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{{formula:ff0192fa-f03d-4066-89a0-ef4c903b6589}} and {{formula:9d01e47c-f165-4288-92d2-f206a5dd6b5a}} being the Laplacian and the divergence associated with the measure {{formula:6faef50e-66e0-41df-be20-8e9092a990d8}} . This is result was demonstrated also by Otto {{cite:e42a6d1980ed6a8523695b5ca493a1a93106b84d}} in the case in which the reference measure {{formula:b6a2bc73-060f-4250-bd3d-34a48a2bd1c8}} in the {{formula:98ae57f6-9655-490c-9368-3d92ccb7519f}} -relative entropy {{formula:f81d0df6-833c-4a48-81c4-b864daec8c35}} is given by the family of {{formula:41b0b09d-9e02-4712-b6be-3b7e1601bcb5}} -Gaussian measures, which is in turn closely related to the Barenblatt solution to (REF ) without drift (see {{cite:2c1f709f0c44eaf19d5e337eef24ebe109cfd0d5}}, {{cite:ab62ec415c74203188d5447669915326ee74788d}}).
| i | e336ce022d983382bffa0f8ff5f3d977 |
For our next extension of the Friedgut–Kalai Theorem, we need the
notion of {{formula:6027db01-dc62-4c81-be7e-a68e62635e37}}-regularity, (see
O'Donnell {{cite:7e4af7a952500071b1042bd66d432794eeda67ef}} for more about this notion).
Let {{formula:7c3050e4-bd96-4a91-bbaf-2396d86d65cc}} be a subset of {{formula:eb80fad5-5e59-41b1-8039-b7b121e89cd4}} and let {{formula:1e859d70-bf34-4cc5-8641-629ed1b8f0a0}} .
We write {{formula:664af995-b2b3-4568-8ed8-c738d86b6979}} for the Boolean function on the domain {{formula:16c1d75a-370e-42e8-a1ce-f158068c9fb5}}
defined by {{formula:a2e994c5-dfc1-42e5-af32-1d1483ea908f}} where {{formula:b6b6746b-72fc-41c3-810d-eaf93e6e6b7d}}
is the vector whose projection to {{formula:44cf6e3d-c144-4a0b-904f-0537fbd55951}} is {{formula:3f9fed35-5838-43b4-aa2f-bb7e9b94ac25}}
and whose projection to {{formula:c35eafb4-2194-4364-8eec-8d0979c465a6}}
is {{formula:fc2f75fc-7779-4395-89ad-3dd6fbd03fef}}
| r | bc56adaa219d76458299f259c995cbaf |
A related idea is that strings and black holes can transition into each other under certain circumstances {{cite:9a696f6b34ff54260cc1cfe00d74cc875791de78}}, {{cite:be8ed4a92500e36c292427d856af40a616307590}}. The arguments supporting the possibility of such a transition rely on comparing the entropy, temperature and scale of the black hole and the string. These were extended to charged systems in {{cite:5ef78c7da95f8c4bf430274f1679e592117694f6}} (see also {{cite:3d463525034074bdfeeeea5b75dac7d8c9945d74}}). Motivated by the idea of the black hole/string transition, Horowitz and Polchinski (HP) {{cite:bb7744cd835759d483b53248bb53590b173ffa99}} found a solution featuring a quasi-localized condensate of closed strings that wind around the thermal circle in Euclidean signature. This solution has a classical entropy and its Lorentzian interpretation involves highly-excited, hot and self-gravitating gas of strings.
| i | f4783efe5389161aab93b31a598eeba5 |
First, we show that the IPF sequence is well-defined. Note that {{formula:0a5f249b-2674-4a1b-aad7-34b86cac3f96}} is
well-defined since {{formula:c8395365-8904-4e04-be7b-cdb75497ddb1}} . Assume that
{{formula:ba2add39-be49-4300-9d72-d7ace638eeb0}} is well-defined. Using {{cite:42a53b61876c671545d48a4ce210639f7d880bf8}} we have
{{formula:890fc0cb-5921-4130-8381-889d0a56f64c}}
| r | fc44d9a3719098bc5c589883eca1bd06 |
Before analysing the magnetoresistance of Ca{{formula:e27ee2de-c90c-45a7-b6a1-da2253d2ffe2}} Ru{{formula:f81288a8-47b1-4b6f-8c84-dc3309150967}} O{{formula:6017cee1-0abd-400e-8cd8-48980c03d658}} , we start by illustrating the use of the aforementioned solution properties. In doing so, we consider results obtained on hole and electron doped cuprates. For hole doped YBa{{formula:bb728a6b-d0ce-4189-90e7-51bfe3da951a}} Cu{{formula:657dc33d-2ad8-4672-951e-9cf030755d71}} O{{formula:c98a62be-10c3-4f5e-8cfb-b0112dd67e67}} and YBa{{formula:0c85ed49-338d-4054-a99c-22a2c7f844bb}} Cu{{formula:29fe17b7-48cb-44ae-9df3-d75aded8c281}} O{{formula:5de96c61-0e81-4dbf-84cd-f0c9618d581b}} , quantum oscillation experiments yields an electron-like Fermi surface sheet with an area corresponding to {{formula:37bf4315-2714-4f6b-8657-bd2c38020322}} mm{{formula:4a10ee55-acce-4cb3-bb34-c86aa808a8e0}} C{{formula:408e7eeb-61c3-418a-81eb-d69f843ba0be}} {{cite:2ceaf6b0dd7e8d98dd57f994e690ff37f37deebd}}, {{cite:7ffd88ab9a3de4077eb745d6cd58cedad9ca219f}}. Since the system is hole-doped, we expect {{formula:053e14ab-3d53-4a28-8ac5-043b30158ede}} . Hall effect experiments suggest that {{formula:84fca26b-681e-49fc-b79a-9e0de3b81ee0}} {{cite:f25254df0cc2ea1c0996cca9a2d0f0b3a975890b}}. Already here a contradiction emerges as {{formula:cbfddbe8-5df7-439c-922f-4b9d4d1f7f2f}} is impossible for a two-carrier hole-doped system. Evaluation of {{formula:403ea05d-82c6-487d-bcf5-decac15b2579}} positions the system in the {{formula:72401f25-44b3-4cb5-aafe-3a4148707467}} region where {{formula:2a13e619-53fd-4ff0-8d96-8b518858516a}} {{cite:6d7d49fa7a240280d1f6fde4bcec7125731536d6}}. In fact {{formula:c864cfcb-9c69-4041-bc00-ec377baa9fe5}} is not strictly satisfied. A plausible explanation for these contradictory results is that the Fermi surface structure contains more than two sheets. Since the Fermi surface is likely reconstructed by an incommensurate charge-density-wave order {{cite:5a26c0eb85e2c10d747d606dcd88846e96421ba9}}, {{cite:5f40eee7beeec8cae2ccc672e77b163887f75e3b}}, {{cite:6924ff9131f946f96ab852bdf022ab3d53af14ee}}, a more complex multi-band structure is expected {{cite:94bb6996715faf54dce2c3b347f1a7976d3fcd9f}}. On the electron doped side ({{formula:1774354a-3be5-4684-9666-b8880c071fd8}} ), by contrast, the Fermi surface is expected to fold around the antiferromagnetic zone boundary leading to a two-band structure {{cite:c1dec1e260f14f88acaf46b22a8d42b64a8f3d02}} (see Fig. REF a).
{{figure:67178bcf-14d6-493b-b73b-3c81e68c668b}} | d | 4022ff945518d952822857b1454b7b2a |
Refs. {{cite:92f0439e22d92d8fa4b7a349806708ba58c74d76}}, {{cite:0698948f57861b843fb200867246a43b0932be3b}} have also argued for an analogy between {{formula:fbf50239-8cc1-4d71-8a99-b74531870cde}} and {{formula:ec2dd0ff-baa6-449a-a53f-8c09d40a5bcb}} (as {{formula:c7f1cd6f-684c-450b-b876-5c5961d6f6e2}} molecules), and {{formula:9ed06889-ec25-449e-80ee-dc3173c0c833}} , {{formula:e815d658-51e6-4ddd-9026-a5acb9dd140d}} and {{formula:6a4eb82b-8bf4-4d46-a121-9ca88e95c150}} {{cite:4ff3962e83e7428a4c84f028f4fa79cdf762a10d}}, {{cite:528f9e4e0b86ef4549d2755e62dd1c6d0ba19ab1}}, {{cite:a4eb8b4ec980bcdd6b6e3acf7ebfbaba2388d962}} (as {{formula:1afad834-4225-4dcb-9b58-c76180bb1057}} molecules). The analogy is misleading, considering that the corresponding potentials are neither related by {{formula:b276bb7b-a15e-4bdd-8fa6-4f86207c8541}} flavor (as {{formula:c83c25de-1ef8-45e3-b7bc-525d51c9fef8}} and {{formula:70d7d94b-03fb-4e9d-b1cc-2b383b1fada6}} belong to different flavor multiplets), nor heavy-quark spin symmetry (as {{formula:6b50bb5d-6bc8-410e-ac08-c6d30c03c098}} and {{formula:ceb0391d-d061-46d6-8bb3-c3f7d464c2fc}} have different light quark spins) {{cite:561e6a083f58bbf2f990055f2e96ea01119d27a4}}. Indeed, heavy quark symmetry implies a completely different pattern of states in {{formula:3c56215d-6f99-4c90-b365-fdb43a97ed07}} systems (where the potentials are spin-dependent {{cite:b73404c201ef0f00ef06c845859fd54e0eabde5e}}, {{cite:978c2468d6b66a0aede6e95a9beb39262f3c63f6}}, {{cite:4a1d11448b2484cfb4cbe68da5b88069ddcff438}}, {{cite:aac80ef4cb509843fcc45302cb6bdc7f8513eefd}}, {{cite:444f455f324c69547fe4530d6849394a8c956b58}}, {{cite:6bf6ce25fd08c462fde6e4da7fd221421f5aab99}}, {{cite:a896a8bde47c6f97bb49c9b4e68962ae04abdb20}}, {{cite:f310b943a7eab35da6dfba7b8605972ad61a6a25}}, {{cite:5bef3027ea716bfa17bde9ffb6a332ddd4b79270}}) and {{formula:06b3c915-542e-43b1-bda0-7a2b20e950ff}} systems (where they are not). Moreover, the analogy relies on the assumption that {{formula:835f1966-caa4-4c74-a17c-9c68f4abd43c}} and {{formula:223a2dfc-e196-490f-935d-34ad54e57984}} are both {{formula:4d1f9787-7d75-4254-894c-25ee58fd13de}} molecules, and we recently argued that this assumption is not consistent with experimental constraints {{cite:916718a63b5503cdf7167585ffea2d1752770454}}. (Scenarios with different interpretations for {{formula:09b2f7a3-f0a4-4d23-9bf6-49353e04c40e}} {{cite:0a5b818cd63b9209b7d4512950f40a2f61c03818}}, {{cite:c41dc63578d4c55ac4032faff2dc40e1a4a8891b}} do not have the same problem.)
| i | e06dfb47b14863533805e73a2795a632 |
The solution of the sub-problem (REF ) requires iterative optimization algorithms like the ones discussed in sec:iterativeoptimization with the significant difference that the model function can cheaply be evaluated.
Importantly, these sub-problems are usually solved concerning a sufficient decrease condition.
If such a sufficient point can not be found, the so-called Cauchy point (that can always be found with the gradient-descent method) can be used instead.
For more details, we refer to {{cite:83f72078e53f52650e77e4e76c6c680147d6f197}}.
| m | c55890a8624c9b73ea5a572d069d70f5 |
Stochastic cargo binding simulations were implemented in MATLAB using the Gillespie algorithm {{cite:01664bc9c60ae3c74c62556c385181e45237a12f}}, {{cite:6527d7930cba6bf8f2f18b170a1e5b11a4f50ec6}} to probe the system state in continuous time (see Supplementary Methods). Molecular dynamics simulations were also implemented in MATLAB, but instead updated cargo positions according to the Langevin equation defined in eq.(REF ) using a forwards Euler scheme. Wiener process displacements were calculated using inverse transform sampling and binding dynamics were introduced by randomising the position of the cargo within the periodic domain with rate {{formula:de2cb807-6a8b-4a59-ac46-a6df5e892310}} (calculated by substituting the local binding and unbinding rates into a previously published average dwell time formula {{cite:ccaa26e98882007c466503552ee4f0ea129916a2}}, {{cite:4c2bfb635a87a56f1927365060f337c91e397767}}).
| m | e406a25c2b9df40be733f6f1dd56de7c |
Considering the prior distribution of latent space {{formula:d10ec1d2-7dd7-4316-ab4a-3b14aaf3a18f}} be a multivariate Gaussian, then the series of invertible bijective transformations with parameters {{formula:73a8d297-fbdb-4fd9-b558-f82cb7723e70}} can transform the posterior {{formula:5ccf1489-3520-4a81-a17e-993d335adfec}} from a Gaussian prior into significantly more complex probability distribution. Hence, we can maximize the log-likelihood {{formula:799eef34-3326-4f23-ab12-212a219d3186}} of the in-distribution samples {{formula:886eed1d-fe9a-404d-b1f7-8709e11db579}} with respect to the parameters {{formula:aa063410-fdb4-4a50-8b7b-448632203f0b}} of the invertible transformation
{{formula:477903e8-51a9-44ec-a0f2-f0830c7764d1}} and use a likelihood-based threshold to decide whether the log-likelihood of a test sample {{formula:fbc13720-6874-4e97-affd-b02d1ad51441}} is below the threshold (classify {{formula:122eac96-4a66-4ac1-88ff-96b985871e93}} as OOD) or above the threshold (classify {{formula:453ff507-68f5-4cbe-a439-ec7e3aa5800c}} as in-distribution). Additionally, AUCROC (Area Under the Curve Receiver Operating Characteristic) can be calculated to determine the performance of the flow model in terms of OOD detection. However, works such as {{cite:0347d2f3287bab2bee5cc6b7952eadcab3eacbb4}}; {{cite:551e9037c6f68b02c471069a0113ebde83a29b43}} showed that generative models such as normalizing flows assign higher likelihoods to OOD samples compared to in-distribution samples, resulting in overconfident predictions on these OOD inputs as shown in Figure REF (b). To interpret this behavior, {{cite:551e9037c6f68b02c471069a0113ebde83a29b43}} argued that these models only capture low-level statistics such as local pixel correlations rather than high-level semantics, due to which these models are inefficient in separating in-distribution data from the OOD samples.
| i | 6291fa06d413447c7c08bcf34de1ec32 |
Indeed, while {{formula:63f3446c-d2fe-403c-be8b-1a08d8f63cb1}} designates approximately the actual size of the shadow, {{formula:ebec2327-5596-4390-a20c-cb5a0ffcc7e9}} connotes the shadow deformation from the circular reference circle. These observables are defined as {{cite:2c9d34e3284c0a939b0baece51415f4e556fca9f}}
{{formula:c03fe5ca-f593-44b1-a3aa-330bb920f2f3}}
| m | dcc7ec9ebb7091ffd69169af9cd785bc |
Our model consists of a ResNet50 variant of Arcface {{cite:680a84b5a4554f957f3bdeae0d3564351c6c2ff8}}, pre-trained with 17 million human face datasets {{cite:611a880bab7daa52cd5046a44c9301521afed9a8}}, as backbone, of which the last batch normalisation layer is connected to a fully connected layer to predict built year.
| m | c6b9034c1fab7d24737bb0e300a42d2a |
In spite of some conceptual issues {{cite:fd29579e3d8cca7d979655252cb82c19049b0934}}, {{cite:074cac6d71018a2a6cee35d1b67866230871ddab}},
the null geodesic tunnelling method has captured some essential features of the Hawking radiation,
and can still be used to derive the black hole temperature.
For recent applications, see {{cite:173773a42d1b56520f6b81df2f05f0845d4b034d}}, {{cite:d5634ce7a6f35df46ff5489c575e90a5cd28bf9c}}, {{cite:8302ae6d647185a74b5c34eace9a958225984786}}, {{cite:d6dedbf0d83ed4e0cac01fb3e9ee2315d73e9f04}}.
| m | 33254ed9cb3201d2227faadaca60af46 |
It may be instructive to compare the results of the discrete self-similar critical collapse (Choptuik solution) with those of the continuous one (Roberts solution). In the Roberts solution, the spacetime at the center is not conformally flat {{cite:0df485f70e93be6393e90986f0a17d85a25e382e}}. Moreover, in the {{formula:a588a6cb-48b7-441a-8cc4-bfebcc957375}} region, the mass is negative. To avoid the negative mass problem, the spacetime of this region is replaced by a flat spacetime {{cite:7993a07e0b33d6bcefc6b98877d822eb72cd6c2d}}, {{cite:45623202f927f60aa3faad422e4906de901df65e}}. The central singularity is weak {{cite:185b63e5bb74a20701c14de08ca8c7bae75df1d7}}. It was commented that this singularity is actually a collapsed cone singularity rather than a naked singularity {{cite:73f9792110b50bac93f190b5229120c1b36a19d1}}, {{cite:1768bf432db73330b5fa3625b537b7d98cce8bd1}}.
| r | 5dcc6c50ed06163c9271aefe36be23da |
Limitations.
Flash-Cosmos has two key limitations that also commonly exist in other PuM solutions.
First, like ParaBit and other PuM proposals (e.g., {{cite:7d6930dbd1db92e2f2153c594e5e8abbab1c905d}}, {{cite:eb5bfe875b713439bd1b28c463632999dfff9370}}, {{cite:75def2e337dc2237da49393987b4a01b406f9a67}}, {{cite:122c48234a39c8719820abe063c458c7cfd9bc43}}, {{cite:308c43bed272eba8c6634b8f323d3ac5c7aa4f1a}}, {{cite:2685450ed0e234c8d0c526b7ec9ac7e05a99a577}}, {{cite:6efcd4d8b0bd3999f4bde44bad4da4d5bfe01467}}, {{cite:d25da6ea2ec81cdad0ce28e707410959d45f8246}}, {{cite:26dedc8651b869758ab82f3fff39a99878da862a}}, {{cite:2fe96f9d89ded8537f1e8691969e14a5168bcdd3}}, {{cite:7fdcda8e17af5c9c3aff0e20246867df3d33be2a}}, {{cite:f0957596987840882f75ac4f3bb198d43cd7ae60}}, {{cite:7aaa2a1016eefbd2b960edf8ccd894e6dc606e3e}}, {{cite:8b14e9cb8d6ba28cd08d8e488ff7513f361e14a1}}, {{cite:71bf26cfa42c496b31f3e51b79f89de8a3bfc455}}, {{cite:667eed54a3aa2f63018683480836a5f53e82ed35}}), it is not straightforward for Flash-Cosmos to work with mainstream encryption techniques (e.g., AES-256 {{cite:dc2cf55eff3fb6c386b295bf5e945c3b4aa01bd8}}, {{cite:e527aff516f38def670d2ba2768fe841b7c58df3}}) that are widely used in modern SSDs.
This is because widely-used encryption techniques have input-data dependence and/or require complex computation other than bitwise operations (e.g., shifting).
One possible solution is to employ homomorphic encryption that preserves the correctness of computation for encrypted data {{cite:214d871ad15e2f7619b179f19d82baca76ee13a3}}.
Although homomorphic encryption currently has many challenges with large computation and capacity overheads, we believe that the development of efficient homomorphic encryption would be a promising direction to solve this common problem of the PuM paradigm in dealing with encrypted data.
| d | 5483ab44bdc9a1e732830382e1c8e5be |
Calculation of transport properties. The transport properties of small lattices ({{formula:7e3852f7-ad84-4ba4-9e54-cf50f3d8e76d}} ) was obtained exactly by diagonalizing the full Lindblad superoperator, written as a {{formula:3f60bd59-90c1-407f-bbc3-e93a82516c5a}} matrix. The NESS is the eigenstate with zero eigenvalue, reshaped as the {{formula:8a7c91f9-e0ca-4f94-b00c-622b9510413f}} density matrix {{formula:43c96140-f2af-444f-a36d-e9e103b3ba8e}} . For larger systems and forward transport, we describe {{formula:024967a0-e9a0-4748-8d5b-1475ab0c95f8}} as a matrix product state {{cite:7c134ee822a1e59bca21d0cfc89f3d60e7fc51f2}}, and obtain the particle current using the time evolving block decimation method {{cite:d7d7d249cfe4d60e4f01289c8b4dc16ed7585e2e}}, {{cite:fa7f6d3ff29ec872706fee98498a139a38c42e46}} implemented with the Tensor Network Theory (TNT) library {{cite:1a2873a27f0a0901d6528eedce4e4fa6ad60741d}}, {{cite:f5952fb01188614b7af5695d5f3c75b3a4db13ac}}. Here, the evolution of an arbitrary state should be simulated until the current becomes homogeneous, which indicates convergence. However, strong tilts and interactions commonly result in a very slow approach to the NESS, making it impractical. Thus, in such cases we focused on converging expectation values on the first site only. This shortcut allowed us to reproduce very well the current of small lattices, so we extended its application to longer chains.
| m | 068d9be97a37d99609a5ea45daf8464c |
In order to evaluate the second criterion of Section REF , state separability, we plot a t-SNE visualization {{cite:cb212a2f4377c0accf5a25c7f2338fffed92177a}} of the state embedding, i.e. of the features after the second hidden layer.
Since t-SNE captures correlations between descriptive variables but not correlations with the network's output, we also compute a partial least squares regression (PLS) {{cite:0a3aa2549ee15441cf55830e766bb3ec43062ab0}} of the action taken by the agent with respect to the embedding and plot the projection of the state embeddings within the two first principal directions.
Figure REF reports these visualizations of the state representation embedding, where we color points according to their optimal action (-2, 0, or 2 with our discretized inverted pendulum).
Graphically, it appears that distillation favors better proximity of states that share the same action.
In order to lift any graphical interpretation bias and quantify this measure of separability, we train a linear SVM to separate the data points in the
space spanned by the PLS projection and report the accuracy of the obtained classifier in Table REF (more detailed results in Appendix ).
Specifically, we train this linear SVM both when using the projection on the 3 first principal directions computed by the PLS and on all 32 directions (which is equivalent to not computing the principal directions at all).
The results consistently demonstrate that (except for {{formula:c6eda854-3c02-48d8-9397-58ccdc32e0fa}} ), the embedding of the student is better suited for linear separability than that of the expert (whether we use 3 or 32 dimensions).
Increasing the variety of training levels monotonically increases the linear separability of states sharing the same action.
{{figure:83d808de-564c-4227-869b-abc2756039bf}}{{table:041672d0-3afa-45b1-8b14-5419424f5825}}{{figure:b96e836c-e98a-40c2-ba9d-278785d411d8}} | r | a0feb56fd09f76559905fcfb9966b0bf |
The second result {{formula:5d02bde1-d660-4270-9b70-253dfb00f053}} states that the Strong Minimal Mass Assumption A3 cannot be skipped for uniform consistency rates and no compactly supported densities. This is in line with the former studies of {{cite:7af0d1018f7592870dbdd56aca5a8bd2818fe5b8}} and {{cite:085f61d047153ae4fe1921f66220797f95e12d26}}. In particular, Lemma 2.2 of {{cite:085f61d047153ae4fe1921f66220797f95e12d26}} takes advantage of some of the positive consequences of this type of assumption. Our proof relies on the construction of a sample size dependent law on {{formula:01ae803a-ff15-4ff8-8dda-2a1c646ce58d}} that violates our Minimal Mass Assumption A3 but that keeps the regression function {{formula:f5cd3e61-83be-4384-800a-2bfbbd5fbdba}} in our smoothness class {{formula:d7cf4fb8-8cc4-43b3-a584-60fa5a7f5f78}}.
This is a major difference with former counter examples built in {{cite:25b4040b7012b1d93d9d27880aa1d866a45188a5}} where the non uniform consistency is obtained with a family of non-smooth regression functions {{formula:9fd41f41-6860-44f9-b7bd-fd1c6c8e671d}} . In our study, we also obtained a family of smooth regression functions for which such phenomena occur.
Even in this case, it is still possible to keep the excess risk strictly positive for any classifier {{formula:872183b4-06bd-45ed-96be-61c746d6c979}} (and no longer for only nearest neighbor rules).
| r | 9a99cc20dc6e02e6d8ec591a59e7c400 |
The theory parameters are determined in {{cite:db523b85cd6b66d1aecbb2753ac2b6e980ab4554}} from the condition of agreement of calculated characteristics with experimental data for temperature dependences of spontaneous polarization {{formula:ea7427bc-6c98-4b17-a769-0b7852c41619}} and dielectric permittivity {{formula:2a9072a7-8ea4-4468-a412-9f00e1ec9d83}} at different values of hydrostatic pressure {{cite:c9463144703a6b294eac42fa7f3253b512693aca}}, spontaneous strains {{formula:a4ec41fe-2a75-49c2-b480-57e08cb8625b}} {{cite:76da77eb7f464c0c2df2b7a7fceca5d1241b625b}}, molar heat capacity {{cite:261629dc4af177f6133e3cda327a974507599b69}} and elastic constants {{cite:bcf5a09995255752a4c70ad699cd75db30d2aae7}}; as well as the agreement with ab initio calculations of the lattice contributions to molar heat capacity {{cite:4491c4b69147154be4cebf4a19bf49adc74cece5}} and dielectric permittivity at zero temperature {{cite:a2570c798420c9db40b75755b6eeaefd7b2be1f8}}.
| r | 8fe40a72c1030a6ef11502d8ce51484e |
The pulse area dependence of the exciton quantum state is rather
obvious, when considering the dynamics during the interaction with
the laser field, i.e., Rabi oscillations take place. When keeping
{{formula:b762454b-23b4-4c71-bf35-be989ce32bb9}} fixed and increasing {{formula:3d1fff35-d79b-435f-9ac0-1e83eb9a0f5b}} and therefore {{formula:c1495393-988b-48c3-88d2-7c136212cf3b}} , the
exciton is excited and de-excited during the pulse more often. In
the picture of the Bloch vector, it performs more rotations, which
means that the rotation speed, i.e., the Rabi frequency {{formula:126da2e5-e576-4f59-81a3-9b2226005bcc}} , increases. Additionally, the coupling between exciton dynamics
and phonons depends on the instantaneous Rabi frequency and the
phonon spectral density {{formula:8521696f-abde-4269-82bc-3adffa783c22}} . For the deformation
potential coupling between the QD exciton and LA phonons
{{formula:5a0dbd47-57b8-482b-8ce0-198457196e33}} scales like {{formula:af34ecf0-6d15-4a0d-8df8-9d3cfa6fe092}} for small
phonon frequencies {{formula:48559aaa-287c-4605-8488-5e4aa42c10c3}} {{cite:f7b8b60a7f63cb8dc9c7092ea845ba59cc4e16dd}}. For
typical self-assembled InGaAs/GaAs QDs with sizes in the range of a
few nm, the spectral density forms a broad maximum at {{formula:bb08277f-f39c-474a-851c-12861feeab5f}} in the range of a few ps{{formula:8b4388e1-bec6-4845-88f4-fcda14dcc07a}} , i.e., a few
meV {{cite:1edd2a94b825e379decde878351895b2b304ebdc}}. The upper cut-off frequency is roughly
given by {{formula:aa8e5beb-d908-47ec-b0e4-d490d87470d4}} , where {{formula:e2921eb5-defe-48f5-b87a-5f07436b4c5e}} is the sound velocity
and {{formula:1793ee7a-97df-40eb-8edf-ffadb13af34c}} the localization length of the exciton. Therefore the
strongest interaction between exciton and LA phonons lies in the
range of {{formula:1935f21b-6b53-4c46-9af9-fdae9e5a9427}} .
| r | 1ad0245a4d7c343390ef2bd08165a8c4 |
For a general CM elliptic curve {{formula:3c6235c3-ed39-47a5-8413-16a258e83efa}} , one cannot reduce the Lang–Trotter
conjecture for {{formula:962a0585-cb3d-485d-8073-3641db78ebe6}} to the Hardy–Littlewood conjecture
for a single quadratic polynomial. One would need many
quadratic polynomials in some sense. This part is doable but can be
cumbersome to work out everything explicitly, as it may depend on a long list
of case by case computations of higher reciprocity laws. This is the main input of Chapters , and . Our main idea here is to use
the orthogonality of characters and analytic methods to keep all cases
remain reasonably compact.
See also Q. Ji and H. Qin {{cite:c057c2075e3d203ddf767927908eb3ccb2c47948}} and H. Qin {{cite:1bdefadbafe0c2a229b943cf0afa0d84af0305ee}} for earlier partial results with more explicit polynomials via a different method. In particular, H. Qin {{cite:1bdefadbafe0c2a229b943cf0afa0d84af0305ee}} considered the curve {{formula:fae28348-c981-4f9f-b37d-c3b44a116823}} , which has CM by {{formula:2c88d5ca-11c4-42c4-9e73-1b2f85e67c3c}} He obtained an asymptotic formula for {{formula:f12dda4c-f7dc-4616-a33d-0c488a5da821}} with {{formula:170f24c1-8547-470a-9fd7-c37620806e17}} under the Hardy–Littlewood conjecture, and the case {{formula:ab3ce293-b810-4bea-a3ba-3895f13d480e}} corresponds to anomalous primes initiated by B. Mazur {{cite:c78aa9f5376e4dd6c77a5ece32adc45d955ae1f5}}. Quite recently, H. Qin {{cite:b2d85f8e0299c4387ef2b22a04fa347a140f0a8a}} turned his attention to the curve {{formula:aca02503-d007-4a2c-bc0e-db410ab14b72}} which has CM by {{formula:09b6a839-1492-4414-acd9-56581c328499}} , and was able to treat all non-zero {{formula:7cca76ae-b982-4d80-ba9e-d6df61704c42}} . In both papers {{cite:1bdefadbafe0c2a229b943cf0afa0d84af0305ee}} and {{cite:b2d85f8e0299c4387ef2b22a04fa347a140f0a8a}}, Qin started with combinatorial congruences for {{formula:332806a7-7c54-4e28-9538-7a7931c8be83}} and reciprocity laws to determine Frobenius traces. It would be interesting to compare our analytic number theory method with Qin's and combine advantages from the two approaches.
| r | c78a5803947063dd6d755198a4cc2a43 |
Proof:
By Theorem REF , there are two continuous bifurcation curves {{formula:0589320e-a8dd-408a-91ce-f8abde452bea}} of solutions of (REF ) such that
{{formula:e2abeae1-fbd3-4d77-b3f5-5ca63f3d3a9d}}
for some {{formula:6001308a-b764-4dca-bbd0-68a068b19eae}} . From (REF ) and (REF ), it is clear that for each {{formula:d768510b-bc0f-4f65-bac7-73c6178109b2}} and for each choice of {{formula:d0d2937c-5e73-4bff-966f-00aba7667cd7}} ,
{{formula:af96d859-1638-4124-b7b3-4a04576ab7ca}}
is a stationary solution to the Euler equation, where {{formula:49047960-f15d-48e8-b0c6-3ee26076bcfd}} and {{formula:21944e46-2a71-4b5d-b35c-c4944dfa18b9}} are the bounded domains determined by (REF ) with {{formula:4143e875-b852-4a74-9831-3474a3e7f6db}} . Since {{formula:57d97997-8c0e-4483-b5cf-6619c2734ad9}} , the boundaries are analytic.
Now we consider the kinetic energy of the solution. From (REF ), it suffices to show that {{formula:18cf9d5e-3445-4798-858f-79266ea8fd48}} . By the continuity of the bifurcation curve, this immediately implies that {{formula:6a13e696-b96e-4c20-94f2-50da9174a6e3}} for small {{formula:0a7c62b9-e951-4cb8-bfe7-e77b18b914dd}} . Then it follows from Lemma REF that {{formula:0d87cb2d-dae8-4338-8039-4c6dcba5c058}} , hence
{{formula:8c39db77-7623-4452-b2c5-708ec6221f60}}
This completes the proof.
{{formula:9bc9513d-628b-4a78-bcb5-35e2c5a06c1d}}
The rest of this section will be devoted to prove Theorem REF . The proof will be divided into 5 steps. These steps correspond to check the hypotheses of the Crandall-Rabinowitz theorem REF for our functional {{formula:ccbcd4e5-d103-484a-8d4c-9ddc30135606}} in (REF ). The hypotheses in Theorem REF can be read as follows in our setting:
The functional {{formula:df755b80-23cc-435d-83b6-0c77ce41c3b0}} satisfies
{{formula:9026e3de-743b-4922-ba8c-8e07fbdd4ae9}}
where {{formula:a2f969fc-9f9f-4a15-9015-5083cc71a5c7}} is an open neighborhood of 0,
{{formula:cabb226c-f3ab-439b-9e9e-3236529c55ce}}
for some {{formula:33759c3c-751b-479d-9aab-28a62fa6bf8a}} and {{formula:2ca73db1-8dfe-4bd8-a8db-f1f5073471d9}} .
{{formula:94f2e271-1be8-4c92-9eb4-89ad897a944f}} for every {{formula:309eb463-68eb-487b-8f0e-a9bf2584d9fa}} .
The partial derivatives {{formula:f67304ea-d946-4a67-8190-91050b15d153}} , {{formula:4887ae8b-ea20-49c0-9d33-bf7c7f3334a4}} , {{formula:c5821275-981d-4610-a11b-0dd76139a995}} exist and are continuous, where {{formula:6bfe6670-80ba-4fd5-beed-6b2bd13cf317}} is Gateaux derivative of {{formula:793f1d7f-e1e2-47f2-9d80-44b8df475f11}} with respect to the functional variable {{formula:64e14c35-47a5-4b6d-8e05-adc2b7070bef}} .
Ker{{formula:b3100b62-6f9f-469d-ab6f-05b02f61f1a8}} and {{formula:160e168c-371a-450e-abc2-54da5b65d85a}} /Im({{formula:010770bb-89b2-4973-ab97-4ab4b35bc47c}} ) are one-dimensional (see Proposition REF for the definition of {{formula:566bf151-abbe-4e56-ba9e-b1e0fd17c711}} ).
{{formula:524c18e5-8d18-4863-8bed-155e5afe65fc}} Im({{formula:c6604354-15b7-4de9-a6bd-f06d5dd6e906}} ), where {{formula:663bd040-655f-4951-98a8-14008f5742d6}} is a non-zero element in Ker{{formula:08d78d7b-417d-474d-8ac6-08221adc3374}} .
Remark 3.3
We remark that if {{formula:cef68b52-4af0-4d3b-be30-df5b39ce02a6}} then the functions inside the logarithm in {{formula:530e6542-bf2b-4d69-b264-dea2a697a21c}} in (REF ) are uniformly bounded from below in {{formula:e1df360e-0ef5-47e5-8959-0abd13a33218}} for all {{formula:cfc18621-4478-4298-8cb4-c6c476a9592e}} by a strictly positive constant depending on the parameters. Then we can analytically extend the integrand in {{formula:f6e80dda-8a5d-4a5d-8114-2641f32a9a91}} to the strip {{formula:0dbc5f9e-3bd5-467e-8293-d617527c893d}} in such a way that the real part of this extension stays uniformly bounded away from 0 for a small enough {{formula:ae4c6bb0-5cec-48c8-9f3b-7ac3c4ba6775}} . The case {{formula:17bd8896-3fa2-41c0-a72e-759d8e47f11d}} can be treated similarily as in {{cite:fd63d60ff2018f8875b4c511ef669bde972878e0}}.
Proof of Theorem REF
Steps 1,2 and 3: Regularity
In order to check the first three steps, it suffices to check if {{formula:1f240be6-9517-406a-a680-bd42ff1c78b5}} in (REF ) satisfies the hypotheses, since {{formula:0c16d397-911b-422d-8ac2-a49b6acea160}} is a linear combination of {{formula:96a1e836-88e1-4455-8b5c-950eb67c2d91}} . As mentioned in Remark REF , the case {{formula:3c2122fc-58e8-4dc8-99ea-98ff7b8261d4}} is trivial since there is no singularity in the integrand and analytically extended into a strip in {{formula:1e467c1a-6aa5-45c9-a756-6680808ce963}} , if {{formula:40009445-be70-40d8-8428-f5037d2cb067}} is in a sufficiently small neighborhood of {{formula:40929cc6-6dba-4d35-a7ee-7e3bcc5f5490}} . For {{formula:458e39c3-6473-44b5-bfdb-7d42b5cd1c35}} , the first three steps with slightly different settings were already done in the literature. For example, step 1 can be done in the same way as in {{cite:44577c8dfe787eb5bc9faee2099b0a543f6228a8}}. Step 2 follows immediately from (REF ). Existence and continuity of the Gateaux derivatives for the gSQG equation was done in {{cite:3b2778b50abd89d8d17fa7e4ff7df3988354b403}} and the same proof can be adapted to our setting straightforwardly.
Step 4: Analysis of the linear part.
In this section, we will focus on the spectral study of the Gateaux derivative {{formula:8a10ffc1-86b4-4e14-ac24-31de3ae80f19}} .
Calculation of {{formula:41db7b50-9051-4e4e-8256-2b516f7642bc}}
We aim to express the Gateaux derivative of {{formula:9aa941e9-8962-4515-b359-af1da0235044}} around {{formula:d55b7bae-c321-415c-93db-21c0ba1e9aeb}} in the direction {{formula:098fdf2c-3b48-47c2-a4f3-40bdf9ff5af9}} in terms of Fourier series.
Lemma 3.4
Let {{formula:a765b5ed-9f5b-4a8d-82dd-6b5cfc2a4aa2}} be defined as in (REF ). Then:
{{formula:4a2fc92d-3a4a-4658-b980-e4b5cf46c0e4}}
Proof:
Let {{formula:3e44ff8b-0183-4cf5-ac82-e83a450592c7}} (resp. {{formula:6e2bd872-e1e1-45e8-8ea4-d9a4ee371845}} ) be the contribution of the first term (resp. second term) of (REF ) where the first factor contributes with a {{formula:2d48973b-3a1e-48d3-bd24-1db46f194c39}} and the second with {{formula:8a9ae0b4-3f26-469a-b242-2b0f4257082b}} . We are looking for all combinations such that {{formula:0e334637-fb45-4590-a3ae-a945bd82d30d}} . We start looking at the first summand. We have that:
{{formula:c951c28f-95de-4f6a-9d92-f272b0900604}}
Similarly, for the second one,
{{formula:72c88689-7a0a-435b-bfdc-ceb078cc8233}}
where we have used Lemma REF . Integrating by parts in {{formula:01406e50-d000-45f5-be05-88d16dbfbc69}} :
{{formula:4050bb79-d567-40ee-a60a-9e180a128941}}
Finally, adding all the log terms and the non-log terms together:
{{formula:c8433658-dc25-4d92-9788-7133534d512a}}
as we wanted to prove.
{{formula:cf4c595a-c14a-4e29-95ed-e6522b6bc18c}}
Lemma 3.5
Let {{formula:3a00f719-5069-452c-89b9-3bcc9cd59c3f}} and {{formula:31947682-6ace-4263-b904-4233f37d3de4}} . Then:
{{formula:772384a7-9ed2-4b64-8fc8-b485ab8e40d4}}
Proof:
From Lemma REF and Corollary REF , we have that
{{formula:2e7e5d3e-3a2c-4d12-828a-99efedf767e8}}
Adding the two contributions gives the desired result.
{{formula:83d32c95-a0b2-45b2-8e90-198a8c8ffab5}}
Note that the functional {{formula:186b7483-83e0-46cd-ab78-3ea5db84348a}} is a linear combination of {{formula:a41362c0-fb54-49c5-a377-1a597c79f1d4}} (see (REF )). Using the above two lemmas, we obtain the following proposition:
Proposition 3.6
Let {{formula:709ec1ba-ebc7-4033-b33b-2e04aea34ceb}} {{formula:8c9f8ca1-fec8-42b5-89e6-4f1d6a10aafb}} , then we have that:
{{formula:0caf7df1-8d02-4983-a5dc-882549887d7c}}
where
{{formula:a9c0ac63-956d-4f48-866e-4c4fde3ca43a}}
and the coefficients satisfy, for any {{formula:05cd47d6-c837-4981-a0cf-1a85b0dada4d}} :
{{formula:08556c9a-c0e6-4c86-a541-bf39b5841b78}}
Proof:
It follows from (REF ), the definition of {{formula:80042d99-8bab-4ea3-a5f4-d8f563c01f21}} in (REF ), (REF ) and Lemma REF .
{{formula:1e8ca6a5-25bb-44e1-9e2a-2bcd3097a2a7}}
One dimensionality of the Kernel of the linear operator.
Our goal here is to verify the one dimensionality of Ker({{formula:70afe32b-9505-4771-bb54-48dd9c9a90e8}} ) for some {{formula:b3de714c-24c6-40aa-b857-7a5dd324ea0e}} . More precisely, we will prove the following proposition:
Proposition 3.7
Fix {{formula:17b5005d-cce4-4964-8934-852d34d41a9d}} and take any {{formula:9db42c1c-5174-456c-98d5-226aac10f622}} , where {{formula:bba51a46-e064-42c0-8232-be77d0a35d2c}} is as in Lemma REF . Then there exist two {{formula:78d80999-164c-4ded-8c96-7ead10543eb8}} , such that Ker{{formula:ab923ece-add6-49de-b3cb-e702af3965b7}} is one-dimensional. Furthermore,
{{formula:d7845a92-1534-4942-b162-c82b3b65afff}}
The proof of the above proposition relies on the analysis of the matrix {{formula:3a753c6c-b5a1-4141-86f6-c326532667c8}} in Lemma REF and REF , which we will prove below.
Lemma 3.8
Let {{formula:42578531-c813-437d-a75b-9a82cb19b616}} be
{{formula:ec80ae4d-632e-4502-b3bc-8399c7144149}}
Then, for any {{formula:0c229e5e-06d9-49b0-a6bc-8539ee7d7647}} , there exists {{formula:a6a4c24f-f9ea-4a7f-b4d9-f51fcca4152e}} such that for any {{formula:4cde4ec2-64fd-483a-98b6-18cc09aaaf1a}} , there exists {{formula:c6673e0a-c30b-473e-b563-275850034d25}} such that {{formula:5bb7fded-360e-4240-8173-6af6452bd7e2}} . We also have that rk{{formula:f2d35166-d95f-423e-9efe-65a6005f934d}} for those values of {{formula:ba730d84-3c47-4ef2-b9db-c20b3be219fe}} , {{formula:315edb2e-4eb0-47b0-9fb1-980522d12a4d}} , where {{formula:e40afa4a-8026-41f5-b607-1c6624d58bff}} is the rank of a matrix {{formula:4db547b0-d0e8-4512-953f-d1e1a8f86883}} .
