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We develop the theory of hEP and prove that this allows computing exact gradients locally at synapses from finite teaching signal amplitudes of adiabatic oscillations.
We numerically quantify the accuracy of our estimate and show that it outperforms classic EP, especially in the presence of substrate noise and in deep neural networks.
We demonstrate learning with an always-on oscillating teaching signal, thereby alleviating the need for separated phases.
Finally, we show that hEP achieves the same performance as BP in deep CNN trained on CIFAR-10/100 {{cite:cfaf0f909bf245bfc3f1ce690104803bb6464841}}, and ImageNet {{formula:c7aa2c75-3ca2-40be-86ca-dcd87f180524}} {{cite:b717832369ff2188478eb07fa59eca0f27a3aea2}}.
| i | 4d6a6468bf6a6400cfb5e56c94a069ab |
Machine learning has shown great success in building models for pattern recognition in domains ranging from computer vision {{cite:325a34f74e2e83e28960fd0bca20fbdc5b4a02b3}} over speech recognition {{cite:085cacffe2adcde13c3b256dbec1d1b8edcfecf3}} and text understanding {{cite:19bed7236392743bf306fdb7e62b8ad73a7242b9}} to Game AI {{cite:32b7aa75d4656048ce9c03e56093a289c5bc75db}}. In addition to these classical domains, machine learning and in particular deep learning are increasingly important and successful in engineering and the sciences {{cite:91d2486033ad6bc416b1084e78695ffc84a2fdc8}}, {{cite:a59f2f7c6bf9963b55e0eec50ab0b09d9e0df052}}, {{cite:e395de5f8de2fa8cc8f743a7d1d1902a680c50ef}}. These success stories are grounded in the data-based nature of the approach of learning from a tremendous number of examples.
| i | 201e26ec42ed4c3d803e1ddcea1c64d3 |
Many stain normalization methods are based on generative adversarial networks (GANs). In short, GANs are suitable for conducting image-to-image translation {{cite:58a82d16606957719ca84823ef3f72a00d91c7ac}} tasks in which images of one group (i.e., a "style") can be transformed into one of another group. Note that GAN-based style transfer is a kind of unsupervised learning, meaning that images in the two groups don't need to be paired. This is a major advantage over many traditional methods in which paired images are needed. StainGAN {{cite:e4f86585cdf474a1f03e89f86c7cc1a9de7c3458}} trained a cycle-consistent adversarial network ("CycleGAN" {{cite:c35a4a40fb66c71edaf0e452ef783e4fe875da05}}) to transfer H&E stain images from one scanner style to another (i.e., Hamamatsu Nanozoomer 2.0-HT to Aperio Scanscope XT). Similarly, Runz et al. {{cite:2973e494abbbea7859da35340f6d048e467872d1}} investigated the potential and limitations of using CycleGAN for transfering H&E stain images. {{cite:1232df57d5b0faa51852ac5455ce5eb237c21114}} and {{cite:332c12d5babcff397b78c525a779208be6aa283b}} used GANs with disentangled feature presentations (e.g., DRIT++ {{cite:fee5239ec69acaec00b9763fb0cc26a307a84c7f}}) to solve the stain normalization problem. Marini et al. {{cite:b663b5af4771f3be4ab0278ceb1376bbd2888e9c}} proposed a novel convolutional neural network (CNN)-based approach that aims to tackle highly heterogeneous images.
| m | ba23e013a6e05cec83697bd494eed1ea |
The existence of a suitable enlargement {{formula:8d7d1105-3fde-4ff1-a517-61641af72046}} such that {{formula:3297d96a-d386-49e8-badb-a97be1d19ff6}} is guaranteed by smoothness. In particular, following the analogous setup for ODEs in {{cite:502e90f6369ab40a8e05fa81010bd21761a20dcb}}, {{cite:dbe5fcbf7cab45c1b36950cdab21b3f186b612b7}}, {{formula:0d3802d2-bf9e-4a89-a5ac-cd78f11d0d8d}} can be chosen so that {{formula:02c86a84-9a05-49d7-9f10-43b5cf31ca02}} is defined/extended over {{formula:42662c43-73c9-42b0-8a5e-fb5c35e1123c}} as a {{formula:75d54c7b-81fd-42f2-bcd7-f212fe84abd3}} smooth graph
{{formula:c48b8913-deb8-42c2-8e8b-87dff058c799}}
| r | b995a905b3cffdf9a07ae10bd80f7f9c |
One important concern about this proposed scenario is the comparison with previous transport results. Albeit previous reports show that even bulk Bi{{formula:780001b2-a111-420a-960d-c93c5b685b55}} Se{{formula:a8113b86-1145-4cbe-81d6-91ada7ba0c4a}} samples have a WAL effect at low fields, and thinner samples have a more significant critical field ({{formula:02d70754-557a-4db1-bb06-67109042d213}} {{formula:06e4166a-f8a5-4288-ae5b-3e1f59c5ca4b}} 1 T) {{cite:f6c0630cd843cf7c6b43aa38bb40254c6bd442f6}}, we must pay attention that all the results report WAL effects at low temperatures. In order to understand the origin of this discrepancy, it is important to understand that transport and ESR are different techniques. While the response in transport is a macroscopic, global property, in ESR we obtain a microscopic viewpoint. In ESR measurements we have two distinct relaxation channels: the relaxation through the spin-phonon process and the spin-spin relaxation, which is faster and involves the carriers of the system {{cite:0c93d00c3b39abf36f42de840a729796bb9393ce}}, {{cite:63d2f2dc2aca4a27d727171b911ff0d9cd2f0ebc}}, {{cite:1d9aedda8b2b07789e0fb02be376a1038a275676}}. From our Gd{{formula:b9da471a-8682-4fec-867b-fc54c33c8254}} spin dynamics, Fig. REF , we clearly obtain signatures that the coupling between the 4{{formula:665e92d3-a32d-4060-b795-ae533f659ef1}} local moments and the carriers are relevant, and they should dominate the relaxation of the system. Concomitantly, Gd{{formula:54d257ce-2a53-4239-84ff-d00abe221f00}} has {{formula:e9d50810-0b9d-4239-8980-90d90ffc7ebe}} = 0, which means that the spin-phonon coupling is rather weak. That means that any potential scattering involving phonons, which would mask the WAL signatures, are going to be heavily suppressed. Such locality also help us to have an insight of the difference of critical fields. For thicker transport samples, the cusp has a critical field of only {{formula:e343a389-50f3-4655-84cc-74126534c74e}} 1 kOe, while our X-band measurements occur at fields of 3.5 kOe. Again, the phonons contribution are going to be heavily suppressed, which means that it is understandable that the critical field for the local WAL effect might be comparable to the fields of thinner samples. Nonetheless, it is worth to note that Q-band measurements have fields applied at the order of {{formula:45aaa498-ec4d-4b12-9aac-7d555b187cb7}} 12 kOe. An alternate scenario could also rely on strain effects, which are originated from surface effects, playing a bigger role in Q-band measurements. In this scenario, the central line could have a smaller line width and we would be able to describe the data without an additional collapsed spectrum. Two different results indicate that this is an unlikely explanation. First of all, the data is better described with two resonances of different line widths at the same {{formula:c404b9a9-927d-409f-a6da-00f824fd2cf5}} -value. Even with we do not take this fact into account, the intensity of the central line should be, at least, double than the expected value to fairly describe the data - which is inconsistent with the crystal field Hamiltonian. Another point is that the skin depth is {{formula:a2c103ae-c057-4a9e-8bd3-ba71e2b3dd3d}} 11 {{formula:c8950153-793a-4804-94fb-3db3db1c58c1}} m and 8 {{formula:44ccee52-459e-4578-ac39-f20763536cb3}} m for X- and Q-band measurements, respectively. With those skin depths, as expected, we do not see any signatures of surface effects in our data. Therefore, as far as our experimental data shows, the bulk dominates the ESR signatures and strain effects should not play a role.
| d | ee4e68980d39a556d0d307ae1243df9c |
XLM-RoBERTa
We also applied the XLM-RoBERTa model {{cite:cb527f03b5cef2922a2c94e8133e4ca3354e3a36}}, a multilingual version of RoBERTa {{cite:88034746e8fc21d4a7b0b277c3990be164a32306}}, within the Hugging Face Transformers framework.
Similar to BERT, it was adapted for the NER task, by reinitializing the output layer and fine-tuning.
The model consists of around 560M parameters and was pre-trained on CommonCrawl data containing 100 languages, including Kazakh.
| m | 393cceb4f3ec6e3b0f10e9fa6960b8a4 |
Opposed to the behavior during training, we do not randomly sample from the probability distribution of the actor during deployment. We instead take the mean of the distribution for the continuous case and the mode for the discrete one. This is reasonable and best practice {{cite:0d4aad5b1ae6fd28d1c98c9b04af0d5b309399b4}} since during deployment we do not aim to explore the possible choices anymore as they were already learned during training: Only the optimal choice, which was learned, should be applied and no exploration is necessary since the actor does not train anymore. Furthermore, the critic is only necessary during training for the reward of the actor. During deployment, it can be disabled, saving computational power.
| r | 9f1dc8155ea981cddfec49427b69b6c9 |
In this section we will show there is no {{formula:e7d69686-7ea3-4368-a723-55896f0fe551}} -size estimator with {{formula:03d0b49b-1a05-4251-b573-a0a9b9dd59b0}} that can achieve the minimax rate when estimating {{formula:ddae052b-1376-4543-88be-dbdc9f8075e4}} . We first recall the metric entropy of a Sobolev ellipsoid satisfies (see {{cite:b5e64f08a59257f470b38bd9b809030e72bf177d}}):
{{formula:bb7c2fa6-92a4-4e29-8c77-e9a3d18088a0}}
| d | 7dbff015502b5442e242b17c1dec2e2f |
({{formula:182c4715-420d-470c-8a18-46cd8aa09da4}} ) that indicates when the group intersection {{formula:f5680e91-14bc-401b-b6c7-549350bdb8cb}} is faulty.
Building upon these new abstractions, we identify the weakest failure detector for genuine atomic multicast and also for several key variations of this problem.
Our results offer a fresh perspective on the solvability of genuine atomic multicast in crash-prone systems.
In particular, they question the common assumption of partitioning the destination groups.
This opens an interesting avenue for future research on the design of fault-tolerant atomic multicast protocols.
System model
model
Basics
We assume a distributed message-passing system composed of a set {{formula:35f1ed08-16c9-4b64-873c-679331778879}} of processes.
Processes execute steps of computation.
These steps are asynchronous and there is no bound on the delay between any two steps.
For the sake of simplicity, we assume a global time model, where {{formula:c0376602-bed5-4084-9e18-15d59be68195}} is the range of the global clock.
Processes cannot access to the global clock.
Failures and environments
Processes may fail-stop, or crash, and halt their computations.
A failure pattern is a function {{formula:6545db92-f5be-450b-b6df-ddd0d04469ab}} that captures how processes crash over time.
Processes that crash never recover from crashes, that is, for all time {{formula:66959b94-d110-421f-a828-8521b4baead1}} , {{formula:3aa7c33d-5ed5-4bd0-bbd9-449fa3f2028f}} .
If a process fails, we shall say that it is faulty.
Otherwise, if the process never fails, it is said correct.
{{formula:8066e6be-55a8-44cd-8905-71aec89ce35d}} are the faulty processes in pattern {{formula:f24754fd-f92b-4a31-b40f-5da8a289181a}} , and {{formula:912bd57c-314f-466f-8eaf-0469a41bff53}} denotes the correct processes.
When failure pattern {{formula:35488724-f4b6-4e3b-a12e-65464f41bad7}} is clear from the context, we shall use respectively {{formula:9f11ec69-6ae5-4b24-bc15-f9ee0eceea18}} and {{formula:a1eb63e3-8e47-4d8a-92e8-1c69c9f57b17}} for {{formula:02323f2f-611c-449f-bfff-b7669d289383}} and {{formula:c0a1c1b1-c783-4dba-a965-4904343d43fa}} .
An environment, denoted {{formula:0ace388d-ca5c-4c58-8484-70b158cc90bc}} , is a set of failure patterns.
Intuitively, an environment {{formula:b6d0ad7c-ca47-4499-83c2-215e89bdd4cb}} describes the number and timing of failures that can occur in the system.
We denote by {{formula:314cc552-2a72-48de-b708-60e432fb5bc6}} the set of all failure patterns.
Failure detectors
A failure detector is an oracle {{formula:5ae90c62-bec7-486e-9db6-228291c7e655}} (also called module) that processes may query locally during an execution.
This oracle abstracts information, regarding synchrony and failures, available to the processes.
More precisely, a failure detector {{formula:ce8924a3-bdef-42e9-924a-4936724875d3}} is a mapping that assign to a failure pattern {{formula:7520abb5-1884-4ed7-b392-a615d2e16627}} , one or more histories {{formula:805dfed4-9d0c-4f3b-a619-ae752f36f0f0}} .
Each history {{formula:a764e03c-948b-40d3-91a0-5c70dd73dfdc}} defines for each process {{formula:eb541295-76de-4dde-8fec-f96b1291ad6e}} in the system, the local information {{formula:4ac296b9-d0c1-43c8-9c97-7b2dd720fe65}} obtained by querying {{formula:8b0b2b01-9c8c-4a24-9c04-453073af394d}} at time {{formula:0a07ea00-caae-490c-bc37-cdfba613e0b5}} .
The co-domain of {{formula:eebcc99b-05e7-454c-8dbf-764895a77c06}} is named the range of the failure detector, denoted {{formula:a637c6ec-37fa-482b-b0fa-aee54cf33c43}} .
A failure detector is realistic when it cannot guess the future {{cite:b700583d03cf681f9e919ef7f4d8f730b11e1cfe}}.
This means that if two failure patterns have a common prefix, a process might not distinguish them in this prefix by querying the failure detector.
Formally,
{{formula:c86c4da5-6f3b-4d31-965f-d1127dddf956}} .
Message buffer
Processes communicate with the help of messages taken from some set {{formula:70d38e24-4a26-42fb-ab7a-fb21e6305d26}} .
A message {{formula:3c62a152-08c4-47a1-b040-b75f76cba51f}} is sent by some sender ({{formula:084ab515-dc08-45ef-b5c8-459802e19969}} ) and addressed to some set of recipients ({{formula:09f1a851-bd73-4aab-affe-c045c3d9535d}} ).
The sender may define some content ({{formula:3cf8f969-55b1-45af-a6ce-eb2df15cd573}} ) before sending the message.
A message buffer, denoted {{formula:6596ca22-35b5-40d3-90f7-a12f612bc736}} , contains all the messages that were sent but not yet received.
More precisely, {{formula:6937dae8-0cc3-4b51-be76-56d04d653113}} is a mapping from processes to elements in {{formula:4936cdce-a522-4779-a340-5d4d1274f554}} .
When a process {{formula:d5e0e663-1feb-44ab-84c2-8c24261baef8}} attempts to receive a message, it either removes some message from {{formula:6ce6465c-56f8-4521-a4e3-4aa8c7d87056}} , or returns a special null message ({{formula:1857e871-eca5-4746-9c66-bea8a2863726}} ).
Note that {{formula:755d1546-65a9-4ea3-bf18-6e49b336e875}} may receive {{formula:105f80d5-e3c8-4cb8-a961-50c8fc00cd90}} even if the message buffer does contain a message addressed to {{formula:cb1db83c-caca-4763-9977-e0e29246fd14}} .
Algorithm, step and schedule
An algorithm {{formula:189b38f0-4027-4327-8592-86bd1045cc73}} consists of a family of deterministic automata, one per process in {{formula:91ee4206-d85e-4519-a198-dc5846a894ea}} .
Computation proceeds in steps of these automata.
At each step, a process {{formula:db3961e6-5ae7-4a83-a9b5-24f8a07c9dbd}} executes atomically all of the following instructions:
retrieve a message {{formula:f0895eee-5f8c-4ace-aaa8-9941f8781838}} from {{formula:576e555b-0920-46d9-8d5a-fd4561aaf04c}} ;
retrieves some value {{formula:042a507d-09d4-4f94-8bfc-488c3fd72594}} from the local failure detector module;
changes its local state according to {{formula:641da305-5b5c-433b-b4d8-b2380805fa7a}} ; and
sends some (possibly empty) message {{formula:b8e1585b-f254-46cf-986f-6037638c770e}} , by adding {{formula:3bd5b0ad-6219-484a-9577-7579a6bbdab0}} to {{formula:b712dfbd-4ce5-41f6-930b-08b0daf97fc1}} .
For a given automaton, a step is fully determined by the current state of {{formula:5fa476c4-254f-4e33-9274-7e5af0aa749f}} , the received message {{formula:d78db72d-addc-443f-86ec-ea82499bd6c0}} and the failure detector value {{formula:cb92a4c6-42b3-457a-b08c-f42655440868}} .
As a consequence, we shall write a step as a tuple {{formula:d33d9950-9e7c-40ff-8740-8bab0df0484b}} .
A schedule is a sequence of steps.
We write {{formula:df8ce2c0-c917-493d-8e0d-320ab7bd08dc}} the empty schedule.
Configuration
A configuration of algorithm {{formula:928a0d15-f1ca-4c11-bb59-e9c7536941f8}} specifies the local state of each process as well as the messages in transit (variable {{formula:a9151aed-bae8-4d04-9e40-b34022c6bd8c}} ).
Given a predicate {{formula:32e0b4bf-5e67-452c-ae6c-2c23f4cb4b3c}} and a configuration {{formula:b950cc4d-0478-4fbb-b94b-b709ea109bfe}} , we write {{formula:922e6013-2344-400f-8603-880d109cc092}} when {{formula:33513db9-c9f4-4ea1-948c-736c6064bdcf}} holds in {{formula:e8d220cb-a1d9-4ff4-94c6-bf2e3d21e2be}} . In some initial configuration of {{formula:e455f6ac-551b-4505-a16e-bb903b831a12}} , no message is in transit and each process {{formula:1ee8e696-104f-4cf1-9faf-626693e75b6e}} is in some initial state as defined by {{formula:c130fa6c-c28d-4a3e-a1d0-a1c3be8bbab9}} .
A step {{formula:1b783efd-6af8-4f57-9021-fd5c85658f8c}} is applicable to a configuration {{formula:bde84ef6-fcf3-491b-8abf-045d6a58d2b0}} when {{formula:781d626b-19fd-461f-b57d-815af3f36f22}} .
In which case, we note {{formula:7367fd77-c0be-4ff4-b879-763356000587}} the unique configuration that results when applying step {{formula:29c1b7ee-6867-49f8-8e3d-8bc551638b61}}
(This means that in {{formula:600ab815-7ac2-415b-83c9-069c2f6d7a9e}} , {{formula:c2133545-ac96-41e7-8320-7179389bc306}} executes the code of {{formula:5c916d05-0526-485c-a520-23f0f8792905}} considering that {{formula:b19b9440-b16c-4788-99c3-61eefd9bd731}} was fetched from {{formula:04f773f3-8790-4618-a855-a70f4a110df7}} and {{formula:a4f29145-ae87-41af-80ce-54507ce9c5da}} from the failure detector.)
This notion is extended to schedules by induction.
In detail, a schedule {{formula:d60f315d-0634-41a0-96b0-dcb9672dcab6}} is applicable to configuration {{formula:0bde0b10-687f-4949-9252-3210a1d95aa0}} when {{formula:f1271b14-b540-4692-89e8-a93493e0da13}} , or {{formula:ec44f38c-e769-4a5a-8feb-30acec2c666c}} and {{formula:711b7db0-62b2-430b-af72-34ddf99062da}} is applicable to {{formula:1584db1b-4348-43a9-bbf3-520c603bff30}} , {{formula:6e0ffca5-4c43-4ccf-89b8-4aaa04d1e3cc}} is applicable to {{formula:3a1d78fd-0363-457b-af2f-41cb40c9a001}} , etc.
Run
A run of algorithm {{formula:8e14b86a-6995-4bdf-9827-9c1d1f2ca7d6}} using failure detector {{formula:8ae40107-8fe9-4176-b191-350143651b2d}} in environment {{formula:bd8d7c99-528b-4db2-aa41-d5771617cebd}} is a tuple {{formula:59fe11e2-16f0-4b34-b209-d667a6a5123a}} where
[]
{{formula:781a6e13-e237-491a-b376-a1c9e5970999}} is a failure pattern in {{formula:754d6634-6f07-4680-99ab-5a3cf0fd8eae}} ,
{{formula:8b358370-b221-4b96-a135-8f9b7c3cdd17}} is a failure detector history in {{formula:f62ca007-fe62-4e2d-bef9-2a25f9aa0e33}} ,
{{formula:45034148-b1aa-4f00-b879-d9bb62636eee}} is an initial configuration of {{formula:d0d7e53f-10bc-4181-8013-b5380aabeb59}} ,
{{formula:d5023c20-a4a6-4814-855e-89afd834ee10}} is a (possibly empty) schedule, and
{{formula:5c08fe54-9383-4a06-9caf-79093b5af652}} is a growing sequence of times (intuitively, {{formula:d1ee16a0-814d-4430-af98-d2450446c191}} is the time when step {{formula:84aabef2-e26c-4d6f-9c30-2e24b7d8a348}} is taken).
A run whose schedule is finite (respectively, infinite) is called a finite (respectively, infinite) run.
Every run {{formula:d9ff82b3-5d2c-40d9-a9d2-ba4645e85849}} must satisfy the following standard, or well-formedness, conditions that we shall assume hereafter:
No process take steps after crashing.
The sequences {{formula:62260c63-bfc8-4611-a340-34626833ce2e}} and {{formula:fcb1c1b6-4ffa-4f0c-be29-1598c8b0ae2b}} are either both infinite, or they are both finite and have the same length.
The sequence of steps {{formula:6ffcd263-f052-42ce-beef-baf6abaa3ff4}} taken in the run conforms to the algorithm {{formula:22c83e11-a5b3-46c2-9301-ce7f2a24d518}} , the timing {{formula:e7bd3323-9c97-48a2-83dc-516f18f2f9a2}} and the failure detector history {{formula:7efbc2e3-0d8d-4ab4-98ae-6ca5fdec6216}} .
Every process that infinitely often retrieves a message from {{formula:6bd9f23b-e247-4123-b6ab-4a339bdc0c77}} eventually receives every message addressed to it.
A run {{formula:d2f6f2b1-3170-4195-b179-7a3a92810c41}} is fair for some correct process {{formula:bfbc00e9-fad9-4664-be19-92982691eea4}} when {{formula:92386c68-2e89-41ad-abdc-1f6bd1726377}} executes an unbounded amount of steps in {{formula:9d0d95fd-4d97-48a5-a16e-922258747b7f}} .
By extension, {{formula:1bbc6f16-39ce-4ca8-a272-538ac7b25324}} is fair for {{formula:53d6487f-46cd-4508-b6b7-650999c10109}} , or for short {{formula:b09dc0fd-f5d1-4ee4-99c5-5169af2fc11c}} -fair, when it is fair for every {{formula:6a1e4f25-bb10-4bcd-9dca-812c77f9844d}} in {{formula:60ad93cb-21e2-45d9-aec7-541b6ffb371b}} .
In case {{formula:aa7376b8-bfa3-49e2-bbf4-14dfded74ae1}} is exactly {{formula:56041bdb-99d0-46dc-a461-e05a7088990f}} , we simply say that {{formula:1bb1565b-c2e3-4a31-b933-766ac81bf6a1}} is fair.
Input/output variables
A process {{formula:bb11f8f9-a859-433e-ac8c-5835e6c74614}} interacts with the external world by reading an input queue {{formula:88fab194-50d2-4b64-8cea-c34e4d96a232}} and writing to an output queue {{formula:a20fd5df-efc3-4e79-9c12-93e81c55c344}} .
Both queues are part of the local state of the process and contain finite binary strings.
For some given run {{formula:88d6c407-efe3-42c0-9d9b-ea5f25dcb425}} , {{formula:a7216a81-6a68-4872-a78b-d47fd2042a5d}} defines the input of {{formula:9c079141-3500-4bf9-943f-ada8ea300b89}} .
This function maps each process {{formula:9a718dfe-9e27-4b6f-90c2-53a01f871c52}} to a growing sequence of pairs {{formula:81d533f8-a4db-44bc-a30e-cefa5d506a29}} such that {{formula:7f7f9c6c-8f33-4fb3-b87f-a93d9842723b}} fetches {{formula:c2a8e24b-3ee4-4506-a13d-dd4f291289bd}} from {{formula:fb142b91-9402-425d-891c-c1f668114fc3}} at time {{formula:4bfb1d3f-553d-4dc6-b3ad-7a55791beac7}} in {{formula:d67ba4f2-b661-46cf-91a2-90fa8912faf7}} .
{{formula:032b991f-4ed9-4eca-aade-f6883cc5e1a7}} is the output of {{formula:70184911-da8c-4a1f-905c-1f0baf94c44e}} , and is defined similarly.
Problems
An input (or output) vector associates to a process {{formula:a4cd77ef-2ad1-47ed-b87a-ee89f1a27cb5}} a growing sequence of pairs {{formula:b993a1a7-fa09-400c-a7fb-148d4496ff5a}} , with {{formula:e08b1dc5-c8b3-4d70-8653-e6bd63c0f135}} and {{formula:6d9f436a-cc2e-4619-b66c-b0cdfa6ae7bc}} .
A problem {{formula:09a4b2f5-a727-46ef-a413-6d43434f56e1}} specifies a desired relation between input and output vectors.
In detail, {{formula:c61dda3e-e0af-40a5-af11-92531c96f8d8}} is a set of tuples {{formula:4abf1d53-f17e-4452-95d6-9c76b55c80c9}} , where {{formula:a635c13d-6896-4fec-9747-da5b1cd7382e}} is a failure pattern, and {{formula:7a5240f6-9edb-4650-a487-79777556494b}} and {{formula:198ecfae-9452-4924-9a14-2dda662f15fb}} are respectively an input and an output vector.
Intuitively, {{formula:a1bd17c2-99d7-4fe3-b3a9-0b22ffc8ef49}} holds if and only if when {{formula:cb4621bf-2c70-4856-abb8-695240575c71}} is the failure pattern and {{formula:7f4ebffa-f9d0-417e-926f-97b3161cd4c0}} the input, {{formula:20590300-b92c-4984-af27-0605474f15ce}} is an output that satisfies {{formula:29de3354-e55a-40b2-a566-b73031564071}} .
Solving a problem
Consider a problem {{formula:1389388a-8375-4290-96ff-629662e0046a}} , an algorithm {{formula:b8a7cffa-f455-4742-b935-d0780fbd6b09}} , a failure detector {{formula:c7473fb6-32fe-480b-b54f-257ef8a69345}} , and {{formula:6461c91e-70b7-4aa9-8cdf-d638e03aebfa}} an environment.
Then,
A run {{formula:a5db5302-49c5-4acf-8a9f-4142db970a8b}} of {{formula:c374bdbd-31aa-4a29-b8ba-76909af1c4ba}} using {{formula:73f890a0-012d-4c26-89e4-85803252333c}} in {{formula:a108a9c8-39eb-435b-98da-85151bd8aae9}} satisfies {{formula:1d78e60a-b33f-481b-ad4e-468c5eeecf6f}} if and only if {{formula:1c88bc16-1314-40f3-ad91-2ea30be704bc}} is in {{formula:b9f97836-5bb9-4130-9609-8a1500494802}} .
{{formula:743431f6-4adf-40d9-bd38-f2c5ab77346c}} solves {{formula:a8d1487c-fd5c-4643-bbcd-7ea77e907ebf}} using {{formula:366bbe8a-fcb1-40c8-b5ae-a50937384267}} in {{formula:8cd5bfca-0ff4-47ae-9ea3-b8b66e8dcfdc}} if and only if every run {{formula:606ce489-f0d3-4449-a5a9-5bf5cd203648}} of {{formula:fe39b2e6-997b-445a-b4b9-f489216ac132}} using {{formula:93f664d2-4e92-4ca7-98e8-2091144f81f0}} in {{formula:f5d13e9e-475f-4fe6-a176-40e4910ac4cd}} satisfies {{formula:cf8ae6c7-4341-43d8-934f-41503f267a76}} .
Comparing failure detectors
As observed in {{cite:2913e44a7b9e589eeffea59c2d680513b550abda}}, a failure detector {{formula:99285601-26bf-4dd2-acde-d37fc960b144}} defines itself a distributed problem in some environment {{formula:fc1af6c8-8e37-4d5e-833b-27045b7acf69}} .
This is the problem of building a linearizable implementation of {{formula:1e819188-3f11-4ac6-b9b4-9e9c18314446}} in {{formula:37006749-ddc0-4edb-8db6-6741ff8f8b22}} .
More precisely, one defines {{formula:4434b0ce-d450-4a84-997b-8c40a7429218}} as all the tuples {{formula:b233c554-f471-4019-a1b6-539fd520ebe2}} such that
{{formula:be380326-d7a3-4bfa-b3d1-07ad7d2a93b4}} , and
for some history {{formula:2c3e2a53-97f9-4214-9bf5-0837360a6b8f}} , if “query” is in {{formula:589ab0bd-494d-47be-a2bf-3d2ba3e7aacf}} and {{formula:dbad1742-b89b-4bcc-9eeb-0d66cfd78732}} is a matching response in {{formula:11e8bd3b-e7d3-4b32-8c82-d847a84c26d8}} , then for some {{formula:ebaeddfb-f2e1-4519-b7f8-8889906ec545}} , {{formula:c45bf79c-7aeb-49a7-9733-5a73c86557bb}} equals {{formula:c234128b-1f0a-4913-9330-5824708147a9}} .
We say that {{formula:f2af3d61-1ecf-49b3-a3fd-8cdb7f274fe1}} is weaker than {{formula:0d375693-b375-465c-9997-cd60398be0c7}} in {{formula:bdc8831e-51d9-4744-922c-69057ac08909}} if there is an algorithm {{formula:5ccd48fc-2be6-4a50-ba7f-b65d017e717b}} that transforms {{formula:e0f60ec3-b1cd-4717-af12-ae72437bd3fa}} to {{formula:abf5595d-9865-4ce2-b90c-1b0fb8ff91ef}} in {{formula:7e0d48a9-e192-40c5-8faa-8846899f9f59}} .
This means that one can solve problem {{formula:3d46b86f-417d-462b-939b-80ae9e40ce22}} using failure detector {{formula:5b3ac125-d94d-4e78-af2f-c2fc6dbbc9e8}} .
Notice that if {{formula:31139022-9a8c-4c26-9c76-c4a46c217ebb}} is weaker than {{formula:479710e1-4cf0-42ab-99ae-95e887661c92}} in {{formula:c572ddd1-a70f-40ae-8499-ced070bb941e}} , then every problem that can be solved with {{formula:80fc9802-b486-464c-a010-2a0eadfb00c7}} in {{formula:73d1a7a5-9cf0-44ca-b7b8-179b90e592b1}} can also be solved with {{formula:edcb79e9-ec6d-4f2c-a605-86d2d82b76d8}} in {{formula:f26553bc-6a90-47df-a02d-5d0993b2015b}} .
Two failure detectors are equivalent in {{formula:966a9170-1cc0-409c-9e8a-32f015aa633f}} if each is weaker than the other in {{formula:8d17dcb6-1562-4999-b922-15ca33f831f1}} .
We write {{formula:dc351c27-31b2-4344-91c9-ebffe4d76c54}} when {{formula:efc74431-e2b2-4985-b144-879fec4edd5d}} is weaker than {{formula:0e65a432-5182-4105-98ca-39bf13a16d4b}} in environment {{formula:f59ece79-de32-4866-a6b9-b87557890fcc}} .
By extension, {{formula:a146984d-adb4-4821-8568-08a1f7b6ea8a}} holds when {{formula:a5513569-b79e-47c9-b7eb-1112f7632c10}} is weaker than {{formula:d4c89501-eb2d-440b-8c16-12b4ecdfa6c0}} in every environment.
Weakest failure detector
A failure detector {{formula:946081ed-ed85-42f7-b2d9-a2fcc7dfc6cf}} is the weakest failure detector to solve problem {{formula:f4d66d1d-6160-453a-a447-b8d1b3097ae1}} in environment {{formula:c60326b2-c399-4377-beb7-e81585fd4352}} if and only if the following hold:Strictly speaking, one should talk about “a weakest” and not “the weakest” failure detector because several detectors may be weakest yet not identical.
Indeed, as pointed out in {{cite:2913e44a7b9e589eeffea59c2d680513b550abda}}, if {{formula:635277e1-505e-4813-ae12-a8fc5ba85fea}} is weakest then so is any sampling of {{formula:ddd5c86a-fa19-465f-b789-fd36da228191}} .
However, as common in literature, we do not distinguish a failure detector from its equivalence class.
(Sufficiency)
{{formula:f8ab2c78-6430-4016-8c70-ba60c2327af1}} can be used to solve {{formula:8769b5e9-5ef1-4a96-8480-91135ca3b93d}} in {{formula:981ff946-03fc-469c-8fad-3ff95c85ac4f}} .
(Necessity)
For any failure detector {{formula:d289fb75-ddcd-400a-ba8b-de8b4cead78e}} , if {{formula:465d4bc8-6b98-489b-9b61-ffb5fe96bf36}} can be used to solve {{formula:c82470fc-c39e-4427-b4a7-42acd2f29051}} in {{formula:0542fdb0-1069-45cb-aa97-bb5577116f62}} , then {{formula:843788e7-1904-4a8c-a89e-f5c17b5667d2}} is weaker than {{formula:50f756e9-2280-48e7-bbd8-80144388bf29}} in {{formula:0b23bd29-47cc-4f2f-b72c-42d930e9f792}} .
In {{cite:2913e44a7b9e589eeffea59c2d680513b550abda}}, the authors prove that in every environment {{formula:496ee712-5045-4b0c-8297-fe68671aba6c}} , for every problem {{formula:9f565454-dc9a-44b8-81f9-eca75b6553a8}} , if there exists a failure detector to solve {{formula:aa4f8e66-d195-47af-b0d9-2cde1e32b8d6}} in {{formula:5554ed56-fda7-4e20-9276-e0440797c66e}} then there exists a weakest failure detector for {{formula:f25dcb44-a7b3-496e-860a-e38b6d09ad46}} in {{formula:2343178b-bd1c-4e67-a418-62a7241db298}} .
Sub-algorithms
An algorithm {{formula:79777a10-0dc8-4e06-bc26-0e9ef2ee3123}} is a sub-algorithm of {{formula:c7bea405-c007-44d6-8181-20f41bf93075}} if for every process {{formula:2a6f2c1d-f424-4f4f-991d-589b921ebce9}} , there exists some automaton {{formula:4eacee77-b430-474a-a64e-4de4b1817849}} such that {{formula:e1f44a23-528e-4835-aa4f-54427bf897e3}} .
Given a schedule {{formula:06dee187-48a5-49e4-b8b8-298376567a35}} and an algorithm {{formula:2e032a78-f9a4-4787-b676-a74b3a93fb8e}} , {{formula:99a2d609-0be1-44de-995e-2df1080b5734}} is the projection of {{formula:0bc742ad-04fa-48f2-aaab-3ebc4ff99b8a}} over {{formula:96789ecb-de8f-4c1e-9939-1ed3ceafa492}} , that is the sequence of steps of {{formula:c5fd65b2-6c08-401f-8fac-2746f9285c13}} in {{formula:f3ca3bfa-8139-4bd7-8536-7069ae55166d}} .
In particular, such steps include reading and writing to respectively the input and output queues of {{formula:d04625a1-8365-4a66-91eb-17c00a05e3bd}} .
Technical Lemmas
model:lemmas
The results below are derived from the model detailed in the prior section.
Their proofs is left to the reader.
model:1
Consider that {{formula:70317422-b7aa-478a-b6f0-c07e59d58071}} is a sub-algorithm of {{formula:ed00f0b7-14d4-4e02-a9ec-89a4edfd5b6e}} and pick a run {{formula:c1478a1f-6d0d-4616-971b-a8493665d40b}} of {{formula:d905f144-8cd6-46bc-82a1-a3d080f9ea08}} with steps {{formula:1ea15c5c-f544-4e80-bebd-23676d00c1e7}} and history {{formula:9ad65fba-c44a-477a-b670-402964bf986c}} .
Then, there exists a run {{formula:517dd02b-1801-473c-9d10-515ca337c212}} of {{formula:ceefc110-9cb6-4cd2-be5e-0a41659709d4}} with steps {{formula:89813fab-1fa6-48e5-8b11-f90fef25a980}} and history {{formula:d4ae741b-cf55-45fa-812b-7bcdef8b1634}} .
Given some schedule {{formula:a6fcdf9b-1c31-41ea-9cbb-29654458831a}} , relation {{formula:1d853c12-2c71-434e-b8e1-ff573228358c}} denotes the happens-before relation in {{formula:355fd1c1-75e4-42a3-a627-8fde3b6f7d44}} .
We write {{formula:f29e1baa-5956-4cca-a112-5576515b13b4}} the projection of {{formula:c6769687-bc39-436d-9245-97cc78257e56}} over {{formula:0a79dffe-120b-4e80-8834-88379a169a75}} .
It is sound iff for every event {{formula:0ff6f3d7-ed6d-4f31-8e2b-ac7f30633059}} , if {{formula:700a5aac-ee3b-4632-9f40-c0d40809ea05}} then {{formula:f62afdd7-4555-4fd9-9a40-78d4fc53006f}} .
[Indistinguishability]
model:2
flp
Assume a run {{formula:d6cad667-cd6d-4a96-a645-14c00f788679}} of {{formula:420ed001-f353-4ea1-9a1c-3a0d1191d1eb}} and some set of processes {{formula:8c4b8ed8-930f-436d-86b5-c15a00b28c03}} .
If {{formula:25ad9034-b9b4-47ad-9a3e-c2f4872d09f3}} is sound, then {{formula:49fc542a-b7ef-4a08-b8c3-c2c4e5eafc75}} is a run of {{formula:8bb02afa-f9d0-45e1-a19a-17a8629c02e7}} .
In what precedes, we shall say that {{formula:afec372e-ca8d-43e8-8736-c09a9908753f}} is indistinguishable from {{formula:c3f97449-eeac-4be9-be29-071b42669cb6}} to {{formula:b315f35c-5c25-43ce-b530-e07e2476801c}} .
The two lemmas below are analogous to Lemma 1 in FLP {{cite:e7802161ac6ee84eff3f4724d9fbecca38b42344}}.
They establish that if two runs with the same failure pattern and history are executed by disjoint sets of processes then they can be glued together.
model:3
Let {{formula:27a091e0-5de4-4099-818f-614f4f66cf85}} and {{formula:26090057-e70a-449c-89e9-a142a5d6323e}} be two runs of {{formula:a1e1f526-0a29-48e9-affa-3e21f765c27f}} .
If {{formula:b7a4c652-349e-4816-a4a0-e166caccd036}} then there exists {{formula:5a8fd706-a11b-47f5-a29d-c277470a2b34}} and {{formula:d7da6aee-e21a-4bdf-b050-bdcc1b49dac5}} such that {{formula:8c87bb12-2186-4623-80e6-7507fda6a135}} is a run of {{formula:25ac8c18-9261-46cf-93fd-50a648c02dc5}} and {{formula:7e26adbf-1605-4667-8c46-c1ded66985b2}} .
model:4
Let {{formula:86a53a48-a542-4180-a758-ff8467f6f8da}} and {{formula:f386b8ba-c49a-477f-a016-12c75125f8d6}} be two runs of {{formula:fd85721c-d891-4b69-a2ce-d1df8ac3e90e}} .
Assume that {{formula:c6fac396-f3e5-436b-9d06-9d7caea8d410}} and that the last step of {{formula:955b0131-fa34-4d6f-adf5-4120c5ef1f70}} occurs in real time before the first step of {{formula:2e9d3849-5e3a-4b33-a010-2a594f45c48d}} .
Then, there exists {{formula:ec27935b-fe85-4e2f-b5c6-66cbb53ab025}} such that {{formula:0d2c4602-5469-4b35-b2d7-6851ae91d570}} is a run of {{formula:6e7b52a4-432c-42b7-ad6f-9e29e27f9aa6}} .
Emulating {{formula:4c095b6c-aae5-4dac-b10e-123ab71e9d44}}
omega
[!t]
Emulating {{formula:87e84de3-97c4-4485-98f9-16ca35e254f0}} – code at process {{formula:a5805115-daeb-4149-9afb-b5b1ab0b06b1}}
omega
[1]
{{formula:0bedad73-4152-42d1-8b28-d9932506e1dd}} omega:var:1
{{formula:8ca79c82-896f-4230-8dd2-8ac1efb0b2a9}} ; {{formula:30134746-e5d0-47b8-9ed4-ba0b08497491}} omega:var:2
{{formula:c3e13f5f-d302-4095-af4e-ef1653d4fbcb}} omega:var:3 {{formula:346fba4e-aec9-49e9-828a-d66db0ee2a6f}} is the empty schedule
{{formula:9687f5f9-cdcc-4cb3-a1d8-f879f9383007}} omega:var:4 {{formula:b016df5c-d56d-4d75-a0fb-98a63def2439}} is the universal set of paths
omega:0
{{formula:727d272e-b589-4742-bb83-2712492f11bd}} omega:1
{{formula:173ca2e5-b71f-4458-ae9f-a969b2398ef4}} omega:2
{{formula:a8018a52-ecd1-4442-8b7a-0a9f3f2fdfee}} omega:3
{{formula:c1053adc-fdad-49e5-a47a-eeb6c4abc33f}} query:1
{{formula:20f1ad1d-6db1-4082-98fb-97e221332fb6}} sampling:1
{{formula:3b63588b-b927-4c19-a4d5-9ee1b3ce9d64}} sampling:2
{{formula:ca1ffd10-ce7a-4fb9-97d2-07ad6fca7601}} sampling:3
{{formula:7864c37f-4ac0-4358-ad22-d14d5bb29b76}} sampling:4
{{formula:58866af2-f331-4a2c-a8a8-22e6be61fccf}} sampling:5
{{formula:d9702f31-12f4-4ec6-88bf-377f9b249eac}} sampling:6
{{formula:a1e0c830-a185-4fe7-939a-ec0180762abc}} simulation:1
{{formula:8bebcae9-c1ca-4f50-b05e-34fe26b73b70}} Following {{formula:1110dade-b991-4e44-aaf3-fad90540c6dc}} , non-empty ({{formula:d29aaea7-20da-4d17-a62f-8e7a02201469}} ) simulation:2
{{formula:bf0ce59c-7559-42fa-b785-221d9897918b}} ; {{formula:77a9afa3-4d18-4fa2-825b-952f5676f12f}} simulation:3
{{formula:ded6dab9-3ba1-4d7d-a747-0376a2ed67d0}} simulation:4
let {{formula:4edc6273-da17-4bb3-8d46-0fa0ec356eef}} simulation:5
{{formula:70b90107-2c1f-4b2f-80b1-13d86008c8d7}} simulation:6 {{formula:b37ec6a8-2f8a-4071-881d-15d91ba64ded}} is the null message
{{formula:0b244908-0a41-4374-a1bf-9538ffce1360}} simulation:7
{{formula:49860aff-5ace-46e0-ab4b-258e4433af01}} simulation:8
if {{formula:ba962138-ba8f-4e30-a7b3-829d21500552}} then {{formula:39ada3f5-651f-4b78-b55b-d7fffd7ebc66}} simulation:9
{{formula:bf889b90-84da-469c-ae73-65806e9030df}} tagging:1
{{formula:a594faae-5bae-4231-8617-4b8bbe593867}} tagging:2
{{formula:8ac87365-b3a4-4176-8874-4d659ac93236}} {{formula:5d76a800-88ea-4156-b79a-286eea6c1128}} tagging:3
{{formula:6ae85ee8-5d86-4d1c-822a-3d2e6ab65ea2}} tagging:4
{{formula:042c3f64-029d-492b-b7dd-488e56861ff9}} tagging:5
{{formula:2472f110-bc6d-42f6-b3a7-b9d5b6939ce0}} extract:1
{{formula:91bbc76a-bd9c-4cd4-a506-661e0494c9a4}} extract:2
{{formula:31bec6f9-d2e2-4340-a7d4-0f31dfcff078}} extract:3
{{formula:cb4c7a16-8ce1-4f83-b17f-023639f75834}} extract:4
{{formula:b7428eac-3ed9-4239-b8f5-688d0ccf29b2}} extract:5
let {{formula:064f4c24-84d4-46d2-9b4c-2ca71aeec612}} extract:6
{{formula:23f1be39-292d-4de5-be8b-17b49b442c83}} extract:7
{{formula:c88b8c98-3635-42c2-a9fa-4ffce8113f6c}} extract:8
{{formula:d6db5987-d750-418e-86af-48dcde3f8fbd}} extract:9
{{formula:4a499b67-06d2-4af8-b660-d85af81c3f82}}
{{formula:18873789-3331-4a63-929b-f52fe1d9e02a}} ; {{formula:a908d3eb-18d2-4664-8fb9-09b2c8d62d25}} locate:1
locate:2
let {{formula:0dfbde85-ad0c-4846-bcfb-3d6cbb5ce8b0}} , {{formula:56b0ca2b-0533-46d2-9a4a-1b3b06a4bd25}} = oldest message (if none, then {{formula:1b914586-1795-42f3-82d1-ceb0d08af6fb}} ) addressed to {{formula:d12337b9-7232-4a8d-a8bf-444bec80a052}} in {{formula:947a59bb-4b14-41a8-a7fd-156c05c85c5a}} locate:3
{{formula:8a4b53ac-e4de-461a-96f9-6653162cbd73}} In order {{formula:7274bf0e-cb89-433d-b699-f6b3cabdbecd}} locate:4
let {{formula:f9fbde5b-fb4f-4ff4-882a-25f3ab7a5fe4}} locate:5
{{formula:b6813dc3-4c7b-4a30-90e6-80a28c5641a0}} locate:6
{{formula:b3324668-104e-4391-a490-0b6cbed61607}} ; {{formula:a3ab93c9-c338-4cf0-955a-3d8b6114922c}} locate:7
line locate:8
locate:9
{{formula:99833559-cdc2-4b23-bb12-f8b69ecf0396}} In order {{formula:8cadc831-f667-4eeb-9185-f3a6c6826f2a}} locate:10
let {{formula:57bd6aa9-2c3a-4b74-b271-bf716048b14c}} locate:11
{{formula:a6c70110-3f82-40cc-9af5-5ef5b3d7a89b}} locate:12
{{formula:4c6af04d-01a0-492f-8f7b-3e70dbc98091}} ; {{formula:11dd772d-fee9-41c6-a3d3-e883bf1beebe}} locate:13
line locate:14
{{formula:7b381ce2-0ba7-4cce-b189-a28b0b2c664a}} locate:15
{{formula:1066a686-a163-4b12-b33d-7845460dd5eb}} locate:16
{{formula:f65577f6-5b87-4514-afd0-82533c77b236}} locate:17
Consider an arbitrary environment {{formula:6b1228f0-72c7-4a3a-802b-30dee58ab394}} , a failure detector {{formula:6a583bb9-b4db-4ec7-896c-4e465a648c6f}} and a strongly genuine solution {{formula:07dadbfb-00c0-4fae-bcc5-0473d48fc72c}} that uses {{formula:c470cbf0-da06-4ddb-b66a-9aa89441c373}} .
Given {{formula:f327c750-f868-4dd8-8c8d-47ec08783878}} , the construction of {{formula:5460a715-04ea-40ff-a8a4-727e6e115f22}} from {{formula:466ad0dc-ef9b-4cfe-9eaa-26c5cb487691}} and {{formula:fec8514c-32d5-4977-b88d-bfbab31b6b4a}} is depicted in omega.
This algorithm follows the general schema of CHT {{cite:ea9cd8cbad59f4421feb8537fbe7d920746b4b75}}, with some differences that we detail hereafter.
omega consists of four procedures that are run in a loop (omega:0omega:3).
Procedure {{formula:3741d883-07d4-4274-aa9e-8ea0d82cff39}} is a collaborative sampling of failure detector {{formula:5aae0d67-b61d-468e-be61-6fb16f828feb}} .
Procedure {{formula:435dca87-3556-4557-89aa-64af551a4bd4}} executes runs of {{formula:84b8568d-e127-42d7-9974-434fe38d78e6}} using this sampling and multiple initial configurations.
These runs form a forest which is tagged appropriately by the {{formula:b8f96be4-b466-46d3-bdb7-c90956683c25}} procedure.
Based upon those tags, {{formula:967cd55a-e2e5-4fef-ab9c-3564d4c4af91}} computes an eventual leader for the group intersection {{formula:7d909b89-3aba-45e5-a432-48314662f0f9}} .
The sections that follow detail each of these procedures and the guarantees they offer.
Then, correctness shows that omega implements {{formula:b0f5cbef-a3dd-447c-8b65-1965b3c4589a}} .
Collaborative sampling of {{formula:d72edd6d-c948-4048-9f4c-6cff3bbd9b9b}}
omega:sampling
The first procedure of omega implements a collaborative sampling of failure detector {{formula:4140455d-a6ce-457a-89d5-38a79f38e9b3}} .
The output of this sampling is stored in variable {{formula:750113b4-44c8-4860-86ef-839d43c5be2d}} which contains, at each process, a directed acyclic graph.
Hereafter, we use selectors {{formula:fdb88dc0-df72-4c60-8f99-a0ce4ad231db}} and {{formula:3e32040f-5358-45f6-b226-de66b3c7b8e6}} to refer respectively to the vertices and the edges of {{formula:913dd9ea-c4e4-42dc-bee4-7f4061922040}} .
The {{formula:00701267-59a9-450b-b2f0-15638abd329f}} procedure works as follows.
When a process {{formula:6d765d5e-f7c1-4bfc-9299-ea01d63334a1}} executes this procedure for the {{formula:be1de75b-c3bc-4a4f-afb7-c7d9129fee01}} -th time, it retrieves a datum {{formula:fa888775-498c-4b18-9273-b4bcdbd6f9e4}} from failure detector {{formula:6c88865b-4cd6-4fa6-9e77-42e15373f0e9}} (sampling:2).
Process {{formula:a1ecbf5e-a228-4166-b98e-52b31f9003b1}} then adds a vertex {{formula:b54e1a8d-98f4-4eb4-b7f1-1cce5e48d29f}} to {{formula:2997f538-7ab5-4505-8031-8697f3876bb1}} and an edge from this vertex to every vertex already existing in {{formula:865a524d-3631-4472-8660-40793d906c37}} (sampling:3).
Then, the new value of {{formula:bfc1a6d3-d14e-402f-9075-b31e37d7c0f3}} is sent to all the processes in the system.
Upon reception of such a message, every process merges this sample into its local graph (sampling:6).
From the above logic, we may observe that every path in {{formula:30e9f703-3165-4a3d-9041-cebc66c6dfe2}} at a correct process eventually appears at every correct process.
This simply follows from the fact that links are reliable.
For the moment, we shall assume that the procedures {{formula:55150856-2bae-4d89-bf2f-b6f0ed17a2b4}} , {{formula:8549711f-b740-40dc-b840-a4286803a410}} and {{formula:a434ccb7-5fec-4349-9c5c-cd87b82a1195}} always return.
This result is proved later.
sampling:1
Consider a correct process {{formula:0b950a7d-d966-4f40-b495-cf53f8146d6a}} and some path {{formula:ff91317b-5924-4e15-87bc-85235b03eff4}} that eventually appears in {{formula:f6821b36-7c13-45dc-8bfc-7d5cda5c9768}} .
Eventually, {{formula:153ecdf1-e002-41b5-b01d-5555d11de652}} appears in {{formula:b7453811-5f0f-4974-aef6-08b290ed8bb1}} at every correct process.
First of all, observe that variable {{formula:cda9e21e-024f-4f25-9a38-5aaa9e350413}} is monotonically growing over time (sampling:3sampling:6).
Consider a point in time {{formula:a2c8c9ff-79cc-4be0-9a72-0861ae1cacd5}} at which {{formula:46c3a36a-c57c-49d7-b65b-14456f47352c}} is in {{formula:0fbcfb31-536f-4da2-b09e-008dd9dd492c}} .
Then, {{formula:a09098e1-96f2-4a7a-8e9f-2545dc0efb17}} is in {{formula:201603d8-634f-4056-b118-db767a1e89c5}} at every later time {{formula:9045f84f-d0a8-40a8-bf28-e99fc3388fdc}} .
Consider some correct process {{formula:d9ebcad6-2f8b-4f89-adb4-033034df9923}} .
By assumption, process {{formula:56b3d096-1bef-465f-ad04-b9526612ea2a}} eventually executes sampling:4, sending {{formula:943a4c9c-7b04-4125-aa66-5b0b8dbcd615}} to {{formula:ad01a462-7e51-4e2a-892c-163db72d8d1c}} .
Since links are reliable, {{formula:c47cafe9-5f8c-4aaf-914a-b8e96b88c60b}} eventually receives this sample and merges it with variable {{formula:49ff604f-11b7-485a-898d-dd754df7abb4}} (sampling:6).
Now, let us assume a run {{formula:10c76bda-8ccc-4142-ab9d-caab13e37b98}} of the {{formula:bd4c5b36-702f-4fd6-a4d7-e80a11fe3096}} procedure.
Consider a sequence {{formula:fa273d8c-9a42-453a-ba45-a9592bb4c506}} of tuples {{formula:acd3bef8-2a05-43bd-9f78-9bf520e9f6bf}} , with {{formula:cc37a939-693c-4fa9-ade7-aab7d795e9da}} , {{formula:1219b9f0-c3c4-494a-bb68-394b3061d84e}} and {{formula:f274c942-bedf-4656-9757-6a068bc0f662}} .
We say that {{formula:1cbf45f5-f6de-45ef-8471-0923051dd350}} is a sampling of {{formula:02c73ed1-412f-4c81-b604-113474ae047b}} in {{formula:ec65f6f2-a8a3-4549-8530-c9a011af1cfd}} when there exists a mapping {{formula:37cb6dd3-9b48-4800-a4b3-5eb70dca0ad2}} from the elements in {{formula:4d8caad1-4ad5-4646-8319-4ca1295909f3}} to {{formula:3bb1e3d3-0765-4d58-b5ef-690d7bf1b9e6}} such that for every {{formula:052a24c0-12b2-4806-96a0-2f60052997d2}} in {{formula:bcc85e38-dff1-420c-a3bf-99bf81a76dd5}} ,
{{formula:2339b164-7500-43c3-8be5-cf4f572d39d0}} is not faulty at time {{formula:e3f9f089-a485-4e2f-88d9-6c20c0c28361}} , i.e., {{formula:61196baf-5ed1-4fbb-93f2-df5ffce55337}} ,
{{formula:76740b52-182e-42e6-bbfb-ea181ffecb67}} is a valid sample of {{formula:e9a3836d-c096-406b-a052-382b43b6bbdf}} at that time, that is {{formula:2db44f3e-5b76-426c-9984-3dcaf1ef61e3}} , and
if {{formula:d0964ac8-de8c-4ea7-9e45-6d7df4e78873}} precedes {{formula:0e911ce3-9c29-4624-af8d-5a20b365a107}} in {{formula:586d2a2d-c1d1-41bf-945c-d1018accb7a3}} then {{formula:ec1d92a3-c67d-4393-a642-912dc4164e37}} .
The mapping {{formula:93724597-afb0-431f-9e89-3275cc074d24}} is named a sampling function of {{formula:a1825af3-b114-43af-a4b2-fd7b2397bc7c}} .
When the context is clear, {{formula:ddeeb108-364f-41f4-b969-5f40b8e1dbde}} is simply called a sampling.
sampling:2 establishes that every path in the graph built with the {{formula:4f8d2318-1380-4b16-aa66-9eaf3fafb84e}} procedure is a sampling.
sampling:2
Consider some point in time of a run {{formula:f599ef19-b94a-4a7a-94e7-e7372c588870}} of omega.
Every path {{formula:b1024bf1-54ba-45c4-b595-19452af12b98}} in {{formula:f83a5f33-1db5-4fec-a00a-0128e69ba2ab}} is a sampling of {{formula:da0f6721-03ad-4e8c-a342-fc9ae0485e29}} in {{formula:89b91824-0427-4775-ac64-0c450c025de4}} .
If {{formula:8374af17-9b46-4fb5-8ab9-4693d0a8956f}} is the empty sequence (that is, {{formula:e53a4e15-79d0-468e-92a7-b7063a057482}} ), {{formula:777718f2-0e5b-4b27-8c8d-d8eeeb9082b2}} is a sampling function.
Otherwise, {{formula:759e97ab-21a6-4c83-8439-4ad1de762018}} .
For each tuple {{formula:05620420-3e24-4946-99c5-32da175ccf8b}} in {{formula:ea07202d-1ec3-4297-841f-a2cc92d966d3}} , {{formula:b48415f9-0bea-40cf-9f3b-7bf4778f2a8d}} is set to the time process {{formula:e959999b-b621-4697-9f96-ebbf166b4eff}} executes sampling:2 to retrieve {{formula:b4f3fd21-b639-4f4d-b794-a732e46e2327}} for the {{formula:fb0e4089-f964-4358-aa53-ce3283fda99d}} -th time.
One observes that at time {{formula:f6344423-44fe-479f-aa6e-b925c7fbe189}} , process {{formula:b68dd900-201a-41da-870f-24b91ce9e77c}} is not faulty and {{formula:864b5f50-0057-4acb-b04e-8bdbb30e1cd4}} equals {{formula:7a9c8792-87c0-40f9-be69-e9183139597e}} .
Moreover, according to sampling:3, if {{formula:39bdc89b-b1c0-4105-996a-e89a5b11ab82}} precedes {{formula:08a52baf-57f8-4a33-abe5-44438221ca41}} in {{formula:5310679e-4723-4932-a91f-1ef0a29a0958}} then {{formula:0de1af8a-6d6c-48bb-99c8-2533e15fd930}} .
From what precedes, we deduce that {{formula:b53741a7-77ce-4588-9420-c279453d2342}} is a sampling.
In the proof above, {{formula:f4724840-cf87-40bd-a158-41a9935e9f42}} is built by taking the time at which each sample in {{formula:7dd6866e-38eb-428a-9f2e-e657ea67fa22}} is computed in {{formula:dae1a3eb-0591-48c2-8f94-4dbcc87d192f}} .
Hereafter, the sampling function of {{formula:504a7210-076d-48a9-81d3-63388d02c39a}} refers to this unique way of computing {{formula:bd57af02-d0dd-46e5-aeef-2a2c4812f12a}} .
A family {{formula:36c3f7d9-c519-4e1d-928c-3b1be29630dd}} is a sampling sequence when for every {{formula:3561a152-7d9c-4ef5-b7ad-25a7b85a3f00}} , {{formula:4fea61ce-cb93-4af6-b9a4-efa259e576da}} is a sampling and {{formula:31682a3c-241c-4903-bf8a-2e8c75ecbe0e}} strictly prefixes {{formula:ded299f7-3c06-49fa-a521-9019649d3152}} .
The sampling sequence {{formula:f52387f6-700a-4da0-92aa-6511703384ab}} is replicated when for every correct process {{formula:614860e5-f28c-4ccf-b32d-8bb71f753b3b}} , for every {{formula:f43a32cb-c83b-423f-9708-83a3270108ba}} , {{formula:a6637f2d-2172-42f6-8e17-f629f46bead5}} is eventually always in {{formula:ccc175e7-2b3c-4ce5-b916-c1c455a98b7f}} .
Given some process {{formula:c31df14d-767b-4f44-ae2a-a4d087a712b2}} , {{formula:aaefb128-aa4c-46b2-aec3-13aace22042a}} is {{formula:9f56d310-e308-4e97-ac53-c1aa8a7e5ac3}} -fair when for every {{formula:4e09199c-eb34-401c-8b55-6be1da8c58f4}} , there exists {{formula:a2edd52d-28d7-4a6f-8ec8-99f0a3c51fdc}} such that {{formula:83174925-22d4-4da3-b086-df8a47d8c379}} appears {{formula:38425b13-cbb3-4154-92fe-9a3196c8d019}} times in {{formula:313716f3-cb00-4bbe-a589-0685503dbf90}} .
By extension, {{formula:c38e7dcd-66d7-4cc7-ace4-6cfee04e28d6}} is {{formula:1bf26a91-1a15-4557-bf61-c07d7aad729b}} -fair when it is {{formula:65a6dd63-5316-48e5-a9f3-3d6b87fd3dae}} -fair for every {{formula:0aca0061-f234-4a7e-9462-42cbc42c71ff}} .
When {{formula:d0d4c954-f514-46c4-bed3-b1e431d2b1b6}} is the set of correct processes, we shall simply say that {{formula:7bb1b63c-8d6a-4bee-b467-116bf26172f9}} is fair.
sampling:3 below characterizes how replicated sampling sequences grow at the correct processes.
sampling:3
Consider a fair run {{formula:db6727fe-76cf-4148-a443-8143abd8ec01}} of omega, some correct processes {{formula:70ca2d7b-4e67-4bff-92f1-ef601d62d909}} and {{formula:2acf5f55-55af-4bcb-b996-a5646b9a85bd}} .
Let {{formula:9bae36fc-4dc1-4997-a6e4-7f32dc840277}} be some point in time during {{formula:fa84f229-4821-4ad2-b752-e3263a37183c}} .
There exists {{formula:afe79aaa-f17f-4a3b-861d-c736db01373b}} such that
for each {{formula:5902dc1e-58b3-4d4b-98ce-81876f30226a}} , only processes in {{formula:b02c5b56-d07e-4877-ab17-d1eaf4d1eb98}} appears in {{formula:3843e7c8-217a-4b8e-9446-6fd4e541da8a}} , and
for every path {{formula:ff444093-06d5-4422-a5b3-936120bd50d7}} in {{formula:710d9bf8-3eaf-4f1f-9f55-b2685b40400d}} , {{formula:ba1f66c6-21b6-428b-8b51-37841e5bdc22}} is a {{formula:298caf02-3f8b-4bf7-88eb-136189b16437}} -fair replicated sampling sequence.
Graph {{formula:23bfd6e0-09b9-405f-b43e-13af631737cc}} is initially empty.
Hence, by a short induction, when {{formula:4b7dff68-aa27-430c-99ce-b42a8a9617ed}} is extended by executing either sampling:3 or sampling:6, this computation terminates.
It follows that every correct process executes infinitely often the loop in sampling:1sampling:6 in {{formula:ac5c081a-072c-4163-a4a9-abba8b982bb4}} .
Consider a moment in time {{formula:8ee95952-fb6e-4f68-a520-2a78785e78f3}} at which some correct process {{formula:5874195f-5e31-455e-ace2-a8ac36b1628d}} executes an iteration of this loop.
Since links are reliable, it is true that:
[(1)]
Every correct process eventually receives {{formula:78da3b1e-9a7c-48fc-a812-77c4b2f6ce78}} and merges it with variable {{formula:993f2e87-ae01-497e-b90e-e4e40ffe916f}} ; and
At every correct process {{formula:205d3233-a939-486b-860e-abfe641fb5cf}} , for every {{formula:6442102a-3d25-4ce4-a4d6-c1f72b467561}} , there exists eventually some edge {{formula:a5f60fd6-8d1d-4904-b953-f603efe09d89}} in {{formula:ce81cd3a-60de-441a-8cb0-32833a82f35f}} .
We build inductively {{formula:6db99c78-f95d-42ac-8a14-9198d662b58d}} using the above two observations.
In detail, {{formula:93482951-337c-41e7-96a1-dfc467b0cc5f}} is defined as {{formula:4b80c7f6-550e-4fb6-8f96-f7e7b39f9379}} , the empty sequence.
Then, for every {{formula:8aacce63-426f-4e48-aee2-8a26772cd907}} , {{formula:efb2726b-cb76-4c9d-bfb3-21decf97f461}} is constructed from {{formula:20cf2595-dcdc-4fed-9998-c9325fc35cf3}} as follows:
replicated
Initially, {{formula:b92d0e68-2e3e-48c5-bede-e8c61d4764fd}} is set to {{formula:813f765a-fb0f-4a25-b6ad-9338d767153d}} .
For every correct process {{formula:c7a3654e-df6f-48af-889e-2bbc33089aae}} , starting from {{formula:584f90b2-c2fe-4c17-85f3-86ff000cd751}} at time {{formula:6d584981-a93f-4768-a733-8ef15bd22a3f}} , waits that {{formula:34319c1f-7892-494d-86b8-141e139bf1de}} the last element in {{formula:44620778-794a-4ee5-950c-a1db2347e5e8}} appears in {{formula:d5723cab-c80b-4ba0-a18c-545832bd30b3}} .
(If there is no such element, this condition is vacuously true.)
Then, let {{formula:1ac733eb-303f-4701-be68-b819a1ec1ac4}} be the next vertex added by {{formula:bd885705-a0c2-4449-86ab-8d439db6fa86}} to {{formula:b1fb557b-5ebc-4ef0-8c95-2d37b4fb61a6}} (at sampling:3).
Append {{formula:c15d3717-d3e0-4520-ab9f-df462aca3d31}} to {{formula:8b5db10f-ef88-4a48-b9f9-de113d884d8d}} .
Observations (1) and (2) above imply that, for every {{formula:987e37c2-5dec-45ea-8e14-576f31ad1168}} , {{formula:721aba36-33eb-4413-913b-51115a71e2e7}} is indeed built with replicated.
From the pseudo-code, if {{formula:f37352cc-cea8-44f7-a2f6-5fb8238d20e3}} is in some {{formula:8414929d-c15b-4577-9a9d-58e8468b3b08}} then necessarily {{formula:a6af90d3-a7eb-48d9-98d6-cb2ee31e11d0}} .
In addition, for every process {{formula:6f552178-11e5-4167-8407-90d7c6fb76e3}} , for every {{formula:57aeb834-9a8e-41b6-a5b9-dd2537c6b75f}} , there exists {{formula:371d0756-422b-4c17-a0f0-df240063213c}} such that {{formula:29747c35-c6cd-495d-9681-03a18382966d}} appears {{formula:40e78eff-88ce-40af-bac0-d755e5355b33}} times in {{formula:c940f3d0-c008-4ea2-8530-da4489ea5ffd}} .
Then, consider some {{formula:c60738c3-1d66-4297-b727-8e54af1c374a}} .
For some {{formula:e7fd56b0-1bdd-4abf-8307-beeb74c48f9c}} , consider the process {{formula:517ce1e5-6237-47b3-b8aa-99e5208f93d5}} that creates the last vertex {{formula:eaa31a17-e2a5-4461-8641-e12764254e44}} of {{formula:db109028-97a0-40bd-87aa-79a4a7abe28d}} .
By a short induction, {{formula:78743508-dac9-485c-a4e6-34406ba914c7}} is in {{formula:36fc29a6-8775-48a9-a0be-90f3d1625bb1}} at that time.
Thus, applying sampling:3, it is a sampling.
Moreover, as {{formula:7da94e90-adb0-4d10-995f-62a092de9c68}} is correct, observation (1) implies that eventually {{formula:95329f96-7e6c-4560-a690-73770b92d400}} is always in {{formula:3b3c0e77-82a2-4df9-a93c-c23a3dc6a261}} at every correct process.
From what precedes, {{formula:50b9c34e-0f16-4071-a5e6-df13482843bf}} is a {{formula:f9546726-994b-4e5b-a323-40145468d384}} -fair replicated sampling sequence.
Building the simulation forest
omega:simulation
A key observation in CHT {{cite:ea9cd8cbad59f4421feb8537fbe7d920746b4b75}} is that every path in the sampling graph induces a valid schedule for the current failure pattern.
Based on this observation, each process builds a simulation forest to explore an unbounded yet countable number of runs of algorithm {{formula:2a2a21d8-6f81-4192-87c5-d3adb0a98ed6}} .
In our context, the runs of algorithm {{formula:032c636d-3600-4023-bb0b-b31ca5c37fb7}} that bring interest satisfy that
the processes outside {{formula:dbcd201e-611b-4b50-8729-cef6a13f3895}} do not atomic multicast any message, and
the processes in {{formula:7085095f-b202-4a63-bfe9-c3e853adb356}} multicast a single message to either {{formula:e3dc7268-9151-4339-9b20-e02d7d0f231e}} or {{formula:c3e60598-211d-44b2-b508-384771cac22b}} .
Let us note {{formula:6546705f-0fc8-4489-b1d1-d534c1a121bd}} this set of initial configurations.
Configurations {{formula:8ba105f9-75e2-4939-bcf5-ede85be1b672}} and {{formula:312f8607-8f48-4774-9e0d-fca379d03b27}} are adjacent with respect to {{formula:2b082dc8-e67b-40dc-8508-fdd46f71c1bf}} , written {{formula:17384e8d-609b-4d45-9396-4df681dc78fa}} , when they differ in at most the state of process {{formula:fb399567-047e-422a-8403-adf8f058c816}} .
Procedure {{formula:b846218b-3e65-41fa-bdc6-06e6f75e5626}} constructs a simulation forest {{formula:42115fd0-39b0-4705-b201-6dc3f4d27566}} from the schedules of {{formula:a69e0b08-397f-491c-8d58-01bad4e72e86}} induced by the sampling graph {{formula:8c45cfc7-f634-49cc-a2fc-72d00d979fb2}} and the initial configurations {{formula:dfe432c2-af2f-478b-b0b7-07ed4310eef3}} .
More precisely, each vertex in the tree {{formula:ebe69c74-083d-4737-8cdb-ad38f3508d33}} is a schedule starting from the initial configuration {{formula:5f917dcf-b080-4205-8ecc-b380bbb41ecd}} .
The root of {{formula:963c1c67-a315-45f9-b2ad-50e13f2dc4f3}} is the empty schedule {{formula:06b4e598-52c7-4ed3-a8e1-f8cf33bc4160}} (omega:var:1).
There is an edge labeled {{formula:b42843cb-0417-42e9-a19f-8fc9259b3492}} from {{formula:44bf67f1-f7bf-4c32-b81f-4f86fe6def6d}} to {{formula:a9cfce41-b09f-43f7-b1da-40bc9dd3d368}} in {{formula:07c0f938-c83d-42a3-8d90-86281fd5fe1a}} when {{formula:f8f991e6-1bbd-4632-a8d6-310822986678}} holds for some step {{formula:0cf2d8e2-0a1f-47a0-b61a-6ee3f058ea8c}} .
Procedure {{formula:1cb3b8d4-f822-4787-92a6-c1578e60508e}} builds the forest {{formula:56faabf6-7390-41de-854d-369e55c05a5c}} by considering all the paths in {{formula:9a5f0153-aaff-42b8-8563-9f8933b017e3}} (simulation:2).
For each such path {{formula:a82c2d34-e1a2-4a48-8edf-8467ae2ab563}} , {{formula:0b0fe549-d909-4175-9fe2-7c695250f1ea}} creates in {{formula:c683845b-50b4-4be1-8c93-6e07fe9f4a00}} the schedules induced by {{formula:6a74a60b-420d-4bf5-83fd-63e0c789bedc}} (simulation:3simulation:9).
To this end, the procedure relies on a queue {{formula:d4e3597d-9fb3-48b0-8bad-eaf0e0ac7215}} of pairs {{formula:470afcd5-bcd5-4ec0-a873-9ec4540a2443}} , where {{formula:852f0942-6404-4543-a846-ab897cb1f6e2}} is a schedule of {{formula:1caabab0-fb7d-4328-9410-5d4fa68f070b}} and {{formula:6da8d95a-64bf-44ad-8657-ae3b156dcd35}} tracks the next sample in {{formula:54d2d4d8-dcb5-4523-b51a-f5a0fca6445c}} used to create a step.
Initially, {{formula:427a15d7-75fd-4d71-9125-e29274602b4c}} contains {{formula:93b166d6-b7fe-4892-a047-bb03c5a978cc}} , the empty schedule (simulation:3).
Starting from a pair {{formula:f3891405-0342-4932-a7fe-9a9067dd7ba5}} in {{formula:ec4d32f6-2796-47c5-a785-2d0f742a5390}} , {{formula:aec3e69f-c171-4833-b1be-a4dd49769e9d}} considers the {{formula:e84aca54-fa3f-43e3-b026-587d85834459}} -th sample {{formula:a06180cf-3744-4cc7-9a17-9f4f9426ae5b}} in {{formula:d2235342-b69f-4518-8edf-c24e16171114}} (simulation:4simulation:5).
For every message {{formula:83305897-8491-477d-b299-c4b5cfc1bd7e}} addressed to {{formula:d93633c8-2ccb-4086-83e5-16063d0f22ff}} in {{formula:a951cc8c-f576-4e18-9033-0e8fb9897853}} , the schedule {{formula:12efb3f1-970b-45c5-b77f-0b5ad9d70d8a}} is added to {{formula:fb4031c7-08c8-40d7-ab53-cfd1e75071eb}} (simulation:6simulation:8).
This process is repeated until all the schedules compatible with {{formula:3e75b95f-6eee-44e7-8144-277b6c6b99fd}} have been explored (simulation:9).
In detail, a schedule {{formula:b52ea9ff-5e22-4b2e-9a72-feb27dac46fb}} is compatible with a path {{formula:e86c8414-b25a-4557-98ec-2c958f516202}} when
{{formula:679eabb9-f0c6-46fc-9588-3603aa94c83e}} and {{formula:a6f4f1c3-9364-45df-8581-2d90529c59e5}} , or
{{formula:c706ecce-70d0-4320-bf09-2fc6bf524835}} and {{formula:4acd05cf-f552-4e31-b387-a5c34403fa23}} have the same length and denoting {{formula:0499ae22-7297-4d0a-9fb8-8dfe6b435b31}} , there exist some some (possibly null) message {{formula:b559c15f-7def-4243-a381-830ee4300a50}} such that {{formula:3aec01d5-c0e3-4f91-8bed-63b60bc95e03}} .
The lemma below proves a key result regarding the simulation.
simulation:0
A schedule {{formula:ff5c4381-ac7c-43fb-bdf4-20ac5bfa78f0}} is always compatible with some path in {{formula:3412d82d-23c6-4a52-b2d2-7481330a10fb}} and applicable to {{formula:81f558b7-6f9a-42d9-8e05-9a1faea13744}} .
Conversely, if {{formula:a205f117-e704-4f00-ad3e-7ace7c5c91eb}} is applicable to {{formula:7a559c42-5330-4dc6-a698-b1442f378977}} and compatible with a path {{formula:673516b9-56c7-4a7c-b2cd-1ebaaabbc8f2}} in {{formula:fdc65646-979e-42f3-9cbf-18a50413974b}} then {{formula:0718887a-3637-48fa-a644-0274ba001774}} is in {{formula:90aac443-a67d-47dd-b322-af69909d0744}} .
The first part of the lemma is obtained by induction:
In detail, this is trivial when {{formula:58499ac0-db8b-4ed2-a753-922bacc449c1}} .
Then, consider when {{formula:fc2e41c7-0c2a-4b70-ae1c-9c67ed56a2fa}} is added to {{formula:ab998525-08db-4d95-98a0-b9d9001b4d90}} at simulation:8.
We have {{formula:02416602-7200-43b4-8edf-07bcb0116a3c}} .
By induction hypothesis, there exists a path {{formula:996ffc5b-c065-4f3a-b474-5e51d7e1c082}} such that {{formula:e0545ce5-c6d5-4cb7-8ea4-5e3e08192d11}} is compatible with {{formula:99c92cc8-f413-44e0-8146-c34d6cfb4ec2}} and {{formula:1315b6cb-e84d-4ed8-9439-dd524a1eb065}} applicable to {{formula:bfb1ac71-df90-42ea-a149-6769a67e0773}} .
{{formula:16880e86-bfa3-485a-9c23-2152d0916783}} is compatible with {{formula:5f831a51-3c89-4744-8d69-41b3220ce70a}} , where {{formula:57bc679c-647f-4da0-aab5-3d61d1be6772}} , as written at simulation:5.
As {{formula:7166a38f-21bc-47bc-a121-c08f36d8a8a8}} suffixes {{formula:e94b0150-7bec-4c48-ae79-62de5cf0f483}} and {{formula:e56eaef2-49a1-413f-8a6e-087befbf489b}} is applicable to {{formula:d93d88a3-322d-4b6a-8580-3aa6e1ccffc3}} , so is {{formula:a5e4d8b1-1f8c-4b32-9a04-c6b0312d2bcc}} .
Regarding the second part of the lemma, assume that {{formula:bee39557-283a-4295-8e4e-c3eab4ed72ee}} is applicable to {{formula:a22ddddd-c39d-482a-9f9c-01cbbe8b987d}} and compatible with {{formula:c6e07661-f190-4074-8193-1024789b1f61}} .
If {{formula:67925b75-1a63-4980-907c-5b273d73ee5a}} then according to omega:var:3, {{formula:a5f4254d-6b27-4c2e-93ec-70d90e8e0d04}} is in {{formula:3bdc3e97-d328-4357-a763-4b2cf5d0a231}} .
Otherwise, {{formula:2b548df6-1f31-4bd7-baf5-f3c16e8d2a52}} is added to {{formula:5ba04701-3ae8-4397-9a3c-2bec48c95820}} at omega:3
Hence, simulation:2 is executed afterward with path {{formula:ee2254bb-5cbd-4520-94f5-de90b4568fb0}} .
(In our system model, this happens at the same logical time.)
When simulation:2 is executed, by induction on the pseudo-code at simulation:3simulation:9 and using the fact that {{formula:7c84829d-23fb-4918-b116-c51c36c01cb7}} is applicable to {{formula:90d9db33-ae7e-4ac7-b158-915ff0c73a86}} , {{formula:78558769-287f-444d-99a5-10442e3bec9b}} is added to {{formula:3fb8095e-b18e-4a25-93f8-f4237889f732}} .
From which, we deduce easily the following result:
simulation:0a
Consider a correct process {{formula:db934b3c-883d-4486-9ab4-c7e7f88ac82b}} and pick some schedule {{formula:77e9f91f-c824-44ef-b00b-a69f12b7cb1a}} that eventually appears in {{formula:0a81b3b0-51f3-4d88-ad3d-0f6af51cf297}} at {{formula:0782a07d-60a7-4ce6-9817-2f92d8c4f8dc}} .
