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Fig. 2 shows the angularly averaged spectrum of turbulence {{formula:6c10cc81-dec2-4499-bf0d-bd4214792592}} at different instants of time. At times {{formula:13117406-af27-4fe4-8a79-4062c4f42105}} one can see
the steady (compare with Fig. 1) spectrum with the Kraichnan behavior {{formula:aa49e77e-4ca6-42e1-b8f4-8d017159ea75}} which is seen almost on
three decades. At the steady state
({{formula:c3020a07-7b3a-4ab5-a33d-66d91c1165cc}} ) by calculating the enstrophy flux {{formula:3fea345c-a625-4844-9874-4937d26f98f0}} as the integral {{formula:9159c1cb-4910-4c42-a31d-f41d9355bb0d}}
we verified that the steady energy spectrum {{formula:094d0191-26d5-4cfd-b878-177065f4e139}} with a high accuracy coincides with the Kraichnan spectrum
{{formula:08f113ce-98b9-497e-8a36-d0d3c4bef1b7}} where {{formula:9fa031f8-ee40-4ecb-93b5-f0024647446d}} the Kraichnan constant, in correspondence with the previous simulations {{cite:4aa7e10b70cf3da631da8bde6ec9b4954b834840}}-{{cite:284ea71a984c55258f98f2553d902e9980f07dc0}}.
| r | c9e32419b52a20263ca97661bc158a4a |
First, we present the results for density profiles obtained from a
solution of the reduced 1D and 2D equation and compare with the full 3D
results. It is known that the densities obtained from the reduced equations agree
well with the full 3D density, as the nonlinearity tends to zero and/or
the trap asymmetry is extreme {{cite:337beb4cbd6f1fe7fa45639435d8e4103c75eb0c}}. Hence in this study we
consider a moderately small trap asymmetry and a relatively large
nonlinearity of experimental interest. In the cigar (1D) case we
consider {{formula:a063aec8-72b8-4cab-9d9e-d0de9afda5f5}} Cr atoms with {{formula:84eca460-4928-464d-ad9b-58a0e97f0b68}} nm, and in the disk (2D) case we
consider {{formula:1e969e1b-17ad-4055-a970-1ad22d273d5c}} Dy atoms with {{formula:cc3dc9db-1ec1-4096-9334-907e57e0fe17}} nm.
{{figure:b367451e-9b13-430e-b834-ad834b8df131}} | r | 70b380dc877506882c3685ed8fffe4d9 |
To address the new fairness requirement, we suggest a privacy-preserving pre-process mechanism for enhancing fairness of collaborative ML algorithms.
Similarly to the pre-process fairness mechanism suggested in {{cite:916f24ad4260f86710e608c44244ecd81e656993}}, our method improves fairness through decreasing distances between the distributions of attributes in the privileged and unprivileged groups.
As a pre-process mechanism, our approach is not tailored to a specific algorithm, and therefore can be used with any PCML algorithm.
In contrast to {{cite:916f24ad4260f86710e608c44244ecd81e656993}}, our method was designed to allow privacy-preserving enhancements, which are obtained through SMC techniques.
| i | d74a0dd833a1f01956f798df04a3da9c |
Results.
The current state-of-the-art performance for pixel-based semantic segmentation on the benchmark data set ADE20K is obtained by an UperNet {{cite:22ad981b1a52eb999159f7ff89af3c187bd3b91c}} with BEiT {{cite:57a30908f245b0c4da6de51b37091948651fe2a0}} backbone and a features fusion strategy, with a mIoU of 56.7.
In our experiments, we compare SenFormer with UperNet architecture for a variety of CNN- and transformer-based backbones (see Supplementary Materials for results on additional datasets). As we can see from Table REF , when using the same standard Swin-Transformer backbone, SenFormer consistently outperforms UperNet regardless of the backbone size. The same can be said for the BEiT backbone. In fact, SenFormer outperforms the current state-of-the-art and achieved a new score of 57.05 when a BEiT-L backbone is used.
The performance gap is even larger when using convolutional backbones (+3 mIoU), suggesting that our transformer-based decoder successfully capture the long-range dependencies missed by the CNN-based backbones.
| r | 754bc92686af764a9cf47afc71432417 |
Finally, to fully classify the global gauge group structure in 8d {{formula:ec915c99-1585-475f-aa2a-aa49a44ece47}} theories based on the 1-form anomaly (REF ), it will be important to have more stringent constraints on the possible levels {{formula:910f3943-8c06-4d3e-bfad-fe0840364ff9}} for given simple gauge factors {{formula:1b951e46-7c69-411d-bb31-167f00f1a6a1}} .
While for rank-18 theories, (REF ) fixes all {{formula:8f15aea4-ebda-4fa4-93d5-6d52ab3fd9d5}} , they cannot be fully determined by this method alone for rank-10 or -2 theories, and will require new tools and concepts to predict these independently from concrete string realizations.
Perhaps, new ideas can arise by establishing a connection between higher-form anomalies and the Swampland ideas {{cite:88060c0ced4ebd99aab49f443cbe846ba088354f}} that also rule out certain non-simply-connected gauge groups.
These insights can hopefully lead to a complete understanding of the global gauge group structure, and prove full String Universality for non-simply-connected groups in 8d.
| d | dbc45808189c0a567c4788d7974f847d |
This issue could of course have been handled through alternative hierarchical exploration methods, however, for a fair comparison to the conventional agent we chose to simply generate training data with a flat policy (as shown in Figure REF in the Appendix). This again showed that hierarchy generally improves upon the flat agents, though the effect seemed to become less pronounced with additional hierarchy levels. In a way, our results should emphasize that {{formula:31e0a6d4-544f-40a2-a7da-4e9b7a222cf7}} -based exploration becomes more and more unreliable as we decrease the temporal resolution. To effectively leverage the benefits posed by hierarchy, agents should follow a more principled way for exploration (c.f., {{cite:669022dbd0c3e5ee50b56a22b6ab03a0c0fb2a0d}}, {{cite:e3e163ce450aea6aec2d6ef96a78bba0440888cc}}, {{cite:351e25aa8c438c82b40b26c01a4ba0b38d3ab523}}).
| d | ec70d0f4ab546f3f42292fea53877ca0 |
In image processing, the active contouring method is one of the conventional methods applied in order to detect, smoothen, and clear the boundaries of the features of interest {{cite:4150d9a879115bb3bdf7b1080d35daeabe647c62}}. Instead of the conventional active contouring approach, which requires distinguishable gradient information within an image, we adopted the Chan-vese method {{cite:4fd04f834600323a4b13cabed359e461889155ec}} which makes use of the intensity information. This has an advantage especially for XRCT applications, as in this case, which contain low signal-to-noise ratio data due to the inherent noise which can potentially induce some inaccuracy in the segmentation.
| m | 1d401a5207c6c370480d35f350b535b8 |
but converse of the above chain need not be true (see {{cite:2bb854b74cbc021ab1a4160c1a76df1fec9a15b6}}, {{cite:85c99fcf9f1073aed106cca8615b3085ed15e808}},{{cite:8cb9ffab9ef063549c17ebc5f990490bf90a35df}} and Example REF ). Also, we show that direct sum of pure extending modules is pure extending (see Proposition REF ).We call a module {{formula:51ae469b-ccd4-4af0-9a69-a423406fa1ec}} , {{formula:12caec73-436e-4c22-aabe-72ef7c10cf4f}} -pure extending if every {{formula:b6f7c059-2434-46b8-b237-7a44febdcef8}} -pure submodule of {{formula:74762253-7ffa-48b9-8300-7421d2feb7ab}} is essential in a direct summand of {{formula:bd71b4bd-67b4-43a2-ae4d-14e1da8d76e2}} . As we have seen above that not every {{formula:40184238-046e-4f44-b897-f24004e62523}} -pure submodule is pure submodule. We show that every {{formula:b7b35c35-2297-4f79-80c8-7755f87a0426}} -pure extending module is extending module and converse need not true.
| i | 9e04f7898bf8743560d4f5f8c2d7d317 |
Model complexity: It is represented as a number of trainable parameters in millions and per-image inference time in millisecond (Table REF ). It also depends on the base CNNs types (e.g. standard vs lightweight). Given the number of trainable parameters (9.7M) and inference time (3.5ms), the performance of the lightweight NASNetMobile is very competitive in comparison to the rest. The role of secondary data has improved accuracy in {{cite:9f23b3f437f468730b767094aa9acb33a48c3977}}, {{cite:0fb29720f62765bb5fc23fca6109b157610e7e0e}}, {{cite:9f7cc463055c9b1cb9cfc9aff81e105b679938a4}}, {{cite:f82c964689f2c392a95167b4164c4a09761e2b74}}. However, such models involve multiple steps and resource-intensive, resulting in difficulty in implementing. For example, 3 steps in {{cite:9f7cc463055c9b1cb9cfc9aff81e105b679938a4}}: 1) object detection and instance segmentation (Mask R-CNN and CRF), 2) complementary part mining (512 ROIs) and 3) classification using context gating. The model is trained using 4 GPUs. In contrast, our model can be trained on a single GPU (12 GB). The per-image inference time is 4.1ms. In {{cite:9f7cc463055c9b1cb9cfc9aff81e105b679938a4}}, it is 27ms for step 3 and additional 227ms in step 2. FCANs {{cite:6b861282ace693fd529b92308c5c8adf7583e0ae}} reported its inference time as 150ms. Using 27 integral regions and ResNet50 as a base, the training time for the Aircraft is {{formula:2674bdd5-6d73-4451-969f-b9f2575c7736}} 4.75 hrs for 150 epochs (12 batch size). It is {{formula:f96c80ed-d170-41ff-aa79-795d97e6b4a2}} 5.7 hrs for Cars and {{formula:55a314a2-5b9d-4811-8386-c092514612a4}} 8.5 hrs for Dogs.
| d | 499fd61505874be203f7bc4b5b083de5 |
Example 1.5 Consider the elliptic surface {{formula:bd00c94f-c632-42f1-857d-543cee92e2fa}} . We have {{formula:2040b216-1af4-4c8e-abe0-951b26d6225f}} {{cite:3519775906f78ed6cd30884e03bdbbaad25c2577}}, so {{formula:24ffb0a9-1095-4ff3-a2c1-9a65f99a9055}} . Suppose that {{formula:fd43dcc7-f87a-4363-befb-76023b4091b6}} is orthogonal to the canonical class and {{formula:ff77810e-00a3-4222-a219-87241106e8c8}} . Then according to {{cite:4f6dc2628b57d7deb39966339e05eab8f843391f}}, {{formula:b5dac9fa-039c-4d93-b2b9-29c1fef37295}} can be represented by a surface of genus {{formula:6b89d62c-4463-4195-88f5-e1a8ae452959}} , where {{formula:33fe4c32-0b55-43b4-9547-091f9130e7b0}} . Thus Corollary REF gives
{{formula:61c11da3-58af-4cd8-895e-cda26d82a43b}}
| i | 59ee60847e5f262ff0bbc39e79cd18d2 |
MUST is a simple unsupervised learning approach that adapts a pre-trained open-vocabulary classifier to a downstream task using unlabeled images.
In this paper,
we consider vision transformers pre-trained by CLIP {{cite:7c9cb004e38308841fe67e8b5cfc2ca3b031f0d1}} as the models to be adapted due to their distinctive zero-shot performance.
CLIP pre-trains an image encoder and a text encoder with a contrastive loss such that paired images and texts have higher similarities in a shared embedding space.
In order to perform zero-shot classification,
CLIP converts a set of class names into text embeddings using ensemble of natural language prompts (e.g., a photo of a {object}).
During inference, it takes the dot-product between an image embedding and all text embeddings to produce the prediction logits for that image.
As shown in Figure REF ,
we convert CLIP's non-parametric text embeddings into weights of a linear classifier, and directly finetune the linear classifier together with the image encoder for unsupervised adaptation.
| m | d78f03d6637367acabf56ad90d437e67 |
The real time quantities in the transport coefficients are comparatively poorly determined. The two real time parameters are the order parameter relaxation coefficient {{formula:1cbc37e4-075a-443d-aa6c-00559fa54b21}} and the diffusion coefficient {{formula:46d97f7e-c482-4ad3-830c-a34d78c42bfc}} , which set the critical
relaxation rates, {{formula:391089d5-f2f7-4ca6-9426-274c2fa84f7b}} .
{{formula:3f93fb5e-a01b-419b-bbac-086f299399b4}} is regular near {{formula:3cdbd329-cf67-49a3-9ea4-6de4256493f4}} and determines the charge diffusion coefficient well above {{formula:ded86be6-7910-42b3-8434-e020c7b91d22}} . We will therefore adopt the strong coupling estimate, {{formula:930ad14a-9548-4056-ba3e-8ceda3cb5a61}} {{cite:7d5c5718d4afc28cd346d41d9e5c23fd5a12f47c}}, {{cite:997bc921b7d9a9a9572c5ab4935507003a677050}}, {{cite:e076670f25a19dd251a40da23f6bb25d7b6c6521}}, and we take {{formula:7e20bc67-32ab-4d45-a218-4d5f916e7727}} and {{formula:3d7dd606-0fb6-49f0-b35c-ad6ad14c521b}} as in Fig. REF (see Sect. REF for further discussion).
{{figure:36612f2b-9e09-4b47-95b9-d96135eb36bf}} | d | 267062bdc86ad102d1fd3a004130f91c |
We studied three hydrogen-terminated silicon nanocrystals of different
size, Si35H36, Si87H76 and Si353H196. An LDA
functional {{cite:e0e409e5833c72ae9960a39fa27d0ebfdadbf50b}} was applied with norm-conserving
pseudopotentials {{cite:0c98d4dba657cde12d1ba3dd5caff695cb5d4589}} using the Kleinman-Bylander
form {{cite:b0d8aa532d544bcb7d278a5e919379976f929b70}}, and we used the Martyna-Tuckerman
reciprocal-space method for treating long-range interactions {{cite:6be0ffe71aacb26c30bf7ef3c2264c19dbe60e3a}}.