Proof:
For fixed {{formula:cbb9b779-5eae-45a6-a3ce-113cdcef7e3e}} , we study the polynomial {{formula:9bc1016f-5f59-4343-b9e9-02e648b945c0}} .
We need to solve
{{formula:6cc5a7a3-5f2b-4ef7-8fca-22925f7fc364}}
Since {{formula:6b8a3995-6270-4887-8a09-831b78204531}} is a quadratic function, we only need to show that the discriminant is positive. We have
{{formula:7351c803-3cff-48be-ba23-149e85fa0a2f}}
Since {{formula:fda7ac26-72d4-45e1-8064-cb651badd337}} , {{formula:4e1bfbbd-41b1-4d22-a86f-b5fa21975779}} , we have {{formula:f56a6044-15ad-46b2-8d3b-a3547550a338}} . We also have
{{formula:e9531bde-54e9-4949-b155-2adf7dd65133}}
{{formula:7a11f7da-1427-4973-993a-fd7f9ca1ba4b}} is decreasing in {{formula:d1abd8a3-20d7-4069-a689-63925f07de37}} when {{formula:c6d57309-9979-4c6d-9ec1-9719e342c9d1}} and {{formula:3ef05902-bfed-408b-bf0c-f6ac41f83846}} , {{formula:0f207cab-46e3-4699-913f-3ee8b86336ed}} . Let {{formula:ef1f5486-2ca1-4a81-9b69-0f10125f3fac}} be the only zero point of {{formula:850a578c-56b2-4013-a73d-84326196f4ba}} in (0,1). If we take {{formula:b4570554-ba60-4fa1-b09f-df120686124e}} , we have
{{formula:f4da1b3e-f7b2-42a4-9333-3cbb99dbd980}}
Hence {{formula:14388c70-71cc-4a04-beb8-7f87fc73d327}} has two different solutions {{formula:c58b3795-def6-4246-8f43-96a55d96f609}} .
Moreover, the matrix does not vanish when {{formula:21349c30-6a16-4dde-9f4e-1a39e6a575e4}} , implying rk{{formula:6731ab18-a851-48ac-ad51-4d5c35d31223}} .
Now we are left to show {{formula:64c2cd6c-74c1-48e1-8b27-367ed8d72906}} . We have
{{formula:e61e0642-2c10-4868-8790-c57e400e0294}}
and
{{formula:a93ddcdb-9615-4ddf-923a-cf6c3f9e097c}}
Hence, {{formula:4d69b36c-b284-4e7c-99d7-16fdd6c6e1b2}} and {{formula:cc5abafd-948e-435b-afe1-dcceda356a6f}} . If {{formula:5037e779-d666-46c6-b99c-7f360cc37e29}} , then {{formula:611b9b43-9ab0-49ff-9808-a3c3244e3071}} . However,
{{formula:e60c7e91-375e-4efb-99a3-d1f58d6f70ca}}
Therefore {{formula:69c965d1-af1c-47ee-9b56-98b8c02d7e1d}} .
{{formula:6a637e56-187c-414c-9f96-1f4b37305d58}}
We now show that {{formula:e3eca9ce-dc0d-443e-8f41-920be19c4ddb}} for any {{formula:a3badc69-24b1-4426-b311-965727b06203}} .
Lemma 3.9
Let {{formula:6a5dcb85-e75d-44ac-bfee-ba05c5c56698}} and let {{formula:eba2dcfb-3c88-4a97-81a6-a2dc81353619}} and {{formula:3c581dc1-294d-4103-af9d-4636a1cbf16f}} be defined as in the previous Lemma. Then {{formula:5266df7a-f06e-460c-a43c-e11ed501965c}} . Hence, {{formula:87e5446e-fa51-441c-8933-e07d6cae99f7}} is non-singular for all {{formula:55527541-33b1-422e-9f7f-af7f8388be9e}} .
Proof:
Since {{formula:376c21c3-a7da-45bd-b020-7e5655414b9a}} solves the equation
{{formula:7173b81a-1fd3-478e-8385-72db842ff7aa}}
We only need to show {{formula:a939ffc9-c725-497d-b430-dba4987ffbff}} is strictly increasing with respect to {{formula:c8f73306-d6ef-4320-b08c-25ef1128bb5b}} .
We have
{{formula:62918894-816b-49d9-bb22-8f69d90dbc52}}
Thus
{{formula:0e36ea77-3816-4572-b64a-32dca3f8d0c2}}
It is easy to show the last inequality since {{formula:8c6c627d-2951-434e-a25e-5596f0faebfc}} .
{{formula:1d12387b-d613-4e37-860b-2d4e9e203ff5}}
Proof of Proposition REF :
Let {{formula:ec0ddab1-1bd6-4d14-b1fe-f13f5917bcad}} and let {{formula:35f7c317-292e-4c91-87cd-460fa1d9085b}} , {{formula:0fec93ae-2987-42e2-b89e-75c9b29d6636}} be as defined in Lemma REF . Assume that {{formula:09de6667-9728-4499-a348-2bf030fe04cd}} and {{formula:04cfebe8-bb09-448c-9187-7942b8581650}} satisfy
{{formula:f0b3097d-bc0f-4444-91b4-a3e76ce17678}}
Then it follows from Proposition REF that
{{formula:b8a8b597-b38c-439d-b28b-fae83f27c936}}
For all {{formula:537fddff-5845-4fa3-b86d-7b5ec7012a96}} , it follows from Lemma REF that {{formula:309025d2-ab72-4920-84fd-d1dc6a35aa4d}} is invertible, thus {{formula:75a94b01-152f-46b9-90f9-d163c35615cf}} . For {{formula:44cf3e9e-69eb-4a87-84a3-197c3d784ebb}} , Lemma REF tells us that {{formula:46cbf770-67ba-4f2a-90a1-e7b1a5ccd559}} . This finishes the proof.
{{formula:34dd7c77-2358-492f-a1d4-6dd874ac54d0}}
Codimension of the image of the linear operator.
We now characterize the image of {{formula:22ab975d-2d25-4afa-aa8c-0c525663d97a}} . We have the following proposition:
Proposition 3.10
Let
{{formula:08b61dce-df6e-48aa-8acd-d5f31de7360c}}
Then {{formula:ba3f609a-2b3d-4323-a52d-972950711c67}} .
Proof:
In view of Proposition REF , {{formula:b3fd1e65-791b-415d-957a-6024a964f351}} is trivial, since {{formula:c9d209ac-6474-4b4e-95ad-31e0fc220674}} is non-singular for {{formula:e428f9cd-c16f-49b8-885f-f33dd41e115a}} , and {{formula:ec9aa35e-307e-420c-9681-83e7852aedbf}} In order to prove {{formula:9da848a8-bdd2-4055-b7a6-e5616664f804}} , we need to check whether the possible preimage satisfies the desired regularity. To do so, we have the following asymptotic lemma:
Lemma 3.11
For fixed {{formula:8c31d440-b4fc-4927-a3c0-a35fb895bf5b}} and {{formula:e5af52ce-ee2b-44b8-b33c-f31661af7828}} defined as in Proposition REF , we have {{formula:f59f3a44-5492-4510-868b-d2eec617af12}} and
{{formula:8ca8a34c-738e-43ca-8844-2abc5e952159}}
Consequently, we have
{{formula:24cef377-d4d0-4c1c-b267-75948286251a}}
Remark 3.12
As shown in the above lemma, there is no bifurcation curve from the two-layered vortex patch with zero-average. This is due to the fact that the radial vorticity {{formula:447ed90f-a7a4-4e0b-a395-00f8759d689f}} determined by {{formula:17dd4685-d536-4d3c-80d7-fe0b2ede2ce6}} and {{formula:31e9c7a5-83dc-4f75-bf99-543edfc1dc90}} as in (REF ) satisfies {{formula:f405185e-9109-453a-9d4a-15d7fa9370e9}} . Note that if we require {{formula:24659ddc-781f-4dd4-8d15-d08cbea321ea}} to ensure {{formula:1f878887-6dc0-4d52-b126-4968d93e6f61}} , it follows from (REF ) that {{formula:407f6c5e-3d71-4136-a3f8-cee00c5d2a32}} , which does not vanish for any {{formula:465c3609-b36d-4cdb-9fa0-34acb828cb52}} unless {{formula:2ee0122e-1e92-4694-ad14-c523fe89ba41}} or {{formula:b39bd1b7-3d06-4191-992c-2767009ef040}} . Therefore, for any {{formula:72093ca0-8f9b-4153-9d17-a841e7e62120}} , the linearized operator is an isomorphism and the implicit function theorem shows that there cannot be a bifurcation.
Proof of Lemma REF :
First we show that {{formula:00cdeb6e-fda7-4585-8bac-fc2645002dca}} . By Lemma REF , {{formula:9ff70692-b3ab-4575-9716-98c3644a915d}} and {{formula:afc09e88-22da-4792-b5ce-c0d6ad34a009}} . Thus {{formula:b2e8be04-8e35-4dd7-bed8-3554ba297980}} . Therefore the first assertion is proved. The second assertion follows directly from (REF ) since
{{formula:cbd92ffe-8002-487a-b243-0f469bc97e54}}
Lastly, by choosing {{formula:b7fc46a7-8a3a-40c2-b92f-4f8f9c22aafc}} large so that {{formula:e808585b-77fb-46c7-850c-f3f98fa5c3b7}} , we have
{{formula:3d5ad343-d587-4223-bf73-e64bda5c8185}}
which proves (REF ).
{{formula:5fb67d7c-51c2-4acb-9071-ff9832abfd7b}}
Now for an element {{formula:122958e9-3c7b-4de3-93a7-719ca8d9c883}} , let {{formula:b3a4ef30-e95d-42ef-8904-a0c6ee35552c}} be such that
{{formula:355cddd2-2617-4eac-bf71-973446e77c9c}}
with
{{formula:a077a66e-1948-4fbf-8c72-4ec17f91f873}}
It is clear from (REF ) that {{formula:88f3bf7b-e47b-455e-b9b2-4582e7bb30d5}} We will prove that {{formula:4e2ac57c-f9a9-4243-8486-98220aa890ed}} .
From Lemma REF and the fact that {{formula:69c974e9-e78d-4a60-9a6b-f72a7cd7948c}} is nonsingular for {{formula:5f694350-b398-428b-ad49-9ba62abef7d2}} , it follows that
{{formula:a617d3bd-e45a-4350-8f88-8fa83c8e7995}}
Thus, we obtain
{{formula:b5e25b62-0fa0-40f9-95ac-200237e3eef7}}
This proves that {{formula:b6825303-cbb1-4769-bf60-4c2fa24d478c}} , and therefore {{formula:2a7ed682-f596-44eb-a6a5-f5e26c8bc7e3}} .
{{formula:7488309a-fc44-4a9f-a808-47f8e0757a9c}}
Step 5: Transversality
Proposition 3.13 We have that
{{formula:ee1be497-4381-4bba-aa1f-08a28927cd58}}
where {{formula:f6cc0ef7-bd91-4d50-a218-930646918e70}} is as given in Proposition REF .
Proof:
For {{formula:b24bf105-28e3-4bd0-9406-50eb8116d89c}} {{formula:6190fa6f-c44c-4111-898d-b55c7ce9c94e}} , we have that (see Proposition REF ):
{{formula:b35933cf-9542-4e8c-841b-60c4499898bd}}
where
{{formula:de0c01f8-9778-43f8-85d4-d65400a1c942}}
and the coefficients satisfy, for any {{formula:1db37ebc-a9ca-4d29-a317-a43f69dacf82}} :
{{formula:f9c6fe60-9c0f-4292-a6f5-18086aeaf3a5}}
Letting
{{formula:dbf0b7ba-a77e-4c52-8fdc-2e3c7149a4bd}}
be the generators of Ker{{formula:1983854e-bc45-4893-83c3-9ddce430c46b}} and Im{{formula:25aad0f8-e21e-450f-94e3-39852212f3fb}} respectively, the transversality condition is equivalent to prove that {{formula:2b1302f5-514c-484b-917e-130d671fc0ab}} and {{formula:e7e31aa8-bb47-4936-8139-f96cabfef600}} are not parallel, where
{{formula:167d3095-9031-4008-8085-9650bb62cce4}}
This is equivalent to prove that:
{{formula:777dfd92-875e-43ef-8657-84d5fb4adb5f}}
We prove it by contradiction. If {{formula:29b7f928-82ad-4056-9c03-9af54701cd0e}} , we have {{formula:9d47a868-53c4-4b91-a52a-0dc511c3a7fd}} .
Moreover, by (REF ),we have
{{formula:b487bd01-41b0-413f-a66c-92ddbf062a19}}
Hence,
{{formula:97154fc3-f10a-4b6b-bf12-b1f92450c1d4}}
Since {{formula:12eb5065-df58-41fa-b796-8fb98cacdd7a}} , we have
{{formula:5747e03e-6acc-4d31-8b77-f6ace4123580}}
implying a contradiction since {{formula:ef92faf3-5133-4420-aac9-f7fc8d73f704}} and {{formula:67aef6a4-65fb-441f-82bc-a339ae2b2089}} . If {{formula:e7d61b5d-04eb-4242-ad31-5c79e5dedb15}} , we can follow the same way to get {{formula:2bc004ad-0b77-44f8-8c92-484e2b0458a3}} and get a contradiction.
{{formula:7044bf25-64ea-4829-a24b-ecba628c909a}}
Proof of Theorem REF :
All the hypotheses of the Crandall-Rabinowitz theorem were checked in Propositions REF , REF and REF . Therefore the desired result follows immediately.
{{formula:5337f268-79ab-4cd1-a0b0-72a32437c23f}}
Existence of non-radial stationary vortex patches with finite energy
In this section, we aim to prove that there exist non-trivial patch solutions with finite kinetic energy, {{formula:0cfaeab5-e4f6-464f-837e-ceebe5bf33ca}} . As mentioned in (REF ), this property is equivalent to {{formula:9a629a41-5fb8-4455-a2c2-2b3894959eb8}} . By Remark REF , we can not use two-layer patches and instead we will consider three-layer patches.
Main results for finite energy
We consider vortex patches with three layers, that is, {{formula:bbd91a9d-d2c9-40ab-90e1-1098d10e2be9}} in the setting in Section . The total vorticity that we consider is of the form {{formula:577a4b01-c5da-4e6f-8c1f-606f03ad9d16}} , where {{formula:15a3cb84-482f-439a-8165-c6031b7b14a6}} is determined by {{formula:6bf6b1cb-34dd-4658-b1ae-2007fe9cccfc}} . We will look for a bifurcation curve from the radial one, {{formula:e3915c87-4d39-4a04-b45c-968b288b1c3b}} , where {{formula:f3817517-5123-435f-9c80-8286a76a475c}} denotes the disk with radius {{formula:0ce25150-6ee9-4631-856d-7f3ef8fc436a}} centered at the origin. We have the following parameters and functional variables:
{{formula:d4a11409-1b6b-4bc0-8951-170066ac929c}} : the radii of the different layers of the annuli. We will have {{formula:604d1597-5dda-4173-88f1-47bd900391e7}} .
{{formula:d16eb449-b7e3-4c2e-8d33-31edac0af363}} : the vorticity at the different layers. We will choose {{formula:3a898c03-483a-4c21-9f97-25828e3c5938}} .
{{formula:e2141085-2004-4e6f-af6e-9817ef47b7eb}} , for some {{formula:91375bba-4975-40b0-9ea3-560a14163d9a}} : the functional variables that determine the boundaries.
In the rest of this section, we will fix {{formula:bcee1e90-2f29-4e3e-a309-0b8b6b0370c3}} , {{formula:c2f8cb69-1091-44d2-82f6-f169ffadb076}} and {{formula:55eadf7b-b458-4a16-aec9-d0b3b2751711}} so that for {{formula:fef874e5-86b7-43e1-96f2-5ff12a1b2051}} ,
{{formula:4b24006c-5441-4a66-86b0-f94279b14147}}
Given {{formula:82109042-3838-47c6-bb79-ead73d6d3922}} , {{formula:20291f26-8a56-4d9f-8454-dab9f511ab8d}} , {{formula:76abe6fd-2655-41c0-8f04-d1002e0d0a45}} and {{formula:055a273b-d71a-4553-87dc-6db715ebb44d}} , we choose {{formula:7294f761-499c-4bd1-8968-1374134e4a6c}} so that
{{formula:40aacf31-8a47-4d60-bf03-a2cce4864954}}
Since {{formula:f81ca60f-7e4f-4cf2-ae9c-0464ea4c4b51}} , {{formula:6cdef915-67e1-4d7b-9ba1-b78ff64c30d2}} and {{formula:2a51d1c3-e269-43df-8026-d8601c8d48e7}} , {{formula:a2eb6ab0-668c-4421-9aa9-ebc1441c1c93}} are fixed constants, (REF ) implies that {{formula:73e70cf6-7036-4d5e-909b-cf9fba6fcddd}} is a function of {{formula:a795691b-7610-4c4c-873d-c63113f994e7}} and {{formula:7a1ac788-befd-4076-9c08-c0d310bfca23}} , more precisely,
{{formula:ce7f70d4-ee64-4c97-a56a-5daaa63be389}}
If {{formula:16a6cb12-e711-4be6-96dd-72575e75e08a}} , then its derivative with respect to {{formula:e2592fd3-ecbc-4a51-b338-c78200df1fdb}} is given by
{{formula:0264abae-b4ac-4331-b2e7-ff98b8048461}}
where we used {{formula:db72b68b-61e4-473e-bb30-49f5482a3ab2}} for {{formula:fc1bc14f-d0ad-4ba3-b9f8-d2eba9ba808f}} .
Note that for sufficiently small {{formula:3bd48483-dfcf-4dae-8e7b-9e58bf0d01ab}} and {{formula:02e64259-0c6d-4366-b42b-4f05776afac4}} , where {{formula:75bfa4eb-102c-4b2b-9c53-354f6ffef28d}} is as defined in Lemma REF , we can choose {{formula:c6ad3671-a547-49c9-a462-4f3400ec22ae}} so that (REF ) is compatible with {{formula:a21e1799-8323-4d7b-b6c7-598616161362}} (see Lemma REF ). Therefore, a 4-tuple {{formula:ff8d065f-2144-4f7c-82f7-7e3c35e85df9}} uniquely determines {{formula:fc92e0e0-ff7e-4b9d-8165-c0dc361efe3e}} such that the boundary of the {{formula:f58b7e7f-5c7c-45b0-b82f-3c0370522a20}} th patch surrounds the {{formula:65980270-b71f-40b0-9b4b-45f527075f31}} th patch if {{formula:3ba2c384-5927-435e-9417-8af1ef87e080}} and {{formula:1f50acf6-82c8-4ce0-8508-09ad399e218e}} . In the proof, {{formula:b82fb060-eb24-4431-ac80-8a3d4301207a}} will play the role of the bifurcation parameter and we will look for a bifurcation from {{formula:6ad00e8c-7a56-4101-b5e9-a3e3e6bc1d7f}} .
With this setting, the system (REF ) is equivalent to
{{formula:0734dbfb-75fc-4a46-8b87-69678ec80a15}}
where
{{formula:0111cd4d-04b8-448c-b575-3a504f1e6cc4}}
Now, we are ready to state the main theorem of this section:
Theorem 4.1
Let {{formula:8167afa1-8b63-421c-8c5d-352283573fb9}} and {{formula:b82ce3da-def9-44e4-9e22-0559374e853b}} , {{formula:884003b6-5d2e-4b47-8b1c-f33aa747632e}} and {{formula:0404142b-f0c0-4cee-8a00-f50a098ff5d4}} and {{formula:b8a22817-d357-418e-9ad5-e3d4869ba570}} as in (REF ). Then for some {{formula:534dd3bf-5db3-4704-8163-87245646e3a2}} , there exists a bifurcation curve {{formula:eac8ca7e-a7d3-46a5-afbc-78b2797ad798}} such that for each {{formula:5bed1874-d8f3-4409-81f1-e5cea7e2b908}} , {{formula:8dd1eb30-dd17-44b2-81d5-6dd6b5bbf962}} is a solution of the equation (REF ) and {{formula:3a95809c-4b51-4b81-87f7-e3a97af42682}} . The bifurcation curve emanates from {{formula:318ed06c-28a5-4894-a794-c84632e208a9}} , where {{formula:d611ba07-ead1-40a3-9374-2f54019c9bb5}} is defined in Lemma REF .
Theorem REF immediately implies the existence of non-radial stationary vortex patches with finite kinetic energy.
Corollary 4.2
Let {{formula:a3aa21f1-e139-4708-9ffd-989034702e82}} and {{formula:e2e4653a-35ba-4285-a948-6b95dec5a637}} . Then there is an {{formula:317ae526-7dba-49c4-bfea-3ed3f5e2eef3}} -fold symmetric stationary patch solution of the 2D Euler equation with {{formula:e3e394e1-de22-49e1-9462-733240d6a434}} -regular boundary and finite kinetic energy, that is
{{formula:3c66e970-01e8-40f4-9112-fcf4c52e1038}}
Proof:
By the definition of {{formula:e9e7eed9-4d4d-428a-823d-e5ca2c795588}} in (REF ), each {{formula:333a6577-9833-4c5d-a69e-949051dcf998}} which is determined by {{formula:19a0fa00-95c8-49e8-89a9-782914439def}} for {{formula:55a38c11-aef0-4a0b-9be3-af9d05d9bc49}} , satisfies {{formula:cb390f03-23af-439d-898e-68c3f85b405c}} . This is equivalent to {{formula:f441efa8-f020-4dee-b5e5-6ac4a23329cd}} , (see (REF )).
{{formula:0051fb44-a10e-416c-ad6a-f7f6385764e9}}
The existence of the bifurcation curves will be proved by means of a Nash-Moser iteration scheme (Theorem REF ). The proof of Theorem REF will be accomplished in Subsection REF by checking the hypotheses of Theorem REF .
Compactly supported velocity
In this subsection, we digress briefly to observe an interesting consequence of Theorem REF . Thanks to a simple maximum principle lemma, it can be shown that each stationary solution on the bifurcation curves has compactly supported velocity.
Lemma 4.3 (the key Lemma) Assume that {{formula:9b98241f-9cf6-4ba7-b082-04b906e89392}} for {{formula:53a3874c-db91-4e86-9b5a-ca205668d6a0}} is compactly supported and let {{formula:a5adb545-aa0e-46c4-b0e0-d6df1986be71}} be the unbounded connected component of {{formula:fd76ed2e-95dd-4f83-b958-c67c732a3748}} . We additionally assume that {{formula:44f3655e-3fdd-4e1f-9c64-1cba1aee72ab}} . Then for {{formula:cb9f9fcf-eef3-409a-a9b0-e66bd53641c5}} , where
{{formula:a4b134df-6811-488c-b8d4-7cf4a6629fd7}}
it holds that
{{formula:78fcc83b-5c75-4c2c-b6ba-ca7c05341d52}}
Consequently, if {{formula:408c8a21-5e1e-4da6-b45d-bf7536c2dc78}} is constant on {{formula:5fe76596-48c1-43d5-9f44-7b7c423f454d}} , then {{formula:6c24ca5d-0e04-4f31-bb80-bdfddc978e40}} is constant in {{formula:1294674e-5d41-4cf2-98fd-7d50f0945523}} .
Remark 4.4 The above lemma does not hold without the assumption {{formula:0c193e70-27e8-40de-a237-9b72b6398539}} . For example, {{formula:4a93cc50-bcad-488b-8e96-4e3814c6bcf3}} is harmonic in {{formula:88713ca5-dc9e-49df-a0ae-400f8561a9b1}} , while {{formula:58182d2d-f664-4010-a1c6-f3364fe92b6f}} is unbounded in {{formula:075e02b1-be37-4e6f-821c-7c99878661a0}} .
Proof:
It suffices to prove the maximum part since we can apply the argument to {{formula:7bb74099-8247-4660-bf36-758ca6dc906c}} for the minimum part. The proof is classical but we present a proof for the sake of completeness.
Let {{formula:11c43b09-f45d-4786-8d4c-5a20dac9c656}} . Then {{formula:f7ac9406-d42e-414b-a717-7749a3050749}} is relatively closed in {{formula:6b77cb19-211a-446b-9836-611e47e80380}} , since {{formula:c026a209-ef4c-4f22-aefa-d0c7e60554e6}} is continuous. Furthermore, since {{formula:271a0bff-6bba-485e-877b-8785fa4effbf}} is harmonic in {{formula:0d4150b1-46bd-4b1f-a3ed-0d5f832299c8}} , the mean value property yields that {{formula:3ac411fc-22ad-4ff3-84fa-ea11044ef4bd}} is open. Therefore {{formula:537f2259-14b3-42b4-85e3-2d80e37b102f}} must be either {{formula:1babb84c-90a8-46fe-ae28-c8d8ab474687}} or {{formula:ce0a2e4c-d48a-4763-bfda-996474e1f7e9}} since {{formula:8bd94a8a-2ab9-4554-abbc-3ee53b74c297}} is connected. If {{formula:a0e529ed-458d-4e5b-bf9b-8a83df58e5cb}} , then the result follows trivially. Now, let us suppose that {{formula:a30d4da3-45f0-481d-86fa-fe43268917b8}} . Towards a contradiction, assume that {{formula:f2e1ab21-5d3e-4bde-ad5a-e8814a1be458}} . Since {{formula:2b1efddd-3f7f-42e6-af5b-2513e8be7a46}} is bounded in {{formula:89c96c1a-eda4-4d35-864a-98c84e878d86}} (this property still holds when {{formula:21bf05b8-ad8d-4db9-af67-d5bfd8bb156d}} , thanks to {{formula:19f2b0f0-14b6-4001-b961-e592f3b584fc}} ), therefore the only possible case is that
{{formula:64ce441a-9fe3-4696-b19b-62ee847329b6}}
Let us consider {{formula:f3b76491-3763-46da-864d-48984b3c6bbd}} . For any sufficiently large {{formula:f678b3dd-4674-4bd4-b00f-65c8c40fb745}} such that {{formula:724e7021-312c-4eb9-911b-c6385bff6fc6}} , where {{formula:a79b86ed-efeb-4101-8186-c981ae559b05}} is the ball centered at the origin with radius r, it follows that
{{formula:2906a536-8232-4cbe-9849-dbf25eb837b8}}
Furthermore, (REF ) yields that
{{formula:f48534b1-d0bd-4a09-a5b0-8391f98dcfad}}
hence, we have {{formula:fef52cce-a343-4968-a6fc-30383a90b7f5}} for all sufficiently large {{formula:0878712f-39df-440d-bc62-e7d587fe71d2}} . For such {{formula:29276327-fa9f-4aab-81b8-15809078be79}} , we have
{{formula:be79a2c6-c373-498c-91c7-0e4e0cbac920}}
This implies {{formula:ba0ea512-adc3-497e-bf2a-1013d5de3ce6}} cannot be empty, which is a contradiction. Hence {{formula:bfe11b2e-7752-4287-887c-8087e31a6de3}} .
{{formula:93d03dfb-0f0f-416c-b388-c0ca2936cadc}}
Corollary 4.5
Let {{formula:8b550afc-3389-4b2a-a1e4-c7d93c7d301f}} and {{formula:9f2fefe7-f9ea-4eec-bf86-18806899cb68}} . There exists an m-fold symmetric stationary patch solution of the 2D Euler equation with {{formula:2736275c-0e40-48cf-bc55-951a76237238}} -regular boundary and compactly supported velocity.
Proof:
Let {{formula:6f74de35-ef95-4ae4-9895-74c6985a6408}} be the vorticity determined by {{formula:a846daf9-e2b6-4f3c-8ef6-0dd9dcb94984}} for {{formula:87054575-0c5a-4f24-bd50-20bc7e551150}} in Theorem REF . From (REF ) and (REF ), it follows that its stream function {{formula:80bf0b1f-f9ee-43e1-8f41-671961d8a41c}} is constant on the outmost boundary {{formula:957ccaf0-a994-4631-84e4-5e95a4690dc2}} . However, Lemma REF implies that {{formula:c692cdce-af9c-4292-a931-7b12e9140016}} , therefore, {{formula:c3503208-32b5-4dbb-b696-fc6fc9d3cbc0}} is constant in {{formula:4ca508f5-41e2-4388-ac64-10cd3347a9a2}} . This proves that {{formula:be73843e-9298-4f85-8dee-3e5475498a5d}} .
{{formula:b4ccc315-c317-4ed6-82ab-77585805fd9f}}
Remark 4.6
In this paper we focus on patch type solutions {{formula:66dcaec4-24ea-40a9-98c6-2ec2516c8027}} . Lemma REF is still applicable for smooth {{formula:30a95825-d7c6-4783-ad3b-f44294401f6a}} as well as long as the boundary of {{formula:520a4fd2-2833-4017-9bc6-dffaf414f998}} can be approximated by regular level curves of {{formula:158f2cff-261f-4a15-88e3-d3bd34c63fba}} (See Figure REF ). This is due to the fact that the stream function of the smooth stationary {{formula:c59ea1f8-fb97-457a-a93b-951885063806}} must be constant on each regular level set of {{formula:4cc9d382-71e8-4e8b-b495-246a32b52a75}} (see {{cite:e63b70a43dd8ca27aa97cb981005e5285271baa4}}), hence the stream function has to be constant on each connected component of {{formula:f437ff73-cab1-465c-959b-39e6a2b9d370}} . If {{formula:34057b7f-1541-4938-8b3a-1eb597d993ed}} has only one connected component, then the velocity vanishes in the unbounded component of {{formula:eb4aa168-8cd2-4747-96b5-4284c9f6866d}} .
{{figure:09faf8ad-1c5d-415c-9353-a77a57acb94d}}Nash-Moser theorem
We first prove a bifurcation theorem using the Nash-Moser scheme under some assumptions, which will turn out to be satisfied by our nonlinear functional. We follow the ideas from Berti {{cite:e362605c3a9871c93f1e04a1e619490c004461a5}}.
Let {{formula:fce45e34-fffa-4bb0-a110-1c94628c0348}} fixed. We denote
{{formula:23a8c7e6-1846-4d38-9e9e-1f1c998c1f84}}
for simplicity. Furthermore, for a Banach space {{formula:b6c6d6b6-9484-44e1-9c9e-b98185e680aa}} and an element {{formula:9cd8e16a-02c7-4acf-a97c-c218672546f9}} , we denote the norm of {{formula:6ea75db8-9824-4b10-b0ec-2d12e7daa6c3}} by {{formula:5418a0f4-1859-4894-844d-bc23241c73ee}} . In addition, we use the notation {{formula:cbbacb45-501a-40ea-ae32-10cdbf6ec81e}} if there exists a constant {{formula:f8c72096-ddf4-4a4f-b333-b1eae199c53f}} depending on some variables {{formula:2411a257-5b7d-411b-a6dc-06dfb8b5e81e}} such that {{formula:9e8bdd70-ae8c-4246-a09a-4553e2bd46c1}} . We also use {{formula:8035d459-6c95-411f-a8d9-b41a7dc4b92e}} to denote universal constants that may vary from line to line.
Theorem 4.7 Assume that there exists {{formula:ce550c96-f6ad-48f9-85fa-6c91adc0c5d3}} and an open neighborhood {{formula:9470d6f7-f0e4-45ff-8bd1-bc5563994fba}} of {{formula:675a7e7f-703f-475e-adcb-e3222e49c390}} such that for each {{formula:741f93f3-f7a9-4609-a1f1-7ad6fa08e63d}} , {{formula:8f10f6e0-3b6e-49c5-8167-40d1c925e198}} satisfies the following: For {{formula:531b5aad-616a-4809-842b-4110e7d2de78}}
(a)
(Existence of a curve of trivial solutions) {{formula:10889a92-44ba-4759-a54f-17574cb5bb5e}} for all {{formula:7b908b5a-e1cc-4e41-84c2-e15bc96b37fe}} .
(b)
(Regularity) It holds that
{{formula:11a4f393-8c92-43bf-8195-9b8e646142ec}}
(c)
(Decomposition of {{formula:30468aeb-3b93-4155-b914-3d68365ffb65}} ) {{formula:d481d403-b775-4620-82f5-a3b604b38175}} has the following decomposition:
{{formula:432014da-5b70-49b4-bfb1-949f3c1b156c}}
such that {{formula:71032d16-0fb8-44b3-be7f-7e4203385eca}} , {{formula:bc3da3be-4237-4fec-a22a-ecc842cd0904}} . Also, there exists {{formula:03d009e0-4981-40c2-8f6b-dec009aed097}} such that if {{formula:db1cb0ab-a05a-430c-9a4f-dc7a473fe8a4}} , then
{{formula:164904a8-1510-46ce-93f8-3367136a7ac4}}
({{formula:df70732a-9087-412f-9537-59d1db550f94}} -1)
{{formula:b3c4d490-d17b-4fb5-a2ce-8c07f87e090b}} is Lipschitz continuous. That is, if
{{formula:bb30ae38-dc81-458e-a5d2-92bb85d05ccf}}
and {{formula:754ba234-4f82-40b1-ab83-fc20cc3cbbc0}} ,
it holds that
{{formula:70414a75-c960-42ce-9bc2-e92f625a870d}}
({{formula:c295e851-f643-4827-a605-d727530ebb3d}} -2)
(Tame estimates) There exists {{formula:54193552-8b45-406c-920f-5245a4558745}} such that if {{formula:28e6d634-2e6c-4cb3-a9ea-4352a2779d4d}} , and {{formula:92361bb7-b731-4198-9ae9-a281b7f8b4f5}} for some {{formula:217c12b1-487f-4c68-b25c-eb141ec32e65}} and {{formula:44a2826e-53be-4474-9307-1436bc5aa9c5}} , then {{formula:b7c55ddc-1123-4af4-b06c-00a1337783be}} . Furthermore, for any even {{formula:fcbffbf9-e7e7-45bc-9e14-4fd892562b78}} , it holds that
{{formula:b65b0154-422b-48e5-9b7e-8f260103038b}}
(d)
(Fredholm index zero) There exist non-zero vectors {{formula:9d6e0d0d-67d7-49fe-adb0-60cb347de794}} and {{formula:2bd8fcf6-71e1-41ad-bec5-837b456c2388}} such that {{formula:2ce1a063-0ea9-45be-b82e-e14f9428ed2d}} and {{formula:056eb07c-1976-4b37-a387-6ca37030609b}} are supported on the {{formula:b45a9ed4-c876-485f-9331-0f04b4b8d456}} -th Fourier mode and
{{formula:c83b0409-1349-4ddc-86fa-35fb2adec79c}}
(e)
(Transversality) {{formula:6c8148ac-a1f6-4c7f-997b-fbf4696ea284}} .
Then, for any {{formula:a5619d28-2a36-4537-912f-a55ebbbbce8f}} , there exist a constant {{formula:e36bd604-a3b1-40bd-b1b3-ad86119b9677}} and a curve {{formula:5fab0299-66fb-445f-b992-eb830fbfeedf}} such that {{formula:6b0b9969-6a99-4555-8c21-5f7f9fba1331}} and {{formula:42834ab6-9761-4fa4-a776-49778ff1a34d}} for {{formula:98535adc-df1c-4042-90a1-3b38ff22c524}} . The curve emanates from {{formula:4665cbd7-57b7-417a-9e96-dbf05caf040f}} .
Remark 4.8 Note that the evenness of {{formula:d6a904d4-6635-47cc-9989-3170cd98bd1e}} for the tame estimate in {{formula:6397db4b-41df-40cf-af11-252099482efc}} is simply because {{formula:e0362d05-0325-4227-b259-2c815d5ab932}} is a space of even functions and any odd order derivatives of {{formula:44636c53-7fc1-4041-a3c8-994e91a28171}} are even.
The rest of this section is devoted to prove Theorem REF .
Towards the proof, let {{formula:e4e4dc58-eb09-464a-bcb9-78825cf201c0}} be fixed. We define the projections {{formula:bce247e4-bcce-4e11-98fb-d38f99feaae4}} and {{formula:eee6193b-ac0a-4516-be93-65b2b0c04206}} by
{{formula:7dc97c22-7b3f-4a21-8f8e-0935f6c71ae5}}
where {{formula:c3863626-da75-4029-87d4-39f487a6023d}} denotes the usual {{formula:249ff181-401a-4ce4-b940-61deb1e39b37}} inner product. Note that from the assumptions {{formula:1aea55b9-9b2d-4b52-bb44-f06eb1c6c254}} and {{formula:6e5bedb5-a1d6-4036-ad90-111e0a016690}} , we make an ansatz that for sufficiently small {{formula:55744db6-a0ac-4604-a7f7-f008769f6823}} , the bifurcation curve {{formula:4787fd69-4ea7-4250-93af-52090ea3ae52}} can be written as
{{formula:4460db5a-5751-4666-acf3-6df45fb6c668}}
for some {{formula:22466fce-b7d7-45a0-97ad-9e43a2331e81}} such that {{formula:35afc7a5-c5db-450e-a764-7b7424480088}} . From this ansatz, we define a family of functionals {{formula:cd1075a0-fd52-4ba1-b8c5-e609facb9d60}} by
{{formula:8f9bac94-b804-4086-9703-13aaadd69963}}
and look for {{formula:76ff001a-3749-4510-a55f-f64d923dfbc2}} such that {{formula:7f1694a5-21c7-4c64-b0fe-3c783ea5e81d}} and {{formula:17536a89-cf75-4282-86be-a7d0693494ec}} for sufficiently small {{formula:ff4250e1-c4d3-419a-abec-82eadb38cde5}} . This will be achieved by Newton's method, where the first approximate solution is {{formula:27babbb1-fc21-457a-b272-ac824894cb62}} . To perform Newton's method, we need to study the linearized operator {{formula:31d1dfc1-6234-47f1-bdb4-2ba8b42cbd46}} at each approximate solution {{formula:68696b0b-a5fd-4e33-8436-acfc7af6630c}} : For {{formula:5df58950-31a4-47d0-aa23-b51a4d1d3279}} , which can be directly computed from (REF ),
{{formula:9785742a-8ef5-486a-9435-ec5c8511e357}}
where
{{formula:d4a4168b-b390-4f28-a7e3-2c2ad810baef}}
The above decomposition of {{formula:244b620a-e595-48fe-a4bd-f504e2927a9a}} into {{formula:3989fd24-b1e2-45a8-8e36-fb07136858ef}} follows from the assumption {{formula:dc990edf-ec5c-4284-9728-829e6be5fed0}} . Recall that Newton's method relies on the “invertibility” of {{formula:a0846901-378c-4d04-872c-f94f1f9c3d6a}} for small {{formula:6f0c6081-e276-4095-b01c-728ff96371eb}} . However, as we will see in the next lemma, {{formula:6f7e2048-1a99-4373-adea-b742caa04818}} is not fully invertible between {{formula:0d33bb91-1e98-4bda-a58b-941eec46fb1f}} and {{formula:68d69d52-1e71-4778-9d75-8d685749e4e3}} , because of its loss of derivatives. This is a motivation of adapting a Nash-Moser scheme in our proof. However, (REF ) suggests that the inverse of {{formula:a31ecec6-5082-4798-a048-412f19a7a7f6}} is a good approximate right inverse of {{formula:00a71b4f-163b-4da2-800c-f6d6cbf8ef71}} . In the next subsection, we will focus on the properties of {{formula:e7d48a39-541e-4cf2-8c40-c0f39853dbb7}} .
Analysis of {{formula:25fa4e6f-1236-4749-910f-2eb35c132f9b}}
We will look for a solution {{formula:de15c3ca-415a-4c0d-8b31-3891484b1b51}} to {{formula:b610303d-92da-4fbb-8cf5-b24e509e56b4}} in {{formula:e34a4745-e609-48ea-8cf2-d26361a55b39}} for small {{formula:cb677371-e726-4c81-a86d-f5ceb7861193}} . In each step of Newton's method (Nash-Moser iteration), we will regularize the approximate solution {{formula:4fe71585-6e34-4961-b9e0-d136335d31bc}} (see (REF )). The theorem will be achieved by proving that {{formula:d1951cb7-e0bf-49f4-9633-b0dc59610ab3}} converges in {{formula:8fbbf987-05ea-4b9c-a3ce-4484bdd76745}} . However, we will also obtain boundedness of the sequence {{formula:c9b035d0-4e67-4d6c-8815-93fa05d617ed}} in the higher norms, which is necessary because of the extra regularity conditions as in (c), {{formula:20e4718c-2ee4-45b5-b55b-cde7a6803797}} and {{formula:83993fa3-7fc3-4f6e-8fd2-b8ec684066cb}} . For this reason, we will establish several lemmas assuming that an approximate solution {{formula:283b9aed-65c0-43e8-bd8e-74f6def4edb5}} is more regular then {{formula:ffb2fd90-3e4d-434b-b07f-4b43f99e8499}} , which will turn out to be true at the end of the proof.
Lemma 4.9
Let {{formula:9a30a720-a6e7-4b3b-8477-e035fa54b2d7}} and {{formula:1d1ad144-b422-4942-85d6-7da52d7fe0c9}} be fixed. There exist positive constants {{formula:84d4b7c8-ba4c-45b7-a7ae-ba760dc67c13}} and {{formula:b0e0a1bb-a325-4b9a-893d-5a8dd51c4da3}} such that for each {{formula:c7757325-1739-4ff5-87ab-e0a03cbdbab5}} , the following holds: If
{{formula:2ee2709e-486d-4344-a9aa-1b9805490e08}}
then
(A)
For all {{formula:4bff3e57-aa61-42f8-af1a-530a24fed93d}} ,
{{formula:e3b78ac6-b62d-4f92-b65c-ab8bb7d58e49}}
where {{formula:952d86c8-a831-46b0-a022-b79807837a4e}} is as in Theorem REF . Therefore,
{{formula:aa188c8b-4015-4a16-8ba5-b629fa4bc7b3}}
is well-defined.