Then, schedule {{formula:6b0f2d77-9518-48cd-85f1-8309e8996dde}} is also eventually in {{formula:219dcb5d-0272-426d-b237-d3cdca5c9df2}} at every correct process.
Pick some correct process {{formula:489b52f0-a227-4688-b6b8-4dcce0764c07}} .
From simulation:0, schedule {{formula:51f7ed42-82df-4933-a5ce-9ff7c08034b7}} is compatible with some path {{formula:dddb2081-73b6-4f40-943b-123e15ce4dee}} and applicable to {{formula:4a3f90e2-4a70-467d-963c-be8dc05caf84}} .
By sampling:1, {{formula:6a3eda72-26ab-4884-bf82-68028acd9b22}} eventually appears in {{formula:388d9230-a097-4a34-aaff-60981e909666}} at {{formula:8beeabd7-c85f-4e71-8c6a-20fd26c842d4}} .
Applying again simulation:0, {{formula:d6c5930b-211c-48a4-af5f-b1b02074b2cc}} is eventually in {{formula:efcf362f-e779-46d4-b4c1-7a11a63a53b9}} at {{formula:6d1d7efc-614a-4255-bb3e-ceb0cc6b3b8f}} .
We now explain how each schedule in the simulation forest induces a run of algorithm {{formula:ee474647-c4d3-4ecd-91b5-f46d29179b3e}} for the current failure pattern.
Further, we prove that each {{formula:7b110196-4ae9-4696-a48e-a7f5c7085b02}} -fair replicated sampling sequence induces a sequence of runs of {{formula:6a3b6fd5-9b98-45f8-8764-c45a99469377}} that converges toward a {{formula:7c89a488-2a1a-43cc-b591-58c32e76771e}} -fair run.
For starters, consider a run {{formula:05b1fe2d-6b0b-4cfb-b191-85187f53cd4a}} of omega and a correct process {{formula:50853935-2a41-44de-b16b-f197a7e27f7c}} .
At {{formula:2c44fc16-b36d-4137-ad25-f2a3e4243c16}} , each schedule {{formula:9bf5261d-d9a8-4e7c-a638-1828f896e851}} in a tree {{formula:08e331d6-01f8-4490-95ab-3a6665a2e313}} corresponds to some run of algorithm {{formula:3cf2008e-1a48-45e4-b45f-f2cb02a63d4e}} .
This run is built using function {{formula:63e509f2-9878-44d0-9618-0db42d9c236a}} defined as follows:
run
Let {{formula:68c3cd13-ad58-4d26-8d6c-3e4d39a18b79}} be a path such that {{formula:80e0e999-53e7-4fa6-9bcd-a485e21aeeac}} is compatible with {{formula:8c0c9af4-954c-4abc-b263-47058833a187}} (by simulation:0).
Let {{formula:96486a15-378d-416b-9aa3-df9bc237bab9}} and {{formula:26fc219c-f4c2-4204-8d58-837042fcd1e9}} be respectively the failure pattern and failure detector history of run {{formula:5c59ffa2-ce98-461d-9380-b92df3ea2ee1}} .
Define {{formula:4727a7d4-371c-425c-a917-b2fc62f64a7d}} as the sampling function of {{formula:231f28ac-3a5e-4b57-b4fd-c7450f04b689}} (by sampling:2). Let {{formula:44d4b8c6-0dad-4455-94d1-3abfb8b7eec2}} be {{formula:ac7f09d7-d7d3-4a72-ae10-e6577bfcb441}} , with {{formula:003b2106-0f5d-4ddb-b003-189d8652928c}} the usual total order on naturals.
Function {{formula:1c25b7fc-abdd-4aa7-941f-2d67f84256f6}} maps {{formula:a071340f-cea1-413c-8fef-2556ac8f9113}} to {{formula:b3f691f4-126f-4d35-8865-67d04f367fad}} .
The result below establishes that {{formula:fefac81f-47fe-4ff2-9acd-075ce2b93b3d}} is indeed a run of {{formula:07550974-c876-4e00-a5c6-3fae468978c2}} .
simulation:1
For every schedule {{formula:463eae29-6e16-4f9e-94db-10e62705064f}} in {{formula:52d6de51-2d0e-4f85-a43c-1c05a2170771}} , {{formula:ba8008c1-1518-45cd-b5ef-778235760978}} is a (finite) run of {{formula:1afc6324-14a9-4bd6-8fd7-872adb9bbd77}} .
Let {{formula:c24d5208-524a-4168-a1c7-1a8ac6020379}} be the value of {{formula:a8ba786b-9f98-43e6-af56-66c1d4271ba1}} .
According to run, {{formula:348ab394-81f1-458c-94d0-fec0f000c001}} and {{formula:83ff0ed5-baf9-49d7-b18c-c2dc41ffb021}} are respectively the failure pattern {{formula:e1439531-1228-4a8c-838a-95ab2907a227}} and the history {{formula:b6042f9f-63ff-4b67-bca8-b44d448dbf1f}} .
{{formula:e4063554-5080-40f3-836a-9db5aa3677fe}} is an initial configuration of {{formula:b883d5eb-5bbd-40e4-80a7-2375b90453cc}} , as defined above.
If {{formula:d959c8ce-3325-43b2-a142-c8fa859b6c40}} , then according to run, {{formula:c8e32d0e-e912-47f9-a681-41d794c146c7}} .
Otherwise, assume inductively that for each element {{formula:da054472-c96c-4e1a-9ec5-38924b000b87}} in {{formula:9364b0d2-ff73-4aa3-9c4e-cee092d5613c}} , {{formula:3ebc253a-152f-4a18-9120-9e66065c0a85}} is a run of {{formula:951db933-7e15-4b3a-ba99-86d463f04312}} .
Schedule {{formula:77028d56-d396-41c3-8093-b4668f812f94}} is built by concatenating a step {{formula:997ac4e6-7e7c-44fe-9a2c-0c94c675c585}} at simulation:7 from some {{formula:af164bd3-a668-4621-967d-a3648c0db22c}} in {{formula:e3fc7d91-d698-452d-9caa-62681626826d}} .
By our induction hypothesis, {{formula:826858ab-c056-4264-910b-1ed0f99b5c18}} is a run of {{formula:8366ccda-eb7f-4e81-a992-0417ef7f5584}} .
Denote {{formula:3c1fd9cc-d230-4db8-b36a-3f414e500a78}} the value of {{formula:ac37f2b1-d2a1-47c6-b2be-90e840287544}} .
According to run, we have {{formula:8d18613e-43b8-4021-b325-28fe2e55fc76}} .
Furthermore, {{formula:5ab6cbe4-ef05-40e3-af90-228679e4d56a}} , {{formula:c69b2d79-5d6f-44a4-9201-5d67a82e2b4b}} and {{formula:914d5997-b036-4626-8938-03c9a9b00423}} is not faulty at time {{formula:06b06778-f5d8-4bfb-a8f8-811d77b29edb}} .
From the pseudo-code at simulation:4simulation:7,
{{formula:0217adfb-c131-4042-a6ad-a311a2e666e9}} ,
{{formula:5d24232a-9e26-404a-ac63-35dd7928434c}} , and
{{formula:689a7b23-3c54-4487-ac8a-a55b3a2fdd7d}} .
As a consequence, {{formula:cf582519-149b-47ba-b253-b44d6e9d64df}} is a run of {{formula:7fd2986c-299f-4a6e-a6e5-53c50945e785}} .
Consider some path {{formula:f14535b6-c0d9-40c3-9540-b4f49aef833c}} in graph {{formula:0da986b2-90c5-477b-a06d-41b36b16c69a}} at process {{formula:a4e5920b-d54f-4a52-9873-50ab06c71229}} .
In what follows, {{formula:6c304359-b916-4a6d-b1c9-14c410771e51}} denotes the schedule starting from {{formula:5b05decc-3058-4b3a-91a7-c0f4dbd55984}} built from {{formula:18b5d80e-52b0-456a-9195-720ae7f1809f}} with {{formula:a63c342d-2231-41ff-af94-49f6fa07efea}} when at simulation:6 the oldest message addressed (if none, then {{formula:6fc1f250-5f5d-43a9-91be-423d1f0d425f}} ) to {{formula:4d04cab6-6b5f-4ffc-b647-c0f7c95fc5d5}} is always retrieved from {{formula:178439a2-8c8e-43d7-939d-5866ea51415b}} .
Notice that by construction, {{formula:cb293228-a294-4d35-8b86-6cb871a57e11}} is a function of {{formula:e0764c56-5513-4dd4-9c4c-0b45e61e646c}} .
simulation:1
For every path {{formula:6c370907-0f31-4500-b700-e27ec6e8e0ab}} in graph {{formula:ce36378e-bde4-47e3-8143-616d19c2fcaa}} , for every initial configuration {{formula:6ca4a727-4836-43ed-a810-fdd50659a32f}} , {{formula:e2612073-0d6a-43e9-adaa-1f079fbae61e}} is compatible with {{formula:0714cbac-2af0-4389-b2b3-a1d81dd5d49f}} and applicable to {{formula:558e2419-d622-42b2-9985-419bbbbbad4d}} .
We proceed by induction on the length of {{formula:5e2cb074-df33-4e53-924e-63ac956eba89}} .
If {{formula:3acede63-8b58-464e-843d-7cccbeb6e9be}} , then {{formula:6871328e-c8f5-48c2-a7cd-ff3387319714}} .
By definition {{formula:b7f84f0a-5431-46f9-bad2-853496e4e1bd}} is compatible with {{formula:2dbd6432-20f3-401c-8806-25b2ee131744}} and applicable to {{formula:0402f548-6a4d-4bf4-ab39-d20596a7e2b4}} .
Then, consider that {{formula:e3fa4507-e3aa-4fcc-8c5a-05f7945a1737}} and {{formula:a72b9760-5819-4c38-9895-2b51de9f6a16}} compatible with {{formula:fb053925-dca3-4376-9265-e2a7ac781b23}} and applicable to {{formula:83767733-305e-415f-b315-235857c91412}} .
{{formula:0db61876-4043-48dd-ae16-89c98cc688c3}} is computed at simulation:6 such that {{formula:b52cc3cd-203d-48cf-9ad1-f1ee37d31882}} .
By a short induction, at that time,
variable {{formula:b853c6b8-f081-4711-92cd-72507ebcb04c}} equals {{formula:0e7fe0ed-1419-4a42-b9e3-1863ebd02f13}} ,
the tuple {{formula:2431e467-8aa3-47ad-ad42-938016e43a02}} equals {{formula:114db2dc-34cf-417f-95ec-df113bb8566f}} , and
variable {{formula:811184c5-0958-4902-bedc-e7b9ae91c2d9}} is the oldest message addressed (if none, then {{formula:17bda6af-d35c-4524-8c95-d7424253aa10}} ) to {{formula:11ca9ef5-d5aa-4863-a861-16c589b01b39}} in {{formula:deff10b3-1ba7-4081-adce-ede4eff74183}} .
It follows that {{formula:8fbbfbde-e47e-4255-8c7d-067a37983ed6}} is compatible with {{formula:0636debd-34e4-4e36-b045-a60687bf7e3f}} and {{formula:a1420d07-fcf7-45bf-b9d1-5bdcaaba3758}} is applicable to {{formula:8c584796-33b2-4701-9cdc-84b9ca43a4e3}} .
simulation:2
Consider {{formula:800a6618-0c22-4580-9db4-f92e5d2c14d4}} with {{formula:c5e8c357-75b2-4118-b862-498a83915932}} .
Let
{{formula:87e55497-8665-4aa4-8ca7-f1a2ec52548e}}
{{formula:e7fdcee7-ce27-463c-8440-8ca4310facc7}} ,
and {{formula:ddfea3e9-2faf-4c08-8a0c-03513747ede8}} .
It is true that {{formula:f9abe6ca-6d07-41e0-9549-e659ea773f72}} .
By induction.
simulation:2
Consider a run {{formula:23585d70-4489-4237-90c1-93fa382a6502}} of {{formula:1182f93c-91fd-4dde-a1c3-6d7cb2b79f01}} and a set of correct processes {{formula:3ac07151-cd0a-41e0-8032-5ce23e730f5f}} .
Let {{formula:b61356c6-71e8-4f7e-be2d-46e9a21429fb}} be a {{formula:8135cd92-298c-435d-97e1-38791f1ab7e0}} -fair replicated sampling sequence and {{formula:4df323e7-6d75-4ffd-b4d7-7d0e3d15e7fd}} some initial configuration.
The family of schedules {{formula:62789a58-21ed-4152-8bd1-1364d1401bee}} satisfies that
{{formula:0b5b32e8-254b-40eb-bfb8-2f9073895aba}} strictly prefixes {{formula:46eb92ad-a656-41a6-8871-acba0aeb3095}} ,
every {{formula:a7d6c3fa-6733-4618-899e-815d373a1a7b}} is eventually always in {{formula:0f9cfb94-9cb9-489d-9cc6-b4ef58a15cba}} at each {{formula:ded51745-853c-4b70-9fd1-0da7840c662e}} , and
{{formula:b0f71ab3-fdae-4fc8-9a96-e62c18b4e287}} converges toward a {{formula:7857982c-00ca-4a35-9646-6eed08760b18}} -fair run of {{formula:e8bd5b5d-f84d-450c-a0c6-dd5117266c22}} .
Consider a replicated sampling sequence {{formula:4a0a1982-9118-40fd-8524-51c0ad39b74a}} .
For starters, we observe that:
for every {{formula:2043933b-19fa-4472-ad25-dce6efe70700}} , as {{formula:3c8ca500-cc8e-4363-8ff9-0f17a6b531d4}} is eventually in {{formula:4134b7b1-8fb9-4114-9453-6c97b1ab4b9d}} , then {{formula:78d3d6f6-18d4-4f90-8d97-46d4e1fc561e}} is eventually always in {{formula:d284debd-19ee-4f32-8f8f-b50ad710c0e0}} (simulation:0simulation:1),
since {{formula:f6040a6b-07e8-4c3a-a56c-7f96fbde10a9}} prefixes {{formula:823031f0-a251-4454-9ab3-0bf41461feb6}} , {{formula:03dbf108-962b-4636-8f55-0d4ee8c69bdc}} prefixes {{formula:519c38c0-0caa-4e5a-9a5c-73257063db86}} (simulation:2), and
for every {{formula:10ef8795-3ec2-48c0-9712-6d2aab0ba740}} , {{formula:ede6132f-6223-4885-8be0-8f1dc62a9175}} is a run of {{formula:7cd69c8b-ffde-4f65-bd3c-dc5d9d3d0986}} (simulation:1).
Let {{formula:dc88a183-4a89-4890-a49e-f8bf99c5368a}} be the space all the {{formula:f968ed5a-dc41-4cb0-b79b-3e9da646ba7c}} -words built atop the alphabet {{formula:5a74d3cd-6cde-4acb-8865-5d7d4ea4732c}} .
{{formula:063be5f2-8b6e-4337-9541-e373f9478b86}} is a metric space for the usual distance function {{formula:7e1ea27a-30dd-4f8b-9709-1b0e2eb23cad}} .
Define {{formula:fa378d03-a37a-4d6e-a7cf-5cab576b3c52}} as {{formula:f680917d-2bb4-49f1-9726-dbb32b0b4af3}} .
Observe that {{formula:d3e44800-e0d2-4916-8372-e6aeb0ae4c2e}} is a growing sequence of {{formula:52c252a2-6a7d-4712-abc5-a747fcd2217e}} -words which satisfies that {{formula:b76c8615-2814-40c8-b54e-91c5fa0f2698}} .
Hence, {{formula:7bc41819-0f86-40bb-8d37-880090eac367}} converges toward {{formula:13646614-1aa8-490e-9676-a42e26fb7be8}} in {{formula:f40ed495-d2be-496e-99cc-42690ef6057d}} .
The very same reasoning applies to construct {{formula:d4f9388b-a95f-4a74-b241-08b6329f24b9}} the sequence of times built with the sampling functions {{formula:152674c9-80df-4806-b018-5eeafd131e79}} of {{formula:7e4240f8-1a09-4187-8050-24530bbe158d}} .
Thus, for some appropriate metric space, {{formula:36ef5929-39dd-484c-a473-4eeeaa07f77e}} converges toward {{formula:d7f22def-d572-4ad5-a64e-423bda60949e}}
In {{formula:fc473f4a-8072-4051-b8cc-ce2dc4cbf14a}} , only the processes in {{formula:7acfc96a-227c-4be0-9db1-fc1a4b9aa1a8}} takes an unbounded amount of steps.
As {{formula:9e2d157d-20ee-459f-8fb1-44b921e0ec37}} , this run is {{formula:266b6003-4f13-42ea-8ffa-c08e380b2e59}} -fair.
Moreover in {{formula:581df907-508b-443f-8d59-48098b71fe8c}} ,
No process take steps after crashing.
Both {{formula:43d7f0c0-454b-41e1-becc-0266d687f4ba}} and {{formula:aa7636ef-7713-4ad4-bf7f-74810fb721fb}} are infinite.
By induction, {{formula:9dfd3b8f-78bf-4c3b-b6a0-2e1756ee5f8a}} conforms to {{formula:0d168320-004f-40c6-95b3-a128733c57fe}} , the timing {{formula:79285fca-de8e-49ea-870b-91c412e43a51}} and {{formula:d6f8f763-cfff-4264-847c-753259ca8058}} .
Every message addressed to some process in {{formula:6f1a2d7a-0d5e-4ef3-a45b-e56098dd6ae8}} is eventually delivered.
It follows that {{formula:104a5555-0dbe-4326-8b73-b15b0f2dfeef}} is a (well-formed) run of {{formula:2b22110b-7e79-40a2-ad28-3b279fa83de4}} .
simulation:3
Consider some run {{formula:ccc5d525-b7d1-4fce-bce1-2f26dd60605d}} of omega and {{formula:7631d2c2-d102-45c6-aa62-020ec157d795}} a point in time during {{formula:1b5bccc4-d87b-4163-8764-b8b174c66e47}} .
Let {{formula:94b7a327-2778-4d48-a361-43fb6aecbdc9}} be a schedule in {{formula:9c1ecb26-aecc-48ed-9626-20ad9220bdca}} and {{formula:5e683520-0eaa-4d8b-a896-f41af0f6e84b}} be a message multicast in {{formula:8d332f3a-bbba-4dfe-abc8-a2177dbe5643}} by some correct process.
If {{formula:799fd14b-ff3b-4913-9c87-b7050b6e504b}} then there exists a (finite) schedule {{formula:b9efbb5b-ae95-486e-8627-de8db03b43a3}} such that
{{formula:26010e48-a002-46e0-96c9-eafa091eb1dc}} is eventually always in {{formula:418734d0-d557-413c-b58c-1fd2a64d2d73}} ,
only the correct processes in {{formula:afdaece4-f91b-4e8f-8a7a-027b0d6d91fd}} take steps in {{formula:b46d4a04-8037-4656-980c-5d3a817a67f1}} , and
process {{formula:942325f7-9b09-4ffd-a5a6-fff9441d94e5}} delivers {{formula:2b940c4f-8cd7-4d0d-b632-90bffc3ccdaf}} in {{formula:22a849b2-2d62-48cb-908c-9eda615948dd}} .
Furthermore, for every schedule {{formula:306eabda-6217-45a1-b63d-221024aa366d}} , if {{formula:697c616d-a5cb-4c62-80ff-e9307dee4d38}} is applicable to {{formula:0e9479e2-953f-4f5d-aafe-d89faeaa8f10}} , then {{formula:757efb4d-a3a3-49e5-88b0-1f39814b2fdc}} is eventually always in {{formula:b19992f1-84f3-46e7-bf58-4c879a424637}}
Define {{formula:200d6703-1605-41f9-a1d8-634ee2075eaf}} as {{formula:b319b5c2-6407-4b2d-bcb0-537b2c7a7fa7}} .
By sampling:3, there exists {{formula:f07ddf80-16aa-406c-aa6a-fac7bfc25d54}} such that
for each {{formula:d445e72e-ae77-4b11-b0df-8efc32bdd1f6}} , only processes in {{formula:97422170-e275-45e1-bef7-4e911cca8183}} appears in {{formula:d854661c-e58c-4b6c-9f25-afe3d7b3dc0f}} ,
for every {{formula:d81287c3-9fd0-4333-9435-17e316439de3}} , {{formula:17ceb025-c14f-416c-a841-3298d781c347}} is a {{formula:0cc66e09-c178-495d-8861-4a5b419ac697}} -fair replicated sampling sequence.
Choose some schedule {{formula:222865de-ba3f-48e1-b9ce-eb26e67de992}} and choose {{formula:5540ac75-363d-4e9c-bad4-e609fe8086cd}} such that {{formula:c880565e-642b-439e-a1bb-ac2fadfb47fa}} is compatible with {{formula:9ac211df-e75d-4bbb-a9c7-197a45c8f522}} .
Applying simulation:2 to {{formula:597174bf-9887-4b43-94e2-d75491c13599}} , the family of schedule {{formula:efa7ddc8-b3a5-42fb-af1a-6571fa30999e}} is such that
{{formula:49a55d4e-0499-45b8-80ac-c6088fd62e2b}} is eventually always in {{formula:3501eb9e-88e0-49f6-86ba-868de9725b81}} , and
{{formula:0c259974-b88d-4c28-ae85-435a02d048de}} converges toward some {{formula:c2f1e79f-62f8-4fee-b3ee-6f2ac694c8fe}} -fair run {{formula:18055fab-79e6-4546-8486-4c7ffa91784f}} of {{formula:be64f6d1-96a2-4a9f-a18e-13995663759a}} .
By strong genuineness (see variations:strong:definition), all of the correct processes in {{formula:6ad26c93-5a51-4429-aba0-e96c0db943ce}} deliver {{formula:b81d611b-ba4c-4673-9569-f4b70a5ac825}} in {{formula:4088a5ad-09b9-4300-8731-f4ccf1d0facb}} ,
Thus, as {{formula:88e91e62-dd55-4bc2-9165-cc8722300f4b}} , for some {{formula:00981b4c-3a54-4a56-b41b-5227b710d7f0}} , this also happens in {{formula:2449ca1a-cf8c-425a-a2ae-1c5c1d54360e}} to process {{formula:896a59c9-2cd0-4037-9fb2-b593a0bf022a}} . Since {{formula:f106fa24-c0e8-409b-b133-0259341b91d8}} , simulation:2 tell us that {{formula:f65cb8f0-e96b-4966-9a8c-78c7f743c4d5}} with {{formula:ddb44776-cffb-4fc9-a426-73b9ade6f195}} .
Now, from what precedes, {{formula:43b5b90a-b943-4f40-be4a-0995f8c6a92e}} is eventually always in {{formula:905c2bb3-f6c2-4308-abec-cfee41587e4a}} . Moreover, by definition of {{formula:b0666ee8-1ce2-4cdc-aa88-8d2f49fb7213}} only processes in {{formula:37b28574-7531-476c-891b-5932186ae73b}} take steps after {{formula:0f7a1110-6935-48c9-b87a-1d556d2f68f7}} in {{formula:5ed3185a-1418-4748-ba19-8d9635e029bf}} .
Consider some schedule {{formula:982aad87-36f3-4357-9d29-ebc3db72dadb}} .
Pick {{formula:f304b5fb-3796-4e4a-8ca0-d05d49a3e197}} such that {{formula:8f7a3800-5af8-4ce3-8af4-609c689aa00f}} is compatible with {{formula:a092286f-04b3-4a5d-bac0-717566bebff3}} (by simulation:0).
By definition of {{formula:24399ba6-9517-4a8a-8ec8-ea781bd8dc79}} , at some point in time {{formula:29fb15ac-69fa-412a-9759-600c8b7300a9}} , {{formula:13487ce4-8d6f-4971-9ec1-9bcbf914a632}} .
Then consider the next moment in time {{formula:66d8f3e7-b0e2-4c72-b68c-02a23f316591}} at which {{formula:bc982659-8e08-467d-95cb-2f1a4a08f67d}} calls procedure .
As {{formula:840d82e5-e9e3-41f7-aa84-90cccaadcefb}} , {{formula:05e5a7ed-eea7-4920-9db3-45cc538bc9dc}} executes simulation:2simulation:9 for this path.
By a short induction on these lines of code, if {{formula:746c8f61-1858-49e4-83c3-77bc30a4ead2}} is applicable {{formula:437bbff8-7c71-4841-a0c7-dc542e3a80bc}} , then {{formula:1b0842fb-5ef1-41bd-a08e-c68c3b7cf75c}} is added to {{formula:632863c1-31fa-4d78-843a-2e6494792f0a}} at that time.
Tagging the forest
omega:tagging
Procedure {{formula:042a6774-f85a-4c64-8d2d-b35d0a017bcf}} associates to each schedule {{formula:eaa697fd-43a0-48e6-9501-2cc3603d924f}} in the simulation tree {{formula:356acd28-af85-4a58-af2a-9d2038b4fe13}} a set of tags {{formula:f56d73a0-a4fa-40a0-bdc9-ef265038b4bf}} .
These tags correspond to how messages are delivered in {{formula:c25d718e-7394-4605-8053-0d6524e3ab6a}} from configuration {{formula:9b903cb7-7e53-4065-9c50-aea97509791b}} .
More precisely, {{formula:b485131d-5246-4174-80a1-afff6a000b1d}} contains {{formula:df0c1d21-43c3-402f-ac32-2a76fc2e1b5c}} (resp. {{formula:b31a2531-9227-420f-90a6-39fa4edde413}} ) if and only if for some successor {{formula:b603bff1-240e-4685-993f-5816efdd0a72}} of {{formula:3e81113a-d173-4b4a-a43e-883a41382a7d}} , a process delivers first a message {{formula:1a67571d-c496-412f-96a1-4bfcea561782}} addressed to {{formula:8b3f2c76-6bb7-4413-a1b0-2da8d43bdd2e}} (resp. {{formula:2de31a6c-88cc-4a13-8cad-d3ec8ee6e5b6}} to {{formula:b9b14043-be42-4629-9c3c-8b09b99bca22}} ) in {{formula:8f9ea289-dfb2-4227-ab55-370d72c05eb8}} (tagging:3tagging:5).
Notice that at tagging:3, for simplicity, {{formula:30720f6c-8a3a-4ba3-ab26-a1a3ec44692e}} equals {{formula:36c5f134-8c57-4619-9f36-52939f156ce6}} if {{formula:9493f811-e357-4e97-903b-9b746fce5479}} holds, and {{formula:5ec3841d-e9d1-4a4b-929d-db7a73985046}} otherwise.
A subtree of {{formula:e842ae52-d7aa-45a5-8d80-6aa87e41ef82}} is stable when every schedule in the subtree has a non-empty set of tags which does not change over time.
tagging:1 proves that every subtree in {{formula:9e263293-ed98-435f-97fc-9fcf601f02d3}} is eventually stable at the processes in {{formula:55590fc3-7b83-4100-87b0-3301f9907e33}} .
tagging:1
Consider a correct process {{formula:249ac816-9c20-4403-ab49-b3eb54c3125d}} and some schedule {{formula:7c1f46ae-c972-4a75-bb05-1fa3c01127a5}} that eventually appears in the simulation tree {{formula:2a6ebf4a-db9d-4c13-b668-1daa51736cce}} at {{formula:4551f2b7-a3b8-434c-aa3a-3e466ee104eb}} .
At process {{formula:4c139b73-218b-451d-9eed-75053db898bd}} , {{formula:8e458e05-f5cb-4d12-ad7a-eb4a7674ed9c}} is monotonically growing and converges over time toward some non-empty value.
Consider some run {{formula:54b7e682-b23a-498d-a9c7-1760b4588eaf}} of omega, some schedule {{formula:2a6bbaf7-c0fe-4d58-b50b-c27e3da38c8b}} eventually in {{formula:6486a403-e240-4549-b6ad-e2f89f4acb5e}} .
Function {{formula:04e922d6-1dde-4e9a-9a94-e429e48c6af4}} is monotonically growing over time (tagging:5) and clearly bounded;
thus, it is convergent.
Choose some path {{formula:b6224463-95de-4f4d-b018-2aedd658cce6}} such that {{formula:ba3c64f5-3a87-46bf-b79d-be36fb0c3c1e}} is compatible with {{formula:c6a48fef-cd57-46f2-91c5-49aa18ed4b67}} (simulation:0).
Name {{formula:8e4cc9dc-bd54-4287-bf57-264802b0e96c}} the message sent by {{formula:7108e15c-13cd-4510-8d42-c5c3a76fd489}} and addressed to some group {{formula:a6ea1031-a8c4-4430-8ce8-462dde8b91f0}} .
Applying simulation:3, since {{formula:5ccbcdd0-955c-4501-87b7-33375c47e081}} then there exists a (finite) schedule {{formula:5e2d4c7a-6605-419c-926f-09cb615102aa}} suffixing {{formula:6c4d5f6b-fc8c-4d94-926e-eed8284416ab}} such that
[]
process {{formula:18ac6d76-a009-410e-8fba-b678b04c1c49}} delivers {{formula:931c2ad0-4dca-4ef5-b559-e98f32cf10f4}} in {{formula:b5afbc53-d331-462e-9020-856d30afab12}} , and
{{formula:ec55c34c-46bf-445b-82e5-3bfdbe761ed4}} is eventually in {{formula:a1ee42d8-23ff-4d00-a008-1c2e7ac49244}} .
Let {{formula:7704baf9-8260-4751-b2e8-d081e882c84c}} be the time at which {{formula:a59b8004-c659-4df5-992e-aa69e0b22ddb}} is in {{formula:16eee6c0-e8a8-4df9-a34b-4721925bac59}} .
Without lack of generality, assume that {{formula:5c380787-c538-45db-acae-b35e27f4da14}} .
When procedure {{formula:abbb78f8-0dad-4de9-aebc-2dbe16fc26b4}} executes after time {{formula:38a03866-400e-4851-85da-e2d122aa862d}} , {{formula:6df5d1fb-6237-4651-b7fc-fd7919fe7048}} is added to {{formula:faf17333-ff07-4180-a1ab-29ad69c5e225}} (tagging:4tagging:5).
From which, we may deduce that:
tagging:2
There exists a non-empty set of tags, such that for any correct processes {{formula:1b7aa787-083d-41ae-b6db-5e7f03451602}} , {{formula:de4d4da7-a63c-4b5f-9726-c86cf96ecfd7}} converges over time toward this value.
Consider some correct processes {{formula:c76b66bd-4be0-4227-87dd-7e5120d841e2}} .
Applying tagging:1, {{formula:4d31e2b2-e9fc-40ce-92cb-57d2a039c3af}} converges toward some non-empty value {{formula:86092cdb-27f5-4a6d-89dc-af1608ed9a59}} .
Consider some group {{formula:77b5cb52-5fed-4559-9194-1b1b7492fb84}} .
According to tagging:3, it must be the case that for some schedule {{formula:989f14ce-665e-4600-b496-5813ca55fcbc}} suffixing {{formula:776131f3-3abf-4a42-9929-d4d1eac220f0}} , {{formula:7e51f1b4-5a07-477e-b5d8-0b6fb10036ba}} .
Applying simulation:0a, {{formula:5f17d446-adbc-4abd-95d6-a889002e60bb}} and {{formula:e6b96755-20b0-401d-9037-055119262044}} are eventually included in {{formula:85e4dd94-e05e-47f7-bb92-d063d4c7eec8}} at every other correct process {{formula:8b29749a-20c5-4100-8b55-0a5de11a2881}} .
Assume this happens at time {{formula:715151ae-d358-450a-b1f9-2278abd2813f}} .
At time {{formula:48738093-6514-4cab-aca2-7f129efcc60e}} , when {{formula:62e102cb-ef53-4351-bd9e-b3a44f2b5c89}} executes {{formula:73403bc4-966a-423f-ac96-793c2df1a543}} , it adds {{formula:49241304-5b1a-4b7f-a95e-41a8b1b21cc6}} to {{formula:127039a4-5375-4b59-a4b1-333b0c2829b1}} (tagging:3tagging:5).
Extracting the leader
omega:extract
Outside {{formula:f32b7a3f-3b6c-42f7-80e5-1f9a76efdb7d}} , a process returns the value {{formula:9ce672a9-68c9-407b-9610-378588840be1}} when querying its failure detector module (query:1).
In the intersection, a process traverses the simulation forest and eventually locates a correct leader in {{formula:2a9627a7-c57b-4c16-ae8d-df6d5899575c}} .
To this end, it relies on {{formula:f54ea493-d7fe-401f-b1c1-6d236e747781}} , the last building block of omega (extract:1extract:9).
As in {{cite:ea9cd8cbad59f4421feb8537fbe7d920746b4b75}}, the procedure {{formula:9143e6ee-dde0-43ec-9383-b650ab3b57d0}} uses valency arguments to compute the eventual leader.
A schedule {{formula:ab6466f9-3f42-465d-b6a1-330a171a7c18}} in a tree {{formula:3ad531ea-5a66-4c5c-8d1e-7bdf3516de78}} is {{formula:fd36a432-bb4e-4831-a078-d10540d70d65}} -valent (respectively, {{formula:0f93541a-efcb-4148-acf6-e3c86ca09ba8}} -valent) when {{formula:893859d2-4f55-44b7-b635-9d06226c3d68}} equals {{formula:45e484d0-bbdc-4028-9eaa-fd0e58a3ff33}} (resp. {{formula:6a668de6-6576-4e48-9bcb-382d3ddb3e7d}} ).
In the case where {{formula:7ae159d3-4b97-4940-907f-ba54261b4fe9}} contains both groups, {{formula:120be277-e8c1-4c5a-a861-3f38f2d88859}} is bivalent.
When {{formula:aa81676b-6d98-42e6-8dbe-d82a37d3e95b}} is not bivalent, yet tagged, we shall say that it is univalent.
From tagging:1, eventually the tags of every schedule in {{formula:121918bb-af29-47bb-98db-d4e60136df67}} get stable.
An index {{formula:07d748c0-c134-44d9-9aa8-b5eb152ecb73}} is critical when either
the root of {{formula:cdfef85a-9c43-4672-a660-98343139273e}} is stable and bivalent, and for each group {{formula:3c58ba97-0eb6-488e-b18d-bec613b547db}} , some correct process multicasts a message to {{formula:ea6662eb-fcff-4cf4-8f3c-7f054ff929b9}} in {{formula:5c5144e2-1f19-4e42-a84e-fcdf5a3b6c50}} , or
the root of {{formula:f84fe31d-60f1-4e23-a79b-a0ce03de85df}} is stable and {{formula:18fc2e10-3cbd-4aa7-9b40-d0ec8eee30e4}} -valent, the root of {{formula:f093a203-cd7b-46a8-a1f5-b146e3da6492}} is stable and {{formula:a71ab989-76ee-4355-9a5c-509001146a09}} -valent, and {{formula:87fe5fc8-1246-4df1-aa44-eb0f2b82a65e}} and {{formula:e6dd4a9d-2997-4a6b-8ecd-53ba82798701}} are adjacent.
In the first case, index {{formula:05d1e6b3-2594-46bc-8f18-1e576cd45b09}} is bivalent critical and in the other, it is univalent critical.
For a process {{formula:0e2353e0-f279-4758-be9d-a683c812557a}} , {{formula:4d2f071e-e52c-4f2b-87bd-9e21bc5c75ec}} iterates over all the trees in the simulation forest (extract:1).
If {{formula:d9a0be94-1132-4574-8d43-834d8bcbfbef}} is univalent critical, then {{formula:caa6f011-0d51-4b92-af82-b9d5e1e56c5c}} returns the process connecting the two configurations (extract:2).
If now {{formula:c942d23b-218c-4fd4-9a2a-b14f5c381354}} is bivalent critical, function {{formula:58f7a17f-c9e8-44d4-a564-38b3cbc4e7e1}} is executed for every subset {{formula:3d5f287d-90cf-40e9-a6a7-9864457590bf}} of {{formula:535fe6e5-1267-4cce-89ce-5afd3677e204}} (extract:6).
This function traverses {{formula:86b5492a-b2e1-4602-98ed-0d8eec05551a}} to locate a correct process in a particular subtree, named a decision gadget (locate:16).
If the traversal fails, or the valency of the root of {{formula:a301f0cb-3076-41c9-bbc7-f12c0093532f}} is not established yet, {{formula:db2e59b0-6c58-4e82-97f7-b51d8d6cad29}} returns the local process (extract:9).
In extract:1extract:6, several results are established regarding the behavior of {{formula:3623366f-53bd-42ba-bdaf-2ebc6bc3ef8d}} at a correct process in {{formula:8773d22f-adc6-46f7-9764-dfb36e7a9ea5}} .
Based on these results, correctness proves that omega properly emulates {{formula:3377ae81-b207-4c01-a949-b9fcacc96783}} .
In detail, consider the simulation forest {{formula:c4a30a4a-8385-4af4-a0a2-ed8f7216ae30}} at some correct process {{formula:85ef4752-fce8-454e-ab1e-6813692d427e}} .
extract:1 proves that eventually some index {{formula:3dd17cfe-4985-4f36-b82f-2f06a51fb020}} is critical.
If {{formula:83f5ff62-f4f1-4d3f-bfe9-659ef069422d}} is univalent, extract:2 shows that the process connecting the two configurations must be correct.
The case of a bivalent critical index is considered in extract:3extract:6.
In particular, extract:3 explains how a correct process in {{formula:e7fc145f-1e20-492a-a022-a157c92675c2}} is computed from a decision gadget.
Then, extract:4extract:6 prove that a decision gadget eventually shows up in {{formula:5df7d75d-ca62-463c-bf2e-a22be5baed88}} .
acircle = [draw,circle,minimum size=3.2em,inner sep=0em]
{{figure:437045a5-1f73-47b9-b5a1-48a9ee7bb801}}extract:1
Eventually some index {{formula:aee33a99-1279-4684-bd35-1eab051445e6}} is critical.
The proof is illustrated in critical.
By tagging:1, {{formula:a4de0bd0-5eca-4f79-905d-e19281f65ff4}} reaches its limit valency, say {{formula:404b8912-558e-4418-b4b6-a45423cc406e}} , after time {{formula:e8376a93-8b08-4dc6-bbc0-5ad37a5076e8}} at process {{formula:ea395994-b341-43af-8d55-252ee53d764b}} .
Hence, after time {{formula:43810c15-ea3b-4e2e-9e1c-ab4e9719696e}} , every root of a simulation tree has reached its limit valency.
Assume that {{formula:40c86ff2-96c5-4f9f-9591-de92688a0a68}} .
Consider the configurations {{formula:f998fd05-9c78-4f83-921c-e14a95168bc1}} such that in configuration {{formula:9e480a39-af7a-41e7-a619-69e1b814226f}} , every process {{formula:40d7dcb2-61e3-4629-9056-fe7d15170975}} multicasts a message {{formula:f48398a7-853c-4f6c-93a9-b6242cbde3be}} to {{formula:9134c02a-ac5b-43ea-8913-9dc0349ad2bd}} , if {{formula:7f49e884-8809-4946-aff4-c4f7a779ae59}} , and a message {{formula:5e7c75ed-731c-4e7e-810b-51af63acd900}} to {{formula:ec4a667f-47eb-4142-9bd2-f121212fe889}} otherwise.
Clearly, {{formula:a48c24e7-f638-43da-bf33-bdc81783cd8c}} .
For {{formula:d9c68215-e4cd-42ae-96b1-df8f7f5a3018}} , by construction {{formula:bab6438f-f1e9-465f-862c-1bd54148dad4}} .
Since all the processes in {{formula:e1569b2c-554b-46e9-b2cb-acf5b925dc7a}} multicast a message to {{formula:b8ec548a-2fc1-4cb0-9490-0079c384fd24}} , {{formula:5c3c01fa-2008-425b-90ff-dea6c81fa270}} must be {{formula:1c75ef2d-c24d-4a0f-b696-805ecf259311}} -valent.
Similarly, {{formula:a2ebde89-2d47-48fd-b5e7-7040d67e37ee}} is {{formula:d9b17fcd-405d-4eff-bdd9-196a754e5396}} -valent.
As a consequence, some configuration {{formula:64a145f3-5ee9-422a-ae06-eac00aad0667}} is either univalent critical, and we are done, or bivalent.
If for each group {{formula:da83f0c6-e5b9-4077-abee-ef2f9cb2f668}} , some correct process multicasts a message to {{formula:675fbc01-c5f6-4ac6-b52c-77fe9b92a1f9}} in {{formula:f8bd253c-da37-4fad-8390-abf477974db1}} , we are done.
Otherwise, all the processes in {{formula:14e1f948-224d-4934-ad26-2164838d4efe}} must multicast a message to say group {{formula:d0c46dab-de32-4a1b-8f0b-5ed9b491170a}} .
(Messages multicast by {{formula:0bc29bd5-682e-44e1-8e32-2c98a8e2ee4c}} are in green in critical.)
Name {{formula:1a95f39b-afd7-4585-93fa-e3dcc6361aed}} a configuration identical to {{formula:50bc8d34-4e98-49c1-9d86-42f72f56049a}} , except that some process {{formula:72e00d63-e53e-41f6-92ad-0aa12baa5173}} now multicasts a message to {{formula:d0c42faa-e04d-43a3-a555-b7aeca468824}} .
If {{formula:9f52e7fc-147f-422a-8d6c-98ac39940d2b}} is bivalent, we are done.
Otherwise, {{formula:8fd0415d-8216-453a-bf05-563aa84f35bc}} must be {{formula:9e6dca9a-0085-4ce0-ba7f-9ce01389a9ac}} -valent for some {{formula:f9e273e4-dfa2-47a2-a0db-1aa8405b24f7}} .
Consider {{formula:dd31d273-028a-43c1-98c3-ec360a717572}} identical to {{formula:54c3ddc4-2117-4ddf-938e-29f0de91be4f}} for all the processes in {{formula:b0ada02c-4d24-4ed8-810b-47cefb674271}} but where all the processes in {{formula:079c4aba-a424-44cb-8162-beb06cbb7078}} multicast a message to {{formula:03d995f7-9dd1-4c1b-a14e-9703d5593113}} .
If {{formula:4e6a7ebf-7b36-4649-879f-3d9b3dfb8c0f}} is bivalent, we are done.
Otherwise, it must be {{formula:6f0a44c5-0a60-4fc0-ba99-ac254054bc3e}} -valent:
Indeed, consider a schedule {{formula:058ca561-7d1e-49c9-9ad3-9c0a1af2cd4f}} where only the correct process take steps from {{formula:69bd4fa3-54ec-49eb-8b31-03649c55fd1a}} and during which {{formula:98f990c4-5a71-4bac-854e-e9b46c783dbc}} delivers a message. By simulation:3, such a schedule eventually appears at process {{formula:516e8e00-da2a-4b2e-9a9a-8179ca5f0214}} .
Configuration {{formula:65b10197-ffcb-4f31-b0b0-2c445ba54215}} is {{formula:c8b45f68-1481-405f-9a8a-bf25e8f88acb}} -valent.
As schedule {{formula:1ae4224b-ed2f-490d-a5be-3c785725ba44}} is also applicable to {{formula:79e992fa-aee0-4023-b88e-c1cd91d0f953}} , this configuration must be also {{formula:f0e47255-4a8a-4bcc-83a4-3b359a4298a3}} -valent.
We continue our traversal of {{formula:ec49b016-78ea-4222-95c6-5f73ff2543bc}} as follows:
For each process {{formula:a4715d06-8ee2-41cd-8678-23947b33e751}} in {{formula:4890ed4f-93ad-4380-8f20-ca7ec482a11e}} that multicasts a message to {{formula:73da1e96-d00a-4c62-9498-1075431dbc8a}} , let {{formula:173344ea-7b78-40cb-ba09-58286c4e8d9e}} the configuration adjacent to {{formula:0e98cec1-c894-4ee1-b6af-4325d4237ced}} in which {{formula:d8ec435c-2539-4f39-ac77-f13665d1cadc}} multicasts a message to {{formula:3886e6fe-fcbe-4e9f-bb1a-5e2d7a8954ce}} .
If {{formula:1725f3d7-241a-4b71-a802-5477025b35c6}} is bivalent we are done.
Otherwise, we set {{formula:2d37675f-a22c-4705-a7d4-8b6054bc3c4c}} to {{formula:5cbb5492-7e8e-4eb1-8117-35f10b044d08}} and pursue our traversal.
At the end of the traversal, either a bivalent configuration is found, or all the processes multicast a message to {{formula:c55ceeb4-93f8-4bf3-acbd-80f54d6bd9ba}} .
Thus, necessarily {{formula:0bdcbcbe-e63e-4810-993b-cac427ca35df}} is {{formula:70ba4463-f3ef-4f24-82d1-c698a5576805}} -valent and the configuration that precedes it in the traversal is {{formula:a975b661-d544-4b9a-8a32-7619fd9264ac}} -valent.
It follows that there exists a critical configuration {{formula:2986742a-03bd-4297-8ed8-2eeccd9a5b9b}} .
The two cases, {{formula:3fb744ba-a2a8-449f-b088-2b5654a7d484}} and {{formula:41958b10-9954-4aef-9527-5f0d4768c042}} , are illustrated in critical(right).
extract:2
Fix some configuration {{formula:d1cb7460-7745-4e39-bd64-6495e83e5295}} .
Assume that for some configuration {{formula:7137963e-febc-4f2e-ba28-2f5bf08ba526}} and process {{formula:80aaaaaa-1b6e-4f79-8dbb-887c86cdb3cc}} with {{formula:e61fe7b1-e4ea-4cbe-abed-329aa6167493}} , the root of {{formula:3f3ad674-a0d4-4684-9189-7762c34b42cc}} is stable and {{formula:cb0d5137-09e9-4569-bf29-0641a70b8d24}} -valent, while the root of {{formula:918e25db-0773-4894-8da3-72232dd0a1e5}} is stable and {{formula:8277bb28-f533-4abb-a2f0-9a6bce431cda}} -valent.
Then, process {{formula:b8592cf5-ccda-47e0-8470-fbdb4c66cb32}} is correct and in {{formula:2753a5fc-e1ee-4768-ad6d-545484d500dc}} .
By definition, the processes outside {{formula:1a44215c-0527-4327-9407-58c4702ce008}} do not multicast a message in the initial configurations {{formula:8f85d27b-faca-40e2-b3ae-d401bf340ad0}} .
It follows that {{formula:8af5561c-df3d-43aa-94d4-452c4d39831c}} belongs to {{formula:7442eeb1-9b1f-41cb-b411-91d4843df83a}} .
For the sake of contradiction, assume that {{formula:ff6ece1b-54c9-4cc6-ba40-0c97468cabec}} is faulty.
Applying simulation:3, there exists a schedule {{formula:449145e7-75a4-4be4-8744-1dbf4474b337}} starting from {{formula:b0edcb62-07a4-4b9c-8fb7-3675cdb9d36c}} such that
process {{formula:d288729d-9398-49bf-8e39-231bff8c2997}} takes no step in {{formula:d9a5b22c-2ab6-4005-afef-d53e3c368583}} ,
some message {{formula:cfe643a5-5d5b-4759-b09a-9f1b31153765}} multicast in {{formula:28db6576-d7f1-48f8-a133-e54b1717fc6f}} by a correct process (e.g., {{formula:7140c54a-0bb4-48a6-aea1-b6676dd355ef}} ) is delivered at {{formula:fdf74b02-5a02-49a4-b27e-2005dfe94f62}} in {{formula:fcd22d65-d552-4db4-89a8-af8826b3646f}} , and
{{formula:9b8fb54d-4343-4fb9-906f-9dcd893d8bb5}} is eventually in {{formula:ca10ad5c-9dbe-4690-a323-b81d19ce1784}} , say at some time {{formula:4dc037bb-37d2-44f7-ae7d-e835b7a7eaea}} .
Pick {{formula:3682f39d-a24f-4868-9308-7ded07df40d3}} such that {{formula:6ae46f81-4f83-44a0-b765-82c15593deaf}} is compatible with {{formula:400a8cef-4b8f-4020-80a1-9825ce981327}} (simulation:0).
Consider the first time after {{formula:76a394a2-7fcd-4f32-a2e6-e1115e585ed2}} at which process {{formula:4a6e30bd-b97b-43c9-b2ee-30f8e5c820c8}} executes omega:3.
Configuration {{formula:a958bbca-efcc-46b5-8166-8873eb3f6a85}} is identical to {{formula:e7e9d006-d099-4791-99b0-3ca99582df15}} for {{formula:d57a9cb6-e27b-49df-a706-8c442f6d1b13}} , process {{formula:5d5c58ad-e374-4a7b-b47b-8970dde88b2a}} takes no step in {{formula:9e0848d4-b879-4cd0-ab97-bb35d2bdd073}} and {{formula:c0f341d7-f36d-4d9f-8209-df1247c1f843}} is empty.
Hence, by a short induction, {{formula:946310b6-6dfb-4d48-acd8-4e59ade357d1}} is also applicable to {{formula:870852b0-1bd3-4aaa-82ff-3ed682522edf}} and {{formula:16b69618-060f-4776-a342-83f7aa9298bc}} .
Furthermore, by simulation:1, {{formula:66363058-757f-4c67-897b-5454c5bf1905}} .
Index {{formula:31c6e74d-106a-446a-a5f9-3facb2079d17}} is univalent critical, thus {{formula:904d3518-dd1c-45c4-8d47-9fbc1b18122b}} .
Now, since {{formula:43de355e-09ce-48c8-ab48-ce9abd0369ea}} , {{formula:aabf29f1-3f38-4c30-ba84-1bf1d0787168}} and {{formula:2a0926d2-9b32-465d-8244-6aae3ca950ef}} , {{formula:cfa3621e-afd3-4e2a-a18e-daab8548c453}} ;
contradiction.
Let us now turn our attention to the case where the root of {{formula:152ec066-9573-45f8-ba97-fc7f99163ab6}} is bivalent.
Similarly to {{cite:ea9cd8cbad59f4421feb8537fbe7d920746b4b75}}, we show the existence of a decision gadget in the simulation tree.
This decision gadget entails a correct process.
Furthermore, if index {{formula:30b5a0e1-5fef-4929-a759-a6c6f8651f5f}} is critical, this process must be in {{formula:8778f2c9-2dce-44c4-80af-42bfedfe117f}} .
edge = [draw, -latex',Black,->]
dedge = [draw, -latex',Black,dashed,-]
vertex = [inner sep=.1pt, circle, fill]
{{figure:85a90409-d986-419d-b1c4-fba3c38ecc9a}}Consider a schedule {{formula:26c70396-2637-479a-843e-1a199d0fb9a4}} in {{formula:27486079-1a48-4d29-849e-add4fa9b7439}} having two successors {{formula:ce980d9b-e185-4926-b412-149a670306f1}} and {{formula:983e493d-e9dd-40ee-8519-a7810058a5d5}} .
Let {{formula:06d7e101-5a1e-4c79-acfb-884291554a38}} be the subtree formed by the two paths {{formula:89d3f14e-656d-4919-8363-3efed9e83b1c}} and {{formula:acc92541-a5c5-4cb7-92d6-b09d6eda2af2}} .
This subtree is a decision gadget when it is either a fork or a hook (see gadget).
A fork contains three vertices with {{formula:918fb326-d718-495e-91d3-a3418cae6c3a}} and {{formula:5d89020f-5234-4645-8ff9-24952f641c6a}} .
A hook contains additionally a successor of {{formula:ca654d4d-36e1-4535-bdaf-fdcc7dd4d7d7}} with {{formula:0bce17a3-990f-4928-848b-cbd48b584d36}} and {{formula:6f26d10a-fc30-4f41-84df-7e56f33bc03c}} .
In both cases, {{formula:f93e7732-cd6e-49d2-8c37-c4e6181bc368}} is bivalent, while {{formula:d5add9a8-4c05-4590-b964-6e813baaa5d4}} and {{formula:62a6b0f0-1e2e-4b91-b66a-2cf5b91017d7}} are respectively {{formula:1ca8bd46-4f38-4016-b75d-9fe1e1174db8}} -valent and {{formula:40752322-e25d-4bc1-999e-b1dbb5d915bc}} valent, for some valency {{formula:8b9c3957-7ac7-4e7a-9967-33290e69bcb4}} .
A decision gadget induces a deciding process.
This process is defined as {{formula:e816c65a-b20b-499f-816b-c620b9e13aa8}} for a fork, and the process {{formula:e7785842-d5cf-4cf3-84ea-65e501841a1b}} for a hook.
Intuitively, this process takes a step that fixes the valency of the run.
With more details, a subtree {{formula:44d5968a-e2e0-435e-bf35-fabda372f486}} of {{formula:58355928-50f5-4701-abb8-5fb30e324bbc}} is deciding when for some successor {{formula:de4fcdd7-3901-4827-8f24-89ccae1d43f3}} of {{formula:bc5dc62a-f09b-48fa-a333-19e5dbb8faa0}} , some message {{formula:38beb28e-cfda-4ab2-a5f4-46f9f7829dd2}} addressed to a process {{formula:7ba4e529-d0d1-4516-aa17-dcb5bb5c0efc}} , and some values {{formula:49172b78-068f-44ec-b533-09c9c2497f1c}} and {{formula:e4e37626-113d-4db9-9dbf-7ad758c51049}} of the failure detector {{formula:ba08a220-6707-4ece-8f4e-ef7e08d1bbf0}} ,
{{formula:1a280b01-edad-4619-b9fd-3e82858819f1}} is bivalent,
{{formula:a8b645e6-97f3-4045-a31e-b6bd48f50599}} is {{formula:dcc086a1-c3c3-491b-b447-5e809ef82b3c}} -valent,
{{formula:215da90d-0d37-4ca4-996f-5be4f6262532}} is {{formula:14edabde-b43c-4630-9ccb-80c8d8e8e5ca}} -valent, and
for every schedule {{formula:9ec25c93-5bb2-4d62-999b-6f43fa1b2552}} in {{formula:8a85d413-08c5-4b21-9cd3-1440d5747f33}} , {{formula:2e315548-7b7f-4f63-a5f0-ea4e86391fcd}} is univalent.
As illustrated in gadget, when {{formula:cff65ad6-4706-4516-94c9-4a1110e9610a}} is deciding it must contain at least one decision gadget.
Indeed, if {{formula:07877b91-8fec-4771-bd24-464c35ea69e3}} is {{formula:f31734ba-d5b4-47b1-b8c6-702534418f64}} -valent, then {{formula:346be80b-a647-4ed4-b68d-85af7fa0a090}} and {{formula:c0da9f1b-bf4f-4e66-b4af-1e955435a0df}} forms a fork.
Otherwise, there exist two neighboring vertices {{formula:5b01e65e-2316-4626-aa83-deb5c4e850b1}} and {{formula:8b98dec3-fa7d-4e70-893a-ccf31cf8c952}} in {{formula:953d0f1c-2c72-44c1-9ed2-76df37884def}} such that {{formula:6072dfc2-a528-4159-b90d-2327807e77fc}} is {{formula:aef61f41-072c-42ae-8512-f3ebe372c2de}} -valent and {{formula:acbe39f3-4255-43b3-94bc-d9571926ee24}} is {{formula:32b1544d-df6e-4a52-828d-2ce932429f7f}} -valent, leading to the fact that {{formula:76a03b97-67c3-4fc6-a6a9-8b3dcb62a158}} is a hook.
During the traversal of {{formula:9b8ced20-df9f-4286-aee0-4b630716ff1f}} , when the {{formula:e1dbd4b1-e728-4155-88c3-644aa5f992c6}} procedure encounters a deciding subtree, function {{formula:9eebbfd4-74a2-424e-a4ca-9ba42bec5d93}} is executed.
In detail, following {{cite:ea9cd8cbad59f4421feb8537fbe7d920746b4b75}}, since a decision gadget is a finite graph, it can be encoded as a natural.
This allows to totally order them.
Calling {{formula:5b856454-35e1-455a-bc01-dfa191c6f79d}} returns the deciding process of the first decision gadget for such an order in the subtree {{formula:d56d6e90-1aa5-497d-a264-ca813104afad}} .
extract:3extract:6 establishes two fundamental properties related to decision gadgets.
First, that the deciding process of a decision gadget must be correct and a member of {{formula:5ad8e33e-c0dd-48ea-b169-13fdb4f5d6ee}} .
Second, that a decision gadget eventually shows up in every bivalent tree.
extract:3
Consider a stable deciding subtree in {{formula:7676a279-7587-4c32-86db-5dfbd7532b30}} and let {{formula:b64d5945-af21-40a4-9542-18721f6ed0d5}} be its deciding process.
Process {{formula:f118bcdf-6563-46d5-bc74-3eeceb519696}} is correct.
Moreover, if {{formula:c8b2ee2f-ceea-46b1-a580-f991c9e129aa}} is critical, {{formula:b2bd8586-c4b1-4b36-bb75-3f1452406d4c}} belongs to {{formula:0b0caad6-08b0-47a0-be50-71a0e81ec6de}} .
Assume a point in time {{formula:bfef13c8-60ff-4084-9c73-53ee604773fc}} where {{formula:54d285b6-c529-4e95-8dfc-85c56f7d2da1}} contains a stable deciding subtree.
Let {{formula:25998112-3fcc-4be6-8aef-a173bfef3dcc}} be the first deciding gadget in the subtree.
({{formula:f66dcf96-f9c9-4806-9002-e037372154c6}} )
We proceed by contradiction, assuming that process {{formula:89f0ef3d-cab8-4351-9983-9607504b89d9}} is faulty.
Name {{formula:1745096c-bad0-4ccc-9ded-4b484d3a6195}} a message multicast by {{formula:19988356-9ef7-4c36-a203-a1a3e04acd50}} in {{formula:a8d1690f-5d77-46e0-8b75-4c3d72b419de}} .
Applying simulation:3, there exists a schedule {{formula:39986493-1869-4be8-8581-7c19af740f6e}} such that
{{formula:b5b9bacd-1c84-41be-a5b5-a78fac7753a3}} is eventually in {{formula:2f9faca5-9afa-4a02-bd8e-df521e799ebb}} ,
{{formula:4a9c4b69-3379-48a1-9445-e090a34d5d42}} takes no steps in {{formula:3907a10a-ac23-4a84-94d0-096378417d83}} , and
{{formula:9ae21125-ca4c-4881-b6d6-4b1d41d96ba3}} delivers {{formula:d64e3bb4-7ed6-496f-ad9f-11d3015fcf48}} in {{formula:1ddb130e-a7e4-458a-b21d-ea88f5112f2d}} ,
for every schedule {{formula:1a3ef1b6-441e-4030-bce9-7a35ca1988e9}} , if {{formula:d08a97ed-1342-4bca-b000-2a24d416390f}} is applicable to {{formula:5fba66d8-d4ab-4f05-8db3-814e27b4f656}} then eventually {{formula:8010fd45-e732-4254-b16c-deeb64de94b2}} is in {{formula:5e236d29-93c8-40f5-9955-07be1dc44471}} .
Consider a point in time {{formula:196667f7-693f-47d4-a1b9-f9585cfc6bcf}} where {{formula:a3953c9a-301d-4309-b9af-393d721cdf46}} is in {{formula:1de048f5-f2d8-4b60-b5d1-de6615485d0b}} (by (i)).
Schedule {{formula:5bba2434-4302-4d25-bc35-a6a9f436f6f9}} is tagged at time {{formula:122018cd-8365-4708-84c1-e7b7c77f2e31}} (by (iii)).
Because {{formula:d41e3dfa-c0af-4afe-b198-12f9db9820a4}} is {{formula:29bda512-6fa9-47f0-b284-a9f89432cd4e}} -valent and the deciding gadget stable at time {{formula:f935a9ae-4442-4b2c-a6de-da95bd5bb96c}} , {{formula:f1388983-1150-4bb5-b46f-155246b26006}} must be also {{formula:7ed956e1-5bf6-4595-8292-ed7ef5801776}} -valent.
Process {{formula:c2faf4df-7810-4666-84a5-12b89d2712bb}} takes no steps in {{formula:0078504b-c727-427c-b592-403db5d0e1a9}} (by (ii)).
Every process {{formula:cba4b664-3310-4b65-9f4d-067f6291ba14}} outside of {{formula:9eb77d34-7a0c-4e0e-bc10-11e6560abe50}} is in the same state in both {{formula:06cec91d-6c6e-4f59-be7b-6d2fb2ba2a98}} and {{formula:6f8759fa-5a2f-4cc8-88d6-0fa3e5f09820}} , {{formula:def8f4c8-c102-4e7b-83ce-83ebcd3765a9}} (see gadget).
Hence, {{formula:498439e8-98a9-476b-8778-578b975897c0}} is also applicable to {{formula:6e90f9ad-7233-42d8-b98d-f1df457894bb}} and {{formula:70716b82-cf1b-4b89-b4f7-42e70ec96fba}} .
At some point in time {{formula:afc27f1a-8241-4d6f-af70-0ccd36b71566}} , {{formula:a3fccaed-5fa2-4698-9cc6-ad5e9d8b03c3}} is in {{formula:4cf0ebb9-ef8d-4e2d-8651-2d318352b02d}} with a tag {{formula:9e8d762e-44ba-4b1f-a6f9-8f127a2e5674}} (by (iv))
This contradicts that the deciding subtree is stable and {{formula:6f80c4e4-6dc9-4cba-8107-c9b9fe6ccdde}} {{formula:5d34c3bd-1cc3-4600-94f9-6878c3a8c7b3}} -valent at time {{formula:0bf6c185-b2a5-4acc-a930-1fe5d843a552}} .
({{formula:1dde5813-54a2-4f69-b9ed-a599e403b10a}} )
The proof is similar to the previous case.
Instead of message {{formula:69348c6a-e42f-4ce6-ae51-75611597bc8c}} , we consider this time the message {{formula:5893d744-3d11-425e-8c8d-117a7e5d244a}} addressed to {{formula:01e4f8d9-3a45-47fc-8570-b80dabb1dddb}} and multicast by a correct process.
Because {{formula:2fd45cd4-1240-4b08-bff6-9447cf908acb}} is bivalent critical such a message {{formula:a4867cb0-ae00-4c65-9739-2f16f66ac89c}} exists.
Procedure {{formula:99718998-3c9d-4bcc-9b18-41dcb2c6b930}} calls {{formula:22fd299a-bbf6-4058-b234-d98e0573e2a7}} , for each subset {{formula:24e793fa-e965-466a-ad2c-43485c75814c}} of {{formula:7407af69-bc2d-4637-8591-d357de046f4a}} (extract:5).
Starting from {{formula:ec969fc7-66b8-4359-ac95-4c32dd256db9}} , function {{formula:b51dd9af-d13d-416e-890e-c16ce6b78eda}} considers the processes of {{formula:a3e37f3d-632e-4818-af35-26d5d243b58f}} in a round-robin manner (locate:2locate:3).
For each process {{formula:400ad457-0061-46da-b1f5-77367ce16b38}} , {{formula:e64c4d8e-1162-4c78-be87-fecc52e4542c}} tries to extend the current schedule {{formula:aeeda81f-54c2-427f-b686-7ce4327c9ba1}} with a step {{formula:20a6df97-681c-4ff6-be2a-7307ead4c828}} , where {{formula:21236761-1c6e-41b2-aed8-d33af970e735}} is the last message addressed to {{formula:d5a72791-8dc6-4517-82f0-a3475d339b5d}} .
If this step leads to a bivalent schedule, the search continues (locate:8).
Otherwise, {{formula:594e983f-0b97-4933-b5e5-d4f02f0a2dae}} determines if it may lead to a decision gadget (locate:9locate:16), and returns its deciding process.
If none of the above cases applies, e.g., the successors of {{formula:ae2916e3-3a36-498c-abc9-dca395ab1e2d}} have no valency yet, the traversal aborts and returns {{formula:d2606c83-020d-440e-b88e-d2ff5c226e75}} (locate:17).
Hence, the search for a leader in {{formula:ac86267f-1d7d-400f-b6ab-78961ec76204}} eventually terminates.
Function {{formula:6d31381d-38e8-417e-8e2f-195461d3b258}} traverses the tree {{formula:f2a45cfb-dad9-4d65-b47c-862d0c286027}} following a depth-first search of the bivalent and univalent vertices.
The search occurs in an order {{formula:7b1d4a06-93e7-4777-8057-466f6ef2b3f2}} over the vertices of {{formula:beb9aead-66d0-4bed-a5ae-571d2a42e8d3}} (locate:4locate:10), picking the smaller schedules first.
To emulate {{formula:9e12ee36-a4e7-4277-a03f-16d0b71424d9}} , this order should satisfy that the depth-first method eventually stabilizes.
This requires that if {{formula:487ea6e9-d720-412e-b599-bcb57933a0bd}} is chosen at some point in time to extend {{formula:525b5ebb-9caa-4154-9835-9e45c97bb379}} , the set of schedules smaller than {{formula:586119b0-760d-4e2f-9fe4-402aa659b2e3}} is bounded.
This property is easily obtained from an ordering of the paths the schedules are compatible with (by simulation:0).
In detail, for two samples {{formula:7537c11f-8ca9-4070-b121-d64e045f86f2}} and {{formula:2cad6bbd-d212-4585-b81c-4834e24ac243}} , let {{formula:b9744547-698b-4c4f-aa43-092576c75972}} be defined as {{formula:25f88a2c-8f45-4c4c-bd81-bdca76dc25c6}} with the usual semantic.
For two paths {{formula:cfbee576-068e-445c-9e87-7f60c32fea3c}} and {{formula:b70dbd8a-5aef-4b5e-b8f0-1f090baf92ad}} , {{formula:372cdad3-3e77-4bcf-9bf2-bff6f0a45770}} holds when for some {{formula:ea644f34-1f83-4972-8874-0a2dd15f4c6b}} and every {{formula:a6889732-7226-4e23-b62f-314ca5ff1255}} , {{formula:1bc4bbbf-9f75-4843-96ae-d190790a9d37}} is true.
Two schedules {{formula:09232992-dd98-49bc-92bc-a4dfc8c5b6c5}} and {{formula:75c86581-db7a-4c35-a8db-7a2621e71393}} in {{formula:20b47796-7ccb-4a40-b8c1-d37f2e4203f6}} are ordered according to the smallest paths in {{formula:0d96d73e-d4c8-4913-bec8-644634cba749}} they are compatible with.
Consider some time {{formula:69316cfc-9688-45e1-b9a3-ebbf2d384c6c}} after which the root of {{formula:8d427d0e-9624-48b6-874d-18bd240b72c7}} is bivalent.
Let {{formula:a23b44d2-9ad5-4f3c-be0c-af1964c8116a}} be the sequence of values taken after time {{formula:f4fe9d13-4eb2-4bf8-983b-061d1c5959a6}} by variable {{formula:9d05974d-5892-475f-96d6-f9e3d63a2bd7}} at {{formula:da01295a-8cd5-46c7-8f2e-57b3533da0a8}} right before the call to {{formula:18912a32-3019-4b22-be6f-fdc523c43141}} returns (extract:6).
extract:4
It is true that
each {{formula:cad5aa77-e4e6-48d3-9a41-57a1a6893de1}} is bivalent, and
{{formula:ad82b175-93b5-46e1-ad7a-71b8ebaffbd8}} converges.
We prove successively each part of the proposition.
(Each {{formula:eabdef7c-5e8d-4ed2-bcfb-15bed5cfc9cf}} is bivalent.)
Initially, {{formula:6547d1d5-5985-4381-9b27-d51b290d64ed}} (locate:1).
Since {{formula:eab02e5b-e06b-45cf-b034-c102871cb7dc}} is called after time {{formula:cd81fa1f-aac1-41a5-86b3-3ba1db028bb0}} , {{formula:aa868d52-ea73-4638-b3dc-bb031d5975eb}} is bivalent.
Then, observe that {{formula:387c3af8-2ed1-44b2-a660-a0a1c9d1425d}} is set either at locate:7 or locate:13.
In both cases, after the assignment, {{formula:c2181262-b868-4c16-8a0c-932d91ff7e16}} is still bivalent.
({{formula:8fdf537d-7cbb-439d-aca8-215ae8723a3c}} is convergent.)
({{formula:f570aa78-510d-4106-8ee4-b9e6dcd16b98}} .)
For some faulty process {{formula:6290c487-68fe-4a64-907d-823ff5687c1c}} , there is eventually no step {{formula:cf83564b-5833-4af1-a95c-862f67166209}} to extend variable {{formula:05fc5164-7c68-4cef-b953-4f188ce385d7}} at locate:4.
Hence {{formula:8f7d8ec3-1be7-4f1c-bf4d-6b3cb509181f}} eventually always takes the same value.
(Otherwise.)
For the sake of contradiction, consider that {{formula:8415201e-edcc-4de3-a7bd-61be6c918d93}} does not converge.
From {{formula:bd2d5f53-7852-41a9-a1d8-f4aa86003f22}} , we build a sub-sequence {{formula:bafd7e9d-e117-4115-ae77-e866c1d1ca2b}} as follows:
convergent
Schedule {{formula:75ac62b3-b90b-4514-b512-f119822429a5}} is set to {{formula:89088f5a-ab9a-4a8b-a657-3ac6441f2ee6}} .
Assuming {{formula:b4e76b6e-bf83-477a-b18b-f32f92470556}} , {{formula:9d9226d8-fea9-4b1d-a1cb-a6036310560c}} is built as follows.
Variable {{formula:7b2a5317-48a5-495e-b3c2-d06c17c825e4}} is changed either at locate:1, locate:7, or locate:13.
As {{formula:31754f90-102c-4aa7-aa5a-e593a60b9b02}} prefixes every {{formula:4600016f-89a9-4578-a74f-d59477fff609}} , eventually for every new call to {{formula:3ab0c28c-0714-4b41-85fc-35eab0b85e0e}} , variable {{formula:deaf174e-3ac9-436c-9200-ccf2454c54e7}} is set to {{formula:728b8f08-a534-4a1e-b603-0f478c76b8eb}} .
Let {{formula:b6ea64f7-eabc-4366-88dc-7b7ce48f6019}} be the process considered next at locate:3 after that assignment.
Since {{formula:880ac98c-130b-46cf-ac30-f3a7abb80d25}} does not converge, eventually for some step {{formula:94f586c8-8fc1-46b1-8bc9-a2a7c4521f76}} , {{formula:15a4a71c-492d-4bfe-81be-bb42a6642557}} is in {{formula:c509ab89-c760-403b-9433-7a394d8a6010}} .
By definition of {{formula:06f01931-d8fd-4184-9a75-6d0fc5d135b0}} , the loop at locate:4 eventually stabilizes for some failure detector sample {{formula:a39287d3-74e1-4f2d-87e2-75c277d97d86}} .
{{formula:a145d9dc-8c13-4d92-bb23-361ef4c88b05}} is set to the schedule {{formula:c4ebe4fa-e584-4f94-8279-ce2d602d8897}} .
The series {{formula:f74a141d-3077-41ac-9c6d-1e72be94d845}} satisfies that
{{formula:8d38224f-5381-43f2-b468-6f922ea3ffef}} ,
for all {{formula:0745d263-243a-4a24-968c-6fd0f4e54aba}} , {{formula:240436bd-9228-4f48-ae8c-11e6fb186c70}} with {{formula:fb4df0cb-373e-4ebd-9a76-23bae5f590d4}} ,
each {{formula:c8bfd2be-1127-42c1-9b31-e1bf569bb770}} strictly prefixes {{formula:d40d66c6-016e-47f3-a9bd-cf5d736d6236}} , and
for all {{formula:85639682-8f6f-4995-aea6-9ad5341335ed}} , there exists a rank {{formula:b2420547-ddb7-481c-8a97-1b8f35aba552}} such that {{formula:efecc8b4-65cf-4ab5-83b2-91a2510b8be5}} prefixes every {{formula:3495eec7-10f5-4e2e-9773-8a9432b10019}} .
Properties (i)-(iii) follow from convergent.
Property (iv) is obtained by induction.
In detail, this is immediate if {{formula:33da39e4-d9e4-44dd-94a1-7d6458d6e299}} .
Next, assume (iii) holds at rank {{formula:8c44b180-71ea-4cf8-b3c7-b5b63f891bfc}} .
Step {{formula:fd9feffb-8adb-400f-b6af-2777a62ae05d}} is such that {{formula:978a9485-96ee-49d3-bf08-c06bd3e6a061}} is eventually always the smallest schedule at locate:4.
Hence, for some rank {{formula:8749621b-f33c-4e03-9522-ede3c99b2128}} , for every {{formula:36b413c0-5f9d-48ca-afb3-3d09c92f8dbb}} , {{formula:94e69d85-8e45-4915-9530-c603429fcc2e}} .
Let {{formula:ea217e53-fd68-4279-986e-9687dafd05fe}} be paths in {{formula:851be36b-34f4-4d01-961f-938f7612d8a4}} with which the schedules in {{formula:cbadd33d-2daf-4e21-a466-140c3aeaac4c}} are compatible.
It is easy to see that {{formula:7e7154be-22ec-4117-a558-e91e591c3fe7}} is a {{formula:69b491bc-7703-4cb1-8fb9-ee9e771bbf46}} -fair replicated sampling sequence.
Moreover, according to the pseudo-code of {{formula:49853dc7-2b13-4f08-bcc2-6dd594209505}} , {{formula:9e6d5452-85ce-48c1-92c4-a43607bc3031}} .
Applying simulation:2, {{formula:0091ea65-62f3-44c6-8390-ceffa310fd65}} converges toward a {{formula:660d0826-cd7d-4bac-8f54-1ede24f60501}} -fair run {{formula:0d7ccf0e-9a98-4bc6-a1a4-592b04b22ba6}} of {{formula:e087225a-cdc6-401d-ba0d-bc111891469a}} .
Since {{formula:dd7dda5e-9a96-4f8f-a09d-e40612ab2439}} , {{formula:318ced37-7f08-4253-915f-519838f9f6dd}} and {{formula:02ee3078-f530-47e4-bfcd-ab9473955ce7}} multicasts a message in this run, {{formula:462685f9-d73b-4453-9510-3213ebe93413}} eventually delivers it in {{formula:4e1f0f73-112d-4ad4-bced-79159e8c3948}} .
Hence, {{formula:eeeac284-2f9e-4667-a189-a9258d10223e}} is univalent from some rank {{formula:7b673fe6-3b02-40d5-afca-3c6ed203c11d}} ;
a contradiction.
extract:5
Eventually (Q,i) always returns {{formula:e35bcc32-b206-45c0-9f1a-bf14849ab424}} , or for some stable and deciding subtree {{formula:26f5b761-2159-4577-bed1-98dfa5f36725}} of {{formula:34d5cb93-9f6b-4854-9e96-20a6b686f0fa}} , it always returns {{formula:359fcc5f-b43f-409c-9293-a91796c2bd54}} .
Applying extract:4, {{formula:7be03cd1-edad-4585-891a-018e23ef42fb}} converges toward some schedule {{formula:e3502d75-8ac7-4737-b0dc-548d578da73c}} .
Consider this happens at some time {{formula:c3f22ed1-dc2d-4ff3-836e-fbb1b0e754e8}}
According to the pseudo-code of {{formula:3e979d46-5707-4bd3-b4bc-66ab465eda6a}} , after assigning {{formula:2e1621f1-5c05-46d6-b20f-67a1138a6184}} to variable {{formula:73fa3372-0f93-4eba-9116-35e4c27bd5c5}} , the function returns.
If {{formula:e38721d7-c654-498a-a3ae-51cd91e1e30b}} eventually always returns at locate:17, we are done.
Otherwise, it returns at locate:16 infinitely often the value of {{formula:8ef1334f-b78f-4ad2-8e76-39e0e6813a95}}
Let {{formula:3f458f4e-011a-4eb0-8cb7-1701eb134674}} be a time after which this holds.
Consider that {{formula:f5a7f1e6-20e7-4b55-a303-8f76cc6d7633}} returns at locate:16.
The variables in function {{formula:7e307c5f-56d5-4f08-949b-5c977abea2c7}} are such that
{{formula:81f6284f-cc16-4f0d-aae4-0342d1beab5f}} is a successor of {{formula:8843c2b3-9416-48fe-8996-0440abaf2868}} ,
{{formula:bd1446e6-0e5b-4b51-9975-3b9d0f98a17c}} is {{formula:d168106e-34a7-42da-8d9b-f9152774fda2}} -valent,
{{formula:4dc68bcf-d481-4681-a150-00a9d0c06331}} is {{formula:5cdc78f9-882a-47b4-a970-64d797bf1df0}} -valent, and
every schedule {{formula:7e0a8a0a-538c-4d64-a779-ad3912b5c783}} along the path from {{formula:58aeb6c1-ffa1-4fcb-9fb8-85847757a75a}} to {{formula:9717efe4-4837-473b-b5ed-63cc3a9c8cd5}} satisfies that {{formula:689c1ebe-513e-4faf-9e3b-e6ea7e135d4f}} is univalent.
Hence {{formula:bc9a0729-eeb1-4444-9825-cf9bda4f000a}} is a stable and deciding subtree of {{formula:ebfabad5-1cac-4ba3-85c5-7e47a5101cb9}} .
From the definition of {{formula:6d9e81d0-6319-45c1-86fa-1bc5529dd1c3}} , there is a finite number of successors {{formula:13c356c9-8b8d-4c83-b9f6-7cfe7b92af25}} of {{formula:c9bce916-48c0-47c1-963e-0d89f8caa12f}} such that for some {{formula:2724f856-437b-4156-8d1c-bc237835db8c}} , {{formula:7b976793-5687-4d19-9dde-045fa56db82f}} .
Hence, after some time the loop at locate:10 stabilizes.
Let this happens at time {{formula:9e2e199d-98d0-4584-9db6-ecf29764eccc}} .