The grid spacing was {{formula:7b3d0d6c-245b-46c4-a347-88ae0d985e27}} , and the energy cutoff was {{formula:3290a425-2192-461e-a199-6ab18d853292}}
for all systems. To gather sufficient statistics, {{formula:74faabb8-d745-4507-99c0-f458ee7089d4}}
independent runs with different stochastic numbers were used for each
of the calculations below.
| r | a5494208db5a7d6a9314e84ab0a7c4b0 |
Hence, for the case of {{formula:5dfe5e2b-eefd-498a-9bd3-556533802ce0}} splittings, we choose the {{formula:2e633efd-92e9-4ad5-84d8-891a677d9ee5}} process to be the source of resummed contributions from the EW shower. The equivalent FO calculation can be carried out using the {{formula:301fd9e4-3c88-42f3-a8b6-898e5d06ab90}} channels, shown in Figure REF . The corresponding MEs for the FO calculations are generated by MadGraph5 {{cite:0bb9195aef0ee77e604fdf9eb68f426417549098}} while for the resummed computations we use a Herwig 7 internal ME, MEee2gZ2qq {{cite:bd9368344afd6a303b68b59755044624daa92854}}, {{cite:d500c7b7fe6833207225ccf5ef8286e2ff56163d}}, {{cite:df2e965dbe88e2d7fde4569e4f9986be9ac4d90d}}. The produced events are analysed by Rivet {{cite:d9f3906cf910fec1c1d284d41aa3c708ffcc221e}}.
| r | 335857cf8a269f0adee66c563964d75d |
The spin average or the center of weight masses {{formula:cc767875-3884-447c-b7ae-807c661c1538}} are calculated from the known values of the different meson states and are compared with other model predictions {{cite:6a5cf8137907fc845b0559214ddc8e4a49c21fc3}} and {{cite:0a721d54823f2b353ee5e0eae8f0a142e68b9d85}} in Table REF . The table also contains the different spin dependent contributions for the observed state.
| r | e3c3927288103f3e5e7d00b378282ed1 |
Bert {{cite:1114883b2ee9c7e24042b9479b2ae44eb5aed197}}, a pre-trained language representation model, made breakthroughs on many natural language processing tasks, including the sentence similarity prediction task {{cite:9f3be874c2688dd6e556a803da4baf54406751c7}}.
The methods using Bert's embedding vectors were also introduced to MT evaluation, the task that needs semantic-level similarity measurement, and show the best performance {{cite:350674706d84f27ca8c75d45200ac79192d9fcef}}, {{cite:1c9e21e0201d3e0ba57315e1201e04a2ae7e8451}}, {{cite:17c3967ff78eaad0a95505cb742ce8fea5511f33}}.
BERTScore {{cite:17c3967ff78eaad0a95505cb742ce8fea5511f33}} leverages Bert wordpiece embeddings to compute sentence similarity of two monolingual sentences.
When BERTScore is applied to the output and reference, it outperforms Bleu and chrF.
Meanwhile, sentence-Bert (SBert) {{cite:cf87adba37d7f38be243cc47302bcacfc61f45e1}}, a fine-tuned Bert, is introduced to derive more semantically meaningful sentence-level representation than Bert.
From the encouraging results of the embedding-based methods, we would expect the embeddings to catch the semantic similarity of input and round-trip sentences.
| m | 1aa24d12aaef2f4b9c5e310a49c2f248 |
As dynamic networks progress rapidly {{cite:0c844fb80e01c6763c13482cd5576337b727ee55}}, novel solutions are needed for their simulation and emulationWe assume that network simulators mimic the basic functions of network devices whereas emulators behave as real networks, allowing to interact with operating systems, protocol stacks and interfaces; we focus on the emulator analysis and design.. Indeed, current technologies make computer networks dynamic and thus, demand testing and evaluating network components in a well emulated environment {{cite:ddc87d54525854300eafbdc9df7990f78d220027}}; that allows reducing risks and saving time at the final deployment. In this paper, we focus on dynamic networks, i.e., networks whose parameters change; for example, the bandwidth and delay of wireless links may change due to external interference, propagation conditions, traffic variations, and programmability of the networks themselves. Depending on the desired network to emulate, the link parameters vary according to properties that are to be respected by the emulators (e.g., large delays on distant objects).
| i | fe69a893e9675137b965eeede79c9404 |
Xian et al. {{cite:97e0a6775665a7366e9f28b44628b2bafd2edba9}} tested 10 ZSL methods with a new seen-unseen split of datasets ensuring unseen classes are not used during pre-training of deep network (e.g., GoogLeNet, ResNet) which was used to extract image features. They used ResNet as image features and attributes as semantic embedding for SUN, CUB, AwA and aPY dataset. With this exact settings, in Table REF , we compare our GZSL results with the reported results of {{cite:97e0a6775665a7366e9f28b44628b2bafd2edba9}}. In terms of Harmonic based (HM) measure, our results consistently outperform other methods by a large margin. Moreover, our method balances the seen-unseen diversity in a robust manner which helps to achieve the best unseen class accuracy ({{formula:1bb199c2-7b27-4728-bb7c-f2dbaaefa0f0}} ). In contrast, seen accuracy ({{formula:2583fa7b-407c-44bb-9de1-deab420ba992}} ) moves down because of the trade-off while balancing the bias towards seen classes. In the last row, we report the ZSL performance of this experiment where only unseen class test instances are classified to only unseen classes (not considering both seen-unseen classes together). This accuracy is actually an oracle case (upper bound) for {{formula:4f5472e8-5461-4056-b043-91743cb7153e}} of GZSL case of our method. This is because, if an instance is misclassified in the ZSL case, it must be misclassified in the GZSL case too. Another important point to note is that the parameters of our method are tuned for GZSL setting in this experiment. Therefore, ZSL performance in the last row may increase if one tunes parameters for the ZSL setting.
{{table:71635a16-61e6-4d5b-bacb-373f197a1009}}{{table:e8f9e0d1-5f79-4b20-948e-caa1108cd759}} | r | 9b4fb1c217950c428c07aa9cc1f67651 |
As previously stated, a fluid shock is mediated by collisions while a collisionless shock is mediated by collective effects. For a plasma where only electrostatic fields are active (such is the case for an electrostatic shock, the kinetic equation accounting for both kinds of effects would formally read ({{cite:96b2202872005351e0d0d07cf40b508f9f745aaa}}, p. 9),
{{formula:f250e0cc-8322-468a-8b11-7011c0e449e2}}
| m | 10133abb9449496ffe232c86d3bc02fd |
Social inequalities in every aspects, resulting
from competitiveness are described by various
distributions (like Log-normal, Gamma, Pareto,
etc., see e.g., {{cite:72611cef5746e3e45eb336019c904324916aae30}}, {{cite:62675f19347abd79c5e3141b463dbdecddc89a68}}). Economic inequality
has long been characterized {{cite:f53949fe6e65c172812c8bf74c4613fe45533689}} by the Gini
index ({{formula:50a9c44a-36c2-4e61-a1b8-72b09939e682}} ) and a few other (much) less popular
geometric characterizations (see e.g., {{cite:5e97441e38ff090acf255bfa1cf5b784d994d9a8}}) of
the Lorenz curve or function {{formula:4dcbf02a-2957-4e10-9fb5-ecb042ef64fe}} {{cite:4da2a356123ab89afbe6e10c63b41969ecf0b37a}} (see
Fig. REF ). We introduced {{cite:2eabd55c8ac94a0f95f357042764dfc1c0f23036}} the Kolkata index
{{formula:8778bd21-de8a-4f59-8892-7e70a927c2ab}} as a fixed point of the Complementary
Lorenz function {{formula:d664aa49-a918-4538-a367-804e1ce46796}} has trivial
fixed points at {{formula:961b8a55-3522-49fc-93e6-aa5fe2748694}} = 0 and 1). In fact, the
Kolkata index {{formula:bba3369e-c7ba-462d-8d8a-0cf43544bde6}} is also (geometrically) related to
the `perpendicular diameter' {{cite:5e97441e38ff090acf255bfa1cf5b784d994d9a8}}, {{cite:4caf3acb0be3190fabe038506e389214a91af5c6}} of the Lorenz
curve. Unlike the Gini index, which measures some
average properties
of the Lorenz function, Kolkata index gives a
tangible interpretation: {{formula:dc2ac6b4-3792-499b-adcb-b5017cd3a7c1}} fraction of
rich people or papers or social conflicts
possess or attract or cause {{formula:3e550bc4-21b1-4a04-83d6-a1daf7f96f86}} fraction of
wealth or citations or deaths respectively.
| d | aa07a1c37f1ba326f91c4d8360cd0a9c |
Based on Table REF , the model used by Kumra et al. {{cite:308aa48728b0b5e354eb5a6ee03421a987f80236}} stands out because it scores highly on Cornell and also has a high grasp success rate in clutter. However, one should be cautious here. The clutter grasp setting used in {{cite:308aa48728b0b5e354eb5a6ee03421a987f80236}} is probably easier than some others, for example Kalashnikov et al. {{cite:89dbf2d4af4054fa15b3e31d9156fa03dd7183af}} or Satish et al. {{cite:40e24e75d1b6533c3895e81711483f064c36cd4c}}. Perhaps the main conclusion to draw is that the fully convolutional model class (the fourth vertical grouping in Table REF ) is currently probably the most competitive choice for a {{formula:7610c4f9-8937-4752-a366-0d2a60606e1a}} grasping model architecture. Three of the other models – sample and test, regression, and region proposal – all seem somewhat outmoded now. However, policy learning is probably an area that will receive continued attention into the future because of its ability to learn closed loop policies rather than just to detect grasp poses.
| d | fc38cc467001cdaa263dcab3e6678dad |
We present preliminary results for a parameter scan on the THDMa, a two Higgs doublet model (THDM) with an additional pseudoscalar that serves as a portal to dark matter {{cite:cc6cacb08dfcc7f72a9468432ea2be27a0d53dc4}}, {{cite:60a6a38bd1cb332a1be11f5531ef05f6ad869299}}, {{cite:b1f01295822240521b50450dcd8145ff8979c9c6}}, {{cite:37a7d21d3bd2c299ab10cb754974b49a0869fff1}}, {{cite:49b35e39ddc1412b9ed6d11f5021f98425607441}}. It features 5 additional particles in the scalar sector, which we label {{formula:a5cd2fb8-9d82-4b6b-ba16-3037d32b2da0}} in the mass eigenbasis, as well as a fermionic dark matter candidate {{formula:be7a109d-32da-47c6-bc94-7bbf0896307e}} . The model contains 14 free parameters after electroweak symmetry breaking, out of which 2 are fixed by the measurement of the 125 {{formula:6b223618-b5ea-4755-bbca-53eacdd5fca9}} GeV scalar as well as electroweak precision measurements. The parameter space is subject to a large number of theoretical and experimental constraints, see e.g. {{cite:56e8f54c637e65ee8a1ee022a6ecaf303b687b0b}}, {{cite:887f4a8161239eda68f7302375e22ced32f65349}}, {{cite:2f7269b8bdae4d72e0fcff7aaa10eb42a10b8aa6}}, {{cite:3aa9e59980570b2ad5ec47c9bd6e8b6fd89c5a07}}, {{cite:2458b876de0a98504a28793b20018cf30ebf1ad8}}, {{cite:a4bbadaf8c0ea087ba914cc0deb0d1cbefff4545}}, {{cite:7bc506bc15b0805675eb80cd5992d80f05ceb54f}}, {{cite:b30099b2b1bcd748fb1b2c49df0f8ef0527fe369}}, {{cite:027d724c1af1e29caa05532557be7fbd7b818d67}} for more recent work on this model. We also impose bounds from experimental searches, where we found {{cite:1edb45687e2e6e89e30e848cf1d831dd0056d4cd}}, {{cite:8489c37e220a9b6c60fd55a4779c76009b6b2e03}}, {{cite:7c9c83409ba4609437cd606ab264bea361da8a28}}, {{cite:ee9fe8f4a22cbec1de9536699aee5796ed054d32}}, {{cite:677b3b41bf8bbb1ae5ca2989062c6f37ebddd87e}}, {{cite:a110d631e83c72be79d650840deee479b7c08d8f}}, {{cite:ec2fd73b9ef31ce39d1a82e99b2abb58f8a0038d}}, {{cite:4cc0b998e5b7fc5209307f678a80f9f9f1abe803}}, {{cite:f3f113a3ecda859ce13042c74e98a2e96f12fff5}} to have impact on the models parameter spaceNote that bounds are here applied successively, e.g. all theoretical constraints are applied prior to experimental bounds, etc.. This work is a preview of more complete results {{cite:866d5c1d469282c5ec3ec7f3668808d8a7b4eceb}}, which will be published shortly.
| i | 0ccf17c4816efe23551dda907b82eeec |
In Figs. REF and REF , we show the results for two different nonlinear systems with {{formula:a86fbbad-5148-456f-821d-a6da1bd2599a}} given by {{formula:6d93d98b-de37-4c79-b8a5-fdc59e505146}} and {{formula:d6d2ed53-ac36-4411-b721-1887ff9a030d}} , respectivelly, both with {{formula:f5807a32-590f-4aa0-b8ec-712c1b6d5bf2}} . For the same initial conditions, the trajectories show the evolution of the closed-loop system state for both controllers. As before, the nominal QP-based controller proposed by {{cite:888da5c138a7ea6f805eb2ffb109feb4bf65cab5}} introduces the same undesired equilibrium point in the closed-loop system, while the trajectories for our proposed controller are attracted towards the origin. This behavior is obtained by the rotation of the CLF around the origin induced by the QP-based controller described in Theorem REF .
{{figure:38c33266-7c4a-41b1-8318-79eed0ae2dad}}{{figure:6fdf5841-2431-4b69-9491-ece520fcb2ac}} | r | 49779754377608b4462dc9c7baad45c6 |
Overparameterized machine learning models have drawn increasing attention in recent years. These models are usually complex enough to achieve zero (or nearly zero) training error, yet they could still generalize well. One typical example is the state-of-the-art deep neural networks. {{cite:f4b49bb258f119daf28959eb96f3041b342f503e}} showed in an experiment that a well-designed interpolating deep neural network can perform respectably well even when considerable label noise is added. Similar interpolation performance has been observed in other machine learning methods such as random forests {{cite:4dfde63fb27712c8c293b4a5f63740ecaa15b31b}}, {{cite:4f70b91b69413e870d8f7776a14596ce57a4304d}}, AdaBoost {{cite:4dfde63fb27712c8c293b4a5f63740ecaa15b31b}}, underdetermined least squares {{cite:cae9b982692b4e59f95bc8578a1b3b8d7a072972}}, {{cite:714c4b7effbc3b78b3f3cd8d355d0de335f52558}}, {{cite:0172b14979ae884f15603d8a11118a456d122dbc}} and nearest neighbors {{cite:12beeac570c964f8a1af719a21a0c3573b3ad86e}}, {{cite:b8b3ce2c5f6295d77f7231de8a9118e10d3d5b8e}}, {{cite:5a880481b44e6dae3c4aca6a9f46645e65d2d165}}. This phenomenon seemingly contradicts the conventional wisdom that interpolation tends to overfit the training data and performs poorly on test data. Further, it appears to defy the bias-variance trade-off.
| i | f227d92576001e0962fe92674098aeb2 |
An axion in this mass range can also be the main constituent of dark matter, in particular if the Peccei-Quinn symmetry
is restored after inflation. Assuming that the PQ symmetry is broken explicitely by a Planck-suppressed operator
{{formula:623a2099-e48c-402f-b477-bc97d4611fc1}} , the observed dark matter abundance can be
obtained for both DFSZ axions as well as KSVZ axion/majorons with a mass in the range to explain the stellar
cooling anomaliesWe may even consider to extend the DFSZ axion to a DFSZ axion/majoron {{cite:a57564fc3cd905d54557e9b70478c921faf5f2e2}}. Exploiting the radial part of the PQ scalar field as inflaton non-minimally coupled to gravity, the DFSZ {{formula:9538837b-9987-4263-894d-d37cbf8ecb5d}} and the KSVZ
{{formula:1b159a81-218d-4451-8f8d-19e3de0620e9}} models can be considered then as a non-minimal variant of SMASH {{cite:dc3bea94aebeb5304e82c02ffcb914603ac20fdf}}, {{cite:b02a0313d360703d17d2d1bc9286b0e68868c8e4}} – the acronym standing for SM - Axion - Seesaw - Higgs portal inflation – explaining six fundamental problems in one stroke: inflation, baryon asymmetry, dark matter, neutrino masses, strong CP problem, stellar cooling anomalies., requiring relatively small tuning in {{formula:53063abc-b520-48c7-af22-bf7be521ad1c}} , cf. table REF .