(B)
{{formula:7363dc1f-2e62-4090-8895-4a4b5393ed8f}} is an isomorphism and
{{formula:1a00a648-f0a3-4396-8ea4-217f57a3fec7}}
for some {{formula:a1cb85d7-60ad-4e3d-9a58-0ea68ffb258e}} .
If {{formula:807a2293-9c2f-48b8-b245-ead90536d785}} , then for any even {{formula:052e473c-c6dd-4a75-aba2-dc8a0e4b0481}} and {{formula:20d03ea4-1df6-4a2b-b7fa-2e3c91f31e4f}} , we have {{formula:f312814b-2f78-4cf1-b71b-1d6375f84d98}} . Also, we have that
{{formula:8f166c74-ecb1-4939-8c34-74f065dc9011}}
where {{formula:f486766a-e515-4232-989a-8241ab11bf35}} may depend on not only {{formula:09edfe9a-c1ff-4314-9d4e-ccf46428b69c}} but also {{formula:04d334f6-206b-4760-9ea4-2494426ed401}} .
(C)
Furthermore, we can choose {{formula:34c5d2b2-fffb-484f-bf68-4f9fd8937a94}} so that when {{formula:e23e9bb4-9cf6-47b8-9d41-6b96ccbd082b}} , the following hold:
{{formula:5fa04b26-5553-4f30-9d0e-70d4445d4347}}
We will frequently use
{{formula:ee5d85d5-95a6-44f4-bb82-309fd1215b8d}}
which is more crude than ().
Proof:
Let us fix {{formula:74194c7f-1bb0-4b78-8e56-1680943a519a}} and {{formula:b157666a-ca6f-49fe-a5c2-8b5b3b837018}} . We will show that if (REF )-() hold for sufficiently small {{formula:8fbd4da2-60b2-43e0-9ea2-9e26ad42331a}} and for some small {{formula:0aa34d9a-9276-4294-9b97-d1ce6c7b6106}} depending on {{formula:e0df46ec-6b11-4b1a-b169-e35a8f686eef}} , then (A),(B) and (C) hold for some {{formula:ed7209f7-6e92-49f8-9a5a-0b63390e3d0e}} .
Proof of (A). To see (A), notice that (REF ) is trivial. From the assumptions (b) and (c) in Theorem REF , the linear operator {{formula:0a3da46b-5dc8-49a1-acdb-dead6be8de51}} is well-defined.
Proof of (B). The estimate (REF ), follows from (REF ), (), () and
{{formula:a0cf0f98-b289-4346-8ef0-0875f003aa25}}
where the last inequality follows from (). Before proving (), we first prove that
{{formula:11be03df-9ede-44ea-ae47-6d213a87012a}}
for some {{formula:8ebd06f7-c88a-4b1e-8627-64d8da470144}} . The above inequality is a consequence of the transversality condition {{formula:f8211510-d18d-444a-b908-18857d3ebb6a}} in Theorem REF . In what follows, {{formula:9afeb00e-676c-4803-b668-cbc819805416}} denote positive constants that may change from line to line and depend only on {{formula:85836c5f-acc8-4f36-a6c1-4d1159b5bc30}} but not on {{formula:bcbf0613-da54-4974-82b9-39b145b80bd3}} .
By the regularity assumptions of {{formula:b103a0c9-6a94-4674-9d5d-cb9837085c1b}} in (b) in Theorem REF , we estimate the quantity in (REF ) using a Taylor expansion up to linear order: For a fixed {{formula:51559827-c375-4a71-8023-fb446fdad4e7}} , we set {{formula:c022f0f5-b158-4072-a31d-bc8ad65f9bb6}} , so that
{{formula:3e936139-98ee-4bcf-abaa-ca0b1dd6830b}}
Using the fundamental theorem of calculus, we can write
{{formula:891a6745-3248-4281-a46b-5e313d0bc7ee}}
In terms of {{formula:aa8cd208-dc0d-4e5a-afd2-cef1913514c3}} , (REF ) and the above equality give us that
{{formula:21c77ea7-0868-49e3-a0f4-eef7f445ea62}}
where the first term after the inequality follows from b) in Theorem REF and
{{formula:e7a304d7-ab26-40d6-a8bb-68cf7a0fd7ca}}
and the rest follows simply from the triangular inequality. Then the regularity assumption (b) in Theorem REF yields that (recall {{formula:db4d0495-a44d-4ab2-be6b-91132f970ab2}} ,)
{{formula:5767833b-2524-453e-9b32-1d69c30f4c88}}
where the last inequality follows from (). For {{formula:e4e4fbb6-88a9-4b91-b711-a6241cc190ba}} , we use (a) in Theorem REF to obtain
{{formula:da2ec665-df8e-47a7-a133-2bf41a90b4d8}}
To estimate {{formula:099c3479-e91f-4214-915d-19c59ccd8cd1}} , we compute
{{formula:1b74a7d9-2222-43b2-a627-d0829887eb59}}
For the first integral, we can write
{{formula:a79d7dfa-073e-47f3-9d81-fa302f58e9ff}}
where the last equality follows from (a) and (c) in Theorem REF , which shows {{formula:53796dd4-4216-4e20-8f61-98a9554ceefd}} . For the integral term,
using (), we have
{{formula:9cf5bdb4-3618-4f9f-a990-3974df703dbe}}
where the last inequality follows from (). Therefore
{{formula:a0218139-0a46-442a-ba1b-bae7fcb9d1e5}}
where the last inequality follows from the transversality (e). For the second integral in (REF ), we have that for sufficiently small {{formula:33c9abe3-6f57-4560-b659-4277ed6ee17a}} ,
{{formula:50a56a00-bbe9-4932-a125-a50a4ece0950}}
for sufficiently small {{formula:2d8bc6a8-ef22-4de9-bbe5-220cc4efcb47}} , which follows from ().
Hence using (REF ), we obtain from (REF ) that
{{formula:5d2370f6-ecd9-4f4b-a67e-a9dde53b5945}}
Thus the claim (REF ) follows from (REF ) (REF ), (REF ) and (REF ) for small {{formula:43db520a-0cc4-4768-9cb4-63094e867aa6}} , depending on {{formula:5ae96276-b13f-4a4a-bcf5-5ed110dc4cd3}} .
Towards the proof of (), we will consider how to invert {{formula:9f93ee92-ff44-482f-b552-99b96ee07eb0}} . Since the dimension of {{formula:11bf9829-f793-4eff-9f30-cff35ae09ed6}} is one and {{formula:de9e4c69-34c6-49cc-a845-0068806a9e2b}} is the basis of {{formula:5d0702bb-26ca-4e76-9619-86202a8326e9}} , the above claim (REF ) proves that
{{formula:af74264b-7029-44e9-8449-9b7d0f54f352}}
To simplify the notations,
we denote
{{formula:03dbe1a9-bf3f-4369-a99a-22aba6aa52ae}}
We denote by {{formula:ca41e027-b898-48a8-ad95-6f91cbc34579}} the projection from {{formula:8f5ea65f-0290-427f-b852-4d80d44b4fb9}} into {{formula:40d74ce4-0a30-4b1a-b92b-0cb8fd4fbf1c}} . Since the norm of {{formula:245bfe31-1fe3-44ca-a92d-49bfe11ef850}} is small, we expect that the functional structure of {{formula:8f759c23-f4d1-4706-a0af-51e029aca9f9}} should be similar to the structure of {{formula:57b4baab-b4e6-4384-8ed7-58c4ab0b29f4}} . Indeed, by continuity hypotheses (REF ), () and Lemma REF , where we can think of {{formula:6d00358c-c258-4462-85bd-9a3401a10f19}} as a linear map from {{formula:966a1978-4684-42de-909b-540e9e20e3df}} , we can choose {{formula:2273e8c5-0e5b-49c9-8868-4f21604cb737}} small enough so that if () holds, then there exists {{formula:8e3aa68b-a8f3-4b20-9714-d11e79325884}} such that
{{formula:80ea0665-b8a1-4437-be20-9838734b06e9}}
Now, we claim that we can further restrict {{formula:22fc611d-2b57-4b5e-b98d-a9ca43d619b5}} if necessary so that
{{formula:d43f5a4d-7461-4402-b2b7-e27953e1962f}}
In fact, note that (REF )-() imply that {{formula:8541c53e-aac4-4ee9-83b3-a6807828240c}} , thus by (), we have {{formula:b1c58d66-4eee-4407-b124-ca66cac750e5}} . Also we have
{{formula:d769e5a1-ba6c-4dc9-8890-bc146d61d67d}}
thus
{{formula:6a761596-44a2-4ee4-b65c-33c7458c35e5}}
where the second inequality follows from () and the last inequality follows from (). Then it follows that (recall that {{formula:ece90cf5-a803-4a85-b733-df4499ebf4d4}} is the dot product in {{formula:2c7bb434-1c22-4947-a82c-0d339fca3889}} space),
{{formula:d50be427-a7a8-41f8-984e-db1597d2973c}}
where we used (REF ), (REF ) and () for the second, third and the fourth inequality, respectively. Hence, assuming {{formula:88d78add-8191-4b8e-8c14-c66e3eb00063}} is small, we have (REF ).
To prove (), pick an arbitrary {{formula:4b075647-19c2-4299-8e73-49ec1689e413}} . There exists a unique {{formula:8e1e2d6f-8119-4348-b87d-38ca3976bb39}} and a unique {{formula:e4e8019b-4389-43a0-ace0-0d2ac84f9287}} such that
{{formula:f937f896-efbd-46d3-a42f-e7e59d7b9f8f}}
In fact, there exists a unique {{formula:23207db0-cdfc-4bb4-b438-77c99b74e633}} in (REF ) thanks to (REF ) and that {{formula:4b10aa2d-3402-40a5-9d19-41f9d81d27e6}} is the projection to a one-dimensional space spanned by {{formula:95187f16-0a6d-4c2f-a47f-b884d2c528f4}} . Once {{formula:e19f39c0-eaf7-4824-8eae-48aebe3be85c}} is fixed, the existence and the uniqueness of {{formula:b1140bd2-26e7-4687-9a8f-53b300edf821}} in () follows from ().
Once {{formula:426143e0-290a-44d1-a1e2-1e58f175a058}} and {{formula:394830a5-10dd-4274-9c43-2ea27d3470bd}} are determined, it is clear that
{{formula:15e685b3-90f1-469e-80bc-9e8cd5c7f6f4}}
where we used {{formula:b13f4dec-1654-4260-a5a1-191734f92fb2}} and {{formula:cd72666b-0cd4-4aaf-a70d-c4a878204595}} , which follows from the definition of {{formula:8beac99a-d2b2-4dc8-a90f-ae67a262a3c8}} in (REF ).
Therefore {{formula:b0158e08-6d7c-4973-b856-6935b7eb0cca}} . Furthermore we have from (REF ) and () that
{{formula:0397a2e2-cf92-42f2-9fcf-0befa1d491aa}}
Using (REF ) and (), we obtain
{{formula:83fefcdf-f04c-4dff-93d7-da12469e919a}}
where we used (REF ). This proves ().
To show (), we compute
{{formula:f6195dfd-4c24-4a03-98d0-d6ac95aef4f7}}
which follows from (REF ). This implies
{{formula:7f7db1b8-0183-4e14-9edf-940e5c027b3e}}
where we used (REF ) to get the second inequality and used the definition of {{formula:a66dff03-c6c9-4870-adec-48817de39574}} to obtain the last inequality.
In order to prove (REF ), we improve on the estimates in (REF ) and (). For (REF ), recall that {{formula:772f6899-52a6-46f4-94f9-f3e6dc7484e9}} is in a one-dimensional space, span{{formula:7e71737f-4618-48aa-a2cb-24817e41ff6e}} and {{formula:52a0dd65-016e-40d6-ab61-4d1d714ad750}} is supported on the {{formula:8c0e61f4-87fb-44aa-928b-15fbe2efc77b}} -th Fourier mode. Thus,
{{formula:32e94748-90b7-4420-ac98-c82b4758a6ec}}
where {{formula:309824c4-150d-421d-b955-d02a3279f872}} may depend on {{formula:5bceb495-cc41-42c0-a3c4-4669744706a2}} .
Furthermore, {{formula:dcb88356-3288-4f07-813c-7309dd980ee7}} -2) in Theorem REF and () give us that
{{formula:f0264f49-e03d-4518-9691-e130e44da57b}}
Using (REF ) and (REF ), we have
{{formula:6e394d75-f0b7-4532-b00f-da47e1aa20c5}}
For the higher norm of {{formula:ef5d4eba-a691-44f6-8d3d-fbecec8f58ef}} , we use () and and (REF ) and obtain
{{formula:2bbd215f-f2d8-47e3-9b5b-dff2ea32ca56}}
Therefore, (REF ) gives us
{{formula:84b0c1b9-6274-4a31-87e3-8bc092e58532}}
Combining with (REF ), we obtain (REF ).
Proof of (C). Finally, if {{formula:fd9985d9-8a0e-4699-82f1-1f46d09a5aa6}} , then we have {{formula:f7969cec-1c49-4d8f-9fbb-fa596d54a15a}} . Therefore we can improve () and obtain
{{formula:a38155e3-bd8f-4dbe-9cf1-9d6a9d0eb4b1}}
which implies (). () follows trivially from (REF ). This completes the proof.
{{formula:e9972d4b-7800-4e12-8af8-2378d5d00797}}
Lemma 4.10
Let {{formula:70d09b3a-78de-43fc-a398-d44113d73e17}} be as in (REF ). Then there exists an open set {{formula:35a65be9-2593-4273-a9be-a5a2d562efbb}} near {{formula:b28ee249-cf4e-4659-be7f-22e9fa2324a3}} such that for all {{formula:a14051d1-f288-49fb-a5a6-5770139a6cee}} and {{formula:b6d74b3e-689f-4f8e-80a8-4476b68e1ab3}} ,
(a')
(Initial value) {{formula:16bf5bb3-8200-4803-b846-28a9e6ce01e3}}
(b')
(Taylor estimate) If {{formula:95900a76-eee6-4477-b805-672b9a2e9d88}} for some {{formula:97bdb157-c5a8-40d4-a067-e4da5b3dfcf0}} , then
{{formula:7e9d1bb6-44a7-4df6-84c9-79bd7d1ed2bf}}
Proof:
(a') follows from the fact that {{formula:52aaf11d-dcb7-434d-b674-84f6ba2d6610}} and {{formula:0f6ca4c7-5f65-439e-a742-8ba39057b269}} . (b') is due to (b) in Theorem REF .
{{formula:c1d27722-28ff-4134-bb7e-275ebf4031f9}}
Nash-Moser iteration
For {{formula:461da4c6-90f5-4065-b3d0-aba62bcf91c7}} and {{formula:d9c397e2-d3b0-4ae7-9571-50c862220c95}} , we consider a regularizing operator {{formula:8002132d-3c3b-4ee5-8595-92d7518c3c3d}} such that
{{formula:4b9b05cf-6540-4504-88f6-bf85a814c983}}
Note that we can choose {{formula:0383ffde-ba7b-4476-93cb-d20bc638875d}} so that {{formula:a6f7a8ae-d2de-4598-8b1f-6dc9f659fa8f}} , since {{formula:f4f23bcf-e287-4d5e-a17e-7f52f40caf5b}} is supported on the {{formula:2a603b01-0ff1-4367-8053-438c846a78bc}} -th Fourier mode (see (d) in Theorem REF ).
For {{formula:cd589c85-ddc8-463e-97e0-99a30814e99d}} and even integer {{formula:1542d11e-e426-4160-86cb-3fcc91e02fed}} , which will be chosen later, we set
{{formula:a341e763-2a1c-4aa2-8cc6-18cbfbc1c6d1}}
and
{{formula:c37d61e6-36c0-4d25-886e-c5d5a2b66fb6}}
Note that our goal is to show that {{formula:d722c996-120a-4b5e-ac3d-b74a7c96a1f9}} , which implies that {{formula:196fa356-a8a2-4bbd-9f11-080d5e669546}} converges in {{formula:cc2d89e8-ba8c-415a-9788-aacdc1be1dd3}} . In order to prove the assumptions in (REF )-(), we will need the boundedness of {{formula:f3df8034-b5d7-4d5e-89f4-5163402bf24f}} and {{formula:514eb2fe-943c-48c7-90df-d6af30bfbb59}} as well.
Lemma 4.11 Let {{formula:364fd02b-38f0-4488-bbac-d4b8e21c64aa}} and {{formula:7687e4a9-0937-4e2c-a231-6a342c19d531}} fixed and let {{formula:ca8708ee-078f-48d2-8370-476a0429918e}} and {{formula:578e318e-da20-42e7-95a1-23f3de92af6c}} be defined as in Lemma REF . We also assume that {{formula:553fbc33-db6d-42b4-b6ee-f58e072fe9f5}} is even smaller if necessary, so that {{formula:ab338eca-b0c3-4fba-a8dc-469b99fe0e25}} where {{formula:a8de980b-20ed-4f7e-a8b6-48d6acb8dbd0}} is as in (b') in Lemma REF . For any {{formula:f048881e-56a1-4375-86f7-a0b0e2a561b5}} such that {{formula:1d93d5ae-4856-48d7-907a-067ade5302f1}} satisfies (REF ), () and () for some {{formula:ba8ac7cb-5e3a-4f67-bb49-9b6cf5b63efe}} , then
we have
{{formula:45807baa-bd4d-4f21-ad6f-0a7361b7f237}}
Furthermore, we have
{{formula:5312efc7-8bee-404f-9544-506ae1e1cc44}}
Proof:
Proof of (REF ), (), (REF ) and (). We first claim that
{{formula:5ae58a48-8295-48b5-93e8-111ab80a2fb7}}
Let us prove (REF ) first. Thanks to (a') in Lemma REF and (REF ), we have
{{formula:fbad7e8c-0b4f-4245-b6fa-6f0f208c27b8}}
where we used that {{formula:0741a311-b656-47ec-bd23-bef64549230e}} and {{formula:8c2e689e-3b5b-4836-aee8-65abb95f7f8b}} commute to obtain the first equality. To prove (), we use () and compute
{{formula:3ba2d44f-8575-4e34-b053-47c35a7a32d7}}
which proves (). With the claim, (REF ) follows immediately. () (REF ) and () can be proved in exactly the same way as above using (REF ).
Proof of (), () and ().
Note that Lemma REF immediately implies (). Now, let us prove () and (). We have
{{formula:81e04072-f304-4ade-8e59-ae3964b583e0}}
We claim that
{{formula:f113c03b-8c65-4526-8429-40b1b6a30cad}}
If {{formula:e7cdcb2d-b195-480b-89c2-8c5362372bc1}} , we have
{{formula:97b5acff-16f0-4107-b50d-404cf664583b}}
where we used (b') in Lemma REF in the second inequality and (REF ) and () in the last inequality.
To estimate {{formula:47f28021-6772-4508-9d96-5a291f804e93}} , we recall the definition of {{formula:a249f55e-c702-4c7c-b30d-b85eee65a093}} and compute
{{formula:c240e3b1-4b07-4952-8473-aa3369ad2bc8}}
Since {{formula:63330373-4f2a-4921-b652-1c67fdf4ae6b}} , therefore
{{formula:5c454777-0926-4762-9631-7e2d360e5cbd}}
which implies {{formula:0f146bd0-9e69-4e8e-a6ba-57e6b584d6d7}} , since {{formula:d423f381-c50d-47b0-9b46-48d7b921aadc}} , which follows from (REF ). Also it follows from (b), (REF ) and () that
{{formula:917ee68e-a1f3-4faf-921b-3549c921e3bf}}
therefore, {{formula:71dd0349-64a7-4747-b976-f16d70d5e5a3}} .
Thus, we have {{formula:f5c72d72-4620-46a6-a504-63cf5b1c68aa}} , which proves (REF ) for {{formula:2b46e860-a061-4b2b-8a79-05f9fda1616e}} . If {{formula:201f6bc2-f217-464b-bc5f-b8d0473bd00f}} , the claim in (REF ) follows immediately from (b') in Lemma REF .
In order to estimate {{formula:c769810f-1909-4503-82d0-05606a001550}} in (REF ), we have that for {{formula:a8d14242-5a2d-4835-b9e6-e3825680a753}} ,
{{formula:51700755-ff15-4271-b779-a41f43414101}}
Then it follows from (REF ) and () that
{{formula:5ffb29a2-22bb-477e-8fff-9beb883b13ad}}
where the fourth inequality follows from (REF ).
Also we have
{{formula:53beaaa0-a92d-420a-87f9-1e09b5dbe294}}
Hence we have for {{formula:c39b0ca0-8104-4f0c-8f22-ddc15af8ec9c}} that
{{formula:8ff184de-6e08-4240-91c7-113eec406ff4}}
Thus with (REF ), (REF ), (REF ) and (a') in lemma REF , we have that
{{formula:2c00c0bc-9cd1-4f77-8784-5304d92662e1}}
Proof of () and ().
Finally, for () and (), we use (REF ) to compute
{{formula:44a83b8d-ecda-4a2d-a4a2-09638922b6a2}}
This finishes the proof.
{{formula:b9890b1a-639d-4667-afd7-f24443463e51}}
Let us now derive a recursive formula for {{formula:77ca645a-ed17-4618-bc41-fed1f9000db1}} .
Lemma 4.12
For {{formula:5ad359d4-a750-4a88-ab67-bfc6582966f0}} assume that {{formula:12c008f3-fd93-4c27-bdcf-ea592ee2b96a}} satisfies the same assumptions as in Lemma REF , that is, given {{formula:097d5944-0a01-4195-907e-d3188b8ca7b1}} and {{formula:dc2bb689-cfd1-41be-aff6-16379b4429b0}} , {{formula:ddcf49f7-232e-4a61-8f07-761b4d98abde}} satisfies (REF ), () and () and the assumptions of Lemma REF for {{formula:ffd04742-9581-467b-a49a-d9f43c509ade}} and {{formula:53df03c7-0cd9-4d15-9b8c-0fdd9d3df478}} . Then,
{{formula:f83e4ebe-69c0-4e84-9189-66fcbc424090}}
Proof:
It follows from (a') in Lemma REF and (REF ) in Lemma REF that
{{formula:3a82a4c3-09e3-4156-bf90-0e7aa2133daa}}
Now let us assume that {{formula:846bf2a4-d000-4cdd-8bc9-8c50c8e20bc6}} . It follows from the definition of {{formula:a0dc948c-d596-4503-864e-6d4a58b2cb2f}} in (REF ) and (REF ) that (recall that {{formula:68a0c2fd-f56d-4f61-9e4b-64597f2feeaa}} is even)
{{formula:69652625-33db-4e98-9172-0c6760ea0c4c}}
Using (REF ), we have
{{formula:4d105a42-9681-4def-b50d-1db6c62a9218}}
therefore, we have
{{formula:2f3b6eb9-d19a-4d9b-929a-c2c012d7b8b0}}
For {{formula:e0aeb674-8e46-4f5b-9715-53e6a89a32dd}} , we have
{{formula:613b9f06-adad-4589-bf1a-febef95b752c}}
Hence, we obtain the desired result.
{{formula:35e75fe6-cc32-4b17-a287-a8042dd7f537}}
Now we are ready to prove the main theorem of this section.
Proof of Theorem REF :
We fix {{formula:8367c059-101e-4fbc-b3e4-61096ca4b26f}} and pick {{formula:3d7f2bab-6a9b-4077-8220-c7abc2de6a34}} so that
{{formula:3094f323-9112-459e-9ff1-d60b030fcccd}}
And we choose
{{formula:91db83d1-d4d1-49d5-985c-e347265dae79}}
and let {{formula:528f1436-fd10-4f7e-8874-36981834e77c}} {{formula:52e8ceb8-016c-4daa-a49d-bc645500ca33}} be as in Lemma REF . As before, we can also assume {{formula:26ffe692-ba22-4339-961c-5d7a0a844417}} is small enough so that {{formula:8bfb4f0d-ca82-439f-99fb-d6eb41e0cd8b}} where {{formula:c153d173-eefd-45f7-a92c-2aaaa0958665}} is as in (b') in Lemma REF . Since {{formula:7afaacda-e53e-43cd-9cc5-5a98480a4d9d}} , {{formula:6f508c6e-6e3a-4ec6-b753-4816a130833a}} and {{formula:fc71acb3-beeb-4664-967e-8382463f3a78}} are fixed, by Lemma REF and REF , we can find a constant {{formula:6f331482-8673-4457-8c24-3f92c2ce4cbc}} such that as long as {{formula:8eb93d7d-b7da-42bb-84f2-9b8a08b813d8}} satisfies (REF ),() and (), for some {{formula:d4946279-3d54-4cce-b14c-cdcfffd6aa21}} , it holds that for any sequence of positive numbers {{formula:38da1f6a-c7e8-4734-a3b0-117aaade00a3}} , and
{{formula:1f12a0a9-5eed-4c7e-bf7b-bce72aa99920}}
In what follows, we assume without loss of generality that
{{formula:3b5ebd81-73cd-4fc8-ae85-db8f5a3a8c62}}
For such {{formula:9d54bab4-6bfd-4a37-80c3-641dbc76bc82}} , we can find {{formula:9a45dfea-5dac-4711-8714-4be3aada3635}} such that for all {{formula:4449efc4-f9ca-40a5-b0ae-98d6501ae6cf}} (each of them can be easily verified for small enough {{formula:4101e71b-a441-4aa1-a0ee-ced76f20b63e}} ),
{{formula:ba0b76c8-1983-4dff-a234-c78be80dbea0}}
Lastly, we fix {{formula:90a530ba-2c08-4bc4-8b23-f69dcac48301}} and
{{formula:8f3be3ac-7288-4a7e-8420-3da7e09a13f3}}
We claim that for all {{formula:dabb8f26-220a-4cba-9f8c-40a63d68392d}} ,
{{formula:2613e3db-a8e9-4bbd-a6a0-9b8eb6f37dae}}
: {{formula:da7a40d5-441f-4b69-8e44-17d0e19926c6}} satisfies (REF ),() and ().
{{formula:eaea5c6a-5b30-4061-a964-d1802ba8c710}}
: {{formula:cfcdd3ee-6dea-486c-8cc7-04d788da0490}} .
{{formula:1986e815-4923-4b97-ad80-ffb426331cb9}}
: {{formula:a1970f50-2300-47d7-9bd5-08a52cd454ea}} .
{{formula:dbe6059c-1c3f-4e6a-8af3-b29f412835bc}}
: {{formula:b5e8ee65-f573-4b1c-8529-2ba63ae8b15a}} .
Once we have the above claims, then {{formula:3e7bfa2d-5ac5-4c56-86e4-ef2470860cb7}} justifies all the recurrence formulae above for all {{formula:9051e02e-9190-437b-a9f4-ac78bee6d731}} , which follows from Lemma REF and REF . Also, it is clear from () and {{formula:4dbfd320-60ad-4c7e-b321-7c088a5920d9}} that
{{formula:8379c460-d1d8-40af-b918-e9d48686af30}}
Therefore {{formula:94d83b7a-108c-41aa-b849-c2b828d098ff}} is a Cauchy sequence in {{formula:880cedaa-9805-49ad-a911-1e700f1881d0}} , and we can find a limit {{formula:0ca59b54-7628-4e72-aec2-52e919021e0c}} . Then, {{formula:ee3405db-19e1-4d4b-b7e0-eb7147c29174}} and the continuity of the functional {{formula:1fddd8b3-f18d-4278-9e84-0c4321ec0219}} implies that {{formula:6c3537d5-eba8-480a-8f67-9536a651b38d}} . From the definition of {{formula:b9d10bc3-3431-4dc2-a0d4-a9cc87bba4d4}} in (REF ), this implies the existence of a solution {{formula:9f98b90b-1430-45c6-9c70-00dba1e21921}} for each {{formula:2ff7c900-3b8e-4954-99c9-d42cff066288}} and finishes the proof.
Now we prove the claim {{formula:b36b6c93-1228-42ca-9f7a-b0c82d5dc7fa}} for {{formula:16e1797c-39cd-4a86-9df4-9928b604d08b}} . We will follow the usual induction argument.
Initial step. In the initial case, we assume that {{formula:4993c031-1d06-4bb5-a7ef-d51c239ab648}} . We first prove {{formula:91b72f7a-502f-4b67-8d9d-55ebd6bcc327}} . It follows from (),() and (REF ) that
{{formula:9e7ac3f0-fb80-48bf-a94f-e292b57f97b8}}
where the two second inequalities follow from () ({{formula:74f78ab2-d1de-46c1-81d1-949937761679}} ) and the last inequalities follow from () and (REF ). Furthermore, it follows from (), () and (REF ) that
{{formula:1f54cd3e-5949-42e6-84f7-6ac266806bd0}}
where the last inequality follows from (). Therefore {{formula:f52cb34c-8fed-4da9-b763-b6f52ca653ba}} holds for {{formula:b2fece69-f54b-42e7-bc95-b7a14e876570}} .
{{formula:3ac56983-79ea-4056-b772-6d73269c9df0}} follows immediately from () and (REF ). In order to prove {{formula:2fd79889-51c0-4a7c-8cec-0c59b04e563c}} , note that thanks to (), it is enough to show that
{{formula:9b6e395b-a5cc-4d9b-bfe6-f919d2e5e534}}
in other words,
{{formula:82473344-6b61-40b0-be7e-e23a09e33e42}}
where we used (REF ), () and (REF ). By (), {{formula:d322a4e4-ae1e-483e-8445-1d99c1190c33}} is the largest value among the three in the parentheses, hence it is sufficient to show that {{formula:345c0b92-d33d-47bc-a000-76d8190b66e4}} Since {{formula:116a87a6-dc0c-4fe8-9b67-df110b893925}} , it is enough to show that {{formula:a5855155-9562-4620-9178-0a994fcf2743}} , which follows from (). {{formula:93f6ea83-5afa-4dca-a240-01c45b017392}} follows from (), (REF ) and {{formula:520ee8c4-c813-4735-9049-acfdbff8297c}} .
Induction step.
In this step, we assume that {{formula:b6dfa2c1-2c10-41d2-8620-bd1ec374b62a}} is true for all {{formula:d7960577-f35a-47ec-8db3-54ae7d249d94}} and aim to prove {{formula:17531bca-35c8-4003-9a32-10ad34c1cea9}} . Let us prove {{formula:1ba6f14f-889d-4031-8ce2-19a0338ccb3a}} first. It follows from (), (REF ) and {{formula:55b30a15-1a31-4979-8963-fb7e12bf5246}} for {{formula:86395538-5e4a-4cc7-bd78-2f84de8d232a}} , that
{{formula:d0877de7-bfae-4e1b-9d71-553505664baa}}
where the last inequality follows from our choice on {{formula:2527c430-c4aa-4a88-b44b-a9a53870e92c}} and {{formula:0b5a97bc-79a4-43fc-9831-edc5866a3a4b}} in (REF ). Hence, we have
{{formula:1bbfe23f-84fc-40d8-bc59-b7d80be2a628}}
where the second inequality follows from () and the last inequality follows from (REF ) and (). Therefore we obtain
{{formula:14fbab3d-2c43-4a64-9244-22701126db36}}
where the second inequality follows from (REF ) and the last inequality follows from {{formula:5d2da154-01a4-487a-a4e4-a17aa4d5a016}} and {{formula:7e802b14-97e2-4d17-9ee2-aa35136b0e6d}} , which can be deduced from () and (REF ). Using (), instead of (REF ), one can easily obtain
{{formula:d529c5e8-0945-4e5e-80d4-8ef0a3aaf705}}
To prove () for {{formula:e4bb4acd-1913-4200-a8f5-c1db0b52a125}} , we compute
{{formula:f1eb9f42-1765-44ee-9e9f-189f9b065a7c}}
where the second inequality follows from (REF ), () and {{formula:548d9e02-ee51-47b2-901d-de873617984e}} , for {{formula:737966ae-d821-4ebc-a602-d7cac7b3749b}} , and the last inequality follows from (). This proves {{formula:bbdeb559-30b8-4c3d-8e35-a89f07634431}} .
We turn to {{formula:96ee6303-e541-4dba-9773-6ea3c9fe5d03}} . Using () and {{formula:abfe9bbf-7d90-436b-8c72-2c028a6918d8}} , we have
{{formula:ad5f5fd8-5541-4fe7-9f1a-efbb8ee92701}}
where the last inequality follows from {{formula:b267a03f-cb5f-4397-b64e-bc466bb2a90a}} . Hence, it suffices to show that
{{formula:475b62f1-6448-46ae-bc7f-f0a0fd77529d}}
Plugging (REF ), this is equivalent to
{{formula:0076070e-c1cd-4388-8757-13e5ad6afa1c}}
which is true thanks to our choice of {{formula:16b0dabe-c522-4e4b-86a1-01a9878674eb}} in (). This proves {{formula:3db41963-6836-4721-946a-bc0876fb9e02}} .
For {{formula:15f9b9d2-2452-4fc2-9f90-7ee9800dd288}} , thanks to (), it is enough to show that
{{formula:daa3232b-d56e-4a7e-b186-087ae69f3225}}
Using {{formula:871e7c85-11a0-4fe1-bf58-e971bdb8e5d4}} , (REF ), {{formula:49b4a584-be8d-4c6e-a8ee-b3394a557f7b}} , (REF ), {{formula:20418b39-bc78-4df3-a00c-3c33e7871ee0}} and (), which implies {{formula:8fa2239f-5149-4928-bf4e-eedf90a84a48}} , we only need to show that
{{formula:68a02fd9-2982-4dd8-a203-fd12b2257392}}
We will show that {{formula:5fdcebc6-f06e-4fd8-942d-1cc18b97da14}} for {{formula:bbb8484f-fcd9-4cc9-92b1-c1c29266323f}} . For {{formula:aad2e3be-fe4a-43a5-bae5-132ecbff0e4d}} , it suffices to show that
{{formula:793078a2-bc64-4278-b031-a54d1bd4748e}}
equivalently,
{{formula:d3f2131a-d330-4b55-a0fa-7ed11e659ae9}}
and this follows from our choice of {{formula:fe52ac43-9041-4c20-83c4-1149dadbda56}} in (). For {{formula:b21a8f6c-d708-4c61-9b54-57ce428f76f3}} , we need to show that
{{formula:3ccd988d-548a-4df5-9c36-610d894a5602}}
eqvalently,
{{formula:a4e617b7-3a58-4544-af86-003230452671}}
and this follows from (). For {{formula:565bac73-64e5-4669-8590-d3bf3d42386a}} , it is enough to show that
{{formula:b968c30e-05e3-4ac1-8cad-22d8b8d3a02b}}
equivalently,
{{formula:480b3460-b550-474a-84fa-884a56a28356}}
which follows from (). This proves {{formula:93ac2464-720b-4b35-947c-2dab5fdfa88f}} .
Lastly, {{formula:362ca915-bbc9-4110-985e-31f8772656d4}} can be proved by () and {{formula:25515b53-4e9e-45ed-98fb-c600e77ae867}} , that is,
{{formula:127a4778-901e-4753-a3f4-e327f6458ad5}}
which finishes the proof.
{{formula:c6e3ce53-7725-4fcc-8cb2-8a45fe7270e7}}
Estimates on the velocity
In Subsection REF , we will check whether our functional {{formula:d38d5fe1-d899-442f-aa81-90e5e9f13d27}} in (REF ) satisfies the hypotheses in Theorem REF . In this section, we will derive some useful estimates on the velocity vector generated by each patch. Recall that
given {{formula:65956020-6a3c-4c83-9d8c-70ab1b1e97ee}} , {{formula:9c2e2bd0-c3ef-4fd2-8065-56b6bdce05de}} , and {{formula:1f751bca-1fd6-4782-abc5-2f0c34880b5b}} , we denote for {{formula:aeace8a2-67aa-4c43-80e0-0684a59a1fce}} ,
{{formula:af8051c8-e0b3-4ef5-8524-d9c1d7f9c76f}}
Proposition 4.13
For {{formula:ccfc2c3b-8ca6-4d06-85d0-a19bd526bab1}} , there exists {{formula:7b2de9cc-8e51-45bb-996c-503fdea64706}} such that if {{formula:70cd4187-2822-420e-84aa-3bd4eaaa2d17}} , then
(A)
{{formula:0654b3e1-9d74-4ea3-9d83-e2c3ac7579ee}}
(B)
The Gateaux derivative {{formula:d7501eca-0aee-4a0a-ba49-dd43bc88c287}} exists and
{{formula:6baae789-dc52-496f-8310-91e1881beea6}}
(C)
The map {{formula:ec198ea3-821e-4a79-b5b7-cc4b8d1da998}} is Lipschitz continuous, in the sense that for {{formula:f3c2461f-16e4-402e-9a3d-5e3166cb71b1}} such that {{formula:840548af-c6a0-456b-865d-612b8a7dbfe2}} and {{formula:aa1fc662-e125-4ab8-8478-31e2da2d7e94}} ,
{{formula:f5b49251-9c0b-4aeb-92b3-bad4c3bb85b9}}
(D)
For {{formula:90020fcf-d8c5-4741-b65c-7166b1f4fe06}} , there exists a linear operator {{formula:701550bf-daba-4b5d-bad1-6ec4ec1645af}} such that for {{formula:07e0c949-db8a-4c8f-9d95-0f42a9a89866}} ,
{{formula:53b5a6fe-ed0b-4bc4-a3bf-26468f68040e}}
where {{formula:4224a3d9-7b61-4990-9b30-03864463bdc5}} satisfies
{{formula:59603ae2-b1e9-482b-941b-03ec653b6995}}
Remark 4.14 It is well known that roughly speaking, if {{formula:55258597-942b-4fec-966f-f80484b3f4ee}} is {{formula:fe4d5d88-5a3c-452b-b712-c3c2086a0a5a}} -regular for some {{formula:890aeba5-6b5f-4839-b365-a234e11ac046}} {{formula:d798ba8d-d35d-4297-a8db-4c10b5dabd9f}} then the velocity {{formula:79ef39ec-f00f-4a63-a805-fd5eefacaa02}} is also {{formula:6fd433c3-2ed5-4d2f-b325-ae799be74a9f}} -regular up to the boundary. In {{formula:df9baef2-5438-4778-b236-7e179b481d37}} in the above Proposition, we do not aim to prove the optimal regularity since it is not necessary in the proof of the main theorem.
The proof of the proposition will be given after Lemmas REF and REF . We will deal with the case {{formula:02fb640e-b8af-4e93-8d42-5e396de9767c}} only. If {{formula:6ea87026-3c28-4103-add1-e9bb0ce8c2c8}} , then the integrand in {{formula:fc423c54-9390-4b37-97b5-cbf5e2dc2088}} has no singularity thus the result follows straightforwardly in a similar manner. Hence, by abuse of notation, we denote for {{formula:3a0f4b97-0d95-4437-8c52-9fc3e7c233ce}} and {{formula:ff72a039-503e-4605-9281-190d1e22d2fb}} ,
{{formula:4e79d2ee-b626-4032-a9f8-991c9561e289}}
Let us write {{formula:6dc7cbbb-be6f-4ef3-b0a2-a88bd8a0652f}} as
{{formula:4b080fb5-5a15-44ef-82fa-3bac4a459fc9}}
where
{{formula:685c6f0e-d30e-4e7d-a3f7-ef7b6e14587e}}
If {{formula:76c72bab-e3a4-4abf-a0a8-15449d9748a2}} for sufficiently small {{formula:210e5b28-e469-4e6e-8a72-0279e4e504ed}} , it follows from Lemmas REF and REF that
{{formula:498b1c0c-d3dd-4020-809e-da406274b3ba}}
Therefore, for sufficiently small {{formula:4ee950fe-40b8-4c64-9b33-0d348ef12c91}} , Lemmas REF and REF imply that for {{formula:cd068773-b43c-4b0e-a52c-1c325d45b64e}} ,
{{formula:9c4a1056-010e-4748-97bc-801bf49bab3a}}
The next lemma will be used to prove the growth condition of the velocity in (A) in Proposition REF .
Lemma 4.15
For {{formula:489a9751-48de-45af-b188-a4ea6905f1a8}} , there exists {{formula:b23f6a91-117f-4f65-8184-2ccec27bdb4a}} such that if {{formula:84e17c01-6334-4d02-a283-a800246dcb9f}} and {{formula:3384dd80-7d8e-4e8c-817d-eed914d543e3}} , then
{{formula:d3f6629e-7b08-4c32-8339-f880e653b920}}
Proof:
For {{formula:0014090a-9375-4277-a464-ab2babce0df4}} , then it follows from Lemma REF and the fact that {{formula:7e615654-2317-4551-8a5e-50985a51ca3b}} is linear that
{{formula:9f2b37b1-f2ca-4386-ac84-4bd278c89131}}
Now, let us consider the nonlinear part. We have
{{formula:ba476479-0c01-484d-be91-3601872d4134}}
where the last inequality follows from (REF ).