Based on what precedes, there exists a stable and deciding subtree {{formula:6cdb1ba4-1b74-494c-93c0-85b4806585a8}} in {{formula:40314f0c-2a07-4892-ae2d-ebc536948d7b}} such that {{formula:93cdab08-1ff7-48ab-949c-19d73402831d}} always returns {{formula:0e7c4885-5d1e-45d3-8f3e-8408c8c13116}} from time {{formula:e68938e8-75ee-4dce-837d-6432da9573ab}} .
extract:6
For every {{formula:3ecdbef2-a376-40a3-b60d-8a3dcc60c88a}} , {{formula:a696bc25-3a83-4850-9793-75bdfb742cb4}} cannot return {{formula:50a341b6-e77d-46d7-9ffe-5de2428de0b5}} forever.
Assume {{formula:2cf7ba65-b17c-4aa6-8df0-000ee92cdd5e}} returns {{formula:ee4e221f-cd0d-4139-b8a0-4aab1e4c9357}} at locate:17.
Let {{formula:6c6d17f8-c5cd-4526-9e99-583b3dacb3d8}} be the next process to schedule in {{formula:32b11c77-34f0-427c-b6eb-52757763fce8}} and {{formula:1ec8d6fe-f498-4df5-8fa7-24237bd3327e}} the oldest message (if none, then {{formula:26ec6624-e50a-4788-870e-5909c792d7e3}} ) addressed to {{formula:f1bb8f00-191a-4b85-b842-5780ce3474fa}} in {{formula:bb7b6115-9cb2-4b6b-8c3f-6442c531c60b}} (locate:3).
As {{formula:966bac25-3ae4-4caf-8812-e95a9fbed525}} returns {{formula:e0a57b07-48a6-41ff-a9fa-7a555880e548}} , either
(1)
there is no sample {{formula:6d8e2be9-0dc1-4f17-98df-66581abd7436}} , such that {{formula:2c7a22f8-a8c5-492c-8781-c17ecf702eb6}} is in {{formula:7387d8a7-5eba-4dd0-b300-774799301529}} and tagged (locate:4), or
(2)
{{formula:44b467cd-2309-40c4-8946-21bb31083de1}} is {{formula:d6615cae-1f76-4551-a27d-a61192ced2c6}} -valent, and for every successor {{formula:f334841f-6e96-49fc-aefa-440f35b232ad}} of {{formula:a629f7d3-dcba-42e1-9240-e649ae3b74d6}} , either
(2.a)
no schedule {{formula:267f4a17-d77f-4d83-8cf4-71d88a5a1ac2}} is {{formula:f76ceba8-f623-4438-a076-41eb203ea734}} -valent (locate:10), or
(2.b)
some schedule {{formula:803e1e32-be42-435b-8276-9f9d675f73ff}} is {{formula:45b5db92-f5c7-4945-ab13-5d9ee3a4b357}} -valent, but for some {{formula:d4b50621-f739-4885-872a-8d4bcacdfd28}} , {{formula:ba9a175b-b119-45d0-baf6-7320c0770892}} has no tag yet (locate:15).
Applying sampling:3, process {{formula:fe05e3d9-18d0-4431-9ca5-3ec651a7b95b}} eventually takes some step {{formula:437c8e39-078a-421b-8643-390728e10209}} from {{formula:90194a10-ff19-4307-8a72-f2d2b1035ad6}} .
Moreover by tagging:1, {{formula:7b3a25a9-ffce-47b8-9714-06dca384c97e}} is eventually tagged.
Hence, case (1) cannot happen infinitely often.
Consider that {{formula:0eed8d38-abe3-40fc-ba91-cc045addaaea}} is {{formula:4f3f690c-9842-4623-be54-f571ff500a5d}} -valent.
Since {{formula:9014d421-e1de-4e1c-be11-970bc4c13197}} is bivalent, then some successor {{formula:f3da5342-5de0-4232-99ac-ae8b57f355dd}} of {{formula:70b119ab-9e64-4eba-b2ec-e728486eb533}} is tagged with {{formula:6b5e0125-c29c-428e-a06b-a311d5628891}} .
Hence, case (2.a) cannot occurs infinitely often.
Then, applying again tagging:1, for every {{formula:862fbf3f-744e-4dd2-af20-14e7836a9f89}} , {{formula:d84a81a8-fc4b-4e8d-b79e-14fe17161026}} is eventually tagged.
From what precedes, (2.b) cannot happen infinitely often.
By extract:5, there exists a limit for {{formula:d1919cc1-ebc2-4992-ba54-ebb5ce101e0e}} .
It follows that locate:16 occurs infinitely often.
A procedure is stable when from some point in time, it always returns at the same line with the same values for its variables.
The result below establishes that this happens eventually to {{formula:fc84292e-92fd-4bba-9bee-3a4eda005ff4}} .
extract:7
Procedure is eventually stable.
By tagging:2, the valency of the root of {{formula:cdae11b4-1ec5-4434-b2ca-3cde5e208720}} is eventually stable.
Applying extract:5, for every {{formula:0b4e3110-f085-4e52-bc01-11cec2956035}} , {{formula:ddc1edf1-0fd6-486d-b203-39569f8552ed}} is also eventually stable.
Thus, procedure {{formula:d5757f90-ae17-4ca6-8c41-dc3acbf0cc0a}} eventually stabilizes.
extract:8
Eventually always returns the same correct process in {{formula:54a3623f-3b83-4f4a-a9e0-d5ef47539339}} picked at extract:3 or extract:8.
Let {{formula:2f1bfa0c-c4c6-4102-a103-ad8946d37942}} be the first time at which {{formula:91695eed-a996-4b46-91cb-172627ce49f8}} is stable (by extract:7).
Applying extract:1, there exists eventually a critical index {{formula:364df2c4-da1c-4c6b-bbef-489fdb44c0d7}} at process {{formula:79530871-47c6-4ef8-a94a-c5206c227bbd}} .
We note {{formula:5dbe9604-e09a-47af-ba1d-3261c13f89e0}} the time at which this happens first.
Then, consider some time {{formula:371d90a4-71d3-48b1-af52-21a24a431958}} .
As {{formula:11e542e8-9926-4882-9ac6-81e8aafcc6a9}} is critical,
(Case {{formula:7d697290-4159-4757-acf0-7d0535b641bc}} is univalent)
By definition, index {{formula:1ade72b3-6af8-4c47-b769-775933aa24e3}} must pass the test at extract:2.
(Case {{formula:8acfa327-b140-4cef-886f-d01b49d33b65}} is bivalent)
Applying extract:6, {{formula:c260f140-cd45-4bf4-8010-2c3b512a754d}} eventually returns some process {{formula:45d563cb-2e3d-4fdd-9852-a571aec960d3}} at extract:6.
By extract:5, there exists a stable and deciding subtree {{formula:fc5e76d0-d3c5-447c-b9ad-e370fb5c0966}} of {{formula:732a0ac7-b759-4db6-8f9c-333a9d1d217f}} , such that {{formula:f12e7f57-58fa-46a8-befe-9dc91ac81d05}} .
By extract:3, process {{formula:cd526ee6-b886-4c7a-8603-8208324192c7}} belongs to {{formula:d8ac5b5a-85d8-471e-82cf-6d85719615cd}} , thus it passes the test at extract:7.
From which it follows that {{formula:e1dc977a-ba24-4563-89c3-4be6deeeec4f}} eventually never executes extract:9.
Assume now that {{formula:7cec7b82-7425-4b6d-aec9-f28fa5ca5ec4}} forever returns at extract:3.
By extract:2, {{formula:17157c02-b639-4a1f-91cf-03b28fc30227}} is correct and in {{formula:414ffd63-b3a5-4b33-b5a5-2d34e016fc7c}} .
Otherwise, it returns forever at extract:8.
Then, by extract:3, process {{formula:1fff6d49-555e-468b-845e-c6a216a1af70}} is correct and since the test at extract:7 was passed, {{formula:2d7fedec-e02b-44e4-b683-6b571aeba9e5}} belongs to {{formula:9e868ac5-c96f-489e-97ac-c7e47726caff}} .
Correctness of omega
omega:correctness
Based on the prior results, we may now establish that:
correctness
omega implements {{formula:cff7663e-1976-499b-9d84-58d4b24ad7c9}} .
Name {{formula:3266dc92-9135-49b9-a1ec-7129a5ae4904}} some run of omega and let {{formula:2a7e6f5a-f5c9-4ff4-b3cc-b7288bf8ed97}} .
For starters, we show that the range of the failure detector implemented with omega is correct.
If {{formula:8de1fbef-4d9c-4c57-9465-515bf76cc639}} is outside {{formula:ad6617ff-5241-4f07-8b44-c5de0fd3fd23}} , a call to {{formula:0bd2fdf3-7b1a-4258-8c07-a35a87057fc4}} returns {{formula:8dfc65fe-53d1-4e82-bd14-8e35f7418e9a}} at query:1.
Otherwise, there are three cases to consider in the {{formula:86d5e620-fe11-4826-91e1-c2fc3998eedb}} procedure.
As detailed below, the procedure always returns a process identifier that belongs to {{formula:4c5d83c2-1aad-4052-a2a7-f00ab021b740}} .
(extract:3) the call returns some process in {{formula:dc8b198f-a4fe-4bb0-aa90-a593e8d4eec3}} ; by extract:2, {{formula:2290f95c-50fb-4588-8ee5-c5ca9d0a15b3}} .
(extract:8) function {{formula:602db52f-2b00-4f0c-a13f-86179cdddb2d}} is called and the result satisfies the test at extract:7.
(extract:9) {{formula:11afb738-474d-4e9a-b953-25e8a2ef3263}} simply retrieves its identity.
By extract:8, every correct process in {{formula:5eba5826-bbbf-427d-9ec0-e405de75253f}} eventually always elects some correct process {{formula:5c8d9c4f-dc3b-4602-b4bd-8d5e904d720f}} .
Applying simulation:0atagging:2 and extract:7, for any two {{formula:91253a37-96d0-4a2f-822e-c57bb42e548c}} , {{formula:61a0760a-a735-4f4e-8d61-c03d7e8d023d}} .
| d | 16ea0cae2dd11e2009dc7f447cad0d43 |
Generalization.
The generalizability and difficulty of datasets play a crucial role in
both training and assessing different algorithms {{cite:e07fa09d07dffd39016e9a86f9cf5529606ce4c9}}.
Hence, we study these aspects for existing COD datasets,
using the cross-dataset analysis method {{cite:1f0b38e4e67f81afda07161bd4b05492376ddfb8}}, i.e.,
training a model on one dataset, and testing it on others.
We select two datasets, namely CAMO {{cite:a0ed28a18558793fc707f905ee170c0da592448d}}, and our COD10K.
Following {{cite:e07fa09d07dffd39016e9a86f9cf5529606ce4c9}}, for each dataset, we randomly
select 800 images as the training set and 200 images as the testing set.
For fair comparison, we train SINet_cvpr on each dataset
until the loss is stable.
{{table:16c6a23b-bdcf-4abf-98ef-f72ef4fac94e}} | r | a94eff57d843dea2de4d624e9798cbb7 |
For ease of notation, distributions of the type {{formula:7d58a95c-be46-424f-a198-948a661c01cb}} are simplified to {{formula:193c882b-f9de-4dfb-b3b5-a7be91a24d77}} throughout the remainder of this paper. Since other metrics that promote diversity were able to give satisfying results with only the first two moments of the target distribution {{cite:3241113da5b81bc5169f94e195cdb89727a9f680}}, {{cite:bda96ded52fb7b09b976f32146ad6ba2ee3a0cea}}, it is chosen to use {{formula:c06af940-57f0-4687-b27e-d15b5ad73435}} . Since {{formula:db8cd65b-8ed1-4a36-8323-cd1271fd60d9}} is a high-dimensional vector, the estimation of the full covariance matrix of {{formula:c008a989-c33f-49e2-9608-f4e96cf35196}} may be intractable. Therefore, the variance of {{formula:3bc2ff28-53a3-4959-8e58-07b630aabbd9}} is approximated as a matrix made only of its diagonal entries. Mathematically, it can lead the entries of {{formula:7f8c9b51-06d5-492f-b816-9a8482c8f273}} to vary independently of one another. However, the results reported in Sec. REF suggest that the discriminator successfully inhibited this unphysical behavior. An interesting extension of this work could consist is investigating the effect of a non-diagonal covariance matrix for the regularization of the cGAN. For turbulent flow applications, a diagonal-by-block covariance matrix may be constructed using the integral length scale of the flow.
| m | 4bb73c3a9f78b79173ec55b7d620afaa |
Toward this goal, several methods have attempted to manipulate images with a text condition which conveys the desired style. Using pre-trained text-image embedding models,
these method usually deliver semantic information of text condition to the visual domain. However, these methods often have disadvantages in that semantics are not properly reflected due to the performance limitations of the embedding model{{cite:a75040c891b1bdc68e20576f9c39c0de6e9b3cf2}}, {{cite:c367047cbe3becbf0390d51b288cd2f337911269}}, and the manipulation
is restricted to a specific content domain (such as human face) as the method heavily rely on pre-trained generative models{{cite:5fd228e52318f9df4d2f2bb14aa5b04784ab34ba}}.
| i | d951cbaab31f41d736dd975472e1300c |
For training purposes it is also essential to be able to generate the {{formula:dae47367-9068-4280-9aef-fa0cbdc82583}} matrix associated with a given density matrix {{formula:66ccafe8-4a24-4db6-b117-211b29b03f85}} .
This can be accomplished using the methods of {{cite:3977e0379b595fcb4bfdae778bcf395867b8d1e0}} given by
{{formula:5cab5845-4592-4453-890c-fd9d593dcdfe}}
| r | 7a9b819181238fd80f734939a0753d2d |
This work presents a Python library for XAI enabling neural networks to solve and explain a categorical learning problem integrating elements from deep learning and logic.
Differently from vanilla neural architectures, these models can be directly interpreted by means of a set of FOL formulas. In order to implement such a property, such models require their inputs to represent the activation scores of human-understandable concepts. Then, specifically designed learning objectives allow them to make predictions in a way that is well suited for providing FOL-based explanations that involve the input concepts. In order to reach this goal, LENs exploit parsimony criteria aimed at keeping their structure simple as described in recent works {{cite:a4777dcdc814d8441e6dc1d78b72db0216102ca1}}, {{cite:2496549f499ce6ddbf350c5502c5cab795cc72f5}}, {{cite:89f0e86e7a0c83e6cc76e02d35f0187772a6c08b}}.
| i | db1cdfd0d4c31abbab0558a7965c5034 |
Metric-Measure Fields. We study metric-measure fields ({{formula:4dfb25cf-5fdb-4648-8488-f5335ba3510a}} -fields), that is, 1-Lipschitz functions {{formula:dc026dfe-6f83-4105-91eb-d32b500d2c3f}} between Polish spaces, where {{formula:ca981333-0ad1-40f8-ab6d-715d054b0a57}} is also equipped with a Borel probability measure {{formula:432e64c7-8576-44a3-8851-d97fbc3fe453}} . These are denoted as quadruples {{formula:bd1bd0b1-c3d7-42c8-a59c-1c7b0a7c86d0}} . We develop a field analogue of the Gromov-Wasserstein distance that has been studied extensively for {{formula:ec88b358-f8c2-426f-a171-f40cc04d63a7}} -spaces {{cite:e4372f06a2b9410cf650738f8ae462fec0f1a79c}}, {{cite:6b1a9535378f9f478f3aa8181cf68f0633f2d492}}. For fields {{formula:cff13087-70cd-4cd1-a9b0-47380e6c2beb}} and {{formula:97ae0742-9a2d-4654-a8a4-fe13721e9d31}} , the Gromov-Wasserstein distance is denoted {{formula:bc1b4bd9-98be-4bbe-9c4f-05c603aedd96}} , where {{formula:18edd0f2-2e2b-4a79-8964-e23b765e4268}} is a parameter, and is used to address convergence questions.
| r | 2c4ba732e396cb7e146f8a34dd9f1c87 |
To overcome the limitations of traditional methods, QSM reconstruction algorithms based on deep learning have been investigated.
Yoon et al. {{cite:35289318aeb4cfffd8b81b977f8caf614795dbf6}} and Bollman et al. {{cite:5a53d048c1f843c3dc22cd75c567c68f267da194}} proposed QSMnet and deepQSM, respectively, which are 3D U-net {{cite:ac852a9f03c3e4da574f1173975494ac77de6058}} structures designed for QSM reconstruction.
By using of COSMOS images as ground-truth QSM labels, they showed comparable results to those of classical approaches.
Chen et al. {{cite:3fdae619bf747bd531f44d47111537c1568d5b3f}} suggested QSMGAN where the adversarial loss was utilized, and Gao et al. {{cite:66f2c169e3e6e7f3706cdff127449d1ed22d68f7}} introduced xQSM where the octave convolutional layers were applied.
Polak et al. {{cite:bb2b91067d5f597cb133e2704977a517b4da6ab0}} also introduced nonlinear dipole inversion with deep learning using a variational neural network, which combines optimization of nonlinear QSM data model and the data fidelity term.
| m | ca299d39afe50fc2dcde08ccf7d12f1e |
Given the complexities that influence human mobility – heterogeneous distributions of population and services, transportation, geography, etc. – one would not expect urban commuting to obey universal patterns that can be explained by a simple microscopic model. This universality is likely due to mobility constraints that are rooted in human biology{{cite:171ea5246fd65bbb468be7ca58abe3717f736605}}. Interestingly, similar constraints exist not only in mobility behaviour but also in psychology (e.g., Dunbar number{{cite:cab318f05f3b3773221262c4e7f1f236accfbd1f}}), social networks (e.g., six degrees of separation{{cite:b9f9de6ee0a69de557138a4f68c9ef92e76cb952}}), and economics (e.g., the natural rate of unemployment{{cite:f756f37abc5962645d8d9a9ecb100da5e4e0d115}}).
| d | 3138581c11513952c6cadef8cf861fff |
Performance of the proposed methodology is illustrated through computational experiments on a 3.6 GHz Intel Core i7-4790 CPU with 32 GB RAM. Optimization tasks are solved using YALMIP and Gurobi {{cite:11c20ac86d7b90486cc7bfd14460d8c71f8dea38}}, {{cite:373f6ea2a5c88e0663b888f1950cce5392bdc1f1}}.
| r | 4cfd406bdbd849d39a1a2414c9442a71 |
The obtained value of current Hubble Parameter ({{formula:723521fd-c80c-4278-a86e-ce9c53558a4c}} ) differ for the two dataset, further contributing to the Hubble tension. The {{formula:ddcbea48-e62f-4a9b-9203-8467b218f143}} data supports lower values of {{formula:966d01f9-3542-429a-8a7b-b11a36fafb55}} which is in concordance with the Planck CMB results while the Pantheon+ dataset support higher value of {{formula:160a96ca-949b-462b-904a-44cfd8f4d9a7}} which support the earlier supernovae results {{cite:2beffd8ce3c3b423f3d35ad8ff7ef6c443502f0b}}, {{cite:6242f0666e50d2ee6ea39d31b3443270b68e789a}}.
For the Non-Flat models considered in the paper, our results suggest an open geometry ({{formula:44d0f996-30a8-49a5-8ba7-bf7640a45cf4}} ) but are consistent with a spatially flat universe within {{formula:c58c87e9-43a0-4c2c-8107-e71b26e0b034}} limits. Similar observations of the curvature parameter can be found in {{cite:fe4e3ef649d0d3a2be671b3acb73e283718ab766}}, {{cite:4dcf2d1ee0e3692833296c59e44b0f043838cce2}}.
For the dynamical dark energy model, there is mild variation in the equation of state parameter ({{formula:a209f9be-0fdb-4bab-a40f-638cce7757d1}} ). Nonetheless, the {{formula:8dcf0096-5773-44a9-8c0b-c84c5fc0a9e8}} CDM model ({{formula:9b09edbb-cfe0-49c1-9eb3-fe2e6605a116}} ) can be easily recovered within {{formula:2ecb026e-23d2-4d49-bb39-d25364c8e13e}} levels. Our results are consistent with those released recently by the Pantheon+ compilation {{cite:cc4f5b14c8209ce77f119beec6adc3840cd2e769}}.
| d | b0a578142bf517f27feaf3582826dea2 |
GRB 170817A was 2 to 6 orders of magnitude less energetic than other GRB {{cite:d1ba3d50f1cb453945499d3d8a756fc61854138f}}; the low luminosity of this source, together with the evolution of the X-ray and radio light curve {{cite:271fe6b7ff3e8966ebea367db2af1b089397a5cf}}, {{cite:f6165deffb960b3e1b481d0408c667222e701d07}}, {{cite:f0604bb10ac803a4bf006e63e4d10b60ef737eb5}} suggested an off-axis GRB with a relativistic structured jet or a cocoon emission from the relativistic jet shocking its surrounding non-relativistic material. Subsequent very long baseline interferometry observations allowed constraints on the source size and its displacement, indicating that GW170817 produced a structured relativistic jet {{cite:661d98c0b74f05854e9d235bc8a652c162dab33a}}, {{cite:88ba1303025678a1dafb91e97c95c0a39e1e9046}}.
| i | 48a2a582fb026b19ead2a2c4981b07d4 |
In Section we compare our results for ALUE with earlier results about a chiral ensemble in {{cite:2739372d03b15fbd7e0d1e1f063fff54dcc0eae0}}, {{cite:42366c44f1fb8491523699b420c46d963bd94467}}. We also relate our results with some other point fields appearing in {{cite:7e58f403af8a3ddba9d943e8218814a6111e94c6}}, {{cite:7c9e4e4da2c29f9708b66b09210a67adf663b6cd}}, {{cite:3fd931e94b0a62fcbf0fcf869f5b883683032c97}}, {{cite:676043a04ff935c4dd6dcc4d098c644787862bad}}, for example.
| r | 1fe805502965ee54475109fd187c7fb1 |
Lie superalgebras have applications in many areas of Mathematics and Theoretical Physics as they can be used to describe supersymmetry. Kac {{cite:cb84e009970eae4a22cb817524bbbea7357c09dd}} gives a comprehensive description of the mathematical theory of Lie superalgebras, and establishes the classification of all finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. In the last few years, the theory of Lie superalgebras has evolved remarkably, obtaining many results in representation theory and classification. Most of the results are extensions of well known facts of Lie algebras. But the classification of all finite dimensional nilpotent Lie superalgebras is still an open problem like that of finite dimensional nilpotent Lie algebras. In the last few years, the theory of Lie superalgebras has evolved remarkably, obtaining many results in representation theory and classification. Most of the results are an extension of well-known facts of Lie algebras {{cite:d9d141c57cae9d91c79aa3d6bdd7a87e3c010fa0}}, {{cite:ad5704eaa9fb7bb5fd368edee84e5df10d62eb54}}, {{cite:a25dd55d4fe75d093dce421192738f0ba8e16d16}}. Recently, García-Martínez {{cite:a25dd55d4fe75d093dce421192738f0ba8e16d16}} introduced the notions of non-abelian tensor product of Lie superalgebras and the exterior product of Lie superalgebras over a commutative ring. In this paper, we determine an upper bound on the dimension of the non-abelian tensor product of nilpotent Lie superalgebra.
| i | 3eda3a440839872f6b176c1f6a493c17 |
Three decades after the concept had been proposed, weak value amplification (WVA){{cite:d7779e3eeae2b9078d4c71af14a5e9beaa8ac05b}} has recently become a useful technique for optical metrology{{cite:b1d4bc9d52a9e2909fd6280a78877a4e5b3f05ac}}, {{cite:2638d4435a6df3a64b5c39eeb3dc3412612b95ac}}, {{cite:2ae5ee0ea2d5e337397494c0559f2fbafb895fc5}}, {{cite:ce643cbad5357f5ec7697fe4aee474da31bba8df}}, {{cite:7141646a886f7ac4d14bec068c3525ece8cbb5ea}}, {{cite:458455b09f5a9f0667cb62748c2467c5628f2d41}}. Taking the advantage of suppressing certain kinds of technical noises and practical imperfections{{cite:3277cfe08307d3668cdc103e1715eb4026f2ab32}}, WVA can efficiently improve the signal-to-noise ratio in practice, therefore provide higher sensitivity comparing to the standard techniques{{cite:cb74f4af251c64069fa35915579ffbe2c96a8009}}, {{cite:9bee189f36aa8dcbb27ae1b8eaf686f55616f16c}}, {{cite:783ff9d1fe5e4b251db173bba166e6f9f74e1616}}. However, most of the proof-of-principle WVA experiments were designed and performed on free-space platforms, e.g Refs.{{cite:b1d4bc9d52a9e2909fd6280a78877a4e5b3f05ac}}, {{cite:2638d4435a6df3a64b5c39eeb3dc3412612b95ac}}, {{cite:ce643cbad5357f5ec7697fe4aee474da31bba8df}}, {{cite:458455b09f5a9f0667cb62748c2467c5628f2d41}}. It is hard to directly transplant them to real-world applications due to their cost, scale and stability.
| i | 9947b238af78a669c195983fb91eb841 |
Table REF summarizes the accuracy results of the proposed
framework on ETH-80 database. In order to perform a direct
comparison with the methods employed in {{cite:26e1686d8380705913b1da0bc5c4a6ce77b5dfb5}}, the
same setup is adopted. Precisely, the same 6 categories
(apples, cars, cows, cups,
horses, and tomatoes) are adopted. For each category 4 objects
are taken and for each object 10 different views are considered thus obtaining a
total of 240 images. From the remaining images, 60 per category
(15 views per object) are used as testing examples. The results are achieved by baseline Logistic Label Propagation ({{formula:bb44b28d-3935-4e3f-8185-b42bc09be9b4}} ) {{cite:30fbae42868f8310acddddb45f3ccd009ce5758e}} + Bag of Words (BoW) {{cite:6605a645080eac08bd42de38de0bfb824e42d98f}}), and those obtained
in {{cite:26e1686d8380705913b1da0bc5c4a6ce77b5dfb5}} by employing the approaches proposed in
{{cite:6780f4056039be450aa6e311c1e6dd021039aa6a}} (gdFil), in {{cite:66548a75790f9180481c91597e0b7428289052a7}}
(APGM), and in {{cite:9fdaae879a3fbbc8d2717bdef0e6c1fbbed5a2ab}} (VEAM). As can be seen in table REF , ARSRG embedding, adopting {{formula:152a776d-4a52-4b4b-9928-d53ab60f4666}} classifier, outperforms the results obtained by the other approaches. These results confirm that ARSRG embedding correctly deals with object view changes.
{{table:333d1970-4c9b-4ca4-b3ce-592fc75cfb14}} | d | 780987e20ff1e87f1cb4a2367a9dc9f0 |
The first experiment considers OMP, BP ({{cite:879ab749d7085a797b0ab824f19645bba01e70d9}}), and Compressive Sampling Matching Pursuit (CoSaMP, {{cite:564854cd1a6ca323c5b15b542469070fa94df246}}) as baseline algorithms. Results show that a few (five) iterations of LiRE increases and never decreases the average percentage of exact support recovery, for a non-trivial range of undersampling-sparsity operating regimes. In particular, LiRE reduces the number of measurements needed for perfect support recovery via CoSaMP, BP, and OMP by up to {{formula:31e4c617-ae8d-4f62-931a-f7dfa0e415dc}} , {{formula:061fc850-e651-42d2-830d-9ee7e531f726}} , and {{formula:052c657e-7a71-4b7b-b587-1802e0c54909}} , respectively, depending on the the sparsity level.
The second set of simulations compares LiRE{{formula:debcfef7-3418-4432-ae70-b44eb2c6de92}} OMP against BP and shows that even though LiRE{{formula:756b61ce-c64d-4295-b9b7-0493a1cd9cbd}} OMP has a significantly lower complexity than BP ({{formula:56fbaff6-f77e-440e-a30c-4d0dee286227}} vs. {{formula:2cbb0b37-4ae1-48cc-8ccb-1015b3bcdf09}} , see {{cite:c697f70fee7d25e13d9150a608d152cc6177c9a8}}), it achieves an average percentage of successful support recovery that is at least as large as BP, and sometimes larger.
The third set of simulations evaluates the robustness of LiRE against noise in the Gaussian additive model
{{formula:fae4c26e-838b-4a07-aec2-2515b8298b58}}
By repeating the second set of experiments but now with LiRE{{formula:2e682a0a-9c61-42d3-a482-054ae2c4c89a}} OMP against the LASSO solution, we observe that LiRE{{formula:d6514fd7-dfac-4fcc-aa0a-8d7b6390f24f}} OMP is superior to LASSO as the noise level increases even though computing the LASSO has a computational cost that is at least quadratic in {{formula:857d0b4b-006c-4171-9e14-0e5a70236df6}} {{cite:b50833397048230b1bfa89e29e858efdaa4ddb3f}}.
The fourth set of simulations evaluates the performance of LiRE as a standalone support recovery algorithm with a random initialization. Results show that in terms of percentage of exact support recovery LiRE lies between OMP and BP.
| i | 048ec95cbe2667c82dbf3f64e040b37b |
Our proof of Proposition REF is closely in line with the proof of Proposition 4.1 of {{cite:d3c6540e9698c53d208f4a51262fc67b0f95a6a8}}. It will use a Laplace transform formula for {{formula:6b39f420-883e-4cc3-9392-bfe4a5feec88}} proved in {{cite:6b9d6bfd2a62a67f638e7023e5d2205540558aec}}. It connects {{formula:9e5cc3ab-d243-447c-86fe-65fabeba57ce}} with the Airy point process {{formula:894edcb0-5513-4d65-9602-6ea7d3a472ce}} , a well studied determinantal point process in random matrix theory (see, e.g., {{cite:e0b97839490d5c1fd4810bcf4cdb3bb2a3ce5b6a}}).
| r | ed66b39375d7fac800bf789ab64e6ce6 |
This section presents the implementation details and compares our method with the state-of-the-art frameworks on benchmark datasets for various tasks, including ShapeNet Parts (object part segmentation) {{cite:56d03801c5d79a52ddd7988fea472fb77bfae16f}}, S3DIS (indoor scene segmentation) {{cite:edab05cd6f1b05f00010f2afce99543e77d617fa}}, {{cite:341a3cd29ddde6d45a19a49cee187f1d53704670}} and KITTI (3D object detection) {{cite:de4c6216da43376698145def1cd12bd9fd14a366}}. We also conduct ablation study to study the performance of MPVConv.
| r | a998e6c2bf02e2227eb12baf8470e0e4 |
We employ our transformation on several datasets of different vision tasks to validate its advantages and generalization ability.
The implementation of knowledge distillation methods are from their source code except FitNets{{cite:5bacb1b42c3b4705554e603b9e12c7752cef914b}}, and we reproduced FitNets for comparison.
{{table:35583fcf-4da9-46f4-b69c-a391cc66199e}} | m | e65962c7e174176d03eeb2e7e2b9f9ed |
Random projection dynamics. There is a general consensus that the ESN model operates closes to the optimal situation when the projection is close to the edge of chaos {{cite:28e6b760ac759a58c2cc34b1c2adc2c6f66a9816}}, {{cite:a506c125966ab15a9bfd2d045f7828944b216956}}, {{cite:92989e01ad2cd94df2a7b349a7ec5724da9376a0}}. There are some works of EAs applied to the Lyapunov exponents optimization. However, it is still possible to provide insights on the specific area of Lyapunov exponents estimation using EAs (for the particular case of dynamics coming from ESNs).
Quality of the projections. When the model is used for forecasting and learning, then the projection should preserve some topological properties from the input space. A preliminary work was presented using Sammon mapping {{cite:a9e49d6a193bc8329db6cc1747adedf6efd66ec7}}, but we believe there are still a lot of approaches to be investigated.
Multi-objective approach. In most of the works in the RC community, researchers apply EAs over quadratic errors as a measure of the prediction accuracy. However, there are other metrics for evaluating the good characteristics of the model such as memory capacity, robustness, complexity and speed. It would be an interesting direction apply multi-objective for optimizing several different metrics.
Self-organized reservoirs. Even though was studied many years ago in {{cite:4ad618e2b0c53cfe2cb315043c40b91c82a6c15c}}, {{cite:50de23df8128940b9360c54bb694ec1833f38a8a}}, {{cite:58a8474e16e3ed2d7fa7a9e53d2af8c0e6454831}}. The area related to self-organization has grown up in the last years due to robotic applications, therefore most probably new developments can be done in this area.
Lifelong learning paradigm. ESN model has very good properties for being used in continual learning. Parameters of control for the catastrophic forgetting problem, and other optimization problems of the lifelong learning area using RC techniques can be studied using EAs.
| d | 1f8a3d41dc82d5c305978e6e3c4f8acf |
The statistics of the radiation field, or photon statistics, begun with the seminal paper by Hanbury-Brown and Twiss for stellar size measurements {{cite:4fd19dc7adf5a71d492e9f931971d1c5ee73be7b}}, {{cite:8b3185c75f492a42cf430eab9a0d2ac0e16fdfb7}}, has evolved to become one of the pillars of physics {{cite:1aa27b2b447ca1f43524d1cb41a3a8301d97a9f5}}, {{cite:db9c2811cc5ed0abae1d1390f7c45f802d656824}}, {{cite:09332c5dd764289673f607fa4e7ce94377377aa3}}, {{cite:325db1981e1d0c6829cd8969b4bf97bd023b492e}}. Its applications run from the detection of squeezed states and antibunching {{cite:bbb7e92674073d1c80ff9da674ec3254cd401671}}, ghost imaging {{cite:950d2d7ab9205396b949c2daab6f7bdb1220c143}}, {{cite:ac0caec1011924db347f567ea3f5471ddc62942e}}, improving the sensitivity of gravitational wave detectors {{cite:10ad122d7a3c1f2d1c1d38347753cfbd0a43ebe3}}, particle sizing {{cite:35dbf99ad0623cc66235acc373c12b009a04141b}}, dynamic light scattering {{cite:096d92f1ede5573575d15df5aa23cef4726747cb}}, fluid mechanics and velocimetry {{cite:9d87da6ea529a46b61b103ad52924912cdff93cc}}, {{cite:db981eb21a4ff3c06a4bd8180fb46d3f92ac994c}}, quantum information {{cite:d688423cec05a6d051c160eb642b90b65b2c5154}}, {{cite:d0cd67d949546099ba611f20ae24b8a88b1bd602}}, and advanced measurement techniques {{cite:4544a612cc425c9b4447df1db049e3d802468011}}, {{cite:3475cf1fbc28759dd1af2a2e11cfebef8148a85b}}, to name only a few.
| i | c18bbdea14f28ae0509a97e8cc034e03 |
We obtain the noise bound with the AC1 by using its implicit solution. Consider that each weight update round is {{formula:2e43647c-4224-4bc7-ba82-8a226ba55004}} LDP that results in {{formula:85d2c665-ca06-41a0-9533-29092a273747}} LDP after {{formula:e69b5a5b-ae33-4a78-b329-762c34020ee5}} iterations, which satisfies {{formula:cee9a991-5759-4a14-8603-2c1354678fc9}} and {{formula:c039c98f-5a9f-4328-b255-d558e7838300}} by {{cite:550237af58a630e4ce1b772307fde96c826c0c07}}. We obtain {{formula:f20344d0-0e8f-4c9a-a02f-3577ed75c3dc}} from given {{formula:39bb01c5-a4f7-41dc-9f9a-201118eec610}} by choosing {{formula:fcf12375-f31c-4121-afc5-4efeecd7cac9}} and plug it into the noise bound of a Gaussian mechanism from {{cite:550237af58a630e4ce1b772307fde96c826c0c07}} as
{{formula:6410c3d2-5af6-4d1a-af4b-f866266db016}}
| m | 92e8b6ad47de9e830669a36caf36f870 |
Second, in most existing works, terrestrial IRS is usually deployed at fixed locations such as hotspot or cell edge to enhance the communication performance of its nearby users only. Moreover, for the IRS coated on facades of buildings, its coverage is further reduced by half which is effective for the users residing in its front half-space only, as shown in Fig. REF (a). Furthermore, in complex environment like urban areas, the signal from transmitter to receiver may need to be reflected by multiple IRSs to bypass the obstacles in between, which results in severe product-distance path loss {{cite:929db83867d256431fd68f72f046b85f1ceb7e65}}, {{cite:f8f5b5deeaab0b39e6253ce789228ecc44b2c34e}}. To resolve the above issues, an effective approach is by mounting IRS onto the UAV to assist the terrestrial communication {{cite:f8f5b5deeaab0b39e6253ce789228ecc44b2c34e}}, as shown in Fig. REF (b), which is named as UAV-mounted IRS (U-IRS)-assisted terrestrial communication. It is worth mentioning that, to provide reliable power supply and stable control for U-IRS in practice, the UAV (e.g., balloon) can be tethered to the ground base station (BS) or other movable platforms such as vehicles with reliable power supply. The aerial platform of UAV endows the IRS with {{formula:a3a52a56-b446-4699-89a5-273009bfd243}} panoramic full-angle reflection towards the ground as shown in Fig. REF (b), which can assist in the communication between any pair of ground nodes provided that they have LoS links with the U-IRS.
{{figure:074e23b9-8fd4-4b07-bf7e-4ec999d106b0}} | i | b61c48deaa1d2d08ffc8c58f4ee234dc |
Followed by the backbone, the feature map will be fed into two branches. The first one performs Horizontal Pyramid Pooling (HPP){{cite:d887dd7404f03737b249b45370ffca78576c16e9}} on the feature map and the Horizontal Pyramid Mapping (HPM) {{formula:9e598ca5-92b7-474a-b001-e8f707d1c7bb}} will be obtained where {{formula:8df64c62-7180-4c4c-8ab5-a152d97c6e15}} is the dimension of output feature. In the second branch, the feature map is pooled to get the view classification feature {{formula:c65d115b-ec37-44e6-8113-5f98cc688f3d}} , and the projection matrices {{formula:8d4e3493-6d7e-4ec7-9044-48677f2b90fe}} are selected according to the predicted view, where {{formula:9bca6b44-7511-49cd-b24c-b5114bfc34da}} is the number of strips cut in the HPP Module {{cite:d887dd7404f03737b249b45370ffca78576c16e9}}. Then for each feature in HPM, we will multiply the projection matrix of the corresponding view to get the final view-invariant feature.
| m | 08c8824f92a497fb15ed335c536c2e54 |
We explore the copyspace problem utilizing frameworks for object detection {{cite:cf38745a6e4458a15839dc313cdebfe0e8a67ced}}, {{cite:50d7b32239b728b09a6a146b160bca1aea771b7d}}, {{cite:cb007ad68f366c012bd2b33f240777304ea51a8f}}. The Yolov5 Github repository is cited in lieu of corresponding publication in arxiv.org, a controversy we will not delve into beyond providing more model inter-comparison results.
| m | 46f7de7bfbd3ab8d2677fd1471cac8ca |
Our 3.5–4.5 PEARLS IGL values are {{formula:4a3789a8-00d5-41d3-95f8-f071fe8c00b5}} 40–50% below the direct EBL
constraints in Figures REF from MAGIC {{cite:1d8d13ba62b5e6d9cc534fcf767c75ef677a1c3d}}, {{cite:98157a2e923f44baaa246686f66daae8993b0652}}, {{cite:e2ae2c23a0fc476d57aa67ef948d59cebf17d4ea}}, which are estimated from how intervening EBL photons
distort the {{formula:12cc21ac-4508-49dc-80bd-fcf66c14baf3}} -ray spectra of blazars over a range of redshifts. More
recent {{formula:f8e926a4-cbcc-4856-8e05-f9e0bf8f0341}} -ray blazar results are converging closer towards the total IGL
values {{cite:af262bf903380e6345ba2cc741c275276948ad6e}}. Various sources of diffuse light may cause a
discrepancy between the IGL and the {{formula:2e167afc-792d-47c0-9441-4af6abd5dfe5}} -ray constraints, as discussed by,
e.g., {{cite:027d7ea981ac0857fb6959a206247c9e0c67ce39}}, {{cite:727094a545b86f7bfd5df108e697bae43ad85201}}, {{cite:0805e2c4009fd511910c7bd733c8fb410e993212}}, and
{{cite:2a14be46942977e06e8c597c8f10082b402b3ed6}}, and references therein. A possible source of Diffuse
Light are tidal tails of long-lived stars pulled out in galaxy interactions
over the entire redshift range where galaxy assembly happens. For instance,
{{cite:57f6d60bea9e759de8a843de911a3869fc5618ae}} and {{cite:47ad5575f3f00c92bad6911d44332105577eeb28}} analyzed ultradeep (32 hr)
ground-based LBT U-band and r-band images at various stacked seeing-FWHM
values, and find r-band tidal tails in galaxy pairs up to z{{formula:2c93a0b4-4119-4049-b73a-72a59c1842de}} 0.5–0.9. They
suggest that {{formula:6be1d07f-dce5-4d7c-9b4a-0e046bcf7bcd}} 10–20% of the galaxy light from brighter galaxies
(AB{{formula:0c515606-de2c-49d8-a899-593561b5ac17}} 20–23 mag, which cause most of the IGL in
Figures REF –REF ) may be at large radii to SB-limits of
AB{{formula:6df1a30b-c76f-400b-88c0-c2c31ee94554}} 31–32 mag arcsec{{formula:65301b48-e6e6-4a1a-8bf4-711508587287}} . It is unlikely then that tidal tails between galaxies
produce well over 20% of the IGL. Remarkably, though, JWST indeed sees tidal
tails between galaxy pairs and groups in the CEERS images of
{{cite:3de011edd713c76186cc758cf47e4bacab4494ea}}, some of which can be also seen in our JWIDF image
of Figure REF here. If tidal tails consisting of older stars pulled
out during galaxy interactions are common place, future JWST imaging should
find many more such examples, and be able to better quantify the amount of DL
present in dim tidal tails of faint galaxies. At the median redshift of these
galaxies, NIRCam 0.9–4.5 images are ideal for such a study.
| d | 965c8e35627679e310f4e2f4d3c4b68e |
The skewed distribution of the data further entices the use of a two-step approach as implemented by {{cite:cafa0ffdea0fb9f319eecc510c0be034d6117a8e}}. This approach suggests making a classifier to filter the “commenting" and non-“commenting" class labels, and then use another classifier on the minority class labels. The two-step classifier was implemented as a way to tackle the skewed class label distribution of the RumourEval dataset {{cite:2f34b7614616c30ce9ef5d9ef5f2b6db03f7bcd8}}. Therefore this approach is especially interesting for the case of dast given the particularly skewed nature of the class label distribution.
| d | f478ed0d78118ed96ed2c13013e70b28 |
First, let {{formula:bfa05c5c-aec5-4fda-afee-b2a57f17761b}} be even. Hence {{formula:fdac0e78-1571-4087-a172-f46b751b129b}} , by Theorem REF and {{cite:49964771c01e876833b88764188f86ce9c04424e}}. Since each color class contains at most {{formula:9ad7bb5e-de79-4f5e-8ca9-eeb99fa4840a}} edges, we can color at most {{formula:140727cf-163e-464a-8924-bed442f25f25}} edges with {{formula:3c574713-24f2-47ac-b05e-bf0eeb869c02}} colors. Therefore, if {{formula:d86cbd1a-4068-4611-a67a-83b4a4b04c3a}} , then {{formula:3f17e3e2-1186-407f-8cc1-bb20102a9508}} . Together with Lemma REF we get {{formula:69120875-8c31-4432-8076-5c0b9686c6d4}}
| r | 34fb46b55226de86e8b66abe8d7ceb4e |
The DEtection Transformer (DETR) {{cite:1b0da93f78278465896052a54310aec2ecd29dc0}} uses a transformer encoder-decoder architecture, wherein the self-attention provided by the transformer removes duplicate predictions. Spatial position encoding is added to the feature maps obtained from the ResNet50 backbone inside the encoder-decoder architecture. Prediction feed-forward network heads decode the output embeddings of object queries independently into the bounding box coordinates and labels. A bipartite set matching loss is used to train the model, however in our implementation, we replaced the original cross entropy classification loss with the focal loss {{cite:d1e8a9611000920bdd0f8a23a9bb5f15f3a261b6}} to overcome the class imbalance problem between positive and negative samples, and weighted it with {{formula:84b93ddc-70fb-45c9-baa0-f0554663f48e}} .
| m | a07ab5232785fbb058eb133699f9a714 |
Figure REF shows the contribution to the gamma-ray flux received at Earth from the CR flux of SNR G57.2+0.8 and SGR J1935+2154 for the three spectral indices. The assumed gas distribution was the one from the Fermi-LAT collaboration {{cite:a5a1c4cd19cad40f59b1085d2ac7e71e0085bf7c}}. Observations of the diffusive TeV energy gamma-ray emission from the GC from H.E.S.S. {{cite:abf5f548f1ce9cb4a55fba399309ce7f2505f2c5}}, {{cite:40ab4282ad84c07311c08163d53e9875a902612a}} and MAGIC models {{cite:c523d4d6c89ef79c6be2db959daf2c350a22bde5}} are presented along with a GeV gamma-ray flux emission source in Kes 73 and its molecular environment using Fermi-LAT {{cite:80da814327438d1f9ac1eace69060943cece8a0e}} as a reference to determine the reasonableness of the parameters used and to test the model. The young SNR Kes 73 (G27.4+0.0) hosts 1E 1841-045. The model in {{cite:80da814327438d1f9ac1eace69060943cece8a0e}} also provides the GeV gamma-ray contribution from this source considering the interaction of the SNR Kes 73 and the surrounding MC. The gamma-ray spectrum obtained can be explained by either a pure hadronic or a hadronic plus a magnetar emission. Therefore, one possible scenario is that SNRs hosting SGRs/AXPs amplify the gamma-ray emission produced by hadronic and leptonic processes and contribute to the diffusive galactic plane emission, in addition to being identified as possible sources of PeVatrons.
| r | 5e8912f03e99aa58dd0dff5543895a31 |
One is also recommended to {{cite:95941a31aaf878e1de561da7672fe2a6474d9180}}, {{cite:b5a9f17bda4a98d212b62ac7b573830d184592e6}}, {{cite:f61dd7b375f658e3e3691f594761174f1d144443}}, {{cite:c45998f640b04a6fe45cc64a7b9db13fbc02ae07}}, {{cite:9dda8ab4de6c50be392d0083a4001e51f5e56cc4}}, {{cite:186d4e7449a42bffd0b70d133ed2c296e9e3a405}} for more related research topics on the notion. Denote the convergence in this sense by
| r | 65bc2dbcce084dea067b4e0396384dce |
Provide a generic SGG dataset class that supports both VG and OI datasets and is easily adaptable to customized SGG datasets. We have integrated evaluation metrics from official Open Images relationship detection challenge for OpenImages Visual relationship detection{{cite:02806bac081008461b198a84c912545563429d3e}}, and implemented widely used Visual Genome relationship detection Recall calculation for scene graph generation on the Visual Genome{{cite:313f1a7faee7e2fa7173efbb118675e455be4cdd}}.
Support multiple popular scene graph generation algorithms. We have integrated five widely used scene graph generation algorithms into our framework, including iterated message passing (IMP) {{cite:7a4a540b37ed8857efaa4a6b87a97bc38e925d8d}}, multi-level scene detection networks (MSDN) {{cite:981f405f2f12ba34d68326824251e15c3676fd0a}}, graph R-CNN (GRCNN) {{cite:08bfc1bf525a73145b2cb0f296637b8ad0d5110e}}, neural motif (NM) {{cite:7ef593cf21fd909dab0e19f08be375145bc51ce1}} and RelDN {{cite:efe0905388e283e2de2d3d5a4b753bd74dff8658}}.
Modular design with decouple-able object detection and SGG modules. We have decomposed SGG framework into three components: object detection, attribute classification and relationship detection. One can easily replace or insert new components. Our framework decouples object detection and relationship detection so that one can do relationship detection with bounding boxes and labels from any object detector.
Fast and portable dataset format. Our code base uses Tab Separated Values (TSV) data format, which is portable and flexible for data manipulations. We have also developed tools to visualize this data format directly. We will release the visualization tool later.
Large models with state of the art performance. We have trained large models with ResNeXt-152 backbone for both object detection and scene graph generation. Our large models achieve state of the art performance on both Open Images relationship detection and Visual Genome scene graph generation.
Image feature extraction for downstream vision-language tasks. We provide a easy-to-use pipeline to extract object and relation features from our pretrained models. The features exacted from our large models, when used as the input of downstream vision-language models, achieve SOTA performance on seven major vision-language tasks {{cite:511b7b6373c4588ad82325dbc5c0ec1eb0d60d40}}.
| i | ec708b4261e18749280f191f5b1da36a |
The modeled triangular nanobeam photonic crystal cavities have high {{formula:4ff67bd7-7f90-4d78-aeaf-544ef466a535}} /{{formula:b4fceb4f-2ad7-4ddc-802d-df901b60cf48}} ratio with promising applications in high-speed and indistinguishable photon generation. For the same triangular geometry, altering of the lattice constant is found to tune the resonant wavelength without reduction in the quality factor, meaning that many different high-performing devices can be simultaneously fabricated on the same chip. Interestingly, in literature on diamond and SiC nanobeam devices, the fabricated quality factors fell 2-3 orders of magnitude short of their theoretical predictions {{cite:6d72de9a0e2f739c91948a6516c5950eaf77123a}}, {{cite:5a50055bbd8b801c6d30e45214c1a6e0e8fd0f85}}. The reduced performance can be attributed to fabrication errors such as mask alignment imperfections {{cite:5a50055bbd8b801c6d30e45214c1a6e0e8fd0f85}}. This indicates that additional efforts in nanofabrication process development need to be invested to realize the full application potential of these designs. Purcell enhancement of the zero-phonon line emission would play a crucial role in realizing quantum systems that require two-photon interference and quantum entanglement, such as quantum repeaters {{cite:e74c106bde14954c7d8894c4e0536b8251e7ddde}} and cluster-entangled states {{cite:a14f143a55c96156c23f5678275209c3fcd35ac2}}, {{cite:8e266b2d3734a51164447d16cceac57eddc641af}}.
| d | 610ba07999a6dd9eb1a5078f7e5537bb |
for {{formula:97bd08a1-8467-4c48-b004-8f3f1041aeb3}} . Motivated by its central importance in random walk theory {{cite:1b9be70ec5ba8a5d11d1218d43175fa10a8c39a3}}, {{cite:54d450de9733fdde8c0d000eaf6161a90c76eba6}} and its applications to data smoothing algorithms {{cite:f3f5f06e3dc71fdff5253533e33df0dae0e360de}}, {{cite:4bb7521b49f7e0813c7fc5bbaf4c5aac61e3cc70}} and numerical solutions in partial differential equations {{cite:1f5df527f1291eaf52b6247a4c8e70cd9f632600}}, {{cite:61c7fdb4361c57b2d91d58b050d6bfe6b9c6e36c}}, {{cite:1e454fff4aa78fe35d661a05aeca46388b8ba30d}}, we are interested in the asymptotic behavior of {{formula:b0c16ede-14cd-40f7-b815-3a52a42de909}} as {{formula:9b3755bf-4f07-4c93-899d-0d725a9c2292}} . Following the articles {{cite:47cb24b1db3831faaf6b47ee963cd29dbee03b81}}, {{cite:ed573b7e392f3114d35fcf037736ca72859f6737}}, and {{cite:1a3a17c044aa2a449c0091a98d7c0096dfc66b35}}, our goal in this article is to broaden the class of functions {{formula:dc9fa169-3429-40b4-b565-0216716d8d1f}} for which it is possible to obtain “simple" pointwise descriptions of {{formula:7f938afe-439b-4780-8143-0111ae8bb561}} , for sufficiently large {{formula:453518cb-3cc9-49fb-a062-7b824bcb17c0}} , in the form of local limit theorems. To better understand this goal, let us first discuss some background whose origins are rooted in probability.
| i | a3cb31c5b5d93cc3479df725b48bb342 |
In the QCD sum rules for the hidden-charm tetraquark and pentaquark molecule candidates, we usually choose the continuum threshold parameters as {{formula:e5d4dcac-c0a2-4e6d-b092-363eebd94ef0}} , just like in the QCD sum rules for the traditional mesons, again the {{formula:2868ffa0-4ed9-462e-a7eb-848f7e4f725c}} denotes the ground states {{cite:e1847811eea41400e23b368359978b88c85fff91}}, {{cite:0f6c9802e42cc5a65e43e987ef6bb431bb6f3eac}}, {{cite:cddb39f49510a45ac4dc32fce7e155344a1b539b}}, {{cite:6dd58a0740fa1325361e18037cf3a215597e37fc}}, though the hidden-charm molecule candidates have not been unambiguously assigned or determined yet. In the present time, as the multiquark spectroscopy are poorly known, we have to obtain predictions based on assumptions in one way or the other, then confront them to the experimental data in the future.
| r | 4684d1ccd46ef1d3e006b6e980b3e36c |
Topological insulators (TIs) serve as an excellent bridge between traditional electronic bandstructure theory and a more modern approach including topological aspects. TIs emerge from a class of narrow-gap materials whose strong spin-orbit coupling leads to an inverted band structure and the formation of helical (i.e. spin-momentum locked) states on their surfaces {{cite:b586607bf850023634bc637933bd7183c078aa1a}}, {{cite:2098357c7c7fb8af719384a8f754c80035d562be}}, {{cite:380ba3a34ea1e5eb9c82e771813af814f2804481}}, {{cite:d095f3bcae4c03a78a89c9357570a4c7302e7a70}}, {{cite:12b186e550a005231831147108cedca90f4a4b98}}, {{cite:e22cd7daa6a1f80c263173cb0924cb08a7a141ef}}, {{cite:7a17dc61ace8882b36980fe09bcdfd30695a912b}}, {{cite:55f49258771ecb7b921468c819615823e07402df}} that are protected from nonmagnetic perturbations. Growing attention to topological insulators is fueled by a fundamental interest in solid-state spin-physics and the prospect of designing novel electronic devices {{cite:a37695619e706eaba5399b03019c9729ee079911}}. Topological surface states have been observed in a number of 3D compounds, such as Bi{{formula:cb01fb2b-9240-4006-a527-b946be9069c3}} Se{{formula:2c835d34-0cd5-4a37-b2a3-382203845453}} , Bi{{formula:82adc9c7-a963-4f08-bd4b-97b34eeb4d26}} Te{{formula:f29134ba-168f-46e1-9a54-d886cbe93d9b}} , etc. {{cite:f57346e7bef241c00325cdb6e4b1156a385abfad}}, {{cite:af256206d9e8cc0f5d7e12ae5b491c15eff3de30}} Examples of 2D TIs with 1D helical channels at the sample edges include HgTe/CdHgTe {{cite:f3b161bad200e3375e11b488ba1d0c153b4550ad}}, {{cite:d5b748478128f85050050b80c2f9e086596d1589}} and InAs/GaSb {{cite:350ff11097ab1abba3e597e16aa02628c080024c}}, {{cite:904dca5192f50084a59eb3de75855253e675e5f5}} quantum wells of certain thicknesses. Experimentally, surface states in 3D topological insulators are usually revealed by the angle- and spin-resolved photoemission spectroscopy {{cite:af256206d9e8cc0f5d7e12ae5b491c15eff3de30}}, {{cite:c8605b4436c8f4381150d47ee6481b6912b21448}} and magneto-transport measurements {{cite:810f0aeb275544a6925d618ea056d439c399e0cd}}, {{cite:62108fa66949e7dcacada2becc7e3e0fc5fc3e02}}, {{cite:4881e3d7f8d4001b420b5442867f5101edfbe187}}, {{cite:444732b909e57886ba80a15060a5f22892e21ea2}}, {{cite:da79c8d05d6b261523ee51ac4a80ee395986dd61}}, {{cite:9d4cbe92342ccf60085ba42da4479d5929f68240}}, {{cite:21d8aee2789f958369c7d49607cb56ae7a9288db}}.
| i | 9c55cb061ee2246185c569827aa3c139 |
In addition to the qualitative comparison, we also quantitatively compare to previous methods through computing the P2S, Chamfer-{{formula:08188120-c877-4a02-a899-4ee85ad9f773}} and IoU of results by different methods on the testing datasets of CAPE and Articulated. Table REF and Table REF demonstrate the mean values of the above metrics of different methods on the testing dataset of CAPE and Articulated, respectively. For the CAPE, the results of DeepHuman {{cite:02be4f40139b6419a8fc8d2768282c23a7799128}} are the worst because the CAPE is a synthetic dataset, but the trained model of DeepHuman is based on a real dataset. The SPIN {{cite:b3f83a1dc882684e26b36ecdbabd0412464d34b2}} is better than DeepHuman, but it is still worse than PIFu {{cite:39b184ec8f627476bc28b979a2938116443d1322}} and our method because the estimated 3D models of SPIN are naked and the poses of the estimated 3D models might not be accurate. Comparing to SPIN and
DeepHuman, the results of PIFu are better because PIFu uses four-view images and represents the 3D model through learning implicit function. Our method achieves the best performance among these methods because VSR can refine the coarse results of MF-PIFu. Both MF-PIFu and VSR in our method extract multi-scale features and learn the implicit function from multi-view images. The coarse-to-fine manner is an efficient way to obtain better models. The P2S and Chamfer-{{formula:05a79c4c-0318-4029-a348-3b440fbe10c2}} are the smallest in our method, which means that the results of our method are more accurate. The IoU of our method is the highest, which means that the estimated 3D models are more complete.
For the Articulated dataset, Table REF shows similar conclusion. The SPIN and DeepHuman achieve similar level on the real dataset and PIFu is better than the above two methods. However, our method also achieves the smallest P2S and Chamfer-{{formula:452e6277-7fec-437f-991c-793cc0c790f7}} and the highest IoU on the Articulated dataset. The two tables demonstrate that our method had good performance on both synthetic and real datasets.
{{table:69b14260-23dc-475e-b598-06df3dbbfec3}}{{table:34a1e41c-9c69-4e6d-a6ae-b9643cba90f2}}{{figure:75438782-e561-4da7-be87-1f6bf713a68e}} | r | 0e21989110a0682b0aca41c49f49b153 |
In this section we state the interior regularity results used in the body of the paper. Even though the results are not new, we provide the proofs since the claims are adapted to our specific setting.
For overview of the theory of parabolic second order equations, we refer to {{cite:4aa48704e09a1653b9a391c92f2fb7961c658a98}}, {{cite:da5326c06e249801767b4aeff89757189a5cd6a6}}.
| r | 920fcb4de0d94ea75335db11cdd119d0 |
Most current methods to train object detection systems assume strong supervision {{cite:f4f896b7ead257866a89c918e39fe64ce5f58087}}, {{cite:4906abbc97f35d8b2c0e3f962363ead73cb7523e}}, {{cite:b444eda5815ea84b0eca6ce433e93cb07ced0e01}}. Providing both the bounding boxes and their labels as annotations for each object, still renders such methods more powerful than their weakly supervised counterparts. Although the availability of larger sets of training data is advantageous for the training of convolutional neural networks (CNNs), weak supervision as a means of producing those has only been embraced to a limited degree.
| i | 2120fa94a4b0a0646b9de1c47e4c75bb |
where {{formula:b9780992-1e74-4356-a8da-ca674a2f82bb}} is the iteration number.
Note that the update of {{formula:5661b5b3-bbfc-4da3-aa76-f54b3919ff23}} is a gradient ascent on {{formula:dc5b0bfb-5eeb-41b0-86b6-89d2250eb5dc}} with stepsize {{formula:ec3b5465-0421-4646-992b-b7a7878d100e}} . Eckstein and Bertsekas {{cite:8ab4a3803208070a3359b269074bcd598ced3846}} showed that ADMM is equivalent to a firmly non-expansive operator, which enables the application of theorems closely related to the well-known Banach fixed-point theorem.
As a consequence, despite the split, ADMM converges for any {{formula:ee65503d-a47d-4388-b7d6-178c89511dad}} {{formula:ec571bc3-0569-457f-99f5-1c5736b4105d}} 0 under relatively mild conditions. In particular, there is no need for {{formula:00b72c3f-91f3-4891-b8ff-c6311fc3f531}} or {{formula:ad6f857f-d64d-4a24-a772-34dbdceeda2c}} to be differentiable or strictly convex or even finite (i.e. they can assume the value {{formula:ed9d55f9-0af6-4296-b1e5-b953372d1ad9}} ) – they only need to be convex, as well as closed and proper. Further, the {{formula:a8b52dc4-78b3-495f-9cce-5374f8245699}} and {{formula:33346396-0b83-4277-86ea-1ee81ac8c369}} updates do not even have to be exact but can be approximate to some degree. Since the initial proof {{cite:8ab4a3803208070a3359b269074bcd598ced3846}}, convergence results were greatly extended, see {{cite:21a06586dec846d0896da4eea50177e92d012c82}}, {{cite:4a5f17c82add91a61a154f94011be5736569dbd4}}, {{cite:a684328dc621c5440ffe37425007413732dd299c}} and references therein.
| m | 3704c2eee77c8fc8a2f6368856aa3ac4 |
CBC is not just a possible boundary condition for AdS/BCFT, it is actually
a very interesting class of boundary condition for a good reason.
At the quantum level, one hopes the boundary condition of gravity to be
elliptic so that
it leads to a well-defined perturbation theory of `quantum gravity'
{{cite:71f6914d1a1214bbd2507ced20c2dec6506cc871}}. According to Witten {{cite:71f6914d1a1214bbd2507ced20c2dec6506cc871}},
in general DBC is not
elliptic and does not lead to a well-defined perturbation
theory. It is better-behaved if the extrinsic curvature of the
boundary is
positive- or negative-definite. This additional
condition indeed plays an important role in AdS/BCFT with DBC, which
helps
to select the correct solutions with positive brane tensions
{{cite:35a18ae3c79dae0cc7a35e5c5a8dcfa340653ebb}}. On the other hand, CBC is
always
elliptic and leads to
a well-defined perturbation theory {{cite:71f6914d1a1214bbd2507ced20c2dec6506cc871}}. Thus it is
interesting to
consider AdS/BCFT with CBC and investigate the properties of the gravitational
perturbations in this context. This is the main
motivation of this paper.
| i | 4ea30d1e310cf4b8ea863f9abadfb197 |
We can also see in Table REF that the coverage probabilities for the multiple imputation approach offers a marked improvement over the alternatives. This is particularly apparent when we average across the 201 pointwise values of the ERF, although the coverage probabilities are still imperfect given the previously noted limitations of LOESS and the initial dispersion underestimation characteristic of BART {{cite:d9225400a43645eae7389092c8e623b6ff78cd00}}, {{cite:7baa253e14b06366e671d44b9cd1978adcd46911}}, {{cite:d102844d59ffd40ce7b57aa31fc2ce88ab13f68b}}. The coverage probability improvement is also relevant for the single pointwise evaluation at {{formula:b6968ed5-76c4-45c0-95c9-f36e1b6b8497}} , particularly when {{formula:8536234a-c501-4e10-b7f2-fcb6649650ce}} and {{formula:b777856a-aa2f-448e-8b59-f81ed7a11961}} are large. The improved coverage probabilities produced by the multiple imputation approach demonstrates our method's ability to better compensate for the added uncertainty introduced by measurement error.
| r | ee89732ba46f5b193e02739932517032 |
Therefore, the well-known purely Jordanian twists {{formula:443be81f-4b47-4d4a-b437-eae9eae8f2fd}} and {{formula:af16ef29-2cb2-4557-a291-8233480c273b}} remain only unitary with respect to the conjugation
(REF ).
This situation can change, if we use the method proposed by S. Majid {{cite:c89d41cd4bbd7af533836d0b892cff2f4ac8b717}} and admit deformation of the original {{formula:4ec595b8-ceaa-496a-a530-d80ae6cbaf81}} -structure. Before, one needs to check if the twist satisfies the following condition
{{formula:7c1acb04-33ec-49c0-bc74-7d864ff532f9}}
| d | 06742372c3744023ac2653c232f84bc5 |
In this work we use cycle number ({{formula:a671d416-fbd3-4967-87fd-2770fc12e4ad}} ), cell current ({{formula:006d4ad5-c876-45d2-b660-3a0f96def516}} ), cell terminal voltage ({{formula:da8ab583-ed25-4173-a61b-f7ec6b3fabe9}} ) and temperature as inputs. Although it’s typical to quote SoC as a degradation stress factor (usually because at rest, {{formula:85f120ba-1595-479c-94b4-f0906c7ed229}} is taken to be equivalent to open circuit voltage, {{formula:1278b8f9-fb3e-4883-ba4a-366c23c784e0}} (SoC)), at an electrochemical level it is the electrode overpotentials ({{formula:c39241b3-6c3a-4827-9ae4-2a46166397e2}} and {{formula:4e7c3568-f6d9-46d7-b2de-d81eb31d302f}} ) that govern the growth of the solid electrolyte interphase, the dominant mode of LiB degradation {{cite:1dfe9141124fb1c0ab86c3aaf65091c725e4a45e}}, {{cite:90b3bf2e941e1d18b54178fefeaa543be9a20f62}}. {{formula:e25b0025-32a8-4d20-92b8-4f21fd421d01}} is defined by {{formula:378910dc-31af-4c81-bce7-53b771fea149}} and {{formula:c5d5a594-f3d9-4629-8092-2f9e3b02f15a}} and so it is appropriate to use {{formula:b88a17d9-34f5-4e13-a909-ce87fe5c3a88}} as an input.
It is worth noting that while open circuit voltage {{formula:2809a71c-018f-434c-aa9d-2cac4bf1831a}} can be used as a proxy for SoC({{formula:f1614d45-e805-4b35-8474-d387a7662a53}} ), {{formula:f45f4527-2ddc-4ce4-9194-f31cb52d0e59}} cannot. This is because, even if we were to neglect diffusion and polarisation effects, the correspondence of {{formula:e1cd59af-4697-4913-98fa-fa07ae39521d}} to {{formula:8edc6cd7-c5e9-4b2c-965e-fa7af9eca3c6}} is convoluted by a nonlinear {{formula:fbe57b5a-ff71-4d4d-8bfb-2e1c15ff361c}} term (in a circuit analogy framework), where {{formula:3e1166d0-05bf-4894-bdff-b564fc7d47ba}} is the cell’s Ohmic resistance. Given that the Bole et al. {{cite:d5f629538170ba00bbca158b1125600de240765d}} dataset includes slow rate discharge and charge data (which can be used as an approximate for {{formula:b019f7ed-1557-4e4a-8e3c-6f05ae462a51}} (SoC)) and current pulse data, an equivalent circuit model coupled with an associated filter can be developed to estimate SoC {{cite:48d223e36d3af911bcaf77615759370148d2a3d1}}.
| r | b15a7915bebdf4723fadf5cc6c0f62e1 |
From Table REF , we can observe that under the similar amount of parameters and FLOPs, our FASeg consistently outperforms the SOTA methods. Specifically, FASeg with conditional {{formula:ba55d81b-f0fb-47a3-b607-36eabf5537f2}} achieves black48.3%, 49.6%, and 56.3% mIoU for single-scale testing and 49.3%, 51.3%, and 57.7% for multi-scale testing on R50, Swin-T and Swin-L backbone, respectively, reaching a new SOTA performance. We can also observe that FASeg achieves
significant performance gain over Mask2former {{cite:17092ec639e92a2a464eca3773f4b7b1c58f45f5}} with only marginally increased number of parameters and FLOPs. For instance, FASeg with conditional {{formula:c02a4408-7fa0-46d1-bec9-34315f544654}} and Swin-T backbone outperforms the Mask2former counterpart for single-scale and multi-scale inference respectively by black1.9% mIoU and 1.7% mIoU with only 1G extra FLOPs, demonstrating the superiority of our FASeg. We also visualize some qualitative results on ADE20K val {{cite:adb34ffd531484d96fc676a075b8808c3b88376a}} (with Swin-L backbone, single-scale inference) in Figure REF , where FASeg shows finer details and more accurate classification predictions than Mask2former {{cite:17092ec639e92a2a464eca3773f4b7b1c58f45f5}}.
| r | e787f900a92b5502ace03b0c2dbdfc85 |
The work can be considered as an extension of {{cite:742c0faa4feed361506d3c4e26b581acf990e5b5}} and {{cite:e5649b82e8a21081b99cd9aebfd84803eb32ba8e}},
where the authors found that the density weighting scheme can significantly
increase the amount of statistical information of the standard two point analysis.
We show that the gradient weighted scheme can be complementary and even more powerful.
| d | d56eaec1b8e7c24aa7b886fa40215608 |
The loss function should be differentiable for effective backpropagation with gradient descent {{cite:764cc639e612f61f196bc4df9ae76c4483e8e388}}.
The trade-off between model genralisability (small batch size, {{cite:280c5e2473d586cd227d772244a5577e7b92bcb5}}) and output distribution evaluation precision (large batch size, {{cite:ee99bff7542b0616eb2a879bf047b2fc9a8ebf57}}) should be considered.
| m | fcdf3c992b0dc912be401c77330c629c |
where {{formula:3a057430-7a3d-441f-9488-8bbd78a0fe2e}} represents Meijer G-function {{cite:7faeb959e07908a43148ffac03045ff1ee2c0c9c}} and {{formula:b5ec4936-857c-4d69-a4a0-47c549af8bdd}} .
| m | 1d631aed19f9ee0bf51491e4c06920ba |
For the 2D experiments, we split the dataset into training and testing sets including 5 subjects each. Each image frame in the sequence is treated as an individual image, yielding a total of 150 images per set. Note that typically, a portion of training data is treated as a validation set utilised for early-stopping {{cite:c9dc06a5bf7bf4ee87429c9e4e4174f6e168ebc4}}, where one halts training if the validation error starts to increase. Initially, we used 3-2-5 split for training, validation and testing. However, even after 3 days of training cascade networks, we did not observe any decrease in the validation error. Therefore, we instead included the validation set in the training to further improve the performance but fix the number of backpropagation to be an order of {{formula:406c73c5-341f-4335-bbe8-d9f7f7803510}} , which we empirically found to be sufficient. For the dynamic experiments, we used 7-3 split for training and testing and an order of {{formula:46fc403e-0911-4a81-b534-a771958c19f7}} for the number of backpropagation.
| m | d8acd1ebd1726be8de5d1795aebdb8bc |
Our results with DeepSNR demonstrate, for the first time, the feasibility of using ML to enable the potential discovery of previously unobserved GW signals in offline detection pipelines in experiments at LIGO. It must be cautioned, however, that DeepSNR is only a foundation for a fully MF-free offline GW-detection pipeline. Additional machinery is required to perform the ultimate goal of significance quantification as it is done in current LIGO GW searches. A helpful illustration of the PyCBC offline detection pipeline can be found in Ref. {{cite:1798d60a64d56a0bb82ef266912b90ce217bc1dc}}. In particular, the ML classifier of DeepSNR has no mechanism to ascertain the compatibility of candidate waveforms from different detectors—the so-called coincidence requirement {{cite:75722251383576b281698526dde1571eac93b2c0}}, {{cite:03c516c3cd98a578b3e48b7b75c47541e282e510}}. Even though DeepSNR exploits information from both detectors for signal classification, it does not explicitly require compatibility. Waveform compatibility is an important and robust layer of protection against false alarms provided by the MF algorithm. The matched templates from the MF scan serve as handles for comparing both the arrival times and BBH parameters of candidate waveforms between detectors. Moreover, the coincidence rate of matching templates between detectors is used as a baseline for measuring the real-world significance of a GW signal candidate. An ML-based counterpart for this step can be achieved using a parameter regressor similar to those developed in Refs. {{cite:cafd2a40e71089cf8569c9326aeb27febd6f4cd6}}, {{cite:2dffb4cc424ab8bda85a7c21d6cd5fa9d2f535d8}}, {{cite:c3730a9f3da8b3fd5332b6581033fc0e961ef6f1}}. Signal candidates identified by DeepSNR would then be run through the ML parameter regressor to determine the arrival times, best-fit parameters, and associated confidence intervals of the waveforms from different detectors. An equivalent coincidence requirement in the ML-based scheme can then be achieved by requesting consistent arrival times and overlapping confidence intervals for the regressed parameters. A version of this workflow was described in Ref. {{cite:0131a2dcb5e8dc599ef9a9b9b743a65d0ac07c59}}. After this step, clusters of identified signal candidates passing and failing the compatibility requirements can then be used to build a distribution for estimating the final background and thus the detection significance. We expect that existing LIGO analysis techniques developed for other SNR-like detection will be usable without major modification. The investigation of such a fully ML-based offline detection pipeline is planned for future work.
| d | b90a1b4730595915d119558dd782656c |
First, we consider the reference SPGE propagation model and the source parameters which best describes the energy spectrum and {{formula:32b5a488-d480-4b44-ad85-f8b75e9827c9}} distributions measured by the Pierre Auger Observatory under LI assumption for that scenario: {{formula:3c4ea279-f640-4173-a595-cef222b9defc}} , {{formula:10ace0f8-2878-4abc-9cc5-db2a3a12b579}} , {{formula:1612f31a-ace4-46ef-9390-27b05fdf9fc3}} , {{formula:389133ed-9672-4b38-ab6b-dce31831ad17}} , {{formula:ceff0912-3ab0-469b-83cc-e5906c056850}} , {{formula:f8c85319-f21e-488f-8bff-36adfb3b6203}} , {{formula:cbf6e983-49e5-4bd3-9260-2ccdf168d103}} and {{formula:ec6fba3a-b3d5-40d4-90d1-b99cb7087ffd}} , as found in {{cite:93c4a6aa44fc15aeea7320896def04747cf30f1f}}. Using this UHECR scenario, we simulated the GZK photons arriving at Earth under LIV assumption.
| r | fce2ec0829514340e95d860f8a6215e4 |
Recently, several works {{cite:5b52f83c420b163c7d08b5d128a9626cbe40a7e1}}, {{cite:d0f7b1e4893ae66967707b7075221d1fa5c91800}}, {{cite:4b354e5d01e59c4eb88f765e7136ee6eed9b0370}} have focused on hardware friendly quantization schemes.
Namely, that their quantizers are uniform, symmetric and with power-of-two thresholds.
Such quantizers optimize computational costs as they allow integer arithmetic without any cross-terms due to zero-points and floating-point scaling {{cite:5b52f83c420b163c7d08b5d128a9626cbe40a7e1}}.
| i | 0c6987b28abcbcda8a4107794a5a58bb |
There are no primordial binaries in our models.
Observations have shown that massive stars in open clusters are most likely all in multiple systems {{cite:b95e997c7b24785b531dc55e641cda0c3a948d0b}}, {{cite:715679f97ccf30ef53e597b2e61acaf7d63e7024}}, {{cite:25459d2c5f0f63b81e88c5df8cf9cac39330ae24}}.
The dynamical interactions of these multiple systems can result in ejections and mergers {{cite:83f3999c19d0059e5a0ed545c3dd0f4da9b8fa0c}}.
As a result, the numbers of retained OB stars and formed BHs decrease.
In addition, the primordial binaries can also attend the binary heating at the early phase that affects the timescale of core collapse {{cite:a4e5d2f0380edf25ff85e41e52a7a6c52cd000c0}}.
All of these subsequently affect the long-term dynamical evolution of clusters.
In the future work, we will also consider the impact from primordial binaries.
| d | b8bbdcdd625bd8ca8dd541796efcc9ab |
{{cite:b89625f3a3dd36b4914af173315fc6ecce2b2c26}} The proposed method: CROPTD Compared to: Faster R-CNN, DA Faster R-CNN and Strong-Weak ResNet-101 {{cite:1ae4ed72926986e4b5142deb4b1df4cd2ce0bf3f}} QuickBird, Google Earth: satellite imagery P : 90.06%, R: 90.87%, F1-score: 90.35% Detection of oil palm
| r | 5dfee08b80eb411f6d6168b50f481ac7 |
Unlike (REF ), the left-hand side of (REF ) will not be zero even if the random variables {{formula:c686e3da-5f23-47f0-9255-4a5264be6fd8}} are
Gaussian, as long as {{formula:d87e7ad4-1845-4a28-9c57-e199557d8b47}} is rich enough to capture the infinite-dimensional nature of the problem. In fact, for certain {{formula:209e56f4-ba5f-457e-bcdd-be3e13c7afde}} , one can
use (REF ) to define a metric on the space of distributions. For example, if {{formula:f7a1392a-3415-4276-a8d2-cd1be606fa3d}} is the collection of bounded Lipschitz continuous functions,
then convergence in the corresponding bounded Lipschitz metric is equivalent to weak convergence, that is, convergence in the Lévy-Prokhorov metric; cf.
{{cite:4f1e48c97bf2725877b655459dcea5d063ca525b}} or {{cite:475d52cfcaab5e0e69851ff73fe4ac7be3deb283}}.
Removing the boundedness condition (for example, to include linear functionals) leads to the Wasserstein-1 metric, which is the subject of this paper.
| i | 6f653085b6ae2a144b74ddd589764e13 |
Let {{formula:3532feec-3216-452a-9a42-b2c19bdd4c10}} be the family of quadratic vector fields given in {{formula:1e1b8afd-71d9-4481-a361-6bfe4565d210}} and consider the period function of the center at the origin. Then the following assertions are true:
{{formula:ce5c83ee-a91b-4695-84cb-5ecd776df675}} for {{formula:17e9ef07-9776-457d-8926-dc84268cd527}} .
{{formula:4ce48d87-0077-411a-b6fc-b9a95edf3ac0}} for {{formula:9f2a21fb-6fef-4a0a-b6ef-3bd83888163f}} .
{{formula:3c4e2d0e-5c93-414c-86af-0e4cfcbee3e6}} for {{formula:d45fddff-ae69-46ea-a069-a6c8d999f9a5}} .
There is a curve of local bifurcation values of the period function at the interior arriving to {{formula:71fdecea-0a29-4f03-97b6-bf4d43c1d72c}}
tangent to {{formula:85338698-1fc1-4752-90c6-cc96cda79d93}} .
As a matter of fact to show the existence of a uniform bound for the number of critical periodic orbits of the reversible quadratic centers it suffices to verify that {{formula:2e84dbad-d060-4dbd-8c0c-c1f4deca37b0}} is finite for all
{{formula:d13afd2a-1617-43ab-9d35-d32e9fe364c6}} inside the segments {{formula:09e171dd-5754-48de-8949-0daa08ad30d7}} and {{formula:91b5e88f-b8b4-4967-b3e3-87257115c25f}} .