However, its direct detection will be a challenge. In this connection it is of interest that in the mass range of
interest, IAXO may indeed measure the mass of the axion {{cite:e1dbc45fcc8725c0abdc86927d3a520332dce805}}. This information could help to design a dedicated
direct detection axion dark matter experiment, for example based on the idea of a dish antenna {{cite:73389f83d61236d36bfdf9f595ee53fc6113f70d}},
exploiting the axion-photon coupling, such as BRASS. Alternatively, direct axion dark matter experiments
employing the electron coupling, e.g. searches for induced atomic transitions {{cite:66ba1ecdb01cdb03e6f5874055a2e7cfbe8185cf}}, {{cite:eda06a2022de8105fecdac35ea86982b8b0be602}},
searches for electron spin precession in the galactic axion dark matter wind, such as QUAX {{cite:88d39c18d05b15bf6f9ca9a95f522ddbc93994b5}}, and
searches for axion dark matter absorption on a conduction electron, followed by emission of an athermal phonon, in a superconductor
{{cite:5ae12cae4e1c68821cd37cb14d9b1383e76fcbc4}}, may also ultimately reach the required sensitivity.
{{figure:d29bd8ba-1143-4442-b5c9-49c94bacbb6a}} | d | 35161107810421c4d788f0b3a59aefaf |
A more phenomenologically relevant direction of work would lie in considering the case in which some or all of the Goldstone bosons of the model have some non-trivial masses. Depending on the mass spectrum generated by, e.g., soft symmetry breaking terms or a feeble gauging of all or part of the global flavor symmetry, leading experimental constraints on the scale of the CP and flavor-symmetry breaking scalar VEV's can be considerably altered. For example, Goldstone boson masses {{formula:18a6741c-72f5-4e64-9139-e8ce8ba0433e}} will generally have much weaker constraints from supernova energy loss than the case in the main text. Furthermore, depending on the spectrum of the Goldstone bosons, the dominant phenomenological constraints on the model parameter space might stem from a variety of precision flavor measurements, similar to the studies on flavorful axions {{cite:4bcfd9c288e9e30e07e1ec3dd56220be118bd1f6}}, {{cite:5629814a69864439daeaab41c0d6c47d569360fa}}, {{cite:b73e12d6129c12e511194fcc775a5b22210d8307}}, {{cite:df1eb725ee56f73eb1b320d02fe56eea45256046}}, {{cite:e24be9d20bf76bb5f5d66f1c010e2dd8c6f2ab13}}. The cosmological implications in the event that the Goldstone bosons have significant masses will also depend strongly on these masses' magnitudes and relative scales—sufficiently long-lived Goldstone bosons may still have an effect on {{formula:4a30e995-57f0-454c-b024-6984398a7bab}} , but may also affect measurements of the sum of neutrino masses or large scale structure formation {{cite:3243ff257e4c916522707890dc65f0f7a4bfbeb7}}, {{cite:ba296d0410d957a7a9a236af51f75eeab4ae0bfc}}.
| d | 661d89532f599681cbd7791ad7d6f63f |
Let {{formula:6b70bcda-c730-4115-9066-222023ac6da2}} be a linear complement of {{formula:1a97d43f-37f4-40fd-9557-a07876de390c}} in {{formula:43941003-d641-4ce3-88a0-d6348d6e670c}} . Define a linear map {{formula:df2064f2-41bb-4a6b-a4cc-8ead8e74b921}} by {{formula:4e10cdb3-0440-49a7-ae9f-a29f2ccc39f0}} for {{formula:a4acd6d4-0d1a-4ccb-a871-0c839e7d36d8}} and {{formula:2b57f536-998f-421a-8a9b-11233cb148cc}} and define a multiplication on {{formula:a72fdeec-a586-4a92-92a0-d620d5be2aa6}} by {{formula:8466f452-1427-42e3-b951-6a87e7800b85}} for {{formula:f20391e6-1646-4f23-b95d-3f56036fe50b}} .
For {{formula:3411b94c-54b9-412b-97e8-0bdea61f2c02}} then
{{formula:a4b009f9-cd4c-4000-aa55-96ce637b8b76}}
Since {{formula:15726ce0-0510-42d6-b9cd-e25761f88500}} is a homomorphism then {{formula:21b72442-bdf7-4356-ad62-55b7e10be2b5}} is a Zinbiel algebra and {{formula:496c13be-38ae-43dc-bd16-587c252e4c2e}} , which give us the uniqueness. Now, define the map {{formula:3b97c9b6-e004-43d6-be43-cd7eb59cbeed}} by {{formula:81ebe9fc-bd3c-466a-aed6-2af142a07b8f}} . Thus, {{formula:61267ffd-5dfb-4f22-9dc3-d998a67a0e06}} is {{formula:1de38ef8-c77c-4867-a99e-eb4b038a0896}} and therefore {{formula:54e36517-4591-4473-86df-d35c845f36a7}} and {{formula:6a312120-9731-435e-a3ee-212d09579539}} .
{{formula:6da4d7ba-5dd6-4e57-911f-9be00215a735}}
However, in order to solve the isomorphism problem we need to study the
action of {{formula:a384c2ef-0cf4-49af-8e81-6bc5d8f78043}} on {{formula:fcffbea3-309f-47c4-9cac-367c0a9f9f77}} To do that, let us fix a basis for {{formula:ec23d604-efd1-4866-9c9d-29b6524a998e}} , {{formula:33a48aa9-35e0-4e57-9767-72e6aabb24e8}} , and let {{formula:dce1270f-af0c-4618-935a-2afeda606fff}} . Then {{formula:74ec81d8-6793-4ce0-aea1-1e9991c73df6}} can be uniquely
written as {{formula:0cc7704c-4b95-47f1-95a2-eb534fe0d5f6}} , where {{formula:1922ea5f-d4f4-4f09-9938-d5c8b62613d1}} . Moreover, {{formula:91626a21-ee43-4c8c-8635-69ccf8b949c6}} . Further, {{formula:8217650d-6ea8-41a2-9243-e60a7fc3fc99}} if and only if all {{formula:cdef00d2-2396-49c9-a293-bcaf5c5063a6}} .
Definition 2 Let {{formula:7009db23-d50e-4313-9047-68839efc19b1}} be an algebra and let {{formula:9876ff8e-d8ec-49db-b994-fb057e8c1247}} be a subspace of {{formula:e0a34208-352f-4cf7-bba6-041afadf2b9e}} . If {{formula:ec848f63-3d3e-4136-b3ac-e23d7909ae34}}
then {{formula:81c22a0c-9d72-45e9-942e-e2fdf32b0246}} is called an annihilator component of {{formula:5dac4f4e-6462-4ab3-9e34-feeac6d1e4a5}} .
Definition 3 A central extension of an algebra {{formula:6589b394-d6cd-41cc-92ba-a9101b10e780}} without annihilator component is called a non-split central extension.
It is not difficult to prove, (see {{cite:c215498e94ddd27fa2b9740b4695f720daf6d051}}), that given a Zinbiel algebra {{formula:2b9fd359-28fa-4a32-af02-d54056df5a59}} , if we write as
above {{formula:7bd820ea-2874-4a6e-8815-231d46b72597}}
{{formula:96471c3f-630e-4f34-9225-d817b236b5c1}} and we have
{{formula:a0b97905-f2c1-4a6d-b94f-94db94777f99}} , then {{formula:16bf9a6e-0927-4110-a71b-32cba824ef96}} has an
annihilator component if and only if {{formula:b1ed0a6b-d1d4-468e-9e8c-e3770d0ad7c1}} are linearly
dependent in {{formula:7dac6c3f-3856-44ae-b082-86847a09e04b}} .
Let {{formula:a3cdc6cd-f3f0-451b-8ae2-276fb67ab60b}} be a finite-dimensional vector space over {{formula:57b5a4a8-ff86-4e8a-b9bc-e4b1d631bd4c}} The Grassmannian {{formula:071ec5a2-7d40-4711-ba27-c938ff009ba7}} is the set of all {{formula:4687c99d-1f43-4607-9234-739e80eac361}} -dimensional
linear subspaces of {{formula:37ae3cb9-9b82-4826-b88e-b3b852963294}} . Let {{formula:93786dba-7e80-49ec-b887-87fffb91daed}} be the Grassmannian of subspaces of dimension {{formula:2525d897-c06c-48b1-87c7-c4cf3ce4ff28}} in {{formula:6055f64b-13d4-4763-a877-5a0a44d8e761}} . There is a natural action of {{formula:602afc4a-b7a4-4bd9-8cb6-609a254c1687}} on {{formula:99b80430-c46e-4994-9d01-c068c1065e0d}} . Let {{formula:ca4ad332-01d1-4d7a-8f65-d1e9464d7d88}} . For {{formula:5fc9819f-875c-4796-a6bc-77038c503147}} define {{formula:874d153e-fac3-4f8c-8a22-263fb1140af0}} Then {{formula:3d91f122-c11f-41ba-aa06-958a9d8b1e70}} . We denote the orbit of {{formula:c8967c55-d84d-4700-b1b6-e87fa22193b9}} under the action of {{formula:ca4a0311-66a5-4431-9f60-5b124c71907a}} by {{formula:e9af6a7c-ce8b-4f12-b6cb-fada56a12da3}} . Since given
{{formula:2984bd71-45da-4b02-8666-ab44a02f0b37}}
we easily have that in case {{formula:2218477e-6b89-4e22-8a84-8e6358bb704f}} , then {{formula:8b4f711d-7c84-4af4-8df3-557817f8139e}} , we can introduce
the set
{{formula:c64c0ae1-5674-4251-8803-360ec9bdf605}}
which is stable under the action of {{formula:c20098a4-cfca-4bed-a7df-3f3e4983c9bf}}
Now, let {{formula:34a2e32c-7cd1-4bc4-901a-b5550032db57}} be an {{formula:382cae41-ee3c-4d06-890c-4b7b8026bcc2}} -dimensional linear space and let us denote by {{formula:acbcbe4f-0ddd-464f-89bd-e81157009ecc}} the set of all non-split {{formula:8f5b4e69-659f-44b9-b78c-9baa1c0ef50d}}-dimensional central extensions of {{formula:904ae1fb-fd55-44f8-927c-e72175ec6272}} by
{{formula:c4d3f2b9-852c-49d1-9d20-4c19cabdc376}} We can write
{{formula:da23cfd9-67f0-4aa9-ae70-513cd04da0ab}}
Also we have the next result, which can be proved as {{cite:c215498e94ddd27fa2b9740b4695f720daf6d051}}.
Lemma 4 Let {{formula:76ed70d3-c1cf-4822-af0b-4c206b6f0a05}}
Suppose that {{formula:f1d33e3b-0554-4f84-bee0-5962eb811ffd}} and {{formula:ba6e5108-e0c2-4692-b3ab-0c8822d515c8}} .
Then the Zinbiel algebras {{formula:f6e1b1fe-5ea5-4ac3-b0c4-0e0059b67f97}} and {{formula:6e0a13f5-0324-4814-97e8-b93502f2ae55}} are isomorphic
if and only if {{formula:234b5a17-6cf4-4ef7-a6be-7d3fdfdd6008}} .
From here, there exists a one-to-one correspondence between the set of {{formula:e3dc0255-bae6-4ae3-a098-58074edbd9ab}} -orbits on {{formula:570c0257-1b08-4d63-acc2-baaa15d06094}} and the set of
isomorphism classes of {{formula:78ce8584-098c-4953-81dc-cebd9353c9bd}} Consequently we have a
procedure that allows us, given the Zinbiel algebras {{formula:139109a9-ece1-4aa3-a64c-60647d01445e}} of
dimension {{formula:69489062-30a3-42ac-858f-4fdaedf2d6e7}} , to construct all non-split central extensions of {{formula:0c160f90-b269-468c-b168-75561023feac}} This procedure would be:
Procedure
For a given Zinbiel algebra {{formula:c0077a05-480b-4769-a67d-80af41e3a772}} of dimension {{formula:ea82e4ec-ef03-44fb-a6f2-0ee2a4e2380a}} , determine {{formula:3d1d8f8b-0bb6-4d9e-864b-c315efb2d65b}} , {{formula:7bf09e5e-64b6-4084-ba83-84d53911fe43}} and {{formula:87084625-ab86-492c-9fcf-5e9c612187bc}} .
Determine the set of {{formula:96ffc629-050d-4a8b-afbb-8d8aaf1fdd66}} -orbits on {{formula:c7a00f5c-c846-4c9a-a8be-04c4094ca9e1}} .
For each orbit, construct the Zinbiel algebra corresponding to a
representative of it.
Finally, let us introduce some of notation. Let {{formula:65ac0b0f-7876-42c1-9de4-622827947586}} be a Zinbiel algebra with
a basis {{formula:baad9379-d152-4470-8448-ceb617b6ee60}} . Then by {{formula:aafea8da-cb62-432e-be55-1f912bdd20f8}} we will denote the
Zinbiel bilinear form
{{formula:c5666b2f-61f7-4c76-ba3b-bb5fa4302be3}}
with {{formula:efb909dd-5015-40e7-a7c1-bcccff48f177}}
Then the set {{formula:26238f07-85c8-4f7c-b538-7ff7cbfc3029}} is a basis for the linear space of
bilinear forms on {{formula:5f998ca1-7356-4c97-9cba-c661d04275c3}} . Then every {{formula:6ce97b38-bfb5-4b99-90ca-0795bb631575}} can be uniquely written as {{formula:b7cfef12-1272-4242-8e9f-5a77105e5e90}} , where {{formula:3fd47808-68de-4caa-907e-baf1a0954e66}} .