{{formula:c8509356-e90c-46f5-a523-8a57f635b92d}}
Now we turn to (B), (C) and (D) in Proposition REF . We will use the following notations:
{{formula:c872ac94-7cbe-4d65-8dec-c369399ece73}}
where {{formula:c735e6df-e000-45ff-bef8-acaac7cb8086}} and
{{formula:5796b774-423e-43cf-8abc-35f59f3d05e3}}
With the above notations, we can write the derivative of the nonlinear term as
{{formula:d70d7d85-b507-4f69-9d10-3e21ebce4cf9}}
Again, Lemma REF and (REF ) yield that if {{formula:12baaa5f-c1f4-47d6-84fc-941dced6eb1a}} for sufficiently small {{formula:8eb1f5e2-0fb9-4f2d-ab4f-9fb054e41f8c}} , then for {{formula:ac4753ab-44ab-48f9-aae5-1a84af60c206}} ,
{{formula:e26ed7f9-2014-4271-af4b-1af7aa16d122}}
Also, if {{formula:ae1728b8-ba2c-49a9-bef8-5ebc361cc027}} and {{formula:42a59aea-1519-404f-9584-99fb9bbf36a4}} , then from Lemma REF , it follows that
{{formula:abfc3794-1422-425f-87d1-ad28b8f313a7}}
For {{formula:2d2a6769-eab4-44b8-b220-d190b5c130e0}} , we have
{{formula:79fd4731-8468-4a80-b4b9-5436609baf06}}
We decompose {{formula:5cd47dfb-faf8-4b1d-8d9f-a616bc588422}} into
{{formula:ac99e9bf-b9ba-4448-ba8f-ca050da459e7}}
Then it follows from Lemma REF , (REF ), (REF ) and () that for {{formula:890cdb1e-038a-43b9-b623-a479ec21078d}} ,
{{formula:0daa3500-fa2d-4697-8e4d-af0d4090ee94}}
Furthermore, it follows from the definitions of {{formula:246dd7b6-c4af-4067-a7a7-42e413605721}} and {{formula:ec4fda51-7201-448b-9357-668113c6e4ab}} in (REF ), (REF ) and (REF ) that
{{formula:e11714b8-108a-4a44-aaaf-3948522d645f}}
Lemma 4.16
For {{formula:d654a441-1078-4dc8-8dc8-e4acdaa7c6d1}} , there exists {{formula:706b2f24-11bd-40f5-b04f-42635d7acafa}} such that if {{formula:3b1ed01b-b266-40f3-bc05-5dea8b4ed935}} and {{formula:27eac2e6-e466-43e8-ab59-073cce9f2b9b}} , the Gateaux derivative {{formula:480e971b-9bf4-4d9c-bf2c-84bb5f81a79d}} exists and
{{formula:415be678-ebed-48d1-9aab-69812382b770}}
Furthermore, if {{formula:2e3bf8d7-8f83-4e49-ad9a-2750efcfcfd3}} , {{formula:5d8537d6-dac0-4f88-9704-12c642ecd3fd}} , and {{formula:f147c7c7-c347-4658-ae93-63b244033d18}} , then
{{formula:89be2bc9-88d3-416c-aa0a-ed9dfbe3c0c0}}
{{formula:e5af84bc-8dc1-4fb8-9c30-9edbf5b192e5}}
Lastly, for {{formula:c3276f00-0d27-44cc-8764-0c3d5efeb53b}} , there exists a linear operator {{formula:4e1a32b7-73d0-451b-a10e-236ceaf84720}} such that for {{formula:8ee7b30b-45c7-4ed8-95ae-2a6abde6e049}} ,
{{formula:f5170586-94d0-4174-b149-c4af24ff8e06}}
where {{formula:49aedc5b-4a0f-4e13-9f92-b2e927b2bd4e}} satisfies
{{formula:4db690ba-617d-4449-8c82-24e72081e903}}
Proof:
We omit the proof of (REF ) since it can be proved exactly same way as (REF ) with (REF ) in Lemma REF . In order to prove (REF ), we deal with {{formula:ab65bf96-d5f0-46fc-8812-ecd99a6871fa}} . Since it is linear, we have
{{formula:18dfc8b5-d8ea-4a2c-9b17-edd4da76c923}}
Hence, it follows from Lemma REF that
{{formula:efea0d9e-63eb-4969-aebc-aa84e16ebf30}}
Now, we estimate {{formula:a79951bc-5884-42fd-a00a-e00fd82bc305}} . Recalling the decomposition in (REF ) we will estimate {{formula:f8c97ae5-50a2-41dd-afde-33960e1e3f55}} and {{formula:7a3408a6-b0a4-4cd4-ba8e-64523343582c}} separately. For {{formula:0863ff7d-3367-45b1-a5ea-75108156d67a}} , it follows from (REF ) and (REF ) that
{{formula:448fe31d-a5a8-4500-baa1-2d6931ed598d}}
For {{formula:6b244173-1981-4aa3-8155-b78a350222fe}} , we recall the decomposition in (), and use (REF ) and Lemma REF to obtain
{{formula:a2f80926-b03f-4afb-bb7d-fa0609316049}}
For {{formula:2e287ecb-0c6a-469d-a3ac-b685805fd089}} , we use (REF ), (), (), (REF ) and Lemma REF to show that
{{formula:42fb18b3-6907-4716-a0ad-0be730695dda}}
Therefore {{formula:afca83b4-b17a-4233-8414-34d802ba661b}} . With (REF ), (REF ) and (REF ), the desired result follows.
Now, we turn to (REF ) and (REF ). Recall from (REF ), (REF ) and () that
{{formula:d47595f5-a92a-41ce-b759-e5db9e445648}}
where {{formula:0bfb46cd-efee-4181-9cd0-0398a24e8ef9}} are of the form {{formula:2561557b-4fee-4e7a-baee-c9ebc1a5f8dc}} for some smooth function {{formula:89fcf1ac-21ba-48c7-a33f-251c34f20787}} . It suffices to show that for each {{formula:5e8d3e68-d7f7-4340-b966-6343235602ce}} and {{formula:2814d216-49cc-4b87-87dd-c6fc4633e62b}} ,
{{formula:8c80a8a9-44ad-4d60-ae83-9a92bde5ae0e}}
for some {{formula:474fe8e0-11a7-4aca-a66b-af2ed25f3c0b}} such that
{{formula:3a352e94-7a76-4770-9f22-335ff7268574}}
We only deal with {{formula:f4e964f8-7c64-4454-a0c6-eed26d677463}} since the other terms can be done in the same way. We also assume {{formula:7ee3ee71-cff4-4431-8170-ed19ec6221f5}} since {{formula:3c87e2cb-8398-4219-a064-8bca2eb92a36}} follows trivially.
Let {{formula:9a7a4671-21da-4278-9cce-57107d86c5f2}} . Then it follows from (REF ), (REF ), () and () that for {{formula:43a4f4b2-0548-4314-a405-0645f2a34118}} ,
{{formula:7a9e7e23-960e-449e-868d-5014e2dc8214}}
Using the change of variables, {{formula:d13a9a14-abf3-42cc-9e14-d10d125025e8}} , we have
{{formula:ed3c0dea-c1b3-439a-b732-439c7f61d06b}}
where {{formula:e5f9c5b8-cfee-44ae-bced-1a223d655486}} is some constant and we used {{formula:3c47bcb1-8a00-40cf-a44e-deb6ba45eaee}} . It suffices to show that {{formula:05a8d7c0-b1bc-4a21-9ea3-ad392c14e649}} satisfies (REF ). Let {{formula:e2e92b52-fc67-48ed-af5e-a659627156a4}} .
Then it follows from (REF ) and () that for {{formula:d175bf0f-88d7-4be0-bd0f-9a675b281ee1}} ,
{{formula:1c0f299f-e5b3-47da-a517-ffa025808a7a}}
Furthermore, it follows similarly as (REF ) that
{{formula:785390b8-dc35-491d-8e2a-f46a933edfa0}}
Hence it follows from Lemma REF that
{{formula:968b87db-a4c7-4439-bd5b-f89e3439cd70}}
where the second last inequality follows from (REF ).
Now, recalling that {{formula:3e15eabd-9bda-475b-b08c-52d20efa2b84}} and plugging the following inequalities into the above estimate,
{{formula:4235b369-b72a-4635-ad1e-afa26708bfa8}}
we obtain
{{formula:476d8b8b-3d18-4628-ba23-539ae5e22248}}
For {{formula:42d321e6-5bf9-4200-9b73-74bf289d8b9b}} , note that {{formula:7b9fb195-8a51-48a1-a6e8-b8f86bab66f4}} and {{formula:536a3c30-ea96-4b17-84e0-622f6baf2c3f}} , hence we have
{{formula:df77ada7-52c0-427e-b44e-a9ab58a8e6dc}}
Using the interpolation inequality in Lemma REF , we have
{{formula:267537df-39d7-457e-8421-737e549deaff}}
Thus, we have
{{formula:6583636b-26ed-4562-9852-18e45722c546}}
where we used {{formula:31f0b064-812d-4b61-a106-48fdd472ec13}} and Young's inequality. Plugging this inequality into (REF ) and using {{formula:98aeec75-62c7-45f5-9a16-dca7d8d703f0}} to estimate the second term on the right-hand side in (REF ), we obtain
{{formula:73b4eba4-e66d-42c1-ad8a-72423ad95592}}
Similarly, we can obtain
{{formula:d9448356-9ee9-4438-868d-9914aedfd7ee}}
Recalling (REF ), (REF ) and (), we obtain (REF ). This proves (REF ) and finishes the proof.
{{formula:a94701bd-84a4-4f0c-8314-ae1ff17a1949}}
Proof of Proposition REF :
If {{formula:de218828-5942-49d1-8ab6-683aa50deadf}} , then the results follow immediately from Lemma REF and REF . For {{formula:3a161f1e-416c-4da5-8cd1-42777658eec2}} , the same results follow straightforwardly since there is not singularity in the integrands.
{{formula:9c494d80-afe2-49ee-8ca8-c447037a218e}}
In the next proposition, we estimate the derivative of the velocity with respect to the parameter {{formula:7f39ecde-7e26-48a9-8ba1-d03f2918c309}} in view of (REF ).
Proposition 4.17
Let {{formula:2ef84df3-1959-4a99-8561-c9ebf95b2131}} , {{formula:bd574ce5-189f-4c2a-bc98-9f3f69a9d336}} , {{formula:54c01891-aeb4-4e18-8bd5-f79634aedd61}} , be as in (REF ), () and (). For {{formula:0a4dc678-835d-4249-85b2-9a50fd7103ef}} , there exists {{formula:e93a6b6e-13ff-49f3-9ce9-4f4422fca8be}} such that if {{formula:29140c93-2e30-4703-9476-f0a175feda25}} , then
{{formula:2c8dd23d-be77-448e-a7f0-b20db631f732}}
Also, if {{formula:c9f818b3-fa8a-4f4b-a8c1-171c2e183849}} and {{formula:74bd5f0e-66c8-4378-8339-eddb192b4cac}} , then,
{{formula:e5e1df9a-e6b9-44c0-841e-1d246ada3238}}
If {{formula:239c81dc-4475-43de-8fea-5d407beddaa5}} , then (REF ) and (REF ) follow trivially, since {{formula:6c8b5dcf-8e34-46a4-b3e8-18c539e0950d}} is independent of {{formula:fbc5891d-667f-44af-8c48-b6998bfaccd7}} . As in Proposition REF , we only deal with the case where {{formula:9a1aa2d7-099b-47f4-a38f-3d648d1660e3}} .
Lemma 4.18
Let {{formula:b9fcf427-c007-4ba2-8426-6a622be05ffa}} and {{formula:68f61da7-aa9d-4e5a-9bba-d698c1660619}} and {{formula:f734c8b7-ea34-4a22-8ecb-5d85b7c0712b}} be as in (REF ) and (). For {{formula:ab0fbf6f-30e9-4bd9-901f-36cfd96aef56}} , there exists {{formula:02438156-03a9-4ffd-bf22-f9a7110409f7}} such that if {{formula:685717df-c237-4137-8bc1-682aa48b47a2}} , then
{{formula:a45ed8a3-e564-4f2e-9c33-4be044e944a6}}
Also, if {{formula:88c6be16-e7a5-4949-8b4c-22b006f15a54}} and {{formula:a7825acb-80c1-4078-bcc3-93354ed4f2f3}} , then,
{{formula:0bd751d0-0693-4def-a329-fee380922546}}
Proof:
From (REF ), we have
{{formula:8c3ad3c0-6b82-4c68-b548-051370c6043c}}
where
{{formula:b49af862-a31b-4d27-890f-a9f054d554e1}}
Again, using Lemma REF , we have that for {{formula:ee1585bb-adc3-4c72-8165-7f113f026630}} for sufficiently small {{formula:36121c32-cbf4-4aa7-8b5a-8d4a30f9c401}} ,
{{formula:8cf0858e-1e6c-4cd2-963e-be7c6c4abcac}}
Then (REF ) and (REF ) follow immediately from Lemma REF .
{{formula:f672b88f-de74-4e2b-886e-04db3fa15b22}}
Proof of Proposition REF :
The case {{formula:e2316399-c37f-4d17-8d73-356ef32ed5b4}} follows immediately from Lemma REF . If {{formula:95b75f4f-a361-4434-a6c2-c4c384be1302}} then the result follows straightforwardly in a similar manner.
{{formula:be569934-b5f4-4e68-9ae8-28d6a656fd59}}
Regarding (), () and () (note that since {{formula:025a8b0e-d13d-491b-8908-53abf29871b5}} in (REF ) depends on {{formula:478ae5df-580f-4788-879c-6380d1b14e3b}} we need to estimate the second derivative wit respect to {{formula:7c8dd4a0-7adc-4085-9ef8-7e054668b1b4}} of {{formula:7f827bff-339e-4ae4-9ddb-3d0158f833b2}} .), we need to estimate the higher derivatives of {{formula:b481f053-07d7-4a95-a2ff-70867f2e6b31}} .
Proposition 4.19
Let {{formula:436ea99d-e3c2-4378-82a0-acf6da2c8dd0}} , {{formula:0134b2d0-60f6-446b-9f27-91b6c2f2d0e1}} , {{formula:c1425a56-d5f6-47b9-b5c1-060715bc784b}} , be as in (REF ), () and (). For {{formula:5c38ee4e-3728-46c8-88c8-0ed21c3e5782}} , there exists {{formula:b7a43e7c-6f9b-4024-9d71-d829209dead4}} such that if {{formula:699a564e-bce4-41a3-a7b1-9b1104cf07f5}} , then
{{formula:0d6cba79-ca09-465c-bb54-fed4d9b4c40c}}
Proof:
We also consider the {{formula:7eefacff-8c0f-4526-b64b-4ce3fea1a60e}} case only and briefly sketch the idea for (REF ), since the proof is almost identical to Lemma REF and Lemma REF . Adapting the setting in (REF ) and (), the Gateaux second derivative of {{formula:feb72156-0418-4ccb-916a-b52d31165ddf}} can be written as a linear combination of the integral operators of the form
{{formula:77780060-62b7-4764-aee0-966e86aa1298}}
where {{formula:f8a18c66-5d16-4f71-a90f-18ca46d65268}} for some smooth function {{formula:914512c0-8fe2-4b48-a2b2-ffce3403dc33}} , and {{formula:af75433b-715c-4f5a-b4c9-b77a9e193ee0}} is a product of two of {{formula:84210349-a5ae-4d47-8fee-eb909480dac3}} , {{formula:a6b33c00-c8be-4a23-86c7-343642cb289a}} , {{formula:d3e2605b-9ba9-41dc-baca-756bee870268}} and {{formula:d3be0625-f43b-4c46-92c1-9bfc7d7b03da}} . As before, {{formula:195721ff-8405-427f-8e4e-7d13d358de21}} satisfies the estimates in (REF ), () and (). Let us consider {{formula:864d2508-b6d1-4926-89c1-2b8ad7e4b11b}} only, that is,
{{formula:d2089d38-10fe-4543-a31f-b1dc09e95829}}
Therefore, it follows from Lemma REF that
{{formula:5e4127b3-d875-4ce7-8724-30c6bfffef96}}
where the last inequality follows from the fact that {{formula:b383d22b-da2b-4ff8-ab1c-b0de6fccd830}} satisfies (REF ). As mentioned, the other terms can be estimated in a similar manner and we omit the proofs.
{{formula:9fa110bf-3340-4d1e-970f-7a5c7ba15400}}
Checking the hypotheses in Theorem REF
In this subsection we will show the hypotheses in theorem REF are satisfied. We will always assume that {{formula:457da207-0511-4246-b6c5-25a4c2c355db}} is fixed. We denote {{formula:e985947f-0ad1-40e2-a7e7-f43c46d20c3f}} . Throughout the rest of the paper, we assume that
{{formula:40acd2f1-0278-4025-baa3-1848b6a7ed10}}
for some {{formula:8ede6711-3bc2-42d3-8c33-d1c043d3a2b9}} , sufficiently small. Note that from (REF ), (REF ) and Lemma REF , it follows that
{{formula:c2347352-111b-414a-9457-47cc07e3993a}}
for sufficiently small {{formula:80dcce1d-493d-427e-a2db-4ebba2f88e67}} in (REF ).
We recall our functional from (REF ) and (REF ) can be written as (since {{formula:8925c667-a151-4532-a2fa-22f1670f6772}} are fixed, we omit their dependence but mark the dependence of {{formula:0d18ae29-4c99-44fe-b64f-577b38b14787}} in the notations)
{{formula:4a955655-a938-4ec9-8bdc-f901012ef18a}}
where for {{formula:cd682fa7-1f95-4bbe-a5de-9290835c015f}} ,
{{formula:ae8c18f8-e482-4ce7-83a2-7be9ab3a2d0e}}
and {{formula:a6f03f93-57f3-4490-a3aa-f88d66b8a932}} is given by () and () and {{formula:35c32fc4-077e-4af9-87e9-3699c5c8bc0c}} is given by (REF ). We also denote by {{formula:fb720cfe-7986-4bee-a054-ec2797277d0c}} the corresponding vorticity and its stream function, more precisely,
{{formula:44d01ce3-0361-469d-a80d-6ed94fc6bb66}}
With the stream function, we can write {{formula:02aa32d4-546a-41cb-8508-48b9545eba26}} as
{{formula:a5c50e64-8550-4ca4-b954-f54b27fa7deb}}
The derivative of the functional will be given as
{{formula:dfdf1a81-4da5-4fc8-ae5c-3593f2e8bfb6}}
where {{formula:0488d9fc-2354-4081-bf4b-6e05c1b59cfe}} is given in (REF ). We define the projection {{formula:f23586c7-0157-4d72-8add-d165fd15c451}} as
{{formula:7b1f26f4-8fa4-4c74-8d50-d999ee478fc4}}
and the linear maps,
{{formula:3ae2647d-5702-4d2b-9119-a55658323fc8}}
Now, we start checking the hypotheses. The hypothesis (a) immediately follows from (REF ), (REF ) and (REF ).
Regularity of the functional {{formula:cf193ccc-891f-4688-8e15-0fa10449d73a}}
In this subsection, we use the estimates obtained in Subsection REF to prove that the functional {{formula:b27ca7cf-9aff-4383-9a5e-27a24eccdf3a}} defined in (REF ) satisfies the following proposition:
Proposition 4.20
Let {{formula:b650bf8f-1b2f-4025-acd1-7a14de292370}} and {{formula:08fd987e-e0f2-4604-b1c6-16224b4fd7d3}} and {{formula:ea6520a6-d0c9-48c8-9870-a3fd669b0937}} satisfy (REF ). Then, {{formula:6d1d734a-3274-4fb9-b23d-dadd61908ab5}} is well-defined and (REF )-() hold.
Proof:
Note that {{formula:a4e9d421-18a8-4b97-b8b8-2affc22d57e0}} is the tangential derivative of the stream function on the corresponding boundary {{formula:20e2be2a-a179-4999-bdc1-86cf471c19f8}} (see (REF )). Thus, {{formula:fc003e29-7b88-42fd-a33c-4aefa81b2e9c}} , that is, each {{formula:e4cbfd02-81cf-4f00-b6ef-97673dc338b2}} has zero mean. If the boundary of the patch {{formula:8dfcdf85-0c16-4578-b6d5-920eebd10ad1}} is given by by {{formula:7f99d493-5a9c-400e-bee8-ea678e185112}} , then {{formula:dd88b093-b79d-4947-885f-e373313b0cf7}} is invariant under {{formula:73bda246-8fde-49e4-8636-cb8b568e056e}} rotation about the origin and reflection, therefore, {{formula:6fe0f8e1-1f2d-47ce-bfe1-124ea6580c2a}} is also {{formula:c0f61827-856c-433d-a21f-954eca6cbd6f}} periodic and odd. Furthermore, it follows from (A) in Proposition REF and Lemma REF that
{{formula:58fc29db-c5ae-4992-9d0f-f6a454bf92a8}}
which proves (REF ) and that {{formula:f3eb121d-b81c-48c2-bab1-9276eabf1ff9}} is well-defined. For () - (), the results follow from (A) and (B) in Proposition REF and Proposition REF and REF straightforwardly.
{{formula:c749dd4d-b12f-44c2-93cc-037c99157a41}}
The Dirichlet-Neumann operator
Here, we aim to prove that the linear operator {{formula:e9b4fa92-bdac-4d0a-a181-4626e16450ab}} in (REF ) satisfies (REF ). Thanks to Lemma REF , we know that at a stationary solution, the corresponding velocity outside the support of the vorticity must vanish. In the following proposition, we will prove this quantitatively by using the main idea of {{cite:36b482467d7795be33b47a689c176e365290de7e}}, {{cite:e27b4c87f6efd00e0db7588b59e38ebce876347d}}.
Proposition 4.21
There exists {{formula:8c63bf1a-4d8c-40d1-a3f4-52e094dabe32}} such that if {{formula:7c89d4b2-ded6-4631-98c9-8406765daf58}} , then
{{formula:5dceb17f-1107-4152-9459-a76b7f73a6c4}}
Proof:
We adapt the notations in (REF ), () and (REF ) for {{formula:af7ea0b4-6f68-48d5-83c1-720a966990d3}} , {{formula:a22f604b-b48d-44d0-8f63-86677c2d9252}} and {{formula:a24d9574-25f5-4710-9e9e-307c8799ba81}} . Clearly we have (we omit the dependence of {{formula:342fa0dc-0909-4110-a66e-fa9c36ba2323}} and {{formula:112209e3-b21d-4a05-a243-633051a70677}} for simplicity)
{{formula:f12452ad-d6de-4707-9fd3-d2755ed1deee}}
where {{formula:13620d82-67f5-4a2d-8762-d8dc4cc53839}} is as given in (). Note that {{formula:1e2a4b3b-a307-49f2-a2e6-9446dcd8b726}} is harmonic in {{formula:f5a17d7f-ace1-495b-a149-b014dfb06de4}} . In addition, it follows from (REF ) that {{formula:a033ee5f-9e84-4205-92d6-280a00a5d2e2}} , hence, for {{formula:f8f4996f-1d49-47ef-8afb-fc36371140ee}} ,
{{formula:5745739a-0da9-46c0-97e1-db7255edddbf}}
With the above decay rates, we can use integration by parts to obtain that for {{formula:c3d14f29-0030-4316-b27b-83c73aca63eb}} ,
{{formula:040aa2a8-d43d-444f-8ea9-c4bd05ea7a43}}
where {{formula:83978658-b543-4a2b-8a7d-1820f6ed60d1}} denotes the outer normal vector on {{formula:e9e8e9f2-3109-41ef-9eda-25f73466231c}} . By the change of variables, {{formula:1a1198e1-fc67-459a-9e5b-a6cf740d6603}} , and {{formula:e2dce356-b76d-4ff3-8cd3-27f002fdd94c}} , we obtain
{{formula:3a554149-59de-424d-8fe6-27d25071746f}}
Similarly, we have
{{formula:3c798c4f-3310-4d13-abb8-1d0080f898e0}}
Hence, we obtain
{{formula:29ce652d-91bd-4378-a321-7fd8658fd955}}
We claim that
{{formula:0ef3410c-5f6d-4a58-a078-4c0b844cd9aa}}
Let us assume the above claims for a moment. Then it follows from the claims, (REF ) and (REF ) that for sufficiently small {{formula:1689bed8-4464-4d79-b5df-e4f2b91d9bcf}} in the hypothesis of the proposition, we have
{{formula:f8c667c5-5ed0-44ac-9a9b-7b66c2d002aa}}
With this inequality, we can obtain
{{formula:fbd52884-c554-4cd1-b102-3624f5130d02}}
where the first inequalities follows from {{formula:dde711d0-c16f-4d96-aaaa-9f459a19fec3}} for some {{formula:b3983248-b6c4-418b-97cd-418d7143023b}} and the second inequality follows from {{formula:6e0e1221-8ac6-4e2b-b391-d832da7e15a1}} for some small {{formula:86afc26d-f673-41bf-b695-85223b36692e}} . under the assumption (REF ). Finally, recalling (REF ), we obtain the desired result.
To finish the proof, we need to prove the claims (REF )-(). (REF ) follows from Lemma REF . To see (), we observe that
{{formula:72995655-6c95-4bc4-8bab-035438476392}}
where {{formula:7bd85948-309d-4c0f-8aff-0186a3a4d137}} . Thus, it is straightforward that (thanks to (REF ), {{formula:1855931e-a7b9-4906-a9e3-5d02ccef8341}} for some {{formula:ef48d24d-1cad-4724-88a7-7cacfee456f6}} )
{{formula:37e76515-3bd5-4232-92b2-882d4bed9dbc}}
where the last inequality follows from Lemma REF and Lemma REF . Thus,
{{formula:c5f1db8a-acad-4ab2-be59-b33c38253332}}
which yields (). Lastly, in order to show (), we rewrite {{formula:ce3db93c-adcb-42be-a60e-fd1c29873c26}} as
{{formula:7021092a-4d5d-44c8-bb5a-818e4c410109}}
where
{{formula:d996a295-f4db-4ebc-bdef-2e717de257a2}}
From Lemma REF , and REF , we have
{{formula:82b55e94-7ff4-40c0-a453-8badda3b1cb2}}
Therefore, it follows from Lemma REF that
{{formula:043b1f23-9574-450e-b7d2-c29f46c9b4c4}}
where the last inequality follows from Poincaré inequality. Hence () follows.
{{formula:0119a29b-6306-4d2b-9e0a-b27c1d898307}}
Estimates on the linearized operator
Our goal here is to prove that {{formula:df6d121a-5107-4c25-817a-6181faf56ab7}} satisfies the hypotheses {{formula:d5c88197-6fec-427c-9142-c81e9dafe880}} , {{formula:f43c60de-930b-45bb-b200-55253d2d18a5}} and {{formula:71dd8695-41cc-4010-a9f0-e71326736069}} in Theorem REF . More precisely, we will prove the following proposition:
Proposition 4.22
Let {{formula:54e4d6e0-59cd-4908-9bf8-9bfffa6472e5}} and {{formula:12398acd-cfa0-4a74-a96d-54195eea9f06}} be as in (REF ). Then
{{formula:6e9c93a3-9d0b-4f15-b87b-fd3764c4dd83}} , {{formula:6fbc81e4-6e76-4d8a-bf1c-0d38f9502498}} with the estimates (REF ) and ().
For {{formula:cb7cc1fc-1b4c-43d8-bea0-7c914836f1df}} (REF ) holds.
For any even {{formula:62d238a5-eaed-449d-8e2c-f1e76efe8b15}} , there exists {{formula:07fe4644-d651-4872-afd0-6d0ea1e9f992}} such that if {{formula:c3de4ecc-5ede-4e34-a4ce-142a501ee90f}} , and {{formula:aa62a3a1-0516-4c15-bb72-8f956eb9c656}} for some {{formula:b6e188a6-a530-4232-9e55-f3f3af9c15dc}} and {{formula:8f5f066f-7b38-4e29-97e8-a4773cba2315}} , then (REF ) holds.
Proof:
Proof of i@). By definition of {{formula:2953569e-fbdf-422f-b96d-d5ad11b2742e}} in (REF ), we have {{formula:714a6381-b7c7-402a-94f2-b0f5fd387910}} . Furthermore, clearly, {{formula:b3244dfc-d8a8-4a39-9b8e-8be859b64f33}} is a {{formula:7a2f6fd8-ecac-4026-a269-1d8c8f90d8d5}} -periodic function and odd. Using the invariance under rotation/reflection of the stream function, it follows straightforwardly that {{formula:44b4af43-b297-4b6b-8d32-136ba7a3323f}} , which is the radial derivative of the stream function on the outmost boundary, is also {{formula:35d0e354-0b07-4e48-a814-95ce8c0fdb7e}} -periodic and even. Therefore, {{formula:8a89d3d3-bc72-4bae-b583-c6eed622484c}} is {{formula:9c4ce9e4-dab0-461e-b8fc-0de021e85962}} periodic and odd. (REF ) follows immediately from (REF ), and {{formula:d0f9954f-64de-4e59-8a65-969d5ad3d802}} . Similarly, {{formula:1b0b28fb-60b3-454f-af97-81b7951fbe55}} and () follows from (B) in Proposition REF .
Proof of ii@). Thanks to (C) in Proposition REF and Proposition REF , each term in {{formula:752af0cb-22ce-4d2f-99ff-cd3b784a01dc}} is Lipschitz continuous with respect to {{formula:40c11714-26ce-4c2d-81e0-722ab29ed457}} . Furthermore, {{formula:7a6eb0c0-aa13-4915-b17d-4fb938abb57b}} and {{formula:0f715525-454b-409a-89d6-216c856f87d1}} depend on {{formula:f33cf433-ce8a-43b8-89d3-5224abfff1d0}} smoothly, therefore the result follows immediately.
Proof of iii@). In order to prove {{formula:0fa23029-bcf6-436c-bd14-57d8bedbdc55}} , we first claim that for each even {{formula:6a96f357-cf63-4284-827a-8bc13a5488ab}} , there exist {{formula:e1b5a457-8965-492f-a237-170130a1522b}} and a linear map {{formula:1f4a2b24-6dca-4ed2-b1f4-be6697256a6d}} such that if {{formula:633e8f18-9ad7-45bb-8b75-dd36ed29b1c2}} , then
{{formula:d3f71479-7255-45f0-a66a-c49f1a3080a1}}
Let us assume that the above claims are true for a moment and let us suppose
{{formula:693427d7-01d4-4942-b241-14faa08e566f}}
From (REF ), {{formula:1f90adc2-c9e0-4f6e-999f-20ed135aa8a6}} , Lemma REF and the assumption that {{formula:51ff7c43-7738-46e0-8454-52ad710bc422}} for some small enough {{formula:20d2dc38-00d0-4f00-a1f8-9a1a9d16cc42}} , we have that {{formula:85c4b6d6-f34a-489b-8dac-dd3dcb962146}} is invertible and
{{formula:e947af95-c75e-43c5-8757-69c8d055b1c9}}
Therefore, we have
{{formula:d7721953-f5c8-4e53-8573-cdf44ade15f3}}
Now for each even {{formula:eb0fe271-dc69-4618-b2b5-06ef895129c8}} , it follows from (REF ) that
{{formula:3b1e58e7-0613-453f-925c-2eea6cc7a8f7}}
Thanks to Proposition REF , we have {{formula:1e85a181-fce3-4a77-ba2a-9493c06e29aa}} , thus, it follows from (REF ) that
{{formula:219b6484-c97f-4283-9036-b0d522998968}}
and
{{formula:b89ca709-2428-407c-a896-3567a1d94d26}}
From (), we also have
{{formula:4d12c7f0-02ea-4024-a0f7-884cf773757d}}
Using Lemma REF , we have
{{formula:90c51a1f-5d68-4499-a26e-d23926ec88be}}
for any {{formula:a80aeb34-5214-40de-95d2-463c0ef37c08}} , which follows from Young's inequality. Plugging this into (REF ) and using (REF ), we obtain
{{formula:f2df3d1a-67c5-4009-ab50-e1db7dbdc828}}
Hence we choose sufficiently small {{formula:1b608a52-b9b5-42fa-a9b0-441e4d1b4db7}} depending on {{formula:8298248b-4950-47fc-b24f-b69266d5b51c}} and {{formula:4daa260e-f523-40d8-a9ed-1afeb040dd3a}} so that (REF ) yields that
{{formula:2340af6b-6c38-4fac-9de9-05028c3ff620}}
With this estimate and (REF ), (REF ) follows.
In order to finish the proof, we need to prove the claim (REF ) and (), we note that each component of {{formula:58bb4595-4b1a-48a8-bed9-d512511f7854}} consists of linear combination of the following forms (see (REF ), (REF ) and (REF )):
{{formula:774e9b21-fd66-4008-aa3b-82c4d6c2145e}}
For {{formula:e1522135-fc7d-4763-ae39-712cb752b403}} , it is clear that
{{formula:4654d124-c629-4005-bc5d-b8cb4b6a863a}}
for some {{formula:81452d19-84ac-4c00-863c-cffcc80006f7}} . From Lemma REF , we have
{{formula:d50e8165-3cb9-4c0d-86f7-3340f51938c1}}
where the second inequality follows from Proposition REF .
For the other terms, we only deal with {{formula:61f4260a-5028-4814-aed7-918c77a15b9d}} , since the other terms can be dealt with in the same way. Thus, we will show that there exists a linear operator {{formula:038c3047-cea6-465c-b884-096a5b9eee43}} such that
{{formula:92dd6425-9e37-457b-a92f-d7e26da4802e}}
It follows from (D) in Proposition REF that there exists {{formula:43324221-7b8a-463d-84fe-89054f5fa513}} such that
{{formula:a5bbcdf7-4ab8-4eb3-9932-4de6ecbb2776}}
and
{{formula:cb1813ef-e126-414c-a65e-7df838e1c493}}
Therefore, from (B) in Proposition REF and Lemma REF , it follows that
{{formula:37588d60-a0ce-40cf-9414-91d90fd2ecd3}}
where we used that {{formula:635888f9-fd40-458c-bcfb-f0c50ef1b7da}} is a Banach algebra and that {{formula:0e435551-afe6-4b61-a781-d60f5a53bf16}} for some {{formula:8ddf8f5f-1766-4105-b2f6-68b7bbe59647}} . From Lemma REF , we also have
{{formula:8affe955-cc6e-4584-aa09-9dab520e4f4a}}
where the last inequality follows from (B) in Proposition REF ,which also tells us that
{{formula:762e1ece-a5e6-4442-bea2-2efa345c81d8}}
where we used {{formula:972c9c3f-de55-4044-87fd-12f571d9b00b}} .
Therefore, we have
{{formula:30125306-aab6-4ac1-ba1a-59a1de779fc3}}
Combining with (REF ), we obtain (). Since every term in (REF ), () and () can be shown to satisfy the same property as in (REF ) and (), combining with (REF ) and (REF ), we obtain (REF ) and ().
{{formula:b2752eed-956c-4a4d-808e-dbe03a308666}}
Spectral Study
In this subsection, we will verify the hypotheses {{formula:b96c8b09-1244-4859-a392-d2e018b71c39}} and {{formula:8fed959b-2be6-4947-b13e-f4bce54b8842}} . They will be proved in Proposition REF and Lemma REF respectively.
Proposition 4.23
Let {{formula:9ff9a4df-30c6-4d4f-a624-a88edff47ddd}} We can choose {{formula:b951e688-d37f-4ef3-ba30-91f47bb225ba}} and {{formula:1ad8a542-183a-40cf-96d3-a726051d194d}} and {{formula:cbb2e171-8878-467a-954b-925174d7c071}} be as in (REF ). Also we set {{formula:df92fe8a-f105-4089-b2c2-9a6b7a065757}} as in (REF ). Then, there exists {{formula:6f769978-63c2-4aea-85d6-fc28dda4cb54}} such that {{formula:10472d0d-592a-4ba1-a4fd-0422ae49d743}} has one-dimensional kernel and codimension one. That is,
{{formula:6bd002c3-b01e-4e0e-9f42-377f4c50d746}}
Furthermore, {{formula:74cfeeaa-c6fa-4124-869d-00d42183cd94}} and {{formula:1d787bea-da58-4e7f-bd4b-4d8d95e38993}} and {{formula:5bb02ae1-8ba8-4684-9982-2708fc5b09d4}} are supported on the {{formula:d719ff50-a83a-4390-8866-5d1c6e8b2fa1}} -th Fourier mode.
The proof of the above proposition will be achieved after proving several lemmas.
Let us first compute the derivative of {{formula:fb733bd3-5969-4ba0-b1b9-d6c1cfe27fb3}} at {{formula:bf3e4c61-db7d-4f60-8403-1b6d2dc5d148}} . Since we have {{formula:cb633f93-ac1f-4f1d-bffb-408411228891}} for any {{formula:da2d96a6-eeed-42f9-91da-80d3f97ed648}} , {{formula:5277bf98-ff95-4213-8590-405a473f3b63}} in Proposition REF yields that {{formula:d440624d-32c7-4f8f-8527-0a79fdb5a74b}} for any {{formula:3c1f74bf-9add-4503-bbb3-e3d0392de00d}} . Moreover, it follows from (REF ), (REF ) and Lemma REF that letting {{formula:91ea07d5-38cf-4e18-a33c-2ec06bdd4297}} {{formula:85e0a013-188b-4b72-979e-ade3e32ce1a3}} , {{formula:ddf95fda-9bb2-41c9-9d2c-29b07154c427}} ,
{{formula:7dab8cc5-5c24-4429-93e9-ace112a0f9f2}}
where
{{formula:e23675cf-ad6a-41f8-971a-47e093cb548e}}
where the coefficients satisfy, for any {{formula:e28d984b-3279-494a-bffc-cde16f038fa4}} :
{{formula:986861ea-15c9-4c42-abb9-9f939456217f}}
with
{{formula:74200c37-64bf-46c9-b780-99142a63acaa}}
Lemma 4.24
Let {{formula:a8b8d991-8916-4d1c-9b05-a1a930835327}} , and {{formula:a0617565-daf8-489c-87a5-f8b0efa4deb5}} {{formula:0dd35e0c-c8ce-4b76-abae-370cd805c222}} satisfy (REF ). Then, there exists {{formula:46628d3f-0c04-4e10-86d4-bc1abe6a398e}} and {{formula:2ad93dfd-ddb2-49f6-8d6c-3ce27f837222}} such that {{formula:9c01742b-e445-42e8-9779-a26c08881f4d}} and
{{formula:acaa6aaf-3abb-4d7d-99af-a080ec32e4ec}}
Proof:
We first write {{formula:f0a22ea6-5df3-4675-9365-e691fe7a149a}} as a polynomial in {{formula:d1a56932-7325-438f-a7ec-072639df4a69}} . Under the constraint {{formula:8a456f4e-bdd4-4787-8f58-7ed79902090e}} , we get
{{formula:ac286640-ffa8-4849-8e1e-fd15d55bfaf3}}
Therefore, multiplying the first row by {{formula:afdda733-5bc0-4a3e-85e5-fe058d6d004d}} , multiplying the third column by {{formula:39fda532-2fff-4e8a-a849-19aa3537938f}} and dividing the second column by {{formula:56642865-7c98-4eaf-9694-33833fcc7aac}} , we get
{{formula:a234cfb3-693d-412d-8a57-c12d5cf1b746}}
where we used the condition {{formula:837d2d79-cde2-436f-a23f-c4ddf2997ddf}} in the second equality to get rid of {{formula:4e572972-5dd1-40f5-8e16-e064d70a2beb}} . We further compute
{{formula:9b7b7420-406f-4d8d-acd5-5b1172153e4c}}
where {{formula:4fab8746-0757-432a-8b51-022f667dd506}} is the cofactor.
Let us write
{{formula:113469d9-c82f-441a-9cd2-bde451c613cb}}
where
{{formula:c3e0e208-7341-449c-9981-acccdcd67b6c}}
{{formula:cec9bc10-fffe-4fc4-a25d-c34d69d2d1f5}}
{{formula:3ba7be8e-2dd4-49ac-9696-acefb5673ed5}}
Under the condition in (REF ), we have
{{formula:319d81c6-d5f1-4e59-8063-31483762f88f}}
So {{formula:5e4b1490-9179-4e42-a357-df2c61223fc4}} , {{formula:5629baa1-ec07-409a-a30c-986a354534d9}} , {{formula:4b49d467-5beb-4efb-b96a-68b0afd16378}} . We then can use the Descartes rule of signs and show there exists a unique solution {{formula:95097ffa-e301-418b-9a09-577439f8d58f}} . Since {{formula:463350f3-7bb9-4889-b258-b7fd81202604}} , the Descartes rule again tells us that we only need show that {{formula:6ac695a2-5b1c-4901-a33f-be3edff52897}} to show the unique solution {{formula:8c95c0a7-0622-4f4c-8aab-65e6ceeca1c1}} such that {{formula:3ed73f6f-e451-438c-a8a1-17c21e508a4c}} satisfies {{formula:c188b3d6-81a9-4e5f-ba27-588afbef7090}} . Once this is done, then {{formula:b24804eb-e810-4edc-9ae2-4fddae496978}} can be uniquely determined by {{formula:6a91ccc7-7b71-4b55-b679-92d81799f2aa}} and this proves the existence of {{formula:68827d74-67dd-4a01-8034-ba2c0280c10d}} and {{formula:2c8663be-141d-400b-aff4-54550defaae4}} satisfying (REF ). Therefore we compute
{{formula:2a508026-adbb-429f-b175-2615146dda96}}
Now we are left to show (), that is, {{formula:9ecf4351-c721-4993-88fd-93e65d2f1679}} .
As before, we have (replacing {{formula:966a2ec4-f9dc-48a4-8fc8-b769a2a2461c}} by {{formula:b1ec7eec-401d-4485-b758-338d7e5b057e}} )
{{formula:6f4c9fe3-b304-4ef5-81e2-73d491d02674}}
where
{{formula:0cb089ec-117f-4d13-a135-5ae1bc6c534a}}
{{formula:61db9a8a-2642-4d8b-8732-5bc7dd889aaf}}
{{formula:98e341ab-ce38-4385-9e1d-2988b35277ee}}
{{formula:5f4eb658-b8e8-4ebc-9efa-21a252054713}}
Since {{formula:7f3cf14a-175c-42bc-aeb3-ae7d6931c7de}} , where {{formula:16c9a424-ad54-439d-a83e-90198e4088dc}} is as in (REF ), that is, {{formula:72a8a38f-24f1-47ad-a49b-5162963c355f}} . Thus,
{{formula:aaa1f14a-53c9-46ea-8692-f578787285d8}}
Let {{formula:4946fb08-976d-46bb-af59-0aa7ec4ae72b}} . We will show that
{{formula:edc008e9-d075-4f0b-a391-6779d92fc284}}
Once we have the above inequalities, then {{formula:b71c4807-f690-43dd-99c5-8a68d5287e24}} , which implies {{formula:ddeae74d-6b41-43f9-a87d-5795cd562e18}} in (REF ) cannot be zero. This clearly finishes the proof.