To put this into context let us recall that the differential system {{formula:ecb074a7-d176-4eb8-9ded-fbc05214839e}} has no critical periodic orbits if {{formula:c998f2db-46a8-433e-b429-96de11eb9eb4}} by {{cite:15d0b9c2bb9622e61269af4aee35174fc3567751}}. On the other hand, apart from the reversible one, there are essentially three other families of quadratic centers: the Hamiltonian, the codimension four {{formula:ccb13773-bc80-42a5-91df-a8b140995db7}} and the generalized Lotka-Volterra systems {{formula:f2fe7917-2310-43c5-a07d-fe3779e91704}} According to Chicone's conjecture the number of critical periodic orbits should be zero for the centers in these three families. This is known to be true for the Hamiltonian and {{formula:ddfebf5c-654b-4690-8aa3-8da769296f54}} families thanks to the results of Coppel and Gavrilov {{cite:932adc66fcbcb88f87f96d43292d9d7cc85fed35}} and Zhao {{cite:532689916d50add8317f54eceffbba7c9515ae39}}, respectively. With regard to the family {{formula:6b7042aa-3168-47b9-b140-bb13c5cba7a4}} it is proved in {{cite:b83d96bdf15486dc08cdc597b9f37b6922128f73}} that, except for a subset of codimension one in the parameter plane, the criticality at the outer boundary is zero. It is clear then that any contribution to the proof of the above conjecture will constitute a very significant step forward to the existence of a uniform bound for the number of critical periodic orbits in the whole family of quadratic centers. Let us mention in this respect that along {{formula:9e159b98-2c91-4541-9574-ab27534c340f}} and {{formula:3de9c7d9-63d0-4fe0-baf8-6a508aa52a0e}} the singularity at the outer boundary of the period annulus is nilpotent. In this situation the results of {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} do not apply and new techniques must be developed.
The paper is organized in the following way. In Section we recall the definition of local bifurcation value at the outer boundary, that we introduce in our early paper {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} to study the bifurcation diagram of the period function of the family {{formula:fbe18e51-0600-497d-b8ec-09adf1826eb9}} , and we prove several results that relate it with the criticality. We also show how to study the criticality by means of a suitable parametrization of the set of periodic orbits near the outer boundary. Section is devoted to the asymptotic expansion of the period function near the outer boundary, which is the cornerstone in the proof of Theorem REF . To this end we prove three results that are addressed to three different parameter subsets according to the contour of the period annulus. As one might expect the proofs of these results are rather long and technical. Furthermore they are based on previous tools developed in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} that need to be introduced appropriately. For these reasons, to ease the paper's readability we defer some proofs to Appendix . In Section we study three distinguished parameters. On one hand the two isochrones for which we succeed in proving that the criticality is one (see Propositions REF and REF ) and, on the other hand, the parameter {{formula:056b8c50-661e-40ce-819b-2212b9a0dadf}} , which is also rather special because it has criticality two (see Proposition REF ). Due to the novel approach of its proof we think that each one of these results is of particular interest in the context of Hilbert's 16th Problem. Section is entirely devoted to the proof of Theorem REF . Next, in Appendix we prove the results stated in Section that we mentioned before and in Appendix we are concerned with the integral representation of the Beta and hypergeometric functions, which usually appear as coefficients in the asymptotic expansions that we obtain. Finally Appendix is addressed to prove a technical result that is used to study the vanishing set of two coefficients.
Criticality vs bifurcation
In this section we recap the notion of local bifurcation value of the period function at the outer boundary as we introduced in our early paper {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}. We relate it with the criticality, which is a more quantitative and geometric definition, and prove a general result connecting both notions, see Lemma REF . More specifically our aim is to take advantage in the present paper of the results that we obtained in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} with regard to the period function of Loud's centers {{formula:fc71876d-9741-4d0b-a513-0351b3ff490e}} and that are not stated using the notion of criticality. Related with this issue, our goal in this section is also to clarify the usage of a parametrization of the period function near the outer boundary to compute its criticality, see Lemma REF . Finally we give a sufficient condition in order that a parameter is a local bifurcation value of the period function at the interior, see Lemma REF .
Several results in this section are equally valid in the finitely smooth class {{formula:55171495-37d9-47bf-aec9-33807a0338eb}} , {{formula:db965c92-c53d-42cb-ab5e-191ef0624a10}} the infinitely smooth class {{formula:e1ee4acf-123a-489b-b788-abb61bf4fe97}} and the analytic class {{formula:d2d6712d-0683-4f9e-8929-62267a1bd542}} . For simplicity in the exposition we write {{formula:ff3d1ba5-4351-4e60-b1ce-641d18b2f5d3}} with the wild card {{formula:ca749abc-ba3e-459b-9d13-16ef009eac3b}} Our first result is addressed to the regularity properties of the
map {{formula:10799ac5-6459-4826-8b7d-c52f55f67672}} that assigns to each {{formula:916366aa-1464-4a37-a77e-e74f907624d8}} and {{formula:d8d3a22f-144e-40af-ba8e-d576a0fc74f5}} the period {{formula:48c7b713-6e25-4200-81fc-2d459480be61}} of the periodic orbit of {{formula:6bed78d4-94c0-47d6-a936-260f2d630885}} passing through the point {{formula:850acc10-55a1-42f2-b74f-bd1801a178f1}} The result is given under a technical assumption concerning the existence of a continuous parametrization {{formula:b878d346-aa52-4abc-8d19-cecb8169af55}} of the period annulus {{formula:710d1d37-75d0-49ac-8d53-dca553ff50f1}} near its outer boundary {{formula:d41ce2f4-69a1-4365-9aa9-a2d25d4987fc}} . We point out that from now on, in contrast with the notation used in the introduction, for the sake of convenience {{formula:3b1081f9-1441-4ed1-abaa-fd6e397282df}} corresponds to {{formula:018e26a8-a09e-4fcb-9e2e-7f4d55554cbc}} and {{formula:dc3c265f-7816-40fb-bf7c-227875286dc5}} to the center.
Lemma 2.1
Let us fix {{formula:c124a294-15eb-478a-a2d6-2c580c277c9d}} and consider a {{formula:24f64bf4-1bdb-4f41-85fb-ec825a3fa0eb}} family of planar vector fields {{formula:6522e81c-c726-41d0-88a2-10b56e55fc21}} such that, for each {{formula:4bdd5ea4-f8b9-4b15-86ee-a10efb30e2ff}} , {{formula:91b9f623-db9f-452c-b83c-1a733b9f069d}} has a center {{formula:df47f225-dc97-4212-996f-3c5dac2334f0}} with period annulus {{formula:0cf36261-6d56-4c0b-81ef-677fe9f5c09d}} . Suppose that there exists a continuous map {{formula:4a315c8f-252f-4715-a040-0c37a501d7d2}} verifying, for each fixed {{formula:a53dc3cc-5d51-4192-8dcd-a58986e75790}} , that
the map {{formula:9801e1ea-1c05-4a2c-9239-46e57b0f445d}} is {{formula:9511c751-b694-4a45-ae01-47eea19a8e79}} ,
the vectors {{formula:b6ee9103-5957-4057-9cae-23dca5fe7803}} and {{formula:46bba0a5-7f11-402d-b066-5100f05c8b6b}} are linearly independent
for all {{formula:2bcb5ce1-5554-4935-8af5-c0411e2c8d6b}} and
for each compact set {{formula:864b3b7a-6a1a-4dea-b5de-bd5c9a787ade}} there exists {{formula:a754a9e2-ada4-4128-8869-62f58960d92e}} such that
{{formula:ea889016-8761-4e33-b371-9dc2653ea294}} for all {{formula:a8e0ddc7-4c31-49a3-9705-65b3a4b9accb}}
Then the following assertions hold:
{{formula:a42fe502-422c-47d2-8a53-d07e1cea4e6a}} is an open subset of {{formula:1b75c610-2e0e-487b-accd-c0aae4d04a33}} , and
the map {{formula:0c688f6a-377d-4eb3-b39a-88c42eef94d9}}
is {{formula:cdec9e61-52a5-4765-a7c7-ace633807909}} on {{formula:dbf86a3f-245f-47b0-b74d-be85ca7c441a}}
Proof.
We consider the family {{formula:70756ab0-688b-44f0-831e-c65285fdcfca}} as a single {{formula:845903e4-606d-467b-847d-df70863287b8}} vector field {{formula:5036afd8-fc60-47d2-a699-18d78cb02bcd}} on {{formula:bbdd3098-ea41-4b98-8a1f-92db940eb8d7}} whose trajectories are contained in the submanifolds {{formula:314b5f9d-7800-45a0-a58f-979691e97afe}} . Denote the flow of {{formula:431f5510-36e8-4b47-a5f9-6ca8a5a6e8f6}} by {{formula:64b3e7eb-0e64-4fd5-9011-2905b8c4790f}} . In order to prove the first assertion, for a given {{formula:3aa57b62-357c-4518-80c9-17454ac60ab1}} we must show that there is an open subset {{formula:0c53c060-43ee-426f-863e-9399fcf2a799}} of {{formula:10cdc86c-24e7-482f-867b-5e5751e34a47}} such that {{formula:19bf9570-b7e1-4c1d-be0e-31249d4aecf9}} We claim that this is true in the particular case that there exists {{formula:c81473b5-acc7-4de9-8bc9-76e4611ab4ef}} such that {{formula:ad5aad3f-b581-4355-bc23-15e441a46fd8}} Indeed, due to the assumption in {{formula:42e7ac24-fa33-4a30-8887-2dc579459564}} , note that {{formula:be0223c4-ce89-4a10-a150-c5ff411b57d4}} is transverse to
{{formula:a51aee67-2b5d-4325-a783-7410d113705a}}
for all {{formula:8f278c08-a0b1-433b-9b23-9f0f23d39b1c}} small enough and that {{formula:4d8f94c4-e1ac-454a-bd63-a74879c8c70b}}
Then, since {{formula:6e9e4e80-0f8a-480d-a16e-da5f8fafbfdc}} is continuous, by the flow box theorem (and shrinking {{formula:b425a419-82c2-4a5b-ac8c-ea0157025c59}} if necessary) it follows that
{{formula:cc90613c-8ca4-4c48-8174-2bce4541a962}}
is an open subset of {{formula:b51f888d-438b-46d3-af16-dbb1ee2d3403}} . Furthermore, since {{formula:8d8c5330-6e61-442d-8232-f59da79256be}} is invariant by {{formula:7d23e277-c499-4ee1-ae84-f2bfd6e95bcd}} and {{formula:610b54a3-c635-4313-a12a-675cb1c439ce}} by construction, we have that {{formula:360ffd07-a422-458b-a8dc-5204789d7d09}} and this proves the claim. Let us consider now an arbitrary {{formula:61a13360-d956-4cbf-a24b-646c0c520a91}} . Denote the periodic orbits of {{formula:e940d738-a59d-4a50-a596-034dd29889e4}} passing through {{formula:2c8b8775-7d45-4428-81f8-481f7ff0f9bb}} and {{formula:4f7fb412-8bff-48c2-ba37-751e173d7dbb}} by {{formula:9d4dc8f3-40b5-427e-865e-dca3584527ed}} and {{formula:080a1d73-f5a3-4527-8ba6-a119d01cd5ec}} , respectively. For each {{formula:02442835-5fe7-4cb8-9f4b-99d5844f4347}} we take the orthogonal vector field to {{formula:df9f36b7-ba14-471c-9d6b-5eb34fd5538c}} , say {{formula:915af4c5-d9a6-49b5-8b7c-b727a4f74602}} , pointing inward the periodic orbits in {{formula:b7680241-59cc-4d04-8791-8bb0326aa91d}} . We consider the family {{formula:e1260a49-c84e-4f55-b107-2c7e2a9a543e}} as a single {{formula:04f09fae-f13f-442f-b89d-a54f8961529c}} vector field {{formula:558b74f1-7beb-447d-b945-1104631443ad}} on {{formula:f4bfb3f3-566e-4d1f-909d-51a8a39c1b45}} and denote its flow by {{formula:49d116ac-4d12-4f82-a101-d33e260000e8}} .
Note that {{formula:6497c226-c37a-4879-b0ef-d8635fccaf51}} is also a singular point for {{formula:5d6c08a3-8855-4bbb-b11c-b619e573103b}} that, by applying the Poincaré-Bendixson Theorem (see for instance {{cite:26076f4ccf158d95a2bd6db4360d3d70ae043e1c}}), it is easy to show to be asymptotically stable. Observe moreover that {{formula:a5ad7fb5-c9b6-4261-b4a3-e84985a4ffc6}} for all {{formula:5d82bcbd-e7af-4e2c-8958-9177b9a50f48}} We define {{formula:7c89f0f9-18e5-4706-8108-eab9200e0815}} which is clearly a transverse section for {{formula:1688ec1e-7ab6-4d7e-aa5b-ab54fe34ce76}} , and distinguish two cases:
Case 1: {{formula:81371993-5c8b-4599-9c48-f5a0b6a726dd}} . In this case there exist {{formula:5057d5e7-31cf-432f-b3a3-9d7db9afc0cf}} such that
{{formula:c314f6d2-eed5-4085-8c9d-0923dceb7112}} Since {{formula:d44c2f86-c3b5-44b2-aae6-ae4c6dcba079}}
on account of the claim we can take
an open neighbourhood {{formula:61cbc82d-c0d4-4234-9835-6441da70a926}} of
{{formula:070843e0-5d13-46e3-891c-83d1e4491349}} in {{formula:df85877a-9f13-436d-9b36-949279c04b2d}} with {{formula:c6cd95fb-7875-4271-b81f-27c53492aa7d}}
Then, by the continuity of solutions
with respect to initial conditions, there exists an open neighbourhood {{formula:ab951ff3-2ff0-43c3-b374-18bfa5441f3f}} of {{formula:22added5-2ad8-49cb-be4e-c1e2ca7524ff}} such that
{{formula:f3ad3baa-ebcc-4e36-9382-f3671b0a016c}} Thus
{{formula:00a6dc67-8d3e-4bbd-b800-906585708cbd}} where the second inclusion follows due to
the {{formula:cfd57f7b-b0ad-44a8-a99d-0e0a4d1ea930}} for all {{formula:bcea186a-e54b-428a-9931-353f04e7cb9e}} and
{{formula:50c58ddb-5410-4d8b-8b00-8ba6b4a7b349}} for all {{formula:f662a52f-b74d-4f75-bdd5-4549005a30c8}} together with the fact that {{formula:a7588c6d-9795-4994-aad9-1c3d3ef587c8}}
Case 2: {{formula:23fffb12-e637-4de1-b629-0b837691ce2e}} . Note that in this case
{{formula:ee7e6ebb-c040-4436-a332-ff6015da87ed}} . (Here, given a Jordan curve {{formula:1d07cedc-3df1-4486-b8b4-2096147dbac6}} , {{formula:55332672-a30d-427f-a429-656dae378a9d}} denotes the bounded connected component of {{formula:874a8e59-6884-4b47-a5b8-f35dbdc8d462}} .) Moreover, by the assumption in {{formula:e8fed98d-67f3-4683-99e8-62be3b9c8d14}} and taking
{{formula:54b5c667-1224-4d48-bd09-32b2859ee136}} ,
there exists {{formula:fee62eee-aa21-4a80-8a41-7e734c8ad553}} satisfying that {{formula:b91f923e-3c54-4cf5-bd68-f8ef2fe61285}} Therefore, since
{{formula:0ce88ef7-dc3a-404e-8d3c-28e370346234}} , by continuity there exists {{formula:09e6aa3b-161c-41da-a62d-1a47fa28b2d7}} such that
{{formula:67a3e494-38f7-4e79-a02b-8e5aa9f3ac66}} Consequently
{{formula:544e2905-1eea-4bf4-b9c4-41792d085951}} for some {{formula:7fc51ded-3e74-4c90-bed7-545be35c72f1}} and on the other hand,
again on account of the claim,
there exists an open neighbourhood {{formula:f5dac669-3457-4bb4-9fcf-add9cde8ffc3}} of
{{formula:25dfd776-2934-4152-8914-689d64b38e87}} in {{formula:dbaad152-75e9-4491-9591-39573d45ff18}} with {{formula:724af634-be04-48f8-981a-03aeaa20edec}} . Thus, exactly as before,
by continuity
of solutions with respect to initial conditions, there is an open neighbourhood {{formula:d828c727-15d2-46c7-b560-60771b7b933c}} of {{formula:ae893729-e34a-426b-838f-068f2beb2b81}}
such that {{formula:511c60e4-d37f-4a57-bfdb-b30ad81fc7a1}}
This proves the validity of the first assertion.
Let us prove now that the function {{formula:c9cb83a6-243f-4907-84d9-e041d31e067f}} defined by
{{formula:e9bd73e1-c699-47c6-b863-0892606c7d04}}
is {{formula:7ee156dc-86e3-4e3c-9d71-7e52182e8c33}} . In what follows we shall use the notation {{formula:976db444-b797-41dc-8033-1ca76e28c956}} for the components of a point of {{formula:cc2413fd-f58f-4f00-bdc2-237b5066b1b3}}
We fix {{formula:7b47fd8e-4f06-4464-9c37-4c98c8422e65}} and suppose that the period of the periodic orbit of {{formula:18a2e2aa-f939-4907-9ebd-c1475a01e338}} passing through {{formula:15c0c404-4588-4849-b27d-cc077a41060c}} is {{formula:8facc935-28b9-4777-ba51-a85b73e4430f}} Then, due to {{formula:df402ce4-9ac8-43a5-94f5-5f5feb297e88}} ,
there is {{formula:d58714f2-0448-4f4a-85c3-c197d5648efe}} such that
{{formula:d0f5c433-2497-47ec-a4b0-80f792cec8c4}}
For simplicity in the exposition let us suppose that {{formula:c7e23f53-1a54-4620-afcf-dfc145d97f59}} In this case we can apply the Implicit Function Theorem to the equation {{formula:314efc3b-cd17-4078-b6a4-c1b34b751073}} at {{formula:f42ddc80-6095-48dc-b986-74ac2a90cf2d}} in order to obtain a {{formula:f64701df-1545-4772-94b4-1996dcd786ac}} positive function {{formula:9853c76e-2c00-4a6a-b97a-5a7e72ce2c15}} in a open neighbourhood {{formula:b88e26b3-75e4-461c-ab37-e2a3716486f2}} of {{formula:9cd9042a-c117-4942-9820-390906f9c894}} verifying {{formula:47fd1034-527b-4c91-bc39-14a28f8b894d}} and
{{formula:8b98add8-a37b-437a-9a7e-f70401b4f147}}
Clearly we can assume that {{formula:aad4e361-555c-49d3-bb67-9fe5e4c0be98}} is a cube {{formula:943b24c7-e4e1-4274-a7f3-7af43e976001}} with center {{formula:bd0afd9f-50ad-4041-8ab2-3253fd38a44e}} and edge length {{formula:129e5c90-1e0d-4f1d-baa0-23936ee1cb7f}} . We diminish {{formula:aba1c2ed-368b-42d8-aaf4-d1ffaacca598}} if necessary so that {{formula:1ede8968-7564-4988-9644-40f71996c46b}} for all {{formula:68a4d336-1db8-4168-a95e-75bc42f45a6c}} . Furthermore, thanks to {{formula:24bb17da-ca0b-46e2-b9f2-59994ff465fb}} together with the continuity of {{formula:6e9bb0f9-975b-46aa-9db8-c421f1849e5a}} and {{formula:f5c541c2-cc1c-4f66-bbdc-36c9fce4b929}} , we can take {{formula:8435c26c-6f99-4e1f-877d-f3a56580538c}} such that
{{formula:dab01a7f-dbf7-4750-b16c-771956907581}}
We claim that {{formula:d34f8b78-442c-4c8b-8579-592e195ede30}} on {{formula:4ee5c151-5700-45b3-bee2-40111fff437d}} . Clearly the claim will follow once we show that
{{formula:33b1e3c1-26c4-4799-bea7-76012dd3d95b}}
By contradiction, suppose that there exists {{formula:a5f100fb-9607-43a0-91bc-e04f0a8158ef}} such that {{formula:4ae46770-7c47-498a-bffe-4000275d8d09}} . Due to {{formula:7226cf8f-df23-4d3b-8a0a-23172435c845}} the trajectory of {{formula:083f6e05-e7e3-4da9-a2cd-2160435dc835}} passing through {{formula:702cb483-4d8e-4d23-8a35-3030b249bd89}} is a periodic orbit which, for simplicity in the exposition, we assume to travel clockwise around the center {{formula:c18f9158-b8ea-4d9a-a12e-16c39363000b}} (the other case follows verbatim). That being said we consider the piece of trajectory
{{formula:e9797eaa-e028-4be4-9010-507963430330}}
Arguing on the phase portrait of {{formula:55824333-3b3e-4f40-8caa-f57c0f51840e}} , due to {{formula:cb328355-3493-49e6-ace9-8ca536cb48c7}} for all {{formula:31cea8f9-3f66-4a85-882f-2f175e8a9e25}} , if {{formula:d5dbb588-2d3d-4fc4-8c99-95c7288153ee}} then interior of the Jordan curve {{formula:6060e8da-b3d9-45a5-a470-afe7addfcf30}} is a positively but not negatively invariant subset of {{formula:0d3be789-b20b-421d-95d1-fda60c0d70fe}} Similarly, if {{formula:c8eef789-f3b1-4163-8322-9eaec0398425}} then we obtain a negatively invariant subset of {{formula:552178a2-f262-41ab-8be8-0fd7f2bdc58a}} which is not positively invariant. In both cases we get a contradiction with the fact that {{formula:ae8c2431-5ffd-4b26-856c-38ed738c02f8}} is foliated by periodic orbits of {{formula:49b73a12-1dd7-42e8-8d67-8699d43d5a45}} and {{formula:1ed105fd-5776-45b5-9953-f7c04b43573a}} Consequently {{formula:df50312e-8d69-498d-a292-ef8e2a2bffef}} for all {{formula:4914271e-5b68-4ee3-9394-0d30f873e84b}} and so the validity of the claim follows. Since {{formula:d55f32e8-767b-4cb3-b1dc-321c590cca3f}} is an open neighbourhood of an arbitrary point of {{formula:7b97a785-158a-4894-b0eb-4f903198ee4e}} and {{formula:c0564ed8-2a59-40d9-80e7-f2e990db0957}} is {{formula:40ffab12-395b-4eb2-9979-831a892031ed}} in {{formula:989c3267-19a4-46b6-b2e7-8d9c04b7ab53}} , the claim implies the second assertion in the statement.
The previous result is addressed to a family {{formula:969e0004-f8ad-43af-8590-21586e397da0}} of vector fields and this is the reason why we require the existence of a local transverse section near the outer boundary of the period annulus {{formula:0f993a97-92bf-4ca8-819a-0e01920a9eca}} that behaves well with respect to parameters. That being said, Lemma REF can be applied to a single vector field {{formula:6f1be3e1-2ac1-4d1b-a8c4-b37b5409af10}} without this requirement because a trajectory of the orthogonal vector field {{formula:2ea1c547-9819-4931-928a-7ce0f93b7a1a}} already provides a transverse section in the whole period annulus. Thus in order to assert that {{formula:ced6996c-e013-40d2-9a71-b8335344373c}} is {{formula:952d1a2a-809a-4dd5-b5b0-23fff070fda3}} on {{formula:79b6f8fb-3486-47a7-80da-4356bb5dc528}} for each fixed {{formula:4c6c0e00-b07e-4c57-b39c-98d074a2cf1f}} , it is not necessary to verify the existence of a continuous map {{formula:b231920f-c062-4068-8585-97b33cdf06a2}} satisfying {{formula:a8f55c9a-5b71-4e69-bf4a-a5fc3932aec9}} , {{formula:61b87630-3003-48fc-bdd7-0f4c97ba7098}} and {{formula:49d66534-33a4-43f8-ac98-286bcad03297}}
Remark 2.2.
If {{formula:f1280d12-5584-470e-8bae-d5a904af7e96}} is a {{formula:d60685bf-a421-4719-9c54-2b26285de40b}} vector field, {{formula:ed466685-bc18-4748-b7a2-d96c41422638}} with a center then the period function {{formula:18b249f5-aae4-4c16-b8fb-8fced1ea2e6f}} is a first integral for the flow of {{formula:0908361f-1994-4a6c-a1e3-64faa54bb8a8}} on the period annulus {{formula:3c86d1ff-47e9-46b8-b590-53961a872a33}} that, by Lemma REF , is {{formula:94752ef6-1b17-4b0b-ba2f-a04786cf39d2}} . Consequently the scalar product {{formula:6e9124bc-9793-4306-8c1a-70e287269597}} is zero for all {{formula:f4a3d312-42d3-4206-a3d1-f88c886f5c12}} .
This implies that if {{formula:e8fa09db-a7ae-49f6-aa7c-e85f7a1c2f4d}} is a critical periodic orbit of {{formula:4498a621-d6bf-42b9-ad0a-963f525a4904}} then the gradient {{formula:b1c9f082-8166-4f35-86e1-2b3194723316}} vanishes on {{formula:af161cc8-049a-45d7-b04d-48677ab2227c}} Indeed, if {{formula:5564188e-3b8b-4161-982a-a8def0c584e5}} is a {{formula:1455026b-dbf2-48a9-bf81-a8a2c6787bc7}} transverse section to {{formula:bb537c6a-f0d3-4876-ad86-3241660f3ca2}} on {{formula:21606121-7c23-486a-8d38-7ecd7cd517e6}} and
{{formula:99eb6a46-e91f-43c6-aa51-845af36fd286}} then {{formula:8fbe8058-51b6-4857-9ff9-c705e1a7d0a6}}
Thus, since {{formula:a8217bf4-0fc2-4bb0-b68b-c9bf4a1a8881}} , the transversality of {{formula:49a9ec83-9796-48ac-92d3-a410daaff0fc}} implies that {{formula:bc004ea6-284f-4fdf-8b64-aadf40b703b0}} if, and only if, {{formula:46ab5bda-3b6a-47eb-85f3-95a719735dbd}} This shows in particular that the condition for {{formula:cf95d7f2-020c-454b-b966-cdf987c2d287}} to be a critical periodic orbit is local and independent of the particular transverse section used to parametrize the set of critical periodic orbits near {{formula:4b56a265-04ff-4823-82f2-213ef98d4aeb}}
{{formula:2b38228c-85e3-45bb-b612-ed4121826ceb}}
We define next the notion that enable us to effectively study the criticality at the outer boundary.
Definition 2.3.
Let {{formula:5e34fbed-85d4-4e8c-8506-f2fa1e0ed667}} be an open set of {{formula:95cc7c4f-8822-4889-87b0-6ec2dae32317}} and consider a family of functions {{formula:e0c59061-18d7-4c7e-9e09-582f3dbec741}} on {{formula:1ab40602-6921-4f97-b06c-9594c5531ce3}} Given any {{formula:61bea597-8e6f-4561-9313-3c795ab49454}} we define {{formula:93e378cd-5d35-474b-a8dd-7a7ff333ecc2}} to be the smallest integer {{formula:2bade86b-f596-40ba-b48a-e4459beb8bbb}} having the property that there exist {{formula:507fa392-91cd-45ce-ab0f-7e5a64b3fd54}} and a neighbourhood {{formula:90ca8c5d-34bf-404d-920a-b4ff342eb4ce}} of {{formula:1a3e8b17-30c9-425d-8803-120fea7fe9f2}} such that for every {{formula:29bdbac0-1e9f-426d-9d49-d43b97797015}} the function {{formula:724c2992-a3eb-42b7-8ced-dac3c41207c3}} has no more than {{formula:c2d01872-daeb-4221-999b-e118932214f6}} isolated zeros on {{formula:4a86aca0-6bcc-4fc7-98db-3364bfea663d}} counted with multiplicities.
{{formula:286af73d-45b1-4095-b412-f454cf08d21e}}
The hypothesis with regard to the local transverse section in our next result are slightly stronger than in the previous one because we require the continuity at {{formula:abe9e9f2-6e97-4b91-a4b1-486a2b91c23d}} and that {{formula:e3a5837a-7b2b-4018-a24d-bf336f050d3a}} belongs to the outer boundary {{formula:09aa2416-832f-417b-a0c2-bac6a9188952}} for all {{formula:87812610-ae40-483c-a019-49edd861ba10}} cf. assumption {{formula:1fcce3f5-878a-4aba-b25b-e020ca1e2c1b}} in Lemma REF . We also remark that in the statement {{formula:e3736a54-2cb5-48ae-bd85-0032843dce35}} stands for the period of the periodic orbit of {{formula:2c7954a0-cdcd-45f0-891f-8d8d7844e5c4}} passing through {{formula:8a20813b-7291-4506-a850-4f38d0ee2f38}}
Lemma 2.4
Let us consider a {{formula:261403dd-a50b-48d7-bbb6-1dfd55eaab3d}} family {{formula:5b5f2adc-60d0-4389-aa07-dc78e2d6af12}} of planar polynomial vector fields such that, for each {{formula:3a4fd589-fdaf-44dc-9585-47ad327d3f9f}} , {{formula:4e119b5b-f233-47f7-b66e-2edd50d3bf30}} has a center {{formula:d5274d04-ab8a-4cb5-a255-036e227f91ea}} with period annulus {{formula:60e1775f-00de-470b-baa3-3e01da732daf}} .
Let {{formula:3f738f08-8ea2-4818-bdee-c960daa78761}} be the outer boundary of {{formula:415a97bf-db19-4591-99fe-b25920cacbac}} Suppose there exists a continuous map {{formula:8c17ffcd-31b2-4443-b3d0-5daf84bdfec5}} verifying that, for each {{formula:122d2ebc-bc9f-42ae-b85e-b5c98aa4d497}}
the map {{formula:93c3ecfa-8360-4e14-85f6-d9c46ee90b11}}
is {{formula:dd889c02-0aa6-4247-9f62-cf5e0c800645}} ,
the vectors {{formula:8b112f47-08be-4796-ba57-f25fa633df61}} and {{formula:374ca3b2-756b-4b45-a132-65dd19649953}} are linearly independent
for all {{formula:3645beb6-5332-44c2-b0f8-d5a5ec9d2d1f}} ,
{{formula:ab1171c4-d617-4b2f-a3b2-3260ee126e99}} for all
{{formula:9f1594a3-840c-483f-b8cd-fb9fc4b0b7cd}} and {{formula:6aa610c3-3aa2-40db-af9b-e18972a6a416}} .
Then, for each fixed {{formula:5c10da13-14ca-4293-9175-7024cecc0525}} , the following assertions hold:
The Hausdorff distance between the outer boundaries {{formula:810ca754-ca73-4d9d-8171-51e2e299eadc}} and {{formula:9df8068c-030d-41b4-bda7-216add1fff0e}} tends
to zero as {{formula:b6fe1f70-66a6-4b63-ad80-5ecc25f5bce3}}
If {{formula:035bdb0b-6a4e-4436-9660-466d36d7573b}} for all {{formula:7b1610e6-3710-4e83-88d2-9b85b2f79c18}} , then
{{formula:ee74310c-9b74-4729-96f4-caeb584883df}}
{{formula:f64b987e-9e10-405d-a381-a4ad5fa6ba8b}} if for each
open neighbourhood {{formula:fdc58ec5-68a4-4f9c-8814-747fa3074e19}} of {{formula:581c2eba-aa06-44fa-a372-4c52bd695a1d}} and {{formula:2702590a-3e2e-46bd-8db8-bbdfe3244e9c}} there exist {{formula:7cf23274-e6cb-4032-ad06-d5593bcbc1f6}} different numbers
{{formula:efe62fb5-a4c5-41ac-8971-f04cc7083772}} and {{formula:6ef926b4-e7e5-48b8-965c-a184aaac1ec4}} such that {{formula:cda5879e-aeee-4e0d-a9de-6168ed517e88}} for {{formula:989116b9-81d7-429d-8148-39ffeeaf9f69}}
{{formula:9c5440b2-0735-4d9b-ae02-7babcd8b4c05}} if, and only if,
{{formula:1ca36836-85c2-459d-a209-6ad5a86b81c9}}
Proof.
To show the first assertion note that, since {{formula:1a845892-a411-491d-8c5f-ee20c99ee0bd}} is polynomial, we can consider its Poincaré compactification {{formula:ff99a95a-016c-4b99-81cf-a3c99aeafb42}} , see {{cite:26076f4ccf158d95a2bd6db4360d3d70ae043e1c}} for details, which is an analytic vector field on the sphere {{formula:14ee358e-fe60-45e9-8fce-9654b76a6d11}} topologically equivalent to {{formula:25537170-45fe-49e0-9e76-ffca2ae43a0f}} The outer boundary {{formula:587bcf81-9723-4c0f-8e99-e0ba1265bdcd}} becomes then a polycycle of {{formula:daac0d3b-f525-4051-84bf-b3eab842eb47}} that can be studied using local charts of {{formula:231c143e-117b-40a2-a8b6-8895df32307c}} On account of this, the fact that {{formula:69ae78e2-03e5-4474-bdc0-04c56c36bbe1}} as {{formula:94626678-5804-4fc3-8dc9-dfc30a9054c8}} follows by the continuity of {{formula:01c79305-1790-4cbe-8754-e84d0b75277b}} together with the continuity with respect to initial conditions and parameters of the trajectories of {{formula:38c44a59-6980-4611-b890-ba56f1606c13}} The interested reader is referred to {{cite:44c371fa22acae495f26ba888c8c78a3e7a58114}} for a related result for limit periodic sets.
With regard to the upper bound in {{formula:6977e232-bfa0-4415-a2dc-0f40560209b0}} it is clear that if {{formula:a2f5a861-de78-4cd9-bb99-bd89cc0c16f9}} then there is nothing to be proved. So let us assume that {{formula:8dc65204-c88f-46cf-87e3-16890fa8eca3}} and argue by contradiction. If {{formula:d39f867e-78ae-457c-8d38-4f4180111e78}} then there exist {{formula:97f34b4e-17b3-458a-b0d5-218f686afb52}} sequences {{formula:fccd19f3-22c2-49ec-9bf1-fda0f3d0d0e0}} , {{formula:e4bee4b0-af04-433a-9df8-fa2dd34af5ba}} , where
{{formula:2a18093d-8986-41b7-9d43-bc5fe8cfa9e2}} are different critical periodic orbits of {{formula:fb21e8e1-6118-4b00-b85c-4a015447dc66}} for each {{formula:a39bda9c-6f4d-4735-9acf-e6f6e20b3d16}} , such that {{formula:5a957ed1-37c5-4ccd-8a23-23dbcdf89354}} and {{formula:bd323cd3-76cd-4b74-a2d6-cbbe8e9a86c8}} as {{formula:12421a1c-4e29-44af-8a26-16b44c840795}} . Then, due to {{formula:e177b171-87ef-4d5e-91aa-cdeadb7322d8}} as {{formula:ab7030fd-c34c-47d3-b5b3-6f6fe5c48a4c}} and
{{formula:4271653f-7eab-4145-8807-6d9520a1d352}}
we have {{formula:482b4e1d-9395-4e5a-a81e-aec822b3e0fc}} as {{formula:ea0afebb-fd61-46c4-9c47-6144efd11188}} for each {{formula:255aedf3-62a9-46a1-9693-6c47fbe36608}} Since {{formula:ea752e1f-e271-4e38-a68b-2554f406f1f3}} and there is a one-to-one correspondence between zeros of {{formula:60ee1d60-1357-4181-9019-fd09f823be80}}
arbitrarily near {{formula:1798ec5a-40fd-4dbc-8295-33b10e047df4}} and critical periodic orbits of {{formula:3161c9d7-6f22-4866-a9ad-ce567daaa705}} arbitrarily close to {{formula:79024b00-a1d8-49a0-9137-2b9550f40b16}} (cf. {{cite:44c371fa22acae495f26ba888c8c78a3e7a58114}}), this implies that there exist {{formula:23615cab-4187-46ea-9f28-c58c98e14fbb}} sequences of positive numbers {{formula:ba1b1a9b-3e9a-4ba1-992d-38b914b11b92}} , {{formula:107442d6-1360-4cb2-a503-d2fd1bdc0f0b}} such that {{formula:12065449-96ac-44f8-ba0c-d2d6c7fda79f}} and {{formula:7886ce3e-8422-4279-8d59-e4c3579590b6}} for each {{formula:a432fa14-a139-45a4-9616-f3ad004c251a}} , and {{formula:6c08f1b8-789f-4588-b08a-6e9336130ece}} for each {{formula:4309c53b-e032-4f79-9c45-f30634a8caec}} This clearly contradicts that {{formula:31aa2d38-b546-4d50-b9d7-2ca455f9b503}} , see Definition REF . The assertion in {{formula:f4ff4df0-a0b3-4017-a4ad-753eb0651929}} follows similarly. Indeed, on account of the assumption and the above mentioned one-to-one correspondence between zeros of {{formula:969e96d6-6f09-4fa6-a791-c0436a33b385}} near {{formula:d70e6b22-e0fc-4442-a948-79c864e04658}} and critical periodic orbits of {{formula:3f977327-44a3-4c40-8ce9-3d813f32dfcc}} close to {{formula:dc6ac4ea-a8a7-4ff7-8e41-4184ab7e8523}} we can construct {{formula:13246c60-935c-4b9e-ae51-f0784a038297}} sequences {{formula:466ca2af-59f0-4783-9c1a-9aeafcf361c4}} , {{formula:a8c9d9aa-ac66-423a-be5c-3c0e9343ed6c}} , where
{{formula:71168026-6943-40cc-af68-9129e2311bd4}} are different critical periodic orbits of {{formula:be982e67-0215-43b0-a5a1-a864940aaf6b}} for each {{formula:2ff80e1f-4b2a-4be7-a1c6-a71fb4582dd8}} , such that {{formula:6701c768-9216-4736-b3a7-94844c6e5833}} and {{formula:aee2330d-096e-4f70-a643-f6ceacc99b7b}} as {{formula:3ad7018c-3290-4242-92b3-a3d1b5f27066}} . Then, using that {{formula:73385ac9-2729-4d50-a64f-d5a654d4bcf3}} as {{formula:f9b524d4-f5a6-46a2-993b-67dd4bef7df0}} , we can assert that {{formula:20b2792a-d1e1-4060-83c5-89721ba759b8}} for each {{formula:b4fe8cdc-2c16-4e4a-8619-dc2b102bd4d0}} , which implies {{formula:98fa536b-9e23-4181-b224-6d84ce56a740}} , as desired. Finally the assertion in {{formula:2785a4a2-64bc-4010-aa43-e95e421aaf22}} follows easily from the ones in {{formula:ef61a94c-4ac3-42f7-addb-de539fbd8162}} and {{formula:d0817cac-a059-4111-b9c2-aa315f1d1578}} This completes the proof of the result.
Next we introduce the notion of global transverse section for a family of period annuli. Roughly speaking it is a transverse section, joining the center with some point at the outer boundary of the period annulus, that behaves well with the parameters.
Definition 2.5.
Let us fix {{formula:5e465639-5a97-4ae6-9e31-1c4c223b067c}} and consider a {{formula:ead82e90-d90c-452f-aca9-9cdf64b19e04}}
family {{formula:2f0e4201-5347-4c70-ac75-4d38a2a75aed}} of planar vector fields such that, for each {{formula:88b431c5-e514-4fed-9976-e80335ab8e2c}} {{formula:cfe2bc6d-5d24-48eb-b958-834edc75583d}} has a center {{formula:47e86328-983b-4a4a-b1dd-0dfb86ae7af6}} with period annulus {{formula:f35d0d5c-66ec-4c2d-bc6e-3d8738f9edca}} . Let {{formula:b11ec01d-6fdb-4dbe-a762-889b0a477ca4}} be the outer boundary of {{formula:27c1ae23-e631-4aef-abfe-809c00df1703}} A global transverse section for the family of period annuli {{formula:b3fd1f05-1763-48a2-b654-2ff2d37419de}} is a continuous map {{formula:dbff111e-907e-45c9-9b63-39134b894b3e}} verifying that
the map {{formula:01d320d2-86f5-4cc5-bcf6-7ace0cd9e8da}} is {{formula:190c8059-bfe4-4243-8275-ffe7a0c1e19c}} for each {{formula:d4c97eb9-fae0-4180-9f23-a00ced0055c2}} ,
the vectors {{formula:73e3db93-fb5a-4e9b-9084-efd943a7e89f}} and {{formula:66f9ed82-fa14-4bae-8c50-e8c14626bc39}} are linearly independent for all {{formula:5fb57316-7f5c-4b3a-9726-44ef83827a6b}} and the map {{formula:5aafd351-589d-4bb7-afc1-ec34a0aaa338}} is continuous,
{{formula:6156d884-0cd0-46e8-87bf-05a700fa13cd}} for all {{formula:7e035a1f-4db7-469a-a45a-bc1411638355}} , {{formula:c8ee890d-cd27-4e3e-bbee-a884a1b8b7ef}} and {{formula:efc8078e-4f4f-4573-acf7-d8ba8698c319}} .
When such a global transverse section exists we say that the family of period annuli {{formula:3d658d58-899a-4178-9804-6d134ff7741a}} varies continuously.
{{formula:e8b3d8af-d273-47f2-a0cf-4231c8ce24be}}
Remark 2.6.
The period annulus of the family of Loud's quadratic centers given in {{formula:a97027cf-c7ec-47f8-92fc-7ee978678ea8}} varies continuously in the sense of Definition REF . Indeed, it follows from the proof of {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} that
{{formula:883180c3-5d69-4235-ae58-8bf43d498b4c}}
is a well-defined continuous function on {{formula:fd7e6614-a8d2-499d-816e-0ee955c085ba}} Moreover the point {{formula:9bcae18a-3627-4219-8920-fb3c7a0e7094}} belongs to {{formula:844f728c-5616-4e60-8805-65f754fa52d4}} and
{{formula:13f04dac-b141-4770-a0f3-e1220fd06824}} for all {{formula:cbe387a3-0856-47a9-9957-86a5caec1c9a}} Then {{formula:84525d2f-22cb-4b2d-8ceb-71c4ca3750fe}} for {{formula:8de4595e-6d75-46cb-a0c1-ee6224b0b4d4}} is clearly a global transverse section. In particular, since the Loud's system is polynomial, the outer boundary of the period annulus varies continuously in the Hausdorff sense by the first assertion in Lemma REF .
{{formula:f2e88150-3c38-4248-b568-3bf67b955703}}
Note, see {{formula:d4a9c5dd-f961-4978-8f06-4f34709dcae2}} in Definition REF , that we also require {{formula:4dec04dd-52fe-4d1b-99d3-44a655c1c160}} to be continuous. The reason for this is because if we define {{formula:0f51ee04-a119-4461-822d-f502093076eb}} then {{formula:8151c9f8-5a39-4014-b7c5-70f8981e2280}} is a continuous function by Lemma REF . This continuity is a key point in the forthcoming results. Before that we summarize in the next statement the properties that we get for {{formula:ce75463a-e5fe-45dc-af0b-4f136c1555e5}} as a consequence of Lemma REF and Definition REF .
Corollary 2.7
Let us fix {{formula:363a8710-fcbb-4a7a-b116-4adb5285983d}} and consider a {{formula:aae56ec7-dc37-4943-bb4c-dab095d197ee}} family of planar vector fields {{formula:1604d916-4e07-4915-927f-634c2854fa98}} such that, for each {{formula:04dbbf6c-6178-4f3c-896f-6f4a23af53b4}} , {{formula:dfe7c207-2358-40cf-8bf3-ae3bd11a2699}} has a center {{formula:28acc032-f8e3-46a7-aecb-f2e05f8f04bd}} with period annulus {{formula:6de1d16d-018c-4f6a-ace6-9bed6362fdb0}} . Assume that the family of period annuli varies continuously and let {{formula:bf95f717-673e-4f49-89a1-61275e0f5de9}} be a global transverse section for {{formula:a4bb51fa-e261-423c-962e-6e99472f88af}} . If {{formula:69008770-3e11-4de5-ad87-b3017f899136}} for all {{formula:9f0726c4-6997-4e63-bf4f-670c5da22e74}} then the following holds:
{{formula:d3a773f9-1bdc-46d6-b5f4-9a8545b08528}} for each {{formula:47f0420a-375e-483e-bdab-bd990bbe524a}} and
{{formula:03b35628-2c91-467f-ac48-f3c01fad1a7f}} and {{formula:36f689a3-2059-4e1f-b5c2-2c8706277b1c}} are continuous functions on {{formula:23b1df43-bbca-45f4-932a-c5817c55a670}} .
Definition 2.8.
Under the assumptions of Corollary REF , we say that {{formula:0823ba96-e6db-475f-9791-b6f5c0dde87c}} , which is defined for {{formula:edd12702-d2f4-48d2-b641-8808f413e5c7}} is a global parametrization of the period function. In contrast,
{{formula:69f85e31-10e4-4418-86da-446ee76e5f37}}
is defined on {{formula:7761b2f7-e86f-4984-81de-fedcf6a2e0cf}} which is not so easy to handle.
{{formula:32c63fa7-98f2-4d22-a35d-d85d50cb6452}}
One of the main goals in the present section is to relate the concept of local bifurcation value of the period function, as introduced in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, with the notion of criticality, see Definition REF . As we will see the first one concerns with the qualitative properties of the period function, whereas the second is more geometric and quantitative. In doing so we will be able to take advantage of the results about the bifurcation diagram of the period function of the Loud's centers that we obtained in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}. For reader's convenience we next recall the definition of local bifurcation value of the period function.
Definition 2.9.
Let {{formula:fd618959-98ea-4959-887b-44edf646ee04}} be a continuous family of intervals in {{formula:d6f08d78-c825-4a10-86a9-341cfb4f63e2}} , i.e., such that {{formula:1f747655-7ffb-4acf-aa5d-224b6915de30}} with {{formula:21bae5b8-c53b-498e-9bcb-ae24f1fa1260}} , and consider a continuous family of functions {{formula:17be115e-844c-419f-9800-79695f9da431}} We say that {{formula:fc058cc9-c7c4-4945-a37e-f8e85a7146f3}} is a
regular value of the family {{formula:d8f93ea1-19a1-4532-adac-5f324411bcc3}} if there exist a
neighbourhood {{formula:5abc4e75-e695-4405-a3a2-a60d09d3b229}} of {{formula:835e3b3d-e1f5-46da-8e59-710f99a273b1}} and an isotopy {{formula:893c701e-6334-4ba5-8b04-fbe4175c803f}} with {{formula:1204cd6e-307f-4657-80bb-334e98d8222d}} such that
{{formula:7a83c0ed-9c62-4321-976a-4dd6c5588d5a}}
where {{formula:381dbb79-4a9b-4ccf-aba5-a8f8da7746fd}} is the extended sign function.
A parameter {{formula:9cc6ac9b-ddae-4791-aa12-598d5a2a8c6e}} which is not regular is called a bifurcation value.
{{formula:602f1c94-fd30-4b12-a9a3-efc818e684c8}}
The endpoints of {{formula:dc5a2d44-83bc-4b31-86b2-afb55c2ca0c5}} , the domain of definition of {{formula:2f0d6c2f-1623-42c6-b7ad-bdc627a3197b}} , depend continuously on {{formula:4f9b6b67-a437-4cdd-93db-18d611299d81}} so that {{formula:533c511a-ba07-4d34-8a3f-94aaa7990754}} is an open subset of {{formula:d97d9e27-059b-4741-8f8f-d92537b9b113}} Thus, by a continuous family of functions {{formula:53a9b78f-7523-4523-96ba-c8da1982fdcc}} , we mean that the map {{formula:14560ea7-b523-49f8-9213-b50e83184fbb}} is continuous on {{formula:efa030b7-c803-4ce7-bc56-057477c8f9b7}} . Next we particularize the previous definition to study the period function. To this aim note that, by Corollary REF , if {{formula:78c246eb-cb18-43eb-b434-579d7d96a0f7}} is a {{formula:ae6cd0a7-37ef-4b5b-87b6-d7cb6c753229}} family of vector fields with a center such that the corresponding family of period annuli varies continuously, and we set {{formula:2ca309bf-62cd-495e-abe4-f2e0cd39bd65}} , then {{formula:afc2b55b-f89b-4f52-bf36-e9ce1a887d99}} is a continuous family of functions on {{formula:25b48322-560b-4cf6-af59-b80088cff571}}
Definition 2.10.
Consider a {{formula:6ae383da-8d89-48e2-bf92-9de8cbc69e8c}} family of planar vector fields {{formula:7d47798a-17f0-4c25-8ab4-efcb4ef826d2}} such that, for each {{formula:a08fc55f-8c58-432e-8402-65189182924f}} , {{formula:6c2e93ff-7aa7-4c24-90ab-093c50030da3}} has a center {{formula:ce32db91-9b93-4da0-9f1a-c77b7cefd049}} with period annulus {{formula:82323597-f6b3-4dee-83eb-1a3488f62b49}} , that we suppose to vary continuously.
We say that {{formula:5fc05fbb-4a61-452a-8f76-5bd2eaf9b406}} is a regular (respectively, bifurcation)
value of the period function if for some global parametrization of the period
function {{formula:107f105d-1b85-4c13-a239-9bc4bf7625e0}} we have that {{formula:492f4c49-0d4d-408e-8024-311f7cd872c4}} is a
regular (respectively, bifurcation) value of the family {{formula:a9ec1753-fda8-49da-bd4b-ecdf42ef5759}}
We say that {{formula:23969fff-c425-47a2-8928-4b81e48f13dd}} is a local regular
value of the period function at the interior if there is some global parametrization of the
period function {{formula:6a699086-33d2-49f3-b27b-cb6a16d6b3b2}} satisfying that for each {{formula:a52ef40d-a5a4-4c74-9b4a-c2e8fc0c2fd0}} there
exists a continuously varying
neighbourhood {{formula:e91bf3c2-c42b-4ad8-9cac-ad522da5d2b8}} of {{formula:5c0f2e79-dc77-455a-880d-770887811d85}} in {{formula:ebaf37e5-8da7-4312-ab6d-ba14f9b3b964}} such that {{formula:cede4e30-4231-46ac-9d01-148abf38bd37}} is a regular value
of the family {{formula:7f3878c8-5e4b-4eac-9e2b-c742ba3f4e0f}} A
parameter which is not a local regular value at the interior is called a
local bifurcation value at the interior.
We say that {{formula:4d926440-5614-45de-b46a-13f480acdeba}} is a local regular
value of the period function at the outer (respectively, inner) boundary
if for some global parametrization of the period function {{formula:517a80bd-2aac-469a-853d-d3fd9f498252}}
there exists a continuously varying
neighbourhood {{formula:60c7a1c1-288b-4166-b36f-466165598939}} of {{formula:f5146e89-95cb-40b0-89b4-9a55e700f3e3}} (respectively, {{formula:33356cb0-ee2d-4c91-a227-50c7b345c743}} ) such that {{formula:4bd9fdd3-a597-4e72-b722-58cccd6c3156}} is a regular value
of the family {{formula:e02f7fc2-db96-41ee-a5dc-21808709624f}} A
parameter which is not a local regular value at the outer (respectively,
inner) boundary is called a local bifurcation value at the outer
(respectively, inner) boundary.
{{formula:a8d786f1-c205-4447-8020-a38f8a3ae6f7}}
Remark 2.11.
Let us make the following easy observations with regard to the previous definitions:
One can replace “some global parametrization” by “any global
parametrization”. Indeed, suppose that {{formula:0b2982fe-e7b9-409a-8f28-578f873362a4}} is a regular value for {{formula:db36e5ae-708c-40f5-9855-4b8270a3f1c1}} where {{formula:61653e90-d4fc-4574-99d4-ab83a3ce7b23}} and consider another global parametrization {{formula:0cfd9bcf-6ff9-4396-9ebd-ea0e9579511a}} of the period function, see Definition REF . If we denote by {{formula:60556f27-bc40-4c09-b098-578ae1f2d3eb}} the Poincar√© map from the transverse section {{formula:456100c4-0ffd-47aa-8a32-073da28275b9}} given by {{formula:66b99a30-75b9-486d-ac54-8d0735728223}} to the transverse section {{formula:7cfa644a-59f4-4186-af1c-b9809bd6cc15}} given by {{formula:536f4467-206c-4df3-839b-c116f4791a67}} then {{formula:ea9db132-2f0b-45a3-b263-d1b1c3eaf0fe}} is an increasing diffeomorphism and {{formula:2c31b043-ef11-4fd8-bfc2-28a99963c670}} so that {{formula:4afeeca7-25fd-45d5-8b5f-652c0d3c7de3}} On account of this and following the notation in Definition REF , {{formula:4310d9de-de53-4de4-a555-10b13cd4c5b4}} is a suitable isotopy in order to show that {{formula:449777b9-9239-4942-9fae-e49cd9348d55}} is a regular value for the family {{formula:092523d3-1d08-4329-b20a-dd70ba48a2c2}} because
{{formula:35dbfc13-a8d2-45fd-8d92-001608069d56}}
where we use that {{formula:3ebd17b9-5cd5-4ceb-a482-493115fbc567}}
In order to study if a parameter is a local regular value at the outer boundary it is not necessary to consider a global transverse section {{formula:473b75be-d37c-487d-9851-3c3d501d42f0}} for
the family of period annuli. Indeed, see point {{formula:f1c86701-1e17-431b-9821-0538037a22e0}} in Definition REF , it suffices to take a local parametrization {{formula:a836565f-29db-4faf-a768-474a165f83e6}}. Similarly, to study the local regular values at the inner boundary it suffices to take a local parametrization {{formula:f3ce01e7-a5cf-4741-8f47-fb6ec4ac38ea}}.
{{formula:50d83b5d-7b5f-4071-a4c9-40b7dda0413d}}
As expected, {{formula:0e19d586-e954-4cb2-aeb2-3de426f2cfb6}} is a bifurcation value of the period function if, and only if, {{formula:751a77e8-bd66-44d3-9e5b-a3cec28322ed}} is either a local bifurcation value at the inner boundary, at the outer boundary or at the interior. This is stated in the following result
and the interested reader is referred to {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} for the proof.
Lemma 2.12
Let us consider a {{formula:03aa4b3a-deaa-4547-abec-571b1031d2d4}} family of analytic planar vector fields {{formula:d83205e3-a8ea-4cfc-ad02-4e4adb32b2ed}} such that, for each {{formula:a29b187b-78ad-449c-b054-0751bce9acf5}} , {{formula:3a8f36c5-2c8b-4c86-a7c4-898120d9195f}} has a center {{formula:c94289ba-160e-45b6-a942-1183732547a8}} with period annulus {{formula:ad082c9b-188e-4472-922b-2a1358015255}} , that we suppose to vary continuously. Then the bifurcation diagram of the period function is the union of the local bifurcation
diagrams at the inner and outer boundary and in the interior.
Under the assumptions and notation in Corollary REF , a sufficient condition for
{{formula:38391568-a43b-4e6a-acb7-89ad9d759cbc}} to be a local regular value of the period function at the interior is that {{formula:1eeaf683-001c-466c-ab08-df8318f67513}} for all {{formula:76f246be-0a6f-40d2-909a-5d36263a1d27}} This follows easily by the continuity of {{formula:5d087e37-d51c-40f2-a6d7-032f49a6dc40}} on {{formula:a3d9df6e-02e1-4e1d-a5b1-d9ad4c3cf9b6}} and a compactness argument. In case that this function is {{formula:d1130e13-c893-4d69-9894-3579f4f3294e}} then another sufficient condition is that
{{formula:f1735a2c-fe4d-4b3f-a677-39cd9538a219}} has only simple zeros because the application of the Implicit Function Theorem provides the appropriate isotopies. Hence, in this context, the set of local bifurcation values of the period function at the interior is contained in
{{formula:c7355c2d-71c5-4d2e-83b0-38fab00d728d}}
If {{formula:8a2c85df-35fc-462c-ba89-c8d35f5ee960}} was polynomial in {{formula:6736dc0c-8e0a-4c9a-aab3-2232064b4dd1}} (which is certainly not true) then {{formula:a94192bc-a653-47af-b8bb-5df21b34b800}} would consist of those parameters {{formula:a0ce6b41-257c-455a-ae84-0d325a29033c}} such that the discriminant of {{formula:3bc8299c-5407-4546-8286-4b1cf911512e}} is equal to zero. (Recall that the discriminant of {{formula:d51b82c2-6a6e-4db4-a2b2-a0523bce5e84}} is the resultant between {{formula:7659888e-f14b-47d0-b393-e3a190468a44}} and {{formula:bd0ccc60-1b2c-4bcf-b76a-4e5193aaa6e0}} , see for instance {{cite:a895669f466ded20131e0fc77f273d2705d2ca43}}.) One may expect on the other hand that the parameters in {{formula:c5ab2dcb-623e-4afa-9b5d-bf88375e8225}} are always local bifurcation values of the period function at the interior. However this is not always the case and the following toy models show that some additional assumptions are needed to this end.
Example 2.13.
Setting {{formula:05064f04-a8ae-479c-a00d-9613c22359f2}} we take {{formula:79b49ac2-a680-4c26-b24d-31b36b79b8d0}} to be {{formula:0c414dd5-54dc-499f-b323-d0c532a41e7e}} and {{formula:a28502a6-9fb8-41b6-b8e2-1aebb84ecc61}} Then it is clear that any {{formula:d8747339-ca3e-45aa-9f85-5f2babb54aa3}} is a local regular value of {{formula:a0f1fa68-459f-4f41-b915-5351e12aa60c}} at the interior (i.e., there are no local bifurcation values) but we have that {{formula:63f7bf0b-e0a0-47e1-b90f-2067f62a0a70}} Note that in this case the interior of {{formula:814fa2a4-92a7-4a7f-96c0-7d742dc41d03}} is non-empty.
{{formula:1a10c565-9629-4dbc-bcd2-e7277787adbb}}
Example 2.14.
Setting {{formula:cae793e1-c4d6-46de-ada0-6df1d928cb13}} we take {{formula:9d1dcaf3-c8e6-4afd-b4fe-d3f275690b1d}} to be {{formula:69219bbf-1997-4acb-a667-415f6aaefd60}} and {{formula:069c4c57-43c9-476c-a71b-acf7d4f5d872}} Then again it turns out that any {{formula:e56d6c05-ca6e-430e-a752-134da88b5970}} is a local regular value of {{formula:c155b8c3-28bf-43a2-992f-817236ab13d2}} at the interior, whereas {{formula:cd72ca9c-f8db-4955-ab87-20068c0dcd5f}} Observe that in this case the interior of {{formula:d23e79c3-cd0d-4755-981b-95e32aa85d40}} is empty but {{formula:3728947d-a72c-4642-8f0e-bf17b7b24ff3}} has zeroes of multiplicity 3.
{{formula:9d799667-29e5-43fc-9ee5-47466163e28c}}
The following result provides us with an analytical tool to study the local bifurcation values of the period function at the interior. We emphasize that it has the natural hypothesis in view of the previous discussion.
Lemma 2.15
Let {{formula:d26d13df-7545-4689-8afb-25f262cc316e}} be an analytic family of planar vector fields such that, for each {{formula:5181f081-a8a8-45e5-98a3-d16b8e620349}} , {{formula:d6f095b3-2f48-4d95-85ae-218b9f7fee68}} has a center {{formula:3d2cc3b4-6965-498b-a68d-e83bd7f761d9}} with period annulus {{formula:cec2bd01-41a0-448b-89db-740bf60ec172}} . Assume that the family of period annuli varies continuously and let {{formula:4e63ce9e-c46d-49d9-852d-fdc8b23e5619}} be a global transverse section for {{formula:ef1e13c6-521b-4333-9759-656791dba845}} . Setting {{formula:837d730f-f410-4a44-9a1a-6a769d92237a}} for all {{formula:883706e3-2c4e-40ce-885b-4e8c287dc337}} , suppose additionally that
the interior of {{formula:63521559-3ade-4852-bcce-d6ee37bff437}} {{formula:d710854d-cf55-4eae-859a-ebf1f62f4012}} as a subset of {{formula:f6d4b5f3-eda5-4625-bb01-f77c6fbba34a}} is empty, and
for each {{formula:a004b0bb-6019-4825-9102-6e68ec1de3a5}} , the zeros of {{formula:be29e51a-5d50-4ff1-80d8-6404eb9b93a7}} have at most multiplicity 2.
Then each {{formula:e2ddd182-f1e7-48b0-af39-1d7c820c0d8a}} is a local bifurcation value of the period function at the interior.
Proof.
Note first that, by Corollary REF , the function {{formula:f6cebd51-fdeb-4e1c-9b0f-77e04eb135de}} is analytic on {{formula:bfe0e5bd-15b5-4ea6-b71a-729091217aa4}} for each {{formula:e3f25ca5-10a3-48f4-b889-93e1bf52a337}}
Let us take any {{formula:52ea4f95-a4aa-4486-973b-7eb37dbdef7c}} Then there exists {{formula:d1703d93-c1bc-4c15-b668-259c519819be}} such that {{formula:9bc8b1d7-8b00-4d28-a9f6-9e9c958c7f49}} and, by the hypothesis in {{formula:2429feba-251a-457a-9304-d24ec89bf73a}} , {{formula:3ce5435d-4e7f-43c6-858a-4d09b7180c0e}} . Consequently {{formula:49f4818b-b12b-47d7-b7cb-6a8fcb9dd951}} has a local extremum at {{formula:bc567fae-0c04-43fd-af18-799e4c613a67}} and so there exists {{formula:163e9580-bd67-470b-89d4-6ff1d7ccc52f}} small enough such that {{formula:fcddac6d-29cf-48fb-af6c-03285eb9e280}} has the same sign {{formula:c9b09f4f-c75e-40cb-98b3-7eadd0e838cf}} or {{formula:112dafb6-0dcd-4dc5-81f0-ea99cf883374}} on {{formula:773646d2-6064-4d2c-8f9c-93b0f3cc0f1a}}
Assume by contradiction that {{formula:e91f18f7-751d-41be-9307-0ccd8ba0ad1b}} is a local regular value of the period function at the interior. Then, taking {{formula:51391393-d688-4643-8c83-c9f5de8bc7e7}} in {{formula:cab2de1e-8e84-4eee-9db0-484144d83a03}} of Definition REF , we can consider a neighbourhood {{formula:dafd97ed-933d-403d-abcb-5ce55922261c}} of {{formula:02f9de35-c3d1-475e-8c3b-29b77a6a0bf2}} , a continuously varying neighbourhood {{formula:012c5ac1-e949-4633-8ba7-97c26712f6ff}} of {{formula:0fe73a9f-d9ae-40b9-932b-8723582f35c8}} in {{formula:531f68bc-13a4-4b28-a3f8-19b91f5c954a}} and an isotopy {{formula:2b8601ca-8a94-4da7-bace-f4e8b4e48061}} for {{formula:d1eac767-169b-4746-89b3-444a0d1d0c60}} , with {{formula:ed8de8e0-4af7-4f07-86cd-6a2884b707d4}} , verifying the equality in {{formula:97fd7bbc-0c7e-49c2-b4cc-8b18a830b585}}. Since {{formula:36c51388-4a93-4896-9bbe-e404d4c5babc}} has empty interior we can take {{formula:ff3cf569-607c-4af3-bd64-d7cd2a1ee1ad}} and define {{formula:9991cdce-028e-49fd-af02-06173cf071dd}} . On account of this, particularizing {{formula:7ccb0f38-d13f-43b3-836f-ca8971207026}} with {{formula:a0eb6396-e67b-4c8c-92f2-45f0d8c39825}} and {{formula:5979b08f-a666-4fc8-88d6-7a1faece9e2c}} we deduce that {{formula:e7aa00ac-76d8-4f58-a574-6b2e75470558}} . Accordingly, due to {{formula:ec35fcf9-f6fa-4922-9c5f-bfed4a31940f}} it follows that {{formula:461973af-f86c-4408-9bdf-911b1a015db6}} . Therefore the function {{formula:e0b04446-ed71-4372-8a75-eb54f9f7044d}} changes sign at {{formula:5e57384a-a45e-42ad-96e1-2a35a0f3798b}}
This contradicts {{formula:314f4bb9-4246-4f38-b9a6-3579abee0ca5}} taking {{formula:0029c29e-5b0d-4316-b686-b4410a5e7637}} and {{formula:0c68cc54-9b65-417e-a1d3-84ab37e23d73}} because {{formula:d7e19f30-56ac-4dc9-8441-7d51222c125b}} has the same sign on {{formula:37595b9c-bff1-4316-9c0b-6b60a634c7da}}
In the statement of our next result {{formula:be1bfb1c-aa93-47b3-be74-3128740a1042}} stands for the Poincaré compactification in {{formula:2765eee0-3821-4e6d-9bbb-56eb76ed76c2}} of a planar polynomial vector field {{formula:ee4ff1eb-a7f4-4266-87aa-925b800db566}} , see {{cite:26076f4ccf158d95a2bd6db4360d3d70ae043e1c}} for details. Recall also that any polycycle of an analytic vector field can be desingularized giving a polycycle with only hyperbolic or semi-hyperbolic vertices. By a hyperbolic polycycle we mean that its desingularization does not have semi-hyperbolic vertices (i.e., saddle-nodes).
Lemma 2.16
Consider a {{formula:7a66090b-6071-4511-a85a-f665c0f01067}} family of planar polynomial vector fields {{formula:61da971e-d7a2-4804-be73-4ed8761dbb1b}} such that, for each {{formula:01f7cf11-04fd-497e-92a7-03af24df49ab}} , {{formula:96f2bafc-8b98-4245-93b0-00ca151c9ae1}} has a center {{formula:e42fdb54-33fe-425d-946a-fb8ad212dfde}} with period annulus {{formula:8a51050c-3c18-4177-8274-961d4236a5b0}} , that we suppose to vary continuously.
Then the following assertions hold for any given {{formula:7336c33c-3610-441d-89c4-95773a4ee233}} :
If {{formula:d5102c05-83e3-478c-9831-a3211cf918be}} then {{formula:9b381564-88fa-4747-85ff-b627a2888913}} is a local regular
value of the period function at the outer boundary.
Assuming additionally that the
outer boundary {{formula:fdbcadb8-2e22-4990-bb51-311de143a5f9}} is a hyperbolic polycycle of {{formula:34e0d59b-976f-4f30-970e-4f2538ede7c1}} ,
if {{formula:0162d357-852b-4128-b857-163615708517}} is a local regular
value of the period function at the outer boundary then
{{formula:47eda92c-143a-4498-bd4e-93e024c13f46}} .
Proof.
Since the family of period annuli varies continuously, see Definition REF , we can take a global transverse section {{formula:90c3baa7-11bf-41bb-93ec-f6d01f38e682}} and consider the global parametrization of the period function given by {{formula:b65447ad-25c1-4d69-a679-4c9f133610db}} for {{formula:ef392511-309d-4147-93cd-fa397fb13d96}} see Corollary REF .
In order to show {{formula:0e24d592-6c87-44a1-96b8-f838f18e3812}} note that if {{formula:41abdc4e-0daf-4020-acca-cf8975606510}} then {{formula:b8338974-25ec-4183-a492-bf6883c218ff}} by assertion {{formula:55625bab-2820-4248-8e7b-50a64d1353c5}} in Lemma REF . This implies, see Definition REF , the existence of {{formula:b8c774e5-63bc-4184-876b-a5c68bdb3814}} and a neighbourhood {{formula:f7092ab9-9e01-41f2-b104-1a7b356ffd19}} of {{formula:51aa64a5-e71c-4b20-b707-c4792c5756d2}} such that {{formula:7da7ce24-d29c-4618-8da0-a40b35290e59}} does not vanish on {{formula:07dfba6c-d2cb-431f-81de-964fae7ef7de}} . Hence, since {{formula:dce46055-5a75-4599-bcf2-40a4a8c7b7d7}} is continuous thanks to {{formula:e3dc82ac-64b5-4bd6-a0ae-338864fc3aa1}} in Corollary REF , the function {{formula:78e9dafb-ee84-4c62-a2e7-50d506db824f}} has constant sign on {{formula:213c878d-16e8-473c-9a8a-5938fbc40bf6}} . Thus, see Definitions REF and REF , taking {{formula:8719b918-0180-479a-b7f3-ec3201c42799}} and {{formula:44d0b31a-5912-4c24-8c98-df0260fdd3f7}} it follows that {{formula:1eeb6151-4bbe-474e-aeb2-2d3231610bff}} is a regular value of the family {{formula:321189dd-c34c-4601-a18f-017e48eae495}} as desired. This shows the validity of the assertion in {{formula:ca0b4a9d-d5fb-4485-9bb4-f6621edf8851}} .
Let us turn next to the assertion in {{formula:08da5a32-59cc-43ab-8f4d-cfc9174cc03e}} . If {{formula:352a2469-23ee-42ae-81a8-a0e27bada6e0}} is a local regular
value of the period function at the outer boundary then there exist a neighbourhood {{formula:f9294987-acc6-44c3-a1e2-ba205c01e15b}} of {{formula:8328c55d-19c7-4755-81dd-65d3faf69be5}} , a continuous strictly positive function {{formula:31025d59-f05e-435a-ac64-22677323f6cb}} on {{formula:6f97f9c1-3e43-42d7-b29d-0c94d1a3ba9f}} and an isotopy {{formula:eca3622d-924a-4b75-9813-608528954bfb}} such that {{formula:7d0eb085-e0c3-4364-b78f-084fc7379a63}} for all {{formula:2af2ee82-fe26-4c7b-9393-e53fbc6b237f}} and {{formula:af516bc2-bb11-4358-abad-c31753813624}} . From this point we distinguish two cases:
If the center of {{formula:f3cd52aa-fca2-4c3a-b075-b305105720d4}} is not isochronous then, by applying {{cite:611f9f0639a6c4b65dc618a3fa7652469d94ffea}}, the zeros of {{formula:8132f818-8396-4323-b351-2e5c3f6328c3}} do not accumulate to {{formula:b9ac8254-0a0f-49ba-bf87-b6cadfbde6fb}} Let us remark that to apply this result we take into account that the transverse section {{formula:d99c948c-a55f-4472-b5f9-b3005546d278}} is analytic at {{formula:ab3ead0b-adb9-4b99-afcc-8fc8ebb225d9}} , see Definition REF , and the hypothesis that {{formula:f20761c7-1251-4c17-9a97-e50eb6198788}} is a hyperbolic polycycle of {{formula:b2b11aed-1021-4af5-a231-4fcc5ae44b2f}} . Hence there exists {{formula:63054a70-4872-4972-bfd4-c2b28e40e3f4}} such that {{formula:bdd63a90-7834-47d8-b8a7-d55cf662b2f4}} for all {{formula:4014bae4-6339-41d0-ae93-faa4b64a7bfb}} Thus, since we can suppose without loss of generality that {{formula:2ba61203-9133-40b8-ba92-ee4246e9539d}} and {{formula:6dfc5a4b-1785-40d3-b62f-8ff6116644a9}} , it follows that {{formula:70f9865b-ef3b-4448-b721-a2075f315cb2}} on {{formula:58f2f733-f8a6-4c43-bc0b-bb4596f4556a}} which implies (see Definition REF ) that {{formula:9c4b187f-0a08-4b52-af75-850fe51f7eaf}} . Therefore, by assertion {{formula:bb5aa882-12b9-424e-813a-721619d0dc1a}} in Lemma REF , {{formula:3194f617-d132-415d-ba69-2c4ef07fc3af}} .
If the center of {{formula:ee16708e-7e9f-4aae-973f-72d04b9d81c3}} is isochronous then {{formula:3df5fb68-4462-489b-b0aa-52456348920f}} Hence {{formula:77c28306-1a60-4f8a-a258-3307fa80340f}} for all {{formula:7eb778b3-0bb4-4c12-b4ae-e0fc80f1a08c}} and {{formula:0cbd1a0b-720a-4124-bb1c-c306540c2ef7}} . Thus {{formula:619805da-c29b-499b-b4a2-5f1ecdd5f0dc}} has not isolated zeros for all {{formula:f7f3f7ae-c79d-414d-b207-4e4d368e744c}} and consequently, see Definition REF , {{formula:38d2afbd-ebee-446b-b029-a315a7b83e89}} . Then {{formula:0f71ac35-4f32-4448-8708-608245e78add}} by {{formula:a26f95f0-5a3f-43c9-89da-f6aa7f872a3e}} in Lemma REF .
This shows {{formula:69929946-058a-4f71-b034-30fac078f258}} and completes the proof of the result.
We conclude this section by showing that, as we explain in the introduction, Theorem REF leaves us very close to the proof of the existence of an upper bound for the number of critical periodic orbits in the family {{formula:41934f9f-02fc-4ac0-b08d-5a99ef332fdd}} . In this respect we note that there are parameter values {{formula:67a2683b-dfce-4bdc-a013-e1d85431a564}} for which {{formula:91b5e8e5-c2fb-4b44-899e-994de260c5b9}} has another center {{formula:1d76ee2c-3104-4222-875a-5b21df133002}} apart from the one at the origin (see for instance {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}). The bound also holds for the critical periodic orbits of this second center because one can always find an invertible affine transformation {{formula:389fb0d5-9100-430a-9f19-529f5d18c950}}
with {{formula:b9f46d34-1b34-43ff-9a51-147a82ade6e9}} such that
the push-forward of {{formula:b412bda4-142c-4248-b967-8723f233c0b6}} by {{formula:29228598-8802-4348-8430-8892bfeb6451}} verifies {{formula:8d165067-cb91-4982-8e5e-6c4b1072428b}} for some {{formula:2bd34804-709c-4207-a5eb-7b4405efdf42}} and {{formula:af56b6cb-370f-4163-8924-36ba089189bb}}
Lemma 2.17
Consider the family of vector fields {{formula:429dff01-eee4-4a41-9830-da1c8b1fbc85}} given in {{formula:6aeb602d-8ec8-4349-9297-02e10466d920}}. If {{formula:a9d56a01-8c99-4aa2-aabf-b0d12d78be88}} is finite for every {{formula:cafae784-af9a-45a5-9617-04c467ed0bb1}} then there exists {{formula:ffaa0db0-a27d-4c6b-86ce-0c57a43c9515}} such that the center at the origin of {{formula:af28c985-8e4f-424d-a0b9-bc64886f05ae}} has at most {{formula:b373544a-07be-424c-8f14-3fd9d664d054}} critical periodic orbits for all {{formula:772aec82-fb8d-4b0a-a839-6d1e0fde66be}}
Proof.
By Lemma REF , {{formula:81828bf7-80c6-4359-809b-2833acf1bc8e}} is an open subset of {{formula:43eb499a-127f-46af-b401-d3f8a5ba88d4}} and the map
{{formula:b9214d6c-a98f-4e4e-8594-7bc670fd577a}}
is analytic on {{formula:f50e05cf-2e34-4ff5-bff5-d5a4970b6b5a}} We define {{formula:5efc11bb-990c-459f-89f2-afd6dc25341d}} for each {{formula:7dd065c6-33d8-4c99-86d8-7ff152a0bb62}} , see Remark REF , which provides us with a suitable global parametrization of the period function. Let us note in particular that {{formula:2387428a-653b-4753-b977-7dacbefc8f08}} is a continuous function on {{formula:de76acd7-dc0c-42ab-891a-a7ef232528ef}} for each {{formula:57d55eb7-d792-42e1-8c19-b317fbf0fe16}} .
Moreover, by {{cite:f0d4fc02e805c76a4504eb8202c701fca40d1ed4}}, we know that if {{formula:fa9f0d50-fa92-4c48-9f47-41d2e651dd6e}} then then the center at the origin of {{formula:d9adecd7-0c62-4d64-afbd-87d1172a6538}} has no critical periodic orbits. Consequently, if for each fixed {{formula:b33a68d4-b197-424b-bb73-b74ae4f7ddf0}} we define {{formula:16c39c4d-d112-461f-8b22-bb654b7dcc8e}} to be the number of isolated zeros of {{formula:625b69ee-4c98-45db-9d56-8b9183b901a6}} on the interval {{formula:4465c0cd-9716-4973-8e0b-fc78fb5a0fdd}} counted without multiplicities, the result will follow once we prove that
{{formula:812742ab-5b67-49de-89b5-eeb88589b4fe}}
Let us advance that this will be a consequence of the compactness of {{formula:2ad78145-10bb-4b92-9742-66d2e9ea6213}} With this end in view we fix any {{formula:6c98026e-92e2-453a-88fd-b3550af89dd0}} and observe that three different situations may occur:
Case {{formula:5bc6088f-f06b-4f45-a35a-2d34d63461b8}} . As a consequence of the result of Chicone and Jacobs, see
{{cite:2bf3254451791a68ce817916f287f1704a0d2519}}, there exist {{formula:18756307-e7df-4521-ae35-796a595ea8f8}} depending on {{formula:ed01ee70-0662-49da-9ea0-1a8b6ce9a9f4}} such
if {{formula:a38b2c91-e219-4419-939b-d80d5534b1ab}} then the number of
isolated roots of {{formula:76020ba3-9996-49a2-9a74-a2ce07f87f0a}} with {{formula:43ac2ad6-79d0-4498-a608-d5ade6dad452}} is at most 2 (counted with multiplicities).
Case {{formula:337595b5-c05e-4389-853d-7598d877d921}} . Since {{formula:d3e8d819-3fc8-4419-9bab-d938fde4757e}}
by assumption, {{formula:0918753a-3a8d-49f8-8dcd-5ba82c62dff9}} in Lemma REF implies that there exist
{{formula:94b3de69-2996-439a-88b7-b4277f3a272e}} (depending on {{formula:b6cedb18-18df-48bc-b517-095600df8cba}} again) such
if {{formula:082928da-5f79-45ab-a162-f0f5c595ea7d}} then the number of
isolated roots of {{formula:560d8a3c-3187-405d-a33c-52c88dc4c58e}} with {{formula:8dfc235d-6637-40da-af8f-c6a631a7b65b}} is at most {{formula:78e1fb68-3001-4bf7-a437-2ba8c2b56b08}} (counted without multiplicities).