Let us fix the following notation
lcl
{{formula:67625268-0ef8-4ae7-88a1-c7201dec9ff5}} — {{formula:5721ea32-f2e1-4f1c-a7dd-d1d1e6cf87d4}} th {{formula:beb7f5cd-95a4-48a1-8b6e-2e8597738ece}} -dimensional central extension of {{formula:eec0e1aa-af8f-41be-ae12-32d3d4e68e82}}
Classification of Zinbiel algebras
Thanks to {{cite:a059a11dea37a57557a0d496f23f716b67ff2830}} we have that every finite-dimensional Zinbiel algebra is nilpotent.
There are only one 2-dimensional non-zero nilpotent Zinbiel algebra given by the multiplication table: {{formula:c0487348-e05c-4f07-84b9-ca5a69da3b83}} .
Thanks to {{cite:b2f8f260ca1245711a4833dd3e9b5e8fa9a8612e}},
as central extensions of 2-dimensional Zinbiel algebras,
we can take the classification of all non-split 3-dimensional Zinbiel algebras.
Now we have the classification of all 3-dimensional Zinbiel algebras and their cohomology spaces:
|l|lll|l|
Algebra 3l| Multiplication table Cohomology
{{formula:9beb4166-2556-48f1-9203-5adc9fd6a7b8}} {{formula:22f6e19f-3df1-41d5-97ac-a3f01a57b236}} {{formula:4362a537-fa97-41ef-80a0-c31c2497c5b8}} {{formula:e10cbbae-c7c2-470f-8bf1-b76b0dc04ded}} {{formula:a1a05ba9-012c-4589-94dd-e3060efa7f5c}}
{{formula:4e1d2f93-40e9-490d-a80e-02f26c63aacf}} {{formula:9c538f26-49ee-4460-a2e6-3464dd191db1}}
{{formula:71a40a24-b398-4eea-a589-4dbd0e440ede}}
{{formula:79d15039-3506-4186-aa06-7a6bf1c50e81}} {{formula:bd9e404b-09c0-4450-9073-01f90f2a891a}} {{formula:914eab4b-a32e-4246-9ea3-bb480c40f94b}}
{{formula:a58c39e4-61de-492f-b6dd-34e63bf78731}}
{{formula:dc7cd854-7a15-4bcd-970c-d95c0a449992}} {{formula:1cf61c05-d8bc-4730-b2ab-319c95bade1e}}{{formula:e17df082-2bc3-40f3-9164-cbc3ad72b7b7}}{{formula:fcbf3d91-2f57-49bb-adac-942a6cbefd5a}}
{{formula:f7101631-e59f-4a14-8eb2-a9f9874f22a6}}
{{formula:9a4a743f-908e-479b-901d-3f946a8f5060}} {{formula:7329be51-f07c-41ba-a564-ada33c4f1760}} {{formula:7c6debde-11b7-4b02-8e72-0693acba2b0d}} {{formula:851e0a27-1a8a-4907-a0bf-5c77d3b354f6}}
{{formula:59fa6f78-6130-4ba8-8984-662bfca7c558}}
Remark 5 All {{formula:e7a06246-7cde-4db5-b21c-219c53a84dd0}} -dimensional central extensions of {{formula:644d4367-cf00-4082-8d93-d7ce51726edf}} and {{formula:7b5d2ee2-7587-4995-9c3e-ec82be191299}}
are trivial.
They are split or have {{formula:d7b318b8-7780-4fed-9c89-43316aea2a54}} -dimensional annihilator, which gives them as central extensions of a 2-dimensional Zinbiel algebra found in {{cite:b2f8f260ca1245711a4833dd3e9b5e8fa9a8612e}}.
Central extensions of {{formula:f679a1a6-251c-483d-b3a2-c8520f1baca9}}
It is easy to see that from {{formula:fac38922-e8f9-4c7a-a13a-99811059f3b0}} we have only one non-split central extension of
{{formula:539336b8-815c-4f93-b20d-e4fd1fcd9c91}}
It is
{{formula:e4ce7119-7758-4f55-91ed-1bff19a18fe3}}
Central extensions of {{formula:6c493836-2c5e-4bfb-8fa2-2d549adc51a0}}
The multiplication table of 3-dimensional Zinbiel algebra {{formula:a7313bee-1405-4d57-a27e-9773fd9d09d9}} is given by:
{{formula:19e6ad68-baea-43f9-ac37-209f7f25c917}}
The automorphism group of {{formula:fc61186a-1d97-4156-81a2-3a10dc05c899}} consists of invertible matrices of the form
{{formula:d81b9e8b-4e2f-40ff-bdbe-0e255b9ad17b}}
The cohomology space of {{formula:440c9ed9-e791-40a5-abe8-293a426fa210}} is given by:
{{formula:180a7f10-df72-425c-bd9f-c4b0ce8027eb}}
Since
{{formula:70071638-b589-4ce3-a0ba-aef9a1387606}}
where
lcl
{{formula:ec7b1e10-1cfd-4c19-b438-df293d18524d}}{{formula:54b0aa2f-294d-4999-b079-99128ee82f23}}{{formula:4ae1e2ce-cfc3-4fab-a1a9-dbd75ccaa5b1}}
{{formula:05db014a-99a1-4684-a708-824c02f782f2}}{{formula:da35b21b-05b9-4114-b313-8250a7296f89}}{{formula:a9e2e65e-cb4d-4fc8-acd2-03f7f638e73f}}
{{formula:37ee1eb0-1fef-4b5e-a571-41fc119801ab}}{{formula:128debc0-51ef-4d6d-b5cb-ff0f27371687}}{{formula:dc37d138-7753-473d-9b45-96e9bdd1156d}}
{{formula:3c433a7d-b3d3-42a6-89e0-1e79141792fa}}{{formula:62ee671e-b87e-4450-991a-afd498475b86}}{{formula:f77ab4bc-72f3-4cae-aa79-5c6276c60f9d}}
we obtain that the action of {{formula:f1da3d81-2029-4398-b135-18b992059550}} on a subspace {{formula:ae3ddb6a-3a01-4581-9217-90209b869b23}} is given by
{{formula:dea42dca-dcd3-4115-8f6c-c32b5fd179f1}}
1-dimensional central extensions of {{formula:e40e628f-87d4-454d-b431-a58ec125d4c3}}
Consider an element
{{formula:2afd7a6c-bd4f-442f-89a2-2f858ef4cbdb}}
We are interested in elements with {{formula:9027d618-60bf-4801-b932-c60d3cafec74}}
and {{formula:e83ab97b-ca08-4ea3-b26a-8ab431441e29}}
We provide the orbit of every possible case:
{{formula:d68c7d88-e8f4-4c57-a561-bd396013f1ff}}
by choosing
{{formula:2873cdf2-68af-4517-b55a-60c43b2dcdb9}}
{{formula:2aea4a08-5b32-4ffb-aa7c-9d2f103c9467}}
{{formula:4f1aabb6-ab27-4905-ab26-b1c7251138aa}}
{{formula:666dabc9-ed32-4fbc-af5b-e92cafc8d333}}
and
{{formula:6557e32f-0d60-4306-9c21-278ac09911c4}}
we have the representative
{{formula:9c3e0aa5-916a-4b73-b247-ff9889472f67}}
{{formula:0f55bf97-d9cb-4edf-9ecb-5ff24fd3c16c}}
by choosing
{{formula:caa0014d-8eab-4c3d-a4ea-1ffe54b7e473}}
{{formula:1801f464-dded-4c48-b297-0a3d9f1c1ef1}}
{{formula:3ad65686-2356-4e38-9e7c-0bbe90930ce2}}
{{formula:7ef1dad3-f13f-438f-b790-88a7d68fae1d}}
and
{{formula:b36684e2-21a7-4dd0-95f4-6250f660bf77}}
we have the representative
{{formula:a7b8791b-7d64-474d-95ef-51fed9148836}}
{{formula:896b9520-fb4c-4f3d-bb9a-f57134499248}}
by choosing
{{formula:c163db87-70b3-4064-b253-bf9ffb462552}}
{{formula:402fc9f5-f6d5-4b66-b0f4-224edf45ec46}}
{{formula:cb8cc6ab-8846-4b9e-b820-5dc7c1bc2e4c}}
{{formula:0828398d-a8cb-44ec-8427-769394d2ec2c}}
and
{{formula:464cddf3-b264-4ff0-ba82-bf1eb45b8ea5}}
we have the representative
{{formula:ff86d3e2-8615-4029-b192-755e7e6cab2d}}
Summarizing and noting that {{formula:f8c5e07e-fec1-4a7e-b6b9-ae439581f62a}} gives a split algebra,
we have the following distinct nontrivial orbits:
{{formula:1d204102-3fdb-4579-8e2d-7d22e8c85c05}}
2-dimensional central extensions of {{formula:02a1e612-1ce7-4e7f-b62c-c892c24ffce2}}
Consider elements
{{formula:37baa85e-9942-40c4-9db2-120dcbb7ffe3}}
and
{{formula:1c77a5c9-0be1-4f0d-a005-c42b8c951801}}
We are interested in elements with {{formula:be2c7c4e-7b29-4a5c-aeb1-940857e48f16}} and {{formula:7ae4abc1-83ca-48ab-a623-2ae4bc0755d9}}
We provide the orbit of every possible case:
{{formula:07cfb493-8de7-4215-9587-f18662a14b3d}} Then
{{formula:a6e63a53-6b8c-4792-8cd9-65685bd67168}}
by choosing
{{formula:ca57eedb-2888-4909-8bdd-b3be616bc3b3}}
{{formula:3808c3b6-fec7-49f5-af84-b89ddf259a07}}
{{formula:3772466d-3cfd-47c9-be0f-1ac460cf4a6e}}
{{formula:7af855ab-37d6-41e1-8f64-5e0624d60845}}
and
{{formula:b432ef00-a31f-4440-b110-16420492b341}}
we have the representative
{{formula:83b56941-ed43-405f-b92b-51300e953161}}
{{formula:b3b039d4-2166-42ec-9687-1a6b4024406a}}
by choosing
{{formula:6e021a7d-fe33-4d13-8191-a5c3aef854bc}}
{{formula:fbcf9c08-990d-4b97-b36b-5b347f318f19}}
{{formula:67e96033-e3da-432f-9b5b-2cfea7c5be37}}
{{formula:033e6fb8-7f10-4f5d-b0ce-127c140ebe49}}
and
{{formula:ad00faae-a9e4-41d9-a696-82c749e7e3e2}}
we have the representative
{{formula:49281ce6-4f8c-43d6-9499-bbafc394a28e}}
{{formula:5694abfd-9dc7-45d7-8db8-f6672b2efd38}}
by choosing
{{formula:c920167e-58d2-4f90-bfa6-ed9d13422327}}
{{formula:3ccece9f-cb23-4563-b912-3f4378ee9698}}
{{formula:2d1bb948-4cd9-4542-9964-b7a64dcf898f}}
{{formula:8c5b6d5b-325c-4c50-b784-17eae443762e}}
and
{{formula:3b98cd1b-cbd3-41e7-90a0-71139fa520e8}}
we have the representative
{{formula:15f91f7c-0f9a-40e7-8db9-9408a582c2c9}}
{{formula:8bbfe6e5-875c-4239-b27f-12c00ffbaf2b}}
by choosing
{{formula:f333fcab-b4da-489d-8395-3366765af957}}
{{formula:69d92c42-cbca-4a95-a9ea-06f30da8c8f3}}
{{formula:5f59d22d-445c-4ef8-a900-3a628d6b700d}}
{{formula:1aeabcd3-db57-465f-b2bc-bb4c0b84a0e2}}
and
{{formula:49242b8c-a887-4eb4-b67b-54c698194246}}
we have the representative
{{formula:e16dd0eb-9aa9-467c-9cef-c543e08c1d9f}}
{{formula:e35a4d53-04cb-43ca-8441-129d2c0e0fb1}}
by choosing
{{formula:71daa482-b2b9-4086-9521-7b7cc41eca55}}
{{formula:536a6196-4804-4970-a0dd-ed2ea5a861a9}}
{{formula:b1a9fdc3-ab3f-41ac-bf77-f54f053bd211}}
{{formula:972f5fe6-cd9f-4b40-a598-9b0325009163}}
and
{{formula:4475dcb3-96f7-4037-b6e9-d2fbb9dfa7bd}}
we have the representative
{{formula:22193d35-52ce-47de-a75d-2cc6537cbb98}}
{{formula:d45890ac-7b4a-4046-bfcb-db77b41d83e8}}
by choosing
{{formula:a06273b2-db18-42c5-8179-1c35fd675a3a}}
{{formula:38453344-7ba8-40a6-ae01-c9d62893aac1}}
{{formula:acae875a-198e-4d8f-b64d-1f192129156a}}
{{formula:589662f3-e9a0-43e7-8d92-76fe535725c1}}
and
{{formula:7a41a34b-c2ad-42c6-b748-21340c60fd2f}}
we have the representative
{{formula:cf78d436-de17-4096-adc1-37a3cb04af77}}
{{formula:0c56070c-8ade-4bfb-b1c9-a67401bd3d8e}} Then
{{formula:965531e0-55c0-4320-811b-e7b7bcdfb8f1}}
by choosing
{{formula:73b6cc19-7f2f-4fb9-bc9f-5576143f5313}}
{{formula:40396718-2042-42d6-bb00-2e8b08662823}}
{{formula:ae1b073f-f088-475e-b301-b42a47943628}}
{{formula:0e840f8c-2620-4a9d-932f-f27d4769e817}}
and
{{formula:7c963e22-2283-4bb9-9995-dec2b9f6348e}}
we have the representative
{{formula:8972bc97-aaca-430d-8020-9dcca72b68ef}}
{{formula:f0605126-79fb-4d09-9905-7dc5fb747f65}}
by choosing
{{formula:393169c5-8248-4715-bcbb-f16dc81ba02a}}
{{formula:8dfcb5ea-ae18-4c0b-ab50-bd19ead6f408}}
{{formula:2c505bc5-2a0e-4c4d-8be3-4687ad055510}}
{{formula:fe6ebebc-f4b1-4d86-a716-3f2702a81359}}
and
{{formula:f2edfa58-d02c-4553-8dc0-e33517be0575}}
we have the representative
{{formula:7b7012a4-2026-4710-8837-a5423812a872}}
{{formula:7e0729ca-bc07-415b-acef-601df44dcde3}}
by choosing
{{formula:1714fabb-907f-4331-a57a-4bb40d7b65c1}}
{{formula:58154b8a-2137-4cb8-98b6-d96fb0cacb69}}
