Iteration at the end of this section. A review of the classical results for first-order methods in convex optimization can be found in Appendix REF .
| m | 3aa29b7ec18f0bd964f31343be4e2393 |
is said to be the inner {{formula:784eab9f-aa4a-439e-9d65-8f67873c9a6d}} -weighted equilibrium constant.Similarly to {{cite:b0d682d3a500687410043e3d300e9ea48280455a}}, {{formula:992f780c-3f21-4e04-9a0a-4df33ae3f308}} is also said to be the inner modified Robin constant.
If moreover {{formula:6b27ce5a-a6e5-4e3f-9def-1d573c64bf34}} is l.s.c. on {{formula:51e28a93-7f54-40da-94f1-b0612661daaf}} , then also
{{formula:97f78f36-82e2-4a21-a49c-f91629342176}}
| r | d652b3997edc787672ee07a98edddd99 |
A fundamental aim at the intersection of economics and computer science is to understand the efficiency of systems when the dynamics are governed by the actions of strategic and competitive agents.
A large body of work has established bounds on the price of anarchy of various well-studied games, which measures the gap between the social welfare at the worst-case Nash equilibrium with that of the social optimum attainable via coordination by the agents {{cite:81d129197717948f5688eefce9dfb6d8564c445f}}; moreover, recent work has shown that in many cases, price of anarchy bounds often seamlessly extend when agents employ simple no-regret learning algorithms in repeated games {{cite:dec5ef47f3222895adf0752dac26bdc2f6bd71e8}}, {{cite:edc0465827cc574881d8a0ede73599beb010c8b3}}, {{cite:4666f8ead0d525e3d28258792127ffdb61426b78}}.
| i | 5382a829d0a53864d6b6688fe9abd616 |
Closely related to CL is the collection of streaming methods {{cite:313715afc7ab5c355e5d27d8cb5f8347e08a2650}}, {{cite:f9902bffad848d5e97c37cd3bbeeb53822a3d307}}, where data items are seen and queried only once by the user and then discarded. This is of particular interest when the summary statistic of choice is updated and maintained in real time, for example in the online learning setting {{cite:0f8ae6db9123de8a38dce6060ee99e490d671cad}}, to reduce space complexities. Notably, the count-min-sketch {{cite:b13c0a106a75313aed9106573cb182c36f5e7071}} was developed to query data in an online fashion with the application of maintaining histograms of quantiles. However, these methods in general focus on the discrete collection of objects and database queries while in CL the framework and method is applied to machine learning tasks where typically the signal is question is continuous. Tropp et al. {{cite:e62a711d8498b5ec2571a4175f4e95911db18a8a}} proposed a streaming framework for large scale PCA. In particular, in {{cite:f9902bffad848d5e97c37cd3bbeeb53822a3d307}}, the authors design random sketches for on-the-fly compression of data matrices associated with large scale scientific simulations. Here the data matrix {{formula:47bd70d8-bb69-4b9d-853b-0d7858d3a8b0}} of interest can be decomposed into a sequence
{{formula:f8492425-5a64-4569-aafa-4f0183f9167c}}
| m | 242d870c81330546927a59354df81c26 |
Our implementation involves a multi-head GAT {{cite:1edd85e391d0fe0ce8fa2714d24ae20cb1cac491}} model as the learning backbone but relies on a new graph construction strategy to make the GNN encoder compatible with multilingual data. The graph construction process is demonstrated in Fig. REF . Multilingual data is processed by a heterogeneous information network that consists of four types of nodes – users, entities, hashtags, and messages. The key problem that needs to be solved is the alignment of messages in different languages, so here is where the novel construction strategy come in. Incorporating both node-level alignment and semantic-level alignment, the entity nodes are aligned with XLEnt {{cite:7e96e6ae76e47a58e08291b5bec937a5c09eb9e1}}. We choose XLEnt as a tool because it comprises parallel entities in 120 languages aligned to English. The semantics are aligned with CLWE methods to break the embedding space discrepancy among different languages. Through a cross-lingual module that uses CLWE methods, we transform those non-English languages’ embedding spaces to English semantic space. Note that the cross-lingual module considers both linear {{cite:1c88e9cc3803e49c4e032c6278651f16de22dcd8}} and non-linear {{cite:661b2a3e2ad5077693724d0dadde0ac512a4aefc}} CLWE mapping methods with the method that gives the most appropriate transformation between language pairs selected as the final result.
| i | 7f98beb1fc58a5225bb64ac701d85d00 |
The numerical results presented in this paper were obtained using the Julia language {{cite:758141e5d364eaff7e7ec2584b98931c92652220}}. We would like to thank Bogumił Kamiński from SGH Warsaw School of Economics for helping us to implement it. The program is available on-line.https://math.ryerson.ca/pralat/research.html#publications
| r | 0cab2472a39d17e3c5a3c0e7ab901783 |