Case {{formula:22cab3d9-cde4-4d8e-9411-0a5ecc7843dd}} .
If the center of {{formula:e2825cbe-a659-4d15-8e7b-6923c04ee488}} is not isochronous then there exists
{{formula:3ae3261c-084d-41a9-9f89-bc1fe8fb9be5}} , depending on {{formula:1543b73a-3981-4122-b644-f7586e229b01}} , such
that {{formula:37ffa83a-ba15-4084-b45d-7d2da3d6b57e}} By continuity there is
a neighbourhood {{formula:5f3c9cdb-5842-419a-9242-1bb666d550ab}} of {{formula:66e4b120-bb7d-4eb0-aed2-1a5010487027}} such that
{{formula:77903ae7-787b-4003-a8a8-32cbf22aec71}} for all {{formula:810fe5e9-16cb-407b-bd67-3f15a6e0a276}} Hence the application of Rolle's
Theorem shows that there exist {{formula:d29ce57a-7960-4936-8a33-aa77885e504d}} such
if {{formula:7e91d71e-0248-43ff-a099-4e7ca8cb9c2a}} then the number of
roots of {{formula:539ab840-6aff-498a-a23a-e20c240ae907}} with {{formula:4c31b6ec-5eb0-4fc7-96f2-ba2e922cf96c}} is at most {{formula:b163313e-074d-4daf-acc3-bf9d561394a1}} (counted with multiplicities).
Let us suppose finally that the center of {{formula:48b7b39f-274c-4acf-bea8-b47bb8a25a81}} is isochronous. Since
{{formula:8f674129-bedc-494f-87f8-22037a262384}} , and by taking for instance the
flow of the orthogonal vector field {{formula:9e323963-687b-41c6-9d67-5c814928e1f9}} , there exists a transverse section
{{formula:e17fc9a1-fefe-4fbe-8c0d-05d9cb509d88}} given by
an analytic map
{{formula:6fc978fe-9faa-428b-ba9b-b144d042fa63}}
and such that {{formula:5ceebaab-863b-40bb-84b0-5b8ee3264773}} We then define
{{formula:f7458766-fa0a-40d5-8175-b50b53b71786}} , which is clearly analytic on
{{formula:5458bc4a-79e9-4e21-8062-cf9e70d4939f}} . We can thus compute its Taylor's series
at {{formula:4e8c584b-e560-419e-bb56-e13a6c1d2c50}} ,
{{formula:becd0d3d-f7c6-48f7-a3f7-0ab06d52e791}}
where each {{formula:27ee8a1e-1a70-4016-aab4-d7ac7953e3de}} is an analytic function on {{formula:e40dcfa5-b232-487e-899b-26a6c0f5329c}} with {{formula:816b46f5-1435-4d46-a14b-a7127e39daaf}} .
Working in the local ring {{formula:077f54e6-bfb5-403e-a6ee-1974544f5cba}} of convergent power series at {{formula:feb1b84b-4ab1-423e-ba63-52da0a3bacc9}} , we consider the
ideal {{formula:baf29e82-596b-4749-8927-0f26a6917691}} . The ring is Noetherian and so there exists {{formula:805f4650-454a-4c3a-900c-cbdaad92fc5f}}
such that {{formula:07299c10-499c-49e0-8697-2ef5fbbf6054}} Verbatim the proof of Chicone and Jacobs for
{{cite:2bf3254451791a68ce817916f287f1704a0d2519}} (see also the result of Roussarie in {{cite:44c371fa22acae495f26ba888c8c78a3e7a58114}} for a similar
result for the displacement map), there exist analytic functions {{formula:bdbb4bb2-9820-49b1-9558-5087b4f794fd}} in a neighbourhood of
{{formula:b974ef64-441d-4758-a8d1-09bf43339d8b}} with {{formula:51374891-5350-4329-b1d7-37a7b869f1ab}} for {{formula:b4a8e586-48f2-4ab2-be7f-9f358e88bda8}} such that we can write
{{formula:e738622d-76dc-471c-ab30-373d326804e8}}
Now, setting {{formula:a7f02eb2-85c9-4a6b-9955-d9d5e82d8467}} and proceeding just like the
proof of {{cite:2bf3254451791a68ce817916f287f1704a0d2519}}, one can apply the well-known
derivation-division algorithm and use recursively Rolle's Theorem to show that
there exist {{formula:8304c151-de2d-46f5-b61b-d84577b95b13}} small enough such that
if {{formula:b0509a2e-4fe5-416b-9428-33cd6347cb44}} then the ordered set {{formula:dd5ce3d2-835c-44ae-a630-40edb3a0ac38}} is an extended
complete Cheybshev system for {{formula:464a38bb-df9c-4f7f-bc41-dd36241cbaec}} , see {{cite:8bf7ff9b569dc3a6597a334ac0db290966d80658}} for a definition.
Accordingly
if {{formula:f8c8b378-be25-4e48-85e9-720cef6103d8}} then either {{formula:b0392a20-15c1-430d-992d-d8ad54311cea}} or {{formula:b11546ac-85ea-4db6-81fc-8df3dbe5fc7e}} has
at most {{formula:cd7f9828-2379-4906-b858-9677757602ac}} roots with {{formula:d1a3871d-3eef-4a2d-aff4-c5667954136b}} counted with multiplicities.
Using the original
parametrization of the period function, this shows the existence of {{formula:f9dbfdf8-38a9-4b2c-bc36-499a40fe2057}} small
enough such that
if {{formula:8bed2757-9d80-4423-ae88-8995ae6553eb}} then
the number of isolated roots of
{{formula:257918d7-3fc4-49f0-a2d2-ba2575e3a791}} with {{formula:d1382af2-579e-4eb5-94ea-08a13c8601bc}} is at
most {{formula:8cf28aa2-ed1c-43e7-9692-ba857934648d}} taking multiplicities into account.
Since in each one of the possible cases there is a neighbourhood of {{formula:1b30a3ed-f397-4e73-8b83-a53fb653e5ee}} where the number of critical periods is finite, the result follows by taking a finite subcover of {{formula:87706959-b442-4951-9091-3841678a998f}}
Asymptotic expansion of the period function
From now on we focus on the quadratic family {{formula:64e6cd71-ca3b-4955-977d-210b9c5138bc}} given in {{formula:6103a869-bd15-4e17-b869-14f9abd78fd2}} and study the period function of the center at the origin. In this section we give its asymptotic expansion near the outer boundary {{formula:39526e5b-c32a-4de7-88db-6e1804a0d7a7}} for parameters {{formula:06d8ec4a-bf2d-4389-9bf3-4dce9d33561b}} inside three specific sets (see Figure REF ):
{{formula:f1216670-fd05-4500-8c6b-80367f19ab6f}}
In all the cases the period annulus {{formula:75aec4f6-097d-4d9a-b31e-e3f20382a8a3}} is unbounded. Since the vector field {{formula:bec6d992-c8f3-440d-8885-9d8c9d00ad60}} is polynomial, in order to study the behaviour of the trajectories near infinity one can use its Poincar√© compactification {{formula:caa5859c-8dda-4fdd-8e6e-53e590cf400b}} which is an analytic vector field on the sphere {{formula:d4ca2cf6-ee48-4128-9e6b-104aeddd660e}} topologically equivalent to {{formula:ac4e49f2-a376-4e56-add8-c091404d4803}} , see {{cite:26076f4ccf158d95a2bd6db4360d3d70ae043e1c}} for details. The outer boundary {{formula:2c4b8327-15af-4879-8420-78615585ebe4}} is a polycycle of {{formula:7e39f9d9-5d24-48f2-a01a-2639b7fdd3ff}} that can be studied using local charts of {{formula:7ae8e556-af09-45dc-b4eb-6ba49e98375f}} In doing so one obtains (see {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}) the different phase portraits in the dehomogeneized Loud's family {{formula:110a6a6c-20b6-4979-8dde-86ee66ea88f3}} . For the parameter values studied in this section it occurs that the polycycle {{formula:44db1d8b-7f93-4851-b84c-56f75f7e1c54}} of {{formula:f87d9f14-fd3f-486d-b343-ee32892e71eb}} is hyperbolic if {{formula:62656d95-44e5-4a5c-9b5a-1e21b3172986}} and has a saddle-node singularity if {{formula:109bcf15-a076-40f9-97c1-e90cafd54db4}} With regard to the phase portrait, it happens that the affine part of {{formula:0c011e86-bd18-4da1-88b2-0af980884df8}} is a straight line for {{formula:1851ddcb-7674-4eb2-bdb7-b86d699ec539}} whereas it is a branch of a hyperbola for {{formula:5915898f-c5ef-47c0-947a-4f2b6397e3a7}} These are the reasons why we split the parameters under consideration in these three subsets, which are studied in the forthcoming subsections. Concerning the behaviour of the period function near {{formula:538cbbf6-2106-4fd6-9973-67d610079aa7}} , the dichotomy between local regular value and local bifurcation value (see Definition REF ) is solved for any {{formula:139a3b7e-7b97-4f25-b4c9-f24eb05e017d}} thanks to the results in {{cite:f73aa8df8bfb916a82c4b022d26828630a4e01e9}}, {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}}, {{cite:232b30000fa4402e54bb8562c9f9c09188965a2d}}. In these papers it is computed the asymptotic expansion of the period function to second order, which usually suffices to tackle the regular/bifurcation dichotomy. However in order to study the criticality we need here to go further and compute the third, and even the fourth, order expansion. Let us advance that the asymptotic expansions for {{formula:4efa1ccf-3ef3-4a8d-989d-cec3079592f0}} are given in Proposition REF and the ones for {{formula:d8f8c544-3ae9-4135-8c30-53a961f15541}} in Proposition REF . Being the proof of both results rather long and technical, for the sake of paper's readability we postpone them to Appendix , where we also summarize the fundamental results and definitions from {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} that we shall use here. Among them we point out the notion of {{formula:eda4a019-06b5-4de1-9e33-87263b72f0ba}} -flatness {{formula:65ac16eb-4708-49a6-8c2f-f3d1e0c6a072}} , see Definition REF , used in the remainder, and the Écalle-Roussarie compensator {{formula:076842c9-223a-46b2-8def-39057f95b473}} that is a deformation of the logarithm used in the monomial scale in which the asymptotic expansion is given.
Definition 3.1.
The function defined for {{formula:847c3f6b-d225-4c4c-858f-09666fdbd857}} and {{formula:9e6bc771-9324-455b-bdf2-2189bc1f1755}} by means of
{{formula:006d7229-298d-4613-a634-2e3e7e131969}}
is called the Écalle-Roussarie compensator. In the sequel we shall also use the notation {{formula:6d1196e9-62d4-450c-93d3-23edc20f102b}} .
{{formula:503c8269-11f3-4afb-b388-2251bf5e1ee3}}
The asymptotic expansion for {{formula:02c4fcdd-d8f3-470e-b700-cf1a7ea6da55}} is given in Proposition REF and its proof is of a different nature due to the occurrence of a saddle-node bifurcation at the polycycle.
Study of {{formula:b10376d6-6725-4a7a-b760-668fb40c4eab}} and {{formula:b0746169-b023-444d-bd88-88206e48b8cd}}
Figure REF shows the phase portrait in the Poincaré disc of the vector field {{formula:d5d3dcfb-45e8-40e4-8151-833ace7bfcef}} in {{formula:56956f78-fa6c-44bf-9b55-6ab2f8fc95f3}} for {{formula:77d6f047-a8eb-409a-9698-d9aedfad1f48}} varying inside
{{formula:2d65ca10-e09b-416f-9aa9-871e5279a2cb}}
We take transverse sections {{formula:f416db5a-da35-4eed-be1b-9c32d45cc1a6}} and {{formula:255dd71c-ff3c-400b-89ee-68374abcbd7c}} parametrized by {{formula:7939025d-e4bb-4248-a3b4-80ba5c25cf45}} and {{formula:c0dee86c-92d8-4a74-a725-228e96d02869}} with {{formula:c221a156-eb15-47c1-b6c1-fb59658edcc3}} , respectively, and
{{figure:36452eae-51df-4921-bd50-24c5db2b76a1}}define {{formula:aa996536-ce68-411e-b9f0-fa1f40a3900c}} to be the time that spends the solution of {{formula:b117ef6a-ede8-4568-bf32-f7c19cbc9a63}} with initial condition at {{formula:8aeb61ae-6e4c-4bd1-947c-1185d00f8f01}} to arrive at {{formula:504a4f78-0469-4d6b-aaf7-e6d084fd247a}} Thanks to the symmetry of {{formula:9b71c646-011e-496d-880b-e7781f0834f0}} with respect to {{formula:211ba6cb-ec59-42d6-b1ae-5780da93891c}} , it turns out that
the period of the periodic orbit passing through {{formula:147a665e-9b80-42b1-9335-5c096ca67ed6}} is precisely {{formula:726839d0-1944-4e21-9d2b-54d82321b3b3}} . Consequently the emergence/disappearance of critical periodic orbits from {{formula:79366ca5-19f6-43c7-a00b-6f5659b2fb0b}} corresponds to zeros of {{formula:14d6030b-5eb6-48d3-9672-e61c69559ec4}} bifurcating from {{formula:36696ef0-1dad-4362-9217-08c9b2bcd2a5}} more concretely to the number {{formula:e8767880-1731-4953-a52a-b2cc3a4925f1}} as introduced in Definition REF . A key point to study these bifurcations is that {{formula:c120e750-947c-4c41-96dd-7cbc1c38d3d5}} is the Dulac time associated to the passage through a hyperbolic saddle, which is at infinity (see Figure REF again).
Therefore we can apply {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} to obtain the asymptotic expansion of {{formula:8857c71b-f0da-4291-b853-22e8e5148eda}} at {{formula:14ac20e0-482c-4499-9b9d-75d18f2dc47e}} and use then {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} to compute its first coefficients {{formula:440f2fac-6140-47a0-86d7-46e4a3af6783}} . Next result gathers all this information, where
{{formula:89212883-f4cf-4f3f-9537-bade17446b62}} denotes the gamma function.
Proposition 3.2
Let {{formula:7cc98aec-fd09-4099-8d52-374671771460}} be the Dulac time of the passage from {{formula:6265ad05-0e26-4782-b9c0-89ba49a3849d}} to {{formula:3d672ccf-4940-4fe6-8910-0a9e2af68413}} of the saddle at infinity of the vector field {{formula:cc894a55-c39e-41ca-97b4-6dab6c5195ca}} in {{formula:70a71b17-0404-4181-98c4-e9329b3c9898}} for {{formula:7269b5dc-9271-4c18-9d49-ac088d075b9e}} . Then the coefficients {{formula:6f46f506-8ada-4485-ae63-f00d16e0f7d0}} {{formula:85906472-4c60-4690-9d81-0e568478db02}} {{formula:fbb7e331-b9c0-494d-ab0b-d16b02bddc17}} and {{formula:3b8ab68f-6b16-4c50-bb9b-501839ad3dd7}} in its asymptotic expansion at {{formula:a64bd1fb-fe0a-4b29-a39f-7cea4ca794f0}} are meromorphic functions on {{formula:d33ade07-3787-48fd-9342-fab95269fb5d}} that can be written as
{{formula:5df3bfdd-4c38-4d10-8934-fab403f5c747}}
where {{formula:1f25a1a0-00fa-4ef6-8e52-95d86afa7271}} is the hyperbolicity ratio of the saddle,
{{formula:24d0ec18-7855-42ba-9f02-5c5f3b1f603c}}
and {{formula:902fe348-33ad-4af0-be61-898dd6739fdf}} is an analytic function on {{formula:be15a400-462d-45ef-97d2-e6a8a94836c2}} In addition the following holds:
If {{formula:eaa2a4e9-0cc2-4578-b834-ae79b7bc0346}} then, for all {{formula:b15a8ee8-b3cd-4ab5-b8d6-f393a216fbbe}} small enough,
{{formula:79b57ff3-0a54-4708-9b1e-4ac315d7de5b}}
with {{formula:244222a6-c7fb-4d11-80ce-cab1276826ed}} . Moreover {{formula:f602be50-a2ec-466e-a734-fda358d97a9a}} and {{formula:a467a2f7-75e0-4ac6-9def-dac188b1c5fd}} for all {{formula:6f6245bd-6fd1-42db-89f6-552e452542db}} .
If {{formula:bc3191f0-982c-4963-b94e-11438ddb1142}} then, for all {{formula:6be82a48-4e67-4dec-b01b-f533f9d4acfe}} small enough,
{{formula:3c380e75-c1d0-4c9e-b428-bc936f03b91c}}
Furthermore
{{formula:b2cd258a-d261-4e65-9a94-d6952a827a18}} and {{formula:d1bed6ac-3ba7-40b5-98a3-c308d38c30a3}} for all {{formula:c5da2dd1-c84b-4005-8fb4-ae4b9108d202}} .
If {{formula:f538e2c3-b1f4-4d53-860b-c0832e37131f}} then, for all {{formula:9a346cfb-5029-4ea0-a90e-052f54e8c35d}} small enough,
{{formula:51e5551c-c840-4818-bfc3-d3465587026e}}
where {{formula:bee44795-63c6-4d36-96b5-b0ee3c400e3a}} and {{formula:a0d1b27c-edd9-43c0-85ee-1e334648f861}} are analytic functions in a neighbourhood of {{formula:ecea749b-9a83-493e-8e69-8b0152b0cfd7}} . Moreover
{{formula:2fb62da4-95ad-4567-b05f-8107907dbbd7}} and {{formula:e424265e-1b16-4c67-8f08-35bcb58aeb23}} .
If {{formula:80d13f99-4169-4fa8-8b77-a4416915b094}} then, for all {{formula:e4d163ab-81f0-43ba-b5ed-465689a695df}} small enough,
{{formula:826a38fb-1c64-4fb7-b5fd-df5daf42cc19}}
where
{{formula:cf7282d6-7a47-4a3d-963b-6ed29c9dcade}}
for some analytic positive functions {{formula:ab57cefc-884f-49ff-b36c-d3234c5f115c}} in a neighbourhood of {{formula:4fcc8ad0-5996-4ed1-b64e-70f4179f2aa8}}
with
{{formula:206a7cef-357a-493a-9e72-4c34324e8069}}
As we already explained, the proof of this result is postponed to Appendix . The monomial order in each one of these asymptotic expansions is with respect to the strict partial order {{formula:8c1a9e2f-f053-445a-8049-4d2d90658194}} given in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}. Let us recall in its regard that we write {{formula:c7142576-ad85-40f8-8dde-27dc3525e201}} in case that
{{formula:63afc2a8-5a53-4fd2-a23c-d87ab08d31db}}
For the monomials under consideration this order is preserved after derivation with respect to {{formula:c49613de-2768-41f3-8721-299b47a777f1}} , and so it is the good flatness properties of the remainder.
Thus, as it occurs with the Taylor's series of an analytic function, an upper bound for the number of zeros of {{formula:22e3679e-8d75-4470-bb55-884f92d51e74}} that can bifurcate from {{formula:372c90f3-c1ef-4ca9-87e1-fc32e42c08fb}} follows by identifying the first non-vanishing coefficient in the asymptotic expansion. For the proof and a precise statement of this result, which essentially follows by using the well-known derivation-division algorithm, the reader is referred to {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}.
Study of {{formula:18b7cfa9-0b1b-4904-8384-8da46d89a58e}} and {{formula:3d1e7219-6066-450c-bf05-ce575bb8c1d6}}
Figure REF shows the phase portrait in the Poincaré disc of the vector field {{formula:39675137-0924-4f82-9f5b-4f1fd6f8183c}} in {{formula:69fff2fa-ea03-496a-ad44-c4a1fd126eca}} for {{formula:b1be5670-f902-4357-b8c6-a390501ace57}} varying inside
{{figure:5882f837-cd7f-4a32-86cc-b86431c939d4}}{{formula:67bb10ed-7e39-4309-a30a-3d81bb6adbdf}}
In this case the outer boundary of the period annulus of the center at {{formula:caa2879e-cf17-42bf-88d8-fbb47ee48e4f}} is
contained in the union of the line at infinity and an invariant hyperbola {{formula:96c5e2aa-1022-454e-83f7-1e47ffd82f3c}} , where {{formula:85d5df65-8e60-42c0-8638-cfdb083ebd3f}} with
{{formula:80bf97a6-937e-453f-9fd4-b7cf6eefcf8c}}
One can verify that if {{formula:a1d6efb3-6d83-4f33-8258-a8344bc44c6e}} then {{formula:681469eb-fc5d-4db1-bca9-d1ff4030d0a2}} has two distinct real zeros, that we shall denote by {{formula:50c3047b-6a3d-4940-9cb9-dee3aa0e5367}} and {{formula:c4652610-0085-4bd7-9d0d-b70497e4de2a}} taking {{formula:5cdedfc1-2b38-4538-91f7-ba1ef052431f}}
That being said, we place two transverse sections {{formula:1c0b2582-6f41-4162-bde7-985a16dde8ec}} and {{formula:2ea95398-18d0-4fad-8e5a-08c72ca727ff}} parametrized by {{formula:fe75d611-8f0a-45d1-b5d6-ed59aac57329}} and {{formula:a10a0b6b-d798-4b9c-85a3-6d0e74650966}} with {{formula:ea11cad9-38b7-4b60-bbce-da363385bfe7}} , respectively, and define {{formula:246d0267-1721-404e-82a8-7d615376abcf}} to be the time that takes to the solution of {{formula:c32fa8a8-fd35-465e-ab6b-a61ddeedcec7}} with initial condition at {{formula:0cd00293-9ffd-4272-9e60-dedfdee41040}} to arrive at {{formula:ee0e92c4-277a-4a5c-b3e0-63eda2ae5764}} Then {{formula:4aa671c3-9a66-4181-ae1a-64d7cf9f81d7}} is the Dulac time associated to the passage through a hyperbolic saddle at infinity, so that we can apply the results in {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} to obtain its asymptotic expansion at {{formula:a2582a20-b931-4941-8e4b-95e4d0b949b2}} This is important for the proof of Theorem REF because, exactly as in the previous case, the symmetry of {{formula:2c66ad05-3117-4949-b2d7-7813d6e0c375}} with respect to {{formula:ba80b5bd-a465-49d9-b1cb-d3136fc3b010}} implies that
the period of the periodic orbit passing through {{formula:ca1c09d2-cb94-4a3b-bd36-ba6078ba0be7}} is {{formula:7bef61d8-ee74-4d56-8ef2-a8d3f1bf62ef}} . With regard to our next result we remark that {{formula:bddf3917-ba73-4edf-87ef-3cfec389062e}} for all {{formula:96781c73-fde3-4aaa-bfef-769b1bddedfc}} , which is relevant since the hypergeometric function
{{formula:e1e0180e-fd95-4449-bf67-0c1ff1cb3af8}} is holomorphic on {{formula:f16069c1-300a-4c48-b496-2b8812f4285c}} , see Appendix .
Let us also mention that {{formula:378e3d84-854f-46aa-80df-bd3771f422b2}} is the beta function.
Proposition 3.3
Let {{formula:21c17192-4cea-4454-ad5a-2c15b91d0371}} be the Dulac time of the passage from {{formula:70a13a94-2660-46ca-8d8b-b3aa74c7473e}} to {{formula:266832a0-5552-467d-bdf4-0b3cd3cb5909}} of the saddle at infinity of the vector field {{formula:66e9d06a-2db0-4414-98d5-cff043a5aef8}} in {{formula:1fef0556-58ac-4b6d-9dcd-c5b25d5452d8}} for {{formula:32368166-3b01-4f0c-ac35-13341770e968}} . Then the coefficients {{formula:70667ae6-6077-4b60-878d-61e884191e3e}} {{formula:2d99b24c-03a8-496e-989a-be2680fbbf53}} {{formula:b6f0e993-092d-48d7-b2ec-d6d13c5b822a}} and {{formula:b4860948-db89-4d7f-b68f-2ed62d76379d}} in its asymptotic expansion at {{formula:14773842-85a8-4b29-8411-f7ae1da3b799}} are meromorphic functions on {{formula:a25aa265-5424-4e85-8d84-344e0d52fc44}} that can be written as
{{formula:2d90f4b6-0376-44dc-a0ef-b4ed35039e9b}}
where {{formula:8934fa80-dac1-4928-b6d2-8c451b06d85a}} is the hyperbolicity ratio of the saddle and, for {{formula:1564951e-55d1-4d4e-8155-e338a6dc062e}} , {{formula:8cf3a93c-5609-4c49-a426-497d0aed0bcf}} is an analytic positive function on {{formula:8d95f36a-7b2f-4632-9387-418252538463}} . In addition the following holds:
If {{formula:aaa7c84f-d489-46d9-a065-8f2be4d31b43}} then, for all {{formula:90e5d018-7aee-446b-a15a-391ad4282440}} small enough,
{{formula:df39a745-60b9-485e-bec5-2b7683d6f076}}
with {{formula:871ad27f-3f86-4fee-9cf2-c29990db8dc9}} . Moreover {{formula:ed7767a7-c010-4d26-bebf-dae5b8a912ef}} for all
{{formula:b5060da7-1a82-4b8b-8059-56bb805d3680}} such that {{formula:cf0b4153-0abf-44ca-b5c7-f1ffd2fd8a29}}
If {{formula:def23218-66b9-4c02-8efb-f066ae34a0c9}} then, for all {{formula:4460a7cd-4f37-4271-89d9-cbf6c11d006c}} small enough,
{{formula:4d4facf6-ad60-4f57-a986-acf76d35770d}}
with {{formula:36631dd9-9aed-46af-99fb-69d71a3ae0f1}} and there exists a unique
{{formula:04724ec0-d8fb-4b08-b87d-c46a1ee2daae}} such that {{formula:a531bc49-d96c-4ade-a2ff-441dd06d9e54}} and
{{formula:d22599da-d7dd-4ed6-9783-d2865bc7cda6}} .
Furthermore
{{formula:43b9c643-20c8-494e-a1f9-80251cad7352}} , the gradients of {{formula:feb479ed-803b-46b7-957b-97646133a26a}} and {{formula:360e1885-ad46-4267-83b4-134aff8a8491}} at {{formula:67581eb2-f9f1-4d94-91f3-2cbc992c45b4}} are linearly independent, and
{{formula:aa2066f0-56e9-47a3-9e15-fe2957bbc44b}} with {{formula:995575b9-b7e2-429d-ad96-2f822586b796}} .
If {{formula:78fb3077-24ee-44b6-810d-07ed8cd1171e}} then, for all {{formula:65a177b4-0965-4002-8a9a-3c1e63a34925}} small enough,
{{formula:5902af8c-3222-4401-8c12-1ebeb10f991b}}
where {{formula:8a57c9b9-916a-446f-9047-83e487caa4ec}} and {{formula:e044597e-171a-40fe-b9a0-820daaa410cb}} are analytic functions in a neighbourhood of
{{formula:9a917f2b-c714-48e0-baf0-838e9de7567c}} . Moreover
{{formula:5187ae88-c8ae-4ca2-9093-3a5288e2b6ee}} if and only if {{formula:f82db4f2-efbd-478e-b4ac-903f709b7558}} , and {{formula:2e3de582-37df-44cf-b838-2f71d5d880af}} .
If {{formula:37f682a2-8dd1-4d61-a9fb-a5982bcbb4cd}} then, for all {{formula:20360c9f-f610-4338-a632-8ddac4663419}} small enough,
{{formula:f120b7a8-cf8b-4035-be78-7091b0acd300}}
where {{formula:4d4df8cf-182f-4834-8c7e-426762888f1c}} and {{formula:5495bada-c21c-42f3-9722-4511bfa23525}} are analytic functions in a neighbourhood of {{formula:e8236a23-5bce-470c-ada6-ec835156d544}} .
Moreover {{formula:33b1d2fd-1823-42d4-a45c-5b6a1b1b1d92}} for all {{formula:ecaac2f6-9e7c-401f-9786-4d2a2c3e0091}} , {{formula:1c27c04d-ce36-4fc4-bd11-03467b9fc5d0}} if and only if {{formula:4431ce16-2997-476c-9590-9dca3a089e08}} , and
the gradients of {{formula:436ffe18-d5fc-46b5-b1a0-4410f79e35ac}} and {{formula:7108cef0-aead-4fc2-bdbc-794990c9f72e}} are linearly independent at {{formula:031e5960-801f-4a41-bcfc-6a914be6bc1a}} .
The proof of this result is postponed to Appendix .
Remark 3.4.
The asymptotic expansions in Proposition REF were already given in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} but only to second order. In that result it is given, among others, the expression of the coefficient {{formula:4ba1268f-ccf1-4a74-861f-cf90d513c9e8}} in terms of a definite improper integral. Furthermore, see {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, it is proved by applying the Implicit Function Theorem that the set of those {{formula:85456948-1bc8-46d9-8fe0-f132c37750fa}} such that {{formula:73a55677-07c9-492d-bf41-51c5ebb583c2}} is the graphic of an analytic function {{formula:d5a363d2-c6d6-4ed6-afe1-7c53367006f3}}
This is the function that appears in assertion {{formula:e9c4e490-fab4-4d6e-9f3c-68b114c174e3}} of Proposition REF . Thanks to the results in Appendix we can now identify the improper integral as a hypergeometric function, so that we can write
{{formula:7029ae2d-abaa-467d-b13e-23c1fb5bcef5}}
where {{formula:66325919-a2ba-47cf-aebc-f2dd4b5a41ee}} and {{formula:08fc2bb4-da40-4be9-b0cd-8d2903e280dd}} with {{formula:18449baa-ad89-4c56-8444-47db9fa77a21}} are the real roots of {{formula:a9e9eba1-9d09-48c7-964f-798273f49714}} and {{formula:7256b757-fa39-4034-b602-485e39be24e7}} .
{{formula:7cbe677e-0c6a-48f5-8bb9-0a865e51703f}}
Remark 3.5.
In the statement of Proposition REF we refer to some positive functions {{formula:813345c2-70af-4c6e-b3e6-a4a0aa43bf3b}} Let us mention that in the proof we show that
{{formula:5b7babc8-c079-4fb7-8323-6a7baf04143f}}
We do not use the explicit expressions in this paper but they may be relevant for future applications.
{{formula:d3b79c2c-f94c-4955-9939-70712a6331fa}}
Study of {{formula:1a6c911d-8dad-4c57-a206-92ec863b5110}}
Our aim in this section is to study the period function of the center at the origin of {{formula:f56ec760-19bb-4092-8643-d72bc5963cf2}} for {{formula:994a667b-cd0e-43a6-bebf-d852b100ef16}} with {{formula:64e0d1f2-9fc8-4fdb-a47d-cde1621a7d7e}} and {{formula:a7197960-5cd4-4ce9-8896-114dc094e02e}} .
{{figure:9d13cc34-9829-4bff-9c52-3943ab75a8d4}}To this end we introduce transverse sections {{formula:60a55b1f-e47c-4485-ad73-020611e32e96}} and {{formula:7864f3cd-084d-4293-b3d2-6e5c58d7a4f2}} parametrized, respectively, by means of
{{formula:1e02f597-112e-446c-aeca-f2589dbd8a34}}
where recall that {{formula:32d02de2-bd8a-4b29-a8f5-281bb53f426d}} with {{formula:4e8abba0-3ce4-49d2-8284-9f0fc42a5190}} for {{formula:a105b73d-baa8-472a-8f63-622e403851f1}} . One can also check that {{formula:af1b9c7c-e78d-45bd-8790-8db3565ca484}} For each {{formula:fe91399e-bf54-455a-856e-a91165a0e39e}} we define {{formula:578c08b9-3648-4c60-929d-171c3cfacbcd}} as the time that spends the solution of {{formula:bee5352c-7c45-4be3-a9a5-ee3a4a96a5e3}} starting at {{formula:909994a7-762e-41cf-9478-38b796add4bd}} to arrive at {{formula:22d8855d-c747-4af0-b4e5-e90b319cc74d}}
A key feature of this Dulac time is that the singularity for {{formula:36b06248-8488-440f-aaa6-bf44aee91e2a}} is not a hyperbolic saddle but a saddle-node. Our next result gives the asymptotic expansion of {{formula:ab385669-5757-4a59-a976-ccc43934cd64}} at {{formula:fd2b4ce8-bd70-4a90-ae8c-de625b1dcf41}} for {{formula:75f8b4af-ae5e-40ee-b0e1-a567886966ee}} . We point out that this is relevant for the proof of Theorem REF because the period of the periodic orbit of {{formula:d128ab8a-297c-4606-9c2e-e5ecf094a889}} passing through {{formula:396d8041-d4f3-4c4f-a4e9-275e77042dad}} is precisely {{formula:f41b86ea-863d-4353-9796-c958feaf8f24}} due to the symmetry of the vector field.
Proposition 3.6
Let {{formula:0743e98f-3579-4e2c-8ebd-25b599d5aa80}} be the Dulac time of the passage from {{formula:98059986-be1e-4a88-8e77-004def8452e0}} to {{formula:b2115222-252a-4ade-84c7-e70a82bd83c1}} of the saddle-node unfolding at infinity {{formula:b5f160af-91f3-4cca-97c9-de68e27ea7b3}}
Then there is an open neighbourhood {{formula:bfa0d9f1-bb60-4f19-a15b-ce4621f5eaaa}} of {{formula:9c262bd5-2de8-49b2-b29a-cf14b49af7f2}} such that
{{formula:eb29333f-715a-4285-aefd-4d0f93ee69a5}}
where {{formula:eae31e51-86b7-4602-8669-51b359911700}} and, setting {{formula:1f625c14-9a6c-4f46-a65b-c26938200346}} , {{formula:a248836c-fd4e-48c7-ba93-eeade9a2c101}} uniformly on compact sets of {{formula:9eeee2f0-de8d-46a2-a3cf-975388ceb0e2}} for {{formula:c9dab775-1cb1-4af2-8af0-95c6cf2ba445}} . Moreover {{formula:486cc09d-2952-40b5-9a8a-e369cb9c3404}} if, and only if, {{formula:3833d15f-d879-418b-8545-3be847264d7d}} . Finally {{formula:ca22d5cd-9708-4c89-8e1b-71a317e4feb7}}
Proof.
To study the saddle-node bifurcation that occurs at infinity we work in the projective plane {{formula:d09aa3bf-093b-401e-81c3-b147fc8487a5}} and perform the change of coordinates
{{formula:580b61bb-d6a5-414a-8c12-d971c4107da2}}
The meromorphic extension of {{formula:a2ead40f-fb57-4c87-a7a9-f5cc950c9f07}} in these coordinates is given by
{{formula:51e56ffa-b10c-4a97-b917-7eccad157ea2}}
with
{{formula:4b2391fe-b93a-4fa5-8e5e-bf4b8ebba8c6}}
Our first goal is to show that we can bring locally the saddle-node unfolding to a convenient normal form in order that we can apply the tools developed in {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}} to study the asymptotic expansion of its Dulac map and Dulac time. With this aim,
some long but easy computations show that the local analytic change of coordinates given by
{{formula:dbd0609b-d4ce-4f8b-a671-7bfa2c98a0a6}}
where {{formula:40bfb0bf-0cb2-479c-939c-d910befa1256}} brings the vector field {{formula:518b82da-eff4-4f27-a565-08e6a4adaba0}} to
{{formula:ee33ea50-ca79-4715-b017-bbbaf009dffc}}
with {{formula:6bd10712-0329-4e38-8960-ac58f9423af5}} .
A technical assumption in order to apply the results from {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}} is that for each {{formula:d224781d-0f23-47f2-a050-5948d39e5e5e}} the Taylor's series of {{formula:6c573d89-0151-43ed-b38a-4441ff567fbe}} at {{formula:5b575336-ecf5-4841-838e-3dcacf602f21}} is absolutely convergent for all {{formula:2862cb19-a248-460c-8166-f9c94506bb0d}} . This is not fulfilled unless we perform a rescaling which is only well defined provided that {{formula:913769dd-ff45-4f98-9526-60f50de4c63e}} varies inside a compact subset of {{formula:44dbbff1-e56f-406d-afd6-fee2facababf}} and this forces us to work locally. For this reason, as a first step in the proof, we will show a local version of the statement. More concretely, that for each {{formula:d22355c2-9395-4427-a6d3-f9f60d358e87}} with {{formula:6c014854-28fa-42aa-905f-cc76e6b37ae7}} there exists an open ball {{formula:b69aec99-7c75-4b44-8270-d166217f9b3a}}
such that
{{formula:4113ea1b-bac3-4e2e-9463-4430babc61a4}}
with {{formula:55e0ec9b-9809-4b71-8b6c-8c0f6b628ea0}} continuous functions on {{formula:be05c648-1d46-4742-81e6-2e2df6a4591f}} and {{formula:570137c4-0faa-401d-ae3f-4a8d658e9af5}} uniformly on {{formula:929bb2d9-6b43-47d1-9fe1-1c35e6b05d9c}} for {{formula:2c602d8f-26fe-4b1f-9735-8385b138f2b0}} . To begin with we take {{formula:60f9eb32-b2c0-4537-9697-d8e34c75fea9}} small enough so that the closure of {{formula:87aa0ef2-73a4-442d-b3f6-5a454b355b06}} is inside {{formula:53395a96-2d5a-42b5-bc86-181d437e6a2b}} and define
{{formula:b7a26e2e-48ac-4cd5-8eb0-403df700e4ce}}
which is strictly positive.
The pull-back of {{formula:ae64be71-8bca-4149-8044-10ae69306a5d}} by the rescaling {{formula:81e0a479-f97c-48b3-9eb0-739b03397840}} can now be written as in {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}} because
one can easily verify that
{{formula:8fe62816-c7fb-4eb6-8495-7966cd020cbc}}
and where the Taylor's series of
{{formula:9aa0fe43-12b7-40b4-8530-d482f6e3c7a6}}
at {{formula:ce3b23fd-948f-4c7a-a7d0-65a47cb26851}} is absolutely convergent for all {{formula:db57d184-b020-4b3e-a8cb-acbcf1990fe7}} and {{formula:240e50ee-8ba2-48fc-93c2-695f96e4d129}} since, on account of {{formula:c5bbba9a-685f-43dc-b724-4b562dcd68d4}},
{{formula:1d90763b-7e2b-436f-8b26-82e25e528c94}}
In these new rescaled coordinates, that we still denote by {{formula:1f758ac9-7136-4718-a3cf-78c581a725ff}} for simplicity, the period annulus is inside the quadrant {{formula:db6bcc05-d55b-4689-a109-0d948d9b4a32}} where
{{formula:fbe7f85e-2ee0-4d89-9c9f-4959a17558f1}}
Setting {{formula:f31a9b04-6e69-4b41-9b6b-f4d99118a747}}
we take two auxiliary transverse sections {{formula:37397fba-4db7-4d2d-9fb4-781bcf74e605}} and {{formula:3d37af17-8f45-4bf0-b352-cc6b215f49c8}} parameterized by {{formula:9fb811d8-2e7d-4245-9054-b78be1a32aba}} and {{formula:8c7da859-69a5-47af-b70f-ae62a3c63b39}} , respectively (see Figure REF ). We define {{formula:a61fabd1-fa3d-4b35-a8c3-983a177e06a5}} and {{formula:20741697-9a78-44b5-bced-401217ddfd85}} to be the Dulac time and Dulac map of {{formula:385f0e59-d760-4ce9-80ba-bf79f9bc45bd}} from {{formula:39f47c02-8191-44d3-8d6b-9cba67ca84c2}} to {{formula:b391e891-17b5-4b27-a138-91bd0f100984}} respectively. We remark that, by construction, {{formula:1f94d17a-8761-4369-bb5c-74322fdc80ab}} is the time that the solution of {{formula:7fc59abb-0c08-4ad8-a4c6-5592d5360585}} starting at the point {{formula:4309271a-0bb1-4b5b-9777-ea9dc8fbfee0}} spends to arrive at {{formula:a0027e2f-cc6b-4f88-a8be-6fe7b4b7b3cc}} and that the intersection point is precisely {{formula:86fe79ca-3501-4d68-bc89-7131cbb74187}} . In this regard, since {{formula:740e0dfb-7d20-4314-8b96-38d2678a7584}} is a trajectory of the vector field {{formula:918e3084-f09e-404d-9595-48ab29b2a80a}} , see {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}} for details, the application of {{formula:185bd27d-3c7c-46a3-adb7-f3accd35df35}} in Corollary A of {{cite:12b662988a3a9eea49d343a4c181075cdff9dfb5}} with {{formula:d15a9c79-813c-483d-8513-df1b08822cad}} shows that
{{formula:56ad0e9b-eae4-42e7-b205-d6ea5756ad9c}}
by shrinking {{formula:4e17f73c-b302-4592-a36d-5fdf024df79e}} if necessary. Here, and in what follows, {{formula:487865c0-4dca-4924-ab1c-9a3b6f8cc9f5}} stands for some function {{formula:402d66f2-dbdc-4453-a47c-15d9fba30d58}} verifying that {{formula:d704be68-14f8-4591-ad8d-7c102571feab}} uniformly on {{formula:20b77e1e-930a-4053-929d-b346ca25f6fd}} for {{formula:61070369-804f-45e5-ab50-742bdaf20254}} .
Furthermore, by applying Corollary B in the same paper with {{formula:4c7f66ff-a53d-4d48-9067-65b4c7fade25}} and shrinking {{formula:1399dcce-018d-4236-84fd-0c88286f2b6e}} again we can assert that
{{formula:fa76c3d0-7299-4c0f-95b0-a7cd5399d944}}
with {{formula:89faa598-e508-4847-9dad-13cf4495f984}} for {{formula:aa50321b-1a5f-4a54-a016-3174b3417606}}
Working in the original {{formula:468ded27-8aac-484d-b745-054105cb793a}} coordinates, we consider next the transition times {{formula:c29bfb25-4e0a-4469-ba04-fa9fcf35bf47}} and {{formula:25faf0b6-11d1-45b2-a65b-fc91fd8d69c4}} of {{formula:75af6143-f1d4-4e6e-bee6-a08628368492}} from {{formula:c8602201-1c76-4451-8a5d-83bd55bb1673}} to {{formula:378280e4-8acc-4fa7-ba65-1869207445c0}} and from {{formula:ceb8ce30-420a-4ae8-820e-ed9e786a4342}} to {{formula:42aace07-9ff5-4396-b6bd-4899a49bd883}} , respectively. We define moreover {{formula:5558e5b6-1d5e-4f3e-ad31-eafe55298561}} to be the transition map from {{formula:cd4a75e3-dd10-476d-9638-0fd0ed5b5ff1}} to {{formula:a9fc1075-a0e9-4267-88e9-925fe51f31b5}} . Accordingly
{{formula:cce8a698-440e-42a5-8002-0df142cb00be}}
where we omit the dependence on {{formula:1803f74c-84ed-4d13-b252-446fa01f6a1b}} for the sake of shortness. By {{cite:f73aa8df8bfb916a82c4b022d26828630a4e01e9}}, we have that {{formula:e6594a2f-325a-4309-8408-1db82776f6bc}} an analytic function at {{formula:2df1c7b4-852f-4268-a678-046b1f5bb46d}} with {{formula:df7f474c-cc24-4130-aa99-d8daec1a9ec3}} Observe at this point that, setting
{{formula:12f0d431-1bf5-4cf9-b986-693468c2f841}}
we can write the parametrization of {{formula:38a3f93a-915a-4f34-b6b0-0be8f5a7dc4d}} as {{formula:6f55f65b-ab32-47da-9244-e53a09ea9b95}} We claim that there exist two functions {{formula:e42b60c1-c5fa-4f20-a11c-8a1e588984e3}} and {{formula:2a5abafa-931f-4963-95e4-6de1313a7390}} , analytic at {{formula:ef562eba-23c0-470c-bc24-dbc4b992e520}} such that
{{formula:dc203b2f-2b68-4a9e-960b-65dcfb71329e}}
To show this let us consider two additional transverse sections {{formula:a02e378a-6dee-4121-b38a-7273e8c844ff}} and {{formula:3e1917f6-6373-4ada-9de4-795f1c5079b4}} parameterized respectively by {{formula:99a3069a-7573-4176-925d-cdc643cb9a0d}} and {{formula:20b1ef72-9c4d-4056-a75f-dd1d1aea7286}} which clearly are analytic at {{formula:d7163157-3964-493a-a75e-ad39ca0ce85d}} . Moreover it is clear that they are related with {{formula:ec4a2d41-0554-4702-b1a3-78c9c06d2f45}} and {{formula:ae302d78-c971-49ad-b694-41c0d071d215}} through {{formula:a2bdaa42-ee72-46cd-8e1d-aceb4446b675}} and {{formula:300ff044-7083-421c-890b-c4d1b0897379}} That being said, the claim follows by noting that if we choose {{formula:9b134d18-b245-4055-9596-1dd831ba3bb8}} and {{formula:1c0e2636-191d-4fd9-b68b-fa984e80dc9a}} to be, respectively, the transition time and transition map of {{formula:3aeb567a-1af0-44c5-a482-fdbfbfa77b94}} from {{formula:e5e4fbc1-5921-46a9-b483-30383612baf0}} to {{formula:a17c73f8-0f98-4509-b19b-ab9f3bcf9e1e}} , which are clearly analytic at {{formula:52ec9b16-8fa9-45f3-be3e-2df70963b7c5}} then the equalities in {{formula:94990be0-2a9e-4306-b244-a24822326ee0}} hold. Note moreover that {{formula:746aae2b-9e51-4b7e-a07b-01ff3c1c7da7}} since {{formula:ce8fe088-06d6-4db3-b6ae-4c20bee2669b}} On account of the claim, by considering the second order Taylor's development of {{formula:a0b2505a-6fcd-4d6a-bdfc-434853d301cc}} and {{formula:158cbd27-1306-4fa7-ac1b-65b3d70e22e4}} at {{formula:4de17a1b-e4bf-40d5-9c5f-cac616f656cf}} , respectively, we get
{{formula:685f2939-4c46-4138-99e2-d7caed709fc3}}
with {{formula:5484983e-5154-414d-994c-7e9a42a182bf}} and where we also use that {{formula:562ecd89-7442-4dc7-9924-ce67e69258f6}} is a continuous function. The combination of the second expression above with {{formula:5ed0a387-b4db-44e5-8fb7-217a603e5935}} and {{formula:908ead52-757f-4912-a64e-8858198e7051}} yields
{{formula:65e77e03-52bc-4ea6-9941-1d07b32b6eb8}}
respectively, with {{formula:91d50539-379d-4649-8b2c-151d94ae3ba4}} Summing up, since {{formula:35f782c4-6b5b-44f0-9486-cd739bf311fd}} due to {{formula:b89744c4-467b-449c-a0df-e3965be53f3b}} from {{formula:43cd5d58-aeb7-4851-837a-475ae7620d78}} we can assert that
{{formula:c39ea87c-5828-4131-9d91-be4b1228499f}}
for some functions {{formula:c41fc0e9-ad9c-4f10-ad6d-019b281083ad}} that are continuous on {{formula:faeea7d3-14ab-4ef2-8287-b7d5200fba65}} and some
{{formula:d49a2f67-a78f-443d-9c7d-b0d9b07cd780}} . This concludes the proof of the local version of the statement, in which we remark that the coefficients {{formula:fe3065c8-919e-433e-a433-15ac6f1d8e43}} and the remainder {{formula:bbc36d0c-16b8-4edb-8eee-48d71e265513}} depend by construction on {{formula:9ec37b12-034b-41f9-a59d-b706a4bcb408}}
Our next step will be to globalize them and to this end we define
{{formula:19eb46c8-7281-48ff-8c49-a5e0a54f7905}}
which is clearly an open neighbourhood of {{formula:3e1e69c6-a141-4946-a519-e2bede2065ec}} . Let us consider now any
{{formula:f457d3f2-f30b-4356-b15f-4aa09de30447}} such that {{formula:1705e3b9-29c5-4d01-b3b2-8def84553ef2}} Then, from {{formula:c35460ed-e385-4811-ac6d-3099abeee9c7}}, we get that
{{formula:1d7eccd9-444a-40e9-9e4e-db6098dd181b}}
for all {{formula:a0d2393a-4861-42b3-8784-fb1ccc8db3e9}} small enough and {{formula:58653386-185d-4f1c-bf92-aa7feb1d6932}} . Since {{formula:d188d896-c1b2-476a-9425-98af86637256}} taking the limit {{formula:48a5b8f5-aec3-48ef-a048-f372618fbcfe}} on both sides of the above equality we deduce that
{{formula:17b22236-40ed-441c-bcba-26563a4cfa59}} on {{formula:799c6aca-4bd6-446e-a7ac-408b939acf20}} . Similarly, but taking the first and second derivatives with respect to {{formula:76712b33-d123-4ce7-b4cc-3f253c99d8f2}} , respectively, we get that {{formula:880bcf68-2851-4ca5-ad67-2d0df4468334}} and {{formula:95b14e9b-3e24-42e7-9ef9-bd30fc2f8e52}} on {{formula:00a8e64e-8a36-4909-84c3-0aea4b5092b6}} . Hence, for {{formula:d1279a22-f0a1-45d5-b35f-33b7abc129ac}} the local functions {{formula:20f4f9a8-1431-4e16-b59d-a5774b540969}} for {{formula:f19b9670-1762-4d67-8d4a-2d70ca106d73}} glue together into a well defined continuous function {{formula:9e5b93a4-4c28-442e-97e6-585aa6e7be88}} on {{formula:68794570-543a-4806-8c7d-798a1d7fa2b9}} . Exactly the same argument shows that the local functions
{{formula:54e9ffc2-169d-4862-917d-efb56e2731dc}} for {{formula:c2164836-3806-4220-9244-fa1c33425d48}} glue together into a well defined function {{formula:5adce2af-268d-4af2-b4e2-910805c7a05d}} satisfying that {{formula:85fca6a1-c9ca-4696-bde8-42650cfef775}} uniformly on compact sets of {{formula:45ab5b97-5630-4a43-9475-5006a2a7a9d8}} for {{formula:de17250b-983e-4fc2-b220-54b6c3cfdafc}} . To show this last assertion it suffices to take a finite subcover {{formula:235b9044-1321-4946-85c8-74d51ae24eb3}} of the given compact subset of {{formula:3517cc45-8e9f-494a-8963-c98f16829a92}} and use that {{formula:c3a19211-76ae-4f88-a58e-3cdb0b872a35}} for {{formula:1a117ca9-2565-4741-a5bf-e642d90e96bd}}
So far we have proved the first assertion in the statement. Let us turn to the proof of the second one. To this end the key point is that for those {{formula:837ae835-0711-4114-9ffd-145b6738946b}} we can also apply {{formula:fe31e79c-e9aa-4497-852e-3fd97475963e}} in Proposition REF
to obtain that
{{formula:b1dfb803-a5a4-402f-b1ac-e9c4aa6545b7}}
where, setting {{formula:489e13c0-5504-4824-9bc0-908f5cf00c02}} ,
{{formula:3670d22e-6240-4386-b5e8-b156d9cf7088}}
Hence, since {{formula:df411631-dcc6-4722-8331-8324535fc6b2}} , from {{formula:8ef7e3e5-98e8-400c-aa66-0de847197828}} and {{formula:2eaead25-b565-41a5-ae99-f5f558a1fabc}} we can assert that
{{formula:3c208d9f-4158-449c-b50f-3dd9b4a6e193}}
where we also use that {{formula:cd0beaf1-5d21-431b-8122-4327ba1b3f1a}} . Consequently, as desired, {{formula:35f504ab-bec2-4c4a-8d22-eeb339d6a79a}} if and only if {{formula:17abed4a-df89-425d-a450-812350f30b62}} . The same argument shows that
{{formula:40610055-6233-4057-8090-420464cb9ece}}
and this completes the proof of the result.
Distinguished cases
This section is devoted to study three specific parameters. Recall that among the quadratic centers there are four nonlinear isochrones, see {{formula:6d2e893c-581f-48b3-bb06-9ecfb36f3b01}}. Chicone and Jacobs show in {{cite:2bf3254451791a68ce817916f287f1704a0d2519}} that the criticality of the period function at the inner boundary (i.e., the center) of {{formula:b771a1fa-cf45-4929-81c0-25fc5fd20740}} is exactly 1 for each one of the nonlinear isochrones. In this section we prove that for two of them, namely {{formula:9f79c797-c790-4ff2-a24c-a2643b2e4df9}} and {{formula:9bcb9d0a-b701-4167-ac11-0d3de7d590f1}}
the criticality at the outer boundary (i.e., the polycycle) is also 1, see Propositions REF and REF , respectively.
In the same vein it is also well-known that the criticality at the inner boundary of any quadratic center is at most two, see {{cite:2bf3254451791a68ce817916f287f1704a0d2519}}. This maximum criticality is achieved in three parameter values, the so-called Loud points, which following the notation in {{cite:2bf3254451791a68ce817916f287f1704a0d2519}} are given by {{formula:8f45c236-036e-4051-be7f-f78423b05556}} with
{{formula:a6edf585-5523-496c-9c24-c27ea28f8c98}}
As we already explained in the introduction, we conjecture that the criticality at the outer boundary of any quadratic center is at most two, and that there are only three parameter values where this maximum criticality is attained. In this paper we identify and prove the validity of the conjecture for two of these parameters. We investigate one of them in this section, see Proposition REF . The other one was already studied in {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}} and we postpone its treatment until the proof of Theorem REF .
The following is a sort of division theorem within the class of flat functions that will be used to study the criticality at the outer boundary for the above-mentioned isochrones. In its statement, and in what follows, we use the notation {{formula:e61e3376-f3e3-4822-b650-7543ca971e35}} for the sake of shortness.
Lemma 4.1
Let us fix {{formula:5d89ba51-efa2-4171-8c03-4c5e313962e6}} , {{formula:40cfe138-6873-4180-b361-a131b6cefcb9}} and {{formula:aa69afcc-a05f-46c8-a6f1-8d431273f0a1}} If {{formula:beef3d3d-866c-4c27-9d18-05cf56e12408}} verifies that
{{formula:78b75a2a-f6d3-455e-9828-6bed6f7b938d}}
then there exist {{formula:0c87d21a-0cf6-4881-9b2e-9ae741ac8146}} such that {{formula:1af6ca04-5d85-4030-9bcf-7ba1b283450d}} .
Proof.
We proceed by induction on {{formula:147ad2f8-f2d5-485a-b2c8-e34918ca81f3}} For the base case {{formula:2406bafa-5084-49ce-be80-2b9eeae38a41}} we take {{formula:6c2e4eab-a5e8-48a5-98ad-73a61c0b3fa5}} with {{formula:0255a15d-42de-4c31-abd1-e7dd55909b22}} and define {{formula:f1a97d01-0051-4211-a4a8-29741e6cc5bd}} so that {{formula:baf46517-4278-4143-8af8-18950636f13e}} To show that {{formula:90efc3ca-83c6-4021-9c76-98ecf355f0ff}} we use that, by hypothesis (see Definition REF ), for every {{formula:9f0663c0-28e7-4fbf-8a2c-a74e9adcb587}} with {{formula:7397ed44-19aa-4475-a2b7-56d2a30f5675}} there exist a neighbourhood {{formula:1790c20a-15d8-42ea-ad1e-9ccb4387a85d}} of 0 and {{formula:80290945-4091-4c50-a05d-11f6a5e1652c}} such that {{formula:9d9f3acf-dcf3-4917-b219-214966b42c6e}} for every {{formula:4ed2f3d1-99ee-4ec3-b4f0-d9c134a1281b}} and {{formula:447eeba6-1b0a-4be4-912a-4ddee4716ff7}} . On account of this and applying the Dominated Convergence Theorem {{cite:65ff67a4754ed82c63951fed7353738cb13ff372}},
{{formula:d9ad4b10-87ad-438c-bd25-198568a74167}}
for every {{formula:1c486e14-30c9-4836-81bd-d88c98c75cea}} and {{formula:a785e66c-3770-45e4-ae78-efe1a881654b}} Hence {{formula:f19f9f82-ab63-47e6-a33e-82dfbd9a56e2}} . To prove the inductive step we suppose that {{formula:20ceb1d7-3f86-4c5d-907e-010582b1f240}} and consider {{formula:b6aabbbf-c4ae-4de9-b5fa-b39a02f7291b}} verifying that
{{formula:7e3e7318-40b7-4494-9c1c-60dce5891571}} for some {{formula:cd93e407-a853-4a77-a120-6954c7047714}} . It is clear that we can write
{{formula:6401a4e5-dc74-459f-942e-d68a2481b5c0}}
Similarly as for the base case, taking {{formula:f5a5dbba-cb8c-4e13-9b34-4bcd87137c0c}} into account, one can easily show that {{formula:937814a7-4b81-4e3a-b5a0-def6e7652ed6}} . Since {{formula:61cdd276-79fb-4cb3-91ae-6097f9ac827c}} by the inductive hypothesis there exist
{{formula:6a8e3651-6932-434d-bc0b-aa2e85493265}}
Due to {{formula:cad6d8bb-8f43-47a9-9aab-e55bf01178ef}} see Definition REF , the combination of this identity with {{formula:4d60ad06-ab9f-489a-8300-844a5c69f212}} shows that {{formula:d4a9ecb6-2640-4fb4-af3f-62a47a2fcf7c}} with {{formula:913f1da6-af00-4896-a8e9-09fa7c0c2a2d}} as desired. This shows the inductive step and concludes the proof of the result.
We state next our first result about the bifurcation of critical periodic orbits from the outer boundary of an isochronous center. With regard to its proof let us advance that, after a convenient division in the space of coefficients, we proceed as in the proofs of Bautin {{cite:20e76adc03a88119f74c5df130214f32d62b061a}} and Chicone and Jacobs {{cite:2bf3254451791a68ce817916f287f1704a0d2519}} for the analogous results about limit cycles and critical periods, respectively, bifurcating from the center. Here we tackle the bifurcation from the polycycle, which is more challenging because, contrary to the center, the period function cannot be analytically extended there. To overcome this difficulty it is crucial the fact that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to the parameters.
Proposition 4.2
If {{formula:859c263f-1064-41c7-8149-ec48d8ed6478}} then {{formula:debb38fd-f90a-473e-88d4-bf2e6a1a5d84}} .
Proof.
We show first the upper bound {{formula:1abfc48c-56f4-41f5-9d21-1bfeb05be811}} , which constitutes the difficult part of the proof. To this end, following the notation introduced in Section REF , we define {{formula:6ece3f9c-46f5-4d67-a407-f974369c0a87}} to be the period of the periodic orbit of {{formula:1b15c6e3-1618-43f0-b452-936ba64e5f77}} passing through the point {{formula:eb3507f1-5245-4c31-95be-ff66004da3ae}} . Thanks to the reversibility of {{formula:bc74b577-75a0-4b1a-94e4-d7bc7a5de11c}} with respect to {{formula:a8cc9381-1320-4337-82bd-67f380c3a772}} it turns out that {{formula:ad857acb-542c-457c-8056-8f3f83bbe8d1}} where {{formula:5d859533-f68b-490d-9579-c486a643ebd7}} is the Dulac time that we consider in Proposition REF . Thus, by applying {{formula:10701156-ad6e-4c1a-9365-4d70d367aab8}} in that result
and setting {{formula:45975d8c-102a-41a5-941b-314544fe7e89}} , we can assert that
{{formula:cea44f7a-977f-4771-ae74-3b222dd13394}}
where {{formula:391b4186-06e6-4e30-b403-b46ac6a38ad8}} for all {{formula:dae37a36-b951-404d-b5b4-f6b8f5b644ef}} small enough, the coefficients are analytic in a neighbourhood of {{formula:f7852f7e-5dbc-419f-ad44-20e2b3c70e33}} and, moreover, the gradients of {{formula:f58583a2-81bc-4cb9-8b53-9de784168ca2}} and {{formula:501330e5-f9b9-4067-957f-fdab2500c8ea}} are linearly independent at {{formula:71e2281a-3b5f-4396-984a-4205b124a6a2}} . Since one can verify that {{formula:d00afc4d-56c7-4bab-a02c-46cc438ba722}} , the derivation of the above equality yields
{{formula:253d78d0-881f-48f2-913e-656b73daf0cd}}
where, by using Lemmas A.3 and A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, the remainder {{formula:3f039d94-c675-46f1-af7b-205463ade011}} belongs to {{formula:59dd0996-6362-4c86-8ca7-5ad5c6ece4e5}} because {{formula:b5a3f09a-3385-49db-b6b8-a0ca9644f914}} and, on the other hand, {{formula:af444ed2-e08a-4f74-a78b-e72ddafb2f40}} due to {{formula:06c1ccbb-922e-4137-bfaf-bfe547fca41a}} Note furthermore that {{formula:59d2b762-42e0-4b02-96ee-03e966490764}} is local analytic change of coordinates at {{formula:3da81bb4-f4ae-4e86-b33f-abee94b9c901}} such that {{formula:ad49b794-9a61-42d8-9488-172296f4000e}} . We can thus write
{{formula:90dafa43-fef2-4a9f-a3e1-b33293eb1876}}
where we set {{formula:0837b536-585d-45ab-945c-ad676add433e}} for shortness and define
{{formula:f1472f7e-7815-47b9-8812-7360142d77d4}}
Recall at this point that the center at the origin of {{formula:e9b7899f-2d73-4da5-9034-f6dcdddce4b3}} is isochronous, so that {{formula:93b16f17-9949-4af6-80a8-538468a15f61}} Consequently, due to {{formula:afec61c0-633c-4e3e-b52a-beddbae28465}} ,
{{formula:d16e1243-2f47-41c2-a35d-0733b178f2d5}}
By the Weierstrass Division Theorem (see for instance {{cite:56ebefebda144413fc3e32066a493c4b162e7550}}), the first equality implies that {{formula:61430806-2e51-4247-8fd1-4ad9bd4cbb0f}} with {{formula:ba47c8d6-d687-4cb1-8d01-0348f6c05d15}} and {{formula:a640acdd-055c-4ef2-a462-507b6d4d6411}} analytic functions at {{formula:de75479c-27f3-4913-82b4-9ac7ce312a96}} On the other hand, by Lemma REF , {{formula:57d3e708-acb8-4259-9594-83f0fb9adbe1}} with {{formula:8d174031-efe9-475f-8831-e6bd00ecb8b9}} Therefore, from {{formula:d9771035-c616-45b5-8fe1-4f9f62f28406}},
{{formula:5832e005-114c-494f-be32-2736c526f1c2}}
Since {{formula:12bd81c2-3d84-4b09-a404-c9313c18f8e6}} {{formula:acda1581-72ef-496e-910d-e9d26e1ca0cf}} tends to zero uniformly for {{formula:07533056-ee84-47f3-96ee-eb602e77c7d9}} as {{formula:d68522ed-d094-4610-ade4-5f037f591026}} (see Definition REF ). Due to {{formula:457c41fa-ce96-4ee7-b427-29157aef9f4d}} this is also the case of {{formula:b3101758-62b7-450c-b256-e30dd306ffac}} and {{formula:dcd40a32-5726-4b56-9024-3571ad966936}} by {{formula:5ba653d7-b0a2-48f8-84eb-7410fff06bd1}} of Lemma A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}. Hence there exists a neighbourhood {{formula:2e1582e4-8297-4455-bd72-e1a014baabae}} of {{formula:79346093-ecce-4798-9520-5e532fb18184}} such that {{formula:62f6d2d6-d865-4502-88e8-be75288577ff}} uniformly on {{formula:b08fd3d2-a58e-4a7a-90b9-33c69c00b1f9}} . Accordingly, the function
{{formula:1a961179-f758-4959-9f44-d250dfdf6eb3}}
belongs to the class {{formula:4bdc1e27-9b22-4f3f-8819-eb69dc2582ef}} , see Definition REF .
We claim that, by shrinking {{formula:1859fac5-b188-4e82-b5a4-f1d62f671c15}} , there exists {{formula:a8f71ff0-b5f3-4fd3-99b4-e578dfd6f2b3}} such that {{formula:7621ec60-7972-47e7-98b3-4b5ed0d00e86}} has at most one zero on {{formula:1edb26be-60e7-4886-ab73-b4de328f9964}} , counted with multiplicities, for all {{formula:e4af13ac-5184-4573-a1c7-4548199bb3b5}} Indeed, to show this note first that if {{formula:792ed1b1-938c-4c9a-b21c-32f3d6296c1e}} then {{formula:a03158bc-732c-4bdb-bb8d-2dca1efb9d46}} so that there is nothing to be proved in this case.
Let us study consequently the case {{formula:9d39b619-1323-48a2-829c-eb8766317e24}} . To this end we observe that {{formula:03a805c3-34b0-4dbf-872f-91cb9c5a00c0}} where, using a more compact notation,
{{formula:db06cc80-0feb-4403-b1d7-ace3210f99a0}}
Here we use the identity {{formula:9cc70016-4251-4c30-8278-2904a43b973c}} and that, by Lemmas A.3 and A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, we have {{formula:76238505-8157-4d33-b821-46d571956879}} and {{formula:6d4ccc13-3dd7-4f49-bebd-11efeeaa524c}} for all {{formula:a53e9164-f03c-47c3-8dde-0491afdb4dc6}} small enough due to {{formula:017cd697-7afb-42bb-97e9-f22e6f023b91}} and, moreover, that the inclusion {{formula:76b5ccf3-9c3d-415d-b8cb-462180754a57}} holds. We also remark that, by {{formula:ee6190fc-a3ff-458f-a29f-14809176f99a}} of Lemma A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}},
{{formula:368f3daf-3897-4ddc-9820-374fd8675435}}
On account of this, from the above expression of {{formula:8a070515-cc24-4dc8-8b48-bbec434ff88b}} we obtain that
{{formula:fdd28c8f-5632-4aae-ab41-d6be5707998d}}
where {{formula:6fa66a2f-8cd4-4696-8a97-f818fff066eb}} . Since {{formula:6f4b2a74-afea-4f78-a457-fef73e1a64d0}} , it is clear that {{formula:dc19a289-8b5a-445f-95ec-625d1da61288}} is a non-vanishing continuous function in a neighbourhood of {{formula:3ec98901-5e13-4484-b070-28bb1f069583}} . Accordingly, due to {{formula:c8730750-8c3b-4541-9bc0-a6593aeaf092}} , we can assert that
{{formula:b69e0330-b27b-4fc0-94ac-2b590d3a572c}}
Since {{formula:caaba704-0a9e-4050-a792-fddfb4d9cb60}} only vanishes at {{formula:01a70531-315a-4623-8034-bf9baede8096}} by shrinking {{formula:80c99310-21d1-48df-9d62-28bbb976a265}} if necessary, we can assert the existence of some {{formula:ef00111f-2c99-4c51-ae09-cf6d714cb348}} such that {{formula:6f6def2f-7ada-48e2-9351-19de8301fea5}} for all {{formula:b147e17f-7ed0-4904-a317-86ca356e96d0}} and {{formula:2a8f11aa-2938-4786-80af-f4103cd2d991}} with {{formula:7013fdc3-3adf-463a-8793-6ebcffd37d5d}} . Therefore, by Rolle's Theorem, {{formula:b4e35134-ad19-4194-a60b-7cc56b682b2e}} has at most one zero on {{formula:4b2301a7-c7c4-4fbe-ad9c-4df3f058946f}} counted with multiplicities. This shows the validity of the claim for the case {{formula:48794aa4-1116-4ac5-8ac8-a4d1760b9e8e}}
Recall finally that the period function {{formula:1d4eccb3-b8b9-4f3b-b194-ffa197faf205}} is twice the Dulac time {{formula:d905c149-8e0f-4539-b8e9-7ed1c93d5135}} .
Thus, taking the claim into account, from {{formula:4addb25f-94c0-4369-926a-2bb96a537620}} and {{formula:86cc8183-9461-4720-b034-8430c1903fab}} it turns out that {{formula:f5f7d87c-8a1c-41ef-944b-d030658465ce}} is an open neighbourhood of {{formula:4c8a66e8-c4df-456a-a9d9-c8bd45ad969d}} verifying that {{formula:1b2522a7-549d-4f6f-9126-8ffadd70d4b9}} has at most one isolated zero on {{formula:6090e62f-b473-40e3-8a1b-cabfac425549}} , counted with multiplicities, for all {{formula:2e22f785-fc38-4e2e-aab7-c7f478ce5c92}} (To be more precise, the claim applies for the punctured neighbourhood {{formula:36cb4af8-a1fa-4147-ac98-c0d1fdfc633d}} and, on the other hand, {{formula:9dacb694-9e60-4d48-905b-487ce257d60d}} so that it has not any isolated zero.) Hence, see Definition REF , {{formula:fbae1970-ce0f-40f4-a0a5-60aa2400f336}} . Therefore the upper bound {{formula:5dee94ff-ed92-40c2-a828-049800bda862}} follows from assertion {{formula:7058a94e-529a-48db-8a22-24f8202e2692}} in Lemma REF
since, using the notation in that result, {{formula:fc5823f5-2789-4093-bef0-09b76fa4b097}} with {{formula:2aa2c3dd-4c46-44ee-8273-f4add6633a44}} for {{formula:8e0d9860-3719-4edb-b0b2-bce18ef5f388}} . Thus it only remains to show that this upper bound is achieved. To this end we recall that, by {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, {{formula:08196fbb-e493-4eaa-80e6-fe3125c1127d}} is a local bifurcation value of the period function at the outer boundary, see Definition REF . Then, since the period annulus of the centers under consideration varies continuously, see Remark REF , by applying {{formula:5c3b1a9c-5b23-4f61-a846-db50c3fdb56c}} in Lemma REF we get that {{formula:9d87b18b-5db4-4fc9-bfee-f68b55bab454}} . This completes the proof of the result.
The following is our second result about the criticality of the quadratic isochrones.
Proposition 4.3
If {{formula:24374604-aaf6-4fb2-870f-d597021517b5}} then {{formula:0b0c3e66-1236-4155-a4dc-b85db51b8393}} .
Proof.
We prove {{formula:28375c02-a684-46a7-bf7c-c1cb0bcb8efd}} first, which is the most complicated part of the proof. To this end, for each {{formula:71c356e9-48de-40ea-a116-5f6b931a07fe}} we denote by {{formula:504c594e-6dcb-4d7e-bc0b-77640ade2042}} the period of the periodic orbit of {{formula:bcba4077-932b-4408-8a07-a0665d29b063}} passing thought the point {{formula:053063e0-c5fe-41b8-8aa3-14dcc8a013ae}} see Figure REF . Then, on account of the reversibility of the vector field with respect to {{formula:269b0f14-207b-45f9-ad12-20d5b88791f7}} , it follows that {{formula:8bd08bde-cd29-481a-a842-751e7d7191c8}} , where {{formula:2fc7fe57-1b62-4227-b429-98626cd21ea9}} is the Dulac time introduced before Proposition REF . Thanks to that result we have thus the asymptotic expansion of {{formula:e3fc2798-7003-4b75-8d22-f58b853a93e2}} near the polycycle, which corresponds to {{formula:5f0dd3fb-423a-4b17-8dcb-fe89673acc2b}} On the other hand, it is well known that the period function can be analytically extended to the center (which corresponds to {{formula:1875ca13-9e78-44b0-8548-580963440b0a}} with this parametrization) because it is non-degenerated. The coefficients of the Taylor's series of {{formula:d5f6911e-d918-491c-a79f-963382b7da43}} at {{formula:fe0cd04f-8b7f-4184-864d-dff843a39d18}} belong to the polynomial ring {{formula:274b9d4a-db4e-4139-9dd0-d4dbc89ca986}} . Chicone and Jacobs show (see Lemma 3.1 and Theorem 3.9 in {{cite:2bf3254451791a68ce817916f287f1704a0d2519}}) that these coefficients are in the ideal generated by
{{formula:a3d081a5-5e0f-46b7-b573-146cd036df55}}
over the local ring {{formula:b9ae631f-daf1-4b01-9087-4ec823b9c5e0}} of convergent power series at {{formula:66a22260-a024-4693-bbb3-e626964933a1}} localized at any of the of the four quadratic isochrones {{formula:94343dbb-0b1e-40eb-85bd-a48b86398963}} {{formula:6cebc5ef-12e7-4a7d-a958-e6e2e5337724}} {{formula:dc4635e9-8acd-4ecb-b27e-a27d761245aa}} and {{formula:137a1b18-7244-4ea5-ade0-c4ea3bacf796}} With regard to the first one, we claim that the ideal {{formula:a223d5f4-6ae2-436a-ac05-ef456a5e9b2c}} is equal to {{formula:23793254-d877-455c-892a-ae0b9faa155e}} over the local ring {{formula:ed9ed1fd-e2db-4924-8fff-d2755809f5a7}} . Indeed, to prove this we use that
{{formula:5080456e-4c9f-4ba3-a0f4-fb031c48c022}}
with {{formula:3951b68d-1584-4439-a150-fc84b754962b}} {{formula:49d740c6-2b99-4906-91b3-7681cc86f479}} , {{formula:b6f19ce4-a479-4065-921b-491544d56bf0}} and
{{formula:c7d472d2-fcd4-4b94-ab8a-6ad2f123ad55}}
(The idea to obtain this is that the zero of {{formula:c596f8e4-261a-43a4-ab73-486b276f9c53}} at {{formula:425f4146-f440-4a21-b6ee-a1ef3fe04608}} has multiplicity two.)
From {{formula:b3c84080-08bb-4bea-b032-bd31894b934d}} we get that {{formula:fe79df5f-34a9-4f31-9299-5c314aeeb60d}} over the polynomial ring {{formula:e2fba0d8-37a2-4adf-8a5d-07731481c9b2}} Conversely, since one can verify that the determinant {{formula:04f4962f-3462-46e2-a16f-fe16cb8b223e}} is different from zero at {{formula:c83ea63a-0b96-4b30-bbdd-5246f4e5dd17}} by inverting the matrix in {{formula:54122236-d819-4189-b74f-08a80b5754de}} it follows that {{formula:fb4ce22d-4e92-4812-873a-cd460f5497ce}} and {{formula:ed94c5b2-1c67-4baf-a138-b8e73d17510b}} over the local ring {{formula:d35d889f-c467-4a3f-9ff4-595b40841990}} . This proves the validity of the claim. Consequently, thanks to the result of Chicone and Jacobs mentioned above, we have the following equality between ideals over the local ring {{formula:11185032-51a2-4ed5-8589-7157e03092c8}}
{{formula:65839497-2ddc-4fe4-8a34-2c96acf674da}}
Now the crucial point is that the ideal {{formula:7d8fe0a6-b1ab-45cb-9467-442e9cdfc969}}
does not depend on the point {{formula:87e741d7-3d48-4b79-8137-d2fc20094c98}} Indeed, this follows verbatim the argument that R. Roussarie gives in {{cite:970002d1f04a652201360989dc830e6d5172b468}} or {{cite:44c371fa22acae495f26ba888c8c78a3e7a58114}} to justify the same property about the ideal of the displacement map, the so-called Bautin ideal.
Here we also use that, such as the displacement map, the period function {{formula:5e2e9481-7b33-4488-b6cd-90b16a94c1bd}} extends analytically to the non-degenerate center (i.e., {{formula:f18e7539-b751-4458-9613-3daffaa22ca8}} ). Accordingly,
{{formula:cfd90a2f-ed04-4d5b-8e54-ea97a6019ba9}}
We turn next to the study of the period function near the polycycle (i.e., {{formula:48cc099a-2dea-4481-a889-15bcee3af072}} In this regard by applying {{formula:5f49b83a-2799-42ff-8780-9407da8a88f3}} in Proposition REF we can assert that, for all {{formula:21e2210c-c770-44d1-8e24-8df31379a8f1}} small enough,
{{formula:280aae7b-92da-452c-a1fc-e6a77ec1ac61}}
where {{formula:dd6f3dc4-3b58-42b1-959a-2e94e3363655}} and
{{formula:9e6985c1-42fa-47eb-9f47-bc58f51fbf0e}}
for some analytic positive functions {{formula:1a95784e-8dcb-43c8-9e47-531382ad6105}} {{formula:887e0f19-3d16-49d5-8b18-43d3e0631381}} and {{formula:d0305d95-7ffa-42b6-9ffe-cf4232d4616e}} in a neighbourhood of {{formula:50ccaa93-62fc-4870-ac30-d320c10b8124}} . Consequently, on account of the identity {{formula:d5e41080-b4e3-41ea-a1f5-9beb0e127568}} and assertion {{formula:3fd92524-6e10-4857-aada-2da11e0d78ff}} of Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}},
{{formula:cbd12ec9-8d9c-4832-9bab-c511f8f67508}}
Furthermore, from {{formula:e5e1d0ad-719a-42b6-aa99-ffdcf4d1c187}} it follows that
{{formula:79c11a11-ab77-46e2-9289-fa096338ced8}}
is an analytic local change of coordinates in a neighbourhood of {{formula:c70df5c6-a60d-4ed4-ac40-c9c6baf7e070}} because one can verify that its Jacobian at {{formula:db14ca14-6884-47c8-976f-740903628dab}} is equal to {{formula:17b97c9a-50fd-496b-a112-c0823c643df4}} Setting {{formula:c577dd1d-8ef5-4219-8536-23b6ff248035}} , observe that then
{{formula:bb012811-0f2d-4dfc-be48-4bbd1c28a3af}}
where {{formula:f97ed86b-f60c-4b58-96cd-325b852c5639}} and we denote {{formula:3f441640-63cd-463b-a084-0e1d201caeac}} for shortness.