To show (REF ), let us first consider {{formula:13e97d71-ab00-4847-9165-45d3f24e82b8}} . Note that
{{formula:f6973893-d341-4b6a-86e9-b0b4bce99696}}
{{formula:836cb52e-f8f9-4670-bef2-5df515740a31}}
Thanks to (), it suffices to show that {{formula:05b199ca-db55-4cf7-808f-6d85bbb7ded0}} . Note that
{{formula:8c5f1644-e4aa-406d-862b-1670ea6fe1f7}}
Clearly, the last inequality holds because {{formula:0e6a9552-ee50-4044-86f2-11d74658150a}} , and {{formula:31f0dc18-6cb7-4883-8510-c89aa5e0da84}} , therefore we have
{{formula:56c66626-3f9b-4cbf-b3b3-dee18201aea4}}
Now, we turn to {{formula:2cf3aac5-65e6-42bd-9760-16d1976a3518}} . We have
{{formula:318dd962-5e81-4cb9-b126-9a9f39272fc8}}
If {{formula:ccdcaab2-aebe-4770-95bc-b473b0e9e437}} , then {{formula:0ee9c952-586f-471d-8cde-694dafd07b8b}} is deacreasing and {{formula:7fab691b-6034-47a0-a62e-46a20ef69a97}} , which yields {{formula:ff7c2a2b-0aed-4a9c-b5e4-ab432d4f2eca}} for all {{formula:9d68c9cb-ad75-41ed-87bd-43cbafb1f2c2}} . If {{formula:422bedd9-6e6b-4dbd-a852-2c3ab92e9948}} , it is sufficient to show
{{formula:af6b2acb-7d19-48c7-a6f6-c755fbc1cc85}}
From (REF ), we have {{formula:8692232c-dcbc-4f6a-8749-787c8a2412f1}} . And the lower bound of {{formula:f3c1ea7b-04fd-4289-b2a8-082140483bd7}} in (REF ) is equivalent to {{formula:d76a4873-3062-4fb7-8e8e-e3c36395ce0d}} , hence
{{formula:9fa08376-742f-4ca8-bff9-9ccf87577811}}
{{formula:71910965-6e19-4ee3-a9ea-ce0bfb89e207}}
{{formula:efb72584-2192-4439-a449-9efbe1ad8324}}
{{formula:36302721-1dd3-4138-9887-2a339aad7625}}
{{formula:15ed8bb8-8180-49d2-96c9-f204e561de68}}
{{formula:4ccba465-8441-4d8d-a282-e5724b6895c9}}
Since {{formula:673e95f7-f259-4691-97d0-1af0df7f7bf5}} , the condition {{formula:0bcff8f0-dd93-4dd7-8d77-f81a49cf37b2}} in (REF ) leads to the inequality we want. This shows (REF ), and thus () holds.
{{formula:53a7ff04-a6a4-4e3f-b8ca-5545917873e5}}
From Lemma REF , it is clear that the kernel of {{formula:b3524a13-c862-4d89-95c6-0f78e3003ed6}} is one dimensional. We denote by {{formula:10cc6d29-4f05-4f24-b7e5-71e46480763c}} and the one such that
{{formula:10b25058-0006-4d2f-b278-d04aa4002f77}}
Since {{formula:ef67d9ad-aeb2-4d4b-92e4-93ae61343246}} and {{formula:16f72708-dd2f-4ce2-b59d-f6f448346a0c}} are tied in the relation {{formula:bb31ffcb-0843-44bc-af0a-03d79adbf7c7}} , we now drop the dependence on {{formula:835548b2-e25b-4ee7-ac04-9d95a18ae151}} of {{formula:87dc9a9c-9744-42d9-96e6-08ce742b196f}} .
We now characterize the image of {{formula:38b2a74e-13aa-412a-8e1f-1e1b960ea51e}} .
Lemma 4.25
Let {{formula:d8d670bb-f275-4991-8be5-64dd8e9fe27e}} satisfy the conditions in (REF ) and let {{formula:d209b55b-24d7-4197-bcfb-a8cb8e32ea53}} be as in Lemma REF . Then,
{{formula:226eb4eb-9ade-4868-bc74-da7dc436e4f9}}
where {{formula:854e8d56-6a44-413f-bddc-3dc4a877ed6e}} is the {{formula:14bb0b5c-3ce2-461b-8242-f61f62ec15fd}} th column of {{formula:cfd80a0b-16f2-4fa9-ace0-802957f88b33}} . Then {{formula:ba3b2b82-bba0-4fcb-babb-37581b52c1df}} .
Proof:
We choose an element {{formula:2e47d65d-9c09-40eb-8243-4cb468076d8f}} to be the one such that
{{formula:4a0afc03-cd8b-42d3-8e14-61a5155dce0f}}
where {{formula:d55e0fa4-b037-4dad-b01a-46cce683bb36}} and {{formula:e8f29823-83ce-4e3e-a1e8-58c5c9717603}} are the first two column vectors in {{formula:554d7df3-7b3c-4d90-9f5b-8407fcbe5381}} , which are linearly independent. Since {{formula:b2aea69c-cb75-4042-bfcf-26fc97ab142a}} , and {{formula:8e525ce7-919b-43fc-9be7-0da48400bc5a}} for all {{formula:7591b9f2-04c8-46e3-b038-680250ebbe9d}} , and Proposition REF , it is clear that {{formula:d7e74865-3b0d-4dcb-9d74-d92db6c5b353}} .
In order to show the other direction, {{formula:cff8693e-fc01-42a9-8249-67797e5706ea}} , we pick an element {{formula:7fa4f8c7-6a7c-4c65-81ab-e28456665078}} ,{{formula:aa332f12-50a8-4959-80c6-dd4ba059e88d}} , and
we have {{formula:a53cd1c7-c823-4d2c-b52f-5f50af3176cc}} , {{formula:bb2e1d83-04c1-45ac-80a5-9879263953f5}} such that
{{formula:4233a978-1f0e-456a-82d5-ddee5190a553}}
Thanks to (), one can find a sequence of vectors {{formula:6af7522c-b957-41a9-90ee-659ca1f21074}} such that
{{formula:eb885b57-382e-4fd2-811c-baa286eea5ea}}
Therefore it suffices to show that
{{formula:e787ec4d-e731-4ae2-93c0-ef7460fff4cb}}
where {{formula:0fc913f8-2a9e-42ef-8401-c71756f30e6b}} is independent of {{formula:92214609-a718-45f4-8ff7-edb4bb68313b}} . Once we have the above inequality, then it is clear that {{formula:9417bf4c-42b2-4041-b9a2-8fbaacddad69}} where
{{formula:2878cf55-9c44-43d3-a2b0-664593f27510}}
and {{formula:b860c95f-7ce2-433b-b24d-1ea159dc41d2}} , which follows from
{{formula:091ac657-e0d7-42af-9918-16f2fb8aab7c}}
This proves that {{formula:ac48dca0-2c38-4d6b-8468-547d2c7b6a46}} .
In order to prove (REF ), we denote
{{formula:9d7155e1-d075-4f42-b163-cf4aa3d17982}}
Then (REF ) is equivalent to
{{formula:d014f159-32bc-4eca-8708-077960fe7eb2}}
We claim that
{{formula:6ce0be36-a8fe-45ac-8d23-d9eb67c91045}}
From (REF ), and {{formula:c881a17c-1753-4620-8a03-081fe3728f6d}} , it follows that
{{formula:0dfa002e-365c-4c3b-8c8d-7a7dd5b53578}}
Therefore, for sufficiently large {{formula:cc1bfd25-6dd8-4670-94b4-cd30c574140e}} , {{formula:c71d06d3-c987-42d0-9da7-6e47ae0a2a15}} is a non-singular diagonal matrix, and it follows from (REF )
{{formula:03ca813f-dbb3-42e9-bbe8-88ff02ea9d4b}}
which proves (REF ).
We are left to show (REF ). From (REF ),
{{formula:0df3a8e7-3d2b-42af-a3a8-75d520c4684e}}
For {{formula:e1b4981b-e25c-483c-8670-fea4b3689fb3}} , let us claim
{{formula:84f07a61-fda1-43e3-81c7-d53316215f2b}}
Plugging in {{formula:86194372-d3c9-4a55-bc6b-37fb28ee2c6d}} from (REF ), we can find that (REF ) is equivalent to
{{formula:8e7d9b33-848d-48d4-9ea4-364aec1b6b02}}
{{formula:932c3030-87bc-4a9b-81a2-ea2fa766dea5}}
According to (REF ), () and (), the RHS is strictly positive and LHS is strictly negative.
From Lemma REF and (REF ), {{formula:bc855d78-11a1-43ad-9f78-869d31273d33}} . This gives {{formula:c84dc7a5-5094-43c1-886a-bc36fa1fcf19}} , then
{{formula:e9b2c7fb-c8fc-4a85-b88f-794da79916e5}}
{{formula:549bf5ac-0441-4d53-9aad-12d1b4276667}}
Proof of Proposition REF : The result follows immediately from Lemma REF and REF . Note that {{formula:fb9ed2aa-2a1d-42ff-b9e3-2ed19a0454a9}} and {{formula:e89eee20-6245-4598-a299-d8030936dc2f}} are supported on the {{formula:804af89b-a3d6-4391-8ee6-3b499fd902a0}} th Fourier mode, thanks to (REF ) and (REF ).
{{formula:f5c2c66c-43be-4039-a05d-aaaab4ca5a40}}
Lemma 4.26
{{formula:6dfbc191-9990-4c8e-9236-bde8038d7b66}} .
Proof:
From (REF ), it follows that
{{formula:5183d7f8-6f62-4719-9b12-506e2cb0fc82}}
where
{{formula:44e9d99f-b7c2-40bb-8ed6-8d1234b80ba7}}
Let us write
{{formula:5d38a011-02f4-4b0d-a779-b4ade0031dd2}}
where the vanishing elements in {{formula:971b17c8-7384-4103-84fe-5dd8b1202d8d}} follows from (REF ). Note that
{{formula:494c5347-9ab0-4357-9df0-7889f21f16dc}}
From (REF ), we have
{{formula:448aa029-1563-48af-a351-48962a773c0d}}
where the first inequality follows from (REF ), and the last inequality follows from {{formula:15c46b43-dfa9-4c19-a70d-04069f917b81}} . Hence,
{{formula:80e4d716-4871-4af7-85f2-dd2c2feee500}}
and {{formula:eb1a960a-bd41-4d3e-a563-165f20d47632}} is invertible.
Since {{formula:99c94256-97ee-4170-9c7d-b3d3cd590705}} , we can choose
{{formula:95e85210-d71b-4749-b21b-533e4a69e921}}
so that {{formula:b40cc2b0-7c70-43b0-9f23-c2bc525f2f1e}} , and {{formula:19d49db9-0ef0-45a9-9975-51723d1a2b8c}} .
Then {{formula:1bbd77cc-b564-48c3-8565-e81348a7ac6d}} is equivalent to have
{{formula:daab0f83-1801-4c3f-961b-917b8af0fe98}}
According to (REF ), it is equivalent to have
{{formula:e64a2d26-bc1b-4124-ba87-9bfb93be8929}}
Using {{formula:05616df1-9d1c-4754-b833-89ce06f406ed}} , which comes from {{formula:0f37477b-5f69-41da-aca3-35a18c9ad6a0}} , we compute
{{formula:9e417073-50fd-4561-8eed-61b5fa5564bb}}
Then
{{formula:b380fba5-ad9f-4e7b-a6b5-009e321484b5}}
{{formula:1a16b236-7afe-413e-a11f-5627caed067d}}
So (REF ) is equivalent to
{{formula:44acb667-e67e-4bd3-b721-1012121d82df}}
{{formula:589c6a05-f0f3-4047-b6f0-405feeba623d}}
{{formula:6680e8b2-969e-41c0-b398-249c62302628}}
{{formula:f28a2321-ccad-404a-ac22-c509c619cfb9}}
To show (REF ), let us write
{{formula:4c57b61b-63e3-49c2-b131-ae7b85f7413a}}
then, it is enough to show that {{formula:30c3f740-954d-4012-8349-64b5a98ef120}} .
Note that we have
{{formula:d0f8b397-80c1-4b9d-89b6-b7275b8e29f1}}
thus, {{formula:2670521a-8264-4456-9a1d-1c2efa8f3ecb}} . we also have {{formula:69a79995-1b5e-4b87-a3bf-a60a183df453}} since {{formula:0c186af4-9389-494e-b41e-359e5c589055}} . For {{formula:dab8dfcd-ee4a-4942-a055-10add6aa3531}} , we have
{{formula:210fd71b-ce0d-4704-9fbb-2be14825aea6}}
as {{formula:5fe998a1-c89a-4678-be07-683580e57858}} , {{formula:b3cf9eec-ec66-454b-b8e7-cdce54461d8e}} , which follows from (REF ). This yields that {{formula:9a017f52-87de-481e-8059-3da178bb47d3}} .
Therefore the Descartes rule of signs implies that there is no positive root of {{formula:363789ca-85ea-45ff-9b54-2939f82fd52a}} , which implies {{formula:9d9356dc-112d-4457-88b4-394e0ac438dc}} , that is (REF ).
{{formula:f9df1623-78e9-43b9-a6a7-3784419157e4}}
Analytic regularity of the boundary
Theorem 4.27
The solutions {{formula:7b42e417-3e52-408b-a122-ba23935324ef}} constructed in Theorem REF are analytic when s is sufficiently small.
Proof:
We use {{cite:2b432ffb4214c1a3459eae3776fda1ba08b95c1a}} and {{cite:89c6ca66e5d00d0990ac7a30dfd0582997aa5c5d}} to prove the analyticity.
Let the stream function
{{formula:54fda3a8-8bf7-45c0-a4c7-dce69956379a}}
We first consider the outermost boundary {{formula:e4a5ed8f-ae65-4217-8143-5f87ca7bfd0c}}{{formula:27a8e30a-3869-4d5f-8e76-1a536cf0a6ee}} . For any {{formula:1769c9a2-8790-461d-a3cf-7448ecc2259c}} , we have
{{formula:7d01d17b-1e3c-4514-a33c-082fed0225b6}}
{{formula:e6589b9f-609d-407c-ab65-9d9682ff5108}}
From Lemma REF , we have
{{formula:3821c773-fe6d-46f9-bb09-3b775d00d1b6}}
Moreover, from Theorem REF , we know {{formula:284d4bca-046d-4c66-97a8-b350648e4e20}} is in {{formula:18abe895-0398-4e51-a53e-b0319dc4268f}} . Thus {{formula:b1f4afa8-ed42-4b72-9207-44f3f44beed5}} is also in {{formula:a037c822-906a-429e-8f11-7efb36b63216}} . Then by {{cite:2b432ffb4214c1a3459eae3776fda1ba08b95c1a}}, we have the analyticity of {{formula:e9d4ca62-14e7-4c4b-84f5-5f325ade599b}} .
Now we consider {{formula:b8cfaadc-f2b5-477e-b1b7-7c35b29e333a}} . We have
{{formula:4c1dbf4a-e21a-4c7e-8367-d22052b2bece}}
{{formula:37232d36-5d7b-4baf-86c1-19fb204ac0ca}}
From Corollary REF and (REF ), at the bifurcation point, we have
{{formula:4e6f712d-ef64-4565-91c1-3b3e92763095}}
Thus when s is sufficiently small, {{formula:7aeb3ce6-1052-4f81-b731-6e5af44e99b9}} .
From Theorem REF , we have {{formula:45ba0f8d-4387-47f4-b067-b2051504c8f3}} is in {{formula:7566f391-6730-4497-bd7a-15895554a765}} . So {{formula:b8e67839-1717-466d-82e2-7c682977659a}} is also in {{formula:1830bf71-54e9-4edc-a911-0fb4a6f0a9de}} .
Then we can use {{cite:89c6ca66e5d00d0990ac7a30dfd0582997aa5c5d}} to get the result.
The analyticity of {{formula:cf4537b6-e2a0-4b12-8894-5304eb7f58fb}} can be shown in the same way by using (REF ) instead of (REF ).
{{formula:b71fc91f-e6d5-4887-960a-4386b81d7eea}}
Appendix
Crandall–Rabinowitz Theorem
We recall the Crandall-Rabinowitz theorem {{cite:a2b863d86ca4c5fc47b85aff8165df958fd409f0}}, that plays a crucial role throughout this paper.
Theorem A.1 {{cite:a2b863d86ca4c5fc47b85aff8165df958fd409f0}}
Let {{formula:6e94578f-b706-4dac-9f02-4bc8d8d8fe16}} be Banach spaces, {{formula:cc7ad85f-1d9b-4933-b2c9-474bb9cefd3e}} a neighborhood of 0 in {{formula:39e3616c-3f9a-4073-98e0-c917e7278bea}} and
{{formula:f6e6876c-4342-4fe6-a009-13dc591b984a}}
have the properties:
{{formula:1f86051a-3196-409f-9867-1c1c6a7cad7a}} for {{formula:d966122a-f52c-4f83-a3a2-ccbbe3a1b7ba}} ,
The Gateaux derivatives of {{formula:1b88ce72-8653-44be-b7c8-2d0261f3eb98}} , {{formula:272cadb9-d7fb-4658-982b-91ee37a70dcc}} {{formula:1027f820-c2aa-4386-8b95-90a9c7cf7e66}} , and {{formula:49223666-957a-4853-9379-a928d92e7c7d}} exist and are continuous,
{{formula:6af09a93-0e35-4ee1-acbe-7a8faef2fa88}} and {{formula:4bce2a66-397f-496f-aab1-0663d10b7d8b}} are one-dimensional,
{{formula:fd9f706d-b63d-4b3e-965e-dc459277a262}} , where
{{formula:2e60f3c5-6b6c-4758-ad6b-570b76561891}}
Then there is a neighborhood {{formula:59fa117c-1e25-46a4-9543-24da8cbc1c07}} of {{formula:48671137-b3a9-422e-8b14-b71d47699a63}} , an interval {{formula:fdfd5747-28b2-441f-abfe-255ad978a894}} and continuous functions {{formula:39c0a93e-70be-4a25-8762-a8ef4168f567}} , {{formula:b6ce8d07-b795-41d9-bd91-0455b501b27f}} such that {{formula:a08d2b86-16bc-4472-9ce8-7b3f3e2560e6}} and
{{formula:e95bc0da-7b41-4207-aeb2-bfd47423c060}}
Useful integrals
The following lemmas will deal with the integrals that appear throughout the calculation of the linear operator:
Lemma A.2
Consider the expansion of the function {{formula:571afc9e-b425-4b50-b102-8eb04a3652a3}} . We have that:
{{formula:6de56012-95d7-4749-a48d-9a55225e3198}}
Proof:
Follows from the Taylor expansion.
{{formula:ce2d83cf-f9e4-4a51-b84e-8cf4664c6aa0}}
Lemma A.3
Let {{formula:94118f18-05db-43e3-af98-80930f6f6e05}} and {{formula:ecf3ce9a-f032-4e6d-aa77-d6ce629986d2}} be
{{formula:95d0b91c-6d8f-4c13-a0ae-207225aa4a2b}}
Proof:
We use the residue theorem. We set {{formula:34c692af-f9b3-4c41-a8f6-6158669af415}} to be the boundary of the unit disk and let {{formula:405ea5fa-032d-4abe-80ae-a919758d0476}}
{{formula:02e20f3e-0f2f-48fb-9bbf-72d2a2c65196}}
{{formula:e9f9d965-b873-449b-bc1f-09367c2c3a39}}
Corollary A.4
Let {{formula:b53d7839-c2b8-4694-b91c-9037fa0304a6}} and {{formula:6ad3ad51-13f8-4d59-bca7-110277729485}} be
{{formula:44d3d3ff-447c-4f08-a369-489beab4a528}}
Proof:
For {{formula:0d6f8c44-e435-40b4-be57-440fc1eba5d8}} , we integrate by parts and we have
{{formula:433110b4-7e02-4059-bd31-e6c2c03b8a64}}
For {{formula:7946011a-c115-4cd3-9e43-193674c4b14c}} , we use the continuity.
{{formula:897a9511-d8f9-4f45-a7c9-440434db2e4e}}
Integral estimates
Lemma A.5 {{cite:d3680a5f688403425924d5e5d6ffae8fc3747a64}} Let {{formula:db4bc42e-96ee-4889-b5c4-5bb7006fca5a}} . Then,
{{formula:a138cae4-3330-4493-8540-db6e7d0beb57}}
Lemma A.6 {{cite:e4ea18026879b33d95dddd032766b982c8816c24}}
Let {{formula:2a001986-56a8-47f1-92b4-d57a637c7f49}} . Then for {{formula:1a4745b3-cc41-4ce7-9290-9603e4400651}} , it holds that
{{formula:b1699753-64db-4ba1-a20e-112daaa5a37c}}
Lemma A.7
Let {{formula:b407e883-a9dd-4f74-8c08-049f16b60467}} and {{formula:7ed2ede5-b7f2-43a2-9f91-7a9d7edcb356}} satisfy {{formula:c591e185-3e06-4440-b46c-786ed3ef85b5}} with {{formula:05beff76-eea9-400c-98af-25f833f5a70c}} . Then, for each {{formula:8d5e97f3-1c12-4c00-908b-984054076bcd}} , it holds that
{{formula:0c0ffb8e-6cad-4ba1-a74c-2bfebfa0d882}}
Proof:
Using Lemma REF and the Sobolev embedding theorem, we have
{{formula:a2ea383b-ab60-4f37-99d1-0db993894ffe}}
{{formula:037c9346-49c3-458b-8e85-34159ab6eba7}}
Lemma A.8
Let {{formula:a1e3b6c5-2c62-4c8c-9794-20fcb7d70a7c}} and {{formula:200f2305-2cd1-42f3-b9b5-eeb043ec2f85}} . If {{formula:4c276f98-c906-46cc-aeca-6844c0cec726}} , then for {{formula:f7810c04-9ca6-4134-b076-d31db7a17940}} ,
{{formula:fe4d4447-850e-419a-bfa0-f414976cbe9c}}
For {{formula:2ccba99e-7b7e-4dfd-aeb9-da0e205d9c7e}} , if {{formula:620e0a5b-2c5b-416a-a8fa-b61b58dfaaee}} we have that for {{formula:54c26cc9-6e05-4117-b7db-8b434ad1d472}} ,
{{formula:623126fd-8718-4041-a87c-55f0d61073a7}}
Proof:
Since {{formula:49767c80-727d-4c58-95c8-3350cf9432a1}} . Then (REF ) follows straightforwardly from Lemma REF and Faá di Bruno's formula. For (REF ), we have
{{formula:fe521c7e-2dda-4e10-a371-263f55fa470d}}
Thus equation (REF ) follows from Lemma REF and (REF ).
{{formula:403b10ce-0373-4d2e-b571-7f9180b695ad}}
Lemma A.9
Let {{formula:2b5f4494-94cd-4ff8-a5e3-b26fb4dfff4d}} and {{formula:20caedd5-c13e-4902-b3f5-ecff425c8cd6}} . Then for {{formula:8aa7e535-eb4d-4c28-b701-109a6852f6b8}} ,
{{formula:086c057e-7664-4aec-88a6-3bdfce68505d}}
Proof:
We first divide the domain of the integration into three regions:
{{formula:cb92cabb-c36d-451a-998f-fe17da61129e}}
Clearly, {{formula:2d9491bb-70e4-4867-86ec-a9ba9ccf0a0f}} . We will show only that
{{formula:8b673814-dbdc-4836-93be-5aa1d6ee2009}}
since the other contributions from {{formula:aa0ab2ef-ebf1-47aa-87a1-a58a59c8e8a0}} and {{formula:dc7a75d8-8b25-4347-9826-67d4dba8aec2}} can be easily proved in the same or easier way.
For {{formula:108eaa64-0dad-4820-ba72-766633552fa6}} , we can write
{{formula:32b81c5a-1e9c-4a57-b6dc-f339d4f2ec05}}
Since {{formula:c5e369ad-e9a4-4fd9-8592-d81fe4fe4d29}} can be extended to a smooth function in {{formula:33abf855-dba0-4f3d-a8d9-097d6747fb4f}} , we use Lemma REF and obtain
{{formula:b943fca8-de7e-4877-9245-bac858138a43}}
The right-hand side can be estimated as
{{formula:e55a5b23-a5d0-4e1d-90a7-ccdfd4640a9d}}
where the equality follows from the change of variables, {{formula:44cb0ab9-7c78-4289-9e39-b734985a3f95}} . Using the Cauchy-Schwarz inequality, we obtain
{{formula:c1610404-f2f6-4518-b385-e80531cb2956}}
where the second last inequality follows from the fact that {{formula:79042679-dbc4-41cd-90fd-b7bd45eee6b1}} is uniformly bounded. This finishes the proof.{{formula:93f7da78-d97f-4888-b1d9-2e01f2f54a5f}}
For {{formula:8cd29779-a77e-4e47-8810-d6f6877b528f}} , we denote
{{formula:bbad7fce-8519-4f50-aec1-96d41e2643a9}}
Lemma A.10
Let {{formula:a58f0a51-c3d2-4685-b69f-44bdb8ec626a}} . Then, for {{formula:5b5c0558-2599-4805-812f-3783758dc1c5}} ,
{{formula:4da73a11-79b8-4cd6-b69d-96cafb0d46c4}}
Consequently, {{formula:9a1026f5-efbb-4d83-9c46-eddd0a59ff03}} .
Furthermore, we have
{{formula:9f49c3a3-b10e-4a11-ad2f-86e8665e3c65}}
Proof:
For (REF ), it follows from Lemma REF by putting {{formula:4f55bb0d-caa8-474b-a55b-5d798babf5d8}} . (REF ) follows from classical singular integral theory, thus we omit the proof.
{{formula:6fc70e03-a03e-4555-b34b-cb2279ae7bb8}}
Lemma A.11
Let {{formula:f913b4c8-5251-46bb-853d-d26308dc1043}} and let
{{formula:6e0bc185-48bb-49fe-8933-9730f8f23850}}
Then for {{formula:8973ca74-1dc2-4000-ad5f-86670e9f0c37}} ,
{{formula:5278141d-617b-4900-a199-0af02abdeb8f}}
Proof:
By Minkowski's integral inequality and lemma REF , We have
{{formula:8efd6d3f-6b85-494e-993b-546e4ad94577}}
{{formula:0d6997eb-5ccc-4dfe-b198-9ee544737c9c}}
Lemma A.12
Let {{formula:731b3f86-122e-4b84-887a-8958023251aa}} , {{formula:25cf98c5-d04c-4793-84f9-2b19b747d18f}} and {{formula:bf6d1704-6f69-4063-8e2d-e8721117eead}} . Let
{{formula:b876b9e9-3663-4011-890f-87fd872261c3}}
Then
for {{formula:bceb61a3-655b-483d-b536-2af15ae5ba73}} ,
{{formula:4e030a78-7db4-46e0-b909-4738de693411}}
Proof:
Let {{formula:4fe933a3-2f64-4d0c-a7d2-74d7fd8d5824}} . Clearly, we have
{{formula:cc8360db-7cf5-4951-97c7-50f6f33c1352}}
We write
{{formula:8a059a30-a9b5-4645-979a-a8775e0d8a65}}
We can then get the {{formula:3d110eba-9565-42ba-b221-80ea630118bc}} estimate directly by Hölder's inequality. From Lemma REF , REF and {{formula:91477848-a271-4d12-8e37-30d7926fb5a6}} , we have
{{formula:f4a3a02d-0525-4343-ae46-9acab12f8657}}
For {{formula:2a31dd3c-b632-4423-bcc7-cde97732b2dc}} , it follows from Lemma REF that
{{formula:14aa69c7-7ddc-4d2f-bccd-01233bdf377b}}
where the last inequality follows from (REF ). With (REF ), we obtain the desired result.
{{formula:dd907d7f-6f77-4d8b-965f-6208c243f028}}
Lemma A.13
Let {{formula:b5a328b6-3e29-4780-a39b-e054d5f4d0c8}} , {{formula:8ba005a0-dcd8-4918-8a53-1211bdbbfe2b}} and {{formula:28708fa4-51ff-482d-bd19-4e4812cbd7ec}} . Let
{{formula:928a4dfe-4686-4783-9885-6a75d84f806f}}
Then for {{formula:b291a6e6-035e-4629-bfcf-438533db39e0}} ,
{{formula:792453b6-ef1d-4cc3-8a93-6b21832e60a8}}
Proof:
Clearly,
{{formula:a2f9c6f8-4131-4d6b-bc7c-62d93d9b2d8f}}
where we used {{formula:7a68896c-6ad6-4636-8409-2e07238422f4}} and Lemma REF .
Now we compute the higher derivatives. Using the change of variables, {{formula:80012d3b-88f3-4b79-ab9b-a3d5f2a415b2}} , we have
{{formula:88c7428d-8eae-4527-8883-4835d1f5a360}}
Since {{formula:02fecbb4-e1e5-4341-8e86-258221939fbe}} , we have
{{formula:f778d12b-6a4c-4f99-bc3a-faf843ce0722}}
where {{formula:6479d9dc-d274-427e-b448-8ca8b8cf266b}} . If {{formula:97b4e23f-be32-4eb2-9920-e19176570e55}} or {{formula:ba1cf9ab-3908-4dc1-b155-131453292d37}} , then we can write (assuming {{formula:7b3d2598-41c0-4085-b6e7-20698051ec5d}} without loss of generality),
{{formula:09dc6699-11e7-4ba3-a6e9-14d1d6056e9e}}
Using Lemma REF for the first two integrals, Lemma REF for the second integral, and Cauchy-Schwarz inequality, we obtain
{{formula:acbcc7c7-e16b-4d39-bfd2-2918c1adb0aa}}
{{formula:33a720ab-d509-4cf0-b3b0-1d8b5b04b392}}
and
{{formula:57aaef45-7612-4b00-930b-19e4077c77d2}}
Using the Sobolev embedding theorem, we have {{formula:353ddbdb-0c62-418a-919e-2d8d597db01a}} and {{formula:d5575366-b979-4fbb-8a86-17845a84587e}} , thus
{{formula:511ac683-ca10-4cba-a84c-551ab19d3f24}}
where the second inequality follows from (REF ) and the last inequality follows from {{formula:aa63451c-f1d9-4db4-b125-ec4f0e660af0}} .
If {{formula:a902a6b9-1871-4144-8da2-aaca4591f0cb}} , We use Lemma REF to obtain
{{formula:ba8807ba-624e-43bd-83ab-79560f740386}}
Similarly, if {{formula:b90a20f8-3562-4099-8c80-039a2effe4c7}} , then
{{formula:f4889430-6e3b-489f-8cc8-2b7b6c210359}}
Therefore it follows from (REF ), (REF ) and (REF ) that
{{formula:783f0839-20b9-408b-9c95-b5fb455858eb}}
With (REF ), the desired result follows
{{formula:e079fded-31c8-4ccd-80f8-f0a964e344dd}}
Lemma A.14
Define a linear map
{{formula:e0a128b5-db23-432f-a830-ce93dd525c61}}
Then for {{formula:420434d3-4f1e-488c-aff7-1b3bbd8ce463}} , {{formula:5fa03648-5e70-46ea-9932-6f48352e3cd3}} is an isomorphism.
Proof:
The result follows immediately from Corollary REF .
{{formula:b05bad54-3ba7-43d1-b15d-27b3ae5b501f}}
Stability of linear operators
Lemma A.15
For Hilbert spaces {{formula:e4153f14-3f79-44a1-a2fc-de97ce4edf99}} , assume that {{formula:28fdca3b-ee80-4a4d-9799-b896092c99ba}} satisfies:
(1)
{{formula:3d2339bb-5366-4e55-b090-8c35bb507ae7}}
(2)
{{formula:c6ea564c-5883-487b-9e08-6e90e8e15d98}}
{{formula:15c55309-85bc-4531-8b29-babc48ded177}}
Then there exists {{formula:d4d4f59d-9f5f-4b3e-b825-c7bae2ea9e27}} such that if {{formula:15d82157-d6eb-4de0-8038-e63426424be2}} , then {{formula:885ca288-6caa-4f03-a122-5b1fb2bbf8fd}} is also injective and has codimension one. Furthermore, there exist {{formula:1e1a722c-b99b-4b38-960b-b5693017a56c}} and {{formula:47c7d558-c52b-4513-9bd2-9e7a252365d0}} such that
{{formula:7d52a3df-5c2a-449c-8eb6-b8c9faef5fee}}
Proof:
Let {{formula:815b7f7f-aa8b-4ebc-b598-f89c44c60cf8}} be the projection. Then {{formula:7c8641e9-e8ce-41a7-8288-0e0c7c67acf3}} is an isomorphism. Therefore we can choose {{formula:36b0b768-8e81-4808-9912-1e76bbca0ae2}} small enough so that {{formula:68512d14-3c69-4832-900a-a0133659c441}} is an isomorphism whenever {{formula:c5410b19-4100-4c5e-b282-4e07de75f62c}} . Therefore, if {{formula:159c8b28-1ae3-476a-ab5e-ba59d896457e}} for some {{formula:8e330b02-830b-468e-ab6b-c22a4e12cbff}} , then {{formula:2d2c5848-e3ef-4b5b-b648-3ac9ebdaa7b4}} , which implies {{formula:005afa1b-aa37-403e-96e0-1949a56481d6}} . This shows {{formula:a2f9666d-5dbe-458a-a48e-1643bc31fb28}} is injective. In addition, the Fredholm index of {{formula:8702898b-d760-4d6e-a4f1-326a9179b9eb}} is {{formula:8fb82fee-c1cd-4c3d-b380-a802e4e05b08}} , therefore the Fredholm index of {{formula:c3979621-1abb-44ce-93b9-856f7c6ee6bd}} is also {{formula:56e9e1c4-46f3-4000-97f0-148bb649c912}} ({{cite:6ce054570be23a243badacdf9ae5c80c28ef422a}}) for small {{formula:46087f5f-65a7-491b-a9d9-98559e3e28b9}} , which implies {{formula:b46790c6-7c1e-4f61-8ff6-498bfebd5291}} has codimension one. This proves (REF ). Furthermore, {{formula:17f29838-e312-4394-83b1-ed6c4e75859d}} is an isomorphism, therefore () and () follow immediately. We remark that the constant {{formula:08440e88-2828-4d6e-91f5-f16aecd7b90c}} can be chosen uniformly in {{formula:9955751e-44ae-40ec-8a7a-18efb312c13a}} .
In order to prove (), let us assume without loss of generality that {{formula:afa20024-595e-4e7d-aaf9-4669693b16bc}} . For any {{formula:e3962642-afc0-40f7-a8a6-f375bf2d90f2}} , we have
{{formula:633fd3f5-7bc7-41d1-98ea-992b1cce6115}}
where the last equality follows from {{formula:52d71990-7d10-4488-8621-e1eb6970b1dc}} and {{formula:43f76785-d9d4-4728-aaf2-ae969060763f}} . Therefore, we have
{{formula:e40b345f-6967-4411-9c24-674a9feda653}}
Note that we can decompose {{formula:1287e3a5-8d52-455c-9af7-1a66ad9f2da3}} and find {{formula:647429e7-c137-44bf-94fe-6a1da83f71bd}} such that
{{formula:1b81c989-1256-4e00-89a8-8ad728cfe7b6}}
where the last equality follows from {{formula:8610a5af-f647-4f8d-9df7-af413f1ad58e}} , hence {{formula:d4fafb78-7fb4-4b10-9132-229a45d0fa8a}} . Then, we have
{{formula:ff68aceb-8d12-4de4-bb0d-699f47bc76a6}}
where the second equality follows from {{formula:64d9831c-aab8-49d9-9713-54cab98a1111}} and the first inequality follows from (REF ).
Hence
{{formula:88647df3-309d-4875-a7a7-1d21684e2bec}}
Now, using {{formula:c2bbff34-5980-4c4c-8b4e-4095c47dd280}} we have
{{formula:c91a45b2-ac06-43fa-9851-af12ac0119b8}}
where the last equality follows from {{formula:88dab05e-088b-4f27-b945-350aec69629a}} . Hence, we have
{{formula:2c65155f-f78d-4ee8-aa4b-6d1554e14156}}
Therefore we have
{{formula:db02fde8-e5c3-4556-b8d2-edb702e3052b}}
which implies
{{formula:b6f44a2f-f290-48a8-8ff7-5bb05d029a11}}
where the first inequality follows from (REF ) and the last inequality follows from the Lipschitz continuity of {{formula:182006a0-6a93-4c34-968b-db6f5e22d495}} . This proves ().
{{formula:0f0ddc8e-6e4e-41e0-98f7-fcee7d638e48}}
Acknowledgements
JGS was partially supported by NSF through Grant NSF DMS-1763356, and by the European Research Council through ERC-StG-852741-CAPA. JP was partially supported by NSF through Grants NSF DMS-1715418, NSF CAREER Grant DMS-1846745 and by the European Research Council through ERC-StG-852741-CAPA. JS was partially supported by NSF through Grant NSF DMS-1700180 and by the European Research Council through ERC-StG-852741-CAPA. JGS would like to thank Francisco Gancedo for useful discussions. We would like to thank Claudia García for mentioning the references {{cite:2b432ffb4214c1a3459eae3776fda1ba08b95c1a}}, {{cite:89c6ca66e5d00d0990ac7a30dfd0582997aa5c5d}} to us and Yao Yao for all the conversations on this project over the last years.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the program `Hamiltonian Methods in Dispersive and Wave Evolution Equations”.
{{table:6f9f236c-4ed2-42ff-8f90-4c71092de87d}}
| r | b80a02f3ec18a5a87de9e6b72cae1d09 |
For the purposes of ML model training, patient outcomes are often defined based on laboratory/diagnostic test results extracted from the electronic health record (EHR; e.g. {{cite:e4116831e82c8f7ee16de61207b62d4ae414617d}}, {{cite:01a93e6d14d923d539aae43a3475e18e8a8acac4}}, {{cite:2726008531727d60030dbe5ac27c898f07006bad}}), since clinical chart review on large patient databases can be prohibitively costly.
In doing so, many researchers assign “negative" labels to untested patients (the negativity assumption in positive-unlabeled learning {{cite:b140ffa2e4ef972aa6b826c8b550feb748edaccd}}). For example, many sepsis prediction models derive labels from laboratory test-based definitions, such that untested patients are negative by definition {{cite:786d2b9d2542cb4a04100888d48cf7a73c03815e}}, {{cite:3020e61bb204616cefacf9ab499f998e2e4e0f86}}, {{cite:e69aa2fb690f0f127018c340035b02b4c034a96f}}, {{cite:723c8e0a696e0d479570539fbe2b961e79fae46c}}.
Beyond sepsis, this is also the case when building models to predict healthcare-associated infections {{cite:40a90fb61866b5c03bd0b6a9ec46bb21f5819b00}}, {{cite:a0260c3251fc484580a93d882071aa8c05883803}}, {{cite:1639461d3ba431449f2af4c4163115e68d67662a}}. Researchers typically justify this assumption, since without it, a model trained on only patients who were tested may only apply to the small fraction of tested patients, limiting its utility.
| i | 39b790707240efa1c11496758bc30a7d |
Note that for all {{formula:79e8c5c0-504d-42c3-b7ae-bdd2d1b314fb}} , the second order partial derivative of the functional {{formula:d0e399e5-c745-43a2-b0cd-bc07760acfa5}} satisfies
{{formula:28fce786-7a25-4db7-a934-5d275a6b19a6}}
which shows the optimization objective is convex. Moreover, since {{formula:b0aa3c6d-6e0b-400c-9343-44fe356febc1}} is strictly convex, the Bregman penalty term is also convex. This implies that the optimization problem we are solving is convex and the convergence is guaranteed. {{formula:9ca1a1ff-4d94-4883-9b95-5e16fee82c8d}}
After several iterations, the local belief {{formula:a6e67c36-2b71-469d-9ea8-c6f459d5f9ea}} , {{formula:9763a157-d74a-4e21-9860-2d91375da795}} characterized by {{formula:5c3aeba8-b89b-4e0f-8232-b8c1e863b3c9}} is guaranteed to converge to the global message {{formula:a3c0e8f2-b79f-47ce-a81b-4726f77c967b}} . Compared to belief consensus-based method, it can be seen the Bregman ADMM-based algorithm requires to transmit an additional variable, which doubles the communication overhead.
If the inter-user links suffer from the additive noise, the updates (REF ) and () can be interpreted as stochastic gradient updates, whose variances have been proved to be bounded values {{cite:5ef6c70cff4a1201f6cf001e9c2481091cffd5a5}}.
Algorithm Summary
For the distributed cooperative detection, we assume that each user has obtained the message {{formula:c050224c-32f0-4dd7-a05d-dd4ac5abbb9e}} based on its local measurements, and the goal is to obtain the product of all users' messages distributively. To start with the distributed algorithm, the local belief {{formula:7e23e15d-ade1-4241-a1e3-4266bbb44b1e}} is initialized as {{formula:4580e927-4bce-4d9a-bd6c-d35c3ba38e43}} . Then according to belief consensus-based method and Bregman ADMM-based method, all users update its local belief to reach agreement on the global message. Also, with the advantage of Gaussian distribution, only few parameters are exchanged in cooperative detection. For both schemes, the complexity is {{formula:e2363460-1cac-4650-9cae-9ab1598f4c47}} , which linearly increases with the number of users, making them attractive in practical applications. The proposed distributed cooperative detection methods are summarized in Algorithm 3.
[t]
Belief Consensus and Bregman ADMM-based Methods for Distributed Cooperative Detection
[1]
Each user computes message {{formula:0b768e36-f58a-4d6e-b5f0-b7506f3cce3e}} based on its local measurements.
Enter cooperative detection
Initialize {{formula:38bda381-1af1-4ee9-9dfb-ec83e8a7896f}} as {{formula:6f2acde3-0005-40e2-b90b-9a2bfe1991e4}}
p=1:{{formula:b9073e8d-d4f5-48c7-8403-705a652710fc}}
Each user broadcasts the parameters {{formula:a08d67eb-37d6-464a-bf73-f9159789ebce}} (Belief Consensus) or {{formula:7ba62ea9-fc25-4d1c-a3f6-73b0b81c95fd}} (Bregman ADMM) to its neighboring users;
Each user update its local parameters using (REF ) (Belief Consensus) or (REF )- () (Bregman ADMM) ;
Calculate the message {{formula:a70e08d2-ccf9-4daa-91e0-253bd657694e}} at all receivers;
Exit cooperative detection
Computes other messages on factor graph with {{formula:17c3018d-e4ab-48b0-bc09-eaacbbbf796e}} .