{{formula:f3096fe2-8041-4a8f-af62-426ddc245256}}
{{formula:7bd9840b-b27f-410a-898f-57f4077bf73a}}
and
{{formula:052ec1e5-53d1-4e40-be49-5792207c48c4}}
we have the representative
{{formula:ddf15f80-38e7-4fc5-8793-e44924476df2}}
{{formula:8e2bd8ad-e7b6-431c-9c61-63e3c5e5ce75}}
we have the representative
{{formula:fd949583-2679-49b5-baed-286209bd2003}}
Summarizing, we have the following distinct orbits:
{{formula:14105455-a4e6-4e30-b4d9-6ce1a7772247}}
{{formula:6f604812-23a4-433a-a420-a0437dc6ae7a}}
3-dimensional central extensions of {{formula:fe67a77f-4dd9-4e69-a2b4-10711978d48f}}
Consider elements
{{formula:589fc3e2-cdff-4e25-a20f-fa2b583dfd08}}
We are interested in elements with
{{formula:5c8a3044-7981-48b2-9d8f-a140b8806349}} and {{formula:ebf8e12e-9e62-429b-b0cf-47e782087abe}}
We provide the orbit of every possible case:
{{formula:acca223b-5f2f-4767-bdfe-d69b984b23b7}} Then
by choosing
{{formula:f9fda0a9-b7a5-4bf0-b161-6d57842fbf5f}}
{{formula:b2d57266-ddcf-461c-b8fa-24ceaed8f7e0}}
{{formula:fd6f365d-c88c-4e0d-a153-98351703151d}}
{{formula:d1498292-18aa-4315-b265-bb2acd23d71a}}
and
{{formula:99d633bf-dd65-4a89-8e1b-9a0ae234654d}}
we have the representative
{{formula:1ecf3aac-502b-4b03-ba54-28bd15ce7e27}}
{{formula:a53b6d8e-2bb9-403e-a810-548f311c7459}} Then
by choosing
{{formula:9f715a04-8e75-4f02-92b8-5db5353cfee0}}
{{formula:85782f00-e127-4fcc-b356-b3f27a677d32}}
{{formula:da8cee89-8249-45b0-ac2b-1d31288226d0}}
{{formula:0e38cd6e-6161-4e71-9b01-7df0e6be697c}}
and
{{formula:7aadf904-2e81-4a46-b2b5-e88e5692d694}}
we have the representative
{{formula:5354d41e-80e7-431a-aa2e-d42bc0d34259}}
{{formula:957c8caa-9209-4088-9005-956bd63f9a7a}} Then
by choosing
{{formula:5a307d1c-4a85-4f9a-8c36-e7227ae1425b}}
{{formula:3eab2b23-e353-4897-b21a-48e345d65992}}
{{formula:2058d778-3d2b-42d4-a7a8-303b8970e729}}
{{formula:d889cc1d-a5dc-49aa-8052-6d74d185af5e}}
and
{{formula:8c907126-d0c9-490c-879d-7fd8f1dc733a}}
we have the representative
{{formula:db835434-ba85-4be7-a144-d4b96d0161e0}}
{{formula:5202f55f-e9f3-4ce7-be2d-efa1d1488cde}} Then
by choosing
{{formula:283cd90d-75d9-4891-9513-9e2a9340a06e}}
{{formula:686b328a-f778-4f91-b600-2ddd6e05339c}}
{{formula:419d4675-87fe-4310-b016-04c4c8c55bb5}}
{{formula:6959a344-c9de-49e9-bd9c-875c5f5e4bd2}}
and
{{formula:59bd0a19-4d8f-49f7-b165-70a5c005342e}}
we have the representative
{{formula:0640c748-9d55-4988-a1cb-97c10eb95b97}}
{{formula:6e5b655d-8856-476f-b291-3e7db2ec9317}} Then
by choosing
{{formula:25487bef-7882-4c95-8f6a-c7cd5fa90fff}}
{{formula:a6a90a2a-e281-4407-a186-cc7664f674f0}}
{{formula:790de01b-33b1-4de7-a215-913d1969d1b4}}
{{formula:e862af3d-63a8-4584-bc9c-caa8bdaab6f0}}
and
{{formula:51c76afb-db2a-4d5a-afae-3ab86f72962e}}
we have the representative
{{formula:4e7a82e4-ed24-4316-a794-2cb1fada7d6e}}
{{formula:899b6136-4925-4cb0-a118-6e43b87e7031}} Then
by choosing
{{formula:2eb302ba-10fe-47a1-91c2-f8f5183547fc}}
{{formula:79358a65-c173-414c-bbbd-041a1b407d69}}
{{formula:50f824ca-acd2-436c-b5d2-77b7cf9eeb77}}
{{formula:a80f03f7-6c27-44a6-be7c-ee567212b8ec}}
and
{{formula:40a10de5-c24e-403a-9dca-5879e4bc0073}}
we have the representative
{{formula:d9de8030-d7ac-4e97-b027-f0bc32c98ae3}}
{{formula:f2d6708a-ada1-4e95-82ef-06b771e05d9b}} Then
by choosing
{{formula:841d2868-4f34-4f62-98c5-b3460d6df32c}}
{{formula:d6d71425-e28a-4739-a926-b4705ec2d0cf}}
{{formula:270eeebd-f77d-4691-879f-25cf91ee6d8f}}
{{formula:dee90dd0-c4c2-4256-bd5d-53237c1ec451}}
and
{{formula:e3d19e78-22d2-4326-acfa-cd5a7c595357}}
we have the representative
{{formula:08afc256-b46b-4866-868a-c81a8c8618ac}}
{{formula:5271a759-ff17-46f8-8c41-641d0c2f1001}} Then
by choosing
{{formula:bcab83fe-226e-4f07-b0fe-b345600df0e9}}
{{formula:8c8a2dc2-4684-449a-8ef3-99f642d46269}}
{{formula:484ba2ad-9b91-429e-9834-204e56f129cd}}
{{formula:f4f651bd-ae9c-4094-adbf-548ffde5a704}}
and
{{formula:cc00c486-50f7-48f5-80f6-604eb23533dc}}
we have the representative
{{formula:867657e9-70f6-4569-9c36-e3e20806d351}}
{{formula:839c6c3d-8933-4edb-bf61-bc76d79e57ec}} Then
we have the representative
{{formula:6b19e63f-3d7c-424f-ad04-b12f7278cbe7}}
Note that
lcccl
{{formula:a8954a57-90a3-4dfc-875c-a26e92ec2e29}}
{{formula:4a4fab81-0198-45ba-b3de-95af0c9e8fb6}}
{{formula:458671e4-d478-4a24-b49d-085b35240db5}}
{{formula:99bf3875-93e2-4d6e-a5e0-8540cfcf7751}}
{{formula:e0487006-7206-46c3-be28-50b881c3d322}}
Summarizing, we have the following distinct orbits:
{{formula:24b039bb-f32b-4094-a386-7db1625994ac}}
{{formula:8526b6b0-7640-4db7-a116-a90c786fd106}}
4-dimensional central extensions of {{formula:83453352-40f5-4ea8-b0ca-6a142c909503}}
There is only one 4-dimensional central extension defined by
{{formula:f63b4f77-7f1d-4de2-af4f-beb75b9df16f}}
Classification theorem
Summarizing all results regarding to classification of distinct orbits,
we have the classification of all central extensions of the algebra {{formula:c20fa6fb-55b0-4f99-8146-96fcc6d5ea6e}}
Note that, we are interested only in non-trivial central extensions,
which are non-split and can not be considered as central extensions of an algebra of smaller dimension than {{formula:64d678f4-47bb-43e0-9b33-182cc0e252d9}}
Theorem 6 Let {{formula:d9d2d1d9-1c20-4eef-8e8a-a19f6e530a37}} be an {{formula:31be9d78-6eaa-4cb5-8ceb-3f55580eb991}} -dimensional non-trivial central extension of the Zinbiel algebra {{formula:a9572fdf-bcf6-4d10-88f3-f53a12550abe}} Then {{formula:c0fba9b4-441e-486f-a0fe-2564c77a5db4}} is isomorphic to one algebra from the following list:
if {{formula:64972e2b-e83a-4283-864c-19a2208c41f7}}
lllllll
{{formula:cf385d47-bc35-4bfb-af26-e3865b103c91}} {{formula:a452abd5-de48-4c81-93ef-d25852812e26}} {{formula:367915b4-867f-4bdc-8207-fd89ea0e89b0}} {{formula:8084ea1f-e86d-42bc-a1ce-a1da03810459}} {{formula:b45ca3f8-527e-4c45-84df-f36a5f1460b3}} {{formula:fadfd9b2-04da-4122-89fc-2410573fd6f6}}
{{formula:38727e63-6414-4b1c-b432-213562eb735d}} {{formula:43bbd222-7f40-4f25-aa60-a347b39ea097}} {{formula:4c603cf3-96f2-4d76-bfcf-e764c1a661bf}} {{formula:f7216ec4-54ed-463e-9eef-e687b7bdb39c}} {{formula:18e19611-f900-4676-9019-23fafcf67561}} {{formula:387d1083-0c3b-4e9c-a366-49dd82449d9b}}
if {{formula:2a254601-2c97-4ebf-b9aa-7e7f206da3d5}}
llllllllll
{{formula:46ac7744-42e9-4087-8aba-2e10fdf88e91}} {{formula:2a12328e-1cc1-4a1a-b2df-07064dfdec6d}} {{formula:992a9204-661a-4b6e-b25e-33e2dedaeebb}} {{formula:bee5cbe5-2354-4959-af66-2fbba2a4f9ad}} {{formula:9186ab1b-8bcf-4bdb-8b5b-e0a148bb49e0}}{{formula:6535f40f-5b72-49b3-8e72-1406a2131270}}
{{formula:1a3ba9b9-bf5f-477c-82ca-2508a1481ae9}} {{formula:27e6a8af-5bcf-48a7-8ad5-def8ebd15c40}} {{formula:0a8e5920-b84d-4386-9f0e-08feff0579ba}} {{formula:2eb85c6d-2671-454a-9edb-b11a7ad47d40}} {{formula:1527cda1-7bed-4aa1-a3d2-7008e1659348}}{{formula:d54a93f1-7170-443c-ad28-2d40725dfaa4}}{{formula:6b2e8280-513d-42c8-ac86-b02473bdc7ca}}
{{formula:06c797aa-0260-4664-9743-88e119e9a479}} {{formula:94773f70-bef3-48e4-b23f-e141bea7bc6b}} {{formula:6ca84b79-27eb-4eb6-bfc6-d6d67895361f}} {{formula:7a81216f-0a15-4ed6-bd88-8521c6c517e6}} {{formula:88f26de1-593c-4b8a-ab67-9107b8a2021b}}{{formula:15d69de9-98e5-4a1a-84e4-6f0581031ea7}}{{formula:4ebe0a08-ab7f-4fd4-a4f5-60be2eb451be}}
{{formula:1ee83b02-839d-4de3-8041-e6b85993d9c7}} {{formula:2551f58f-747e-4276-ab68-d07189a65940}} {{formula:863dc433-daf2-4a01-8745-e5c8a2e5b4e0}}{{formula:51cd18e4-96d9-4b86-a06e-1af9ff4082a5}}{{formula:7057e960-1a6a-4d8a-9bff-b0e951b84539}} {{formula:f2155c96-6611-44de-971d-c4ef7b298cb5}}
{{formula:853670d4-96e9-4ddb-b2fc-a4dee0ff261b}} {{formula:76c17eab-5861-4acd-a227-333cd051fa0a}} {{formula:b45bc03e-1401-49bf-a25b-a1d8b170732a}} {{formula:8dc6b932-3336-4fd3-9b31-9ea51245a0ee}} {{formula:8697efc5-4067-4d26-94a7-1342264e6388}}{{formula:4cc56861-3eb5-4c13-b07c-50583d12966b}} {{formula:6b38f3eb-d20e-4456-81cf-be66ae34d30a}}
{{formula:d24cad0e-1d22-4c93-9059-26b46e8d5c6a}} {{formula:587aed37-3854-42f4-8e79-50e8a8199217}} {{formula:ee3891ba-2c00-4e14-accf-166b013ab340}} {{formula:59f9a90b-3d4c-46e3-8e95-17ed38115e7b}} {{formula:ab401e5c-c5fe-4148-9a4d-6be78338b381}} {{formula:64ada6b6-1d87-4967-8184-ca3d2a1d5073}} {{formula:077c2758-95c8-4719-af5e-ba4176681b83}}
{{formula:da2af858-3858-46d5-ae71-8b507e267c57}} {{formula:006444df-c82d-4a8d-baf0-b435b9d64aaf}} {{formula:5a2915df-78db-42f8-86c0-75a772df3ab2}} {{formula:7eeac89a-5def-4ca5-a8ff-e0c7d73e6299}} {{formula:cca67331-108f-43e5-a79a-8b99cc6ef3ea}} {{formula:a7ab907d-5afa-4608-be8b-809969d6aaf8}} 2l{{formula:f5622a1f-00b6-4b92-b6e9-52b287ec51e1}}
{{formula:29587e1b-de96-488e-a3e0-acbe2e4a1fa7}} {{formula:0d13ff3c-0c6b-4f49-9d5a-43edb5d0eaf5}} {{formula:467ff75b-c6d1-42fe-ba16-8038556523c9}} {{formula:170cdca6-1881-4661-820c-fce6edff2a1c}} {{formula:a3ad38cc-61a5-4821-a932-063b2cd3c9ef}} {{formula:b75536a6-eb3f-4f99-aa19-2866e61cf588}} {{formula:85318e57-39d3-4896-bea8-12499883d984}}
{{formula:3d090bdd-87dc-46bc-9490-333f7a8dc082}} {{formula:1dc94aba-5aa7-48f9-a84f-667b746c7e54}} {{formula:efb97e87-4b1f-49b0-9595-4ab0d5e5d96a}} {{formula:a3dd8962-8b0e-4ab7-9752-654c3550e5f5}}
{{formula:7f6e1d35-4e73-4f19-a910-7d9b78a2591d}}{{formula:7591059d-a7c8-48de-8dd4-549c74a75b7d}} {{formula:a3e9e1c0-f6a4-4979-9e96-5a7cd9e927e8}} {{formula:8786539c-5cc5-4883-baba-4ec5fbcee45d}}
if {{formula:a9b47547-fdbe-49dd-9a4f-f4d296058a58}}
llllllllll
{{formula:2b35c06e-ae58-4c57-95d7-0ac4143754ba}} {{formula:3eb3eaf6-e6d7-4dd5-aad0-9634f34a356b}} {{formula:81469ed4-2255-4f62-8b59-0e8e0765fbeb}} {{formula:118991e5-f63a-4ad2-8554-527167c9e1d5}}{{formula:c0f46afe-b7cd-40d3-8c8a-a041e31c8221}} {{formula:d9c33b39-b090-47ed-a7d7-37b2603d417a}}{{formula:e0c5ca13-31bb-468d-a5ef-2d25fd91ed86}}