We claim that {{formula:99d5aa51-99a2-4abd-8e2f-c7cefdb3d910}} over the local ring {{formula:570558d5-adf3-4586-8e41-a8f30070bd1d}} . To show this we note that
{{formula:d87babe7-21db-4d7c-84de-8df67fcdd9fc}}
Since {{formula:5692a191-82b6-48db-9ff8-8d2027d72a23}} by {{formula:134ad0d9-cf2d-4802-b145-e525858d9f18}} in Proposition REF , it follows that
{{formula:f9af72c8-036e-4c5b-9122-f2caf304c0f8}} for some analytic function {{formula:7a3af11e-7f48-4e46-8e97-a16b56685764}} at {{formula:e1671b74-e29b-4a7a-a622-f29006c703f8}} Consequently {{formula:aa758320-8e62-4504-b902-26f9f7219294}} with {{formula:36153496-3961-4412-9f75-e2b3c1450fac}} analytic at {{formula:fb8299c3-97a3-4348-aba2-f120e04dac7e}} Taking this into account, the Weierstrass Division Theorem (see {{cite:56ebefebda144413fc3e32066a493c4b162e7550}}) shows that
{{formula:ddfe6c81-a9ff-4da2-a666-e00de01d89fb}}
for some analytic function {{formula:a290ae49-e55c-46da-ad1b-f18757a0a54b}} at {{formula:8e0c3b0f-c1ac-4bbe-908f-e1b892044f84}} which, from {{formula:61ae3d78-1025-4707-8d3c-40bf7659d496}}, verifies {{formula:359eb0a9-f86b-4e1e-b6ce-87e4ebc52d9c}} Hence we can write
{{formula:895e18ea-d8dc-4152-ba0a-1987df4b56a5}}
where the matrix has an analytic inverse at {{formula:9642bfff-a2a6-4f0c-b624-997b0a55aa65}} Taking {{formula:e6471726-5324-4bb4-ac87-2eed06693153}} into account this shows that {{formula:e0e9b277-3bfb-44b8-b941-b9b33f41d42c}} over the local ring {{formula:710a77e4-1a3f-43ee-8801-3042ea101526}} , as desired.
Recall at this point that the center of {{formula:e33a87a1-5629-4a44-9def-5c7bee81d1f3}} is isochronous. Hence {{formula:974909aa-6c28-46a5-82c0-7b74e4913651}} Thus, taking {{formula:efed2921-e6be-4f93-8487-609982891b2d}} into account, from {{formula:270d03c5-a913-4997-81a9-2ec6e1c63776}} we get that {{formula:064faf94-877b-4ca0-90a6-3d5c8aafd4b1}} Having this in mind we write the remainder in {{formula:0d8c881c-4f0c-4bad-a44c-124236254018}} as
{{formula:abb32186-782f-41a1-8d70-2f949befaa1a}}
with {{formula:6526a027-8382-4d11-aba8-f97a25b712cd}} and {{formula:f3ab1eba-6afd-4a46-a302-bd99b4a6699f}} . Since {{formula:fe79b267-bced-443f-b7b4-9fd1dd26adc4}} , the application of Lemma REF shows the existence of {{formula:2e6429d2-7f38-41ba-96b8-fd20e6548c58}} such that {{formula:6e609891-1f1a-45f0-b628-1cc6a1d6ab68}} . Due to {{formula:c2123a3b-8add-4407-8c67-775d11cb145c}} and again by Lemma REF , {{formula:c5f15b71-dfca-4db7-a71e-b86fc8202872}} with {{formula:29d8224a-e6cb-430e-a3c2-5dc3034ef5dc}} . We also have {{formula:a1a5a40c-6803-4fba-8723-6b3deafc07a6}} because, otherwise, it would exist {{formula:cd129d22-1d6c-44e9-a9c7-896dced7d2f0}} such that {{formula:6792ce38-26ee-40bd-9c2d-8f1eb35aa5fe}} for all {{formula:f79b6538-5849-4786-913b-17f3625767cc}} . In this case, taking the claim into account together with {{formula:2d91d697-9ffb-453e-a4b1-8fc3ca228b3a}} and {{formula:5dd6ef18-6bec-4cf1-912c-3ec0c61e5371}},
{{formula:822b1096-66bb-4238-aff5-e61c2480a5bc}}
From here, since each {{formula:8229011f-66d4-4bc9-9af6-6d4c0b6b3258}} is analytic at {{formula:8b034c19-a81d-4c2c-b140-3d27582130b3}} and {{formula:44a1eaa8-05b9-41cf-b5ad-b70066a2a46f}} for {{formula:ccbdb704-4bea-4550-9e6c-4d9f985dd5de}} , we would get that {{formula:6c728431-cbcf-4030-891a-db85559edc17}} over the local ring {{formula:821d401f-9d1a-4265-a0be-b319673a1f07}} , which is clearly a contradiction. Concerning the analyticity of {{formula:1dc2d3e9-f94b-46a0-bb66-fe76cb0bbbb8}} , let us remark that it follows by applying the Weierstrass Division Theorem thanks to the analyticity of {{formula:36ca4e10-f773-4a6a-899f-607793b09cf8}} at {{formula:3ae7b16d-40a5-45b2-8ff0-325f8b877c55}} which in its turn follows from {{formula:63976b66-2b84-459b-8702-d8fcd263d5f3}} noting that:
{{formula:64ba80bc-1b38-42ef-8bc9-96c76e1b01c3}} is analytic at {{formula:78b1461d-5edf-47b3-9ecb-3cbdc6332e60}} because {{formula:7c556705-31ca-48d2-8984-f66289081217}} is an analytic family of the vector fields and hence, by Lemma REF , {{formula:9a21acb6-d4bf-4e52-879f-88f1f1c74dcf}} is analytic on {{formula:5422c362-4615-447b-9d58-a24f7530a328}}
the change of coordinates {{formula:8c95e120-5bbe-40bf-b19a-95277dcf7a1d}} is analytic at {{formula:1d606e88-aefe-480c-9b3f-fcf959d9afc9}} and
{{formula:acfa6548-3eb7-4dd1-8e3e-d05979c795c0}} is analytic at {{formula:eb48814b-e9ab-4b19-9c44-0a4539df0cf9}} because we can write it as
{{formula:7e73f5af-be6b-41d9-ae7c-e95432512a75}} with {{formula:fdbcee9e-93c8-469e-8deb-2a0ae1161c63}}
Hence {{formula:c8607a69-a2d5-4b53-9e5e-c4b7215c4633}} and, by Lemma REF once again, {{formula:97bcd197-7b2c-4be0-8d02-d08fb01baab2}} with {{formula:147e1da7-7965-48a4-a075-7103d3b86720}} . Summing-up all this information about the remainder, from {{formula:383e830b-2b7a-455e-a4d1-424ec90e1499}} we get that
{{formula:9dedc28f-e76d-4a58-827f-309db70371e8}}
We are now in position to complete the proof by showing that there exist {{formula:1b742087-4029-4256-b220-04144bd35f02}} and an open neighbourhood {{formula:21bc5c03-0792-45ac-accc-d8dfc9108286}} of {{formula:d710beb4-b538-4412-94a1-2f07c8ac14de}} such that
{{formula:40edf984-fe40-458f-ad63-54e778d08e92}}
has at most one zero on {{formula:9a01f391-024b-4134-9605-ac801e8c8f05}} , counted with multiplicities, for all {{formula:ed4b9a4d-a177-4e50-9e6a-f8875a0d2195}} This is clear in case that {{formula:d3b86458-29ae-4ff4-851f-da359fca0c32}} To tackle the case {{formula:5e4de127-3b16-4a08-b47f-68d1686dc9d9}} we compute the derivative with respect to {{formula:4c24ae77-b8c8-4938-85d8-27b83a4caadc}} to obtain that
{{formula:fc77e546-5b82-4e75-b21d-045f23e4d3b8}}
Here, in the second equality we apply first assertion {{formula:5e649251-6f46-4b41-8379-365e9dc273ec}} of Lemma A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}} to get that {{formula:888cc537-9689-47f6-99d0-45132fb3d7e1}} for all {{formula:37845634-53f9-4ec9-918f-704b2a921a5f}} small enough, due to {{formula:1dee8ec3-9893-40b0-82bd-d525872b9248}} , and use next that {{formula:cf0a3dc5-7960-47de-a66e-954a3a8edfe4}} from {{formula:dd11b789-7e43-437c-9ec0-e2c878152ee9}} of Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}. In the third equality,
on account of {{formula:7fd53791-7b3a-49c8-8532-0f351a61bcbe}} and by {{formula:e261dd7c-0c79-44c6-afe9-b36ae2073378}} of Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, we use first the inclusion {{formula:d904ef6e-3a22-4b6d-8dc3-c48ca3c943b7}} . Then, by using {{formula:33b13fe6-b36b-497e-a5af-eb1eaf09df06}} and {{formula:d3b3d24f-11af-4273-af78-392867e4f346}} of Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, we expand the numerator to get that {{formula:a1781bbc-629c-4cdb-bae3-613a23c21734}} . Next, in the fourth equality we use that {{formula:edba0382-9f18-4cd2-adae-492124c8a1ec}} and assertion {{formula:988f1ae0-575c-46d5-af35-bf6ffdf26f8e}} of Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}} to deduce that {{formula:d92854cc-8e4d-4463-b430-b8f8ab59b00a}} . Finally in the last equality we apply {{formula:6b86eaa2-a574-4620-a69f-4d48f050e6c6}} of Lemma A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}} to get that {{formula:bb0a6052-3b98-4129-afae-7c353707a03e}} and we use again that {{formula:e4a0dc13-9533-4983-9632-329c3126bf16}} .
On account of Definition REF we can assert the existence of some {{formula:8cb1eaed-4e6e-4ae3-8089-8c8d04e9f1b1}} and a neighbourhood {{formula:1fbc003a-16bf-4c86-a791-26109be0ead0}} of {{formula:64cea727-86a8-4f7b-a13c-78fdde4ffdf6}} such that {{formula:d0b2ba4d-d85e-4f19-84cb-75efedbe4962}} for all {{formula:529304a6-93ac-470d-8786-333b6e22add8}} and {{formula:723d84e5-9242-4eb4-a645-f08bb1e6608c}} with {{formula:8af794a4-beb0-42ab-b401-f670a7b17218}} Consequently {{formula:645e334b-fa0b-4742-a257-e55c39fb9515}} has at most one isolated zero on {{formula:2b472fa7-e937-4b1b-bf0f-024b50890cc4}} counted with multiplicities, for all {{formula:7102dd2f-e74a-4be4-9e26-1893fcd7e6ac}} Thus, on account of Definition REF and the fact that {{formula:1b6e17e5-5b37-474a-8f5a-983bfbbe127e}} , we get {{formula:023b444d-b143-4cde-8396-c9dbe413a6ae}} . Finally the upper bound {{formula:19ebe01c-7368-4873-ba5c-3ea5958e3d78}} follows from assertion {{formula:1b68a748-4322-419b-a88a-3c4921572c9d}} in Lemma REF
since, using the notation in that result, {{formula:134ed527-24af-42bb-a5f1-70ca33b86d9c}} with {{formula:c7d5bcfa-1e52-4c7f-b3a1-f9e5fae98d17}} for {{formula:627b3660-7032-460e-8fbe-cce0cbd73981}} . Therefore it only remains to show that this upper bound is attained. To this end we recall that, by {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, {{formula:a4c9920c-c821-4492-8689-74ef777f531e}} is a local bifurcation value of the period function at the outer boundary, see Definition REF . Then, since the period annulus of the centers under consideration varies continuously, see Remark REF , by applying {{formula:148a986b-4566-41b3-8db5-ae5b83c1160a}} in Lemma REF we get that {{formula:7b63a1d5-90ad-4c6e-9370-7cd5f57b6ffa}} . This finishes the proof of the result.
As we explain at the beginning of this section, the maximum criticality of the period function at the inner boundary is 2 and it is achieved at the three Loud points {{formula:ba5d746a-657a-44b6-aba5-460fbeaaba94}} , see {{formula:2e5d1109-d4bb-4d37-8bd5-f07022537fa7}}. We refer the interested reader to the paper of Chicone and Jacobs {{cite:2bf3254451791a68ce817916f287f1704a0d2519}} for a proof of this result. In a joint paper with P. Mardešić, see {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, we prove that at each {{formula:9bda19ad-e533-4b1b-8216-10f7b10f0143}} there exists a germ of analytic curve that consists of local bifurcation values of the period function at the interior, see Definition REF . Since the period function extends analytically to the center, this follows readily by applying the Weierstrass Preparation Theorem.
In our next result we identify a parameter {{formula:6e8228c0-1499-48d8-baaf-c63e2e3e4996}} for which the criticality at the outer boundary is 2. Furthermore we prove that at {{formula:5055cc8d-288f-40ee-be32-a535939c7139}} there exists a {{formula:d108b53e-2d0f-45c7-a6fc-3f752e5f3345}} germ of curve of local bifurcation values of the period function at the interior. Hence, roughly speaking, this parameter is the mirror image at the outer boundary of one of the Loud points, see Figure REF and Remark REF . In the statement, following the notation introduced at the beginning of Section REF , for each {{formula:e2b8c784-f375-4cce-8a3a-d4c1de68ef3f}} and {{formula:41eaa2a5-b95a-421e-9df0-e390e5675813}} we denote by {{formula:1181e3f3-15aa-43cb-bfb1-f2fc27149768}} the period of the periodic orbit of {{formula:c430f098-5474-4d93-bf87-ed59826ec04b}} passing through the point {{formula:0f39064c-1c0b-4085-9c0e-1beb2d6c01b7}} . We also remark that {{formula:22b37e5f-4259-411f-bbbe-7ff35913134a}} and {{formula:8d8ca0af-cabc-4b60-8e5a-7aa8a8ab2f99}} are the coefficients given in Proposition REF , which vanish at
{{formula:7faa8534-ed2a-4d35-b003-4cc5eb347a5a}} .
Proposition 4.4
Let us consider {{formula:21d0f109-beff-4a9c-9cdd-0c8b80038ad3}} . Then the following holds:
{{formula:e3e5e755-7e2e-4d1c-be3c-924a57f82d8b}} .
There exist an open neighbourhood {{formula:1ae7c6d7-3ad8-44f7-ab37-8511bda4b55c}} of {{formula:eb921dc1-81ff-44f4-b92c-b2a76e919139}} and {{formula:6e6e8063-eb71-4779-aa08-16bb211f444a}} such that the set
{{formula:37d4634f-7072-4dd4-9e0c-0f48d79860b2}}
satisfies the following conditions:
Each {{formula:9a728a53-9458-473b-ad4e-2d095d0cb920}} is a local bifurcation value of the period function at the interior,
there exist {{formula:32c5d652-9f83-4acd-bea4-a0c6939fd361}} and a
{{formula:ba73b7c1-9481-42cf-94af-6545331af009}} injective curve {{formula:9e50dabb-83de-4da4-868a-671d3a7dbdff}} with {{formula:3ae6cea2-c745-4aae-9419-e66f4e751efc}} , {{formula:f5a90a60-b5db-4339-9958-9c9711c6e2ee}} and such that {{formula:7770e6f6-40c6-4d34-8115-4bfeb53bda86}} is tangent to {{formula:cee57345-238a-4fa7-b66d-e98bd48c0e6e}} ,
for each {{formula:5ba5ff84-086d-49c4-8f6a-bc81ecba95dd}} there exists a unique {{formula:01927bc9-27d5-4404-8f95-330ab00ccb71}} such that {{formula:2d90e65d-1265-4b30-8095-d7fd45b256ef}} and, moreover, {{formula:ce6fcd8c-63e1-4752-8b91-c2100979faaa}} ,
{{formula:e80fa446-5255-468d-a1b9-76927225ecf0}} , and
for any {{formula:b4052635-8c11-4d00-8eb7-9d4c311bf2a2}} and any neighbourhood {{formula:23c390f6-ea8a-4397-a984-d74e8bfd682c}} of {{formula:ee5cbb2f-3578-4c56-92af-f8ed3e0f6488}} there exist {{formula:40ff12e9-f071-4bb9-bb4d-6bd8bc781575}} and different {{formula:780f8614-2664-4d5c-b375-9e6158338bb1}} such that {{formula:e8ec8eb8-816b-4b1b-b1b0-7973ff25ba36}} .
Proof.
We observe first of all that {{formula:645cd13c-885e-4bcd-a520-006f9781801b}} is a parametrization of the outer boundary of the period annulus verifying the hypothesis in Lemma REF . This will enable us to relate {{formula:61ab9585-72dd-44e7-924b-e79c2dfbc782}} with {{formula:d39e252c-41ed-4385-9737-11e6653446cb}} That being said, thanks to the reversibility of the vector field with respect to {{formula:0990abdf-6c92-45b5-9d5d-e75d151d1771}} , we note that {{formula:54f267cd-cc0a-4f1e-9e80-764a6dcb851f}} , where {{formula:8ee5094d-7934-4c0c-9f6c-1d1da37f4ff2}} is the Dulac time considered in Proposition REF . From point {{formula:4721a302-e4cd-490f-a43d-3ba562b1f365}} in that result we can assert that, for all {{formula:c812cb92-7a85-4b15-b4c8-c120577516af}} small enough,
{{formula:389b02e6-7023-4d60-b199-84e9a91e6846}}
where {{formula:148dd1af-3e75-44f4-9c97-7d4159077f6a}} {{formula:aac7d270-43cd-4874-88f9-c2eec214f0b3}} , {{formula:1d3c4f9d-ec70-4624-b180-ef88a0cb719e}} and the gradients {{formula:34c29c94-87b7-4a21-89b4-8aef8863415b}} and {{formula:3a8d1a17-f680-4c3b-80d5-31741313037b}} are linearly independent. Due to {{formula:1bc447b2-1bc6-41af-89b3-ee5a72570e99}} , by applying {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} we get that {{formula:9699cccb-3a50-41a2-8ef3-0077255c08ce}} .
(For readers convenience, let us explain that {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} is a general result addressed to the Dulac time which,
by using the well-known derivation-division algorithm, gives a bound for {{formula:55ef31d7-87c1-41f4-9222-2d5b409fda50}} in terms of the position of the first non-vanishing coefficient in the asymptotic expansion of {{formula:16c575d8-a768-4ee7-9d8a-1b9644ef1400}} at {{formula:60d29e34-0b0c-461c-81a8-248e46de1357}} .) Consequently, by assertion {{formula:91898ea6-2116-4b05-8f37-1bf175f26eee}} in Lemma REF , {{formula:05e1e4f0-ddf9-4409-b69a-f944a8a25d8e}} . In addition, since
{{formula:ecf0d26f-8662-46f3-908c-b41995143427}}
and the gradients {{formula:f3cc9b52-0a22-47f6-8a06-d1740c4bf1e7}} and {{formula:f0cdd4ee-8e02-4f0b-b19e-5d23d8765872}} are linearly independent, by {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} it turns out that {{formula:9bec7f6f-296b-4cd8-813d-a69aa095876f}} . As a matter of fact, from the proof of that result, this lower bound is achieved by means of two different sequences of zeros of {{formula:d8373452-31a9-423e-a77f-a769df228452}} and, therefore, by assertion {{formula:1469baac-049c-419a-966a-e515d9ab38a6}} in Lemma REF , {{formula:dd80de70-02f9-4199-a897-19750f408374}} . Accordingly {{formula:e227c7df-ccc0-4130-b004-4fd28d06f2ae}}
and this proves {{formula:4de925ee-9375-441f-a58e-901cd48be4ec}}
Let us turn next to the proof of the assertions in {{formula:ffe961cc-a025-4062-9f54-37ba89c85464}} For this purpose, from {{formula:7b510df4-cb5c-420c-9dd9-930f2384db91}}
and by applying Lemmas A.3 and A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, we get
{{formula:dfe9a482-ceee-463b-9a63-4b3fe2915c97}}
Setting {{formula:077e0c38-f7f9-4633-a5ba-e456e7ed1279}} , the map
{{formula:4e0f70ef-7448-4506-ad49-606f6afda49e}} is well-defined for {{formula:214e8bfa-4035-4de0-b831-7f0e5a4f67e0}} taking {{formula:fc2cc656-849f-4d04-96df-907f0a9b2cf4}} small enough. Since {{formula:8a0d5d58-c0ce-4494-911b-df9f9204b40e}} , {{formula:0e9f87b0-f6e4-488e-8bae-c6694730f680}} and the gradients {{formula:89d10c2a-d859-4a2b-b032-c54bb7697e66}} and {{formula:8c482fc3-ded1-410d-9a35-af927dd612b1}} are linearly independent, we can assume by reducing {{formula:c7343db9-1527-49e9-9b79-ceeb81798260}} if necessary that {{formula:b0157dbc-4b5e-4418-8a11-c69b5d2bcc87}} , defined by means of
{{formula:c735c5a9-ba7b-4948-9aa0-cbd93228573f}}
is an analytic change of coordinates from {{formula:44607247-d1c4-4cb2-9f80-7afc156eb193}} to the neighbourhood {{formula:3dc53edf-1f3f-4b23-9be6-1d47e49036c0}} of {{formula:4b2abeb5-1c4d-4a05-80ad-8040a2d6a0df}}
Recall that our aim is to study the solutions of the system of equations {{formula:3efbbb41-d717-4267-9d11-2b73d15ddf4a}} which, on account of {{formula:9ee7072e-6197-4590-af8e-a45b044ea51f}} and {{formula:e7a4a7dc-1d3e-4b45-b641-2e2c405cf14d}}, is equivalent to {{formula:d22f5da3-b023-4782-b423-0680370e5fee}} In order to study the latter we first lift {{formula:99edfc96-c882-4bd9-acec-e731d5f04757}} to an analytic change of
variables {{formula:c1559bbe-4d3a-42b9-8002-c36a365d0a0f}} given by
{{formula:4d71cb6b-b683-466b-ab60-819a35195105}}
which (after diminishing {{formula:771a0592-a15c-49ae-8d55-ced54a9e3d6a}} and {{formula:0abd8af6-f50d-404a-99b7-327a4e6dabc8}} if necessary) is defined from {{formula:6f741845-8c22-4319-becf-89dabee9e4c4}} to {{formula:d8b945eb-b007-44dc-8364-fea29943dadb}} and then we consider the map {{formula:cc75e251-be4d-48c3-a5a6-2ee19e8c8028}} defined by {{formula:ce4d8ea7-6988-463b-a38d-1ff087fb7db1}} with
{{formula:f25d713c-191b-445c-8886-56dacb3e9c54}}
By assertions {{formula:5899522c-761c-4d66-bc2b-294257cc4b60}} and {{formula:a20349e4-7514-4625-b072-84d8ce97d6c3}} of Lemmas A.3 and A.4 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, respectively, it follows that
{{formula:7a191a93-243f-4e31-8889-00924d123c47}}
where {{formula:26ac339d-f5f7-4d00-ac10-41410e3f98cb}} for some {{formula:2708b3e9-3089-4aec-907c-d0634127bd2c}} small enough and we set
{{formula:aa6b76f8-1692-4651-82c6-0eb73dc2315d}} for shortness. Here we also use that {{formula:f53554b8-4738-496f-b0a5-bd875736ff7d}} and {{formula:69d22366-c61d-454d-b261-6e3b274980e9}} due to {{formula:20436afd-d2f4-4b96-a80e-d70fa7b26371}} . Observe on the other hand that, via the diffeomorphism {{formula:38338f01-78dd-46b5-8a76-6529ff29b19e}} the system {{formula:25be33cf-7d1d-4179-bb5d-082bbe94eea6}} on {{formula:fa2df8b0-dc87-455d-80f8-aba7bfdead51}} is equivalent to the system {{formula:ca7ca11a-acdc-4f8d-aa84-febd81e3c532}} on {{formula:17e045f4-3eb4-4218-b1dc-2fdfadcccdf3}} . With regard to the latter note that, by {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, the remainders {{formula:134a6444-9bcd-4ceb-9d2a-84130989a792}} and {{formula:8861c24a-e61a-4504-ba7c-f5408920de5c}} extend to {{formula:bbc6d6ae-3988-48af-aaeb-b1f3dfba8caa}} functions in a neighbourhood of {{formula:32da1e7f-52c7-4813-9f56-1fad0a97852e}} satisfying that {{formula:471ff1e1-0806-4ec8-81c1-dfd8e8813dd6}} . Observe in particular that {{formula:003056c4-544b-4497-b2dc-2610766148ee}} extends to a {{formula:04d624aa-d2b8-4fce-9a2d-f1c12164c264}} function in a neighbourhood of {{formula:d8b81f53-3ece-48fc-b588-bc3c213026b8}} . Hence, taking {{formula:cd3dcebf-e291-4b21-b83f-6aca2286d123}} into account, by the Implicit Function Theorem there exists a {{formula:fd54f31d-bf56-4b0f-8e0b-2a6b4f4ff17e}} function {{formula:f547ad7d-7386-4130-9e76-0a87b62db33d}} in a neighbourhood of {{formula:34665bee-c08f-4e01-a1f3-e678aa43808e}} such that, by shrinking {{formula:76060b66-a879-498f-b0c8-329cd019ebea}} if necessary,
{{formula:3bf824ea-7d4c-4f5d-8a7c-77a0659b572e}}
Furthermore {{formula:40d62124-e127-454b-a573-843e3297f2eb}} satisfies {{formula:d18c10d6-ab55-4130-9b8b-51cfe3dc8d6c}} and {{formula:d89674ee-56df-4a6c-b30d-7cf12d06fac9}} . Our next task is to substitute {{formula:70af516c-5e46-4d68-bf3b-9e07185cb96e}} in {{formula:45c8cac1-c4b3-4b0f-a90d-9c3e34af67ca}} and analyze the resulting equation. To this end we extend {{formula:288bb604-1f72-42ff-9048-0e1b0565976c}} on a neighbourhood of {{formula:f6daaf4a-afd1-4528-a99c-917f516958bc}} by means of
{{formula:3db9caf6-d913-4b1b-bfe5-7427d7e16617}}
where {{formula:3f23dc19-be51-4ef9-863b-cc44280e4c66}} and {{formula:3715dded-6453-44e6-961e-5dcb2cfc11f7}} are clearly {{formula:c180e4ca-ad57-4240-932b-3a5ed15e3d3e}} in a neighbourhood of {{formula:db8c7d42-e711-4ee4-81c3-d3fd8ebc97e6}} . We claim that the function {{formula:2cb2ec20-5415-4008-a219-c305d48b8725}} is {{formula:33fabd0e-2a48-46b5-aa35-3e928aa2d9c1}} in a neighbourhood of {{formula:92ba255d-19d6-4f22-9556-2da2d29429ca}} as well and that its gradient vanishes at {{formula:c18bd218-c9fe-45b5-9cfa-eb4a43319b0b}} . To show this notice first that {{formula:4e5ded7d-54be-430d-b221-2047c7da4537}} and, consequently, {{formula:73eade14-d16f-4f03-bec2-31d2c72c09d6}} Moreover, using that
{{formula:1a1446d1-553d-44d1-b75d-e43aae8973ac}} , we get
{{formula:553f1c9b-ba8c-4798-9982-e2320db3e908}}
because {{formula:53c4c5e2-b700-4fd0-9181-bc025b661da1}} is {{formula:4f7fa8a3-ff32-41e4-ba49-0549dae2f843}} and {{formula:0fae8685-3fc0-44eb-a869-f59458d754de}} implies {{formula:f13e9b15-7770-40e1-bf6c-8474818f92d9}} for {{formula:56625d4d-05aa-4ff6-94a6-1c073c31c5e6}} Similarly, if {{formula:d7a0683e-6ca4-432e-8231-c3c764ca69b1}} then
{{formula:7c3f7c95-7521-4af6-b596-7a003dca1f62}}
tend to zero as {{formula:c09d1fdb-0297-4ff3-a1db-55fa7092141a}} uniformly on {{formula:958cc065-6a96-426c-af69-3e23b0d81a7c}} . This clearly implies that {{formula:712f8db2-792a-4a46-ba02-dd16f52f4e25}} is {{formula:3e647c87-2907-457a-bb94-e31eba936240}} in a neighbourhood of {{formula:e17dcd57-561d-4237-bef7-0051ed6eeca7}} and {{formula:76ebe617-2505-4eb3-b042-cdadb0c9047e}} , so that the claim is true. Thus, by applying the Implicit Function Theorem to the “extended equation”
{{formula:47426cf4-7736-462d-a404-34fcd9099df9}}
and reducing {{formula:c69b6966-7c18-413a-9c53-7ee9d921195b}} once again, we obtain a {{formula:abce5cad-6f8e-46a6-beb3-9e8e8fbb1030}} function {{formula:ca8b70a1-3c2f-48d4-b2ad-a5c1de8098b1}} on {{formula:486355eb-6fd5-4426-8082-8f3a363fa755}} such that {{formula:43efc493-3340-415b-89d1-e03920477f53}} with {{formula:dbd616bb-261b-4469-a639-364ec4e935e2}} if, and only if, {{formula:46af2b64-bf31-4109-83c8-8bd9ff17c313}} . Moreover {{formula:e4b85dac-432a-4677-a7dc-9bde18b17cd3}} and {{formula:147131e5-7953-43ee-9989-bb28ee17e632}} . Accordingly, after shrinking {{formula:c0db97ca-a872-420b-b77b-ee3acf68e1da}} once again if necessary, we can assert that
{{formula:3428d5f8-7d09-4d29-b5e8-8b80e637e94e}}
At this point, since we reduced the original {{formula:43809df2-f2e3-45a4-bd2f-69d32af1e847}} ,
we also diminish {{formula:57cec904-015a-4d09-b4ae-1922e9b03617}} so that {{formula:2d22525a-6b8b-4114-af08-d9b4da9a5a10}} is still a diffeomorphism from {{formula:8d9347fa-9861-47ea-a654-841687c6e09c}} into {{formula:06e99fd2-e6d4-4937-887c-26c7f621aa96}} . Then, from {{formula:040c1332-521c-4671-8870-34f3556f2588}} and {{formula:1082809a-0fbc-45d4-b76f-5d6194523b1f}}, the following assertions are equivalent:
{{formula:53a759c2-b6cf-4360-bdb5-17a9d649fcfa}} ,
{{formula:a0aab3b0-d0b7-490a-9c6c-764f7a8838be}} for some {{formula:8abc8993-61fa-4bcb-a9ec-1a8a679f8065}} ,
{{formula:dc93022e-c7b0-4545-8f0a-e50af8dbd5ec}} for some {{formula:8c0370ab-8045-4808-a14e-b02bd11ad97b}} , where
{{formula:dc745431-76eb-4e1a-88c3-170dd07168bc}} .
It is clear from these equivalences that {{formula:d9852fcd-9990-47d6-8dbc-a6a254c95e1f}} is a {{formula:d86e5277-2832-4925-8feb-6f1c1e8cb9a0}} parametrized curve with {{formula:4c23cb04-af6e-4b32-8e14-75acb080c5ec}} satisfying that {{formula:2820b470-7525-470b-bc5c-9e53f542797b}} . One can easily verify, taking {{formula:16b199df-76f4-41fa-aa88-4b19fa5c61f5}} and {{formula:70e67642-52f1-421f-a6be-ab2a7bc05215}} into account, together with the definition of {{formula:7100435d-145e-44ad-afd5-f42fa65df5fd}} in {{formula:0eb2cfe4-deeb-4ba6-b954-340402f455b2}}, that {{formula:0de79535-1f94-425a-ac80-5e3f14160626}} is a non-zero vector tangent to {{formula:4aabcc1c-9edd-4ecf-9c48-1c7d6f82b208}} In particular, on account of {{formula:afad627d-da32-4d76-ba8c-08e7dca72fa0}} and by reducing {{formula:f1a6d27a-5d22-434e-a9c8-b8f4c03b4cdf}} we have that {{formula:e9b420b7-35d7-4bca-a817-1bbb6ef2bb7e}} is one-to-one. This proves the assertion {{formula:6ad91a62-d8c1-4003-a67e-6991ea7b0579}} in the statement.
Due to {{formula:1e8e9197-c23c-402f-ab15-7dca2835fbb7}} , and after shrinking {{formula:96e0d665-1bd6-40b7-9b04-6ae25ff17a23}} if necessary, note also that the zeros of {{formula:1c64063e-5a3c-4055-9585-237a5330e023}} on {{formula:a479e19e-26bf-46d2-8355-0023ee4407c3}} can have at most multiplicity two. Therefore, since the interior of {{formula:9bc20bb4-4070-40d0-a2c9-4e0fcd05289a}} is empty (as a subset of {{formula:c5f71a0b-d2bb-44e2-9fdf-a212270838c3}} ), by applying Lemma REF we can assert that each {{formula:030b4ab2-0f81-4863-878c-ad2a2d0f77c6}} is a local bifurcation value of the period function at the interior, which shows the validity of {{formula:1c6b10e9-7197-4b25-a536-a75349729972}} in the statement.
With regard to the assertions in {{formula:5217288a-df51-4b2b-8f5b-dbdd5aeba393}} , we note that the uniqueness of {{formula:f59f02d0-1ed1-4d87-8c64-b354aac52787}} and {{formula:b085b5d9-7d85-4b07-b132-fbbf8fe9c393}} follow from the point {{formula:c39230d3-b028-48c3-8639-3bfe2ea7a4d2}} above using that {{formula:1c97943e-0f59-4a0e-9768-a06926eb854d}} is invertible at {{formula:9f30ab89-7f18-41a7-8bdc-23eef13b78e9}} and that, by definition, {{formula:8d5f1928-28c7-4ee9-aef7-c858ba1063af}}
On the other hand, since {{formula:e8d7d837-799d-45e9-b94f-97d3b294f3db}} for all {{formula:9eb71afc-88a6-41b3-93c8-c74719ee6f47}} , from {{formula:82c1c7ab-357f-4b12-ba06-bd37793d163d}} we get that {{formula:99232339-3eed-4fb5-96f8-6afa5a993b41}} for all {{formula:56ccc6c7-1032-463e-8d13-02b08948ed86}} . Here we also use that {{formula:1d4b5357-17b4-4bbc-936c-5308afc04edb}} to take advantage of the properties of the remainder and the fact that {{formula:a66e5bf6-a73e-4672-a94c-9aa7c5feb8a7}} By arguing similarly, on account of {{formula:31a4300c-2d0b-445a-aea3-d4c8f8b0aa84}} for all {{formula:4a5a6e5e-6fa7-413a-a52c-5be97777796a}} , from {{formula:179b1273-44bd-44b9-8848-009a28b98670}} it follows that {{formula:aabd38ae-8d7a-4ee6-9531-33a025c922c2}} for all {{formula:bc0fa4ce-26ca-419a-8af7-2354a027e370}} . Taking this into account the assertion in {{formula:b7266ed6-a23a-4e6b-b062-4543987d0a2a}} is a consequence of {{formula:b46a21b8-082b-4c04-bc47-49621c879c69}} see {{formula:597186e3-4652-48ce-8afa-eeb112c5ee43}} in Proposition REF .
Finally, in order to prove {{formula:ab4ee8bd-192f-4a18-ae3c-73f64b81b6c5}} , let us consider {{formula:a5c74357-5338-425f-bfa5-2c064d83c9ec}} and note that from {{formula:af4fb7c8-70f5-4fbb-801e-46a7482c7de5}} we obtain
{{formula:600f34ee-a077-4a27-bd99-430b3bb12970}}
where we use that the flatness of the remainder {{formula:d982a8e1-de7c-4fab-a3dc-88ceb06fe26a}} is preserved after derivation with respect to parameters, see Definition REF . Similarly, in this case from {{formula:121faf4a-cc73-4189-99fe-7982c3182732}} and using also {{formula:594ea54d-2a16-479b-9fdc-a4847b6b3274}} , we get
{{formula:77e4b6d0-8e28-418e-93c8-2970662c60cc}}
Thus, since the vectors {{formula:878d00df-0527-42a0-bfbe-15a8b53e9979}} and {{formula:116a18f4-5c25-465f-9cd0-ef0bc8229be1}} are linearly independent, so they are
{{formula:263a75fb-0d2e-4771-90a8-85676649930b}} and {{formula:c5df8958-3da8-4b01-be12-75392d4545a7}} for all {{formula:bf2a97df-dce7-4355-a1f6-00f565a7a272}} (after shrinking {{formula:7d5cabdb-cebd-40ab-a9bf-b8154019c602}} if necessary). That being said, we fix any {{formula:09689c7c-321b-487b-8d3f-56a97ffe3221}} and compute the second order Taylor's expansion of {{formula:95d138d7-accb-41fd-b5f2-825f4fb9c8d2}} at {{formula:133d6552-ee0d-4d01-83b9-5e591b9f4143}}
{{formula:8a2e555a-be09-4fce-baa6-f102cd082860}}
Then, due to {{formula:d84fc7b0-9e37-4f7c-9b10-e9a607df7283}} {{formula:84a84928-2f21-44d1-95e6-7e66e9b3f31d}} and the fact that the gradients {{formula:f4e8ef4f-3c42-4e03-b026-e07fa17b2150}} and {{formula:f7e83915-cffb-4740-852c-a410fb43f40a}} are linearly independent at {{formula:6b96cf34-c025-4384-934c-e1deeceba6ef}} the application of {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} shows that for each open neighbourhood {{formula:66a1a9e8-ff2c-4d84-a98e-9046035580d3}} of {{formula:cce1a5f1-3cff-4af5-a3ab-b203efe3209c}} there exist {{formula:075cf6a1-b4dc-4a7f-8685-1209cff71770}} and two {{formula:37b455f0-26f4-4f7f-b5e4-5897935c20f3}} such that {{formula:b9d327cb-ad97-4385-bd16-595f6bf551ae}} This proves the validity of the assertion in {{formula:92d2f4a9-babf-49e0-98a2-7c29c5783797}} and completes the proof of the result.
calc
{{figure:009f63f4-47cd-4847-b348-b3a317f73830}}Remark 4.5.
Let us finish this section contextualizing the results in Proposition REF . In Figure REF we display the ellipse {{formula:0eca2ef1-8557-4efb-bb77-96552927d94e}} that consists of local bifurcation values of the period function at the inner boundary (i.e., the center) of {{formula:77141254-81e0-4522-b7ce-b66cf072f5c9}} . It corresponds, see {{cite:2bf3254451791a68ce817916f287f1704a0d2519}}, to the vanishing of the first period constant
{{formula:90a87f28-5e1b-4323-aa56-b653e6bb83f0}}
Moreover the curve {{formula:f99b3ef8-3440-4066-a8a8-c458bc29b9e2}} consists of local bifurcation values at the outer boundary (i.e., the polycycle) of {{formula:c321bd29-22db-4c9f-862a-1da103a89b76}} , see {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}. It is made of the arc {{formula:d6b84370-509e-4aaa-bc66-5ebbd24c120d}} joining {{formula:0e229e74-2552-4b98-90aa-cd12f8e82c06}} and {{formula:9177931e-c55e-4e97-af68-05e0f7601230}} together with several straight segments. According to Proposition REF and {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, respectively, the germs of curve {{formula:cc6e211e-d8a2-41f8-9514-a68f782dda28}} and {{formula:c76c3f0b-7904-4699-ab6c-92badc0cd128}} are inside the set of local bifurcation values of the period function at the interior of {{formula:fc019ea0-06bd-4ecc-9630-0c2b86b6af1e}} . At this moment we do not have any analytical tool to fully characterize this set. We conjecture that {{formula:75b2e02d-894d-466f-97d8-7e2cb9a9c7d5}} and {{formula:9c6d6774-f1d6-4026-bc2f-00eef0c99de9}} connect with each other to delimit a region of parameters for which the corresponding center has exactly two critical periodic orbits. With regard to this conjecture
it is proved in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} that the center of any parameter inside one of the two light gray regions has at least two critical periodic orbits. The boundary of these regions is inside {{formula:b3f22847-7370-4a2c-984c-87bc1934c625}} {{formula:75a4f65b-324f-45f5-8ec7-04b80e70b1f0}} and {{formula:e4d42c2c-6679-4770-877e-24002c95d6de}} For completeness let us explain that the curve {{formula:b6c220e0-8d09-4973-b8d4-611dd6bdb421}} consists of those parameters such that the period function tends to {{formula:2d91b21b-b2b6-4a1b-b154-f32c8210c6c8}} as the periodic orbits tend to the outer boundary.
{{formula:217b0f05-4dcf-442b-8d0e-af52f1e67dab}}
Proof of Theorem REF
Proof of Theorem REF .
The statement covers all the parameters {{formula:3bc64f17-423a-4b56-93c3-78a3459b5404}} outside the vertical segments
{{formula:00128262-a2b6-4daf-9a75-51c49856bfa6}} For simplicity in the exposition, instead of proving the five assertions in the statement separately, we split {{formula:3758d0f2-8ba7-47f2-9f3d-819041873e54}} depending on the result and tool applied to study the corresponding criticality. For reader's convenience we enumerate the different cases that we obtain in this way.
Let us consider first of all the set {{formula:f13a6b46-0061-465a-86fb-3755ff370879}} ,
where recall (see Figure REF ) that {{formula:d3f30ac0-f26f-4ab6-a8f0-a3f637f05da6}} is the
union of the dotted straight lines, whatever its colour is,
and {{formula:9b7dc72a-edcb-4fac-a693-0379107dcc1e}} is the Jordan curve in boldface type. Then, by {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}},
we know that any {{formula:094b2533-c03e-4e2d-8225-0ec36d4d8d7f}} is a local regular value of the period function at
the outer boundary, see {{formula:f673f1c9-7057-4c2d-91ca-4ede19a4348d}} in Definition REF . On account of this, by {{formula:613a5758-dc29-4560-94fb-a0d1fd225cd3}} in Lemma REF we get that
{{formula:4d2ec5ad-7c6d-468e-8e0b-08306e12f6b3}} . Here we also use that the period annuli of the
Loud's centers vary continuously, see Remark REF , and that the outer boundary of
{{formula:2d8ba376-2a86-4c7f-8cf8-0b8274d856f8}} for {{formula:28ea0d61-63f7-45ed-a1e6-e370dc2d9028}} is a hyperbolic polycycle,
see for instance {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}.
The criticality at
{{formula:4ec958a9-2706-4109-97bb-5db70cc86261}}
and {{formula:1b0e288a-a023-47d4-bddd-f1629e5b4ee0}} follows
from the results in Section REF . In this case {{formula:19fceb1b-4816-4674-833f-72bb13fdb816}}
is a parametrization of the outer boundary of the
period annulus verifying the hypothesis
{{formula:d151341a-417b-46a9-97ca-b28d356030ad}} , {{formula:54c1084c-2307-4747-9995-f455e0c649cf}} and {{formula:f8fab717-6572-41f3-8394-e24ecfaa2809}} in Lemma REF . Moreover denoting by {{formula:69140191-7e02-447a-bb51-c2efde60d14d}} the period of the periodic orbit of {{formula:a80fa2fe-92ed-4198-afa3-ba177c630162}}
passing through {{formula:d717c7a3-d757-4c12-a0c2-1ca441abd465}} , we have that {{formula:61376f63-e552-42dd-a1a9-57e280023fdd}} , where {{formula:2476958f-6968-451c-929e-bf4cb02408f2}} is the Dulac map considered
in Proposition REF . By applying this result we know that the first non-vanishing coefficient
in the asymptotic expansion of {{formula:3221c501-fce5-4905-98ea-8709453e3668}} at {{formula:3720a52b-748d-455a-943c-cf06b2db0ce6}} is the third one for all {{formula:ddbc621b-42c9-43c9-a826-7c44c98f5357}} Therefore
{{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} implies that
{{formula:635d1049-1b7e-4faa-b350-dfeaf7017203}} for all {{formula:14d135cc-6d27-43b6-b062-fcbad80c69d6}}
On account
of this, by assertion {{formula:68c1b32f-b6d0-4bdc-ab4b-c81469f1afce}} in Lemma REF it follows that
{{formula:fc5311da-4d0a-44b7-ba9c-4d6b5b48c540}} for all {{formula:1c1474f2-a489-4543-aa7f-b401275930a4}}
On the other hand, due to
{{formula:f44e638b-67a7-4812-99fa-031718559ac2}} , we know by {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} that these parameters are local
bifurcation values of the period function at the outer boundary. Thus, since the period annuli of
the Loud's centers vary continuously (see Remark REF ), by applying {{formula:9eba31b6-3dda-4c11-bae3-b49bca37955d}} in Lemma REF we get that
{{formula:0d1d45fb-ad2b-43eb-ad0c-9cf0c49add8e}} for all {{formula:871477d4-77ec-48cf-a4e5-417c2a2a6284}}
Hence {{formula:bf5b9bfe-6d6a-4988-bc6d-7f4ad36cb9f6}} for all {{formula:9ae99127-e82f-436e-b5e6-c438fdd601a6}}
We turn now to the criticality in the segment {{formula:6cf2f8f0-6dc2-4333-a2d6-f7876307e39e}} So let us fix any
{{formula:5a7ca572-9f83-431b-9c56-218b033800a5}} with {{formula:75017bc2-31d5-420e-8ac5-bd548ab6b6fc}} and note that then,
by {{formula:db490cde-1b4b-466a-8649-1ba0e0ed327a}} in Proposition REF ,
{{formula:4c8860d7-6340-4bfb-b68d-e098b6ac6196}}
where {{formula:24cdf057-a954-4b24-a8b3-d29edd815f8a}} for all {{formula:739a8175-b688-4407-bb57-c57dbe928ca5}} small enough. To obtain
the derivative of the Dulac time,
we use that {{formula:85d848af-f85e-413e-9ebe-49bccd63341e}} and that,
by {{formula:9bfd3924-3574-401c-8530-21f9fe3ff2c2}} in Lemma A.3 in {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}},
{{formula:86306eac-3437-4b11-a3ed-b5da1411b666}} From this equality,
since {{formula:25f8c0cf-5033-43ba-b98c-25b2de466a85}} tends to {{formula:2af73806-40eb-49e2-b71a-5626b3bd4c7a}} as {{formula:87e51899-db12-4666-8a7a-1babe352e68d}} due
to {{formula:f7c40512-63e4-4dbc-9387-0cc1e8e770fa}} (see Definition REF ), {{formula:f2920df4-dffc-455a-b0c5-5488f430618e}} tends to 0 as {{formula:14f7f974-4a44-4bc0-b49f-ad8037864836}} ,
{{formula:dcf99395-7145-441e-bff4-2ddae3a97d50}} and {{formula:7ef5b82e-1326-4f9f-9ad4-4001d764c64a}} , we can
assert the existence of an open neighbourhood {{formula:b19545c2-fc1e-46e7-a1f9-4f390c09e7c7}} of {{formula:88b1e58c-3de0-4c70-b46c-82d32d221268}} and {{formula:19a6d0b0-2a27-411a-88e7-b29f842b7042}} such that
{{formula:444a5801-a178-4f49-b781-e36d528e3b17}} for all {{formula:21a625df-f3fb-404f-a10f-f70cc7a552fe}} and {{formula:290099bb-1117-45a9-92ac-70bcc352cad5}} Consequently
{{formula:8d3f79cb-4dc2-48b3-96c4-da20253c75e0}} and so, by applying {{formula:331ef49d-e8b3-41c4-83b6-1a2bbe6c3db1}} in Lemma REF , we conclude that
{{formula:e932d47f-af54-4a32-b587-81153f294454}}
We turn next to study the horizontal segments
{{formula:61e5ee4c-6bba-44a1-804e-f78f2abefbc3}}
and the curve
{{formula:4bb8b21d-9f83-41a4-b250-9d5070661fae}} Here we set
{{formula:0792bb7e-0b9c-4c32-b7c8-6ddebfe68f07}} because this parameter yields to a distinguished case.
We begin by noting (see the first paragraph in Section REF ) that {{formula:3d98e92f-b173-4381-b72d-adb95e8f75a7}}
is a parametrization of the
outer boundary of the period annulus verifying the assumptions in Lemma REF and that if we denote the period of
the periodic orbit of {{formula:c68a5fee-9330-4a79-898c-fbc70497d0a0}}
passing through {{formula:f2242ed6-e208-42f8-9b4d-ec885a2c274e}} by {{formula:cb747d38-05e8-4494-a37b-39dcd299e36f}} then {{formula:ef565932-2422-464f-bfc2-ef5425f8e13e}} ,
where {{formula:79356c7d-021d-499d-a6ec-a737a56c6737}} is the Dulac map considered
in Proposition REF . Thus, by applying first that result and then {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} we obtain that
{{formula:696458e5-79a1-45ae-bcf7-a07ca087e679}} for all {{formula:298cd2cc-51d9-4f01-8ba6-0268dabbd5ec}} .
Moreover {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} shows that these parameters are local
bifurcation values of the period function at the outer boundary because
{{formula:50f85bd8-3e39-49ea-b397-ed4876cf9c61}} . Since the period annuli of
the Loud's centers vary continuously (see Remark REF ), by applying {{formula:c49a5a1e-44c4-4c8b-a147-84982e2bfa66}} in Lemma REF we have
{{formula:a715bbb3-2f74-4c2e-a6b0-dd91b29d3165}} for all
{{formula:f4713bd8-bf71-46fc-ad06-4ca0087b6344}}
Therefore {{formula:1b8d20d2-5b88-4643-9d5c-12ed2b29c690}} for all
{{formula:5b6b53fa-ab5a-4d58-a9c9-3bc03b46d1ba}}
On the other hand we have that
{{formula:3eac813b-0a0e-4bea-9948-58d15c5f7aa8}} by assertion {{formula:3bd7a156-5c3a-497f-afc8-e51e49d71fc9}} in Proposition REF .
Finally the fact that there is a curve of local
bifurcation values of the period function at the interior of {{formula:70eb37fa-b473-4fc3-b7d4-2dacbe78cf38}} arriving at {{formula:3e836178-b9c2-48ed-9efc-cbf4128ba69c}}
tangent to {{formula:2c20e831-4d47-4733-8ede-b311c894a179}} follows from assertion {{formula:27978aa6-3ed9-4d2e-ac4f-b911c6786c5c}} in the same result.
Next we analyze the parameters in the segment {{formula:b44decfc-411c-47ba-8f9f-a0d5a45a0ee8}} , that corresponds to a case in
which there is a saddle-node singularity at the outer boundary of the period annulus.
This is treated in Section REF , where we introduce the map
{{formula:78f2040e-1241-4f4b-86f2-453a28abe11e}}
that provides a parametrization of the outer boundary of the period annulus verifying the assumptions
in Lemma REF . In addition if we denote by {{formula:dd4f1339-380b-4f18-9b7e-355d6e002dc3}} the period of the periodic orbit of {{formula:e7c6c57d-3a49-4360-941b-19789f4841f1}} passing
through {{formula:a7672dd7-97dc-4a62-a351-faf3c5eca462}} then {{formula:fee1cf71-20fe-4075-997a-8df4cbd1a94f}} , where {{formula:2f446873-81eb-4c0a-9c78-9e526fb2e9bd}} is the Dulac map considered in Proposition REF .
From that result we get the existence of an open neighbourhood {{formula:4538ab4d-8075-4cf7-bb10-4adcea01f859}} of {{formula:262f606c-aad1-44ea-bff5-3e539eb4ef50}}
such that
{{formula:9a7fcf1c-2057-4471-bcaf-3a9d2755f1ef}}
with {{formula:426bf944-9877-4fa0-bb5c-a06b35e20c90}} and where {{formula:14d9532d-4f00-435b-a778-a0cde21c080c}} and {{formula:f178250a-1983-4d09-8129-da1a30c95c69}}
tend to zero as {{formula:1aff19c1-e74a-4d89-bbc1-18b54c9ba9fb}}
uniformly on compact subsets of {{formula:5e1d276d-064f-412d-8316-dd802e074474}} . We know
moreover that {{formula:746d634b-cc49-40ce-b2a7-29818b0af898}} if, and only if, {{formula:5ae13bf9-f2a8-4ef9-b9d4-57ef0a1ce8b6}}
and that {{formula:e40841a0-f798-48bf-acdd-7e7338e3c97d}}
If we take any {{formula:717991cc-cf47-421c-aac5-cc0fd8b5c4c7}} then, thanks to the good properties of the remainder,
we get that {{formula:4915692f-4089-4fe6-930c-2530e320bd88}} and this easily implies
{{formula:f3d11b79-86f0-47f8-bd2a-fcae2cd54946}} Hence, by applying assertion {{formula:5254b30e-1aad-48ac-9ce0-c89f0b5c35ed}} in Lemma REF ,
{{formula:e002daac-c403-4fca-a29c-6d649a775b53}} for all {{formula:58f43838-5d58-4722-8e4f-9982a9175a1d}} .
In order to study the criticality of {{formula:68a326a2-5ca3-4b31-89ac-22c7d0b1ff3d}} we observe that
{{formula:ce875a75-bfd4-4c69-b759-760857dec554}} and, consequently,
{{formula:0823b262-034e-4e9f-a69a-cfb9d1cf3cfd}} by Rolle's Theorem. Therefore, by assertion {{formula:f61e4d4f-d7f2-4e86-999a-d7f229f79d34}} in Lemma REF ,
{{formula:c968904f-fcdb-4126-ae19-3987239cf5b1}} On the other hand, the application
of {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} together with {{formula:f0a475be-cca4-44b1-bc95-b7be07d44c5e}} in Lemma REF shows that
{{formula:dcbbd39c-e654-4563-abd2-27bae5b476e7}} due to {{formula:a2ebb1c7-aef0-47a2-9052-ae73bf718de8}}
Hence {{formula:81f9d8e0-2384-49dd-879d-210378fd8dca}} .
We proceed with the study of the segment {{formula:5e838936-d3da-437d-813e-4217e4703a7d}}
which, as in the previous case, corresponds to
period annuli having a saddle-node singularity at the outer boundary. In order to compute the
criticality of any {{formula:1956b12b-8ccb-440a-8922-f5af577cb5b4}} we apply the results obtained in {{cite:33f03877dbf6e3e78b708de5e0136ff468423f5b}}. In that paper it is proved that
for each {{formula:081d0f69-f320-45e1-b6fd-2346dc29affc}} there exist {{formula:630d82d2-db2e-4bf7-856e-92a8e0e34600}} , an open neighbourhood {{formula:a9a48ad1-8289-4e87-b86a-89b2d8d990d4}}
of {{formula:26b7cd23-4aea-44cd-9e83-b9c17c0d80f7}} and a continuous function
{{formula:a3b25710-5fb5-4aed-8fde-ff66a857ada3}} verifying the hypothesis in Lemma REF . Moreover, denoting
the period of the periodic orbit of {{formula:b4aa40e4-4aad-4b18-a73e-70ec53579d38}} passing through {{formula:4d39c3d1-dfc4-4ad4-aabe-28e279eef390}} by {{formula:ba17073d-daab-41f7-87c4-0db9d7dee474}} , the proof of
{{cite:33f03877dbf6e3e78b708de5e0136ff468423f5b}} shows that {{formula:7334073c-1016-4935-a845-b8c3c046705e}} tends to {{formula:44df2445-eb53-443e-ad45-95ff2539e2ef}} as {{formula:73b9e264-56dd-45a9-99d0-f1715c4810a7}} Consequently
{{formula:f64ae17e-a81e-42d9-a7a2-ec839df011ba}} and hence, by applying {{formula:5e081d3b-e3ec-4b1a-bf2a-3dbc43615881}} in Lemma REF , we get that
{{formula:edb6c252-1107-4019-9671-6af24e7d49ff}} .
We analyze next the parameters inside the segment {{formula:470df06d-9020-4224-a0bf-4b7c0112a94a}} So let us
fix any {{formula:79a30e8a-0962-4622-a307-d63e361c4dd2}} with {{formula:e4c81b15-3c56-4c92-91b9-2c54fea8d8ab}} By {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} we can assert
that if {{formula:cb4f3591-8d89-4bd5-a9e8-649d38d79908}} then {{formula:8da8fbfa-fb83-49c3-aaef-45c1985db032}} is a local bifurcation value of the period function at the outer
boundary. On the other hand, if {{formula:8ae4a176-ed25-40d4-bf88-cd2d1e41007d}} then we can conclude the same by applying
{{cite:3a485cccaa57abdd2f40293534231f244ea4dbcf}}. Hence, since the period annuli of the
Loud's centers vary continuously, see Remark REF , assertion {{formula:44805540-3625-4c98-9245-d6aa2be79b64}} in Lemma REF shows that
{{formula:da78b44f-97e8-49c5-acf1-a6f9ed6cdf63}} for all {{formula:1bd55cbb-8264-45d4-b90a-dae53769f93f}}
From the results in {{cite:f0d4fc02e805c76a4504eb8202c701fca40d1ed4}}, {{cite:9109926e171a4eef1283fc5d92f295372e1937c0}} it follows that
{{formula:5755fe39-581b-4435-b46a-241aa2f258ae}} for any parameter {{formula:5c1dfd1a-7c23-4115-84e3-a181016f1071}} inside the
set
{{formula:dcfff02c-fb5b-4609-b752-8e112fa93e3a}} .
Indeed, in those papers the authors determine
a region {{formula:9e7f1de2-0c6c-40f0-9a32-46634d7ccbc1}} in the parameter plane for which the corresponding centers have a globally monotonic period
function. Taking this into account the assertion follows easily from the fact that
{{formula:f1557871-3502-4433-b7d3-ac250543091c}} is contained in the interior of {{formula:2a730020-fc6e-4fb0-9b30-5b5c7e9b5679}} , see Definition REF .
We consider now the half-line {{formula:ca39d847-770a-49a1-bae7-88be7a880e74}}
so let us take a parameter {{formula:89369e1b-c07b-4544-82b0-34c7258873ab}} with {{formula:6cbb7328-a39d-4d85-a2cf-0dfef669ad5e}}
In this case the assertions with regard to its criticality
follow from the results in {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}}.
It is proved there that there exists a function {{formula:57573e3b-e8f1-4ebe-9973-7db0dd333934}} in a neighbourhood {{formula:c2c9b904-5b8d-43d9-a5e6-0a9df10537d1}} of {{formula:58a6abf3-078e-4ce6-8c6c-7a23bcbe14d7}}
such that {{formula:bd20748c-98eb-4ee9-96fd-7467d8f7b390}} is a {{formula:302f28ee-0637-4892-9ef3-37b6f9db7618}} map on
{{formula:77a5eb0a-040c-4bbb-ba23-d17e07b902bd}} verifying the hypothesis
{{formula:68a0aefd-8cfd-4779-b980-46a44ab4eae1}} , {{formula:155a6a7f-537f-45c1-940e-0fec1fde640c}} and {{formula:805dac6d-ce3b-47d3-8b3e-9391fe11871e}} in Lemma REF . Moreover if we denote by {{formula:230a7a5c-1c82-4135-9ca5-1333aec29d7a}} the period of
the periodic orbit of {{formula:ad59456a-4afe-4a1d-ad03-4c7786a9a4ee}}
passing through {{formula:c5ef1f9b-f031-4129-8d4a-2c89cfe617e2}} then {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}} shows that
{{formula:ed7e0143-c0dc-4f6c-974a-2f10f96a26de}} if {{formula:2821bbad-dfdd-4582-a0c9-63f6918582ca}} ,
{{formula:804c4a2c-43fe-41e6-bbea-4e49565fe730}} if {{formula:2312df4a-5e36-408b-97ec-1d7748937081}} and
{{formula:ad9aeb9d-1ed4-4cd6-baa0-b181ddf72cc9}} if {{formula:7a01f7c2-54f9-41d5-8b2e-13efb4f446fa}} .
In the first case
{{formula:d708c37e-cd88-4c20-b313-908afce59328}} by {{formula:a6a4d1fa-1440-4ec7-9675-b89ba8c6770d}} in Lemma REF , whereas
in the second case the combination of {{formula:42056f1e-9b98-4775-b276-32e920e11782}} and {{formula:485c3af8-0098-4834-ae6c-dfa31a771721}}
implies {{formula:1a73d86b-208c-41b9-a833-3daa7e0386bf}}
In the third case, by applying {{formula:24a0ff18-b8a7-40a6-a696-feb04e98d409}}
we get {{formula:ab4de74c-3120-4709-89e9-f77a31d9e7a7}}
To show that this upper bound is attained we also apply {{formula:0999ffb6-5481-4995-915d-a35dfb61b71c}} in Lemma REF but to this end we must
check the assumption that for each open neighbourhood {{formula:1de856b0-1ca8-48c0-b6b0-001cc1c3287d}} of {{formula:7f02efbd-e624-4a48-8ca0-ee71705c2379}} and {{formula:c482ae0f-e270-4677-8a17-60a56a980d5a}} there exist
distinct {{formula:6db55bb3-e825-4807-88a0-3a9ce44d091a}} and {{formula:18cee820-47a1-4d28-98be-b317662626fd}} such that {{formula:4c38b859-aed6-4c65-90c5-9cd49abda4bb}} for {{formula:70d7d565-25c4-4727-82eb-77bb66ab7708}} To verify this
we note first, see {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}}, that we can write
{{formula:87f01de2-fd41-49e8-a4e0-abc116579ba8}}
where the coefficients {{formula:766ac412-d55d-40a3-9918-c2fd38ba9436}} and {{formula:bbcbf857-a164-4e4d-a762-61c143425824}} are independent at {{formula:ab6ad2c1-a97f-45e0-979a-39098c936bb5}} in the sense of
{{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}} and, for {{formula:16725d35-ab0f-4001-9f41-f971350ee52a}} {{formula:1e5dfc90-f9bf-4706-801c-e8f8f142c67e}} .
On account of this and {{formula:a3ff66e9-26ca-4306-be8c-ad06c1b4d942}} the fact that the mentioned assumption is verified
follows from the proof of {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}. Related with this let us also mention,
see again {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}}, that the ordered set
{{formula:414bc780-bbf6-4773-9255-246f4778ca28}} is an extended complete Chebyshev system
on {{formula:384e67d8-18be-44e7-9c17-db3cd2cf7617}}
for {{formula:0d86915c-9180-48e4-addf-0e75b71957b8}} sufficiently small (see {{cite:8bf7ff9b569dc3a6597a334ac0db290966d80658}} for a definition).
On the other hand, by assertion {{formula:3a0bc82a-bb4b-4ac1-9d32-bb3e596fbbf5}} in {{cite:ff16723c957f9a8d09068b9f56983b0b5ef97e44}}, there exist a neighbourhood {{formula:8e15d268-e79a-4055-acd9-3181f39e126a}}
of {{formula:7af836d4-889a-458e-81a6-e29070909729}} , {{formula:037aa8f1-ffea-4c83-8796-c6d75c792f77}} and a injective {{formula:77b991eb-823f-4d0f-b251-6e93438d1149}} curve {{formula:51984093-18ae-4f35-9403-2c36655d1025}} satisfying
{{formula:297e1247-dc38-445a-9f3f-29b66b3a1560}} and
{{formula:5b792256-dd87-4bcf-85e3-9cb4e15bb478}}
Furthermore, assertion {{formula:0759900b-4304-4b6e-89a8-bf150920201c}} in that result shows that the curve {{formula:13704f2c-a1ce-428d-997e-7c1b1eb8af9e}} has an exponentially flat contact
with the straight line {{formula:44d5bcae-588a-4a6d-b5ef-ede8f80739c7}} at {{formula:eefe6cc8-3fce-4f3b-9516-fe3a293f42ef}} , see Figure REF .
Since the interior of {{formula:e22f94d3-a58f-4a02-a092-24621be3c5ef}} is clearly empty and
the Chebyshev property explained above prevents the zeros of
{{formula:80df9317-7186-4721-9eac-2d3e4b22945e}} to have multiplicity greater than 2, the application of Lemma REF shows
that {{formula:8cf831b4-5397-467a-a6f7-7246099b51f7}} consists of local bifurcation values of the period function at the interior.
Finally the fact that the criticality at the outer boundary of the isochrones
{{formula:115b8f74-94a0-40b9-b584-a0574a2573e7}} and {{formula:b3faa0c4-e1ce-4b0a-a648-f5e23070fbda}} is 1 follows from Propositions REF
and REF , respectively.
Since {{formula:ef1a477d-ac7f-4b8d-83bc-88d0e714149b}} this concludes the proof of the result.
{{figure:a896b107-dd2d-4052-8e09-82f6e402a5b4}}Remark 5.1.
We conclude this section by making further comments about Figure REF . It follows from {{formula:751dbadd-6e52-4a09-8394-b42fe073dc04}} in Theorem REF and {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, respectively, that the germs of curve {{formula:6c8d41c4-ea6e-4efb-87eb-de4ec90921f9}} at {{formula:9f85447a-25b2-4039-8411-3a4fe95fb31b}} and {{formula:661e407e-bbfb-482c-ae5a-baebd7cbae19}} at {{formula:95af54fc-eee3-442f-a59c-771568fbdfd7}} are inside the set of local bifurcation values of the period function at the interior of {{formula:c1ce30f8-1d41-4432-bc36-cc9ea0a7364d}} Exactly as we explain in Remark REF , we conjecture that both curves connect each other to delimit
a region of parameters for which the period function has exactly two critical periodic orbits. In this regard
{{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} shows that the center of any parameter inside the light gray sector has at least two critical periodic orbits. We know now, see point 8 in the proof of Theorem REF , that {{formula:92f81f87-2120-45ec-83a8-9379d48fc967}} is at the boundary of this region with exactly two critical periodic orbits. The numerical visualization of this fact is a challenging problem because {{formula:c4d63e20-b5f1-4b9f-b9fb-50924f786a75}} has a
exponential flat contact with {{formula:ab38e51f-ad63-429f-84f6-5ac40a483d83}} at {{formula:29581feb-958c-48ba-9328-195232cfc8f2}} .
{{formula:ce12c995-4511-4044-88eb-39ab8180963e}}
Coefficient formulas
Previous results about the Dulac time
This appendix is entirely devoted to the proof of Propositions REF and REF in Section . For the parameter values under consideration in both results, and thanks to the symmetry of the vector field {{formula:d399d8f2-9729-498a-bfc8-5ce55171752f}} in {{formula:40670999-1347-42ef-b91b-745294623f7b}}, it turns out that the period function is twice the Dulac time associated to the passage through a hyperbolic saddle at infinity. The asymptotic expansion of this type of passage is the subject of our recent papers {{cite:9fe1c57b33ca2a3bd306c334e7617bd08031a793}}, {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} and in order to prove the results in Section we strongly rely on the tools developed there. For this reason we first summarize for reader's convenience the definitions and results from those papers that are indispensable here. We recap the results in three theorems. In short, Theorem REF will provide us with the monomial scale needed in each asymptotic expansion, which only depends on the hyperbolicity ratio of the saddle, whereas Theorem REF will give the explicit expression of their coefficients in terms of a sort of Mellin transform that is introduced in Theorem REF .
In order to facilitate the application of the above-mentioned results we particularize them to fit in the context needed to prove Propositions REF and REF . Thus, following the notation that we use in {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}, let us consider the parameter {{formula:bd301625-8496-45e1-90d4-aedc84aaed4c}} , where {{formula:890d37e6-9aa9-4ce6-a2c9-fab16a824b21}} is an open set of {{formula:61ea2589-48c1-48d4-b699-1d78a0c083ea}} , and the family of vector fields {{formula:1cd2f711-1883-48db-a1a5-80a83483b966}} with
{{formula:efb28ebf-083b-4d93-8cb7-7563d1fcd260}}
where
{{formula:08ac5a36-bf16-45ba-bc83-d6a9f12b1dbd}} and {{formula:e85849f0-7583-4d35-91c8-1c3787343c65}} belong to {{formula:fda87100-af94-4307-bf11-ff71e8706031}} for some open set {{formula:bf1bbb9f-6abd-4174-b95c-522a79da21f7}} of {{formula:43524209-2493-49ac-9bcd-7ebf6c03d1d5}} containing the origin,
{{formula:922eb3d6-2c4d-4f84-b745-b24a62837de6}} and {{formula:1c70d105-eba7-4eee-affd-bec29e7ec470}} for all {{formula:8bf0703b-991b-4175-a6cf-88c494bd0d75}} and {{formula:ba3b682e-f10b-40f2-bbbb-81bac0f8c39d}}
{{formula:e3a3a42c-667f-4d05-bf51-a3cbdb4057b2}} .
Moreover, for {{formula:5ff7330b-bf54-45e8-9402-637d59636ca4}} let {{formula:19b508b4-725c-4864-9a59-07f1da870b14}} be a {{formula:f2f758a5-b134-438c-a61c-569f981925ac}} transverse section to {{formula:7b013207-dbb6-4112-b2c1-06acbd73f9d4}} at {{formula:b38dd2de-7eed-43f6-8725-27cb9c3a942e}} defined by
{{formula:66e6a48e-610f-4f20-ba80-0380a2e67acc}}
such that {{formula:3412b5bc-78fd-4bef-9b56-1e98b54e7d15}} and {{formula:48735566-5710-4608-b5da-d1d3dc297594}}
for all {{formula:30a008c9-00dd-4d89-92ab-e58d7e59cce4}} Then Theorem REF is concerned with the time {{formula:509b11f3-8f34-4c0c-8f96-e8ea3025798b}} that spends the solution of {{formula:418692f7-ea8b-41a1-92e7-11631e4ca061}} passing through the point {{formula:9a6d8b39-c1cb-44ff-bc7f-6d192b2bb85c}} to arrive at {{formula:b66858bf-67d5-46e8-ad53-bdbdb9d2c153}} see Figure REF . More concretely it shows that {{formula:cfbd3edd-b138-494b-a722-6b230d9dfc30}} has an asymptotic expansion at {{formula:1ec01900-1b2a-4376-a51e-99824df0eff5}} with the remainder having good flatness properties with respect to the parameters. We specify these properties in the following two definitions.
{{figure:951c6220-632c-4f91-83ce-88da5b889d1d}}Definition A.1.
Given {{formula:f5607d73-8bb1-43b4-9155-05dbdce11c66}} and an open subset {{formula:1d093750-6b29-48de-8059-b09bf06c98a9}} we say that a function {{formula:1b2e8087-ec30-4fab-97ec-a1e96e39315d}} belongs to the class {{formula:d5356959-acbd-48d5-890c-82bbc6010f10}} if there
exists an open neighbourhood {{formula:ca0a10b1-7aac-472c-9e3a-406d7a213ca7}} of
{{formula:d70b3f76-747e-4468-a782-aa717da85bda}}
in {{formula:1269a99f-6a80-47c9-a64f-17fd891627a7}} such that {{formula:1251f3e7-cbdc-456f-88be-95d415dee856}} is {{formula:85cee800-5a93-4fdb-a0fe-f1467ba25971}} on {{formula:7bfb6e11-231a-47e0-a5c5-aa322f9b9eed}} .
{{formula:5eb0a1b8-6c8b-4dd2-8336-08c65e83e139}}
Definition A.2.
Consider {{formula:c2a48758-c9e9-45ba-a219-05ede8bb21a9}} and an open subset {{formula:44c21078-021c-441b-9d3e-44f3a1fe2677}} Given {{formula:e123fb1d-3073-4b85-98a7-f8ca38c20d6a}} and {{formula:8093ae7b-4047-4239-bb8a-8e57f44f79ae}} , we say that {{formula:aa57f20f-c953-4cc1-970a-e3bf5ca5e785}} is {{formula:4574c384-7fa0-4147-834f-dbac571a36ac}} -flat with respect to {{formula:1c415292-a379-41f0-962c-fc6a0217b439}} at {{formula:cb75ea79-152c-48ff-964d-883e0482db49}}, and we write {{formula:d4b79a72-a5cf-49bc-bbcd-ba5243bb2fb0}} , if for each {{formula:dae3d8ec-ecec-4cf5-8394-274f5d62ce82}} with {{formula:29f9894e-bd20-4300-9db0-c60719927e74}} there exist a neighbourhood {{formula:b91a0948-f894-423d-b20e-285e2e4e2e37}} of {{formula:f2dbefdf-8618-4527-b212-cedc811b19d0}} and {{formula:aa769079-8cf9-483d-9f27-30347300a475}} such that
{{formula:a4f1c929-a1bd-43cb-b028-ce8c850eb6d2}}
If {{formula:aa662f92-16f7-4d8a-82fb-71dcec6affdc}} is a (not necessarily open) subset of {{formula:9e9993a8-8b9f-4553-8997-c3407e71b7b2}} then define {{formula:6bfd755e-23a5-4407-89d3-8190b66bef22}}
{{formula:dd77bea3-cac3-498e-b304-0f91f6f1169d}}
Next result merges the statements of Theorem 1.6, Theorem 4.3 and Corollary B in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}. Following the notation in that paper, we particularise them to the case {{formula:fc4a9907-95cd-4e45-bcfb-f7bb80c5ece6}} for simplicity. Moreover, for the sake of shortness, we only include those items that will be used in the present paper.
Theorem A.3
Let {{formula:d4a99ec9-9a35-4b7a-aecc-52875489a388}} be the Dulac time of the hyperbolic saddle {{formula:7968f536-0a51-4e70-90e0-3a950ccb7c06}} from {{formula:ad8ca41f-021b-43ce-8a8a-741f426badf8}} and {{formula:071cd673-87b8-4217-b3b4-a34e063b87b3}} . Then, setting {{formula:261a0714-16ea-484f-b487-cae001cb5985}} , {{formula:e96a8686-f949-4ef1-aa8e-5203712ed3c4}} , {{formula:177d0f0a-d514-4da4-a7ff-0a18480f4928}} , {{formula:bb961be1-633b-42e3-8843-3f6c4cf34ae0}} and {{formula:44a8b4b1-b6e6-47a4-acad-134d7ece83bf}} for each {{formula:b3076e11-fa77-4686-8801-f5e0000e99ac}} there exists a meromorphic function {{formula:cdd98b25-6994-4eb1-a7e9-f2e6a5ad36ff}} on {{formula:0b2fe505-07ca-4c81-a6a0-69e9c58151b3}} , having poles only along {{formula:3684be25-14b7-400e-b613-50587aa48cc7}} , such that the following assertions hold:
If {{formula:24acbf4f-7fce-4228-b465-d69a5a67a92d}} then {{formula:ddecb902-d750-49a0-988f-9e6d5a78139b}} for any {{formula:74a33bd0-5f4e-40bb-8dc1-3fecd4f83bea}} .
If {{formula:126634ea-2701-4b06-aacd-6449c85a9ed4}} then {{formula:d8ac3607-8cd2-4061-9811-346f038c1163}} for any {{formula:fe1fc84c-e5f6-4d81-aee4-e75dfb098416}} .
If {{formula:b24ebfad-de31-4977-ad02-40989102fa2e}} then {{formula:777aeef0-4365-45f7-9d16-01156cce17cf}} for any {{formula:516ae2e8-78ac-4ecc-861c-8b1a3c623274}} , where {{formula:65e1d86f-e541-4f7d-89dd-01bcdce9c21c}} , {{formula:da5a8b07-4788-45c1-826b-5dee33b5e497}} and {{formula:72bf1a05-d41d-4391-a2e1-bbeb0b46c94c}} for some open neighbourhood {{formula:98d31182-1920-4907-a50f-a5d28d7d01fb}} of {{formula:94f6390f-a30c-405c-b47d-ab77b4c037a9}} . Moreover
{{formula:d583e714-bf48-403f-aa6f-35740d2a539b}}
If {{formula:e128bd5a-15a1-44f7-a953-ea4fa8cfbba6}} then
{{formula:1594a2c2-734b-43c0-a7e2-3cb1df081828}} for any {{formula:27cf1862-4b05-4d55-8f9e-7b861319ef4e}} , where {{formula:c999fed1-7287-4a0c-84c8-14a885c19b9f}} , {{formula:35383c74-9d5e-43ec-95bf-a999af26b7b9}} and {{formula:47c13b4f-af30-482b-9892-60a8fde58f50}} for some open neighbourhood {{formula:5c5b1ea8-f522-4831-9c0e-1798378899d2}} of {{formula:13698628-89ea-40af-8fb8-c6c9dd91eee9}} . Moreover
{{formula:95e50a82-47a7-4529-841d-95a7a84eab18}}
If {{formula:ede4d69d-7239-4a94-a847-6889deffd598}} then {{formula:8f6d80e8-60b2-44b5-833b-a96bb6cfe901}} for any {{formula:1715fefa-684e-43d2-9869-4191ebea8164}} , where {{formula:afcaef82-1db6-4132-8e22-d13d401e0f99}} , {{formula:89ffdd5b-a9d0-41e7-9b11-6e63eacf6d60}} and {{formula:73274952-3c9f-4a4e-977d-8d7586717727}} for some open neighbourhood {{formula:b0627b88-5ba2-47fb-b577-ddcb3a3dc0a6}} of {{formula:bef4f6b8-0e1e-4960-b85d-d570aa6f0b59}} . Moreover
{{formula:85dd0a4c-4bde-4c24-8b7c-da90dee7530d}}
We focus next on the expression of the coefficients {{formula:cc73379f-6f8c-4f37-b565-e50edd762e04}} and the result that we state below in this regard follows from assertion {{formula:8c587231-ba77-4d04-a276-28a6b71ce4cd}} in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}} particularized to {{formula:313446b7-d1ae-4e99-a241-97a19ac41158}} . In its statement we use the following functions:
{{formula:bc6a7a45-6331-499a-b143-e1230330c71a}}
Here, given {{formula:4be69c31-02d0-442a-91d1-fb8d0fb879ee}} and a real valued function {{formula:83fa351f-d167-4d06-a58b-9e37919b0d87}} that is {{formula:e9bdf074-1fe4-42a5-ab46-3982df2acb65}} in an open interval containing {{formula:36140346-7fa1-4f52-971c-ce5fe6cfed01}} , {{formula:a93a3cb1-77af-42d1-9780-5a701dfc17f3}} is a sort of incomplete Mellin transform (see Theorem REF below). Moreover, for the sake of shortness, in the following statement we use the compact notation {{formula:a0de76c4-fd75-459d-b76c-56297c0a9bc9}} for the {{formula:c793e70a-9736-4ddb-b780-fc50e2fa7932}} th derivative at {{formula:2c2f806e-142b-4fe3-9d88-539754161cd9}} of the {{formula:95e67568-2e14-4437-9a49-36e261cc9530}} th component of {{formula:8d484b22-89dd-4277-a505-65c36f459cf9}} , i.e.,
{{formula:f5e42bb0-3184-46bd-9981-4aba46667547}}
Also with regard to the statement, note that {{formula:649abc0e-8000-4088-aee0-d5217ac4c98b}} refers to the discrete sets introduced in Theorem REF .