Simulation Results
We evaluate the performance of the proposed receivers by Monte Carlo simulations and compare them with several state-of-the-art methods. Consider a MIMO-SCMA system with {{formula:d7eda718-c0d8-4e5d-bb42-36678cc7773b}} antennas, {{formula:ae01d707-3330-4182-afe8-f47143618fa9}} users, {{formula:5e3be212-9c26-4e07-895d-696f857fea52}} nonzero entries in each codeword and {{formula:8b79bcb3-0127-4539-b3ac-7a5df87666cc}} . Therefore, the overloading factor is {{formula:27526b25-3985-4673-a1b5-69ae3e9800af}} . The SCMA codebook is designed according to {{cite:431886b5eba1ce48ca360cbb01f6451dbd437721}} with the indicator matrix {{formula:49d5260d-25f9-4c16-9468-a2516a722cd9}} defined as
{{formula:acb24c33-06d7-405a-9a88-0f5aad808013}}
A 5/7-rate LDPC code is employed with variable and check node degree distributions being {{formula:9f5c20a3-dbe6-4b63-bbdb-f4d92e4c1db4}} and {{formula:5bf588f8-1265-48bf-8b05-39a906ed7970}} . Quadrature phase shifting key (QPSK) is utilized as the mother modulation scheme.
The channel is frequency selective with {{formula:69634f86-6ddb-4856-8743-12797127d721}} taps, and each tap is independently generated according to the distribution {{formula:77f89d06-7573-4316-9e1e-876a2a052478}} , {{formula:1944298b-eb24-41b7-b6fd-965b6cee6862}} . The normalized power delay profile is {{formula:6c8c70a2-1ac9-45b3-90c5-721e51992dc6}} . The maximum number of iterations is {{formula:0dd136ad-e59b-4ea3-8050-03729dc5776c}} . The simulation results are averaged from 1000 independent Monte Carlo trails.
{{figure:998ebd11-aed1-48ee-bba5-3263898a84f7}}In Fig. REF , the bit error rate (BER) performance of the proposed stretched factor graph-based BP-EP algorithm (denoted as `Stretch-BP-EP') in Section III.B is plotted. For comparison, we also include the performance of the minimum mean squared error (MMSE)-based method, Gaussian approximated BP (denoted as `GaussAppro-BP') algorithm and a combined MMSE-PM-MPA algorithmGaussian approximated BP is also based on the proposed stretched factor graph. However, the extrinsic information of data symbols are approximated by Gaussian directly, instead of approximating the belief as that in EP. The combined MMSE-PM-MPA receiver first performs the MMSE-based MIMO equalization and then use PM-MPA {{cite:9efa011c524005c0e12fa06b0c578b912f65161f}} for SCMA decoding.. A {{formula:0f5a9612-3467-423c-baef-89c47fd23754}} scenario in which information of different users is transmitted using different antennas is also considered as the performance bound (the coding and modulation scheme are assumed to be identical to SCMA). It is observed from Fig. REF that MMSE-PM-MPA method suffers from significant performance loss. This is because that MMSE detector can only output hard information for the PM-MPA-based SCMA detector. The proposed Stretch-BP-EP algorithm slightly outperforms GaussAppro-BP and performs close to the MMSE-based method. However, the complexity of the proposed algorithm is significantly lower than that of the MMSE-based method. Moreover, the proposed SCMA system has similar performance compared with the {{formula:0f019cea-ebfa-42a6-8f29-c047625d26db}} scenario, while the former is able to support 50% more users.
{{figure:e3a09065-fdfb-4f30-93f3-c83911802793}}BER performance of the Stretch-BP-EP method and the proposed convergence-guaranteed BP-EP (denoted as `Conv-BP-EP') algorithm are compared in Fig. REF at different values of {{formula:1a74563f-eaa2-4882-9fd4-ccc216273c85}} . It is seen that performance of both algorithms improve as the number of iteration increases. After several iterations, the performance gain of both methods become marginal. By comparing Fig. REF with Fig. REF , we can observe that Conv-BP-EP algorithm converges faster than the Stretch-BP-EP method.
This results can be explained by the fact that Stretch-BP-EP algorithm may converge to the local minima of variational free energy while Conv-BP-EP is guaranteed to converge to the global minimum, which demonstrates the superiority of the proposed Conv-BP-EP method.
In the following, we evaluate the performance of the proposed distributed cooperative detection schemes. Consider six users uniformly distributed on a {{formula:935285e1-d66b-48a3-95a9-a8904b944f16}} unit square. Two users can communicate and exchange information if and only if their distance is less than {{formula:64a6191d-06f7-4f84-a69e-4edc622da6e3}} . The channels between users are modeled as AWGN and set to be the same for all links. The vanishing parameter for belief consensus is set to a typical value {{formula:3d59918c-0aa6-4ed0-85ea-e08d0239b7de}} .
In Fig. REF , performance of the proposed two distributed cooperative detection schemes with perfect inter-user links are evaluated. As a benchmark, the BER performance of a centralized scheme is also plottedRemark that only the measurements from connected users are collected at a central unit for fair comparison.. Two values of the number of consensus iterations are considered, i.e. {{formula:2820c455-1026-46c3-a534-d5d770c6d82d}} and {{formula:9daa89f6-308e-42d3-b45e-51a233bac409}} . For comparison, the averaged BER performance of all users based on their local measurements as in Fig. REF is also included. It is observed that, by performing cooperative detection, BER performance can be significantly improved, which reveals that diversity gain can be achieved by exchanging information between neighboring users. By comparing the belief consensus-based method and the Bregman ADMM-based method, we can see that, with perfect inter-user links assumption, both methods deliver similar BER performance at {{formula:0358fae8-0399-44e0-b8f9-9ebf336603f6}} and {{formula:0d365d9c-7e5a-449e-9fdc-9d732c9559ac}} . Moreover, after 10 iterations, both methods attain the performance of centralized processing.
{{figure:c980fed8-304d-4d05-99e8-72dfa1479e04}}Since the maximum communication range of inter-user link is critical to the power consumption of users, we compare BER performance of Bregman ADMM-based algorithm with different communication ranges {{formula:9a7df33e-80a2-46fd-9c61-5c98bad6386d}} , {{formula:886793be-c389-4fc6-a708-b37e331a8d84}} , {{formula:fd7c04d9-b11a-44cc-9173-792e88b605dc}} , {{formula:1b907aa4-291c-4de6-bc5f-fa5cbf894a9a}} and {{formula:871387b7-3893-4520-8756-690d140e0d34}} . Obviously, increasing {{formula:3da094a4-f940-4967-82cc-605e2a312162}} will result in more neighboring users and higher probability of being a fully connected network. As shown in Fig. REF , BER performance improves as {{formula:c6b85380-4f98-4223-8b11-4d6340ae59fe}} increases. However, the performance gain becomes smaller when {{formula:fa132dd1-8b23-438a-9979-d3efc8506edf}} is large enough. Considering that the power consumption will increase exponentially as {{formula:3dac47c0-d891-48a2-b071-9bd0924db196}} increases, we can trade off between the power cost and BER performance in practice.
{{figure:cdb90f83-d62e-4576-909c-3bf1e089a8ce}}{{figure:68787cb6-8a9f-4d9e-82db-073d51ff26e3}}{{figure:40d277ed-8551-4203-b805-fe71555aff76}}We further evaluate the performance of the proposed distributed cooperative detection algorithms in the condition of noisy inter-user links. In Fig. REF , the BER performance of the proposed distributed algorithms versus {{formula:e21d3d5e-1479-4a8a-ac38-5d70ba5b3821}} is plotted, where the SNR corresponding to the inter-user links are set to be 10dB.
The number of consensus iterations is {{formula:109f0c3c-4434-42c3-bdba-fed1fdca3285}} . It is seen that due to the noisy inter-user links, the performance of both distributed schemes at {{formula:2d8144cc-b74a-4acf-b93e-e9b905030eef}} cannot attain that of the centralized scheme. We can also observe the Bregman ADMM-based method outperforms the belief consensus-based algorithm. To further analyze the convergence properties of the two distributed schemes,
in Fig. REF , the mean squared error (MSE) of local parameters {{formula:ba616e70-74b6-4519-bf1c-77c25de7b78d}} versus the number of consensus iterations is illustrated. The MSE is defined as {{formula:bf2b55d5-9197-4f2f-a4a9-169642c96fe9}} where {{formula:16c30bb0-54a3-4f07-b1c8-e64a15b6ae8a}} . Three SNR scenarios of the inter-user links are considered, i.e., {{formula:e0c4566b-6e23-4d08-8b81-99b8bd927c8d}} dB. As expected, larger SNR leads to smaller uncertainty and the MSE performance is better. Due to the vanishing factor {{formula:893fdf4d-8360-41e9-a8d4-840d33e02099}} , belief consensus-based algorithm converges slower than the Bregman ADMM-based method. Note that the MSE performance gap between them becomes even larger at higher SNR. This is due to the fact that belief consensus algorithm uses the same vanishing factor at high SNR while Bregman ADMM-based method benefits from small noise variance. Therefore, Bregman ADMM is more efficient in noisy inter-user link networks.
Conclusions
In this paper, we proposed factor a graph-based low-complexity message passing receivers for MIMO-SCMA system over frequency selective channels. Since the direct factorization of the joint posterior distribution leads to huge complexity in message updating, we introduced auxiliary variables and constructed a stretched factor graph. EP was employed to approximate the messages of data symbols to Gaussian distribution and a hybrid BP-EP receiver was proposed.Considering the poor convergence property of the standard BP on loopy factor graph, we proposed to employ appropriate counting numbers to convexify the Bethe free energy and derived convergence-guaranteed BP-EP receiver.We further considered a cooperative network and proposed two distributed cooperative detection schemes, i.e., belief consensus-based algorithm and Bregman ADMM-based method. The proposed iterative receivers were evaluated by Monte Carlo simulations and compared with the other schemes. It was shown that the proposed Stretch-BP-EP receiver performed close to the MMSE-based receiver with much lower complexity. The proposed Conv-BP-EP receiver outperforms the Stretch-BP-EP by improving the convergence property. Compared with the orthogonal multiple access counterpart, MIMO-SCMA system with the proposed receivers was shown to be able to support 50% more users over frequency selective fading channels, with negligible BER performance loss. In cooperative networks, it was verified that BER performance could be further improved by exploiting the diversity gain using the proposed two distributed cooperative detection schemes. Moreover, compared with the belief consensus-based algorithm, Bregman ADMM-based method was shown to be more attractive in practical noisy inter-user links.
[Derivation of Messages (REF ) and ()]
Solving the optimization problem (REF ) yields the corresponding beliefs as
{{formula:dd353390-c257-48a1-b8a5-cd61531ba993}}
For clarity, we make the following definitions: {{formula:f13ed82e-c04a-4bdb-853b-c62837d7bd42}} , {{formula:a2ce5b34-a078-4b85-add5-15c13ab6ee9c}} and {{formula:85f4c815-22a2-4d7d-b1f0-9ef5b1c9c6d9}} . Then we have
{{formula:ab58333c-7875-4f35-9c0b-aa8ede56ba0e}}
Substituting (REF ) and () into (REF ) and () yields,
{{formula:4c0870f1-852f-4d24-b2b2-9d1ecd131d20}}
Then we define two auxiliary messages as
{{formula:9a5d79db-4dba-4d71-a84a-64125717447a}}
From (REF )-(), we have
{{formula:468f45e1-7ce4-4318-8913-b1bb9a164399}}
Comparing to {{formula:8ed635d6-08a6-4b5e-bebd-16b1d3c8adaf}} , we have
{{formula:0b916594-6c8a-49ca-9581-2a69f1c2a9df}}
and then
{{formula:2c0ed792-11f8-45cd-ab36-958ed27e1d10}}
Based on (REF ) and (REF ), we have
{{formula:11bce909-5ef1-4b85-8ae9-5b8025dcc875}}
Finally, we substitute (REF ) into (REF ) and obtain
{{formula:018b8661-17bc-4424-8d4e-f860feb629c7}}
With the definition of {{formula:746a0b7c-105e-471a-a4d2-46da752a8935}} , it is easy to see messages (REF ) and (REF ) are the same as (REF ) and ().
| m | 57d22697b40d926d7968e231fa7fbaf4 |
For the PB calculation with MG5aMC+CA3(Z+1)NLO, we set {{formula:94abc278-b82c-4c4d-9b84-fe57f7c12ef3}} GeV, {{formula:1d4f6b67-0853-4f05-b86d-c6798131ea9a}} GeV, {{formula:1f0a22a7-7f32-42d3-90ed-e657f687e720}} and {{formula:13e1be14-b2c5-44dd-af9a-d92adce81dcd}} , where the sum runs over all particles and parton in the matrix element. The hard process calculations are performed at NLO with MadGraph5_aMC@NLO {{cite:b5e970c16854b7536fd301ae5b615100d51c6e94}} with herwig6 subtraction terms and the TMD parton shower is simulated with Cascade3 {{cite:682d11b9355843a425e2d8a54d49cfa207657b1a}}. The theoretical uncertainties are obtained by varying the scale of the hard process is varied by a factor 2 up and down, provided by MadGraph5_aMC@NLO.
| r | e364af3e167c08f37f5435c8a796156b |
Finally, in the hate dataset the interleaved model is able to reach an AUC of 0.92. It is the only case where the precision and recall is similar to baseline, the one presented in {{cite:2caddf6510faf26fd13de9e4c33cce8191ed7174}}, but our AUC and accuracy scores still outperform this baseline. This is due to the fact that we could not find any user or network metadata related to this dataset and, therefore, our classifier is only using the raw text and the tweet-based metadata.
| r | 9b68f2af6a2c3d58f57a5fe198b17679 |
A foundational principle here is that exploiting subgraphs confers graph classifications models with both the ability to fit the training data and extrapolate to graphs generated from a different environment. As detailed in Section REF , this insight follows from the Aldous-Hoover representation exchangeable distributions over graphs {{cite:b967b906c57e21f060becc4287bd4a9186547232}}, {{cite:2e31af147d749f925c416c56af7054ac3cf4d5c2}}, {{cite:d35bd31f02a03cb15d3833a5c1ae46817f89c45e}}, {{cite:6578112e54c9e9991c30ab326d0943577691f7f2}} and work on graph limits {{cite:780fe861e86c16cf24f5d685eabf46d5a0d0e2a5}}. We discuss the large literature using subgraphs in machine learning.
| m | 4bd61510a28b42fd6d9a24d04c8181ca |
Figure REF shows sample results by VisDB and I2L-MeshNet {{cite:04e923a5249019cdfeb09e40ed10820953be887c}} on the 3DPW dataset {{cite:ebcb65b9a20110e9ff28d7469b96a4cbaf0634bc}}.
I2L-MeshNet {{cite:04e923a5249019cdfeb09e40ed10820953be887c}} regresses SMPL parameters from the entire heatmap-based mesh output, which leads to erroneous output meshes on truncated or occluded examples.
VisDB predicts accurate vertex visibility labels, improving both the image-space dense body estimation and SMPL parameter optimization.
The results show that VisDB (mesh) outputs can fit the human silhouettes faithfully, and VisDB (params) further regularizes and smooths the mesh surfaces.
More qualitative results are shown in the supplemental material.
| r | baecf1c5043aa435f4033af5722bf2f2 |
When {{formula:3c935c8f-1314-43ab-89d3-27010012f358}} , then {{formula:db734ec2-8aaa-4719-8484-00dd299e082d}} is a trivial
solution of Eq. (REF ), but this vanishingly small
fraction of seeds cannot trigger global behavior adoption because
{{formula:6f43abaf-5bdc-4b0e-a247-92d41455537e}} {{cite:902079d9d872bd25202056daab8ff2b10f24ccc3}}. To stimulate global behavior
adoption, we must have a finite fraction of seed individuals. Here
{{formula:e3df55b9-f009-4296-b79a-54b74dc6564b}} is no longer the solution of Eq. (REF ), which
now has either one or three fixed points (including multiplicity).
If Eq. (REF ) has only one solution at all values of {{formula:554ca558-1c07-4842-a839-7c9980b99ada}} ,
then {{formula:fccbdd4c-ae15-4396-af67-6ec6eb7d966b}} decreases continuously with
{{formula:70f4b70d-7815-4d10-867d-243b4027a6eb}} , and this leads to a continuous growth pattern in
the final behavior adoption {{formula:66fc6248-2c67-4dca-8510-8e708c2d1d73}} . If the number of the solutions
of Eq. (REF ) varies with {{formula:d91fb317-a646-4694-830d-1a747d316ad7}} , the situation is
different. For a given {{formula:a385b7a9-e6bd-4963-ba83-d00e7e75e22b}} , if there is only one fixed
point of Eq. (REF ), it is the physically meaningful solution. If there
are three fixed points, which are stable, unstable, and saddle points,
only the maximum solution is physically meaningful in our irreversible
behavior spreading dynamics when we randomly select a relatively small
fraction of seeds, since the individuals in the adopted state
persistently transmit the information to their neighbors, and
{{formula:84e7f6e7-f70a-49dd-89d9-f29fb94565d7}} decreases from unity. Thus a saddle-node
bifurcation occurs {{cite:f8ec8ef308a9957390e5e049a701d24610f11e80}}, {{cite:a5258a74d444453a42658dde31d4513188de3bc7}}.
Through a bifurcation analysis of
Eq. (REF ),
we find that the system undergoes a cusp catastrophe: varying
{{formula:b3dd2fe2-60fc-4b17-96d4-1b11ab20844b}} the physically meaningful stable solution of
{{formula:a363155b-7267-4ccf-b80a-57588bbdd00f}} suddenly produces a different outcome.
Therefore, the
growth patten of {{formula:5690324e-9d03-4e0d-a66a-55d80f3655b1}} will be discontinuous because a meaningful
solution decreases abruptly at such critical conditions as the critical
information transmission probability {{formula:1553527c-7b27-48e5-a618-9d0b971f85c0}} and
the critical CCA probability {{formula:1e7855b0-d055-4a84-b2fe-4e9b4f6ed689}} .
| m | c34de6cd801bcab65f61ef7f4c45c1ce |
A desirable interactive segmentation tool should 1) achieve accurate segmentation results with as few user inputs as possible, leading to reduced burdens on the user; 2) have high efficiency so that the user can get real-time response, even when dealing with volumetric data; 3) generalize well to different objects so that it is ready-to-use for new objects or image modalities. However, existing interactive segmentation methods rarely satisfy all these often competing requirements. Many traditional interactive methods use low-level features (e.g., gray level or color distribution) for image segmentation {{cite:8e6498d90a5bf9aa02859cb33e5512786c375c7e}}, such as Graph Cuts {{cite:f99cb775214a4cc42aff78c68957cd58be0a0593}}, ITK-SNAP {{cite:2e6486d5931700df9a2f17ec7203350f48a8684a}}, GeoS {{cite:722b38205b93d078a760d2aff819052365b98c09}}, Random Walks {{cite:548d3891b2cac70ae652653fa811f3bba1ffedf4}} and GrowCut {{cite:e35d7b42d3e1a9bd72b4044eb8c7c6d6188d7ee7}}. As low-level features cannot effectively distinguish the object from the background in many situations with low contrast {{cite:8e6498d90a5bf9aa02859cb33e5512786c375c7e}}, these methods often require a large amount of user interactions and long user time to obtain reliable results. To reduce the amount of annotations required from the user to build an adequate foreground/background model, machine learning has been widely used to perform interactive segmentation. For example, SlicSeg {{cite:c8fcf2316759d6cd6093c835cf65efeae61125fd}} and DyBaORF {{cite:36e948b21abcaf23c5b4ddf34e3dda0ca7757edd}} use an Online Random Forest (ORF) to segment the placenta from Magnetic Resonance Imaging (MRI) volume. GrabCut {{cite:1d7c51fd62773437d63581fee72c33176fc8c398}} uses Gaussian Mixture Models (GMMs) to estimate the foreground and background distributions. It obtains an initial result by a user-provided bounding box around the region of interest and allows additional interactions for refinement. {{cite:950db704bea74de0aaefda76c45dddab23cd3f77}} used active learning to actively select candidate regions for querying the user to obtain much informative user feedback and thus reduced user interactions. These algorithms perform better than traditional methods without machine learning, but they are limited by the use of hand-crafted features {{cite:945443879c82755fa9ad85b9a9c5dcbcbcb514b1}}, {{cite:bba3333eaaeda98aa53411b5caf8c345ba77f57e}}. As a result, they still require a considerable amount of user interactions for accurate segmentation.
| i | d01207dc3d559bf28965d735f7527126 |
The location of the center of 'island of stability' and hence the next magic
number for proton beyond {{formula:288915e7-b5ea-4376-950f-2cc1a3aecf4f}} Pb (Z = 82, N = 126) in superheavy mass
region is debated since the prediction of the existence of long-lived
superheavy nuclei in sixties by {{cite:15247be6913bdce0ee7d64eb140e3c0f94543247}}, {{cite:6122bd027c2f38b9e2cbacc75025c5ae44be9170}}, {{cite:66a56f0dc6f7fb91a16ec7b9809911c120d1da27}}, {{cite:9317cfb0df3ce1bcf1ff7b7d90e13e1cc1da49f0}}, {{cite:0f25224aa6057922614613c6e648f5f918640bd9}}, {{cite:fe49c9baa2946a93c4b92b9544e1e68346acb27e}}. Since then a significant progress has been made in
the discovery of superheavy nuclei {{cite:5e51420c161074af4555299e016e184974240d2c}}, {{cite:cce717619a258f8d2c6ae29ef9b80bffcc38aad4}}, {{cite:8b6f640047d6532604c15aedb65c2da21c817ab9}}.
Experimentally, the elements up to Z = 118 have been synthesized to-date,
with half-lives varying from a few minutes to milliseconds
{{cite:cce717619a258f8d2c6ae29ef9b80bffcc38aad4}}. Recently, the nuclei with Z = 104 - 118
with mass number A = 266 - 294 have been detected at Dubna {{cite:3ebda1a88bdaba75c65eee2aef923e0a83073c59}}, {{cite:41a2e089d8d2149c54b16f4ca81d0bf0895c8731}}, {{cite:674cd41192312f5ddde03a7d93ab497d75b4211e}}, {{cite:afd628da29f0c88ecb58777c76b98a5282ccd42b}}, {{cite:666ae3409105ee9f1ddac1882ab1c529e72f518b}}, {{cite:f554f25aaec12558feead0e759003ce745e47a96}}, {{cite:ccea23a81195916cd88062bbd534995a5c19a41e}}, {{cite:d55aab08d7cecbdc3f3cbb605d6b3bdd810d3474}} using “hot fusion” reactions
with the neutron-rich {{formula:017457a1-137e-4f7e-8b57-cbae48846fe2}} Ca beam on actinides targets.
These measurements show the increase in half-lives with
in neutron number towards N = 184 give indication of stable center.
In more detail, the cold fusion reactions involving a doubly magic spherical
target and a deformed projectiles was used at GSI {{cite:cce717619a258f8d2c6ae29ef9b80bffcc38aad4}}, {{cite:272e713b7510c1cbbcec30c482d7c118842e89e7}}, {{cite:5ce9c07f9db6b88ed45cea7245a07937893ae43a}}, {{cite:5e51420c161074af4555299e016e184974240d2c}}, {{cite:6e6d0246044b4cc52c9a81b8c32b46c6647d6710}}, {{cite:a6ccbcaf65ca4e7400b31f06e1df1deb20e7d7ef}} to produce heavy elements upto Z = 110 - 112.
At the production time of Z = 112 nucleus at GSI the fusion cross-section was
extremely small (1 pb), which led to the conclusion that reaching still
heavier elements will be very difficult. At this time, the emergence of hot
fusion reactions using {{formula:a98ade99-fadc-4b08-b4b7-a1e2b812cf6e}} Ca projectiles at Dubna has drastically changed
the situation and nuclei with Z = 114 - 118 were synthesized and also observed
their {{formula:aa0febaf-bf78-4125-aeb4-ce27a6116f7f}} -decay chains. The element Z = 113 was first reported by
Oganessian et al. {{cite:afd628da29f0c88ecb58777c76b98a5282ccd42b}} and then using cold fusion reaction
confirmed by Morita et al. {{cite:a6635df2a3e4881d91156ba0f73c6fa11ced94d6}}, {{cite:144bc644d656bfc50ed1f94a35c4e95a6710fb7f}}.
| i | 0be0e8f58ae508ae2d1658af216a323d |
Recently, many studies have focused on leveraging the rapid advances in ml techniques and availability of data (e.g., high-quality observations) to develop more accurate models.
Several main approaches that best fit different applications have been pursued, including learning fully data-driven (equation-free) models {{cite:5addf6572c13870de17adf2f4c40c64feda7aaa6}}, {{cite:385b87f35b17d056cac70b202f518d7ba51e4301}}, {{cite:eaf1d387a5719e9b6a9ba0275f88d82f4f6101d2}}, {{cite:9c78aae7634470ca2c057e78a43bed880be0a817}}, {{cite:ac8a50d9e8b4023e72987bc426517df3c38d1fa6}}, {{cite:3d7b4c9f5e27c716e4057434e082c0dfbd69d4a3}}, {{cite:a999d76fc5153d6354710cae833c6d2b6debdf29}} or data-driven subgrid-scale closures {{cite:972f99518acc6365fe1909c4f46dfb4ac8cf6920}}, {{cite:4f87f4938772f1c571bd4b2e1dde0c281e1d115b}}, {{cite:4631e63a643513ec887854b7024b3430cdd2c522}}, {{cite:2bfc45e98b55bce521fad508dc92479808741ff8}}, {{cite:15ab38e563afb83a7f7ae6200e84c288a6297cf1}}, {{cite:d060aaa038da7459366a4f614e4b4fa10c0c4d78}}, {{cite:0b034664078e5e621cfdf8866fc7d82e7389869c}}, {{cite:8870a7f834c5bb5d958a95b68af5fa812548bc13}}, {{cite:dac92dd9d4d19df62cb2ab47fcc1fe72c17f61a4}}. In a third approach, which is most directly focused on reducing the model error, corrections to the state or its temporal derivative (tendency) are learned from deviation of the model predictions from the observations {{cite:84cd224c9aa2243bfb8a5a9b3aba088ce375b87d}}, {{cite:c2d66db7cca740045bd3a43e5605c1ac29c7d4c1}}, {{cite:45c5635eae3a853cfd16cec3ba5eea4102a9f0d5}}, {{cite:ea26359292d80c72d6748cd8262ac85f9ec8349d}}, {{cite:c609515995788975cee6fd8dfdc2d46ebc18699f}}. More specifically, the model is initialized with the observed state, integrated forward in time, and the difference between the predicted state and the observation at the later time is computed. Repeated many times, a correction scheme, e.g., a dnn, can be trained to nudge the model's predicted trajectory (or tendency) to that of the system every time. To deal with observations with measurement noise, particularly in the third approach, a number of studies have integrated da with dnn {{cite:4d9ed033f350073977a77ad85c3ebcb522b458fc}}, {{cite:17ca12743feb15ea27ae4d64230c742fcca07bd8}}, {{cite:b798bcba37f1f14bb362b37cd2ab085064307bb3}}, {{cite:0296eeed74f77e44beb4627801b1278f10c44dac}}, {{cite:0e14c101195fef030c589ff8429205ddd0772acd}}, {{cite:8886613bc0086d3a7bab5ebeb1de185bef5d3e0e}}.
| i | 93b00295ea9abc3db6a7f983c0516c36 |
We simulate our model on a two-dimensional lattice, with global mixing. We implement mixing by allowing individuals to exchange places within the network at rate {{formula:9337d648-9ffd-45e4-a518-08474bcecfd7}} (relative to the update rate). The importance of mixing in sociological and ecological studies has been demonstrated in other contexts {{cite:73f2ad153de0b7a1dcd1f116704491f2eeef3d2a}}, {{cite:53e0e0af4ad6dae828e72c7886dc8d136117aa30}}, {{cite:f2d80c93aaf58febc67d8b2564815734f3ea7f1c}}, {{cite:6136b0bfe7771ff1f3fffdc67d439730f777a0c9}}. Introducing global mixing on a two-dimensional lattice is similar in concept to using a “small-world" network {{cite:4edb88579504abdfa612d734b3ebfe9bf33428d6}}. Both cases have regular connections, and random global connections – the difference is that we implement random global connections by switching individuals.
| i | 91e9c2d992dd442c74142468f2320183 |
{{cite:5cf9bebff653751a13fda5eda7c26a2c7a28df91}} proposed a combination of deformable-model based method for the segmentation of prostate and bladder. The method is performed with different shape assumptions for the two organs.
{{cite:8d0024998ea7a284f6e0f951aa47de78087d8daa}} introduced a deformable model-based segmentation method for prostate and rectum in CT images, where a local boundary regression method is performed on the near-organ regions.
{{cite:97df9eac152e3b8d322fd34f5709883634727b22}} proposed a deformable model-based segmentation method combined with a random forest to obtain the organ boundary. The initialization problem of deformable models is thus alleviated.
{{cite:17e4fa7ad7644193ad84ee6fcf5ddb933d3c7256}} proposed a 3-D UNet based network with dilated convolution layers to improve the segmentation performance of prostate.
{{cite:d299d8154816f7131c4dd713e5f7ebb6725ffcaf}} proposed a fully convolutional network with two corresponding paths, namely UNet, with an encoding path and a decoding path with shortcut connections, so that the gradients in high-level can be better preserved to reach the low-level layers. The network is frequently used as the backbone and baseline in recent medical image segmentation studies.
{{cite:c08e14abf9bde198e30c06e42a6ebf05c9f477b5}} proposed the VNet, in which a Dice loss is used on a 3-D UNet based architecture with residual connections.
{{cite:6c8d6b82beab8301d56d2fbd549d412e4ffc48b9}} proposed a two-stage UNet based network to segment the pelvic organs. Specifically, a novel morphological representation, namely distinctive curve, is incorporated to provide additional guidance for the network.
{{cite:75d663aadd054776d419abc89f9412e094fcce5e}} proposed a general segmentation process for a bunch of medical image segmentation tasks. The method focuses on the choice of pre-processing techniques and hyper-parameters for specific datasets. And the network in this method is a slightly modified UNet.
| m | ab3cd902c344c436bc43c86e47bb7792 |
From Tables REF and REF , the superior performance of the proposed scheme for both, coarse shape-based and fine point-wise correspondence estimation tasks is evident. While the proposed scheme clearly outperforms traditional feature matching-based methods, it also outperforms recent shape-based methods {{cite:f4a7a895731be72639d740096a79ef82bba71a4f}} and deep learning-based methods {{cite:c0b3592c743e3fa68b35dfd8b374e7ac13a09b8e}}, {{cite:508a8d3067696e845f1265c83efb45d81b37bb72}}. Additionally, the matching results using the baseline conv_4 layer features show that the performance improvement is not due to the choice of features, but due to the ability of the proposed scheme to learn the underlying shape representations.
| d | b28a4dc92a8a83755665464883d011a2 |
Standard arguments that relate unbiased random walk models on domains with absorbing boundary conditions can be used to show that the mean exit time is given by the solution of a linear ellipse partial differential equation {{cite:e4d3b1848348960ae78b7f20c59560eaa3ad6362}}, {{cite:297a158bed223d49a60e30a8cfe9b4bfbaa82199}}, {{cite:bb6966808863e5216a27ae45199e5f4638c11c72}}. In this work we seek solutions of that equation,
{{formula:dd343443-0297-4fb5-ab66-47dc2fc494b1}}
| r | 2cad65947cc1d97f734edfd61ab999ed |
Interacting particle systems: PushASEP, ASEP, {{formula:2d0c3620-48bc-46be-931e-21ea907ebf7a}} -TASEP, {{formula:e29961cd-947c-45ff-8b1f-c2dff0fb572d}} -Hahn ASEP {{cite:d40a2a71c08d5ab52eaa533d53be8b083214f12e}}, {{cite:e66eba688d938c5f28e4d598a2f74ed61db180d7}}, {{cite:c5b630e6f572fa224f0c8393d0ee7baf0e5463c2}}, {{cite:33143ee44116d52d614b063241c9a14438f781a8}}.
Random growth models: KPZ equation, stochastic heat equation {{cite:3dbf907ff903b519d33dac3ef6bed93f39b35039}}, {{cite:e2b8801b0ddb676ba34e1269531f6fa4caa9c58d}}.
DLPP and directed polymers: O'Connell–Yor semi-discrete polymer, log Gamma polymer {{cite:e43a855147daa116d2bd9d34f93b4f3d8aa6d8f5}}, {{cite:2afeb3a22e21b8837fab751f7f562f336c5fd44a}}, {{cite:53a3c74443820bb41f469653015814422342bada}}.
| m | 22b5c735d2b120a813ece816c3d7e2a4 |
This work explores the problem of visual non-prehensile planar manipulation, reconciling tools from model-based mechanics with deep learning. Our proposed Differentiable Learning for Manipulation (DLM) approach: (i) encodes the input video {{formula:9ab8ba2d-182c-49ee-bf8d-4187db9198ae}} in a latent vector {{formula:55788443-bea1-4058-a4bf-83a796a7d718}} , (ii) derenders mechanical parameters {{formula:5e59e101-66e6-4cc0-bb9d-a42d8d5caf1a}} for the task, (iii) solves QP {{cite:61a13ae0ae792638a67d6bcf801839ff620e04af}} to obtain robot finger actions {{formula:d4984372-15f5-4ab6-b0c3-9d626336fea2}} , and (iv) evaluates the performance of the model by simulating these actions. We train this model by minimizing {{formula:e79ea5d6-c705-499f-94c6-30c9ff2186d7}} , in order to match the ground-truth parameters, and by minimizing {{formula:c2990049-cc04-4b61-897c-d26d37a00d0a}} , in order to match the ground-truth actions. Moreover, we can self-supervise this approach with new data by back-propagating through the simulator {{cite:5430d2c5ec7d4f56969b3ebbb18c8bdecfcc3553}} by minimizing {{formula:8635758f-b8e0-4216-ab9c-f05ef628722f}} , without the need for ground-truth labels on parameters or actions. We assess this method by learning how to solve planar manipulation tasks given a pre-segmented video showing a desired object motion. Our experiments suggest that, when compared to fully neural architectures, our approach can generalize better to unseen tasks and shapes with the same amount of training data.
| d | 29f1a061bd0eaab99e9da3a5ea492143 |
In this section, we analyse a technical defect observed in two representative GCL methods, DGI {{cite:7b8e6ae3a952f0108b1e43ec16d4069a076464b1}} and MVGRL {{cite:e8c35c517a2d62d8fe09f2b3ce718c260aed8d5c}}. Then, based on the technical defect, we show the learning method behind these two approaches is not contributed to Contrastive Learning, but a new paradigm, Group Discrimination. Finally, from the analysis, we provide the definition of this new concept.
| m | bff50b8b4856c5600d3907ecf014924e |
Lemma 1.2 ({{cite:ea7283196cf64326225bfa27c2358d256ac49145}})
If {{formula:d30f6bc1-8c78-4d6e-860f-781fcbaecdea}} is a GCM of finite, affine or hyperbolic type, then
{{formula:c00228cb-ed11-4a71-a9af-97d6a7da52ac}}
| i | b130ca2140ddcfaca8b2f633b06e80e0 |
Multi-modal vertical recommendations. Pinterest has 3 main surfaces (Figure REF ) that provide personalized recommendations: (1) in the Home surface, users are provided with recommendations based on their past activity, (2) in the Closeup surface, we provide similar recommendations to a pin the user is currently viewing, while (3) in the Search surface, we provide recommendations in response to a query string that the user has typed. Note that in each surface, the query comes from a different modality: (1) in Home, the query is essentially a sequence of pins, (2) for Closeup, the query is a single pin, (3) while in Search, the query is a text string. In contrast with other works that typically target a single vertical application (e.g. product search {{cite:77ca8274a44685ad6f748463954fae155fe60edc}}, {{cite:6d6309af68abeafe935e0b2809a2feb2dbe5b91f}}), ItemSage can provide relevant candidates via approximate neighbor search {{cite:4b55374826c027facdba04eb4a9ddad940513c40}} for all these surfaces and, therefore, in response to queries formulated in each of these modalities. We achieve this by training ItemSage embeddings to be compatible with the learned representations for pins {{cite:63fd1739d9bc0b2194f1f72ff7e6ef3493e16669}} and search queries. Recommendations based on user activities are a more general case of pin-based recommendations where the activity history is first partitioned into clusters and then a few representative pins are sampled from different clusters to generate pin to product recommendations {{cite:aff36f37c08c926ac002947db7cc2705bed48947}}.
| i | c162fb7cefd3847b20e28a27088a883d |
The rehearsal strategy stores data from previously learned tasks with a memory buffer and interleaves them with the training data of the current task to jointly train the model, as shown in Figure REF (a). Incremental classifier and representation learning (iCaRL) {{cite:dd22c15b825b388812cb4aa09dde3c9d942a7830}} is a representative approach. iCaRL employs an exemplar set including the most representative samples of each previous class. These samples can most accurately approximate the average feature vector over all training examples. Next, the approach applies the nearest mean classifier {{cite:1440c5298b6c25131c8173441f43391320f335de}} strategy to predict a label for a new image based on the exemplar set. iCaRL can incrementally learn many classes over a long period; however, it updates the exemplar set and classifier independently. End-to-end incremental learning (EEIL) {{cite:62f5716b1c3e3031f33b40b78307afd887352ff5}} overcomes the limitation of iCaRL by jointly learning the classifier and features. The approach achieves end-to-end learning by formulating an integrated cross-distilled loss based on the distillation loss to extract representative samples from old data and using the cross-entropy loss to simultaneously learn new classes. Subsequently, the imbalance problem between previous and new data is considered. The unified classifier (UC) {{cite:5e60aa585339d35239a990a8f03cf4d92bc45635}} adopts a cosine normalization-based classifier to eliminate the significant difference in the bias and weights between old and new tasks. Subsequently, the approach employs a less-forget constraint strategy to ensure that the features of the old samples in the new and old models do not significantly differ. According to large-scale incremental learning (LSIL) {{cite:164084daac7d5a4d90ecc389be9b8974c3382b10}}, the class imbalance problem causes the classifier to classify an image into the category with a larger amount of data; therefore, the approach introduces a bias correction (BiC) layer to correct the bias regarding the output logits for the new classes. In the incremental learning with dual memory (IL2M) {{cite:ae0f3b2db34ac5cb36ce6d491326d617a1093c94}} approach, a dual memory is introduced to alleviate the negative effect of the imbalance problem. The first memory stores the exemplar images of past classes. The second memory stores the initial class statistics in a highly compact format as it is assumed that the initially learned classes are best modeled. Experimental results show that the initial class statistics stored in the second memory can help the model overcome the problem of imbalanced datasets in past classes and rectify the associated prediction scores. In addition to the sample imbalance problem, certain approaches have been developed to optimize the memory storage utilization. For example, MECIL {{cite:8352fcb6f1b1c5f16cdcb680178135777dd04b4a}} aims to optimize the exemplar set management. This approach retains low-fidelity exemplar samples rather than the original high-fidelity samples in the memory buffer to ensure that the limited memory can store more exemplars. Certain other approaches attempt to enhance the stored items of previous tasks. For example, instead of directly rehearsing the stored examples, the gradient episodic memory model (GEM) {{cite:1281ef54e1338e2ac60ed44dea044f0cefedc71a}} stores gradients of the previous task to define inequality constraints regarding the loss to ensure that the loss does not increase with respect to that in previous tasks.
| m | 30e15bd2b67d48b67d664003efaafc2a |
The particle phase is described in the Lagrangian frame. The motion of the {{formula:fb047756-ac57-432d-9379-971aca3d9a95}} -th particle is given by {{cite:e8c3128e3ecfe471ed250fc558a69190fa6d79b8}}
{{formula:83befe21-ee12-4af6-91a5-e811ecc1711c}}
| m | 3f6685f37bf9a9bc0c91cadd5ce51405 |
Boundary correlators in AdS: There has been a lot of activity about computing and studying loop-level correlators {{cite:ff1c1b579e20fa62a7c3539a4b4b32f1c2c3a98b}}, {{cite:4a67dc3a3ffcf8fcd17442e6a5e0ac30753db7a1}}, {{cite:86dad2f2c8ea1e39eff71460df9c7986ce3607cc}}, {{cite:eb92923fba597d8b30b12a760ba2cb87fad2caea}}, {{cite:03b1d199f4793d2b2621962ef9c90739fabf7db0}}, {{cite:a324ebd5d18f3bb6e0f5dc4a61fdea5b96492d39}}, {{cite:535f041afa4e5e361bcfab6540d387835bbf1650}}, {{cite:4c087b77f1cb5a8dc6cc6e5872d612a524bbda40}}, {{cite:6506b463dc10211e535851c54044ca57214aba71}}, {{cite:f43aad748a65ea30779a66e90456107366c761d7}}, {{cite:9b631489fba05bd79ca2be577847ff06004d36d9}}, {{cite:cf04c46f8f3589df0a3b05aaa0050b8d30ebe522}}, {{cite:d3de3301720341c9580638d75879b938ca2b6c0b}}, {{cite:871ca73770850a80580a22917dcf228ff324a642}}, {{cite:068813a0630823a33920974211da1bf698ac7b60}}, {{cite:98bca08f3cc977da730d4b1bfc5cf99b284131cb}}, {{cite:356e9ad14294d5f918a575e4c35d8a2cb36d3b9d}}, {{cite:f669fe78ecc61478cddbf2b2768dec99dd45d9df}}, {{cite:12f569dc4e3078038c30d09bc0c158ef381ae8e2}}, {{cite:b8647eb5ee4f3a6753e122218bc666a18b2f7b9d}}, {{cite:f8e85e953b92d96901ad6d63c3b456c96201e3ce}}, {{cite:01dd8de19d7e4f510567f4bb39fd0fa940818b02}}, {{cite:f8751708a9d34866470365872c3b4a3ef1da36a9}}, {{cite:1a5d34ac7d572d561357c319f0399ec507edb054}}, {{cite:f7249a0289ff1e4013cf52afb10b83277347f183}}, {{cite:893d8c387b21795e960d5282b82f49246e5d0012}}, {{cite:dd37c638c570b1c29e1673599f9dba307f4a1c6c}}, {{cite:87027d2894b17d82065fb7c5e86c016d9a6a1144}}, {{cite:ed6e549848ff8ae77029e800f5fb79da02129dce}}, {{cite:2caf2f03684c4905005934f0004385ee41806cf2}}, {{cite:2a603c8b58f3d4693f2b963114eea18a51e1b99e}}, {{cite:535e315b9523510a8ce397e707b38b38cda37105}}, {{cite:d24113380a862bf9f49bdde7b1d0346767b8694f}}, {{cite:a0fc49b400eacb5b5c81c73c4a91d577fff7d6b5}}, {{cite:232b97f415a482ae496b1d99afcabd62f1bc9449}}, {{cite:cba6858fd02958eea99dbf8b29b41b9ceac0f5c7}}, {{cite:ea9a59dd213d71eb5b56bfebb6de084a1e891959}}, {{cite:409d01c1cccbf417c67d354810e37beb20d79675}}, {{cite:802f3d503e4df59a5eed9008923d1a2e763b4fdd}}, {{cite:6d214b39a09d8ce2e209d05f8e5f9b08b17e0a39}}, {{cite:03a51cadc1f854f1069a2b0a186edf11600eb0c0}}, {{cite:fcff4cf83da15d2ba32f801f56ec35c717074724}}, {{cite:2b0e7094a58af5d3ddff1dd4de3b6afe5f76efbe}}, {{cite:04e1bbe6ff56a21b1a9c12bf0ea4d27619d85191}}, {{cite:e118c5342094960b2639d048b8a6ca28f2c2f6d2}}.