{{formula:cb71283c-e10e-4ff8-8d93-e6940dc4d538}} {{formula:99897fd4-31e6-4aed-8324-47cb7aca76d8}} {{formula:8db0d456-352a-4b53-8229-741a451b1de0}} {{formula:dfb1da41-4c5e-4bb6-89e6-68a055464505}}{{formula:8d434ec3-5ca7-445c-a53f-1fb686bb8b19}} {{formula:82948aae-93a2-426c-8859-3a06f93fb19e}}{{formula:0768d655-6ede-43ce-babb-7a2c39a80461}}
{{formula:8ca41c17-8894-44bf-8ccc-b4afa42d2615}} {{formula:4b3e9b2d-2e7b-4409-b0df-a80b93e4d6a1}} {{formula:d3fbe92a-b3c4-47b0-ab53-257636e88df9}} {{formula:fc128bff-ddbc-4f9a-a7f8-92f46ea5e74b}} {{formula:e9125c9e-24c7-468f-8542-1fea172444e3}} {{formula:ae115768-1c9b-46a7-bd30-7ac8a0d0ff8f}} {{formula:c67f5dc9-b026-406d-9660-93fa69379935}} {{formula:11502538-cc30-4c03-9741-434533b225fe}}
{{formula:967b3029-c089-4df4-8ae3-e60bf107a477}} {{formula:00b1ffad-76b4-474e-a123-8e1f976a02cd}} {{formula:777c84cd-0dce-4068-bfd2-0549759ee042}} {{formula:5e1934cc-775d-4246-a11f-826563f1e3cd}}{{formula:50b343d1-9008-4d87-8e76-2db7fee1f7a7}} {{formula:389e4ff0-6700-4577-9982-18b7d7d36344}} {{formula:2a14c190-5012-44a6-930a-4ed8010eaab5}} {{formula:2fbe64a8-e52a-4da4-9dba-28583b1939f8}}
{{formula:633777ee-b4c9-44a8-99a4-1d44459a8a2c}} {{formula:28c42422-291d-4291-9312-f6dc6fb8550e}} {{formula:f087c808-55c5-4bac-82ca-f367f13ebcd1}} {{formula:c431eb43-39db-4ed3-ac7a-3b1cc6523e4c}}{{formula:e64e4261-d3f8-41ba-ac0d-5761c8297ecb}} {{formula:b7a4d844-03ff-4cb2-9af9-3850a87f0207}}{{formula:2c9d370c-047b-42b5-b1bb-3031f3bc17ec}} {{formula:76f0a03b-0675-4b9b-8131-740d6a995b90}}
{{formula:f5c46e24-0da5-412d-b120-7a0a8b6290e4}} {{formula:3e3f4f25-3e46-40f9-88b6-1e20e5fbc54e}} {{formula:fe47ff57-b292-4956-823e-44f6cf296339}} {{formula:a6709867-6ad1-4cf9-85c2-740178662c16}}{{formula:86fc85a2-b8f2-4d5e-8a6e-549c09587977}} {{formula:d989a32f-155e-4967-9a64-a8fa01b43925}}{{formula:7792fa03-a9f4-4f69-aad3-c8972644820e}} {{formula:60a0fd56-1921-48d8-b973-876daee2ff11}}
if {{formula:8207e667-01f7-4572-8e6f-b24598dc7a55}}
lllllllllll
{{formula:f335908b-4843-4650-b929-03d5021c09db}} {{formula:2dc023ee-efa0-4733-9e05-20e0f3b7a711}} {{formula:3dd2e26e-bbe3-429f-9d6f-2fa851db8970}} {{formula:f2a408b8-b55b-420b-805a-98a9c55924ee}}{{formula:04f397e4-3b67-4fe4-8370-91e0a78d6e34}} {{formula:17ca2208-867f-4319-8679-66e3d00b3a73}} {{formula:c726d0a4-af28-49e2-a4ff-be3571bd72b9}} {{formula:77c319f2-3cbc-4d56-9181-750147e4e34e}}
Central extensions of {{formula:4188e84b-5efb-4067-8152-582c97b625e9}}
The multiplication table of the 3-dimensional Zinbiel algebra {{formula:0e3673b0-d9b8-4817-920c-ae0ed5db9790}} is given by:
{{formula:92ba4c7c-a5ac-455d-9604-692b660d2a72}}
The automorphism group of {{formula:5e2bed36-7736-4f3e-9827-38594a5caf58}} consists of invertible matrices of the form
{{formula:20fa79cf-3f35-4eb6-9404-b7e01a16d773}}
The cohomology space of {{formula:63c88483-5a38-4b37-a46a-8972ea9f5d26}} is given by:
{{formula:bfc87553-609b-45e2-acff-f3d7ccf99024}}
Since
{{formula:a60efd66-1144-48cd-ac2a-af4c3bfa2e6a}}
where
lcl
{{formula:55be3e48-0e2f-40ee-8dd2-4999ac4617db}}{{formula:358cd47b-8d26-41fe-92ae-e9a2c253108f}}{{formula:c043e973-3263-4c83-8a36-4dea9cc56938}}
{{formula:e7b10518-b3b8-4737-86ce-7810e4b3bc33}}{{formula:d07057cf-2fe0-4b66-990c-0964ad5678df}}{{formula:6ed33a44-aec2-4508-b78c-ce239c2ca737}}
{{formula:a04a5ff1-1857-40df-9619-43c8ab8f8985}}{{formula:09c346d9-0809-4b91-9e60-b225426cf798}}{{formula:7a19d924-4c78-4acb-85f3-9ff32b576226}}
{{formula:68a7838b-bb26-4042-a81c-2b27da4de55e}}{{formula:270c3dd6-9815-49e8-8ca7-a16b87ca8b3f}}{{formula:4b9872b3-0191-4194-ba1b-c533757fc751}}
{{formula:b61c63a3-30fa-42fb-ba0c-fdecd591cb3a}}{{formula:e187c2fe-b597-4d06-ae7d-b79de60317a6}}{{formula:9f257254-15da-4749-9d23-a2959b704dc4}}
we obtain that the action of {{formula:aec569ea-db55-4dc7-be8e-a09539a6b1c1}} on a subspace {{formula:314ba05f-e40a-459d-8743-1faefce80cc0}} is given by
{{formula:c090701b-4d4a-48da-a0f6-2a83533dcf7e}}
1-dimensional central extensions of {{formula:347c941e-6973-4724-87f2-409c32436c3f}}
We are only interested in cocycles with {{formula:6044dd4f-d2ab-4036-9c40-3936918cf318}}
Lemma 7 The 1-dimensional subspaces {{formula:59577100-ba2b-41a4-8eac-247bbb065124}} and {{formula:40bcb335-5b3b-4afe-bdcc-e49a3352c3be}} generate all pairwise distinct orbits with {{formula:3bb751ae-bbfd-481b-b6c8-696ff3192f47}} .
It is easy to see, that we may assume {{formula:ae3ea836-2671-4bf9-a405-269042c543aa}}
By aplying an automorphism {{formula:141e061a-94b2-42a2-b97f-1757ffc1332f}} in {{formula:86a5d015-c2ad-4e8e-9a80-8878bbc064ff}} with {{formula:a3505778-d438-449b-9d03-671e28052405}} , {{formula:f6358024-9977-424c-b3fd-5abc30ae7b9f}} , {{formula:641476e9-aa6c-4240-a2db-9239b684b36b}} , {{formula:3694cb5c-b5ed-463e-b11b-a3dc0f164822}} , {{formula:a5b3e370-6f35-410e-93ff-b6e3f158c71e}} , {{formula:6e21dbd9-cc6d-4ab7-9d29-414f09fd75db}} , we obtain all subspaces {{formula:599738ad-ce33-4237-859d-3b01f9056b0f}} with {{formula:e5c6f6df-8d66-4cfd-8911-78566f41f2da}} .
By aplying an automorphism {{formula:844dd8f5-0fd7-411b-89c1-e38a73d8f8b3}} in {{formula:fc77d198-9e05-4ac1-a126-b0408a81d540}} with {{formula:bbdea08e-0130-46a4-8807-91cfc2332472}} , {{formula:b68324d7-f87e-4fb8-963a-4c88c5539d72}} , {{formula:948c7085-d2b5-4c6a-81ce-c53b624ace81}} , {{formula:067a4a6b-0a89-4c12-b717-474cbd6c95a4}} , {{formula:522d987b-63f0-440e-8652-412c1cdcefc8}} , {{formula:b727c52e-82d5-44d1-b578-f7eceaef25ef}} , where {{formula:973eecd2-65e6-48e6-b0f5-4c0e72d7bbbd}} is a fourth root of {{formula:9cba1b96-f601-40bd-a626-1594c8ad05a8}} , we obtain all subspaces {{formula:d9caa7fb-46f7-4a62-b5a2-d50542d1e575}} with {{formula:063302b2-7e21-4c70-968b-eb288582d6f0}} .
All cases have been considered and the lemma is proved.
2-dimensional central extensions of {{formula:37fb6124-4f23-484e-9939-3551a89c3439}}
From the previous section, all orbits are generated by two-dimensional subspaces of the form {{formula:8c477d28-4028-4976-95bd-d271a59267fc}} , where {{formula:7c2b3c1c-ac59-4707-bca2-fbc538c0e147}} and {{formula:4d68004f-1aad-4af0-ac84-2bbd6ba4a91b}} or {{formula:465e7398-4a83-48d7-83f3-f76b9086c56b}} .
Of course, we may assume {{formula:097bf88b-5d0e-4ab5-85bc-ba706a01adc5}} .
Lemma 8 The following subspaces generate all pairwise distinct orbits.
2
{{formula:50db08cf-c3bc-4bea-8c0e-b97f82a3dff4}}
{{formula:178b72b3-2ed1-41a0-babb-cef6d2113e2c}}
{{formula:cc7cbdf2-1ce4-44be-80c3-4b90ee679f20}}
{{formula:3f8bf5a9-b267-4cf8-b398-f850ce5861f9}}
{{formula:d9f45e09-b1c6-4536-88c2-b44e6d2dd4ed}}
{{formula:df5c851e-78d1-4981-a7be-75d247ea280c}}
{{formula:cd1c87f5-fd3c-4ca5-a7f2-6362d2548ffe}}
{{formula:6181eb05-2e94-4b90-86bf-33b373599fd6}}
{{formula:006d8fca-163f-4567-91ed-e7eb2f19df2f}}
{{formula:8727abb5-a380-4395-9bb6-b4e245195799}}
First, we observe that any orbit is generated by a subspace contained in one of the sets below:
{{formula:115ee475-c093-4760-a67b-79e5298a2b6e}}
{{formula:26935f72-b42c-4611-af0c-f15b99e2485f}}
In order to prove the result, we need to show that all elements in {{formula:7b086e2e-2c60-4377-b9ae-8138b2333e58}} lie in one of the orbits {{formula:d413fc66-aeff-4f09-9c7f-d885a3ab080d}} , and that these orbits are pairwise distinct.
Applying the automorphism {{formula:f342ab42-eb44-4404-a3cf-fee7e67d4e8c}} in the subspace {{formula:be2541a9-9681-4d5a-85a1-4db4b2bab87f}} with {{formula:7ca1f767-0767-4ded-a2e4-e11856936e2f}} , we obtain any subspace in {{formula:dd762871-bc70-4bac-b374-36aa52957109}} with {{formula:106c5626-5d44-49ba-bdf2-3a87035d6799}} .
Applying the automorphism {{formula:028cd3e3-4a25-4937-8acd-d6627fbfbba6}} in the subspace {{formula:d202f984-6dd0-41e7-a2bf-b8a1d41c57b3}} with {{formula:f8cab42e-9ffe-4e04-b710-6ab1848df6c4}} , {{formula:83251459-13a5-49e1-85f2-b3d36f5a6494}} , {{formula:3b67db0b-5291-4762-8571-585341288d61}} , we obtain any subspace in {{formula:e0e72e7c-efe7-42fe-87c8-769b9c6fcb2f}} with {{formula:42e44044-6d8b-4fa6-843e-40063633b0ea}} , {{formula:44138b95-3688-4996-8f28-79f772532a16}} . Moreover, any automorphism {{formula:549a0393-fd13-49f2-a5ec-e4cef811fd00}} applied to this orbit does not contain {{formula:dea3827e-bdfa-43d4-934e-ed301dde789b}} . Indeed, by applying an automorphism {{formula:c8b0f27c-35ec-443f-ad64-e1f349e376bb}} in {{formula:5c11096a-38bc-4e7d-88ef-048daeb10c16}} , to obtain an element of {{formula:b32fea3c-dd1a-48e7-9d43-3959ea7542b6}} we must have {{formula:b0d97c44-6e72-40ad-ae8a-ec0d9fd07736}} , {{formula:2c79031a-027f-46ed-8707-3e3971635c97}} and {{formula:52968f2a-9408-45ce-9903-3805f04d25ef}} , and this yields an element of the form {{formula:34fb4183-0c55-4ed0-9075-164239428afc}} , where {{formula:a00c08b3-de2f-4e48-9c5f-63f0672d70de}} must be non-zero. In particular, it does not contain the subspace {{formula:c6bf0c2a-3653-4e08-985a-cea0475fe11d}} .
Applying the automorphism {{formula:07373d2c-4e97-4a07-a409-379de3b92046}} in the subspace {{formula:cb3c5b78-129a-453f-ae52-eccbdc439e87}} with {{formula:46bf605b-36fc-4b71-ad98-d44bc6ae1751}} , {{formula:ec8ba27d-7753-4732-b8e7-d347f05f5329}} , {{formula:b986e66a-c792-4116-a19f-689d5800c81c}} , we obtain any subspace in {{formula:3cc145f6-1db1-4bd0-a8cf-b87e00a7db10}} with {{formula:3020b3dd-2b15-4e97-b256-19be8c784401}} , {{formula:62226b52-558f-431a-ba60-eda4d4fc495d}} and {{formula:821bb80a-9801-458e-9513-a11560c9e3bb}} . Arguing as in (REF ), any automorphism applied to this orbit does not contain the subspaces {{formula:b9464ac4-4d6c-439a-a518-9e4e29624a9d}} and {{formula:794fe3f9-f659-4110-91f5-d4b3d95c378b}} .