Theorem A.4
For each {{formula:b6ab9468-423e-44fe-9db6-4718aefe52df}} , the following expression of {{formula:de1a17dd-a870-46df-9c08-188c6cb1d1d8}} is valid provided that {{formula:9b2bcb69-4837-4d46-9665-111c13e0366b}} :
{{formula:38e50b4d-eaea-44e2-8c36-b26ddd6d748b}}
As we already explained, the following result (that merges Theorem B.1 and Corollary B.3 in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}) is the third ingredient needed in the proof of Propositions REF and REF .
Theorem A.5
Let us consider an open interval {{formula:464b8ab1-a2ec-4a50-a188-b46322e296fb}} of {{formula:b0c72e20-452a-488f-9235-6948f314e8b7}} containing {{formula:418aff6c-8c81-4283-8d56-8cbecdb2f11e}} and an open subset {{formula:7a30f4ba-1528-4780-badb-acfd50b2aa2a}} of {{formula:f5fbe2cd-6bdd-4db6-9e1d-24d9943a5dfc}} .
Given {{formula:b1eae5a5-d19a-4d72-bd2e-42d314540d7e}} , there exits a unique {{formula:129b63c9-4aa4-4bdf-a940-3b4a1fd04fc5}} such that
{{formula:19ba55c4-864b-4f91-a0ec-f3329e680059}}
If {{formula:ea85abcf-cc75-4909-9744-f69febe9af03}} then {{formula:a03b3ee2-0b52-43b5-82c5-a572e6b0e130}} and, taking any {{formula:3317b9b4-da76-48e1-92bb-2d5c48f8d0be}} with {{formula:438dbee2-d0e0-4ab0-9c7d-2568fd4be4d3}} ,
{{formula:c06a607c-bcd7-4839-8859-68655041cddc}}
where {{formula:9271ea89-c674-414e-a6e4-4721aea229cf}} is the {{formula:33546609-3dfd-4925-9aa0-10dad3e5d00c}} -th degree Taylor polynomial of {{formula:1c64be16-e8a8-4eee-a743-a106bcd8a8f0}} at {{formula:59fc0b7b-63bc-44f9-938f-4928e93e74ca}} .
If {{formula:c4e1c845-d3f7-4def-b1c9-03f60d66eebd}} is analytic on {{formula:b302c412-ec71-467a-9a74-2adeb6529ab4}} then {{formula:eef3eeb2-e4f5-4930-ac5d-b170b75597c1}} is analytic on {{formula:4dbade08-d941-418e-bbca-d38fa2301558}} . Finally,
for each {{formula:c57f22d5-538c-4b55-b39f-5748b4f05ea0}} the function
{{formula:5508b986-0197-4749-8420-2f2bf216fe53}} extends analytically to {{formula:1e8ca9ed-29b2-4f9d-85ce-41f5cde13be4}} .
If {{formula:a1f8ccc6-809d-4b4f-8d3c-de4d741fc939}} with {{formula:1218da99-2c58-441a-a421-b4f3165c9848}} and {{formula:4eff5e39-90e3-4c97-976e-09e3ef79e1b2}} then
{{formula:c97d6c89-f3f4-4e7c-b30e-6447bee36097}}
The following simple observation will be useful in order to study the coefficients of the asymptotic expansions that we shall deal with.
Remark A.6.
If {{formula:6d7f0da4-28d2-4865-8390-b843654ec38e}} for all {{formula:19beca77-3d06-449f-90d0-c9aa6173a363}} where {{formula:f8ea7f4d-87f3-443b-82ea-724de39abacb}} with {{formula:c8833ecd-6e70-4c1c-9320-d2aaf383fb08}} , {{formula:766e9a9a-ca85-414f-90f8-ddd82897aaa3}} and {{formula:e5e8d896-378b-4eeb-b36d-7ffbd1225e1f}} then {{formula:648bcd9f-c11f-4097-b89d-1b915cd3bc84}} .
{{formula:a362c15a-d0c0-4cc8-88d1-2911b14fe0a6}}
We are now in position to begin the proof of the two first results in Section .
Proof of Proposition REF
Proof of Proposition REF .
We follow the approach in {{cite:f73aa8df8bfb916a82c4b022d26828630a4e01e9}} to take advantage of the general setting developed in {{cite:bb4b2e2a088f548865c66f199c257c574e5339ce}}. To this end we will work on an extended parameter space {{formula:d2883ee9-22b7-4847-90d1-7baac1c44534}} that we specify as follows. Firstly, introducing an auxiliary parameter {{formula:288bfad4-73f4-45c3-9f48-433d4914bdf1}} we consider two local transverse sections {{formula:a0b51696-79dd-4d65-8527-5f009dff2e66}} and {{formula:edacca91-691b-4cc3-8bc8-4d6c9b22a842}} parametrized respectively by {{formula:4bb91738-bccd-4edf-8986-41f447247a65}} and {{formula:174bcf63-71a5-413f-ae4b-02af4d81c8ee}} , for {{formula:2ed20bab-5b14-4b4f-96da-20ac5013a4ec}} , cf. Figure REF .
Secondly, taking any {{formula:aaa7ec7d-ddf5-49b4-a328-e4c4ace4b603}} such the straight line {{formula:463636f7-e8f9-41ad-9ef5-6c18c82cfc9d}} does not intersect any solution of {{formula:73bb3d02-2adf-47ea-b541-e0820c780822}} while traveling from {{formula:622d0e2e-219e-43a5-b0bf-12bb88575cf9}} to {{formula:bfc1adab-f42d-4a0d-9d70-dc43351a48d2}} One can readily see that a sufficient condition for this to hold is that
{{formula:e0f51ab2-13dd-4b14-b07b-a85e524af230}}
Then, setting {{formula:5f5f4017-49d5-4ffa-9119-e3bfcdef29a9}} , we will work on the extended parameter space
{{formula:83fc57cc-61dc-4eee-8ac3-b99cfef60625}}
Taking this into account, we consider the projective change of coordinates, see Figure REF ,
{{figure:40910dfe-c045-411a-82aa-475d8509d62d}}{{formula:75926d97-ed5e-45f6-aeea-93b8f6830cc7}}
One can verify that in these coordinates the parametrizations of {{formula:b819dec3-95a1-491e-b1f5-b5aa9d230cf6}} and {{formula:bf4d3dda-d62d-4b09-bcd5-756b120290d1}} become
{{formula:a21f7c9f-a7cf-498c-85f5-b5e131e0391f}}
respectively, whereas the vector field {{formula:4e43ad7e-dae7-4389-82ba-dfaa05cb1d51}} is brought to
{{formula:3008e199-ba54-4583-a7e5-d213b907b035}}
with
{{formula:96f1b486-2011-4174-98f8-e29af258d47b}}
The reason why we introduce the auxiliary parameters {{formula:ddb824b1-e8e6-40ef-a07b-ff1ca9934eeb}} {{formula:327a728f-3567-4406-9892-1ba6869e5fb0}} and {{formula:5754f3d9-d3f4-41e4-be6c-371b6b3db0e7}} is because the computations are much easier taking the projective change of coordinates that sends {{formula:2fdebeaa-e4cb-4df5-9877-bb5c956807bc}} to infinity (i.e., with {{formula:6f48a095-c832-4562-a69b-915c30c1e6cc}} , which is not compatible with the placement of the original transverse sections (i.e., with {{formula:ca45a58a-de15-4cb6-ac3f-afe4fbdaf642}} Since the parameters {{formula:ab6cb774-b2ed-4123-813d-3f40b0d69daa}} with {{formula:08b97f20-e046-45fd-9dae-36ed3497c560}} are in the boundary of the admissible set {{formula:e626c538-8843-46de-8187-d3a83e7be98d}} , we will work in the interior and then make a limit argument. By introducing these auxiliary parameters we end up in a setting where the assumptions to apply the results in Section REF are fulfilled. Observe in particular that {{formula:e0089a6f-0d1b-4e43-9058-64fac9e54def}} and {{formula:4a21290e-0a27-41dd-9f0d-112839be5620}} are analytic on {{formula:b5608cbd-5576-4c5a-b293-187d17d99ee8}} . Following the notation introduced there, note that the hyperbolicity ratio of the saddle
{{formula:a1bd7dfc-a4c1-4d9e-8310-81c6f05cb364}}
depends on {{formula:63e0cc6f-642b-419b-8597-4822ec357b5b}} . We denote the Dulac time of {{formula:fefcbf64-7c74-461c-a7cf-cb8495446638}} from {{formula:9798ac1e-e03a-4afe-a071-202bdd3bd52c}} to {{formula:9e68e15a-9373-40fa-bfd3-b765a1810f9a}} by {{formula:7e3b7626-09a9-4819-bcf4-8f4c7e739477}} . Note that, by construction, it does not depend on {{formula:93373414-0360-403a-bde0-e70d494d2cb2}} and {{formula:f487407d-14c6-48e7-97c0-dff9a5c86b9e}} as long as {{formula:33b6ad00-9e44-4eeb-b76f-bdf326f56e4b}} and {{formula:9faaf9ad-9cae-4020-a463-059beac9b532}} holds. Moreover, and this is the key point for our purposes, the Dulac time {{formula:d14bceda-63a0-4b26-99bc-263b595e3d5e}} in the statement is precisely {{formula:05c0de6a-2087-44fc-83c3-81d8de1206bf}} for {{formula:9c8713f4-3b7c-4df4-b705-06771184bdea}}
Let us fix any {{formula:1525d15e-00f5-498e-8478-d44846bc73e5}} with {{formula:5768cef4-1ec7-41d8-8ec0-92017e8efd59}} . Observe that {{formula:5cd36fe7-45fe-4483-8870-ca12599100d8}} if {{formula:e792284a-dee5-4653-9b5c-ce9914c938f9}} {{formula:1e98b4ac-c806-4c31-ba37-b69598692d2e}} if {{formula:4f276b41-1a98-4836-81aa-f5035106d676}} , {{formula:7c20fa71-77f2-4403-9676-3e4e54d8fd8e}} if {{formula:1c3f5b8f-b887-4fb8-aa59-60f1b43c38a3}} and {{formula:5f3dc587-4042-4db0-9cd7-23e539533c96}} if {{formula:98096198-0b2f-479d-9ce0-389d917e0c22}} . Consequently, by applying (2), (1), (5) and (4) in Theorem REF , respectively, we get that
{{formula:d5871a79-5c17-4b9b-90db-1bff3886acc9}} if {{formula:c081be15-235a-4451-a335-2ad4c8cf0096}} where {{formula:a4b60006-d4bf-4b8b-951a-3c659db6be24}}
{{formula:506e6d61-b012-4ecf-8d2f-ded8f67d2dac}} if {{formula:9dd255e7-3494-4ca3-a046-8150b30edcb0}} ,
{{formula:25f3fe3e-8db3-4946-b8d6-e31cf62608e6}} if {{formula:3d829a11-b4f6-40d1-918e-53eb044987f5}} , where
{{formula:d8d37690-9b00-421a-896b-70d5059ea279}} and {{formula:73517381-f339-4375-b4f8-940c8c446810}} are smooth in a neighbourhood of {{formula:507cb192-2e92-4c04-bda9-dd68ee4aa48d}} and, moreover,
{{formula:a32b3a06-d9b2-4a3b-ba24-c814918364c8}}
{{formula:5b499a0c-d266-408d-8e60-24c16a200ad5}} if {{formula:49121534-3bd2-4bea-b2a1-d21d91ad62f4}} , where
{{formula:cc4ecbe7-dbc4-4f1e-97ac-e82b83df0880}} and {{formula:835c657f-433f-4c07-aecc-d4da1e7c7658}} are smooth in a neighbourhood of {{formula:b8b405d1-cdc1-4152-80c2-b5e6c6f55088}} and, moreover,
{{formula:61440a4f-710d-4ba4-820d-86a952810109}}
Here {{formula:8101f82e-a032-43a5-a7fb-bc647e003acf}} is a small enough positive number depending on {{formula:ea490fa4-91d3-41b5-a139-572b6e47244f}} Furthermore the coefficients {{formula:6f1dad6d-93c8-4ebc-81ff-2d8f4b4fdf38}} are meromorphic functions on {{formula:b9a8205d-0edd-4bd5-ac39-82a92e62ecc7}} with poles only at those {{formula:f6e63db3-f7b8-46a2-9228-51800c8a50bb}} such that {{formula:1ad4ce37-e3b7-41a9-a357-4a06f85d2b7b}} where {{formula:90fffd81-1ca3-41e2-938d-c45f873cff4f}} , {{formula:e6d89c16-134e-42ff-90bf-a6a843f68630}} , {{formula:547b5eb0-51bd-4722-980c-177bafddaa22}} and {{formula:37e4a959-8779-4918-afee-4f182645de23}} .
We claim that the coefficients {{formula:3043e8e0-6ccf-455f-9e8d-b2ba8f901db1}} do not depend on {{formula:7af3b362-0d04-4134-8923-fbd5f4ed558d}} and {{formula:6703b4b9-5d65-45fe-a12f-1b7aea7b009d}} . Indeed, to see this recall that the Dulac time {{formula:7b5abe3b-57ca-4958-a3d9-af47516c4607}} does not depend on {{formula:398de91b-363a-4b20-9ef9-15b35df8aec7}} and {{formula:9490fdcd-96e7-4dcd-bc77-c5b4f5327602}} provided that {{formula:191c8624-c28d-49c4-9fd0-5ec54e89a32f}} and {{formula:eb218732-63ca-4f19-8291-a05a6b4f8ff4}} , which is verified for {{formula:445e8a5f-8c20-4db7-830f-b2e8837ed5f0}} Hence {{formula:0781084b-4c41-4a64-b355-d9aa2f2e2406}} and {{formula:d17a39c4-6c21-45a3-b788-cba6bb951d4e}} Thus from {{formula:f52c30bf-5a7a-488a-91ae-82f81cfc25d2}} we get that, for each fixed {{formula:8fd95e69-4a56-4663-b35c-0862b98b4c94}} ,
{{formula:5cc93478-5c04-4afd-a335-96a0a15239a7}}
where we use that the flatness order of the remainder is preserved when derived with respect to parameters (see Definition REF ). Then, since {{formula:df8da865-8805-4dbc-a167-12cb7eb12555}} for {{formula:662ece74-8d41-4cb1-983b-a348af67369a}} and we can choose {{formula:03876992-d283-4d58-9730-2f53e118adf1}} arbitrary small (depending on {{formula:5101b111-f550-4091-9f6c-d514c7f15786}} , by taking Remark REF into account we can assert that
{{formula:715a6c61-12d1-4235-9bf2-1a78f1e5b7d7}}
Since {{formula:a1a0d03e-eb73-45ca-b9ce-8aa0906652e8}} is open and the coefficients are meromorphic on {{formula:c2e5b429-0ef7-4c7d-bfbf-011f882aa957}} , by Lemma REF it follows that {{formula:61d674b2-7aac-48e3-85d8-b7c1db5159e1}} , {{formula:417ae193-e2a5-453b-9281-9eb32e7404a2}} and {{formula:b24f082c-f80d-49aa-9ad7-f0bbdf80ddf2}} are identically zero. The claim for {{formula:22127268-444a-4d7d-9e11-bcb80ae48e8b}} and the other coefficients follows verbatim.
On account of the claim and the fact that {{formula:d735578e-e380-46d2-9ad7-8bbe873401fb}} , the assertions in {{formula:3a686439-64f8-4bcc-8e8f-50229479687c}} –{{formula:1fa8abc0-5931-4773-8735-f98b6f1a593b}} with regard to the asymptotic expansion of {{formula:45e936b4-0368-4e8a-80bf-03362501394a}} follow from {{formula:fb27e874-431c-4fc7-be48-50c6c16c4b1e}} –{{formula:c01b59eb-a846-4306-8916-2ea38fdc1794}} , respectively, by setting
{{formula:29883bb7-bd75-46e3-bcfa-7091122384b1}}
Next we proceed with the computation of the expression of each coefficient. To this end observe that
{{formula:008693ac-d79d-4db0-a559-efc1514d6ff4}}
where the second follows by the continuity of {{formula:9242af07-2428-462c-8648-0f374d0b2e05}} on {{formula:992c3d3c-eba5-4f1c-a5aa-79eb11114f25}} and the last one on account of the previous claim. In view of this the plan is to compute {{formula:4b33dd61-e575-4988-be5b-b5979446421a}} with {{formula:4024a2fb-bbd1-47c0-ba98-d2c90dd878dc}} by applying Theorem REF and then make {{formula:09e76c7c-2017-4e86-bfb1-e6e7f10466ee}} With this aim it is first necessary to obtain the functions in {{formula:5356027c-88c7-4b87-994c-1c6e543bd909}}. In doing so, and setting
{{formula:705adc63-5997-41af-824f-d81c8058ea1b}}
for shortness, one can verify that
{{formula:060485cc-e38f-46c0-8e53-86b266a1ed3f}}
We stress that {{formula:39cee88b-6783-47b2-bee9-8870825c6d97}} and {{formula:63c73a51-6e86-4568-af93-ad586ddc774c}} are defined in {{formula:00f27dfa-4bc1-4f50-bf46-ab8ff016e619}} in terms of the functions {{formula:f26d6986-1296-472d-ba53-e9f1a7d7ab20}} and {{formula:5ad84ad6-0653-4360-9bf9-d0b8c752bcfc}} given in {{formula:3667ac8e-0393-4f56-8f05-a196aa7f5ab5}} and that, as we already explained, we take {{formula:e0df0e15-a3af-46c6-b32f-427c52d655fb}} here and in what follows. Similarly, setting
{{formula:560cf654-a679-4b3f-9e4a-7e878ec811f5}}
for the sake of shortness again, some computations show that {{formula:7fd05fff-2cd5-4679-90ce-e6e898a79ad2}}
{{formula:7e469856-1c5d-4d7d-8013-df70d67fda73}}
Moreover {{formula:f9b00d2f-d27f-46fa-8f02-8a36ea62b33d}} With regard to the parametrization of the transverse sections in the expression of the coefficients, see {{formula:d2aa6c01-0f53-438b-9935-6c6316d6c9c1}}, using the compact notation {{formula:8eba4b57-2a31-4f86-a097-de1c6e896cd8}} we get that
{{formula:15b18306-e8a3-459a-8052-8feec21147f2}}
We are now in position to apply Theorem REF to obtain the coefficients {{formula:2f8dadc1-8bb9-4808-a26c-7873bdb0b27d}} . (We omit the computation leading to {{formula:6920ad98-f434-4540-b933-667b0ce53e06}} because it is given in {{cite:f73aa8df8bfb916a82c4b022d26828630a4e01e9}}.) In doing so we obtain that
{{formula:d24fdcf0-5bfd-4c40-922d-21316ec631e5}}
Therefore
{{formula:ddb70018-e040-4ad0-af9a-458b8691929f}}
where the first equality follows from {{formula:4a987afc-4657-4612-a7a9-a0f7c6a86e34}}, the second one by {{formula:5c53cffa-ee24-4b64-ab61-929d5c5d65b5}} in Proposition REF (provided that {{formula:41d13db5-c96e-433d-bc67-0ce3af7beebc}} ), and the last one by using {{formula:c95baeb3-7168-4923-aa42-a6b7af5a505c}} and that {{formula:f467f67e-cd31-4a73-94c7-321e34d7b0f3}}
Since {{formula:28021214-cbaa-4ecd-9984-ea8194a8243d}} is an analytic positive function on {{formula:1161ea13-3f75-412f-be1f-cf6b83157d91}} , this proves the equality in the statement because we know that {{formula:27f294f4-38d5-44d2-bb79-9e87cd78351d}} is meromorphic on {{formula:61f40323-c4c6-4e9a-9488-d379c927f9dc}} with poles only at those {{formula:a18d027b-e759-4390-b7ff-aca02208c4bf}} such that {{formula:0766e081-e3dc-4bdb-bd53-4b2bfb5737eb}}
Let us turn next to the computation of {{formula:a5f08a11-0328-4d1d-b26b-1d4ac027e4bf}} . With this aim we note that
{{formula:3404bbf1-a028-45ad-a16e-2b81e9052d52}}
Here the first equality follows by Theorem REF , the second one from {{formula:6b757dc9-1374-4553-8166-ecc65846192c}} and the last one by applying {{formula:696639e8-de9a-4658-93b3-cb0ddf030d94}} in Theorem REF to the function {{formula:cb52a4f4-229a-49bc-ab41-82b285216732}} , see {{formula:31241b11-031e-4b52-a997-b69c20622e43}}. Since {{formula:4d382428-3c65-4063-9926-6255badeab7b}} , by applying {{formula:df56d073-2c0e-459c-a4ea-f58a2840077f}} in Proposition REF we get
{{formula:309028dd-9e91-42b0-bd86-6661754aef44}}
and, due to {{formula:40839bfe-394e-4d76-89ef-3505d2d5af5d}} , this proves the validity of the expression for {{formula:ee951a18-6d01-46e5-ab19-ee60f43844f9}} given in the statement. Let us finally compute the coefficient {{formula:95c1e232-dd9c-401e-a383-12620b71e46f}} In this case, on account of {{formula:24726d39-727d-47bb-a8cf-a3060593d59c}}, by Theorem REF we get
{{formula:4b958016-3adf-4cd2-b07d-d5fc837efd19}}
Thus, since {{formula:c10f8156-ba3b-4578-a193-a672b1f99158}} for {{formula:ce03f445-930e-4253-9d69-1a799a82211e}} from {{formula:22661dea-e094-4a01-ac6b-98f965683e1a}} it turns out that
{{formula:497a2c59-4ee0-4380-8ad9-6760b3c4c80d}}
Hence, due to {{formula:c46b3613-ed78-4406-83df-07c21229631a}} , by applying {{formula:1185c678-4579-4e0d-bbea-547e779df0ea}} in Proposition REF once again,
{{formula:09eadc63-e031-4bd3-a828-000cffad987e}}
Since {{formula:60e4cf57-b0a3-4af3-bcf9-e2dfd33816bb}} is a meromorphic function having poles only at those {{formula:a84afcb4-cf54-40e7-960d-0bdb8f537790}} such that {{formula:60872ff5-9157-4baa-b9a1-9477030e290e}} and, on the other hand, {{formula:387e3dae-1962-4f6c-8781-1ca92602ccd3}} for all {{formula:c945aad8-795e-4a04-a435-4137df17e742}} , by applying the Weierstrass Division Theorem (see for instance {{cite:56ebefebda144413fc3e32066a493c4b162e7550}}), we can assert the existence of an analytic function {{formula:3fada725-f6b9-4d6f-b87a-1b4868d2deed}} on {{formula:2459d9e3-aa6f-467e-9064-8f6443bc07f2}} such that
{{formula:4429d5c8-7b3c-45ed-b1e8-5e49904876f2}}
It only remains to prove the assertions with regard to the properties of the coefficients in the respective asymptotic expansions. Being the ones in {{formula:5f89fa33-52d3-48f6-b30d-fdd8fe7c9832}} and {{formula:8d4f57f1-d63c-47f1-9fb2-a814f6ddc3b3}} an easy consequence of well-known properties of the gamma function (see for instance {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}), we proceed with the other two:
Let us take any {{formula:cd14244b-11ae-4996-b694-dcf61fca7724}} and note that, from {{formula:d98f5889-0c8f-4e3d-adcf-803d2501b489}} , the functions {{formula:3c312ff3-e057-4677-9294-1ccef6ddfdbc}} and {{formula:226e4c44-61c6-4e7c-ab41-d21c3225f297}} are smooth in a neighbourhood of {{formula:ab40dacd-c10c-4d1f-9ac5-9fd0629fef81}} and
{{formula:15989750-f267-41cc-8208-cba03931d245}}
Recall that {{formula:7ec83a09-1132-4a7f-99af-52c248faf741}} and {{formula:80a13002-423a-4225-b7c2-368f5734abcf}} are meromorphic with a pole
at those {{formula:adcb9231-12ed-459e-9587-db2226508630}} such that {{formula:a034033a-6bbd-43e5-b58d-30407419d99b}} .
What is more, by Propositions 3.2 and 3.6 in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}, respectively, we know that in both cases the pole is simple. Consequently by the Weierstrass Division Theorem (or, more directly, by {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}) it follows that {{formula:86108394-6214-4b52-9026-be5a6760b14d}} and {{formula:e0d462e5-d8d5-47d1-910f-76031151c4a8}} are analytic in a neighbourhood of {{formula:2443a60b-f478-4fab-9c8f-0b83e48c0c3f}} . On the other hand, from the already proved part of the statement, if {{formula:28c04696-916b-4155-8487-019f6a4eae4f}} then {{formula:1ce73186-1be2-499a-abc2-b3cc91a6c141}} and
{{formula:d4d0e94d-408c-4689-8c9d-ffd4343b5690}}
because {{formula:e2f3174c-db74-466f-adcd-e930992058a4}} (see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}} for instance).
Consider finally any {{formula:daa34c46-c731-4b1e-8bfb-127b9c0c7017}} . Then, from assertion
{{formula:af4b44e9-174c-42b1-9fbe-9c975b6034fa}} , {{formula:4518c560-2c3b-4588-a42f-a26582d310ec}} and {{formula:e3404c2b-c682-4773-8009-3651c7cd3d5a}} are smooth functions in a neighbourhood of {{formula:f17c9783-9aa4-412a-984c-773a141787f6}} and, in addition,
{{formula:fd16a373-ac4c-44bc-9c4e-e6038ce2d24f}}
Since {{formula:cde469fb-440f-4b7b-a5fd-b82a639d0bd6}} and {{formula:232e32d8-8d10-48b7-993a-73d057de8a5e}} are meromorphic with a pole at those {{formula:d8cde092-4aed-4db7-a655-deaec57070aa}} such that {{formula:f4edb249-d135-446d-b7b2-12cdbaf59ccd}} the above equality shows exactly as before that {{formula:d891e9e5-177b-4334-b864-53240f54e427}} and {{formula:ed27e246-9af3-4be7-8065-e373a456e899}} are analytic in a neighbourhood of {{formula:6232eb61-1726-470c-93f0-42d51795262e}} . Moreover, from the expression for {{formula:cdee17ee-ca0a-4dbc-b5b9-3fbde8a3459e}} in the statement that we already proved and using that {{formula:da2cceda-333d-4974-aeee-b28500477513}} , we can write
{{formula:0976acd9-c831-4a1d-8f3a-953979973fc5}}
The function {{formula:4300b10a-523b-490a-adf6-d29d5942192f}} is analytic at {{formula:7f4b632b-39c8-4a76-a6bd-27e86e5a38e1}} because {{formula:ad41b3e2-c365-4ec7-a3cf-a2d4eacf5f5a}} and
{{formula:2acd0612-4b05-4b87-8411-07f1b9330250}} has simple pole at {{formula:90684938-9770-4916-bba8-20b3fd591221}} . In addition, since {{formula:56acf2e5-aae7-48b3-a5c4-644ca48ea32a}} and {{formula:33af0a7d-f800-4468-9a63-abf13174acd8}} , we get that {{formula:80492b16-2d65-4b0e-8736-73fe5e970c94}} From the expressions in the statement as well, we get
{{formula:5454eb42-85e6-47b5-a3b6-db5af57d390a}}
with {{formula:fbe94974-eff7-4636-8767-9466476ec9a5}} and {{formula:f8f721f8-87e8-4745-b8e6-d950ebe4085f}} , that are analytic and positive at {{formula:89ddfc79-d3cf-4532-9319-d06ff6aba9f7}} due to {{formula:8bef0acb-761e-474a-a20c-9d0d2f322f74}} . Finally a computation shows that {{formula:efddcdd0-1b11-4eba-871a-7213bd12182f}}
This concludes the proof of the result.
Proof of Proposition REF
Proof of Proposition REF .
We will adapt the arguments in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} to take advantage of the general setting developed in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}. To this end, as we did in the proof of the previous result, we will work in an extended parameter space {{formula:8c02bbf4-f8b0-4e04-9bd2-caf024fa410b}} to be specified. In this case the computations are a little bit more involved because we also need to straighten the separatrices of the saddle, see Figure REF . With this aim in view we first take {{formula:64c65968-5223-4d36-90ad-b2097dbfe305}} and consider the local change of coordinates given by
{{formula:7fa518d7-c2f7-4a89-a90a-ff3346d3867f}}
where recall that {{formula:27d8f612-b760-427e-9afa-534035c2e0c4}} with {{formula:94bf7fd0-4117-4516-bf79-a38640a295be}} and {{formula:a69f56fa-1602-4a9b-b32d-d6071a7a1ef9}} {{formula:5e098cd3-ce29-4ced-91db-820a29e37fc0}} , for all {{formula:18da0e10-a537-4d9b-b25c-d3ae19a36f2b}} In what follows, for the sake of shortness we set
{{formula:70b9df9e-a0e7-4772-9072-7304735b2394}}
One can check that the Jacobian determinant of {{formula:5309e7b6-0374-4f5c-8c5f-21f658dadc57}} vanishes at {{formula:cc633010-e6b7-4a56-ae37-a1aebc724e63}} if and only if {{formula:ed6a15f4-e88d-4349-bda4-b327a922955b}} where
{{formula:2010c5b6-b698-4f10-85a2-c2dc2502968e}}
and that this straight line is mapped by {{formula:7c0ebe99-e92e-43d1-9f86-259645f3d6e9}} to
{{formula:c38de4fa-1d57-41bf-b629-b91ccdf31378}}
We claim that {{formula:6a087ce4-7b1f-4a1e-94fc-8b94250031c6}} is an analytic map from
{{formula:356e5ce2-8efa-4fa9-b646-9daa6503c5f3}}
to {{formula:052efd27-1a67-49ce-a385-fb8a5a210a29}} with a well defined analytic inverse given by
{{formula:3d572bce-2a08-4d99-9daa-288377b33b9a}}
Indeed, the claim follows by checking that {{formula:02c445fe-ce05-491f-952c-702351706307}} on {{formula:66dfaab8-a725-4712-9394-cafb1e6b550a}} and that {{formula:1fe4d224-d73a-4404-9f9a-653bc4f2c129}} on {{formula:0cb99efb-0372-4531-95c5-aadab71684b1}} To show the second identity we use that {{formula:039aac8c-56d0-4b68-b729-9a2d4598a00e}} Consequently {{formula:3a1462bf-7d4e-44a9-8847-54eb4992922a}} is an analytic global change of variables from {{formula:4634784e-3a27-4e25-907f-669de2826ef2}} to {{formula:379ff772-7d89-422c-b640-ea8dbaec1688}} for all {{formula:c68a4400-1564-4087-a702-3127a5907d41}} In what follows we require {{formula:d1e0760a-6118-4532-b41a-01d0ca93cba5}} in order that the straight line
{{formula:c8692da9-e3eb-4572-95b9-c4f838b38056}} does not intersect the left branch of the hyperbola {{formula:1667f988-b9c8-4347-8237-138eff928f5d}} , see Figure REF .
{{figure:a0c845b8-380e-43b6-9738-2ccece9357cf}}Next we introduce a second auxiliary parameter {{formula:b2929ca1-0a61-4577-8b06-19374ac96f70}} and consider two additional transverse sections {{formula:f620a20e-7a0d-4404-8e1f-d3e55ab3e10c}} and {{formula:af31437f-320a-405d-a300-60e52e691ca2}} laying on the straight line {{formula:e377cd95-d4a1-4807-b2da-fa7c830bc3f1}} see Figure REF .
Observe that {{formula:e2df51d7-462f-4cd8-b3e2-1e829ae03c4b}} intersects the right branch of the hyperbola {{formula:b5d08766-97cf-4521-8a33-34efd8c030db}} at {{formula:a28a22a0-a478-4826-bbb4-250e970b7b0a}} for all {{formula:99b8ca12-c102-4984-accb-0755b1764199}} . We require {{formula:a5501305-c9b1-4925-9092-db45460da8ed}} additionally in order that {{formula:4383297e-2ba8-4ea3-83ee-e52577f78ae0}} intersects the hyperbola at a point {{formula:73fb6a40-5c44-4748-b8b8-4a75803597b7}} in the left branch. Then we parametrize {{formula:a24cf644-bca8-48b5-a8e0-bd8bfe8e2d1c}} and {{formula:dcfe5fb1-b789-40ea-a65c-8a1b88aed89a}} , respectively, by
{{formula:7de7a7e5-8509-4e8f-b1a5-4cdf8cdc2a6c}}
for {{formula:bb8596ca-d281-4441-845b-f31af8d61af4}} small enough. We also require {{formula:794d2cb0-1d42-47d5-aea0-f3fc6e8333c1}} so that {{formula:5627d123-bb1f-44aa-8637-e9f1a0855eaa}} Summing up, the admissible conditions
{{formula:5c829d8d-e9b3-4e2b-93af-338450bde745}}
guarantee that any solution of {{formula:12ce12ee-1184-40fd-bd3c-b0be188562ba}} going from {{formula:328c1dbe-2e96-4b2d-bea1-868a896f0b2d}} to {{formula:e9d97bf6-da8d-4d3f-9181-0ef7969493cc}} is inside the domain {{formula:a6e9f5ec-f592-4377-8cb0-a2f22d00241e}} of the coordinate change {{formula:3a26674b-b8c5-486e-86b0-3b9476563a6a}} . Thus, setting {{formula:b9873dc8-750c-4f0e-b6e7-3fa117c6f301}} , we will work on the extended parameter space
{{formula:e919e59f-e10a-496d-b061-574bb3a294ef}}
Clearly the sets {{formula:7d0d4104-1c08-4a6b-837d-d01a58ae2fd1}} and {{formula:56c22d2b-08a3-4455-bce1-de1c53eb9ea6}} are inside {{formula:8c78004d-41ff-4f2b-99a7-dcc85714975b}} , that will be crucial in the forthcoming steps.
At this point we define {{formula:16587a70-6623-4d71-95f1-86548f82269c}} and {{formula:970b4de4-cf5f-469f-a8e2-667f85df1909}} to be, respectively, the composition with {{formula:d7397327-5e56-4982-a06f-c9f4fcd6ea4c}} of the parametrization of {{formula:df028a2b-a342-44ca-8302-09c79802bdc6}} and {{formula:09acc779-9251-4d4c-9451-ab8ef2805b95}} given in {{formula:25ce6b79-3ccc-4b9f-ab14-2f728d228221}}. In its regard one can check that
{{formula:9f2c544c-7603-444d-a65e-db32c4370780}}
One can also verify that the coordinate change {{formula:75d9c052-1b67-48b6-8483-e335f3c67b8c}} brings the vector field {{formula:886dd8cf-cdd0-4623-bac1-00e7401e4837}} in {{formula:9e9e1353-7024-4907-a130-74b111d67fe7}} to
{{formula:3b4f59b3-b96a-4ed0-9318-8a9220c5c3f1}}
where {{formula:1b41ac0f-ccef-4f22-9d53-9eab77f15fea}} and {{formula:4283ae8c-7008-46ae-8ab5-bc4e0a4842c2}} analytic functions on {{formula:69ce6693-8334-41a0-beeb-23c4eca28a84}} The hyperbolicity ratio of the saddle at the origin is
{{formula:4088e8fb-493a-49a7-be6a-bcb5a455d68e}}
Moreover {{formula:fc95828d-69a7-48bf-9821-543df64914c4}} and {{formula:9649119d-46f0-45bf-93ed-049984182b7d}} where {{formula:8af1ef5d-14ca-419f-8bcf-769f828a1089}} ,
{{formula:98d8b420-dec7-4c17-8b3d-bd99d2e4c7c8}}
(It will be clear in a moment the reason why it suffices to restrict to {{formula:5a1fce39-e26c-4361-b337-3a1e02e4c698}} ) For each {{formula:69063788-1dfc-4381-b1b9-752a831be9d3}} we define {{formula:08707ae2-a835-4afc-895f-c8fae6d561ba}} to be the Dulac time of {{formula:49475d60-4761-4256-8485-3b5369a2936a}} between the transverse sections {{formula:ecb0fb08-54d2-4d88-937f-974996647255}} and {{formula:981aaa5b-4a74-4410-847f-5ffa09a471a1}} parametrized by {{formula:c2fc0479-0109-45ec-9255-1bf4449ce2ed}} and {{formula:8db10a75-0e45-4339-a48e-9a4719363bca}} , respectively. We point out that, by construction, {{formula:633f878b-7826-4b03-881d-a80eb2e2fa10}} does not depend on {{formula:cd8a581b-45f8-4de1-9547-adef6931dc45}} and that, furthermore, {{formula:1c254597-dbc5-42fa-bdfe-d88d749d4099}}
Next we will apply Theorem REF to obtain the asymptotic expansion of {{formula:efe1a927-9bc1-4f26-9b66-b0edf03b988a}} at {{formula:31d59b11-a3c3-4fa1-899c-eaee489f27ef}} . Note to this end that, by construction, given any {{formula:7eaba64d-e4ea-4103-92d0-1a48c39b1dd6}} there exists a relatively compact neighbourhood {{formula:59270814-fa21-4426-844e-54de839169c5}} of
{{formula:71dc746a-473e-46b8-b242-5c942d83a1bc}}
in {{formula:f49aace6-170a-4b42-9f4d-cb6d2259ca1b}} and a neighbourhood {{formula:a74eeb14-5a23-4cc9-a594-b3526f57e82f}} of {{formula:f7587103-1e03-4032-88f9-1df1080271eb}} in {{formula:d290c0e1-39bd-4ed1-8549-891796303e31}} such that {{formula:0212840a-6a1f-46d3-94cd-77c81b73e18a}} for all {{formula:681cc3fa-3cac-49ca-956f-c38367a2781c}} and
{{formula:93e18f2d-6ddb-4ef1-a635-502ff62636e3}}
Here we use (see also Figure REF ) that {{formula:cadc2f7a-2ff4-4dff-b873-802e62bd4ef3}} maps the straight line {{formula:374e197c-62b8-4ac7-9dcb-177e4106ba6b}} to {{formula:b2a0f1f2-ea54-49f0-aec3-8f90a1675bbd}} . The above inclusion guarantees that {{formula:6a17fe65-ad13-427f-8c0e-97acb35fcc0c}} and {{formula:5e58e606-9be8-4f81-8655-e4736709758e}} are analytic on {{formula:74e8289d-b3b4-4950-bb90-a54ba8b129cf}} so that we can apply Theorem REF to study the Dulac time of {{formula:21135600-fdeb-4a06-9e2f-8cdf6c483e08}} for {{formula:015bc4f7-25ad-4c3b-8c2a-aff2adec2e05}} . Accordingly, with this aim, let us fix any {{formula:293eeb42-9316-4964-a0d9-f455757ca12b}} with {{formula:79cea31b-23c1-4205-b3ba-b1e13642d13a}} Observe that
{{formula:d96c1ba2-b698-487c-a542-f24ff56bccb3}} if {{formula:38a03b8b-20e3-439e-9c2d-0a377963329f}} , {{formula:d6d1ef0d-a0c4-461b-a38f-c97442722659}} if {{formula:90e6ef6b-4ae1-43f2-8f61-d4733725c085}} , {{formula:3d9ba4f3-dcd1-46f7-b7d5-e9e35c38620f}} if {{formula:d4f8f2cd-337c-469d-a226-265ac1dc8cbb}} and {{formula:959646c5-d2d4-4b8b-8cde-d25ee4473467}} if {{formula:0cfa796c-1cdb-4003-ad7f-aee17a0fa51f}} Then, by applying {{formula:29ec02e6-1e79-46af-a2cb-4a9970709376}} {{formula:f52520c3-8911-43fb-95a9-480c9b23cfdb}} , {{formula:305dd4f5-66e3-4a88-8cdd-c5aa541d61da}} and {{formula:51d212e6-84fa-4454-8a85-28051a0e53ea}} in Theorem REF , respectively, we can assert that
{{formula:1434436c-d04a-4c0c-bf4d-bd8158dabfb2}} if {{formula:f67e6952-4529-4250-8dee-1ae0733be9e7}} , where {{formula:2155134d-87fa-4216-bd13-166a348e48a2}}
{{formula:b28df522-f016-4222-a814-64e3d5e56b5f}} if {{formula:6c0bfef7-6214-4e02-bdcb-3f2924463751}} , where {{formula:385d5212-ed6e-47d6-b786-070ba132bdfa}}
{{formula:fdead2ab-d7ff-4a1e-83fa-f6f8a70dcaa9}} if {{formula:a205d250-4d7c-4168-b171-cdcbd928f086}} , where
{{formula:9cb35207-3de0-4c41-9921-acedb9cb3a83}} and {{formula:4d6f1d1a-3d34-4cd4-b6fd-9dc4b54a2f23}} are smooth in a neighbourhood of
{{formula:ccbc0438-25a5-475a-9eba-b9efa8d73924}} and, moreover,
{{formula:cbc2296b-2caa-4dd6-924c-70d6c89777f0}}
{{formula:d81b1328-1127-422d-acd1-87387c6711a7}} if {{formula:6e19ae7a-eb2d-4d51-b503-6a8d86e4eed3}} ,
where {{formula:a27d6580-cf37-4df5-8023-ed51517ca9ff}} and {{formula:f0e454cc-9450-4612-abfc-4de52e365b3f}} are smooth in a neighbourhood of {{formula:e7e394aa-0e2f-49fc-bb01-eca1fe012400}}
and, moreover,
{{formula:1b7a9bbb-3347-4a2a-a284-6ce565488992}}
Here {{formula:8ff4c780-9989-4aa8-a467-0e7815367193}} is a small enough positive number depending on {{formula:cefb9cb9-5cd3-4a72-b80d-a12212240912}} Furthermore by applying locally Theorem REF we know that the coefficients {{formula:8d97f2ad-e8a2-4317-acad-b92bf74d8efe}} are meromorphic functions on {{formula:c6c44dcf-01a9-4e65-ab44-9a677d00887a}} with poles only at those {{formula:30b29ee1-aaa8-40c8-964d-1a6c3bfc260d}} such that {{formula:1eb31b72-283a-4870-828d-d658c53020eb}} where {{formula:afc929bd-2e2b-410f-a5c2-af03216ffdae}} , {{formula:0a016d1d-1a9d-451e-a8d3-b6cd9bef9b08}} , {{formula:9e1e4f2c-5b28-4e3e-93c2-2a6ab3720b70}} , {{formula:f12c0c90-3c3e-4ff2-9446-0bf5da10fa36}} and {{formula:ac18b2a5-e6c9-4ebb-86fb-a7a5516815c2}} .
We claim that the coefficients {{formula:6e7aef31-1f97-436b-a31d-331cd110b705}} do not depend on {{formula:f50a379d-60ae-44b8-bdce-43e3541a2744}} To prove this observe that the Dulac time {{formula:05bcab17-de05-4b1e-8ff9-2b51a8ec2e78}} does not depend on {{formula:e0f3a7e9-89e8-493d-9efc-5b4aba26ea15}} as long as {{formula:1ad0cd4f-1c33-45cf-9c4f-c59dfeb47dbf}} Accordingly {{formula:c076142c-fb6b-47e8-9e98-37f442378ffd}} . Thus from {{formula:ffff0104-6a80-407f-a109-969812f5ab67}} we get that, for each fixed {{formula:ce33c8ad-5192-4abd-9931-bd15748f5a02}} ,
{{formula:5d43edd6-df2c-4569-b17d-87fa705d387a}}
where {{formula:06c4b8e9-d6f8-41e6-b1bf-fcfca90d30e3}} and we use that the flatness order of the remainder is preserved when derived with respect to parameters (see Definition REF ). Then, since {{formula:bf2936e7-4dc4-4319-80d3-94ae2847cc3e}} for {{formula:cd873c06-3ed5-45d7-bb8d-b587fe406906}} and we can choose {{formula:5ce3355d-2d32-420f-9111-287a2b4aff09}} arbitrary small (depending on {{formula:eb9dfdc8-dc5f-47f1-95fa-def19a466859}} , the application of Remark REF shows that
{{formula:35c4c49b-7f4d-4dc0-bf87-6b560512f533}}
Since {{formula:ea0c8eb9-994e-49c2-99f4-4fc69697908c}} is open and the coefficients are meromorphic on {{formula:f8e85612-a833-47f1-b7ed-565d1317f025}} , by Lemma REF it follows that {{formula:7af4121c-bd92-4750-b331-8dd65420dd7a}} , {{formula:625665d5-7aa5-47d9-af93-f96ea7a0c414}} , {{formula:b8bec2a2-25c1-423c-9820-0ecde07e7f1d}} and {{formula:9c8b2471-1851-46ba-94aa-860a198128a4}} are identically zero. The claim for {{formula:27b26385-a2a9-4964-ba3d-bd04b682c463}} follows verbatim.
Thanks to the claim and the fact that {{formula:9cbfb38b-09d0-4d5e-891e-64842d969001}} by construction, the assertions in {{formula:8a1c1db2-e564-4681-b38a-9de9454bcd5b}} –{{formula:67b43b70-1e66-44f8-9d52-04dc12b87ddb}} concerning the asymptotic expansion of {{formula:908fd92b-21dd-42db-850d-bee9d724604c}} at {{formula:ae13dd81-a629-4662-b80d-69bbf5d56068}} follow from {{formula:10f9b531-ed7d-4071-b2e8-78ea7c2373f7}} –{{formula:736a8b2a-56cd-472c-a248-bbceb6262935}} , respectively, by setting
{{formula:ae3ff8db-142c-4970-a12b-d853643fcdf9}}
We proceed next with the computation of these coefficients and for this purpose the idea is that if {{formula:f7f22baf-f9b4-462d-87bd-40613c5e71c5}} and {{formula:3b99e3a0-f384-46e9-8594-20a60ecaacdc}} then
{{formula:ead37d0c-9d60-45ae-8bed-5b2907d3e515}}
where in the second equality we use the continuity of {{formula:b19ea607-4241-4d58-9e7f-cd195966aaf5}} at any {{formula:3b4a1eee-1a0b-46f8-af9a-16bb9158f475}} with {{formula:2cb11b6e-cd55-43d4-9ea7-cbb362a90275}} and in the last one the fact that {{formula:0c694947-f3f6-4b75-99f3-929cf70cb260}} does not depend on {{formula:b3562d96-5b8e-4f3f-8895-758caa8827b8}} . Hence our first goal is to obtain {{formula:4b65d10e-cd82-4a74-8ea4-69f4db0e6211}} for {{formula:d0710481-cc66-4fef-a0e1-225ca322ee02}} and to this end we shall apply Theorem REF . (We point out that from now on all the computations are performed taking {{formula:f779f5c1-c742-432d-9179-12b6294df75f}} and {{formula:87a41095-c1eb-4ae5-b0b9-3e94d64aa5a1}} .) In doing so, and setting
{{formula:b70b1683-bf54-4d62-880b-001693e9a5c5}}
for shortness, from {{formula:b2d7c710-f370-429e-9b69-3046e844853c}} we obtain that {{formula:a082fa08-8481-4a5b-8c44-33ec0cac45e5}} , {{formula:f78cf6b5-1b36-484d-a139-c78026aa8862}} , {{formula:3f9fca31-603f-455e-beac-7608608c45c0}} and
{{formula:0234e5a1-d747-4305-898e-0bf08985bcb3}}
From {{formula:e0493c42-275a-475b-a62a-7a2d796efad0}} and {{formula:23590176-df8c-4a8b-b55e-ee90d662c6d6}}, the necessary information with regard to the transverse sections is the following:
{{formula:a02bb1aa-fec3-4d3f-bfdc-8e7f1e7994e2}}
Taking this into account we obtain that
{{formula:f127b542-74d9-4a9c-9f87-7657b06387a7}}
where in the second equality we use Theorem REF , in the third one we apply {{formula:d318d6c4-82c6-4e75-a37f-b8c94967a105}} in Proposition REF taking
{{formula:ebab9472-945e-4f59-a52b-70b4634ecc3d}} and in the last one {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}. Since {{formula:9b181247-4ac5-42a7-b511-1e6e59bf9f37}} , this shows the validity of the expression for {{formula:da5c80c2-9977-408a-bf2b-3ea24893f747}} given in the statement.
Similarly, from {{formula:5860d120-0056-4dce-8834-59ee9c8cda99}} and {{formula:a247a199-4837-4880-9c84-4673df663f08}} again,
{{formula:1a47b7a1-7e0b-46d9-8052-d5772de001c1}}
Here the last equality follows by applying {{formula:5c4ca06a-1522-4e50-bf51-b410450e7e37}} in Proposition REF taking {{formula:20317ce3-1233-4725-bd5a-47a1f4e73a76}} , so that {{formula:d557bd36-e65d-4026-9eef-b81a4e6fb450}} , and noting that {{formula:8df6f848-a0ec-4831-a8ca-264626517caa}} , see {{formula:3b747faa-173c-41dd-b191-ce2baf27851a}}. Since {{formula:4508ce17-9f4b-4dfd-8805-b97db35c6812}} is an analytic positive function on {{formula:67a1ddde-8c9f-42db-8f96-955859f20d40}} , this proves the the expression for {{formula:1ddcfb03-425e-4dbb-a43e-5a8fb8adfaf1}} given in the statement.
Let us study next the coefficient {{formula:54c672aa-e3de-4b71-b2e3-4c60dd2a3830}} . For this purpose we apply Theorem REF , which on account of {{formula:a590cab8-7517-46a8-a042-408b4c0a9e40}} shows that if {{formula:1a480c7f-c361-4d39-b73a-9e29e66d0257}} and {{formula:0e7f7a2e-cdb7-4174-a883-af4bdfdec167}} then
{{formula:9ccaa5ea-e21a-4e68-ba82-61df0ef7dc2d}}
Following the notation in Proposition REF , see {{formula:c5f685e1-5a8f-4e9c-8843-7e0452c2b7bb}}, we can write {{formula:129a60dc-79d6-42b9-83c0-f4979a0a24ce}} with {{formula:73dcdae3-ee64-471c-ba6c-108480bc5f18}} but we cannot apply it to get the limit of {{formula:e3f5befe-5a12-46cc-b962-bda509a89fe8}} as {{formula:0fad41e7-9401-466a-a3ae-440eeab5e2f5}} because the condition {{formula:9d4c2cc3-7c60-4104-8ad7-c4c180696856}} is not satisfied. As a matter of fact this is coherent because, since the first summand in {{formula:ff7513a2-4d8a-43db-9138-97e67c7b58d8}} is divergent as {{formula:824d0910-2908-48a5-b549-7dfc0f1c93f3}} it happens that {{formula:fde95b89-30b0-4105-9f5c-ed42a0297654}} diverges as well. Hence the approach to compute this coefficient hast to be different. The idea is to take advantage of {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, which shows that if {{formula:ceee699e-df3f-4547-ab20-f37050d49c88}} then {{formula:82157ff6-ac0d-4ac8-8059-9a9ddc298811}} where
{{formula:5696373c-2cf1-405f-9e4e-5091d0e506f5}}
with {{formula:5650781d-a742-4b48-88e7-23ffa49524f1}} By applying assertion {{formula:9c4b4ddc-4fa7-42c9-8149-1913f43e04a1}} in Theorem REF taking {{formula:33b0be2b-8271-4f81-a0a7-496ab2555f67}} , {{formula:0c854810-a1da-4c22-ac82-a4d24abe97bb}} and {{formula:5cd5b71a-10ea-4c0b-8851-fef34efe9f3d}} we can assert that {{formula:2e52d821-e4c1-476e-884d-e6f3e0efa3a8}} . Observe on the other hand that, following the notation in Proposition REF , we can write {{formula:3b284b76-40af-466b-9edf-9d84a6bad13f}} with {{formula:96893c59-51d7-4826-9c4c-fb27f04e4583}} Thus, since one can verify that {{formula:32df1032-0efb-40be-8c70-07249cc10b53}} and {{formula:88ed0115-9953-4f98-aab9-63caa1d5556d}} for all {{formula:fbc6f550-e143-4dfa-ba32-d9fef89cad16}} the application of assertion {{formula:6ceaae11-4675-4f2e-9d0b-01ae3e42e745}} in that result gives
{{formula:1e2cbc83-3a64-4b16-9026-71ffbf76f6df}}
Due to {{formula:c68092f2-7d5c-49af-ad16-05a4a694676f}} , see {{formula:37c8fb93-2425-4481-90e4-0642f6ae60a6}},
this proves the validity of the expression for {{formula:614d6eb6-6df5-4620-9d7d-e1540caf33f0}} in the statement for all {{formula:dda4f5ad-a5eb-4f6b-801c-036dab7e8e50}} Accordingly, since {{formula:16870b81-60e0-42fc-a522-2db1c8084a37}} is meromorphic on {{formula:a79e76a7-740b-4ace-9618-3ceca8b827c9}} , this is also the case of {{formula:931ed92b-b5e6-4860-97a2-2f15a66264e8}} thanks to Lemma REF , and {{formula:408ab35c-050f-400b-beda-2b171dba71fc}} is analytic on {{formula:d82f72eb-fcc8-479e-8c4d-058213ba0cf0}} , the application of the real version of Lemma REF implies the validity of the equality on {{formula:759f3a4b-d1fc-413f-a7d8-bb369f1dc93b}} . Observe moreover that {{formula:f5626f23-bd15-49b2-a9b8-5e5203d540c3}} is positive on {{formula:56352d61-a11b-45b7-97cb-66b76076a9fa}}
We proceed with the computation of the coefficient {{formula:5d1397b1-cf53-4ef4-aebd-f594044ccf2c}} . In first instance, for the sake of convenience we shall work with {{formula:a2bccf8e-97e7-4af0-8362-9a21719daf5e}} so that {{formula:2c3fff77-d963-498f-af1e-79a965a3603b}} Due to {{formula:a204731b-1325-4e3a-bc86-a9cef32def12}} Theorem REF shows that if {{formula:fd42143b-4742-495d-8f53-df95fb033f40}} then
{{formula:f20e87f6-ec13-4e73-ab1e-f29f200a1f5b}}
On account of {{formula:1dff6761-6f61-4f24-8307-88b855fdc7d5}} , {{formula:65e9b285-b794-4251-a1a6-47c4a947ea59}} and {{formula:200e1cdd-f068-49ee-aabc-8dbdd3555689}} , see {{formula:32276443-4fd1-41cd-8fad-ddf9f9e5cd2c}} and {{formula:39994dd8-65e2-4897-93ec-a90933aac561}}, it follows that we can write
{{formula:b321581e-8d4b-4b6c-9a4e-8d815e03bf4b}}
where here (and in what follows) {{formula:69e657e2-3b9c-4dd7-b790-dc0228275885}} stands for an analytic function at {{formula:66c01578-c723-4739-989b-62a041b42552}} (In fact {{formula:9b9c6eba-89dd-44cf-b309-619b9891a70d}} depends also on {{formula:0b54cd44-967f-48cc-b90d-b44b0ea6b0e1}} and this dependence is analytic on {{formula:18950d6b-f0df-4f46-abda-3282cbeb39a3}} . We omit this dependence for brevity when there is no risk of confusion.) Following this notation, from {{formula:22c5ac36-cba4-4a40-9fa2-3499d8ca58cb}} and using that {{formula:8903cde7-2564-4e08-9500-691fbed5b530}} we can assert that
{{formula:356d2192-8cdf-44f8-9281-a0e29ec8bd78}}
as long as {{formula:6221bc16-e80f-480d-ae37-c5e49dde628b}} On the other hand, since {{formula:af110994-1eb2-4ae4-a09e-84840632a166}} for all {{formula:23b69d5c-dc05-45e9-afe6-123986c69c25}} by applying assertion {{formula:7e95bb2f-89f6-4d4c-a50a-3113ab3a9dc9}} in Theorem REF with {{formula:9579f342-e3ab-43e6-934a-4b807c6fd39e}} and {{formula:408d1fdc-f699-40be-8531-0f7a2858ec31}} , we get
{{formula:d2b3e24c-8b29-4482-8d0f-7abc10aea127}}
where in the second equality we perform the change of variable {{formula:336c685a-faf6-441e-8d0b-0fe8897c90f9}} and use that {{formula:b9f3118d-8284-421d-a7d1-486613c292ba}} . Next we split the above integral as {{formula:1711b89a-f0e7-404c-a8d2-0e30bada4004}} with
{{formula:895bb397-0b08-467c-b9e5-ec719af51a91}}
where we take {{formula:f4f6d1a5-59d3-4dc1-b71e-4a077dd375be}} so that {{formula:69020bf1-1eab-4aae-bc3d-0d8a08263c5d}} converges as {{formula:1846432f-92a4-4aba-ad7a-74e89d4a626d}} . Due to
{{formula:ddcfad50-3221-486b-a29c-cdafa7f48c26}} we can write {{formula:a6e5c881-2381-45c0-945f-f0b6dc6cb7bc}} with
{{formula:84bc4cb9-71d5-4135-94e0-14b725629e85}}
Consequently we obtain that
{{formula:549a01c8-0cc6-4cf5-a7de-57a5dce1d435}}
On the other hand, setting {{formula:e2a3f97c-2e0c-4d89-aab5-03ce6319491b}} we get that
{{formula:7885e8c3-e0f7-4689-a181-4fa424d229de}}
Here the first equality follows by {{formula:7fe8c0a2-5171-4a0b-a078-808ba91351bf}} in Theorem REF taking {{formula:a14fda4b-e550-4902-85ec-0b4fe757e4ae}} and the second one by {{formula:68c07252-86c9-4919-81fd-d5186ec89b42}} in Proposition REF taking {{formula:0d6da7ee-df4e-4c7e-89a1-7256e9d7c388}} and using that {{formula:78bd7e40-d40a-4380-b7fe-7b71c94d8a5c}} thanks to {{formula:81986083-8766-4817-8b03-af8988374c40}} for all {{formula:72132297-3219-42af-ba3e-85cec03c3dc5}} That being said, substituting {{formula:1b02036f-1e91-4746-84b9-3b4fbf692d57}}, {{formula:d5a75651-7b30-43f7-ba96-87ad862f5e9c}}, {{formula:da54f7a8-f503-487e-9ce1-da598c0b65af}} and {{formula:5fb349ad-8ea1-471d-9b0d-084cd80d468b}} into {{formula:136167ef-843d-42e6-91fa-297336cd6597}} and gathering next the analytic functions at {{formula:40533518-24d2-4479-abcf-944b392cd04c}} we obtain that
{{formula:bf951833-d1ac-4dfc-922a-0a3918144802}}
with {{formula:4d3cd7d5-86b2-45ef-9423-7242fc19db1b}} analytic at {{formula:42497c71-eadc-413a-b8df-e5b9d15a06c8}} . (Here we specify again the dependence on {{formula:72da8ff9-c6ca-4c46-baa5-5a3f785133f6}} for the sake of consistency in the exposition.) Consequently, from {{formula:01364317-d74c-4f81-8612-59c0fa76ec86}},
{{formula:28a6100e-2e29-4974-9b21-e06318f64947}}
where we use that {{formula:64bdf8ab-c6c9-4bf5-96ea-d56d71a6b128}} because the limit must be finite. One can easily check that
{{formula:c0381705-448f-49df-8b10-9e6f6045c9d5}}
This proves the validity of the expression for {{formula:6f1cf4ce-6f9c-4b3f-826b-588b824757d4}} for all {{formula:213b2cf2-6973-4e36-bd7b-f4c3b29a74e4}} Similarly as before, by applying Lemma REF this equality extends to {{formula:b660b63c-760d-4cd0-b0a9-7fed03cd518e}} since {{formula:55ab83d6-bd2e-4f1c-8d69-7e6603935b33}} and {{formula:95b89919-cf56-4c13-962e-f7f01927d008}} are analytic on {{formula:ea40da2d-8f3a-4913-bc57-2dcfec277f1c}} and, on the other hand, {{formula:9a333dab-3c4c-4ed3-87a9-f6e7bb9a6c85}} is meromorphic on {{formula:2257c62f-c908-4f4e-acb2-87a10ccd774a}} by Lemma REF and so are {{formula:2e1ce216-a866-4a6f-a74c-5dbc6793fc9d}} and {{formula:06ca022c-b356-40af-a4c0-bb568c6982da}} . Finally an easy computation shows that {{formula:26bb9ff9-4a2e-422e-8809-29434ebb7591}} and {{formula:a8aa5000-3037-4147-9548-1721093ae29e}} are positive on {{formula:4fb64363-4ba7-4766-8b7e-45cbdb8b9b73}}
So far we have proved the validity of the expression of the coefficients {{formula:07b46653-f687-4872-af69-ed372f1ee081}} that we give in the first part of the statement. Moreover, since {{formula:e963732e-2f6c-4700-8cd0-5548d5b5494f}} and {{formula:0c98bef3-252f-4749-a771-e58eb0d3fc59}} the assertions with regard to the asymptotic expansion of the Dulac map at {{formula:a1b1cc77-9586-424a-a9c6-3e27cfbb11ec}} in {{formula:27aac039-0f09-43db-b847-fcf113e435e2}} –{{formula:8719beba-baa5-48ba-b1c9-cc00dd444e2c}} follow, respectively, from {{formula:e7636e0e-b471-44f3-937f-784929d1a6f9}} –{{formula:fbb73a5b-8ab1-4a48-b27f-989f5981fa88}} . The only remaining point concerns the behaviour of the coefficients in each one of these cases. This is our final task, that we carry out case by case:
Let us consider any {{formula:dc1e755c-f43b-4a96-a704-f91aa59389fb}} such that {{formula:919d2b02-11e2-45af-900c-0b5bc55bd80d}} . We claim that then {{formula:346b8c71-d5c4-4794-b070-a352e683886e}} Indeed, to prove this we first use Proposition 3.11 in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, which shows that the set {{formula:0539c2d3-d0ba-4147-9602-722c7d6057d3}} is the graphic of an analytic function {{formula:7c1f422a-4bfa-41a8-8b89-4011c3986137}} verifying {{formula:e91c661e-57b9-4d5d-85f9-c09578e0415c}} and {{formula:8b5aa529-dfe3-4088-994a-8fd0f2e8ba30}} Therefore it is clear that the claim will follow once we prove that {{formula:df3e7504-8c4d-4342-af76-492a4133786b}} for all {{formula:88126dc6-8f06-4412-8c42-fa5d65df7c07}} In order to show this we note that {{formula:b7db117a-5660-4e6c-8ef8-d505f787f7c8}} and, consequently,
{{formula:67f01797-010c-4924-ba01-c2b2cd9bcd00}}
because {{formula:06ccbf6f-b43c-4e3c-aef6-92d8776a0c25}} by definition and, on the other hand, {{formula:98eb9ea6-b93a-4676-9433-a6cb687a15d5}} for all {{formula:979724e1-d34e-484f-8802-c9f5502fb6bf}} and one can check that
{{formula:6dfa40e9-5640-418d-a3e5-30b45d8188f2}} for all {{formula:6e5866f0-988a-4fbd-9c19-4d55b88a7774}} . This proves the claim.
Recall at this point that
{{formula:dcc8661a-5e36-4843-a56f-21fea622a8b0}}
Accordingly, since {{formula:7dbf2919-127a-4ef4-b96b-89f187dfd792}} with {{formula:7df08019-d4eb-4b13-a250-ef1393ddb23d}} and {{formula:2ae68f46-eef8-408c-877e-17fca506f3c4}} and {{formula:98de272c-5b31-43fb-b763-b71b7ed969d2}} are positive functions, in order to prove that {{formula:70388385-05a8-4b2e-93e2-6d99924cdc1c}} it suffices to show that the linear combination
{{formula:1715a212-6cff-48a6-b19b-fa47036d020f}}
does not vanish on {{formula:79d0ad58-1496-4065-93af-69d825864bd4}} . Since one can easily verify that {{formula:21e90214-ad69-4553-8691-c51f7dd0705a}} and {{formula:34344a30-cca1-48ad-873a-ecf7261baf65}} for all {{formula:489d833f-b9ae-4d6b-b438-eead68086944}} , this follows directly by applying Proposition REF with {{formula:5c247b9d-365e-4424-bffb-67336c79b6f2}}
We already proved that {{formula:c199441b-a0f6-43e9-b866-8f10458c67fe}} with {{formula:fa099d81-8e1a-4318-936b-8ec3a0245dda}} an analytic positive function on {{formula:8476fd93-3fe8-4a60-9f09-a2d8db0b7710}} . The function {{formula:80238ef6-c61b-43bf-93f9-73396fb4bcf7}} vanishes only when {{formula:c1303dec-10ed-4f43-800d-06c6ddd68c07}} and, for {{formula:3e1c7b99-8a22-40f7-865a-80ff045d9b7f}} , this occurs if and only if {{formula:959acc9c-5509-41dd-b8f4-5b51e81758b9}} i.e., {{formula:903089c5-1581-4a41-9c05-4dd61387efba}}
Recall moreover that, by Proposition 3.11 in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}}, the set {{formula:464b477c-c890-4402-9e1a-b226b982bf45}} is the graphic of an analytic function {{formula:3ced7f19-c42c-4a98-8a25-498073ba33a8}} on {{formula:de340d5d-83a4-4486-b516-2e18fc8aa83d}} Consequently there exists a unique {{formula:8aec5b88-f00a-49b4-a01d-ef41611d0f04}} inside
{{formula:06670c96-4ac8-4f23-891e-d855c3f18a0a}} such that {{formula:77b552ea-83d4-4eda-a556-f76a1e7cd2e8}} and one can prove that {{formula:6ce0299b-feb3-4600-8984-68836541ba1e}} . The gradients of {{formula:47e03373-0357-4a6c-ae75-ede32eda6d3e}} and {{formula:bceb41ff-701d-4b30-97cc-a57eb2a3100b}} are linearly independent at {{formula:e35958d7-7a77-4fcd-a435-35e82a811591}} because {{formula:0edbd250-03a4-4a89-886f-59319f62596a}} for all {{formula:cc59d786-cb98-4ca1-b39d-58f64a98deb5}} by {{formula:7ce04a77-d18d-4f09-8d97-e78c6bc02b13}} of Lemma 3.13 in {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} and, on the other hand, {{formula:f48aab08-5827-4649-8d56-c53665c5b130}} and {{formula:4e56cde9-336f-4945-93eb-44be9ae46d38}} since the gamma function has simple poles at {{formula:e99c9a5c-4cd4-47a5-9d01-c4d54238fa5f}} with non-zero residue. Finally {{formula:ecd60b00-5730-4ae0-974b-a3efd1d4e176}} follows noting that, from {{formula:a75935bf-0f40-494c-8aa1-8ba846e7633a}} and {{formula:a0281d92-d749-4f79-927a-897b29774b8d}} , we get
{{formula:da482d49-e26a-4f60-a8e4-8726d294c074}}
which is negative because one can easily check that {{formula:b6bc1bb4-119a-4edc-93af-07c17d52fe67}} is positive on {{formula:5adc8199-a1a9-419e-bc83-4ba75acff940}} and {{formula:812776b7-617f-4e02-8e51-dc04931162ed}}
Let us fix any {{formula:8fcf5957-439d-4b8e-a70a-cb08cd5a02f6}} so that {{formula:cadb0800-87f2-44cf-834b-2d0e2f835d42}} Then, from {{formula:6e2b9819-8c9d-435b-bfea-5e9de3f022dc}} {{formula:da797384-6ada-49ff-99ba-820d21ec211c}} and {{formula:4addcd84-f9db-4317-bf3b-47ed80c8ffb6}} are smooth functions in a neighbourhood of {{formula:0014d134-d186-49a5-b26b-906d434c0439}} and, in addition,
{{formula:609364b3-1fac-4c78-ba60-98309960a6bb}}
The functions {{formula:2901b363-bf54-4b10-a9cb-0f926bdbefe1}} and {{formula:be9ab285-881c-4563-86b0-d9f4b1d2cdaa}} are meromorphic with a pole
at those {{formula:26a3ab84-3675-404b-9a09-ceca51ed3292}} such that {{formula:3e25fbaa-465a-47c6-a19e-fc6b27776b3d}} , and the pole is simple in both cases by Propositions 3.2 and 3.6 in {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}, respectively. Therefore by the Weierstrass Division Theorem (or, more directly, by {{cite:1c6ea8daf0c5dae002d1f2f2388307ad9acd3586}}) it follows that {{formula:730d9e5f-90f1-443e-a4b6-bb04a75f79d1}} and {{formula:0b53eddd-9c1e-47eb-8027-9ba415339cdb}} are analytic in a neighbourhood of {{formula:00d91870-3fa8-4f8a-839c-534b58f51024}} .
Furthermore, by {{cite:f3e7f374f32630ab24569c6e444f9e8c60b73533}} once again, {{formula:a1600411-29be-4a48-8c5a-6bf0095501a5}} if and only if {{formula:29dcaa68-d8ff-4255-85f3-b65d185c8231}} . Finally, since {{formula:893aac97-7f27-4436-ad24-9a8620451b6f}} {{formula:8ac38e97-6023-4d5b-8c5d-bedacc43bef2}} and {{formula:e095f72e-7361-417c-8829-a849ae5a112c}} , we have that
{{formula:c1fb1218-5af0-4b56-bf5b-16b6dbb55360}}
because the gamma function has simple poles with non-zero residues at {{formula:5f6fc3c7-6804-4b9f-9df6-c31260ecf188}}
Let us consider finally any {{formula:3f1f37c3-e0d5-4820-817a-05081c468239}} so that {{formula:cb1e924a-7891-46c6-83c0-6389a06f5cc1}} Due to {{formula:2ba2e45f-0ad9-491c-b24c-cfedd976b46e}} with {{formula:a78a87f1-e1e2-4ad9-8a59-90f9d66c00cd}} , {{formula:05654d0b-c018-4c65-9042-a9d15bad911b}} and {{formula:3eee7e88-ba6d-4465-8051-d3eccd8f1945}} , there exists an analytic non-vanishing function {{formula:aa8d71e3-2594-4a99-8a40-22b5c9b49fa7}} in a neighbourhood of {{formula:bb2b1bcb-6132-4590-abed-a389c8c55115}} such that {{formula:634e910a-7146-47fb-ae27-87399b1d2710}} On the other hand, from {{formula:d63810b0-12fd-4ce7-99fa-9d403bbd3704}} and arguing as in the previous case, the functions {{formula:0843d247-8686-4351-9911-ebf06c873dc6}} and {{formula:46a00336-1a36-4aff-8c4e-e342171ceb73}} are analytic in a neighbourhood of {{formula:00bec79a-a114-47e2-aca0-f73d75d8e036}} and
{{formula:ab96a9a9-cf1d-43d8-a903-6eedd06d61b5}}
In particular we have that the sum of residues of {{formula:e9b8ee94-29df-4f4d-9fa6-ffa08b91d465}} and {{formula:b5712a48-16d6-4eef-9e93-e571202aae23}} along {{formula:c15c9866-3d0c-459f-9845-ce7130579387}}
is equal to zero,
which saves us from computing the explicit value of {{formula:360b9c8d-57de-4ffb-a573-c72629ed7974}} Indeed, since {{formula:159ec89c-d114-43aa-830e-cab76c7dc6e7}} and {{formula:f682a38c-6f84-4b0b-ae36-a73dc956f908}} we obtain that
{{formula:8181a566-9215-4ae5-b8da-70aa14e3bf1c}}
where in the second equality we use the expression of {{formula:c350f893-d119-4f2d-9502-ee6959545f29}} already proved and in the last one that
{{formula:3ca5691b-06ce-4206-9fc7-d73351a3b560}} and {{formula:e1d793d0-e65a-479f-9042-61ec21df0311}} see for instance {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}. From the same reference we get that {{formula:173a453c-8f0a-4c4d-bf59-ac8d9348c7f8}} . Moreover one can verify that {{formula:8c850ba1-6a2e-4c80-a983-916c3f5e10c6}} maps diffeomorphically {{formula:052beec4-466e-4c15-9dcb-6fc1d4e82f79}} to {{formula:4d2144f1-a8a4-4737-bfaa-801249e2d699}} and that it is equal to {{formula:27b1e3fb-77a3-474f-9d2d-b9bd68217ac8}} at {{formula:7c8e1894-820b-46b9-b0ae-6990dda368a8}} Accordingly we can assert that {{formula:97cfd76b-378d-4ed0-adb6-c9855697575f}} where {{formula:e05630a3-38f8-4ef6-8731-56f66f89f652}} is a non-vanishing analytic function on {{formula:66d3422a-6907-4d03-9973-d6885901859c}} and, consequently, {{formula:0c8861f5-952f-4f74-8525-73f5919f04dd}} Since {{formula:57102285-d1fc-4423-8bdb-ee2e47057d1a}} and {{formula:077ae934-a337-49c8-be12-2f85f38fa3c9}} this proves that the gradients of {{formula:96b3b226-87de-4da0-81da-54ae5f284a6f}} and {{formula:24afd7ce-3907-432f-8ea8-d4b89e6f00bf}} are linearly independent at {{formula:d8cc5bfb-3039-4b87-a32c-a4f6c4b4279a}} as desired.
This finishes the proof of the result.
Beta and hypergeometric functions
In this appendix we are concerned with the integral representation of the Beta and hypergeometric functions (see {{cite:e4af7114e75a7861941159b039b603d973c1819f}} for details). The Beta integral is defined for {{formula:08e29f5b-3a88-4366-b3aa-a5eff20bbbc8}} and {{formula:34990b53-3c6d-4ef9-81ae-015b531383b3}} by
{{formula:9a48759f-ba70-46da-97c0-c7e8ccd513e3}}
This function can be analytically extended for {{formula:03f8ccc1-796e-4f91-a835-e5a65010efb0}} thanks to the identity
{{formula:3850359c-05b3-47c6-b1cb-eb9fc2b43bb4}}
where {{formula:04ee3a0c-12ac-45d6-8285-a2c121b6c17d}} is the gamma function. Recall in this regard that {{formula:e47a825f-484e-4e3a-8eb6-e7174239877f}} is an entire function with simple zeros at {{formula:30944c3a-b66b-4f6f-b673-d4a99597de0f}} On the other hand, if we consider {{formula:04ffdaa1-db5e-469d-be57-afd533bdb144}} c{{formula:8cc8603a-5294-4010-9789-a960ef2b2d4a}} 0{{formula:6f508094-4a8b-4769-b91a-f1b8f33d7e55}} z{{formula:ff1e6174-fdb0-433a-b088-aa60f4f0a60c}} D:={z|z|<1}{{formula:d1c94470-431d-47e1-ac2e-7466bfec32f8}} x we use the Pochhammer symbol {{formula:fca3f3f8-070d-474c-bc41-52fd71cccf41}} .
In this section by a meromorphic function of several complex variables we mean a function that locally writes as a quotient of two holomorphic functions. Recall that a function {{formula:d7fd5429-63de-47ec-8299-a955b3e50345}}, where {{formula:6bcfa99e-ad6a-4a24-8a0f-13e6376fc21a}} is a connected open set of {{formula:abb3d386-9848-4a94-b52c-9d71c592dcec}} is holomorphic if for each {{formula:4bff73f3-baf3-4bae-b8de-b0363ae61f60}} there exists an open polydisc {{formula:9a7d7678-551f-43cf-8089-e9828eeb5a68}} such that {{formula:f768860f-27eb-4886-8678-fe69e7d5da0a}} can be written as an absolutely and uniformly convergent power series at {{formula:f018a788-aa8b-414e-9f6f-1aa65250d74c}} i.e., {{formula:c62f89f8-7d7a-4482-af85-e68e72323e79}} for all {{formula:dc218465-d921-4f85-a453-ac0c41557c73}} . On account of this we have the following result about uniqueness of meromorphic continuation.
Lemma B.1
Consider two functions {{formula:2ff8c59e-60fa-4c50-a1fa-73abf535c87e}} and {{formula:430c982e-ded9-4910-afd7-3ac3c5642f2a}} that are meromorphic on a connected open set {{formula:f3fc0b48-dd68-4af2-b3c6-905f1736caf5}} . If
there exists an open subset {{formula:b3d7017f-6901-42d0-96ba-ae7cfd645178}} of {{formula:bc7684ac-8774-4b38-a41f-fb69af9e855f}} such that {{formula:18921e3a-6370-457e-99ff-c8ffdff5be8d}} then {{formula:45faf5c5-2955-4847-a73c-c690d5eab238}}
Proof.
We assume without loss of generality that {{formula:6d4f1ad3-46de-498e-a2c8-60dcf62e5239}} . In doing so the equality {{formula:673129ef-c50e-4374-a536-33d901190491}} has to be thought only at regular points of {{formula:3b692d07-f9e2-40ce-8a11-1f2294559ab1}} That being said, consider any two regular points of {{formula:fc7f1640-0aa2-4eb8-b8c3-34323c4dc6b8}} , {{formula:1995ab64-5e16-45ef-9aea-64d27e8d967d}} and {{formula:bd51637a-11ee-4dc9-81bd-e3db0d6d3823}} and take a continuous path {{formula:fdaf76f1-b466-46e5-a519-df9cd1f02211}} joining them. Suppose that it is parameterized by {{formula:963497a7-a1f1-4eff-8b67-86324835cccd}} with {{formula:7d848c9c-d1e1-4b8e-88b5-b122a7df2d13}} and {{formula:2a929b5c-027d-4c1d-bfe3-02852037dc07}} By compactness there exist {{formula:fe1096ac-e3d3-46f1-b81a-342a9528ca19}} and positive numbers {{formula:4d5418cb-6730-4ad8-ac99-b37a4f896eb7}} verifying {{formula:5493bf12-28a5-45da-b0fb-375e30f9ebb6}} and such that, for each {{formula:7d211e18-d8a6-45f7-b048-97174313861b}} we can write {{formula:3d70d49c-1514-4b9e-abc2-da3eddce5dbc}} with {{formula:4244031d-f476-4cae-8350-9ff87ee1fd28}} and {{formula:22ff99df-aedd-4a55-9f1c-5e7893bdf72d}} holomorphic on {{formula:56132bd4-3e91-4c9e-95c0-6bd004d785e2}} .