However to the best of our knowledge, such studies are always focused on specific diagrams, and not on the one-loop effective action. It seems that the AdS one-loop effective action has been used only in the very specific case of the one-loop scalar potential (namely, for constant scalar field with dimension {{formula:50b950b6-44f9-48e0-a60d-c0f0cff2b1ce}} ) {{cite:cf3e87028034e12b4cf1c69bcd2e66706b626080}}, {{cite:2f459e975c719051cf5569f67156a709915c5b68}}, {{cite:e9067a09011820392fee6da1a9b3420c0546b644}}, {{cite:9372d3b9771e2e34ebdbff121ddc13d33d0c8ff7}}, {{cite:86c3b560ce374fa31ebeca33bf02ad52694d7460}}, {{cite:314b8063102956551d1cdd561c4b6b33c5648b77}}, {{cite:f7249a0289ff1e4013cf52afb10b83277347f183}}.
The one-loop boundary effective action, through the heat kernel coefficients, contains much more information on the (bulk and boundary) divergences and on the long-distance EFT.
| i | 3bce488470b9fc037fe3810a4b441185 |
Recently, with state-of-art performance, deep learning methods have dominated the field of image processing. However, deep learning methods require a massive amount of well-labeled training data, and the majority of deep leaning methods are sensitive to the domain shift. Therefore, transfer learning (TL) has been introduced to deal with those two issues. In this paper, we propose a novel medical image classification algorithm. Moreover, we implement the proposed algorithm on a COVID-19 diagnose application. Generally, medical image data sets are difficult to access due the rarity of diseases and privacy policies. Moreover, it is difficult to manually collect a massive amount of high-quality labeled lung CT scans associated with of COVID-19. Thus, the performance of machine learning models can be unsatisfying with insufficient training data. To overcome this obstacle, artificial and synthetic data can be used to expand the volume of the data. However, these methods cannot handle the performance degradation caused by the distribution mismatch between the training data and the testing data. Therefore, transfer learning is considered to be one of the most effective ways to solve both problems at the same time. Theoretically, transfer learning algorithms aim to develop robust target models by using only a small set of target training data and transferring knowledge learned from other domains and tasks. Previously, {{cite:13ab0e6e8e603375a06d380b688c943465fd920f}} proposed an adaptation layer with domain distance measurements to transfer knowledge between deep neural networks. In general, conventional transfer learning algorithms assume that the source domains and the targets share a certain amount of information. However, this assumption does not always hold in many real-world applications, such as medical image processing {{cite:f4245bdf2789e69c784d420e2ab1d73724c2fa02}}, {{cite:a421a2c77ce620e567c0f24dd13ff8919d660458}}, rare species detection {{cite:c4db34484042e0ac325e571b7503d30f7cbba043}} and recommendation systems {{cite:ed355ae15017ec0af4357b12853d713458db9ae4}}, {{cite:be8f75934a56668fc0650c4eef09ea02c4903830}}. Moreover, transferring between two loosely related domains usually causes negative transfer {{cite:aace417c5f2c8026ea8e51e1a1cca1500582e6e5}}, meaning that the knowledge transfer starts hurting the performance on the task in the target domain, producing worse performance than non-transfer models. For instance, building a dog classification model by directly transferring knowledge from a car classification model would likely to lead to negative transfer due to the weak connection between the two domains. Therefore, it is not always feasible to apply transfer learning to areas where we cannot easily obtain enough source domain data related to the target domain. COVID-19 diagnosis based on lung CT is a typical example where we cannot easily find related source data for training, so conventional transfer learning can lead to negative transfer.
| i | 4da3ebc6249c12b5a1bd4e62debc0d81 |
in agreement with our sketch. The theory at {{formula:5d54a61b-038a-4ca5-b8ec-516fe0c4b861}} should instead be described by a complex CFT {{cite:353888da350e3c2123427b8e7576bd1bde4a6a5e}}, {{cite:4787a9ca41dc32aa55adb23ec68428e1cdf79a3e}}. The same features present themselves in all of the figures in this section, except for the left panel in figure REF . In this case, the two curves cross rather than merging smoothly, and the lower curve is allowed to proceed to the left towards the flat space limit. The reason for this exception is simple: the relevant operator {{formula:2cc974d0-e12b-4b5c-940d-fb90f462abde}} is {{formula:62170ef5-c805-4d12-a462-f2d328f667fe}} odd, while the bulk perturbation is {{formula:8cf6ce14-5d05-4f4c-91d2-9deaf05155c8}} even. No fine tuning is necessary in this case, hence no dangerous marginal operator is turned on at {{formula:134d91a7-ecc9-4c8a-8f6d-6db25fd1afc5}} . Finally, this discussion suggests that turning on the relevant boundary operator allows to flow between the theories which merge at {{formula:687a862a-1752-4fcc-8d8c-b8bde1cb5fdc}} .
| d | c6aebf191e4592ae5c4b1f36bee70002 |
The approach of mathematicians is to consider the real Jacobi group {{formula:e31f3524-cd02-4639-a783-4168a920aa2b}}
as subgroup of {{formula:f215f551-e1fc-4f67-986c-0bd41f1d6f61}} . In the present paper
we follow the notation in {{cite:98a797d249a1a19d09f6d2b7d10ff7dbf8c7e31c}}, {{cite:3d08fa63dfe05b47ab2386699c160d8b417455b6}} for the real Jacobi group {{formula:3e15796e-fae6-4b93-863f-48865b636745}} ,
realized as submatrices of {{formula:ab9d0d61-4d35-44ab-ab5b-9bb9efea656e}} of the form
{{formula:d4973777-618c-4905-8c44-46c1064b0fb2}}
| i | c71bbf299ce4f688f545a8988889964d |
Recent progress in engineered materials with novel electromagnetic properties is paving the way for technological advances on several fronts, ranging from non-reciprocal devices {{cite:cfc5da78cff383fdca68723873086c7f6a7653de}}, {{cite:890460042f1d7cff0cac690b2e5f2933870026a2}}, {{cite:c36589de721263937f1885b245b652282aa45f2f}}, {{cite:9f0da5d3949587c95e29f54a7fee9a96913192b0}}, {{cite:b160117e454b8f88def8f5e8f9a77503f433d9bd}}, {{cite:a4ed245426ab74e8a1944df52c8d57836c9ce3a7}} to hyperbolic metamaterials {{cite:2f6f5a0fe899e279098c699db275e4ca65df7eb1}} and photonic Chern insulators {{cite:64df257703157abdb713b4a9f124b62fe446f284}}.
Three major electromagnetic wave phenomena, namely the (i) non-reciprocity, (ii) photonic spin, and (iii) hyperbolic topology, are at the forefront of next-generation devices, with all three posing unique challenges and opportunities. Interestingly, gyromagnetic materials exhibit all three phenomena (see Fig. REF ). Gyromagnetic materials such as ferrites have been extensively used in non-reciprocal devices at microwave frequencies {{cite:518213efcedb62da1ec8b5d972c9afaa22ba1d07}}, {{cite:da6a236aa355f97a99cc11f53d1b45129a45f4c8}}, {{cite:2b9549e6884f10f863ac009eb7dffaf2ea3a5556}}, {{cite:b055ffa8b427df1cc69ccfd969477872c086ea93}}, with microwave isolators and circulators being their primary applications. Conventionally, ferrite-based devices were bulky and not suitable for system integration. However, recently researchers have integrated ferrite in substrate-integrated waveguides for non-reciprocal mode conversion {{cite:293fa8a4170e23657b7cc6929d1aa436250354e4}} and filter applications {{cite:6fe382e4a25160386af1a657e5b20a8a973becaa}}. The application of ferrite nanoconduits for nanometer-scale RF magnonic interconnects has also been reported recently {{cite:cc054b47e2c3be1a4ba1497bb3bbc005b95dbb97}}.
| i | bd96ce2019c82bce30bed174b5a7c366 |
As the case of periodic solutions with prescribed periods near an equilibruim (cf. {{cite:710b30ad3eb676eaacbdc5c069a1d7d8fc90604b}} and {{cite:918921b57abed338fcbba77fae6b246c19e90122}})
we can derive from Theorem REF :
| r | 874bb1a5357b0d89694de6eca19d6520 |
The Krawczyk method {{cite:1266557729313d45ba71c7a55dee7b2da277cc5d}} is a known algorithm to generate increasingly precise element-wise interval enclosures of solutions of linear systems of equations {{formula:8c1a4f67-b5fd-4c39-a0ad-7624200f4132}} , where {{formula:124dbeb6-35b1-4fc6-8724-8c1a7a64919e}} is an {{formula:0e00019d-b66d-4633-9ebd-40eec4b432d8}} -by-{{formula:d7b0a70f-70d6-4d5b-8ab1-b2490d5d0c70}} matrix and {{formula:573d091d-41d1-409d-a614-94b6cc1568a3}} is an {{formula:5aee452d-c231-48f7-9054-ec36f0122738}} -dimensional vector.
Letting {{formula:aa5332bf-3465-413f-8d60-32d3ecf52789}} be an initial interval enclosure of the solutions, the Krawcyzk method uses an update formula of the form
{{formula:b66505c7-2670-446f-8672-91c6a599a727}}
| m | 2ad7b6df8f0678ffbfc213c05baf2761 |
By previous results on eigenvalue concentration (e.g. {{cite:3ef815d7cba5fd0484e3e3134b925c98a989c745}}, {{cite:18ffdb2c563dfaa2c4b995f7fdc44e9b71a218ae}}, {{cite:51dbcc612b17d4864823990cd5c2a603d45a430c}}), we have that
{{formula:79acef25-238e-4460-bce4-e8a7234cc6fd}}
| d | 36af10a4afcdf769937338997e4fdd3e |
The property of dense nuclear matter is crucial to the study of many phenomena in heaven, such as that relevant to neutron stars and supernova {{cite:c234291c426212f96356803b67ca37b201fe6116}}, {{cite:f0b82db3afb58ce4a80de9491898be2561d55f86}}. The property or stiffness of dense nuclear matter is also crucial for studying the phase transition of nuclear matter from the hadronic matter to the QGP matter {{cite:21ef4e2b33540f8ec6329774a6e8aa7b62732c4b}}. To study the properties of dense matter, besides using neutron stars as natural laboratory in heaven {{cite:105e9b0b85b46578713461ea703c829c3d154508}}, {{cite:24e7c2ad09590f777025921814e8017218bcf7a1}}, {{cite:5f4e2977097a74b2cb07ac2134fdf75efce5d0df}}, one frequently uses heavy-ion collisions to form compressed dense nuclear matter in terrestrial laboratory and then probes its properties via emitted final-state particles {{cite:df02592030568cbb574828f6dfa6e1379e493493}}, {{cite:7766ab7b85382095f8017d6fdb62483ae9f6eb0d}}.
| r | 267053a16dc8c298cda803b4de0ea4e8 |
One of the first mathematical models that was used to study the spread of viruses in communities is the susceptible-infected-removed (SIR) model. In particular, Kermack and McKendrick {{cite:d4e1b527f0ab297e9ce831a662f27406990961f0}} presented in 1927 one of the first models to forecast an epidemic. According to their model, a population is divided into the susceptible population ({{formula:7db90b95-2028-426b-9fe7-5971aee0635e}} ) whose individuals can become infected, the infected population ({{formula:31733a04-853e-4f5f-9f46-3852506547b0}} ) and the removed population ({{formula:2cd3cd68-a5cb-4f8a-8c07-7c1d5807f0a5}} ) who are individuals who have either recovered or have died due to the virus. The SIR model is given by a system of three coupled, nonlinear, ordinary differential equations (ODEs) which describe the dynamics of the disease in time as individuals move from one compartment to another. Since then, the SIR model and modifications {{cite:f4e24880d27508a316653120437c7e83a137c2c3}}, {{cite:61466959e47b620cb88e665dd0923cf1497460fa}}, {{cite:e3a01985b7482b43f087ea9460adb43785a0d315}}, {{cite:543c6673789dc678263e8b66280fe29acb9d6295}} have been used {{cite:bd1028334ef456a89d435e137bfe7388f56b221b}}, {{cite:d4404fdaa408117ffb19c58d6e7cec083f54c07e}} to model the spread of COVID-19 in communities {{cite:9afa1dd688d62aa6d988edbd993aabae096b9da7}}, {{cite:802e2f3508afe35577bae1d6d134d6632fd22dbb}}, {{cite:4402bb7afb56f4943472affd472c769447897c8d}}, {{cite:d9f5ba6660df1d14b85a093cbd9bb10323a89111}}, {{cite:cad98295b60cdda1f5081d1badc5fe990e6cc442}}, {{cite:fb0c90cb13c377dd6ff5b0e2d33e5387d1e7c31f}}, {{cite:ade6f65cfdcbf25c583fbecc698d490632d948a5}}, {{cite:a15ecd6792067f2b910896b1735a92630306e376}}. The main goal is to predict the duration of the pandemic and to find how interventions such as social distancing, immunisation and vaccination could reduce the number of infected individuals. Importantly, mathematical models can be used to evaluate the effectiveness of control policies against the spreading of infectious diseases, such as COVID-19.
| i | 1f5d56c2d5cd1304d8e9bbc9ec349c81 |
As discussed in Section , human annotation of the semantic variables faces the scalability problem when applied to large-scale conversation data (e.g., over 3.3M utterances in our dataset). To address this challenge, we provide a preliminary attempt by using automatic annotation with a careful selection of training data. Although we have provided a human verification of our annotation, such annotation unavoidably introduces the distributional shift between the classifier's training data and the conversation data. Future researches may explore semi-supervised learning methods, e.g., variational inference {{cite:de198b975aa0cc6a69bc7cd1838d35365624fd6d}} and domain adaptation methods {{cite:31ac23070a832c6f043cb04c4082c8ac7b8c1418}}, which helps address the distributional shift problem between datasets.
| d | 1dca0ee7b85c81fa3c07e77b0645941b |
There are many directions for future numerical studies of black bubbles.
The most crucial would be to relax spherical symmetry to explore
stability to non-radial perturbations, and if stable, accretion of angular momentum
to uncover the rotating solutions. The fact that the bubble surface
is within the photon sphere of the spacetime suggests there may be long
timescale secular instabilities {{cite:2a46877a3d96e5d9acd9e3cc6dcd685b87971fc6}}, {{cite:80299f96bb758c30e68fb6f46b994a58237420a3}}, {{cite:6d07fa3b14524c73706f0cae28dffe275dd6858c}}.
Classically, this might be analogous
to the so-called weakly turbulent instability of AdS spacetime {{cite:5a83b385f1a3eef0b0e5b62263b89ed0bbfea853}}, which
certainly is also relevant for the black bubble interior. If so, the consequence
of the instability might “merely” be that trapped energy could eventually
form small black bubbles that merge with the larger one. For rotating
black bubbles, similar instabilities could be associated with the
presence of an exterior ergoregion {{cite:6c4b0e0e69e49135d998a6fa53aaafc8da0f44d7}}, {{cite:40f5dc739c444ba07510ae591850e9a9a7ad7e43}}, {{cite:47d65bec3cb35dab17ffe754c2b9b2c2feb3550e}}. Also,
it would be interesting to investigate whether in such cases there could be
superradiant extraction
of rotational energy, which may lead to similar observational signatures
as the presence of ultra-light particles around rotating black holes (see e.g. {{cite:85f486493b68cf0580ebb77f3579c7bc3f5f8fd7}}).
Rotational energy may also be extracted if a Chandrasekhar-Friedmann-Schutz instability
operates in fluid shells {{cite:b9b7fba649d4f1f066a0ccec3fa548fe0e5edb51}}, {{cite:5fff1f16ad494464281159152a92b1ee06a4da29}} (it is
generic for rotating fluid stars in general relativity).
| d | 77b50da2212b529cb84b7b43f5e71ae1 |
The dilute atomic gas is amenable to a systematic analysis mainly because of the length scale separation inherent to this system {{cite:96e85938a60377a920b7ae9018b833e4d585da27}}. The following length scales are involved in this problem: (i) The low-energy scattering length, {{formula:465377b3-b3ba-401d-96aa-eaaec588998e}} , which expresses the strength of the atomic interactions and is positive for repulsively interacting atoms. (ii) The mean interatomic distance, {{formula:738ac4fe-fa76-46f5-ab01-9643c848c4c9}} , which is set by the mean density of the gas. (iii) The de Broglie wavelength, {{formula:09fa7590-84e8-49e8-92f5-3a54751be082}} , of each atom. For many experimental situations, it is reasonable to assume that {{formula:ab92629c-9434-49ca-ac47-2ae2c398ad5d}} . If a trapping potential is applied externally, another length scale is the linear size of the trap, which can be of the same order as or larger than {{formula:0ea2a08a-3ece-4ac4-85f9-5de685696255}} . The gas diluteness usually amounts to the condition {{formula:db168bea-2113-4d76-81f5-62b0dec498a2}} , and a macroscopic quantum state may exist if {{formula:2e379709-0271-4f7a-95e6-fc753973dcf5}} .
| i | 8d49d55fb29afd3e950d010aa185b142 |
The current result may be extended in several directions.
First, for manifold with smooth boundary,
the random-walk graph Laplacian recovers the Neumann Laplacian {{cite:7e710d452278661cb50e4b1bc943e9e1e8078ff8}},
and one can expect to prove the spectral convergence as well, such as in {{cite:5587ee43d6b6168c127ca974a92098768f0f7095}}.
Second, extension to kernel with variable or adaptive bandwidth {{cite:d91d8fb5c606831b1da7436e0e80f2c916177bf8}}, {{cite:e3bed5e7ce479cb3a22202ceca2a36fd09cf32d4}},
and other normalization schemes, e.g., bi-stochastic normalization {{cite:ee015b44d8bcebcae5d93c931e54ed5bf6abb040}}, {{cite:853eaea39be0d9efaa0f350d2a5feb657fb5862a}}, {{cite:c6c970f4027dc894219545f466c099eb82741c71}},
would be important to improve the robustness against low sampling density and noise in data,
and even the spectral convergence as well.
Related is the problem of spectral convergence to other manifold diffusion operators, e.g., the Fokker-Planck operator, on {{formula:62123fa1-8306-46ee-bdb6-a5cfdc83f110}} .
It would also be interesting to extend to more general types of kernel function {{formula:ea535b83-dcf0-4e64-8f3a-700ffd714ac0}} which is not Gaussian,
and even not symmetric {{cite:908ecbb3ab09e4dcad8044b2a8e12c65cf07a1b8}}, for the spectral convergence.
At last, further investigation is needed to explain the good spectral convergence observed in experiments,
particularly that of the eigenvector convergence and the faster rate with density-corrected graph Laplacian.
For the eigenvector convergence,
the current work focuses on the 2-norm consistency,
while the {{formula:4e082508-e2ef-409b-b525-7ed543fe7589}} -norm consistency as has been derived in {{cite:67426ed04e00f4209e7805452034d57205a1e3d1}}, {{cite:f56fa387bdbea5bbda9d36a7e787d6afda1ba780}} is also important to study.
| d | 8d293b2ae73ad678d8d500e2feebdda8 |
Navigation and collision avoidance are fundamental capabilities for mobile robots, and state estimation and path planning, which are integral to such capabilities, have undergone much development. Deep reinforcement learning (RL) is another method for realizing these capabilities, where robots learn to solve problems through trial and error. Extensive efforts have been made to formulate training methods for deep RL models using various types of data, such as those from RGB cameras, depth images {{cite:907a3c6659172af52ab6d60531bddef81a1ca5e2}}, {{cite:6038ad0eb8f8a8516aab7c7e5f097512f2a9ce24}}, and laser range finder (LiDAR) point clouds {{cite:7ce8b770ba2718e055cee33a7d78fe3c071e5816}}, {{cite:a757b5f4c95b20103b2a2ad064299844c9f5bbc4}}. However, optical sensors are often useless in adverse weather and environmental conditions: LiDAR inputs are degraded by rain, fog, and dust, and cameras are sensitive to variable lighting in dark and sunny weather. In search and rescue missions, for example, adverse environmental conditions are commonly encountered and result in degraded sensing. In the DARPA Subterranean Tunnel Circuit, there are challenging environments in unknown and long coal mine tunnels with smoke filled adverse conditions that impair commonly used camera and LiDAR sensors.
{{figure:13c547c5-c330-4a63-a922-0525dc458ae1}} | i | efb9bd5e6d37af42e89f54abbcd83f44 |
Iosevich and Rudnev proved in {{cite:367d1e502cee89bc14be1fd5d7a35bd553937d31}} that {{formula:d31534db-295c-4569-917e-fa0514461d7c}} . They also showed that the exponent {{formula:1c99c36f-dd4b-4230-abbe-db89e2fc043b}} is sharp in odd dimensions. In even dimensions, it is also conjectured that the right exponent should be {{formula:f9185c70-c3bd-45ae-bb28-f439621c16e1}} . In two dimensions over prime fields, Murphy, Petridis, Pham, Rudnev, and Stevens {{cite:7bc3b953bad245cdc1985f9ae30ce8505957a3ca}} recently proved that {{formula:ec533004-0674-4798-ab9e-8e9c90630432}} , which is directly in line with Guth, Iosevich, Ou, and Wang's breakthrough for the Falconer distance problem. We also note that Murphy et al. actually proved a stronger statement, namely, if {{formula:2ac4b977-c4ab-4e90-a0de-4d5c0f98905c}} , then there exist many points {{formula:4ac25ba5-e1df-4f0a-a22b-3a5bc5ac8496}} such that {{formula:df807fff-472a-4629-bdba-307a3bb4a9df}} , where {{formula:b4fa3cd0-3d1b-494b-9ef0-7d7755c27bb3}} is the set of pinned distances from {{formula:b27a2109-f1c3-457d-9950-df38bb7a188e}} . In light of Theorem REF , this result settles the case {{formula:e131c676-26a2-49cc-a11b-865a0f8a57f1}} . For {{formula:cd473a94-675d-45a1-aa92-c14923644af8}} , we have the following theorem.
| i | 5760e4ec4f4a10f2ba470e556f181260 |
The modulation spaces {{formula:6adbb4e6-94c4-4074-a19d-68dd4b2842be}} are one of the function spaces introduced by Feichtinger {{cite:58072fcf2c0f438dc985cb37dff5e4dc04522774}} in the 1980s using the short-time Fourier transform to measure the decay and the regularity of the function in a different way from the usual {{formula:c7f8ad12-e658-4b6a-bf07-c47b13e0693a}} Sobolev spaces or Besov-Triebel spaces. The precise definitions will be given in Section . Roughly speaking, the Besov-Triebel spaces mostly use the dyadic decompositions of the frequency space while the modulation spaces using the uniform decompositions of the frequency space. Therefore, these spaces may have many different properties in some cases. Also, they have some properties similarly. So, the relationships between the modulation spaces and Besov-Triebel spaces are important.
| i | 314640731b8f495fe521c2f7bc13b0fb |
In this section, we focus on assessing the pre-training performances with cross-modal retrieval tasks, in both zero-shot and fine-tuned settings {{cite:c7acdc4046167b921c421550a541656f5a70eb3d}}, {{cite:2a5cd5dd226df75e8b7d4c711a75dbd76087faa5}}. We name our method as “ZeroVL”, where “Zero” means the motivation for designing a strong baseline with limited resources.
| m | 25b3838ac77a2c815a6ef79777a3b763 |
{{cite:de8a76db8ad64c4deb1d17e5cf7b5d3d2b77d8c8}} famously made a distinction between two very different cultures for data analysis, contrasting statistical modeling with machine learning. Whereas machine learning is optimized for obtaining precise predictions that can be profitably put to use in industry, statistical modeling is concerned with understanding the data, and to this end builds models that could have generated the data. LDL is closer in spirit to statistical modeling than to machine learning. It capitalizes on simplicity, in the hope that this will further clarity of understanding. Undoubtedly, this simplicity comes at a cost. Although many mappings in morphology thus far have appeared to be well approximated by linear mappings, nonlinear mappings are likely to offer further precision. Interestingly, exactly the words for which LDL lacks precision are the words that are likely to be more difficult to learn also for human learners, and that hence will have greater lexical processing costs in the mental lexicon. George Box is well known for stating that all statistical models are wrong, but some are useful {{cite:63d4a0b6a00dd126f363c46f5a55d4503fcc050b}}. In the same way, even though vector space morphology as implemented with LDL requires many simplifying assumptions, it nevertheless provides a useful tool for probing the mental lexicon.
| d | adfd3cbbe870a8a1efb0cce7b45831ef |
Several DNN-related algorithms have been proposed for automatic music tagging too. In {{cite:7359e7233e1651bf027cf7712d676e6b13e5bdd6}} and {{cite:4b9bf269f9445fbac98aeeac55177ed7a28c6a61}}, spherical k-means and multi-layer perceptrons are used as feature extractor and classifier respectively. Multi-resolution spectrograms are used in {{cite:7359e7233e1651bf027cf7712d676e6b13e5bdd6}} to leverage the information in the audio signal on different time scales. In {{cite:4b9bf269f9445fbac98aeeac55177ed7a28c6a61}}, pre-trained weights of multilayer perceptrons are transferred in order to predict tags for other datasets. A two-layer convolutional network is used in {{cite:ef84555c18db1cc2050f104b7c2c7e9dd4707d96}} with mel-spectrograms as well as raw audio signals as input features. In {{cite:fde9443d2d7e8dd09b781b36fd0c751f9e1489d5}}, bag-of-features are extracted and input to stacked Restricted Boltzmann machines (RBM).
| i | d3df4432bc1e69ace47ff6d357b600cc |
Remark 1
Assumptions of this type are generally essential for the theoretical analysis of Q-learning. An assumption equivalent to assumption REF , of a 'covering time', a constant time window in which all state-action pairs are visited, was also assumed in proving the convergence rate of Q-learning {{cite:177b27fc85468817efd1062fcace251ca2ff260b}}. Similar assumptions on a minimum state-action occupancy probability, mixing time, or covering time were assumed in {{cite:585ea409720dd0430158e4638376fe2644ba3111}}, {{cite:f6a3885564cfe35f448397734608b57088800f0c}}, {{cite:307a8bc781fc1532c5c4346ba334cc045709d556}}. {{cite:4d63d99734a7d67bb1be4369b326dc4a66b30e3c}} show that under-representation of state-action pairs in the buffer can lead to sub-optimal results when using experience replay. Hence, this assumption is critical and standard for achieving stable convergence to the optimal policy. We note that such an assumption is satisfied if we have a generative model from which we can choose a sampling distribution, or in highly mixing domains where {{formula:d349c3b2-93e3-4cf2-ab4a-3d6f276e11a0}} is related to the notion of hitting time.
| r | 0c99286d799f8c1f1a59b0ead20118fd |
We mention that our work do not calculate all the lens configurations in the spacetime.
If a light source is at the different side of the wormhole throat,
images are formed inside of the throat.
In this case, one may use an exact lens equation investigated by Perlick {{cite:d31d74d22b8b229e0e372fa88a7d3aacf4f24dd7}}, {{cite:ff2c0ff1fb8d410081e05ebec67e407af0aea8f3}}
or one may use an approximate lens equation with the deflection angle in a strong deflection limit {{cite:fc6379bba54fb3f93ba1c64c4b6d24108baf8b27}}.
However, the errors caused by the approximation under the lens configuration has not been studied.
| d | 571f4d8465b31900feacf10378f4ccf9 |
Since the bonds in all these systems are weak, and consist of a mixture of covalent and
van der Waals bonding mechanisms, first principles calculations tend to be quite involved
and highly sensitive to basis sets, degree of correlation included, nature of approximations, and other factors. Density Functional Theory (DFT)
for van der Waals systems is often less than satisfactory,
typically showing a varying degree of overbinding without any obvious systematic trends. Several new
DFT functionals with corrections for dispersion interactions have been proposed
very recently; see for example, {{cite:83dc25bdf4bc3f110f777dd485c001a53318925b}}, {{cite:6d5780b72e6b1f901d5fc5cc7b9f56fb973920d2}}. However, thorough
testing and benchmarking of these new DFT approaches is necessary before
they can be reliably applied across a variety of systems. Dispersion interactions
result from transient-induced polarizations between the interacting constituents, and
are therefore subtle many-body effects which are difficult to capture in the functionals
framework.
| m | a56e9e57947c7579fd59c49f4043cd8c |
The success of our proposed shared autonomy approach can be attributable to three main factors: (i) domain randomization, (ii) simulated user modeling and (iii) training efficiency. Transferring models trained purely in simulation must pass the simulation-to-reality gap, where visual perception accounts for the most significant portion of the gap due to difficulties in replicating photorealistic images, leading to simulation trained models to fail once transferred {{cite:273d223a30b4e09e8067af7ad9a4ece9dccd605f}}. Domain randomization reduces the need for photorealistic images with the aim of making the trained network invariant to unimportant information i.e., illumination and background textures, which has shown success in other UAV applications {{cite:bdaec5390d1ab4c8b1ce2a870c5c4742f031490c}}. By training the perception module over a wide variety of simulated scenes under intense dynamically generated noise, the perception module becomes robust to visual disturbances by only encoding fundamental information of the scene. Domain randomization is also a key consideration when developing simulated users as human pilots vary greatly in terms of strategy and proficiency.
| d | c6dad241dbe1084ffb90fdf3d806851d |
Noteworthy is the fact that insights for asymmetric Hi profiles
along a ring-like structure as caused by the warped gas orbits has been reported in {{cite:ef7ae0f5e57d0a9f7e7d69a2bff4e73a0800bb78}}.
The location of that ring-like structure found by {{cite:ef7ae0f5e57d0a9f7e7d69a2bff4e73a0800bb78}} corresponds with that of the external arc-like structure, but not to
that of the outer Hi annulus we evidence here. Furthermore the Hi annulus does not share the same orientation parameters than the external arc-like structure
(Fig. REF ). Two different ring-like structures thus seem to coexist in the outer regions of Messier 33.
{{figure:77c5804e-e58c-434b-81c7-26742463d4dd}} | r | 52267f7bdd711048fc401b1aa6cf56ea |
Related Works:
Recently, several autonomous models are proposed/designed based on DL solutions to assist in the rapid diagnosis of COVID-19 from other types of respiratory infections {{cite:3d9d62bacaf627556d0d94ca6ac9cb0fdc8bb9a0}}, {{cite:c2add6c518e6c544d349c496cfb20242055a3b13}}, {{cite:3f7d70e8fe97a58f295d07efde54dba7ee9ad702}}, {{cite:8a1315fac5e3a284b9240579cb049bf217f6f834}}, {{cite:6612e380ca84cc40b909233b0ebdb840148dea15}}, {{cite:3bb491ff3734b7d2e0dc086e1504ace730633415}}, {{cite:f642a511665641e8353f8e5ee736bdcd8a6fb736}}, {{cite:4f86116d398981603748e813621b5c69274ed425}}, {{cite:0f49f06b9b9ca58ad0690b6c970320402601ad5b}}. There are, however, fewer works on developing DL-based models for segmentation and quantification of COVID-19 lesions. Segmentation models designed based on DL models are mainly developed based on CNNs. Such DL models are image-to-image networks containing an encoding path for extracting high-resolution features from input images and a decoding path for generating masks indicating the regions of interest.
The majority of the COVID-19 lesion segmentation models have been developed upon U-Net {{cite:790905635697ebac87df59ad7d5e6e6978a2aa29}} due to its superiority for the task of medical image segmentation {{cite:71738411261ac00c1c725b66965b0b620031e298}}, {{cite:39028ac17586f773ea4b7c3466e4edc40f1016da}}. For example, using a U-Net architecture integrated with DenseNet blocks in the encoder path, Chaganti et al. {{cite:2a0e3b767bfa52769e8269344336262769f936a8}} proposed a segmentation model to quantify lung abnormalities caused by COVID-19 from CT images. Similarly, Zhou et al. {{cite:a36ea7a9be59bf44b09fe87ec41446e6a1e6b7ec}} proposed an enhanced U-Net segmentation model by incorporating spatial and channel attention mechanisms. Segmentation networks can be designed based on 2D or 3D CNNs to segment COVID-19 regions of infection either on slice-level or patient-level basis. It should be noted that 3D segmentation networks that can segment infections on the whole lung volumes are more desired from practical point of view. However, they need a large amount of 3D annotated lung volumes for efficient training and are more computationally expensive. Reference {{cite:39028ac17586f773ea4b7c3466e4edc40f1016da}} trained both 2D and 3D variants of the U-Net model for segmenting COVID-19 infections from chest CT scans of 558 patients confirmed with COVID-19 pneumonia. According to their experiments, the 2D U-Net model achieved a Dice Similarity Coefficient (DSC) of {{formula:ff240b2d-e6f1-4e89-a0fd-1c4e34ceebae}} % while 3D U-Net obtained DSC of {{formula:addf11d2-2af7-4bac-ba75-277e939b695b}} %, demonstrating that on small datasets, slice-level segmentation can achieve better results.
| d | 03a3ae938e75d63c968588bd3c51f2a1 |
We collect {{formula:c6ad62e2-63de-450f-9a28-8510febc3d79}} demonstration trajectories on three Mujoco {{cite:5e6cf5193a9990e4e99ec635ebc7c3d81691424f}} simulated environments: Hopper-v2, Walker2d-v2, and HalfCheetah-v2, using a SAC expert trained for 1M timestep, using {{formula:e77f4ce8-0c54-47d6-b8f2-7ed0690c1e15}} random seeds. To evaluate the performance with varying amounts of demonstrations, we use a subset of 40, 20 or 10 trajectories (respectively {{formula:884e37e0-380d-45b4-bc39-9550aa0becb8}} data points) as the training dataset {{formula:a2c99f8f-2f89-42b5-aec1-3660bb16f107}} for the density estimator. We compare IL-flOw to the three variants of the f-IRL {{cite:e18e9892c427a7f9cb501cc7d3e16794169bde39}} algorithm, as well as a state only version of MaxEnt IRL {{cite:3ed669703629279c4b62b351223a10fd8c57d613}}, using the implementations provided by {{cite:e18e9892c427a7f9cb501cc7d3e16794169bde39}}.
MaxEnt IRL minimizes forward KL divergence in trajectory space under the maximum entropy RL framework.
f-IRL is an imitation learning from observation algorithm that operates by state marginal distribution matching, through optimization of the analytical gradient of any f-divergence (JS, FKL, RKL). It also learns a stationary reward that is reusable, although the imitation agent still faces a moving reward function in training through their iterative training process, while for IL-flOw the reward learning and the RL process are sequential, to convergence.
| r | 7ab416279d16b63f315c83afd0f4724c |
Field-induced resistivity upturn in the XMR materials can be
explained by semiclassical multi-carrier model in systems obeying
modified Kohler's rule
{{formula:383fead3-74fb-4eff-b4f5-64d58f4f73e0}} {{cite:c8993e992c411a9832b6d2bfbe9cc6e66d4d5ea9}}, {{cite:73d8f0c8184aba517a64db8b944d1fa4f9717f9e}}. In case of
perfect electron-hole resonance condition, {{formula:20960ee7-8961-4167-aa19-c38c22597420}} =2 {{cite:59b3eea4c66adbff85429fa1be9cda490441c80a}}. For
systems where Kohler's rule {{cite:8d9f54836ba982ffe5c85e98e9d9a8a39c22c3e1}} is nearly obeyed
(m{{formula:e00c5fc5-0f5e-4925-8345-a58925b2df06}} 2 and {{formula:93d8e341-b77d-4d98-9daa-7bc77decb564}} - independent), MR is found to scale with
{{formula:bfe03dbe-7570-42ca-91b0-91b1a3c3db32}} {{cite:4545a2fa13fd4c5815191c08fa1c2a9a8f792cd3}}. As shown in Figure 4c, MR in Ir{{formula:5b710b50-6c77-4a3f-a964-fc2b5c412772}} In{{formula:894ab525-c7f2-472c-a594-614e3ec91350}} S
systematically deviates from Kohler's rule above 10K (both {{formula:72ea084a-7e4d-41d4-bef3-69eeb24f5339}} and
{{formula:bb1b86b1-f309-4299-85cc-c215caef0f0d}} varying rapidly with {{formula:47dfedff-539c-403b-802f-13cb82adf4b9}} ). At 2.5K, {{formula:f4cc5894-c197-4590-9d43-0135b651cefd}} =1.58 (see inset in
Figure 4f), similar to other type-II topological
semimetals {{cite:4545a2fa13fd4c5815191c08fa1c2a9a8f792cd3}}, {{cite:88220aaea64706108654195284c1d3c89e281242}}; deviation from the
quadratic field dependence ({{formula:936b4089-b5e5-4c37-a354-c6e2c1cd9360}} 2) can be due to un-compensated
carriers or anisotropic Fermi surface or field-induced Fermi surface
modification {{cite:003a70141c875ad5fa2d5f8a8f07803ab744c01f}}, {{cite:2feefeeab6ded5475a3858b4a29318a80c28b9ac}}. In case of anisotropic
multiband materials, relative contribution of different FS pockets
also vary with strength and orientation of the magnetic
field {{cite:a0dee696b9cd5ab6c369919cc70ec364ac498ab9}}, {{cite:03f19864a24b110094adabc55f76f569f2f5beea}}. Variation of {{formula:ed7dd473-94e9-40df-81a6-94b74550e9ed}} with temperature,
when compared with field-induced resistivity change (shown in Figure
4e), indicates a rapid change of carrier transport behavior below
50K. This is further verified by the observed variation of the
exponent {{formula:758db53e-f522-4522-9375-cb5c9cf93194}} as a function of temperature (Figure 4f) indicating
electronic structural modification at low {{formula:142f6759-8f6b-4347-9146-d90c69611555}} . The field dependent
MR thus follows MR{{formula:081a116d-24db-474a-88b8-09bb9099ec31}} , where {{formula:c5515185-f715-4710-bb21-dd6a73fba258}} is the average
carrier mobility.
| r | fea3980de0d98cab355e2e83a309e77c |
This section presents numerical results to validate our analysis,
followed by discussions to shed new light on the performance of SCNs.
We use the following parameter values, {{formula:9ac4b76f-a07a-435f-bc1b-a02db19f5810}} , {{formula:4ff85b4d-6d66-482a-a2b7-87a225972ac8}}
, {{formula:5890028c-20dc-473a-a580-ba275e669102}} , {{formula:36c692ea-49d5-4830-b1ca-4eecaeaab648}} ,
{{formula:89bff0dc-8931-43b0-a6d8-99a952db6d0f}} , {{formula:a59ce71c-133f-48e3-b0c2-cc837cb9bf52}} ,
{{formula:59a87fc4-7277-4910-9d41-f6359e951d46}} , {{formula:62671bbe-dbf2-4d72-af81-5d6696937ec4}} and {{formula:3a6a4823-9c2e-4ab6-9dfd-2400fd1e8f00}}
{{cite:6489247c5c06d846f11e337994e20724a604f7f5}}, {{cite:6bfae44d96e0981fd25fbe9a64b0b052cc0661cd}}, {{cite:c8c7a3860d2f1cec2378f98e12bfd347b4642213}}, {{cite:82d4f7e01f8c842c6740ad30c3280822d80f7df3}}, {{cite:4bbd391208b48a876552d7c5a220f085bf34d48e}}, {{cite:176c4a62e115d040d256a9719024a2e5b198a6da}}.
| d | 38d77896b3c56d4fc3fcbdc5af72fd81 |
A guide to the presentation of our results is the following. The various fitting analyses and contour plots under different conditions (to be discussed in detail in the next sections) are displayed in four fitting tables, Tables 1 and 3-5, and in seven figures, Figs. 1-7. The main numerical results of our analysis are those recorded in Table 1. Let us mention in particular Fig. 6, whose aim is to identify what are the main data responsible for the DDE effect under study. Bearing in mind the aforementioned significance of the LSS data, Fig. 7 is aimed to compare in a graphical way the impact of the {{formula:945896fb-1071-48be-b619-4db4259e15f5}} and weak lensing data on our results.
The remaining tables and figures contain complementary information, which can be helpful for a more detailed picture of our rather comprehensive study.
Worth noticing are the results displayed in Table 5, which shows what would be the outcome of our analysis if we would restrict ourselves to the fitting data employed by the Planck 2015 collaboration {{cite:1e0a05971d4affd43d8a0aa8511855833d334be1}}, where e.g. no LSS data were used and no DDE signal was reported. Additional details and considerations are furnished of course in the rest of our exposition.