Applying the automorphism {{formula:33a178ed-80ea-404f-86f6-5d371147594e}} in the subspace {{formula:08fb248b-c47a-4bee-b949-a9e02bc788e8}} with {{formula:1c456111-547d-41ce-a608-ac931ef70de9}} , {{formula:dcc9e81d-a6cb-4b83-bee9-f545b1f20676}} , {{formula:e83a1a2b-4163-4c0f-9392-29a4bcb84b1a}} and {{formula:ec0f54a1-0f3e-463b-b6cc-599b9e7decfe}} , we obtain any subspace in {{formula:0585f1db-33a7-4bb1-8c95-df8785d57910}} with {{formula:df092b65-92d4-4449-af7d-274f6a4d3edc}} , {{formula:cd605f5d-801b-48c8-bf25-0c91f958a5b1}} and {{formula:d52dc321-ad33-4702-a137-11473e9ada8d}} . As above, one verifies that this orbit does not contain the subspaces {{formula:67d89141-e8a0-460b-b893-fa1bccb29ce9}} , {{formula:478c0531-beea-4299-b442-dab0bbba5f7d}} and {{formula:846509d7-f8c7-41be-bf36-cf4f94d5205c}} .
Applying the automorphism {{formula:feca1ddf-e096-42a0-82d9-29f14671874e}} in the subspace {{formula:929cd6ee-9321-4745-880d-91a7202b0f96}} with {{formula:a617a583-9bb1-4749-83b8-ba75def8e9e6}} , we obtain any subspace of {{formula:c2f1c72b-77b1-46c2-866d-c10c870c70d4}} with {{formula:926f3d0d-b477-4ef4-85f0-e683388e5fb6}} and {{formula:13bf1854-3163-45a2-ad90-4eb6c9104fdb}} . Moreover, by applying an automorphism {{formula:76cd984a-72fa-4012-a9d7-3b7c8bd3bfd8}} with {{formula:8dd784b2-042a-4a5d-8a36-183bf8f29381}} , one obtains an element of {{formula:9c1c92f2-2e3e-4571-8372-d73e56a977ef}} if and only if {{formula:2d503f28-da86-4375-b519-d4465d0b3a7e}} , and the resulting subspace will be {{formula:04928bd5-671d-4aac-9629-a29d6346e10a}} . On the other hand, by applying an arbitrary automorphism {{formula:74e0d9ce-ba52-4564-8a89-c19f53e7d50a}} , with {{formula:225b7f00-3d09-40ab-a399-b9af5f526202}} ,{{formula:b519f159-62f0-4548-ad80-a9eb8e9db3f5}} , we obtain the subspace {{formula:c6355554-af00-4867-9b7c-8058b271a10a}} , where {{formula:a7d24774-4fc6-40a1-8a5b-ba8a141fc8ec}} . This is an element of {{formula:1b61b9cc-8cb1-4e43-ae9c-a5bbf19db644}} if and only if {{formula:2458fc93-1c71-418a-80ff-3d0ad04ff1e0}} , and this also yields the subspace {{formula:942f5273-e749-4a41-b6bd-2d964a938537}} . As a consequence, this orbit contains no other subspaces of {{formula:73e898f0-f148-4d5c-a9fe-3e77a9754821}} other than {{formula:979ece5d-3851-4de7-afa1-caed99a134ab}} .
Applying the automorphism {{formula:57684ec4-afa3-4798-abfc-eb04198610b5}} in the subspace {{formula:5736c1b3-5fb2-40df-b8d8-05273303652c}} with {{formula:1869d63d-c329-437b-aaa2-c723b4071ee1}} , {{formula:4fbbb777-703c-42d5-ae63-e758910f42ff}} , {{formula:0905e2f8-bc02-4e10-8932-807a41f95ed6}} , we obtain any subspace of {{formula:2aeb9d9b-a3d1-4c92-9be0-217915a97fa5}} with {{formula:795d8716-a054-47a8-b7df-1d1886f061c8}} , {{formula:5183736e-b88c-494f-b6e7-5a54b22493f8}} , {{formula:99955b77-eae2-41f5-8588-2c6ac1d42472}} . Arguing similarly as above, we conclude that this orbit does not contain the subspaces {{formula:f19f33f6-5a4b-4adc-acc7-d3cf435d1dc0}} .
Applying the automorphism {{formula:73d044bc-f04f-4b12-a8b6-4f9fb5dae691}} in the subspace {{formula:5678ab9e-f3bc-4c99-a293-48f7f6e34971}} with {{formula:e5270f07-52a8-4bb7-9dab-fa5748b7714e}} , {{formula:ce93283c-d8e5-4b8a-a3bc-ca7c5d3ddaa6}} , {{formula:5d7bc492-1f79-412c-91cd-540c0c2c6a81}} , {{formula:364be9c2-2a8a-4847-a113-0f1e02057aea}} , we obtain any subspace of {{formula:fc42e064-0f1d-4137-a305-93bc78c78bb8}} with {{formula:e697fc88-2edf-423e-9694-d77da5035e99}} , {{formula:e3d70c9a-f629-40ba-b3de-71a263b8d50a}} , and {{formula:a7ee7e1a-984f-460a-8a69-2d244bb1e368}} .
Applying the automorphism {{formula:80fb33f2-0d81-422d-bbe9-cfce8d826315}} in the same subspace with
{{formula:7e0fea4e-f0b8-469e-8f2d-0aed4726dced}} , {{formula:7f96f871-2e8f-486b-9c94-287b0bf1c46e}} , {{formula:ba252dab-0c46-4fa2-acb2-b6929b3f8376}} , {{formula:48757b9a-a2c1-42d1-9697-fbdaeaab3a77}} , {{formula:f5f9e920-179b-4fbe-b004-8d0ddb37ab4a}} , {{formula:607e0736-dcb5-417e-90c6-1bffb9245567}} , we obtain any subspace of {{formula:5eeab09e-4bf7-439f-9b4d-97edc93b6d84}} with {{formula:e2c0209b-ee8a-4217-be33-57f9e74c513d}} , {{formula:0ffed952-8b47-413c-8e68-4cc29646e56c}} and {{formula:07caf512-9992-4011-8ae4-e3de18f3473d}} .
By applying an arbitrary automorphism {{formula:c42f5d69-5436-457a-9f79-a76274972e4d}} in this orbit, it results in a subspace lying in {{formula:d4d5caff-44ab-4dac-8a8c-9ea0944d9d2e}} if and only if {{formula:f1bffdc6-5c7d-44e7-ba57-87d4ac5fdfeb}} , or {{formula:edea7d40-faac-4034-9901-933a463197b6}} , which result in the cases already considered in (REF ).
Applying the automorphism {{formula:3b975efc-e109-4dcd-83cc-c108ae0ddf3c}} in the subspace {{formula:3aebd361-831a-4c57-a22d-fe686368e5b7}} with {{formula:b702bbce-80d4-4ab1-9cdf-d2526acf64b5}} , {{formula:03cc8646-029f-453b-8efc-b8bb56418dc5}} , {{formula:4891c7d6-b3ed-470c-be31-ea1638f37172}} , {{formula:34d053fa-b44a-44dc-8b73-d1a8250f3620}} , we obtain any subspace in {{formula:0d907889-b08e-429f-a832-6ad72db6158a}} with {{formula:49f95d6f-078f-43af-85b8-1f23972cb0fe}} , {{formula:44872b88-153e-4f32-90e3-4973d6d39fb3}} and {{formula:873be7a6-67d9-411f-ab5a-e579d7d7a3b0}} .
By applying an arbitrary automorphism {{formula:b1aa4182-a05d-401b-b793-d3d0c0ff3f92}} in {{formula:8ce54936-e7fd-44d3-9993-1ea5d75f7bf4}} , it results in a subspace lying in {{formula:393e260e-8ea8-4946-a6df-7ebde8c9dc68}} if and only if {{formula:6a1c54be-fefd-45d9-8b6f-a589084a6210}} . And the resulting subspaces are different from the subspaces {{formula:ea0dfc36-f10d-4eaf-a0e9-efaf95d4c6b3}} .
As a consequence of the above, we obtain that any subspace in {{formula:1e35f1c9-aebc-4ec3-a7be-240e551038e3}} lies in the orbit generated by one (and only one) of the following subspaces:
2
{{formula:2fc6dc88-cbd5-464a-8c46-8fda8e2855b5}}
{{formula:66f7208c-0708-40df-8cf6-3db01df02af3}}
{{formula:38b08ad4-e23c-46ab-b36b-4927c75fd8c1}}
{{formula:941a4a35-b364-40a3-afa0-30438a258573}}
{{formula:a921971e-7b2c-485a-a567-79587dfa385e}}
{{formula:cd3233f0-9dbe-4bfc-881a-3a4a39f54583}}
{{formula:5e1b26fe-70e8-43d2-86f5-e3c7b3a638d0}}
{{formula:9515f88a-4466-4117-aa2a-ff02379705c8}}
Moreover, as mentioned in (REF )-(REF ), {{formula:2350d18e-6087-4436-947f-1149ccd42696}} generate pairwise distinct orbits.
Applying the automorphism {{formula:efbe3173-2de0-4a13-8fe9-d6a99928e510}} in the subspace {{formula:d48fe698-19f3-4416-9889-16c1f3891686}} , with {{formula:cb5cff29-6631-4877-89b0-7b0dff592591}} , {{formula:22e84a8d-077e-4c88-8a6f-d4c967ebdd48}} , we obtain any subspace in {{formula:0e4d0657-8b09-4589-b023-3360646850a4}} with {{formula:26df623b-38d8-431a-a083-6b3ecd250cca}} . Moreover, any automorphism {{formula:0172b6bd-5b58-422a-9042-580e790101c0}} applied to this orbit does not contain elements of {{formula:eb923b24-c30a-44e7-88b1-926db2f767ca}} . In particular, the orbit generated by {{formula:b93b6744-c21e-4ef7-8e0c-c73200fd9514}} is different from the ones generated by {{formula:58acbd1c-82d8-4027-abab-ba2555e42142}} .
Applying the automorphism {{formula:7e8512bb-f6f8-489d-95bc-908c1258804e}} in the subspace {{formula:88950869-757b-4ba3-8b2f-ee1e0c4c4f78}} , with {{formula:4a05af9b-1f2e-48d5-bb8a-c5ecbca74476}} , {{formula:0778d658-42b2-43fe-8340-e1f897492e73}} , {{formula:25431b7a-d4b7-4d7c-a6ac-dcd5078eaafe}} , {{formula:d8bb4cc4-9698-4721-8894-12adf6c807c7}} , {{formula:a805d968-c65a-4c72-b1a5-48401fc557e3}} , {{formula:c48fe0b5-c14f-4c5c-bfb0-a087fc006959}} , we obtain any subspace in {{formula:2364e181-f834-4e2b-872f-c82cf1ca7f0d}} with {{formula:a2df5ef2-e048-41da-9e15-4e0bc5c23111}} , {{formula:8ae495c4-b432-4a9c-aec0-6e30da8c382d}} . Arguing as in the previous case, we obtain that this orbit does not contain {{formula:ad61ad80-6bc4-478f-88e3-a47bcd7b6684}} . Moreover, if {{formula:56ef56eb-2f4f-404a-8a2a-b1ee3a4dc850}} is an automorphism, applying it to {{formula:d125bc4d-0e02-4c5b-98f4-d7b9d1ce6031}} it results in an element of {{formula:8976e891-f9d2-4831-bbcd-8aba9e859cf7}} if and only if {{formula:09527084-2423-44c0-91a1-9ff7d81b1c0a}} , {{formula:c0d7b75b-afa9-4860-a8ee-b291bde64664}} , {{formula:ccb69084-b18b-4b7d-9014-9639d1aa7d1f}} and {{formula:0410b19b-3302-4748-8c00-7f29504896ba}} . As a consequence, the resulting subspace cannot be {{formula:7db8d93f-a5e7-4e94-b33e-753b511d5121}} .
Applying the automorphism {{formula:c382e25f-a25f-4f50-97a1-32c4b93e3b6f}} in the subspace {{formula:0fc319f1-6925-4242-8a6f-b577e494b25f}} , with {{formula:60b36de8-d78a-4144-92f6-e6385089ed7b}} , {{formula:ef4fa3a6-6e23-4b2a-b851-43c9901c51bd}} , {{formula:f4ce120a-a9e7-4788-8d74-8146cd22a646}} , {{formula:9866cab8-8ed9-4dfc-bf24-5beb618e0785}} , {{formula:f136240a-148d-496d-bcc7-52782e681cae}} , {{formula:1f15894b-12a2-4e1f-870b-05f1ded067ff}} , we obtain any subspace in {{formula:4f67c2cf-692a-4048-bc4e-e0a2c31353a8}} with {{formula:95a3bbb6-965c-4de7-ad76-e97df576ab95}} , {{formula:3ac03233-a6f5-4cb9-84f5-0dacb4ec0adb}} and {{formula:e23a08d9-db68-4d20-9d3b-b6e3ba7a5e81}} .
Applying the automorphism {{formula:a562b5ac-80b5-400c-99c2-d1648782efb6}} in the subspace {{formula:046914a6-8022-469d-985a-3415b92540d0}} , with {{formula:564847aa-49d7-474a-950a-dc1346eb4aaa}} , {{formula:0b6e5428-380d-4850-8afa-0b21591efd3d}} , {{formula:1fa08031-2b2a-4cad-ab70-e74032920695}} , {{formula:699c85e8-43c6-4b27-abae-c77a76c995e8}} , {{formula:475511df-8441-4fba-98bb-4b2b83df62d6}} , {{formula:8b10922d-1859-4957-9b58-83267dec244b}} , we obtain any subspace in {{formula:885a17c8-8aba-46fc-bc77-daccb4115ec5}} with {{formula:7b2847dd-6d3c-43ce-92ff-d9457e614ab2}} , {{formula:42c4426d-b4d2-44fa-841d-190b9cdd8908}} and {{formula:ee99a2d0-23fb-4a3d-8cd6-0c247a7e6c20}} .
Applying the automorphism {{formula:f6956ec9-4502-49cf-b0f3-b328efe5bbcc}} in the subspace {{formula:53ec4183-2ecf-4d55-af31-3189e2706d78}} , with {{formula:80a246f5-de8d-4116-9350-5693290f5670}} , {{formula:72c061a6-31df-44e8-a297-d210881f95fa}} , {{formula:d4e92435-7c17-4908-b13a-1c286828318b}} , {{formula:36c41961-91e5-4f5a-a952-84389869c0d4}} , {{formula:5a573e11-27a9-48dd-9f31-96c772b641f8}} , {{formula:a72c300c-fc7b-4e4e-a501-bf84bfafc352}} , we obtain any subspace in {{formula:1a90448a-0f1c-436e-a506-2b226a29acb1}} with {{formula:56d9e3a1-70dd-4ea4-bf4f-0df23c030472}} , and {{formula:f75975db-6ec2-42c3-9ad0-ba47f19dc199}} satisfying {{formula:f148441e-b97b-4ab2-9551-32f59d7990df}} and {{formula:27cf988e-1497-4da3-999a-90d3fcd46e9f}} .