Define {{formula:5652b83c-7ffe-41ac-b682-a500e21ba1b8}} . Then on the regular set of {{formula:6c928007-a55a-42d3-af26-a81dd5e3e561}} (i.e., where {{formula:57cc352b-e872-4354-9a87-09c5fce1e418}} ), the equality {{formula:bcac6090-8a80-4d0d-84c5-0548e09da301}} implies {{formula:2630d5fc-cfa2-4383-b7e8-8c13d90225b7}} by the uniqueness of analytic continuation. Accordingly {{formula:b8f25fb7-a974-44b7-80c0-55a31c1a7132}} Next we compute {{formula:5878dc9e-32a8-47f3-9045-c57fb9a82e4f}} again but replacing {{formula:ae9abed9-5cdb-451e-b595-dabc9981112d}} by {{formula:725ac7f3-a3ef-4188-a215-9bbc2fa6e3c6}} and we iterate the process to conclude that {{formula:87bd4181-7827-4309-a7db-a472ae22521d}} Since {{formula:f9b72f81-397f-4fb4-9a4f-41107ba09b0a}} is arbitrary this proves the result.
Let us remark that the previous result is also true (with the same proof) in the real setting, i.e., for functions in {{formula:38119ce6-8f29-495d-9b01-93b1f8fd9d8b}} that locally write as a quotient of real analytic functions. The following result is well-known but since we did not find its statement in its fullness we give it here for the sake of completeness.
Lemma B.2
The function {{formula:98f62d5d-f85f-4be0-b9cc-060f65ed7a74}} extends holomorphically to {{formula:34b70ea7-196b-4fe0-8678-b52f05ea2962}} .
Proof.
Following {{cite:e4af7114e75a7861941159b039b603d973c1819f}}, let us show first that the function extends holomorphically to {{formula:d3586af2-0949-4d37-8fda-ee3f57e7683a}} . To prove this claim we write
{{formula:87d2321f-7e64-485e-aba3-0d985715b95f}}
Stirling's asymptotic formula {{formula:b3327f4d-496d-4542-8b21-19908347723b}} as {{formula:c84cf689-d095-47d5-abb1-0c8318477c21}} (see {{cite:e4af7114e75a7861941159b039b603d973c1819f}}) shows that
{{formula:7a8512ee-b33f-4336-8b43-a2aaeff5b624}}
Fix any compact set {{formula:c4a614e6-bd6f-4384-b3e8-f4a54a4b343b}} and suppose that {{formula:70fcf3af-2b9b-4933-b959-61a4d037d890}} and {{formula:d165ba80-324b-4b55-9f76-631bd73c7915}} for all {{formula:e0f78fe1-db12-4379-8327-03a341b9339e}} Then, on account of the above asymptotic estimate and the fact that {{formula:609d41ab-f847-46b7-9bea-f7bd70dc4999}} is an entire function, there exists {{formula:e8081b36-e620-4bf9-947b-f6027fafca63}} such that {{formula:2a92f47d-3bd0-48f8-bf6e-a4d18083979d}} for all {{formula:333cd485-3ab3-4a6b-9853-ff642a983ba4}} By applying the Weierstrass M-test this proves that the series {{formula:973a10cd-f9a0-4ec2-a345-c3fb9620dffa}} converges uniformly
on compact sets of {{formula:82b7a8c3-59da-4b63-a3c0-0b7cb0f72c33}} . So the claim follows because the uniform limit of holomorphic functions is holomorphic (see {{cite:8547593f08706077ca3a286b84f978da7a4c1e66}}).
Finally the result follows by Pfaff and Kummer's formulas (see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}} or {{cite:e4af7114e75a7861941159b039b603d973c1819f}}) relating the values of {{formula:b04cba85-1ae3-4426-84cb-74f8baaee858}} at {{formula:bdf56e34-bbcb-4b99-a72e-0b5db56a4e14}} , {{formula:b7de0ccc-4b23-42bc-b9a8-a57ed259fed6}} and {{formula:70e0eb57-5920-4218-a3e1-b2a755da3761}} , which enable to extend holomorphically to {{formula:76668b8f-4245-4ae0-8cdd-f32a290df795}} the map {{formula:158cd9fc-26e5-4463-88d1-6750f7f58437}} . (These formulas are usually proved under some restrictions on the parameters {{formula:27477ba2-cb70-495a-9581-2c3413ccabcc}} {{formula:9074a8ed-52cc-46d7-b0eb-c736ca814feb}} and {{formula:8758b367-1f51-4d26-9022-cce809ede655}} but they are always satisfied thanks to the claim and Lemma REF .) This concludes the proof of the result.
It is worth to mention that by Hartogs's theorem (see {{cite:55a4d21bc58b42d4dd460ac15f8c072248c7c81b}}), a function of several complex variables is holomorphic if, and only if, it is holomorphic (in the classical one-variable sense) in each variable separately.
The main concern in this section is Euler's integral representation of {{formula:abbd7a81-ba4c-44d9-823f-4b13d65330bb}} , see for instance {{cite:e4af7114e75a7861941159b039b603d973c1819f}}, that is given by
{{formula:c44ba7db-8795-4d37-a09d-d6659592edb2}}
provided that {{formula:12b2ae0b-f516-4faf-bc1c-2897ec934d43}} and {{formula:0981e6a6-b9d4-4495-8639-876214929429}} . Our goal is to use this formula to compute {{formula:c45a06c1-062d-4e4f-9283-d45929876707}} (see Theorem REF ) for some specific functions {{formula:170de3e3-aab1-4604-af05-3b8f728d5033}} . Next result is addressed to this problem.
Proposition B.3
The following holds:
Consider {{formula:75adcc84-7370-41ce-a252-d7b776daf4a9}} with {{formula:d3372991-08e6-44ed-9e2d-cb66eec4c674}} Then, for any
{{formula:f6e684ab-6bd0-49ff-afc7-888fda3e2ac5}} and {{formula:a8fecbb8-b938-4363-9cf5-45fcb58f5108}} such that {{formula:6308e463-c960-4b92-b860-b1bc7c2ac0f0}}
{{formula:f0d0d74e-6f59-40b0-a765-45b6b2bc1a77}}
Consider {{formula:89beffa4-6533-4a7a-b2a9-6eed4cde522d}} with {{formula:d9a8771a-2512-49fd-91e8-e31567ffb6c9}} and {{formula:36b3a4e9-ee3e-4fdd-8496-9175971f5bd1}} . Then, for any
{{formula:d5a9b968-ef44-4d07-aeea-7e8a51a7ee45}} , {{formula:2c8878a2-cf19-4965-b5d3-b207272b0210}} and {{formula:ca9a8052-bd18-4c17-b0b7-6b1355e2cbfb}} ,
{{formula:64f8cda8-f73e-4890-87ae-f4fb9e489a69}}
Proof.
In order to prove {{formula:a4508e01-33c9-4570-8d00-776632bd407c}} we define {{formula:dd3c8acc-3a5c-4c5f-94b7-3dded4e354c4}} , which is connected. Note then that we must show the validity of the identity on {{formula:8f2f3f71-8345-46cd-979c-70cac95dc226}} . We will show first the identity on an open set of {{formula:0356cd34-62e2-489c-9586-641195d819d1}} and then extend it by using the real version of Lemma REF . With this aim observe that if we work on {{formula:f3f36df8-8601-4307-9ff0-8804fb11009c}} then the application of assertion {{formula:40ab92ea-213e-4ff2-9906-234dbf2bcd37}} of Theorem REF with {{formula:6ee18ded-c2e6-435a-92a2-ced1ae369580}} yields
{{formula:4f20c58b-e5f4-4b0d-8caa-9f956e014d01}}
where in the second equality we perform the change of variable {{formula:042b3616-375d-4c37-8897-e57973219a75}} and in the third one we use {{formula:686b4846-9f4c-4eb7-9e8d-a56e37998eb3}}. Note at this point that the function of the right hand side is meromorphic on {{formula:ed875f4e-fe33-4912-8cbd-5b948615f4f1}} because {{formula:6279093d-d9f9-49c9-9c6e-31cbb6c8a8cf}} is entire and {{formula:baee1d53-c0f9-400a-815c-1d144e889498}} . We claim that the function on the left hand side is also meromorphic on {{formula:5b81dd8a-3e70-4d8a-bd5c-44f9284990da}} . To show this we work first on {{formula:66c5ba45-d34f-4d35-a674-a94c1bf4fe83}} because in doing so we can apply assertion {{formula:12efccd0-ab0f-4e25-a4bf-aaa723fed22f}} of
Theorem REF with any {{formula:d5610b28-17ce-4018-a51c-0c3dd26c0d3d}} to obtain
{{formula:6a57bd9f-e15d-4c1a-a5a7-bdda179b66de}}
Here we denote {{formula:e2e98996-c879-478d-84bc-008c00f7cab3}} for shortness. Consequently, since {{formula:2543ae08-29fa-4c18-b1f7-6eeccf33131b}} ,
{{formula:e81a8742-e0da-4687-959c-35c79ac48e51}}
where in the second equality we make the change of variable {{formula:cc991d93-16f7-46a9-880b-effd01ae0da7}} and in the last one we apply {{formula:0a85f02e-685b-492e-a12e-21ef59918733}} in Theorem REF with {{formula:659c7560-f179-4897-83d2-3f3d5ae2d5d9}} . By {{formula:347e8453-5e25-45e0-8a18-a95411d91865}} in Theorem REF the second summand is analytic on {{formula:211f1f4c-45b5-4eda-851b-fd66ffd0dce0}} , whereas the first one is meromorphic on {{formula:04f2e962-5baa-4b0e-ba1d-f93e2d66f736}} . This shows the validity of the claim and so the result follows by applying the real version of Lemma REF .
In order to prove {{formula:616bc9fc-2e22-43d0-a629-3dd587ffae46}} we fix {{formula:89cf9899-9647-4506-950f-78a422234f19}} {{formula:7ecd85e1-4e84-4642-a03a-fc8f5c2edf90}} and {{formula:47924173-46ab-41fa-8d2a-eddba23796d9}} and apply
{{formula:a6ee841e-10a4-4279-95f4-1d3054136d0c}} in Theorem REF to the function {{formula:08b2bf4e-a2ae-4936-b6e8-468e0a5c14c2}} Then, taking any {{formula:ebb26baa-12af-40a2-8482-3690d3b260b7}} with {{formula:549436d5-be1b-432e-940f-3b2cfa87afb6}} and setting {{formula:48719e77-bf2a-430d-87da-eab11520b02c}} for shortness, we get
{{formula:0715fb26-6f08-41e5-9413-3534210e5dde}}
where in the last equality we use {{formula:a85b5f4d-591d-4952-94b8-4c4273eacb7a}} in Theorem REF
with {{formula:6dbaba89-3c8f-4ab7-89a4-3ab8488ebaff}} and also take {{formula:01736cbd-53d7-48b8-a47c-188b183585ea}} into account. Thus, by applying {{formula:fbb57578-6a57-4b1c-8b84-9de8b0551da3}} in Theorem REF
to each summand in the last expression, this shows that the function
{{formula:1477786d-434e-47b7-82ec-c38978085bf4}}
is meromorphic on the open connected set {{formula:f321eb98-c365-4444-b296-37043b511484}} . Note also that if we consider parameter values in {{formula:9c1b47d1-ab26-4e75-9564-e1cc76019b7b}} then
{{formula:7a977e67-a236-4d45-a69d-9cff5a9bdb36}}
where in the first equality we apply {{formula:4c266cfb-bdc4-4710-883d-9711a0ceb2ea}} in Theorem REF
with {{formula:5ce4ec68-d509-47cd-a2ff-1062242d06ec}} and in the second one we use Euler's integral representation {{formula:e731dce5-450f-47bd-983f-f93a360d6e40}}. We have just proved that the left hand side expression is a meromorphic function on {{formula:261b6d1f-34b6-4199-a462-d3dab5ce2ce2}} . Furthermore, by applying Lemma REF and taking {{formula:59bd9b9e-04c5-45e1-a7a1-9254b0e4bc6d}} into account, we can assert that the right hand side is also a a meromorphic function on {{formula:368817af-05c4-4cdd-ad9e-48103a4c71f5}} . In view of this the identity in {{formula:9198e746-d099-4f01-80bb-b942449ff4a9}} for the parameters under consideration follows by applying the real version of Lemma REF .
It is worth to point out that the application of {{formula:97499c90-3e82-4107-bbd7-51a366e81a3c}} in Proposition REF provides integral representations of the hypergeometric function in a range of parameters not covered by Euler's formula {{formula:b152d106-8bc3-4c04-876f-ce52339272e8}}. Indeed, by applying also {{formula:e20894d5-c254-4eda-bc10-2ff1d969cbd4}} in Theorem REF with any {{formula:ed6a3f40-5ab7-4740-b6d9-8aa543caba72}} we get
{{formula:25aeb331-f9af-4c27-8a60-c298a74b2821}}
which holds for any {{formula:b2a58f62-6c96-47ad-8448-fea23c3bd402}} , {{formula:24f40934-9ff1-43f3-a76d-ea8b2a07b9f5}} and {{formula:2a229f3b-c821-403a-8b11-8e641ba55f48}} , where {{formula:3ff71d41-10bd-45c1-b2a8-5b7321ea1fef}} . We stress that {{formula:19604538-ed4f-441a-9655-d59558033766}} gives an integral representation for {{formula:bf82c29d-dc0c-4634-916c-b2318858350d}} only in case that {{formula:a6690d9b-bea7-48fa-92b2-442326719217}} and {{formula:9c7da634-c5db-418b-bd66-abc72ccf4f40}}
A technical result for the proof of Proposition REF
Proposition C.1 The function
{{formula:55426e60-8730-4fea-a78f-51a0dc22ff21}}
is strictly positive for all {{formula:33421663-30a5-4f0b-8627-0b207f5568c9}} .
Proof.
In what follows given a smooth function of several variables {{formula:a8631665-b3ea-474f-9271-5cc3b00c1a11}} with {{formula:5c465399-c97c-487a-a3cd-6bf830f13ae4}} , for each fixed {{formula:e7c9dd82-7b43-46d4-a6da-5079d47f4223}} and {{formula:7f6d3e2c-e700-4223-9dd6-3f29ae392fd3}}
we denote
by {{formula:beaacd0f-d9b6-40b2-875d-e3035e0d1390}} the {{formula:c5d18cbf-ba78-4eb6-9f3e-f65b4c6d0f24}} -th order Taylor polynomial of {{formula:dc16f080-683b-46fe-8500-b4099f000e4f}} at {{formula:3e9f25a3-a8c5-4538-8352-11d6b991e331}} i.e., {{formula:576f01c3-490b-4f19-9737-88b7ff12d108}} .
Setting {{formula:584b7327-0f3b-4691-ae08-9e8ee673950b}} we claim that the following inequalities hold:
{{formula:eadd6844-cd22-4b35-bc8a-87a1b886d59f}} for all {{formula:82c0fc33-76be-46f5-b544-55dd073be47e}} , and
{{formula:b08d38ea-3917-4f56-8fcf-28806ce44198}} for all {{formula:7312708f-8b4a-4932-a1cf-586831899cb4}} .
It is clear that the result will follow once we prove this. For this purpose our first task will be to express the function {{formula:0f961817-e527-4e27-8597-2f7c9a2ea4ca}} in terms of a definite integral and to this end we define
{{formula:981c8868-c52c-400e-a43b-0e6b333ef733}}
so that, taking {{formula:55c8f410-fc30-494b-bac3-8172363440bb}} into account, {{formula:df524387-cb94-4226-8659-e2525549f6a7}} . Then by applying {{formula:b9e3fc5d-5e97-4f3f-a55d-235a9a60b1c3}} in
Proposition REF with {{formula:3f5cec24-9e94-4451-9ded-d78a1963d7e5}} we can assert that
{{formula:7b746e61-1aa8-4382-8fb2-3d61a21b0ac2}} where
{{formula:b0085b56-6b0c-4364-bc8a-082806650a81}} Next we apply {{formula:625e46d1-e482-4f0a-8570-7d148af96200}} in Theorem REF
taking {{formula:ed445ffa-71f2-47d1-a874-8824edfc1fd7}} to obtain that
{{formula:ab17935c-5f60-4f2f-899b-4199b0c9f7e1}}
where {{formula:2b3f44c1-96ff-49cb-8241-788ba252b1ac}} and {{formula:7b57bdf5-3c66-4663-b14f-ba52b2f19b93}} with {{formula:238c27ac-f2eb-4728-a61f-1585180dabd6}}
Similarly by {{formula:61b04d9e-1b04-42a7-82f6-716490012e7c}} in
Proposition REF with {{formula:ad80e711-d3a4-498b-ab88-e13309b15ff8}} we obtain that
{{formula:bd4c307a-8d31-4c2e-a0bf-51a66f5b5e0b}} where
{{formula:36a6d570-0e35-4947-9ea7-e2a46ac41196}} Next we apply {{formula:f78ba1e2-4c9a-4db4-bbc0-f8bde3cc933e}} in Theorem REF
with {{formula:cf008e6b-6f30-4b53-8f82-9ff2539ff361}} to get that
{{formula:bf054fc7-2199-467a-8c14-3a9783a35bde}}
where {{formula:964af86d-1d09-4117-800c-456528c4ba60}} and {{formula:f62949b7-541c-4295-841b-5a9f731557b4}} . Accordingly an easy computation yields
{{formula:03c8a582-87b2-40be-9e97-5817a9de06c0}}
where
{{formula:16f2c2e4-29eb-427e-b230-1fe1f7cc6bb1}}
and consequently
{{formula:b15a5ce6-d9ee-42a3-b9a9-b51527a192b3}}
On account of this definition and the fact that if {{formula:c5f2a191-9811-449b-97fb-5a6729eba9cc}} then {{formula:d0f76a90-440b-46ac-9748-5c9ea8d000aa}} , we get
{{formula:1c63eed4-fd94-4e3a-a691-c3255e2db941}}
where
{{formula:44ba173d-da55-43c3-9248-3c7d2d69dcbf}}
Therefore the assertion in {{formula:b01a9f84-71d4-46b6-90f7-a47b7d517c66}} will follow once we prove that {{formula:9d0e6eab-4a79-496b-b71b-7b51586ba4ba}} for all {{formula:e4421af1-39f5-4567-a462-95324b23d5ef}} As a first step to this aim let us prove that, setting
{{formula:1aee2606-52ec-43fd-a2ee-19297ef63218}}
In order to show this we note that
{{formula:fe4b5d84-2fa1-49f1-b8eb-349795779761}}
because {{formula:37229657-bb14-417b-bad1-32361e44d5fb}} and, thanks to the remainder's formula in Taylor's Theorem, one can easily verify that {{formula:09d9ec0b-37ff-467e-8541-624858d0305f}} for any {{formula:a9d8b759-3652-44e3-a587-e1672a743aeb}} odd and {{formula:cb621fae-c435-4b90-823c-bb3522abc1ee}} . It is clear then that a sufficient condition for the inequality in {{formula:a51cefdd-6217-455c-8979-31a4d4c7050d}} to hold is that {{formula:9283bdd6-f421-4d1c-9002-f4e7bbb129cb}} for all {{formula:96e7ece1-f9d9-40c1-995e-75d9504a76ff}}
In order to show that this is indeed true we note that, by the remainder's formula in Taylor's Theorem again,
{{formula:5bb84a15-40b8-4564-aa77-ebc868847ca5}}
and consequently it suffices to verify that
{{formula:cbcff421-3811-41fb-bbe2-b9d97643845b}}
for all {{formula:0e52a51b-f53c-40a7-8622-98fe2d250c56}} This inequality is equivalent to {{formula:63e6e597-1385-47de-ad4c-88082e7b92f4}} , which is obviously true due to {{formula:dea09b9f-1980-484f-a6ca-08e47704f548}} for all {{formula:e3381cd9-1847-4f3b-ae2a-0c958941c3eb}} . This proves the first assertion of the claim.
We now turn to the proof of the inequality in {{formula:0bad1ad2-d835-4566-9412-08542a11fbb2}} . On account of the definition of the Gauss hypergeometric function, see {{formula:56b2ab9d-7a05-46a5-96ab-75865eabedf3}}, together with the definition of the function {{formula:93cd3f07-340d-4f84-80bd-962b2fb4f6aa}} given in the statement it easily follows that
{{formula:04e0c8b6-5e16-4870-9561-aae621e57c07}}
Observe that {{formula:807ab270-6a6d-4076-a1b7-74b4b8b6dab9}} is a polynomial in {{formula:4784f432-bf8c-4896-8d44-b3196b5acc4c}} for each fixed {{formula:691875c1-adfc-4c3c-93ad-c6d05cfaa44a}} . In order to prove the inequality in {{formula:e4429efb-ed44-4765-904a-283035169f49}} we consider {{formula:a92a788c-51de-4c90-9fa8-e6320e49aa21}} as a polynomial family depending on a parameter {{formula:b83e4067-ad74-4288-8c8a-382048035c9a}} In doing so it is clear that the following three conditions imply that the number of zeros of {{formula:bd6f1946-3f9f-4bc6-ac6d-0fadb26316b0}} on the interval {{formula:b344c8e5-e167-44ff-a98e-d3509b804784}} , counted with multiplicities, is the same for all {{formula:f2250398-05cd-42ae-abef-37aa663253a6}} :
{{formula:59bdf3af-e3df-4476-886f-1628c0d80168}} for all {{formula:89c3b38b-80a4-4cc0-8612-1dafbbef449c}}
{{formula:dcc61af6-32c3-4bb4-ba4c-e12ec509accd}} for all {{formula:bab11b04-58be-4d0f-a705-e8cb84fd2e34}} and
{{formula:f4348a3e-13e6-49d8-b531-1157059c5306}} for all {{formula:32c3ac68-1385-4f61-962d-0c0244fd607c}}
Since one can readily show that, for instance, {{formula:366674bc-fc37-4476-8a1d-5e5a9ef339fa}} for all {{formula:875d635b-eb44-4232-a81e-ea3285b027a5}} it is clear that {{formula:49d24e03-0767-4770-b0ab-8636e953897b}} will follow once we prove that these three conditions are true. This constitutes our next task.
In order to prove the inequality in {{formula:701eb9bc-b776-47d6-a436-a98a344047ca}} we first note that, from {{formula:a5e7918c-64a2-4007-affd-02033065c08b}},
{{formula:7657b150-5a2c-475f-9e5d-d67aa4144399}}
where we use that {{formula:df610d06-64ba-46a1-9a7c-c1d4ffaae332}} and {{formula:a6d0577d-af7c-4b49-861e-8f6388579412}} see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}.
Taking this into account, the fact that {{formula:a604307e-6622-432c-9698-7850f3731150}} for all {{formula:5524384d-1dbf-49e0-a700-bfbfe7c3f27d}} is clear because {{formula:d6fed25a-8b54-4433-b93e-e83abaa4e3d8}} is negative for {{formula:c3d91b0b-2b5c-46d7-a14c-69c233cf2e94}} and positive for {{formula:dd214151-bd16-40c0-8c20-94bce56c0d84}} and {{formula:e197019c-f4db-41a9-8cb2-c2a6d95095a3}} , see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}} again. To show that this is also true for
{{formula:33b428d4-40c5-44f4-b7cf-c5a392342cd7}} we use that then
{{formula:c576664a-4bc3-4d1b-a1fd-cfb80b724a02}}
The second inequality above is obvious whereas the first one follows noting that {{formula:f2e5eaaf-8549-4f78-9ccf-1f70fdf34e48}} is positive and decreasing on {{formula:aa833ce0-a4ed-4609-9825-28e1768a1c0e}} . In its turn this is true due to
{{formula:e4898599-f43c-4b18-90ed-45b69bd3ffac}} for all {{formula:b679f3e8-97b6-4493-b54c-b957165c65d3}} since thedigamma function
{{formula:c78fe84c-4c16-42f3-99d6-ecfee2003e0a}}
is a well defined monotonous increasing function for {{formula:d3c48d03-409c-4b7c-8867-4f32397cfece}} , see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}. Here {{formula:2e4fb1b0-9ff6-4bf7-89e6-19bb7c7b6165}} is the Euler-Mascheroni constant. This proves the validity of the inequality in {{formula:1c5866e4-8577-48f2-a60e-fead952a9fee}} .
Let us turn next to the proof of the assertion with regard to {{formula:295517ad-9d65-462c-b100-53212a9080ae}} To this end, for the sake of convenience, we introduce the function
{{formula:17f637f5-3ed4-4aa0-9ca4-15f0552594dd}}
where the identity follows using the so called duplication formula for {{formula:a1611919-3e87-4721-b932-df3ac743b2b4}} , see {{cite:6367e8780461f8a7ff3be9d0031ee3b6505ff9f3}}. This function will enable us to write {{formula:63b52ed1-0436-4a26-af79-801f8c9526b6}} in a more convenient form taking advantage of the fact that each {{formula:dadfde9b-e407-49d6-82d2-a8ade083c1e2}} is linear in {{formula:b461dc1d-4c17-4654-9b4a-9fe9609c34e2}} and {{formula:53b7faa7-744d-400c-9576-aa0d27eadd6a}} , see {{formula:ae309fb5-ec55-4eba-8561-7ad6d2c5d856}}. In doing so, some easy computations using a symbolic manipulator (see {{cite:7239ef771aa10d97c33c97107e53e3418f37cd7b}} for instance) show that
{{formula:88013025-d40d-4a5c-9db6-2d658fec8727}}
Thus, since {{formula:495b0e5c-69eb-496e-9e3f-7101043e2e42}} is negative for all {{formula:97ee6437-4da8-4232-9693-9710ee062168}} , the assertion in {{formula:275a3590-afc4-4042-b25b-3d711f9ca4a4}} will follow once we prove that {{formula:60b36a89-0798-4a64-bf61-a76a8c136673}} for all {{formula:588276d7-92d1-414d-8d7e-7950adc716ab}} As an intermediate step to this end we claim that if {{formula:6cc2171f-06b7-4877-b0df-0dc78def85d0}} then {{formula:3f46e892-56cb-486c-a236-8811282b6883}} , {{formula:bea1460b-cb1e-49df-bff5-8fbeca557a08}} and {{formula:983f550d-3e85-432b-a142-b4a421107d5a}} The first inequality is clear from {{formula:fb50fdee-4b81-4b03-ad28-7fb0daa5cf6c}} because {{formula:c1866d4e-7b1a-4902-91b0-2f91c8d5164d}} for all {{formula:8dd36e41-4dbf-4c2e-bc06-bb338150b3e8}} The second inequality is also easy because some computations show that
{{formula:e89fe6ad-dbff-4d0b-8500-4e2c95eb4197}}
and, on the other hand, {{formula:eb63ae2d-8f7e-412f-90ab-edc758e0dab9}} for all {{formula:76b89948-5f02-4555-ade8-a5d571afc52c}} due to {{formula:74eeea45-0c78-4cfa-8cd6-6fe54d295904}} for {{formula:aba234d9-008e-4804-85cb-99ae58889f73}} Finally, in order to show the third inequality we first note that
{{formula:ff703c8b-0e2a-4dd4-9be8-bc173969f289}}
Furthermore, due to {{formula:d3522d6e-0622-44e2-9f51-525c3624daba}} for all {{formula:3cce0079-dc25-438e-a174-97860cf25091}} and {{formula:ef07bfc7-987c-4b85-82eb-c48992f83cc5}} from {{formula:aee6e471-754c-4f1f-a41c-5fcd8638700f}} it turns out that
{{formula:ebe926f7-0464-4af3-b839-f5c8a87d5a14}}
Hence {{formula:46e5e3c3-7cfd-404d-9dfa-a34ef392088b}} is increasing on {{formula:71b52a4e-f17f-4aa4-badd-ffd7492f92ee}} for all {{formula:73e4f54b-0034-4af8-b9ff-6a8faf451aea}} and, consequently, {{formula:30eae1b1-9095-49fb-af90-2d8ba638f232}} for all {{formula:53507d78-e10a-4b45-8e73-b4d9082ce6c5}} . Thus if {{formula:eed588fd-e2c9-40ba-b248-eba043662692}} then
{{formula:d52afaa4-0b61-42a2-9aed-9acd5b0ccefd}}
Accordingly, {{formula:945dd8ff-4f70-4058-b0b4-206224b5b1bf}} for all {{formula:dc1cd229-9865-4f55-bc29-0e246a8a09be}} and this concludes the proof of the claim. We proceed now with the proof of {{formula:275f16d8-eec4-4f85-9533-7a23fc7b926e}} , which let us recall that it will follow once we prove that {{formula:d0fa96bf-b39a-4c6e-bb3e-fcb87427a8a3}} for all {{formula:2f98b6f7-66bb-4891-b545-245fc439f9e9}} With this aim in view we take the tangent lines to the graph of {{formula:7f80a7f8-bb0d-4036-b79e-44f436445fc3}} at {{formula:1d59bedb-d37e-4a9e-9b9d-c38bb14e64b0}} and {{formula:98175e07-348d-4a7e-9e7b-c1ae78698e8a}} that are given by
{{formula:454ee6b6-3005-48e9-9508-6a2c7344bab9}}
respectively, see Figure REF .
{{figure:e63057b1-06f7-4cf2-996a-dc01f6a9077b}}Since {{formula:d8057736-786b-4e5d-bd61-7ce10e385a4d}} is convex, in order to show that {{formula:a8277c8d-5258-4729-b329-95157fbfabad}} for all {{formula:4fed558b-7cc4-4836-a9af-d256a82b33ed}} it suffices to verify that {{formula:1cc33118-91f3-4abd-bdfd-f9bc928173a8}} for all {{formula:eec3de6d-a318-4391-a97d-5e250fa772e3}} To see this we consider the unique solution of {{formula:e69ef6c7-c8a0-4283-bd59-4cfc67dc1fe6}} , which one can check that it is given by
{{formula:1f02bedf-738e-4e98-b022-e0c72af0b093}}
One can also verify that, for {{formula:f3db8e0c-2b6a-42f7-a8da-cf89af478ea0}} {{formula:38ce29d6-1ce3-4738-9f3a-6f497d778bc3}} with
{{formula:0cc00387-c671-4681-9ffa-1cb69b0e7138}}
By applying Sturm's Theorem we can assert that {{formula:b86c8314-1ca2-4adc-aab8-f330dccfb3fe}} is positive on {{formula:2a9e9113-cd83-4976-bad9-630e480b3cfb}} and that {{formula:5ee26ac8-02db-42cb-bb79-b3564b2cfe20}} is positive on {{formula:0164fbd2-8566-425c-bbe2-deb458d2048f}} , which imply that {{formula:ba481209-8e5a-496c-8f9f-fe6c8c81a8ca}} for all {{formula:de017e03-8b27-4e27-97ae-9d5964cb4fc4}} as desired. This proves {{formula:9f5aeea6-d1ec-4b06-aef7-dbb213d6dff3}} .
Our last task is to prove the assertion in {{formula:de4a094d-ed8a-4b0e-96e1-a90c51ab8a41}} . To this end we use a symbolic manipulator in order to show that
{{formula:21fec345-9b33-4428-a24a-ce7df2ce2df7}}
where
{{formula:6225dcea-a6ba-4de7-8222-b836fcff78e9}}
Let us mention that in order to ease this computation and introduce {{formula:48e807ec-54aa-4096-826b-77bd33696cfd}} we use
that the coefficients of {{formula:6a12f7f8-e771-409c-a7b8-ff23aeb054fe}} , see {{formula:edb18121-eb54-49bf-a385-d93ea4caf6f9}}, are linear in {{formula:8d0e5349-4dc3-408c-819a-1da938faf8d9}} and {{formula:ab7af198-e6a1-4b50-b9b7-428160a6e7c6}} and that, on the other hand, the discriminant of a third degree polynomial is a homogeneous polynomial of degree 4 in its coefficients. On account of the above expression it is clear that {{formula:75162031-3cdf-4c2c-a555-853d21ff0f71}} will follow once we prove that {{formula:9606e3da-3210-40f1-a519-b0e327d04d14}} for all {{formula:937999f1-cabf-4afd-8e3c-bfa7a1758055}} To this end we note that {{formula:a8e2263c-de85-4f7f-8a8f-1d13478981de}} and {{formula:d1a4e69e-0360-48ed-89ad-d887267214ef}} Therefore, see Figure REF , the graph {{formula:40be85a1-2d92-43ff-b6f3-0f669fc91a89}} for {{formula:db596ed2-cfc0-4166-839e-58e5d097c760}} verifies {{formula:e94e43dd-0ee4-44d2-9747-f6ce572c070c}} because we previously proved that {{formula:657e14f1-937a-4dae-87e3-37508c7ec6be}} and {{formula:fe93ca92-0b0f-4cfb-9588-ee631442abbb}} on the interval {{formula:ff779ecc-e966-4db1-a717-6dc738424267}} . Accordingly it suffices to show that {{formula:6b18a288-1f16-47eb-a253-f20c2fe9e87d}} for all {{formula:8a717a72-1a2d-4c12-8789-89a6357603df}} inside the trapezium given by {{formula:e558cf2c-fe24-4056-b1de-63073f8e770e}} and {{formula:b166e1cd-db25-430a-8249-f8bd49788105}}
We will prove this taking {{formula:20cfaff0-ac92-46ca-ad3b-77ce5fec17e6}} as a fixed parameter and showing that the polynomial {{formula:8dc715ca-68ad-47b4-85b3-04133b455d46}} has not any root on {{formula:a0dde8ef-85fc-483d-969a-d77e0a59293d}} where {{formula:b11f4882-8843-4c4b-b0fb-9311b861e536}}
To this effect we show first that {{formula:02f6ea3e-bc39-4d30-a4a7-9c7d954e0484}} , {{formula:5c64b29a-705d-4b28-8985-a29b501d572e}} and {{formula:bb4658e8-779f-465e-a31f-384ec47b3df6}} do not vanish for all {{formula:2b7020a7-f1e6-4fb2-a93e-9b9d582375b2}} . This implies that the number of roots of {{formula:ba312fdf-8219-49aa-a086-888e9954ff34}} on {{formula:7d214c81-b8bb-4c65-9f5e-a9dbfbec348d}} does not change for {{formula:97ccd20e-37f5-476e-ae31-b68a2a56ce82}} . Taking this into account the desired result follows by checking, for instance, that this number is zero for {{formula:9d1158b3-5379-4fd0-9ddd-bb5da48df8c0}} . All these assertions can be checked systematically by applying Sturm's Theorem because only one variable polynomials are involved. This shows the validity of {{formula:1eea5cfa-9de8-490c-81b0-fc3d1ad95aa4}} and so the inequality in {{formula:121d65dd-1ec2-49c5-be08-222edc84af36}} is
true. This concludes the proof of the result.
| i | d9ba276f7c821fd2ca0baf7f4233195f |
Lipschitz assumption: In contrast to non-private heterogeneous FL analyses such as {{cite:4cc0e91203e1928a647bed9fe459adf2493a8947}}, {{cite:0a6b725e62378561ef39eff6ddefaf5c366590ec}}, {{cite:2705b346a16f11e93fcba46857b06761e25108f7}}, our excess risk bounds in thm: MB SGD excess risk (and throughout this paper) require {{formula:d31bc17a-1f30-4280-8317-c177cb6c587e}} -Lipschitzness of {{formula:3a76b073-1ade-4c50-bea2-e4c6be3ae852}} , which conceals the role of heterogeneity to some extent, since {{formula:43d67949-121e-4705-9532-ce80ccb1688a}} Lipschitzness is necessary in order to bound the sensitivity of the algorithm and hence ensure DP, and is a standard assumption in the DP literature {{cite:f1eafadd766e2734b3d1d22c5826b8a7f9ddf73f}}, {{cite:edab65d325d35c20b57c4e9dabbe9584a3993d44}}, {{cite:a1568da6f7d5e37b9379cd4966e9d3ac185ceff1}}, {{cite:833465c6a297f9aeb3ef01a24aef8b47e71a1c5e}}. Without the Lipschitz continuity assumption (e.g. if {{formula:dc3920f8-4367-4114-a845-d6decc016532}} is unknown or unbounded for an unconstrained problem), one can bound the sensitivity of noisy gradient algorithms by clipping the gradient at each step and then replacing {{formula:0dd6d2d1-ae0f-4aec-942e-8937d2719366}} by the clip threshold in {{formula:6aebdebe-b9d9-4e66-8f98-57786a885cdf}} to provide DP {{cite:6103c90592d0b492c3e444a03bb89918304c79ed}}. However, this complicates the excess risk analysis and would make our bounds less comparable to existing DP bounds, which assume Lipschitzness and scale with {{formula:8e5b595a-f6c4-4250-9e13-4a2456359bd8}} . Thus, we simply assume Lipschitzness in the theoretical portion of this paper, but in the numerical experiments, we incorporate gradient clipping for a linear regression problem. On the positive side, heterogeneity alone does not impede excess risk potential for LDP Federated ERM, as the final bounds in eq: convex ERM risk bound and eq: strongly convex ERM bound do not depend on {{formula:af4ccbe5-c7cc-4b40-b7dc-dfa37b672214}} or {{formula:48e08edf-02c3-4801-969d-87251d5cb108}} . On the other hand, if the data is relatively homogeneous so that {{formula:dd8e2501-997f-4a0e-aced-bdd609794f48}} or if {{formula:e68ed09f-d766-4a27-92a9-8202e777bb04}} clients are available each round, then {{formula:e5c233f5-f884-4012-8813-c3bacaa03b98}} and {{formula:8333a820-38ec-4e8c-bdda-62029dcedec5}} which improves communication complexity.
| d | 8a6ca256feb2ca4ee13a6f095581437a |
Many quantum algorithms for simulating quantum chemistry rely on second quantization. In particular, algorithms for the electronic structure problem using a second-quantized representation are widely studied as a near-term application of quantum computers {{cite:b0ea89da83e75ca34b22313eac52dec59d340b40}}. Work on this topic has adopted different representations including Gaussian orbitals {{cite:b0ea89da83e75ca34b22313eac52dec59d340b40}}, {{cite:9a1ec0adb67c436fa419f28a4acb65fb4d8066d9}}, {{cite:0d585db44147f1658b82667345c75d23c2aa28c1}}, {{cite:2561b4ee0a4d5e7db88c6e6cf77599f45bb4ff21}}, {{cite:4bd608d08a68b4567fd620ebd996597533c2af6f}}, {{cite:01c3f9697d33027ce7dfc683399fba0626302e41}}, {{cite:9a101cd96c822458c18d1e9273923e5b7aebdebe}}, {{cite:43c2e6592321dcd164e69323c23ccf33938872b4}}, {{cite:5a7c4de7f2b4761f28dcca890c82bc81552a1d17}}, {{cite:88b62bd43ac00048b2291240cf4b3238388e41cb}} and plane waves {{cite:f58cc0d0cf7a4354db8e022fc826b7fc1f2ac2c4}}, {{cite:6741d1c35dec7ad187f78c837e5298e8181edcca}}, {{cite:4526938ab620a38d4ebe5eacc5fa2e23d10cde53}}, {{cite:08cd8252b59ec83b685a4febdf474200575eab76}}, {{cite:63c3bd83aec9465fc57bef26bf17e3013d4368df}} in search of algorithms with lower resource requirements.
| i | 1e63e88399169df0aa7c7b4c3e10d87a |
We begin by providing the necessary notation used in the paper, followed by a summary of a few of the major non-linear models for scalar-on-function regression before describing our methodology. Suppose we have {{formula:c0856620-ef68-42bc-8ee9-e5b914773d06}} subjects observed over a compact time interval {{formula:576f8614-00a8-4f4a-9fa3-a8b33add90a2}} , and for the {{formula:968c30b7-a22c-4c25-9b5e-e89cde31c130}} subject, the inputs are {{formula:5dcebd65-b152-47a0-8465-afa42fcbc0d0}} random functions that can be denoted as {{{formula:4f6d121e-9c53-4d7d-be15-74152b42313d}} . The output is a scalar variable {{formula:83201422-ca09-4162-83e8-f9ba98ad3283}} for {{formula:6a95ef26-2784-46d8-858e-b997d99e46c0}} . We assume {{formula:ce8da456-1a17-465f-89fe-37c7154e243c}} for {{formula:99f82cf5-5110-4d6a-b371-2453d36a0c82}} are fully observed functions, which is also known as dense functional data analysis {{cite:5b188ce63db30ec805ade93e1b7350c96d732ff3}}. The main objective is to learn the non-linear mapping, {{formula:b724810f-8592-4542-a735-55b15d393e14}} from the functional inputs {{formula:437148b7-d120-4e70-96e9-cc42960cbfcc}} to the scalar output {{formula:bde852a3-32a9-4e65-ad8e-77b7deb7b0c2}}
{{formula:1bba89a2-474d-49a3-8d76-4e16cfc647ec}}
| m | 3ea2b4cd5158315df7386bf5e1552692 |
Moreover, we can observe different results between the grid-based and
the archive-based container variants considering the novelty
score. This difference is likely to originate from the fact that the
novelty score is computed differently in these two container types. Indeed, while in the archive-based container the novelty
score follows the formulation introduced by Lehman and Stanley {{cite:0a1657c6857a0bd86c521f706af2656be56f8577}}, in the grid-based container, the novelty
score is computed based on the number of individuals in the
neighboring cells (see section REF ). Both of these
expressions capture the density of solutions around the considered
individuals. However, in the grid based container, the novelty score
is discretized (because it is related to the number of neighboring
solutions). This discretization is likely to have a strong impact on
score-based selection variants using the novelty score because all
individuals in the center of the collection will have the same and
lowest novelty score (because of all neighboring cells being filled). In the
score-based selection, individuals with the lowest score have nearly no
chance of being selected, which makes the selection focus on the border
of the collection. This behavior is not observed with the
archive-based container because the novelty score is continuous and
the distribution of the solutions in the collection adds some
variability in the novelty score, which makes it impossible to have several
individuals with the lowest novelty score.
| r | 2ffe0ce339b84cc32c382acb27b1781d |
In Tables REF and REF , we compare MCAN{{formula:1505f7a9-0202-488e-9e32-58a3918b6345}} and UNITER{{formula:24128807-e163-4173-974e-9dc9fdbf535e}} to the state-of-the-art VQA methods on VQA-v2 and GQA, respectively. For MCAN{{formula:98f30398-866c-4298-bb85-b1e0ab6d8e7e}} , the compared methods include UpDn {{cite:04019ea46f4c3d9f8951b1c927d27ef4682a3c71}}, MFB {{cite:a48cc3e9b973d726219946e690e6d40e718e712f}}, MFH {{cite:0711daac611800b39c4943bda344c69e50233648}}, BAN {{cite:c3ed93d034474ffb54a7a34d51fb48db952d33b7}}, MUAN {{cite:0c022f6bb23f66e4dc1746efba9a52a6af9b2c9e}}, and MCAN {{cite:859f906e02b8e8a045fa43663f57c56a49ea29fc}}, which are the best performing solutions for the VQA Challenge in recent years. For UNITER{{formula:d405f174-414b-4b54-9289-48eb48c51479}} , the compared methods include ViLBERT {{cite:1062f6570c6eed7284af1afb50003a1f4480cdde}}, VLBERT {{cite:b8a26468481f2a59aa095b8a9c5a9da418dde551}}, LXMERT {{cite:0a71286d105fca10ba990cc7da4946df00efe04f}}, OSCAR {{cite:f5c36fb37d70048801d512ac2ec0b46bc10518dc}}, and UNITER {{cite:dc1eddc235f6095cc404f62964cf66cd56998062}}, which are the representative vision-language pretraining methods. Due to space limitations, we do not show the results of all the submodels (i.e., the selected ten submodels by the triangle selection strategy in Fig. REF ) of MCAN{{formula:549a202c-6786-4d51-84e0-46505c8388c4}} and UNITER{{formula:93956391-537f-4181-b9d2-1290f67783db}} in these tables. Instead, four typical submodels are selected to compare with the state-of-the-art approaches.
{{table:b7408c95-dd19-4c4b-a464-adf763fa1936}} | r | 3801325c8f6c6acc612ac63fb3da6123 |
The spin-polarized DFT calculations were performed using the projector augmented wave method implemented in the Vienna ab initio simulation package (VASP) {{cite:cedd9f1466dd536f7a691fdc0e0c8c820994a2dc}}, {{cite:8cbbe75fc4242bf10ec69a6909ecf10e96277045}}. For the exchange-correlation potential, generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) {{cite:4b32aa6994ff3000efb2a9f6f4d83b4da10ad359}} was employed. The van der Waals correction was included by using the DFT-D2 method developed by Grimme {{cite:6ae0a1a34ef92a39e4b1ba1604d2712445c5be5f}}. The kinetic-energy cutoff was set to 400 eV. The Si(100)-2{{formula:185722db-d178-4492-ab44-90fe2031546f}} 1 surface was modeled by a sixteen layer slab with a 6{{formula:838bcad3-4617-483d-9aeb-202b7ff429eb}} 6 supercell. A vacuum spacing of 15 Å thickness was used to avoid interactions between the neighboring surfaces.
Calculations of the electronic structure were carried out for DBs on chlorinated, brominated, and hydrogenated surfaces. The adsorbate atoms were placed on the upper surface to form a Si(100)-2{{formula:19f84df8-861f-4a6b-9b99-687fdb030edd}} 1-Cl, -Br, and -H structures, whereas the bottom Si surface was saturated with hydrogens. The bottom two Si layers were kept in their bulk positions, while the coordinates of other atoms are fully relaxed until the residual forces on each atom are equal to or less than 0.01 eV/ Å. To simulate positive or negative charge states, the electron was removed or added to the supercell, respectively. The Brillouin zone was sampled by a 4{{formula:241f1722-bcb0-425b-b392-a6712fa879f6}} 4{{formula:ca576d1b-00ab-4981-b58b-cbb4962f5f11}} 1 k-point grid. For the electronic density of states (DOS) calculations, the reciprocal cell was integrated using a 8{{formula:0775b026-25b0-43dd-8365-da5b489c37e7}} 8{{formula:fb5aca59-92c8-4ead-ace4-7226ad678c8e}} 1 k-point grid. STM images were calculated within the Tersoff-Hamann approximation {{cite:ab7c0a84495141f3d65f02b8a6b795f37dcfdeee}}. The voltage in simulated STM images is indicated relative to the valence-band maximum (VBM).
| m | 5808b35637f77f585e634e347751ae5f |
Recently, the application of deep learning (DL) to physical layer problems shows promising results mainly when there is a lack of appropriate mathematical models, i.e., model deficit, or a lack of low complexity algorithms, i.e., algorithm deficit {{cite:b66d6abdee880d5bf9536017fca252b57e2ab114}}. Given the fast development of artificial intelligence chips, it is expected that DL will find more applications in physical layers problems.
| i | f1e5e1badd0ab197065da80acdd39b72 |
We follow the same implementation details in Section in the main paper. We test the proposed algorithm on the dataset CIFAR10 {{cite:79316d4b814fc1f067a0eac7c641f58a5be8131e}} and the VGG19 network {{cite:cd0ff357af1f4df7697f3d1ec9d5fdb7263d6d08}}. The results are plotted in Fig REF . From the results, it can be shown that the error-feedback technique helps accelerate the convergence of our algorithm with gradient quantization. However, in this experiment the networks with weight quantization perform better than those without weight quantization. This may be because we overcome the over-fitting issue by considering weight quantization as a regularization term.
{{figure:b11af8d1-cbe5-43f6-9c7f-4c0adfa1c2c8}} | r | 64c0d939edca41c4440a1efc9ea529ff |
We tokenize the captions and utilize Bert {{cite:83c6cf587e0468378c181ac0229fc922a9a2e182}} to extract its word embeddings. Then the word embeddings are fed into the LSTM network. As a network transformation, a linear layer is connected to the LSTM layer. The output of the network is {{formula:b21bafca-cc80-4db8-a171-4366bde9f6f6}} where the parameters of the network are represented as {{formula:4de4d120-971e-45b3-b349-f0efb38248b9}} . The caption embedding vector is defined as
{{formula:3e8c81a0-fadd-4065-b549-db74e228bd93}}
| m | b4ed82785ad8b36be81fc4be6f287342 |
When {{formula:386106e9-98e3-4b56-b8c7-db3ffca0d362}} , Liouville type theorems on {{formula:36a247fc-4e96-49ec-962d-f59908d35b54}}
for the {{formula:a910dc2b-47f5-4ebc-abe3-058f95bddc99}} -Yamabe equations have been established. A powerful
analytic device is the moving plane/sphere method. Let {{formula:a9ee9bd2-bed4-4241-aeb9-4e970f6abe26}}
be a solution to (REF ). For {{formula:8561da1d-7f6a-4252-830a-17f48671c672}} , it is proved
in the celebrated paper {{cite:a29273d2b3a3e00312146e9ca9dc56df96b9c4f9}} by Caffarelli-Gidas-Spruck that
{{formula:0d0f9dc7-8dcb-4e45-b6b3-04cc33be94c6}} is rotationally symmetric. See also Li-Zhang {{cite:279b5ca0729c90f31654751d5afe7ed1f1224885}}
for an alternative proof. Under the assumption that {{formula:2a07415f-9911-4618-86b7-6a295a7f3ef0}} ,
Viaclovsky in {{cite:686330a3ff306e203a0e2788d5e39202961640b2}} proved the Liouville theorem for {{formula:7a56bd81-48b0-400e-ac9f-b5c32798fd49}} .
Li-Li in {{cite:e13112a3401b56663c8971d25d66c13581e9eaab}}, {{cite:ea9157f798fdd6590e41f1ea18acb88bd085f5b2}} demonstrated the Liouville type theorems
for a class of conformally invariant equations including {{formula:39ed35cc-4e5d-4c3a-ab2e-93ba82d820e7}} -Yamabe
equation (REF ).
| i | fb91f638c6cf4d1078884ebc805f917e |
Stability conditions on triangulated categories were introduced by Bridgeland in {{cite:7e30a904daeabc0ab12dbb66f72a70fc465dc41b}}.
A remarkable feature of Bridgeland's construction is that the set {{formula:f35146ba-6d54-4048-9c05-a5e759ba2f38}} of all stability conditions admits a natural topology, and is in fact endowed with the structure of a complex manifold {{cite:7e30a904daeabc0ab12dbb66f72a70fc465dc41b}}. Even when {{formula:5d6b8441-1953-4d0f-a130-85a00efd181f}} is the bounded derived category of a smooth projective variety {{formula:8ca58155-7c80-4bc5-9eb3-c867b0fa003c}} , it is challenging to study {{formula:74bfdf60-e8aa-4ed2-97b1-9e668c14d31b}} , and questions about its non-emptiness, connectedness or simple-connectedness are difficult problems. The stability manifold is completely understood only in the case of curves (see {{cite:c7f1e7d5dbbb59856da3a2ac89a6f89286146338}}, {{cite:cbdefec5c8df9c73b963c27b62e8ed0f53f48497}} for {{formula:d4a2759a-5d28-4b74-bb8a-f95acc6f7a00}} , {{cite:7e30a904daeabc0ab12dbb66f72a70fc465dc41b}} for elliptic curves and {{cite:121b3837fa6e469aa99c3c56a99c24a974cc86ea}} for the other cases). A connected component is described in {{cite:8f5b0b2b8738ed0701e18b4b31e73cf8212ac878}} for K3 and abelian surfaces (and in {{cite:c0fca51655ed0031010ebec8e8302807d17ed9d8}} for their twisted counterparts. See also {{cite:afa5515a14b7693f49f82b9a9dadfd3f8c2d6c0f}} for a more detailed description of K3 surfaces of Picard rank 1). A general construction - recently extended to the positive characteristic case {{cite:ef86920c3b4881fdd240d2552d23e425d08215b1}} - of stability conditions on surfaces is given in {{cite:2a7e680ca172f3e0cc13b4ce54bc2a53d0eef713}}, and alternative constructions in the presence of curves of negative self-intersection appear in {{cite:4741bd6ad0ac46fb388e29876a39d27e948ff10d}}, {{cite:86021493c4cc7e039d4e09ce1a9601d0e5819508}} (these can be interpreted as stability conditions on orbifold surfaces as in {{cite:2978f0e9f2d79f4f26f72a7f91473c8f52c164ff}}).
In dimension three, in a series of remarkable papers, stability conditions have been constructed on Fano threefolds (see for example {{cite:1f0c3beee376ed6979316fad80e023808f9222af}}, {{cite:1af7735a744b9d92ef9693f011d355ca5dcf0468}} for {{formula:9eb82a21-c920-4539-8c4d-9596893972f6}} , {{cite:08e2702460650f91a93d7368918ffa8b5174eede}} for Picard rank 1, and {{cite:ecf30a4b8e478b447d8b6145a52a5f2ce5705e96}}, {{cite:40c6151b6e1e2334a95fd96eaacdf9bfd3268902}} for the general case), abelian threefolds {{cite:c2ad3bf1ffb0cf7586537e24f8d006726a5bdaca}}, {{cite:a117274121661f7d0e277932f212dc0913569765}}, {{cite:4d681ca2d27a58d242661b4fdbde3c0dc8ad3575}}, some resolutions of finite quotients thereof {{cite:c2ad3bf1ffb0cf7586537e24f8d006726a5bdaca}}, quintic threefolds {{cite:10758184ad6777d0b64af1e79a922d25c40a8627}}.
| i | f74dad9bcb36f5690e1d505e2f4a839a |
There exist other extended statistic besides Tsallis proposal.
For example, a largely used statistics is that introduced by
Kaniadakis, which comes from relativistic
corrections to the Boltzmann theory {{cite:4d9a7fb8a5410791e2c49e720d270017be00efc2}}.
It would be interesting to explore how modified Friedmann equations
affect the mass density/pressure content of the Universe
in that context {{cite:77d72438bc070741a2c30f40d70068f1465c3cfa}}. Along this line, preliminary considerations
could be derived by making use of the relation between
Tsallis and Kaniadakis entropic measures {{cite:003614b46476fc7c7b55e7e2332d3231e313d5e5}}.
Work along these directions is in progress
and will be faced elsewhere.
| d | f2a96ba5d7dd73f142b2833cef5a6ad0 |
A recently popular approach to deep representation learning has been to learn disentangled features. Whilst not rigorously defined, the general methodology has been to use deep generative models such as VAEs {{cite:8adcd0d8ba30c6c42f457f98c7eac857545fd2b2}}, {{cite:ffdf8707c5198236172af2aa06806789ac458f56}} to estimate semantically distinct factors of variation that generate and encode the data. A substantial problem with the vast majority of work on disentanglement learning is that the models used are not identifiable – that is, they do not learn the true generative features, even in the limit of infinite data – in fact, this task has been proven impossible without inductive biases on the generative model {{cite:be770edf8e3b7511d6256f107af1e69fae4b9b71}}, {{cite:851670702c61611a30124e45e4ded2707aee288b}}. Lack of identifiability plagues deep learning models broadly and has been implicated as one of the reasons for unexpectedly poor behaviour when these models are deployed in real world applications {{cite:9c6cc8f94111c93903a8583797f0d759324a1fb3}}. Fortunately, in many applications the data have dependency structures, such as temporal dependencies which introduce inductive biases. Recent advances in both identifiability theory and practical algorithms for nonlinear ICA {{cite:df22ab194d0c26ab57ed5f1e01373cd770ee1615}}, {{cite:dc5d6d273985b610656e8f9e242adee0f3ce87e4}}, {{cite:09c04bdbf45769613bcff4db9882f618bd3e38e5}}, {{cite:f6f087312c4455373b40e2d9473bb924f41641a0}}, {{cite:f6734bb31bb9ceffffb620572e6fc051f1395059}}, {{cite:bb05ab51c55d025617bf6cc674525532ac35f709}} exploit this and offer a principled approach to disentanglement for such data. Learning statistically independent nonlinear features in such models is well-defined, i.e. those models are identifiable.
| i | e5d0f7a8979ca26526e911de8403bba0 |
Directly involving users in the collection of a dataset intended to drive ML research does present some challenges.
It can be difficult to reach the scale of web-scraped datasets {{cite:7827a5b70e0a0adedddff80e1793eae832fffb19}}, {{cite:dd7d3d89de032562df470133a333244fdf7ab012}}, {{cite:c681470bb6b6fe478be1e08053a83720e78cdb1b}} and users need an understanding of the potential system to contribute useful data.
Building the system first would address this challenge, but it cannot be done without innovation of the algorithms.
The ORBIT dataset is a starting point. It can be used to build the first generation of TOR which can be deployed and themselves then be used to collect more real-world data to develop a circle of innovation between dataset and application.
| d | a11882c1d184fc929f5e31db0aa6b25e |
The size parameters {{formula:1831f2ce-d021-4340-8589-878026704b7c}} are set to be a geometric progression form {{cite:93ea06d07569308b87260394de575ddcad7a921d}}
{{formula:64daf68b-b720-4061-8642-cbf6041bc682}}
| m | e73449186ccfae4c398a463f5ca55692 |
Data engineering pipelines are used pervasively in both industry and academia, and dataframe serves as a key component in the practice.
As the scale of data increases, distributed runtime libraries, such as Dask {{cite:4ff320556c0d5dafd2c0c335387c826840c36efa}} and Ray {{cite:9c88f5032610ac579c66deef09d7e1512b56c5df}}, emerged and were widely adopted, and they significantly alleviated the complexity of working in large clusters.
On top of these distributed runtime libraries, distributed dataframe (DDF) solutions like Dask DDF, Ray Datasets, and Modin are proposed.
Cylon{{cite:c921b3fa793d2d8139e68dad1b73410361ee57af}} is also a distributed dataframe system but outperforms its competitors in many scenarios, especially for High-Performance Computing (HPC).
| i | 57972c3f7d9facc0ff34f7236b56979e |
The origin of tiny neutrino mass and the existence of particle dark matter (DM) are the two concrete pieces of evidence for new physics beyond the standard model. Appealing pathways that connected these two parts together have been extensively studied in
Refs. {{cite:66e4c5fb844e85224596c5c7aa327bbc4436f3bc}}, {{cite:b4677f00218454fd2d795e4b8e4956b35f16adc2}}, {{cite:90a763b32f4e976985cd37eaef61f1a5c5314417}}, {{cite:81a314e80e361ce2128380fc8f47f07a18decd31}}, {{cite:30bac4dd54dcffc2d45d8c68296e59385406acab}}. The most economical way to explain the tiny neutrino mass is by introducing sterile neutrino {{formula:c9a2e294-ddb2-4a57-aa71-d614af600be6}} {{cite:a451a585a7671e51440cc26e0683190d40ce8b7d}}, {{cite:c4006eda212a06fabcf72567a167c1795e69445c}}. Although a quite high scale {{formula:f990cf41-4a06-41eb-9bb7-314691d597ac}} ({{formula:e0062ef4-d932-4961-bdf7-911877c76b80}} GeV) is required by type-I seesaw and leptogenesis {{cite:150ff038a45a3fe361f13bd03b43577079029e79}}, light sterile neutrino in the range of eV to TeV scale is also well studied {{cite:2de00cb49e5a1318ef9a8a2a083ed48d91c09154}}, {{cite:2667f49c4109f117f3fda47c287af8554f2564ea}}.
| i | db9e2768fac7b9e0cc364daf62ad8772 |
Other approaches decompose the primary task of the GAN into separate, domain-specific tasks performed sequentially {{cite:cf7f2f04a8b61f3a2ca7f2715f9855b262bd8b62}}, {{cite:cdcf40178a1f1c4df5e77fa3a6fb381695d21bab}}, {{cite:4b3ec02e7e3090360517ae2f88a09f03763c482e}}. All of these have focused on image domains.
| d | 42c7ad48101474f8d62456197643d3f8 |
LSR and Flooding do not enjoy the implicit bias property because their optima are finitely located. By themselves, they have to mainly rely on the noise associated with minibatch stochasticity for reaching points of good generalization.It is worth noting that Szegedy et al {{cite:a883ffe2ccf8f9042d9a81b40e7d8a7f194c9fbf}}, which introduced the idea of LSR, also uses {{formula:e7f2a561-8ce4-4192-b09d-6041d0671076}} regularization for training. On the other hand, {{formula:ff0458e1-d7b7-4b5c-9f92-e3fc54832a4f}} regularization, like LAWN, also approximates implicit bias - the former uses the weight norm in the objective function and the latter uses it in the constraints. See Appendix for details. Having said that, (REF ) used by LAWN could be better because of the following reasons. (a) It does not interfere with the formation of the main classification boundary in the early phase of free training. (b) The {{formula:ab4f0787-b56d-411a-b52e-6d9754755802}} values are set naturally and differently for different layers of the network and so they do not require tuning. On the other hand, the {{formula:4a35893f-fcdb-4352-8e5e-9c4e5a5277d4}} regularization constant, {{formula:25579e0e-e695-4319-9592-6d35b5b9a8c5}} is a single value for all the layers and it needs to be chosen to avoid the extremes of loss flattening (which happens if {{formula:d3cbf780-1a9e-49de-836f-cd400eb28fc9}} is set too low) and too much interference with loss minimization (which happens if {{formula:2bee16a3-b38d-4ea1-a9c3-155890a7f1c6}} is too big). In general, {{formula:e9e59465-506f-4e39-a47a-e9a19f7e5393}} has to be carefully tuned, say using a logarithmically spaced set of values, to get the best performance {{cite:0b471258a071d5b294655a23f5d4a13bd7f97137}}, {{cite:51c6c8dfec54961c9855a5a7b0d807477a6247bd}}.
{{table:a52627b8-7be7-4799-854a-70d492d3b9c8}} | m | 146e20353a4ef7ce200f27494ee301af |
where {{formula:31acca26-167e-4196-9deb-8a47f306e07d}} is the Q-value for the trajectory {{formula:1ab75f0b-27a5-449f-ab76-bf96e64e779a}} , and policy {{formula:1a6c7f5e-d165-4ca5-9b09-024bce39ad44}} . Throughout the training, the PG agent estimates the probability of selecting each action and chooses actions by random depending on the probability distribution. Before it learns from experience and updates the policy parameters, the PG agent performs a full training episode utilizing the existing policy. The disadvantage of PG method is that the network policy is updated after the completion of the episode only. This slackens the convergence rate {{cite:55f841f286d6d7c20c1b9ce11d32ab8250279597}}.
| m | 6bccc77f7288fd9ddba6d954cc224cdf |
Lemma 16 (Hoeffding inequality {{cite:3d8526122ecdf407118a0e7752d22a47755d37f7}})
Let {{formula:e1a5667e-9c73-416a-b7ad-33b882d7b51c}} be independent random variables such that {{formula:3c86d2c1-2e2c-4000-9492-df8ad9d1f204}} with probability 1 for all {{formula:ce099cad-13f0-4be0-be9e-821733877346}} . Let {{formula:9c67c7f6-9cc6-419c-89e2-e8a7103fa2de}} . Then for any {{formula:b0f6e8fa-5ed1-4669-9c05-4c5ea134ec82}} , with probability at least {{formula:c96b63a8-1e4e-4cad-983e-e278f634de47}} , there holds {{formula:14c59eeb-4136-4e43-a221-fd264de25e22}}
| r | baf8f12df108745b996f51ddca62f087 |
In this section, we first briefly revisit DARTS {{cite:98f7c5d4a5c52e674e9e3e7dec3d0afecee3b726}}, and then describe details of the proposed methods including three key components.
| m | 0dfbe543f844263608f37974f943e251 |
In this Section, we perform the comparative assessment of our approach. To this end, we use PHOENIX 2014T dataset {{cite:93ef96822f9768074993dc714cf40490aa7037b1}}; this constitutes the most used benchmark in the recent literature. Hence, this benchmark selection allows for optimal and full comparability of our results with the recent related work in the field. The used dataset contains German SL videos of weather forecasts, and corresponding translations into the German spoken language. They are obtained from 9 different speakers.
| r | e9ec34a519f00309e54f938208b414e5 |
Regrettably, minimizing bias is not
the only key aspect to be considered: maximal matching can lead to the pruning of patients, thus possibly ignoring relevant information contained in the dataset. As these matchings are usually not unique, there can be a high variance in information in between matchings and thus conclusions drawn from them
can potentially vary to a high degree.
Hence one needs to find a matching optimized in regards to bias and pruning.
As the underlying distribution of the dataset and the influence of the therapy is
unknown, finding the optimal maximal patient-to-patient matching is difficult.
Based on the foundation
laid by {{cite:dc1a09776dc20a8452666efe602d464d378e843b}} for propensity score matching (PSM), many
different methods have been proposed to deal with this problem, e.g. nearest neighbour matching {{cite:a79a35260a312db442963e89d793b32ba843019d}},
stratification matching on propensity scores {{cite:2e3fec81bae0290cbaddf6e44a6718c12c31e20a}},
caliper matching {{cite:e0955551976b0be19370639f1eeabce88528995a}}, optimal matching {{cite:389d1bd2dbfb277f73104f48aa7665bc10cf0e2e}}, coarsened matching {{cite:a7627a9e61624cdbe9e1b4504636270d6bc27ec5}} or full matching {{cite:9016ede60e39989ea8328af3e747cc88077b7ae0}}, {{cite:135fb9bc1f965f8350c94fcbf4a3750d2e57271f}}.
A comprehensive overview can be found for example in the article from {{cite:e0955551976b0be19370639f1eeabce88528995a}}.
| i | b35a7abf92dd83c68d84c88a3f8f0028 |
In this work, we employ adversarial domain adaptation to extract domain invariant features from the source and target domains. The architecture of a DANN {{cite:6d904f920d206eef7109b588dfb9a7528e71702d}} is shown in Figure REF .
The architecture consists of three networks: feature extractor to extract features from the source and target domains, label classifier to predict class labels, and domain classifier to predict domain labels.
The domain classifier network includes a gradient reversal layer to make the distributions of the source and target features similar. Specifically, for the samples that are correctly classified by the domain classifier, a penalty is applied through multiplying their gradient by a negative factor during back propagation {{cite:0f2c27ea466a62f48f877305219900e1c0dff78d}}, {{cite:6d904f920d206eef7109b588dfb9a7528e71702d}}.
{{figure:7a2f194c-fb0a-4253-b00c-9132eeca0e1a}} | m | bb6467cce31ea3912a190744627307a7 |
We prefer to interpret the massive neutral gauge boson as
the massive photon. This makes the non-Abelian superconductor
more like ordinary superconductors. Moreover, in this case
we can interpret the ferromagnetism in ferromagnetic
superconductors as an evidence of the long range interaction
generated by the massless magnon {{formula:dadcc10e-ff88-4e66-8934-a74026941c2a}} {{cite:f95c18e800b148862205a951913e05ce0a2a2e2e}}, {{cite:e5db0248702802dc7156fcb69d3e659b269f9d26}}.
We hope that the long range magnon interaction could clarify
and explain the connection between ferromagnetism and superconductivity.
| d | 134a6ccc9dbc77c81a838265158b96b6 |
Semiclassical transport theory based on the Boltzmann equation predicts Kohler's rule {{formula:c8a9f9e4-b205-4d4e-942b-64a720c7a025}} (0) = {{formula:2dba1808-2302-44f0-ab35-db18f6ab245c}} to hold if there is a single type of charge carrier and scattering time in a metal.{{cite:1914aa167abea5ce6aa6d80a4f6c2b1c572de7c9}} Violation of Kohler's rule is common in XMR materials, such as LaBi, TaAs, TaAs{{formula:0a3b7d87-1a7e-40e6-8231-224beaf95da6}} , NbAs{{formula:80b380a1-f10c-4b9e-88af-6f3747c96f6f}} , NbSb{{formula:3d1fa59c-5cdf-434f-8f3d-b0bb9dd16783}} .{{cite:54980296d3ab9375fc9b193e5a4ba035f51eb0cb}}, {{cite:d87418d8ccae46300d0713ab1a3db49eab5ab776}}, {{cite:79e48929b5353bee098971c20277ddcf77949efe}}, {{cite:cd647a57c5c193538105c029e0ef7510115bf5da}} As shown in Fig.2(c), MR in WP{{formula:887536b6-9017-4ada-8b55-24236df82fb3}} systematically deviates from Kohler's rule above 10 K. The field dependence of MR at different temperature can be well fitted with MR = {{formula:1037547a-93c6-4445-81e1-328788a51580}} {{formula:57823931-e2df-48c6-a0ff-20a90ed54585}} {{formula:3786566a-0287-418e-b61f-af42a38e8cd9}} , {{formula:eeacb565-d005-4405-a430-4c4ca53025b7}} is 1.8 at 2 K. The exponent {{formula:4b6c9ab1-5539-40b3-8eac-3c2ef417a9b9}} systematically decreases to 1.5 with increasing temperature, as shown in Fig.2(d). The temperature evolutions of {{formula:49ca719c-26fe-4193-9ef5-322305d5b1bc}} and {{formula:35aef64c-b112-43f4-b44c-7b0810dda190}} are similar to temperature dependence of MR(9T).
| r | 0599996f20221fd235ebbcfc727a35af |
We suspect that the classical radius is primarily due to screening of the charge by virtual particles from the quantum vacuum. If it were possible to see near Planck length via scattering of some kind after breaking through the screening effect, one would see the radius near Planck length. The screening effect of interaction with the quantum vacuum “spreads" out the apparent charge of an electron so that one might think there is a “bare" charge if the screening was not there {{cite:fbd829718ae8a4e437e0c0c9d0c17e84353d7606}}. This is suggested by Schrodinger's conception of Zitterbewegung. Accordingly, the point-like entity of the electron near Planck length vibrates around in the quantum vacuum interacting with virtual electrons and positrons with an amplitude of about the classical radius.
| d | 85e1056a0a40a64a444975c4221e7a07 |
Following the classical framework, described in {{cite:eb53a92f9028c222339a326cde7ae3720a447282}}, for each initial data {{formula:d3af7d5f-9a0b-411e-83c8-6821c26e3db3}} , we consider the MFG system in {{formula:ad25cf35-f811-4082-a558-3e5f16cfc545}} with a homogeneous Dirichlet conditions:
{{formula:f5cd8f00-411e-4980-a472-cd2246ba5951}}
| r | cfe19fdea8319b6b6802ac0f4bfe9679 |
Among the gluon TMD PDFs, the so-called gluon Sivers function is regarded as one of the “golden measurements" at the future EIC {{cite:bd05b8fa76b9c30b078cb8fc9b414d0f145a9603}}. The gluon Sivers function encapsulates the quantum correlation between the gluon's transverse momentum inside the proton and the spin of the proton, thus providing 3D imaging of the gluon's motion. Quite a few processes have been proposed to probe the gluon Sivers function at the EIC, including heavy quark pair production {{cite:428c8895c07c0116754b018fe7324ea3d5be7eed}}, heavy quarkonium production {{cite:d7e9f4a8ae64497c16ad9eddf50baa73ab6dac6c}}, {{cite:898744c230d1ecb458243f5546ce2fe6085cfc80}}, {{cite:baf488d25e2b106aaf1c5b068b4c6fb4e01c39b6}}, {{cite:973ef02eac8b9fe1feaa418ac18683796dd05fdf}}, {{cite:c4a15f305453a98557d602df035ef731a52c4db3}}, and quarkonium-jet production {{cite:e4d82a15384f0fc276b27bc84a693a1a958b44c6}}, as well as back-to-back dihadron and dijet production {{cite:b78258c35c762921578370626f196d5929261b75}}. The feasibility of measuring the gluon Sivers function in the above scenarios has been studied in {{cite:1ef9e3eb7e3062e1ee1cbb600e4cdc302af20dbc}}, where the authors use the PYTHIA event generator {{cite:0ba65400447d89521551510fb940ea5835dc8409}} and the reweighting method of {{cite:b6b3e558f589038be948442d3a9ad4db7deb9544}} to investigate the spin asymmetry. They conclude that dijet production is the most promising channel for probing gluon Sivers effects, where the selection of a sufficiently small-{{formula:f9d4a697-0777-44c3-89dd-451d3d91ece3}} value suppresses the contribution of the quark channel and the corresponding quark Sivers function. In this paper, we discuss spin asymmetry in the process of heavy flavor (HF) dijet production, where the contribution of the quark Sivers function is further suppressed compared to that of the light flavor dijet case.
| i | 37a04ecaccf4a3660383d70f40c6fbf8 |
is an accurate local model of {{formula:8ad5517f-73fb-4095-adae-2fed6894d795}} for suitably small steps. Note that {{formula:fb3ba4a4-0ffa-48fe-b65d-278b36753974}} is not the usual, quadratic model of {{formula:1e09c6b0-5661-4cf1-8f04-4489c7dc95cb}} derived from a Taylor series because {{formula:3680c6cc-44b2-4544-a2f4-7d6fd1967bf2}} {{cite:97f580a0dd55b1a899ffbe5866ac7b116d9374be}}. The idea is to solve
{{formula:7f17c8ec-c980-4152-8f98-90423711af63}}
| m | 74d98ba4cf3f9d2435554b54228d161e |
This principle has now been made quantitatively precise by a
bound on the extent to which the two natures could be simultaneously observed
{{cite:fb1493a582a55b24763dac1378e6c4f1aad7a433}}, {{cite:c3a1cc5133bf7dfd6eab989ba5bcbd7865ca4ca6}}. The extent to which one
can distinguish which of the two slits a particle passes through, is given by
a quantity {{formula:7596cbf5-a339-4aa5-8019-31a7f6f34add}} , called the path-distinguishability, quantifying
the particle nature. The wave-nature is quantified by the visibility of the
interference pattern, given by
{{formula:b4700f42-3881-47ff-931b-3756bd583b02}} . The quantities {{formula:98d22dcf-ae4a-4a20-ac1a-f0c5a053b881}} and {{formula:939360af-e15c-46bb-bda4-f7fba6624ebe}} are
so defined that they can take values only between 0 and 1. The relation
putting a bound on the two is given by the so-called
duality relation{{cite:c3a1cc5133bf7dfd6eab989ba5bcbd7865ca4ca6}}
{{formula:0ec90099-f0b0-4755-adef-f335b5925a87}}
| i | 66cfa9896fd3453a6cc37ee2f9e47dd0 |
We also experimented with a version of VDTN in which the transformer network (Section REF ) was initialized from a GPT2-base model {{cite:2db53ca1e2691747ef93db01994eea254f15f4d2}} with a pretrained checkpoint released by HuggingFacehttps://huggingface.co/gpt2.
Aside from using BPE to encode text sequences to match GPT2 embedding indices, we keep other components of the model the same.
VDTN+GPT2 is about {{formula:f182e438-c6dc-4bab-b0e0-daf3dc53937f}} bigger than our default VDTN model.
As shown in Table REF , the performance gains of VDTN+GPT2 indicates the benefits of large-scale language models (LMs).
Another benefit of using pretrained GPT2 is faster training time as we observed the VDTN+GPT2 converged much earlier than training it from scratch.
From these observations, we are excited to see more future extension of SOTA unimodal DST models {{cite:33872df1efaffa23cd1a14f6a6972953a326ef5e}}, {{cite:afffaec3c9964a58316dc40cf5d9fa2946c50460}} and large pretrained LMs {{cite:aec9e97664e2925f2c2ed8d4c236c1c763725135}}, {{cite:c94981565836641ddb71c3604ebc3a0a8839eb99}}, especially ones with multimodal learning such as {{cite:e79156796fa23561d0fb23745664fad24b3f282e}}, {{cite:c8db22a629a1d63235fac0c0a5ea37d067db5e13}}, to MM-DST task.
{{table:bacd157f-5f7f-4c1c-9572-c95b01101d0d}} | r | 7938136ac8913c999e4b139899657b91 |
While the first equality in the first formula of part (a) is a consequence of {{cite:e183518dddb6cd27f2eb0494c18b9b7fbbe1f302}}, the second equality follows from Euler's reflection formula {{formula:d7a55ad8-741e-435a-9514-d5e3c2f3fd04}} which holds true for any noninteger {{formula:9825265c-43a5-4a35-ada4-14d8905c9d50}} . The second equality of part (a) is implied by the first with {{formula:5eea19d7-25ab-4766-b8aa-dbe3351ba424}} and the formula {{formula:227a2dd3-db47-405c-ba86-f6213d95ac60}} . Part (b) follows from {{cite:00d37f7206bbf0acc3dfd96a99f1e401697ea78b}}.
| r | 5b15eed7a87b42bdfaadecd3e13538d1 |
As discussed in the Introduction, a variety of UV physics scenarios may give rise to unwanted defects or relics like monopoles, moduli, gravitino (see e.g. {{cite:307276311182a6484e56775e3fcc6d0096497064}}, {{cite:cc010bd7ca6fb685d87e2c0a27a4cb0199b02fc9}}, {{cite:a8ed84fefdd21a9df7b82d805ae72accc1b24b40}}, {{cite:eddd1c93af0c165e1332e5bc4e037bdb69887187}}). Different UV scenarios can also exhibit a meta-stable high temperature phase in which the universe can remain stuck if the phase transition to the familiar low temperature phase fails to complete {{cite:d38729b844b00bbe138265cdb138269844e42519}}. Reheating of the universe at a low temperature, following inflation with a low Hubble scale, might help to address these issues in a straightforward way.
Another motivation towards low-scale inflation can come from the constraints on isocurvature perturbations sourced by (QCD) axionic dark matter (see e.g. {{cite:628224ca7ea31c06eb16d6cd630b09a372aece14}}, {{cite:d4232dadde8cddd28c0d6c24c85e4ec51679c485}}, {{cite:999b07b3955b10f66d789629fba204918dd3e7e9}}). If the Peccei-Quinn symmetry is broken during inflation, axions source dark matter isocurvature perturbations which are stronger for higher {{formula:35ffad0a-dcec-4cec-b335-f7019ed0eb0c}} (for any given axion decay constant, {{formula:b30022c1-4cfc-48e7-9df3-955bae383ebd}} ), the non-observation of which thus prefers low-scale inflation.
Furthermore, with current and future collider experiments, such as a future
{{formula:e133f16e-8690-40b5-b123-d58075f4176c}} TeV collider, we might have the opportunity to investigate the physics during and after such a low-scale inflation in laboratory searches too, along with the cosmological ones!
| d | 48c7683ab42945511f75e9d3ecf90b4d |
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