{{figure:08a9d049-efbe-47e7-b8a3-e3007d69c403}}{{table:6d85a2b1-d9b8-4209-9f0d-3889d6ea3e1d}} | r | 06f2830140f59a0c37a40f1ec37358b8 |
Last but not least, in subsection REF we have implicitly assumed that there are local gauge invariant excitations in the island region and we want to extract information about them. In the gravitational system (FSC without Bath) as a gauge theory, in order to define a gauge invariant operator one needs a dressing procedure. Accordingly, in order to define a gauge invariant operator for the island {{formula:d6ba1d24-bb68-416a-a5ca-aea42f07a9b6}} , even a spatial geodesic should pass through the complement region {{formula:8d1ef44c-3348-4bbb-b599-34f34931b100}} to reach the radiation region {{formula:54b15436-44d6-4fcf-aea3-77e4879de11f}} , where {{formula:51777f92-444c-40bf-89e3-3af20b1e83d6}} here denotes the overall Cauchy slice. This implies that to construct this gauge invariant operator, we not only need the information of the entanglement wedge of radiation ({{formula:387af64a-060b-4c63-854e-37a19586801a}} ) but also the information of the entanglement wedge of its complement. But this is in contradiction with the known principle {{cite:c32c1efd31d1e8b0fd4b942630eecce7f6717a51}} that the algebra of an entanglement wedge should be closed and commute with the algebra
of its complement. According to the paper {{cite:93cc503fb4b9c55b48bbe72dbd39b33aaebe2f29}}, {{cite:22bfc468c1010dc974322f5246b27b988d7b1213}}, the source of this puzzle seems to be whether or not there is massless graviton in the setupWe would like to thank Suvrat Raju for illuminating this point to us.. If there is massless graviton, we really encounter the problem since in the procedure of dressing we connect the entanglement wedge of radiation to its complement. But, if there is no massless graviton, then the necessary Green function to define the dressing is a decaying functionSome criticisms on this issue can be found in {{cite:7cf1943b8ef9bea9850fdc9051e8282c748dff7e}}, {{cite:73f1e29dd7893495ed0c5339d618332a4451dbb0}}, {{cite:31c33cdf2ed008d79cb68ebc837e629ca54b1e0c}}.. Hence it might not be any connection between the entanglement wedge of radiation with its complement and consequently there is no puzzleMore precisely, in presence of a mass term, the equation which describes the linearized graviton {{formula:7e6f5e33-2df4-47cd-89e7-97bc63b7351c}} on {{formula:891135d2-54b4-4443-b7fb-c9e633f77cb5}} background together with the energy density {{formula:42ce0699-dcf3-4537-8ba0-41f4e1197eb3}} of excitations, {{formula:b198c8b1-4d03-444e-8e84-fb1c15f91027}} , has not a gradient form. Therefore, the integral of energy density over a volume cannot be expressed as a boundary term.. Interestingly, in the similar setup, AdS spacetimes in {{formula:25caca42-3cea-4c99-bac6-6e7fc947443c}} dimensions, the graviton picks up mass in coupling to the non-gravitational bath {{cite:8cdcb83ddc00886c834257968669756cedf69ba2}}, {{cite:9efac428f0db2d80e6aee41d031c04044c92e0ec}}, {{cite:af109dfbdb787ace37d7df5db18fc2530466f0fb}}. The reason is that, the energy-momentum tensor of gravitational system on the boundary of AdS is no longer conserved. It is worth noting that if there is for example an {{formula:af8b96c8-1292-4119-98a2-f59f0f1a0eb7}} charged excitation in the island, there is no problem to associate a gauge invariant operator to it since for such gauge theory we have negative and positive charges together. In gravitational system, there is just one charge with a fixed sign. Our setup is similar to the AdS case, where we couple FSC solution in {{formula:87e6e576-6e1c-4dd0-bfec-1a79414d0ac6}} dimensionsIn {{formula:520cbb34-f157-4a64-88f5-13ef79180510}} dimensions, there is a notion of
“graviton”. to the bath. By this coupling and allowing the modes to travel freely to the bath, the energy-momentum tensor in gravitational region (FSC) is no more conserved and the graviton can becomes massive, accordingly there is no puzzle also in our setup. Of course, checking this guess more accurately needs concrete calculations such as the one for the AdS and we hope to address it in our future works.
| d | 9e6605a15647ae605c2a61cbadaa5d3b |
Database: To validate the proposed PnP-PGD method for ECG CS recovery, we use a subset of the data from the Physionet MIT-BIH Arrhythmia Database {{cite:fcd702048e714c447bbfdc8d836965cd78577b1f}}, {{cite:d8597a775d2744dd819d1d49ed456a4a55b81bd7}}, {{cite:06e34e1432c472ff51161049f4714bd8f7aed642}}.
Every file in the database consists of two lead recordings sampled at 360Hz with 11 bits per sample of resolution. It contains 48 half-hour excerpts of two-channel ambulatory ECG recordings, obtained from 47 subjects studied by the BIH Arrhythmia Laboratory.
| r | 2ffc7e533f7752a202256445e3734c70 |
Detecting conspiracy theories is a complex problem to solve. Finding conspiracy requires identifying the sentence structure and its sentiment. As the language and structure of the sentences are very discrete in a different language, it is hard to detect and tokenize the word in the correct format {{cite:cfe80929f970a05ccdab9c6a67cc046a5be94539}}. Also, current social media users use emojis, unique jargon, and symbols to express their opinion, which is still a complex task to solve for text classification {{cite:8f9933985fdc2d251e2ffc6d1e82c03139cefdad}}{{cite:15598e6c2aaa3de8d3978a9da08e28434672ba1a}}{{cite:746378c9e1caae4803d2fa3c468e3a876965e7ff}}{{cite:8899f04239bfad10be6f0a32a981cea6ac7c4dac}}.
{{figure:2d0623ca-3371-4bbf-86ee-332382258098}} | m | 823b2f01827079c5b44e6617f9bb0282 |
Transformers {{cite:43a0acf0ee65bedfbf5b6c0b5213ea234d1505dd}}, {{cite:8ee31c189f07d58ccc546eecd566c5132148c375}} have recently made an attractive resurgence in the form of Vision Transformers (ViTs) {{cite:6742bd5751fe79fc65422abdb68acab297c6c8b2}}, showing strong versatility in NLP {{cite:a229b2cf0eb80f7fbcf07f68da8723b7df396ffc}}, {{cite:a50b1adccba6de98de383876818815098a1710ee}}, {{cite:b09bad1890d49fc9e62016aad26b98d682df6fd2}}, computer vision (e.g., image classification {{cite:6742bd5751fe79fc65422abdb68acab297c6c8b2}}, object detection {{cite:31e2f4831a59f55524b1b354ae792918df366a64}}, {{cite:81b817b19c570c4d6d968925f9ca9a9ab3292fd1}}, semantic segmentation {{cite:caae81c4c2a5dc910624fc346f0f7d6e3f5b3d27}}, image processing {{cite:4880218188bc8289342c8b0f6354a52b96a19dac}}, and video understanding {{cite:b162a59e813fe1e45102206910567edfaac5bbba}}), and complex scenarios with multi-modal data.
Furthermore, ViTs can be used as effective backbone networks {{cite:6742bd5751fe79fc65422abdb68acab297c6c8b2}}, {{cite:63c0ceaf9bcf77a766efd6e9caae9fc068359562}}, {{cite:e98677bd9d55e1681c8b212624187923b4fe0797}}, {{cite:081e2ad96ac8cc361b17969031728edf78ea7abd}} with superior transferability to downstream tasks through minor fine-tunings.
Seemingly, ViTs and transformers have great potential to unify diverse application domains through common architectures and tackle the reliance on scarce domain data, ultimately addressing the two fundamental problems in deep learning: (i) strong reliance on domain data, and (ii) constant model improvements to serve evolving needs.
ViTs and transformers are largely considered as one of the future dominant deep learning techniques.
| i | 61f1926d46470df901eb7ca7f71af1c5 |
In this section, we will discuss the importance of the paper by Reingold{{cite:b807a775b8808958796f2e559edb73be1db98184}} and some further research based on the paper.
| d | 387fefed4bf6ff290cf4f5aebb41a550 |
We validate the IF-MMIN on the Interactive Emotional Dyadic Motion Capture (IEMOCAP) dataset {{cite:179d16ed13574bb3b1586cb01c6258e88cba7bc7}}.
Following {{cite:806e606d21b30736e1b7d7476f1a2d1292600fe6}}, we process IEMOCAP emotional labels into four categories: happy, angry, sad, and neutral.
The splitting ratio of training/validation/testing sets is 8:1:1.
| r | 000ace09b00667c50c1def68387cfb8e |
In this work, we propose an alternative methodology for fitting SMMs to a time-series of system configuration measurements.
We observe that for a trajectory to have been generated by a given Lagrangian, the discrete Euler-Lagrange (DEL) equations must be zero along the trajectory {{cite:9d467fe885463b987f91bcae08cf579f33749347}}.
We therefore propose to minimize the DEL residual, attempting to find a Lagrangian such that the DEL equations are zero along the trajectories being fit.
This approach is not sufficient in itself, however, due to gauge invariance.
Specifically, if the DEL equations are satisfied along a trajectory given a Lagrangian {{formula:7089350f-4765-43cb-bca5-5296e6001bcb}} ,
then they will also be satisfied given a Lagrangian {{formula:967af0f9-ac6f-4093-a443-2729594e8af8}} , for {{formula:d017d926-0902-4122-83f5-34ff4e3ae5a5}} .
As a result, a Lagrangian that is everywhere equal to a constant satisfies the DEL equations.
To avoid this degenerate solution, we propose a regularization term that ensures the learned Lagrangian is non-constant along the trajectory.
| i | 2c98a42b2be1f729e2b73d34d0e43a53 |
In AdS/CFT, wormholes provide the essential motivation for the idea of emergent spacetime. They are the geometric avatars of correlations in gravity. Motivated by the close connection between (bipartite) entanglement and wormholes {{cite:022dcfc12bbbd88fb5168307d5716769418617be}}, {{cite:c632c9ab5348316911fca5e83382542b2fb1b2fe}}, it is even conjectured that wormholes like an Einstein-Rosen bridge is universally equivalent to an Einstein-Podolsky-Rosen pair (ER=EPR) {{cite:7c1ba1271dbd7a941b07bf8385d95d703f3a70b7}}. However, it is important to emphasize that the wormholes that are supposed to represent entanglement are spatial ones that are not localized in time, whereas the replica wormholes are spacetime wormholes that are localized both in space and time. While the difference between them is explored in {{cite:97f7454072620bf38ff453b782b8dfa53fc49adb}}, {{cite:b3d57b37dfe3d1b1f3b6fca151906138e785e480}}, the de Finetti theorem confirms that the replica wormholes, as opposed to the spatial wormholes commonly studied, only represent classical correlations and there is no entanglement being mediated by them. Hence, the replica wormholes are genuinely different from the spatial wormholes from the perspective of correlations. We anticipate a rich correspondence between different sorts of wormholes and correlations that are permitted by quantum theories.
| d | 786527dcf5667736b90b13b280076524 |
In recent years, the WENO scheme has been widely used in CFD and has continuously improved. Liu et al. {{cite:2ca05e91b4f6e31b96855e7ac6926642a9036b8d}} first proposed the idea of the WENO scheme, to assign a reasonable weight to each flux function involved in the essentially non-oscillation (ENO) scheme{{cite:101352678b4a94e6a1b9290cb38277e790b906f3}}, and replace it with a weighted convex combination of all fluxes. The basic idea of the WENO scheme is that each flux plays the same role in the smooth region, and only the smoothest flux plays a key role in the region with large gradient changes. In this way, it is possible to maintain the characteristics of substantially essentially no-oscillation and achieve higher accuracy in the smooth region. Jiang and Shu{{cite:09fe245fe48d2bb47ee0fa6ec70f4b4a579391a6}} found they can further improve the accuracy of the WENO scheme of Liu et al. And they provided a method to assess the smoothness of templates and a framework for composing the WENO scheme (WENO-JS). Since then, many researchers follow with interest in the WENO scheme and got various related results. In 2000, Balsara and Shu{{cite:9de86f8e090fc3d497d60bbf0d243180a09148dc}} even constructed the WENO scheme with 7 to 13 order precision. In 2003, Qiu et al.{{cite:1a596292ff01aa44db303d7d8da25df42d4b73a9}} got a fifth-order HWENO scheme by using Hermite polynomials. Soon after, they extended the HWENO to two-dimensional problems{{cite:4b4b64e9ce60cd94bde509ea66eac9354e8cb868}} and the non-uniform grid {{cite:efe1870679a2d2bb3ff3778d00199771877ae20c}}.
| i | 364bbdd7ffb5be5bed5cec939173431c |
Since conservation laws have a very wide range of applications and it is almost impossible in general to
get their exact solutions, many scholars have explored and proposed a series of numerical methods and are still
trying to improve the performance of these algorithms.
In 1994, Liu et al. proposed the first finite volume WENO scheme in {{cite:76ebfb8ba388dcf14607a84979b07dc3f8ebd3ef}}, and then, in 1996, Jiang and Shu
improved this WENO scheme to fifth order and to conservative finite difference formulation (which is more
efficient in multi-dimensions), and gave a general definition of the smoothness indicators
and nonlinear weights in {{cite:34db91d132f985e65a9676c8cb27da0a1650544d}}.
The methodology of such WENO schemes is to use a nonlinear convex combination of all the candidate
stencils to improve the order of accuracy in smooth regions without destroying the non-oscillatory
behavior near discontinuities.
This is also the difference of such WENO schemes from the ENO schemes in {{cite:243ee85e35631ef6af07d0960f612dfdefcee006}}, {{cite:d8a856c507347a24a40514412afd6fee15c78491}}, {{cite:d0da14e4f4637c9eb5476d9924aac3bf4d77db3e}},
which only choose the locally smoothest stencil automatically among all the central and biased
spatial stencils.
Thereafter, different kinds of WENO schemes have been developed in, e.g.
{{cite:0028cebdcbf97ea21a42054dba359c32824dd504}}, {{cite:8dd0ac4e0f7a480942fb4b554c6cf20158560ce9}}, {{cite:b7c13f36020d637a75dca70ff05ee397e99f8ce2}}, {{cite:6203f75853ab43275a512fdf22376f8274096b94}}, {{cite:da77cd10e163312a1c22736d132ce8c74f57e757}}, {{cite:171847859884b86dfa3aeeffde312430b71fa6da}}, {{cite:926816a295b567c0b1e08ce1ea399b0717bcc055}}, {{cite:5eb6f9ac11b4e7918ef566d4c6885fe7c8f80820}}.
Although these WENO schemes work well for most of the problems we encountered, there is still room for
improvement. For example, if we want to obtain a higher order scheme, we must further expand the
stencil. This will make our scheme not very compact and will also bring trouble to the processing of
the boundary conditions.
In order to overcome this drawback, Qiu and Shu proposed the first HWENO scheme
for one-dimensional problems in {{cite:d0949fdc5fbc304a9d8cb0221c07b8250b3cc7cd}} and then, in 2005, they extended
this HWENO scheme to two-dimensional problems in {{cite:c4f8c72532e1814f5d91e95df98f910a32b893f7}}, where two different stencils were used
to reconstruct the function and its first order derivative values, respectively.
The main difference of such HWENO scheme from the WENO scheme is that both the function and its first
order derivative values are evolved in time and used in the reconstruction process, not like the WENO
scheme in which only the function values are evolved and used. This allows the HWENO scheme to obtain
the same order of accuracy as the WENO scheme with relatively narrower stencils.
But there occurs a new issue, that is this HWENO scheme is not stable enough when simulating certain
severe problems with strong discontinuities, including the double Mach and forward step problems.
This difficulty is largely due to the fact that the first order derivative values may become very large
near these discontinuities.
Thus, the stability issue may arise, if these large values are used straightforwardly without any
modification.
Driven by the goal of solving this issue, many effective methods based on the idea of the original
HWENO scheme have emerged. For example,
the scheme with a new procedure to reconstruct the first order derivative values by Zhu and Qiu
in {{cite:5dfeddac311f6068915f74e8e2ebcbab7bdcca40}} in 2008,
the scheme with an additional positivity-preserving limiter by Liu and Qiu in {{cite:92488f7f356bcd3d2b346e4ff33d7587463669c0}}, {{cite:07443b1ec2fdac964396d4b6378f48e30ae87903}} in
2015 and 2016,
the scheme with a troubled-cell indicator to modify the first order moments near the discontinuities
before the reconstruction algorithm by Zhao et al. in {{cite:9989ab86f7a9c2c5c67e6ab7a81e3c79463f3ac2}}, {{cite:1d597c8f859469bd747e9cae4d956527a9b27355}} in 2020,
the scheme with a hierarchy of nested central spatial stencils by Li et al. in {{cite:c7e08386ad9f7dc1d0b8c7a1766ae53b575d8bc3}} in 2021
and so on, have been developed.
| i | 6eebda0b6c899174ede0affc726b2045 |
We first outline the training details and hyper-parameter settings. Later, for evaluation, we test using: (1) three shift-based perturbations, (2) three recent translation-based adversarial attacks {{cite:b9ee3697e8248a14951e795f015c1987c5e3774b}}, {{cite:65804c92ffe52114e979b2b657062d6d893fc89d}}, and (3) a range of corruptions and perturbations. In each set-ups, we refer to the following works for comparison: MaxBlurPool (MBP) {{cite:87ac0bc835a692a25aa8ef72c53a853d28576859}}, Spatially Adaptive Blur Pooling (SABP) {{cite:b48601badfeb4603b302c0214e7b5fb30471c87b}}, Wavelet Integrated CNN (WaveCNet) {{cite:fcf5d69c485b995eb8066383c0d0d22ff6005b48}}, Batch Normalisation Statistics (BNS) {{cite:27a8759c8f90455ee4914145222a597c05d363e2}}, and Full-Convolution (F-Conv) {{cite:9189d671b66206091561a226fc924b697ee00265}}.
| r | c20b44f892d001df5a923dfe5119e013 |
Fig. REF shows examples of retrieval results by the MS method {{cite:b685d47fa2508433083d7d193931d199dfb0a26a}} and our CBML method on the SOP, In-Shop, CUB, and Cars datasets. We select the MS method for comparison because it provides the original source code for the algorithm.
In each row, the first image is the query image, followed by the retrieval results. Retrieved images with green boxes are correct ones with the same class label as the query image. Those with red boxes are incorrect results from other classes. We can see that using our CBML method for feature embedding, the retrieval system returns many more correct results (more green boxes) in the top matches than the MS method {{cite:9ef7adec52e42246f29f336086a7d7b9ea72480b}}.
{{figure:28bc2b1c-f7ba-4fdd-88bb-2455733fc44c}} | m | 26d491dd75150094aad26a2acda60a3e |
Our proposed approach is a two step process where we utilize the large pre-trained language model, specifically BERT {{cite:a79a58273f09f35e0e51235ceda391cb0c20bd9c}}. A default classification task using BERT is done by taking the utterance, {{formula:0afc2789-c530-4e73-a9e4-2ce1d73698cd}} in the tokenized form as the input, in the {{formula:d818b649-5b34-4400-8bca-3a7fddad1070}} format and predicting the label {{formula:6b75029a-e243-4f92-83bc-bf0bba44865b}} , where {{formula:291faf29-d870-49b3-8706-717d4732a31f}} and {{formula:5dbc1e03-81de-4b21-9c95-5c0f172c92a7}} denote the special tokens used by the BERT model for beginning and the end of the input respectively.
| m | 337da69ec028d2929c0ed8829322c8d5 |
We also derived column densities for S-bearing molecules related to HC{{formula:e03c0f24-2978-462e-a819-81722d6a765b}} S{{formula:d7c09259-f594-45f5-8d76-2f12057a5109}} using our Q-band line survey of TMC-1.
The line parameters are given in Table REF . For CS and HCS{{formula:8058aad2-a9ee-431c-950e-e95e86800c22}} we only observed the {{formula:d9aebcd7-810d-454f-96c5-d61d5633a42d}} = 1-0 transition
and therefore we adopted a rotational temperature of 10 K. A similar approach was taken by {{cite:d2f6e76835f73ca20e197cd61cdb635caeca955f}} in their
study of L1544. The column density of CS, whose {{formula:0613b42c-67b3-4609-95db-fb7e4ba7f7e1}} =1-0 line has a significant optical depth, was derived from that
of C{{formula:8b35ab19-a769-44d4-8604-898337928e50}} S adopting the {{formula:629d2554-8f22-40ad-b42c-88a0b1947e14}} S/{{formula:7ae8fd1e-0ab2-49b6-b287-34941f518f3f}} S abundance ratio of 25{{formula:50456163-bbab-45a4-aca7-80c01b7e5725}} 5 determined from C{{formula:925d5001-9aa8-478e-a34d-8f6b1f9d1fdc}} S and C{{formula:ddb76f61-3588-45cd-afa3-2a5caa7cea72}}{{formula:b0453a76-03d1-47a2-b76e-48e3abffa890}} S. The derived
column densities for all species studied in this paper are given in Table REF .
The derived column densities for CCS and C{{formula:c81dde04-3b56-45b0-8842-94709d027147}} S are in good agreement with those derived by {{cite:cd61b7f90bdcac5b48f67557cd14436cb3822dec}}
and {{cite:f4f9b1f6ed771e40dc4b6c1e23dfc9aeb9a28e25}}.
A detailed analysis
of the effect of the assumed rotational temperature on diatomic or linear polyatomic molecules is provided in Appendix .
| r | ec5f06923c72afb3892b02a665de71a8 |
In this section, we present our experimental results and analysis.
First, we introduce the applied evaluation metrics in
Section REF . Second, the implementation details and
the experimental setup are given in Section REF .
Subsequently, a detailed parameter study is presented in Section
REF , in which we extensively reason the design
choices of our proposed method.
In Section REF , we explore the performance of our proposed method
on different distortion types and distortion intensity levels.
Subsequently, we examine the relationship between the performance and the
amount of training data in Section REF .
In Section REF ,
a comparison to other
state-of-the-art method is carried out using six benchmark IQA
databases, such as
KADID-10k {{cite:30372a014c98bb6fcb450b987d3e3616b2f14952}}, TID2013 {{cite:dc49064e345e0024704c07476208fa7ccf88d77c}},
TID2008 {{cite:bf719217083bda70f3bc47d0325f62b9dc4dc700}},
VCL-FER {{cite:c44693696608656609170f1a65500d92625b4853}}, CSIQ {{cite:f09e55a89baf98f24569a0864be6b3204e4130d3}},
and MDID {{cite:455f0ad759d11625005b32d52164cecac45f3781}}.
The results of the cross database are presented in Section REF .
Table REF illustrates some facts about the publicly
available IQA databases used in this paper. It allows comparisons between the number of reference and test images,
image resolutions, the number of distortion levels, and the number of distortion types.
{{table:b3d78eb9-aacc-46fc-a3f7-6a937340d0b3}} | r | b90fef9d4d5c5f98ce468ad783508bf8 |
Taking into account our interest in noncommutative rings defined by endomorphisms, in this paper we will focus our attention on the study of ring-theoretical notions above for the skew polynomial rings (also known as Ore extensions) defined by Ore {{cite:4b37eb5f272a102eef6ca33b94dbfd7a8e69f206}}, and the skew PBW extensions introduced by Gallego and Lezama {{cite:a5ba22ec15571c1a8e43be1b9c9b7c40521147ed}}. As is well-known, skew polynomial rings are one of the most important families of noncommutative rings of polynomial type related with the study of quantum groups, differential operators, noncommutative algebraic geometry and noncommutative differential geometry (e.g. Brown and Goodearl {{cite:231741517ee3a788be9dd030d22b95b482a12047}}, Goodearl and Warfield {{cite:399d32903d865299d84aa1fabfe688f361978499}} or McConnell and Robson {{cite:4634f1a87a7990b9c523a518a008b60283139394}}), and a lot of papers have been published with the aim of studying different theoretical properties of these objects. Now, regarding skew PBW extensions, their importance is that these objects generalize PBW extensions defined by Bell and Goodearl {{cite:bc1af65b3ed532c55e3e1e0d0bdf92ccacce8311}}, families of differential operator rings, Ore extensions of injective type, several algebras appearing in noncommutative algebraic geometry, examples of quantum groups, and other families of noncommutative rings having PBW bases. Since its introduction, ring-theoretical and homological properties of skew PBW extensions have been studied by some people (e.g. Artamonov {{cite:3fc44d128411959f3e596346d964cfb5ec9a189d}}, Hamidizadeh et al. {{cite:0ade3c2aabd29bb912215f951f9810a6381d9dfd}}, Hashemi et al. {{cite:0f2e123d5ed21c25b7068b3b71f1f27b9ccef979}}, {{cite:0779c449ef439d6b1609ab2f105e8a8e04a6f928}}, {{cite:1b4f2e575c4418d2a2e5c148115969315e1499c6}}, Lezama et al. {{cite:4245501b3e75cf70b7d72b3565a7f614edbd7717}}, {{cite:354367d277f6742425e54f62064900162fcce899}}, {{cite:d1019b953d2e3738cab00d08ee10dc11154b3d8b}}, {{cite:1318d7a1fe94caff592bf0f87c9f235d69c44e7e}}, Tumwesigye et al. {{cite:53d37df24b6f1824a66f5847cb10f2f215afb9a7}}, and the authors {{cite:863a835b7ad9ae7ed6bfa18efc512cd0af7eccf5}}, {{cite:e6c97c8909e85270b99970620a0125bab9da940f}}, and {{cite:cc8f3827a3568bac684df4acb8b06bae64ac1a38}}). As a matter of fact, a book that includes several of the works carried out for these extensions has been published (Fajardo et al. {{cite:c69a19d17954fae5d5c7bdcc279a72beb9954c54}}).
| i | 6da17eb372d26cd1c5c56b720b8af0be |
Recently, there have been experimental observations that cannot be explained by the three-neutrino oscillation framework {{cite:e37c62a5e299940b67667726191c0ff54eb255ca}}, {{cite:bb63e4c07ed63b9c8ffe602f28b293ff12b9fce8}}, {{cite:2b94bba75ddaa8d868e9b845256e3d5479a2639e}}, {{cite:d626b2bea4fb77559391897ea51fb8a6f330d04f}}, {{cite:09d378b2e5f8ec2269ab62c3ae4fc83251fc9baa}}, {{cite:f600ce6b96d50f0f8d10cc1c97e277c7c88334fe}}, {{cite:fe6aa9c4e724e13a8e812eeb219ce48cecb9a2e0}}, {{cite:a7643370eceaf68aaf5befb63929304b14f9dc9f}}, {{cite:8b622ef7772f7fab8d5f074a3af8aacf02c0cf7d}}, {{cite:5937f9038210b0737a6aa7a873205b919c2cd371}}, {{cite:af84cca741972d0ebca1c1d9609e8e067137169a}}, {{cite:9f885454acc1be233789e92cdc9284f25d17bd5e}}, {{cite:bcd9cf466e7056174f2312196c4cdb2aebf322be}}. However, these observations could be explained by adding at least one additional neutrino (called sterile neutrino) with mass in the eV range having non-trivial mixing with active neutrinos. The mentioned sterile neutrinos are singlets under {{formula:8907c89b-ee89-43d9-8c36-92713ed84731}} which do not take part in the weak interaction but mix with the active ones that can be verified in the oscillation experiments.
Nowadays, there are a number of schemes favouring the existence of sterile neutrinos, including the (3+1) scheme {{cite:64a1f665b77cce60dac1c1173413cae46c9748cc}}, {{cite:d6559dba4119fdcd79bd1c93e4528aba0baa054b}}, {{cite:7bcdefaee5036f7fdc7b5e7269fae2d4aacadeb2}}, {{cite:537428611fa3d47a9b9874b07efa23ee7e3b97e7}}, {{cite:5516109e6d50f4d87cd2e6390d8748d713ef9658}}, {{cite:8a4a9562471d9549ffa7b2a42dc6e64b978f88ab}}, {{cite:90a75f9d647d3aee4a64df59e4556eb2e72b3a1c}}, {{cite:338228e9d5f90bd13b20081ccc4cabf21e5f4021}}, {{cite:ea468cf845a55dc76691527add3e2e951aa60e21}}, {{cite:82f91a87e66131f5e3a83e3844ee32475ec5925d}}, {{cite:3bcbb7269acc5e792a88fd5a2b8ee4055a49ce15}}, {{cite:3fd1d74ab73121e8082769346592edb8c2e53a41}}, {{cite:23054d5c4995c9119a83c5a6e75d93817f5b3867}}, {{cite:1ac0d8af45940c01e69eb2a01354bba3c362ac77}}, {{cite:54486f68ef82fc67512b0cdb1730385fb1a83835}}, {{cite:bcd9cf466e7056174f2312196c4cdb2aebf322be}}, {{cite:7bde7d632721340f63db181444470aeff365041c}} in which one sterile neutrino with mass in eV scale is heavier than the three active ones; The (3+1+1) scheme {{cite:b0b49083f0fe3c36da2127a286462c86f6732740}}, {{cite:2d72ff5d15192cabf9939a1bfdd31883a062ea60}}, {{cite:6373397aa5ceb6896b76c1acc412c7401f7c01aa}}, {{cite:236cc97b667139e4c9c4faff3f437a22c3eac68f}}, {{cite:796ba38942b6047a3090439d30e1cc131456de36}} in which one sterile neutrino with mass in eV scale and the other is much heavier than 1 eV; The (1+3+1) scheme {{cite:960986b09d7b1e159f47ff84d2efd2e6652a834d}}, {{cite:f50322082fd6d879397c88f5d5b6b931c4bd50cb}}, {{cite:1b6d971e2d522f089bac7488be91f9faa6fc71c5}} in which one of the sterile neutrinos is lighter than the three active ones and the other is heavier; The (3+2) scheme {{cite:95d9be6d9ff269c0b8452b784ccb5357a3c0884f}}, {{cite:8c0736f5dcfb09b0e59b9e81ec6f167406b7e20a}}, {{cite:d2b37fc9f5cec8ebbcaabd20220d0d6a341526c4}}, {{cite:967e0e507d8adb29fd1a4b6ba93bf9b64594757d}}, {{cite:afe0f85152d881400e4212650852117caae00ece}}, {{cite:fd20fb3ee4c128cd128401fa8e774524913603df}}, {{cite:a77cd8e05424f2f8715a8b9ce447d4fb63916fdd}} in which the two sterile neutrinos are lighter than the three active ones are added to the standard three-neutrino oscillation framework. Among the schemes with sterile neutrinos, the one with one additional sterile neutrino with mass in the eV range (called four neutrino scheme) is the simplest extension of standard three neutrino mixing that can accommodate the anomalous results of short-baseline neutrino oscillations. Among four neutrino schemes, the 3+1 scheme is preferred because the 1+3 scheme whose one sterile neutrino is lighter than the active ones and the three active neutrinos are in eV scale is ruled out by Cosmology while the 2+2 scheme is not suitable with the atmospheric and the solar neutrino oscillation data.
Currently, the neutrino mass squared differences and the mixing angles in the three-neutrino scheme{{cite:c68a78a179812f8385baeef4e36f77d926a8fd35}} and 3+1 neutrino mixing angles{{cite:a9f293441e16200632b618ce14005c6b93e09952}}, at the best-fit points and {{formula:fe0d311b-1ab3-4488-8a53-f0bde702abc7}} range, are shown in Table REF .
{{table:c1c022c2-a3c0-4743-82be-f7ac711ec656}} | i | 2e71fe9ee52aae05a4ac634ad3a9fa94 |
The optimal control approach is inspired by the success of fluid control algorithms. The variational problem is solved by the gradient decent method. As indicated in the survey paper {{cite:dacb640eca1cce808999b9540bf3c1b50f7d6a11}}, one could add a penalty term to {{formula:3c44d0ab-9cf3-471e-a8c6-52e76b146edc}} for specific application with a small parameter value {{formula:d9275ae1-ef54-4342-a607-abfde035ab8b}} in order to use prior knowledge. We now present a teapot example.
| m | c4eda75e404808a9442a9345c96adb26 |
Latency Reduction as Form of Compression. SNNs with multiple timesteps are trained with BPTT {{cite:cb56d8d83e8a1660f318973f4dbeb21020daf36c}}, like RNNs. If we unroll SNNs in time, it becomes obvious that each timestep adds a hidden state to a spiking neuron. Thus, reducing latency compresses the SNNs in the temporal axis and the eventual unit timestep network consists of a single forward pass, with no internal states. From a related perspective, we can perceive gradual latency reduction as sequential temporal distillation, where at each step, the network with higher timesteps acts as the teacher, while the student network with reduced latency learns from it. Similar sequential distillation (training by generations) has been implemented in ANNs {{cite:3e901646292aeeac2970c197db821f89364a5533}}, {{cite:3c20221936bcf1ed4725d67cb4d69d0cc69e5094}} to obtain better performing students. However, in our case, there is no explicit distillation loss and the sequential training is a requirement for convergence instead of being just a performance enhancement tool. Also, unlike ANN sequential distillation, the student architecture remains same spatially throughout all generations in the proposed method and compression occurs in the temporal domain of SNNs.
| m | 10201bd3fcb3b33a3f597ff32b650258 |
It is useful {{cite:2e932c0922913b16cbd9b4e06ca88eb7805c89f2}}, {{cite:a99f5d71ec38c6e7e4bdb68d45bd6ec91ead3fc6}}, {{cite:c13fa767cf4829ff0b081bc238e4e98e0a97c01a}}, {{cite:61f3e283b135b43197d84bb1e74f00630f3486e7}}, {{cite:a40334e84200234ad9034bd4076814be768fb370}}, {{cite:7bcfd30b396443a63cdd1842a5d04e46214bf7d8}}, {{cite:297a343d07b2c727e4173acb2d25ac6b66f56bfe}}, {{cite:9010eddc52c72b99b2ba429f969ec2ac7a51ab3e}}, {{cite:6f96446ba53e34ab5e183d342e620a9f2836d3c6}}, {{cite:cea2a89d22806519aa9f0a662f2cb6d2e8d9e8b9}}, {{cite:ce0f2890ee741a283c7ac2464cb11a8d2978e3a8}}, {{cite:47345fc271d9b70ae0ffd39887475f505b84a5f5}} to consider
the pion form factor as the boundary value of an analytical function
which has a cut on the timelike axis, of the square of the momentum {{formula:147ef0d0-8d70-4e0b-b808-088977c73bc9}} starting in the branch point {{formula:19239d81-4e26-47f5-8df7-43f4430ec866}} , the production threshold.
Thus the electromagnetic form factor {{formula:9b42e188-391d-4f4e-924b-6e97ff96b1a0}} and the production form factor {{formula:d99d140f-aea0-4b76-aa9d-a0adc4e3b7a1}} can be evaluated from the dispersion relation for {{formula:a1ccdb3a-5d34-43d7-80a8-07d69f6f39e7}} :
{{formula:1f62a174-8d87-4e68-9518-7cf26571c49b}}
| i | cbbe93b632b6f75ac65a055ee7a47d46 |
Albert (short for “A lite BERT architecture”) {{cite:f43a920dcf3b342e46a50d4e341af4e59bf178b9}} reduces the number of parameters, compared to BERT and Roberta, significantly. It uses two parameter reduction techniques. First, it decomposes the vocabulary embedding matrix into two smaller matrices. Second, it shares parameters across different layers (thereby reducing parameter growth as a function of network depth). The authors demonstrate that this model achieves significant speedups without affecting result quality significantly.
| m | 8200bcc45ff91e0b9b9b48dc95b5e6f4 |
Theorem 1.1 {{cite:ca1716cb1c9c3c3528edd71b6a44ac8e2500cd1c}}
Let {{formula:73f2e37b-4eee-42d1-ad39-b03e1df16045}} and {{formula:2f6bcfa6-a2b0-4223-8394-d2aa39a6162f}} and {{formula:cc4dfd27-4a47-44ff-ad87-cc195d8cf7ad}} . For any {{formula:6d949b2c-1c04-4ca2-90f7-03e9354b31d6}} , we have
{{formula:0294a1a2-43b9-424f-ae40-a9e5f03b42cf}}
| r | 371b253011bbfda6e4c01cb1f8c51864 |
Strong inductive biases towards abstract structure allow humans to generalize to novel environments without much experience {{cite:ba97641202e74f053301e516ae85646a7983343d}}. By instilling these biases in our artificial agents, we can work towards enabling machines to demonstrate human-like general intelligence.
| d | 645bdfbe31755740cbffd850c9eae590 |
We show the efficacy of proposed StatDEC by comparing our results with unsupervised DEC {{cite:259c6308f218c876a3343e598eae564ebb25c285}}, IDEC {{cite:6a61a5ab933572840c25d34e4de3e4af01dc66b6}}, and VDEC {{cite:c385eee3267b065e3a8a5849d1e940144abd81aa}}. These methods can be viewed as a variant of our method when the reconstruction loss and network architecture are different. Note that the reported results for DEC and IDEC are based on our implementation and the results for SCAN {{cite:af6deb3e2cec8de4aa35236e7f5561f025c258ce}}, IIC {{cite:61eb50b5a7953e606b308e606c1175d080087ef2}}, and DCCM {{cite:5575a2861db8cfdb0ba98f02a58fca0e44c6f6f7}} are based on GitHub code by the authors. We further demonstrate the effectiveness of the proposed method in handling out-of-distribution and imbalanced data situations by comparing results with and without statistical pooling layer and contrast results to those of two supervised state-of-the-art techniques LDAM {{cite:6f140a2cc3fbf4aab470562cf71f90399871d59d}} and TAML {{cite:496ad2cab9850149a6b35e52e565642cc619c69e}}.
{{table:12812128-28c8-44bb-b923-ee973a06946b}} | m | f294756c8f6264dacd48bafd517d5769 |
The advancement of semiconductor industry has enabled rapid development of AI and deep learning technologies,
which in turn offers great opportunities for novel chip design methodology, enabling faster design turn-around-time, better PPA (power, performance and area) and higher yield {{cite:8e959e49da5ebf3575dca79898c44c4b0b728e3e}}, {{cite:deb511b3932131c210bf4d9ca3274e98629b8572}}, {{cite:0cd2489b37ca853cb6bc5779340435ba52389476}}, {{cite:2f6906b2272749c30d32122634c8c11f3497dfa2}}, {{cite:bd42027ce058ed699a08fdf3ef919594316a4159}}.
Particularly, recent researches have demonstrated efficacy using reinforcement learning to place circuit components (macros and standard cells) on to chip canvas {{cite:8e959e49da5ebf3575dca79898c44c4b0b728e3e}}, {{cite:2f6906b2272749c30d32122634c8c11f3497dfa2}}, which is one of the time consuming phases in chip design flow.
| i | 516bb9ffd79d27b352a8e65a1f5ad5e6 |
The strongest machine learning methods in this study were convolutional and/or recurrent neural nets (CNNs, RNNs, or CRNNs), as has been observed in other domains .
In order to ensure methods could work in conditions different from those in the training data,
various participants explored self-adaptation, in which a trained network is fine-tuned upon exposure to the new conditions (without needing any additional ground-truth information)
{{cite:a11eb3806c29a96da65245012ac089b6b09e8917}}, {{cite:a217b368c50aba1334320b5bc8713c9bb3cc1827}}, {{cite:d99629b81fafb8bdf75633c26b8c8ac5b510b979}}.
Participants reported mixed results of this, some observing no benefit.
We found little benefit of self-adaptation for matched conditions;
however, in cases where matched-conditions training data is not available,
we found that it can reduce the adverse effect of the mismatch (Table REF ).
| d | 077a0db36ee8752b699103e9e75ae336 |
Strongly Supervised (SS): We apply the standard concept drift detection procedure which assumes that all the true labels are immediately available after making a prediction. This can be regarded as the gold standard. The term strong means refers to the fact that all labels are available during testing {{cite:cb50f61e6f4834e8cafada943a00cd23a7bcd970}};
Weakly Supervised (WS): In many real-world scenarios, particularly in high-frequency data streams, data labelling is costly. Hence, predictive models can only be updated using a part of the entire data set. This process is commonly referred to as weakly supervised learning {{cite:cb50f61e6f4834e8cafada943a00cd23a7bcd970}}.
We simulate a weakly supervised scenario in our experiments. Accordingly, predictive models only have access to l_access% of the labels. In other words, after a model predicts the label of a given instance, the respective label is immediately available with a l_access% probability;
Delayed Strongly Supervised (DSS): Labels can take some time to be available. We study this aspect by artificially delaying the arrival of the labels by l_delay instances. After a label becomes available, the respective observation is used to update the change detection model;
Delayed Weakly Supervised (DWS): We combine the two previous scenarios. In the DWS setup, only l_access% of the labels are available. Those which are available arrive with a delay of l_delay observations.
| m | 3ba3ce30fb2948b68e4918897b87376d |
We compare PALT with the following KG completion models.
(i) task-specific models (designed for KG completion): TransE {{cite:471c2b965ca41cd9b2c8a924a01ace56c728705e}},
TransH {{cite:6d513ecca1f38cb30a3e0eab1d3f71d91d2a4e1d}},
TransR {{cite:e4a265d328415de4b6b00622fa407fda275d0c3c}},
TransD {{cite:af6ddc4302958d8a3c4d66eb482f98658c1e06c9}},
DistMult {{cite:cc3d03973763290cef2827b6138030fa5c5bc910}},
TransG {{cite:9533495e0387f953745a105f0fd978e4ee04fd5c}}, TranSparse {{cite:5309c6580005e8494404f53a4f55c89ff1f56612}}, ComplEx {{cite:ef8d54f54fc97cc9ab19ff07400239e66a9587ab}}, R-GCN {{cite:4454e96c06e934b1708c2ab09fc38fbebb415cef}},
ConvE {{cite:94dd47cbd6081abdad2fca09fc7b4a8c434e5894}},
ConvKB {{cite:a47cc2b5abfa57be96a60136a0f5ac4e16196791}},
DistMult-HRS {{cite:99dfd6e50609cae68e14c1e7e56b208a11c96083}},
RotatE {{cite:3f9197bcc383b7db04769f159f893522321393ff}},
REFE {{cite:75bdc4d8cb2956653a8c2d1b5dbc85f5a308df31}},
HAKE {{cite:d9df479a25cadafa5bf0e22647017b3d9153d547}},
and ComplEx-DURA {{cite:a2a9da1ef290cfb01e64c4ff1fb66432b401315c}},
NTN {{cite:ad70a5507005a18ff02b38d3c3440e511658240b}},
DOLORES {{cite:89c150ccd4e10b42a4ea38ff37c946c6d1225e76}},
KBGAT {{cite:523dfedeb392f4d64eef42db6d51a1f72e4d4c26}},
and GAATs {{cite:4848711fb18584a77062655ab3344a76a48f8f79}},
TEKE {{cite:7dfde29512c5f672a60b512bc8017cedd652b25f}},
stAR {{cite:5f46c9d2ea5602844b3ab3073ac16050ad5974b5}};
and (ii) a general model KG-BERT {{cite:1b74f202f759fe7ebcebc7bd1754f714769630f9}}. It utilizes finetuning to transfer LMs for KG completion, and is task agnostic.
| m | 226d5a18a67d8cc952f5fd35f93b57c5 |
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