Applying the automorphism {{formula:dfd9f606-1bef-488a-8559-c3fc7064c52e}} in the subspace {{formula:0327c969-b576-4c89-b1b7-d7e2cdca6510}} , with {{formula:99f639f0-6be0-4c82-9324-4dc8e87f711b}} , {{formula:caa8b441-40c5-4b89-a289-a30e00845365}} , {{formula:4a4bff3e-7d92-4e73-a948-1d0bb0336400}} , {{formula:f965838b-5012-46ee-bf8f-3f0c30cc4ab4}} , {{formula:aa0fc04b-1c54-444c-badd-d9382f89968d}} and {{formula:d5155b17-7b2f-4ab5-ab70-ede07ced89c0}} , where {{formula:45c40448-0eb8-46f3-b314-eb3511498c13}} satisfies {{formula:b4b9b41c-0731-42d5-b475-87ad7ba40c11}} , we obtain any subspace in {{formula:5ab32ee7-10af-4eb1-ab36-f7dd2abe4016}} with {{formula:9d61788d-1623-427e-ac6c-60fadd1d6fce}} and {{formula:6a9c8f43-f368-44aa-abe3-c5ee0cfc8478}} satisfying {{formula:e448aa39-73da-4828-a5a2-a2fc7c422fbf}} and {{formula:89f21709-c6cb-4819-8c93-904bd1d62215}} .
Last, we remark that the subspaces {{formula:9e8acbd8-0f0e-4cb6-98cf-6e6e00afcfc2}} and {{formula:3af175d0-1843-42cd-9c42-3655c5de6c1e}} generate the same orbit. Indeed, by applying the automorphism {{formula:a2745f6e-dae8-4812-b6a8-7ddcf66e868d}} in the former with {{formula:1441aa91-13eb-403d-9ebe-7f8b79dd60b0}} , {{formula:1d65a387-7b2c-40f8-a47f-2db2a849bc46}} and {{formula:c1ace6e7-c66b-4efd-a710-9624f9aca9ef}} , {{formula:7d3d7bbc-08ee-4dc4-9044-bf6dcc06550d}} , we obtain the later.
Applying the automorphism {{formula:a9671cc2-cfda-44be-ab21-c8c3ce04f09f}} in the subspace {{formula:0ec21bdc-dd54-4d73-90ec-3a87f25e0cac}} , with {{formula:854e8d68-2850-499e-aabe-fbec1da5822b}} , {{formula:2adcadbd-8582-4fd5-8893-592cb27a1103}} , {{formula:be2953ed-f40c-424d-8319-5359178e2d1d}} , {{formula:831386ba-d321-4dbc-b70d-70926421327f}} , {{formula:353ecfaf-d887-4601-b42d-e211134adea0}} and {{formula:469155ee-a43d-424f-a15a-14f783f70da0}} , where {{formula:c48256aa-d112-47d7-97a3-16d9673a156f}} satisfies {{formula:67f33669-c3bd-4c54-bb22-aaba8ec2ad5f}} , we obtain any element of the type {{formula:784ea8c2-1fb3-4fbe-9896-210e9f3e921a}} in {{formula:ca752591-1b6e-4f1c-b7a8-3a6c6b368b16}} with {{formula:a0299593-a409-4cdf-b496-e3d85f72e73d}} .
The element {{formula:2142475b-e347-4041-b20f-711d7850dae1}} generate the same orbit as {{formula:3f41ff76-c43a-42fc-a5a4-2114524da57d}} . Indeed, by applying the automorphism {{formula:8b325537-e88f-4d2d-8c6a-accef6ca8fab}} in the later with {{formula:4a5061b1-2920-4e3c-9bce-bf3a5d36c2ff}} , {{formula:6fb7e8b5-5fb5-4c58-b000-ed1e49124599}} , {{formula:6d7ca42f-178a-4d49-845f-efe97c0a9554}} , {{formula:ed3ccdfe-ce40-4166-8bbc-3ed764de5f6d}} , {{formula:17af17a7-55b4-4a9e-a8cf-152aa23c3fd0}} and {{formula:2cc09681-34fa-4319-85f5-5ee4395d959b}} , we obtain the former.
The element {{formula:6e22b32b-1a95-47ec-a00c-b70963c055f8}} generate the same orbit as {{formula:778e2ba9-9df0-4cef-801c-d11823fdd78d}} . Indeed, by applying the automorphism {{formula:be0a7206-9202-4ec4-a648-3eedc6acd11b}} in the later with {{formula:80c815a5-86f4-42dd-9466-6eb2771e7a52}} , {{formula:9c61f827-89e7-4afc-94ad-6e31ea601a4a}} , {{formula:1cc45943-0ff0-4f7f-8677-a929d5069d8f}} , {{formula:a6ee12c7-1239-4ac4-abbe-01c6b8bd7fa8}} , {{formula:a02a393a-e25b-4c48-ae75-707ee2af995d}} and {{formula:0b97b22b-b3fd-4639-95e3-e1aab45b8ad4}} , we obtain the former.
As a consequence, for any {{formula:591e2ded-45cf-43b0-8c9c-613704f1bc6e}} , the subspaces {{formula:1e3c77b6-f52a-4e35-9472-94e39c73349d}} belog to some of the orbits considered above.
Let now {{formula:519f8904-635f-4057-9ffc-d584d081061f}} {{formula:73fb456c-a98f-426c-84ce-96c90090d3a0}} , {{formula:c9f90505-2d50-431c-8613-a294603d71dd}} satisfy {{formula:7643db50-34be-4937-a858-a5bf463e558f}} and define {{formula:a213fa8a-4d5b-4097-bff2-cf72120ad2d2}} . By applying the automorphism {{formula:563681a1-f1ad-4184-8c75-2ce82353fec1}} in {{formula:5909aa77-2b03-4515-8709-ca9ff45f590e}} , with {{formula:ad394b03-b97b-47de-99c3-af9d081b5286}} , {{formula:16941813-4bfc-4b89-875e-fe6dfc3ac516}} , {{formula:684823e8-8261-4df4-bce3-96c0f45a3e47}} , {{formula:20c3fbe0-a245-4ac5-a781-5885e5417c5f}} , {{formula:8fe7ba93-66dc-4fbe-920e-d9232b911b99}} and {{formula:e859f9c4-377a-4a12-a733-186a3affde49}} , we obtain any element in {{formula:e47f3b07-7046-4c90-a024-d5979046918f}} with {{formula:db0ab2bb-7da8-42c2-b77f-30be751f35cd}} .
Hence, all subspaces lie in some of the orbits generated by the subspaces {{formula:bb4029a4-85aa-45b2-9ce9-dbc18219b4b4}} , which generate pairwise distinct orbits. The Lemma is proved.
3-dimensional central extensions of {{formula:0970e359-e392-42cb-908e-71ead9ec5602}}
We may assume that a 3-dimensional subspace is generated by
{{formula:fc70a7ea-a299-4771-8bc6-1aff4dac4626}} and {{formula:5af5940c-5178-4b15-a928-6d4dd0121f64}} , where {{formula:31a79070-32cb-4461-9470-407d5ba21887}} .
Before the main result of this subsection, we need the following technical result.
Lemma 9
Let {{formula:ea6c627f-2b48-4b12-8bc2-56fa9253436a}} . The system of equations
{{formula:e9e8ace1-ccd6-40e3-acc8-96c1cfeab3db}}
has a solution if and only if {{formula:6af2fc51-5e0d-4bc6-8355-86be7e2b6069}} .
Let {{formula:93d8185e-78c9-4bad-bfdf-bf1502ca3151}} be a solution of (REF ). Then {{formula:d330fda9-bb5a-43f4-82fc-f459f8d628f5}} . Let {{formula:2191fcc7-c8ca-4a77-9b11-81f9c65db634}} . Substituting in (REF ), we obtain
{{formula:57f65909-0d91-4bc1-ab1e-d74747161d2e}}
Then,
{{formula:10f9ac31-d676-4ee4-bab9-e30d1c958632}}
and {{formula:989da84c-1d2b-4110-99b1-d3cd54c2df7b}} must satisfy the equation: {{formula:27063905-2166-4460-b7cd-264bef05c9a5}} .
If we denote {{formula:4e085d20-5b2c-42b3-b561-7e494e9d82f4}} , the above equation becomes
{{formula:42883270-bfa6-4107-ad46-8a28a9fce4ef}}
Once the above equation has a solution {{formula:74c83d52-a35d-4af6-b0e6-4c23797d2598}} , we obtain {{formula:2bd101aa-c715-443a-b7d5-03c7d280e319}} in (REF ), and we find a solution to the system (REF ). Of course, {{formula:500a3bfb-fefe-454a-bde7-eb15e8de1b4c}} is not a solution to this polynomial equation, otherwise {{formula:a9c32c2d-aef6-452c-a8cc-40a300fbd755}} .
Observe that {{formula:733fd402-e0de-4f46-8f7f-4309e8059581}} is a solution to (REF ) if and only if {{formula:b51d3cf0-acbe-4fae-ba26-dc1a1b5db0e2}} , and {{formula:363dd197-16c3-425a-875d-1a39f3a73fca}} is a solution to (REF ) if and only if {{formula:91965e5a-0923-4c50-8431-786d8a6ae231}} . Moreover, if {{formula:6e0213c2-2a36-4749-942a-aed025849cb7}} , (REF ) becomes {{formula:fba84f08-b9a9-4780-b9ba-1f7563b283e4}} and if {{formula:ea11ea0b-2f2b-4bdd-a660-32ccbd3cd2bc}} , it becomes {{formula:85f93b98-6165-418b-aeed-a67ac5dfc7db}}
As a consequence, equation (REF ) has a solution different from {{formula:368f4ab9-4dfd-4fef-b4e0-0b266aecb0ab}} if and only if {{formula:1a74a2e8-5688-4579-91b4-1c07a5eff391}} , which is equivalent to {{formula:4f76c7ad-f43b-4e14-b8d1-0a651de32082}} .
The proof of the next lemma is similar to the above, and thus is omitted.
Lemma 10
Let {{formula:5beaae46-05c5-4000-9bfd-31cd92bd7958}} . The system of equations
{{formula:7395de6c-46dc-4429-8c10-4272742398be}}
has a solution if and only if {{formula:8945b655-ebaf-409b-b222-61f9fa4cb53f}} .
Lemma 11 The following subspaces generate all pairwise distinct orbits.
2
{{formula:0c3d9fb3-da38-4a78-b183-7463b12a0c18}}
{{formula:4c129257-5068-4135-8e58-649d890658a0}}
{{formula:7a222813-9dce-4b02-a47b-a8319ac7e00e}}
{{formula:7b3226d6-8a84-4de5-870a-ebbaa0e2c961}}
{{formula:7dd2fbe0-6c36-440d-8099-4b79e029c04e}}
{{formula:ab60dc17-beea-4991-94f1-b81abc7bb187}}
{{formula:dd2b06a5-b045-4438-829a-600127479611}}
{{formula:1970ea40-1f9f-493c-a530-24cd8d1369fb}}
{{formula:75c3a4a1-c2a8-4baa-b5d8-d7de86cc08fc}}
{{formula:4fd8b6e7-2ee2-4940-a04d-e17abf2ebf2e}}
{{formula:2f8e512b-ffaa-4ed2-afce-7bfa82a69fd3}}
First, we observe that any orbit is generated by a subspace contained in one of the sets below:
{{formula:e99565a0-211c-4aa4-9f88-b2b211dfd834}}
{{formula:7e7aa5f4-5df8-4cc5-bffe-b3241caf0561}}
{{formula:b4d0bbf0-fa72-4417-993c-005e4b3a148d}}
{{formula:86c13d59-e1be-409a-8ee5-4c46574b5451}}
{{formula:2bd4180b-187f-462d-8d08-840d7dbb9e03}}
{{formula:65a44be6-e864-48ef-b7f4-efad73f23ea8}}
{{formula:47200d64-050c-4b24-88e6-d9b6e2365f28}}
{{formula:3c325fe1-0175-4656-93b4-07c00c5e4d9d}}
{{formula:ca28f290-8cca-4a80-8eda-36c0d4d2aaf5}}
{{formula:01a3617c-16b8-4eea-80e3-664fc38b303f}}
In order to prove the result, we need to show that all elements in {{formula:266c7855-6ff6-4383-a8a9-106f80547ed6}} lie in one of the orbits {{formula:51f62c35-4dce-49d3-99cb-1e5833c44357}} , and that these orbits are pairwise distinct.
{{formula:cfbfc12d-e1f5-4398-81cc-48d5dd051561}} .
Applying the automorphism {{formula:e98564e7-9160-4015-9955-de05fd89abdc}} in {{formula:77f0872a-fb53-4645-95ed-ab646237f72b}} with {{formula:cf2ce2b7-83dc-4031-a65e-8faead096f54}} , we obtain all elements in {{formula:1da7cdef-4717-41eb-beb2-a1794b325852}} with {{formula:7d102547-10f8-486b-8583-d324f53e046e}} , {{formula:9ad2de87-bb5c-4189-85cf-ffbc40835aa4}} .
Applying the automorphism {{formula:88d39137-0d84-439c-ab4e-674818209938}} in {{formula:268f9876-9f75-43a6-a7c9-7eee6da4cc51}} with {{formula:20bea762-7a44-4408-b652-e252e4a551b0}} , we obtain all elements in {{formula:f20a4cf2-f820-468a-b3c1-73e04e11f353}} with {{formula:b32ebafb-cfe1-4150-96f9-0025483741e4}} , {{formula:4cd406aa-0f4c-4b1d-8003-8f9b8b505f79}} . Simple computations show that this orbit does not contain {{formula:4a3af619-a64b-4400-a254-b218ea7298ff}} .
Applying the automorphism {{formula:0abaea71-9c66-4633-8d11-318644b1bd10}} in {{formula:51e1c90a-eab4-44bf-9e7e-5ad8f7146538}} with {{formula:a1403ff8-96f7-4dd5-8067-8e810c94e8f5}} , we obtain all elements in {{formula:21831834-afe1-4552-93a1-2f5d57b1597c}} with {{formula:f6f84ae9-2402-44f5-b74e-a9f2dd511ec2}} , and {{formula:f5a12f93-0513-4917-91fe-b4662beb6f60}} . Moreover, any automorphism {{formula:77f863de-1ae9-452f-9221-291e354783e9}} applied to this orbit does not contain {{formula:73815cad-ed33-481e-b7d0-25a798a68e9d}} and {{formula:53fa2666-6e89-4b5a-9a55-28ff0232400d}} . Indeed, let us consider two cases. If {{formula:a032c24d-8ba9-494e-9057-61b55b134dd5}} , then one of the vectors is {{formula:32e2eb5e-0aa3-49d4-871c-afcdea7c4ac9}} . One of the other vectors is {{formula:1424f520-feb4-4cf2-862e-ea83d9e4764d}} only if {{formula:156b0c2e-eb3f-4b80-a9d9-d0bc51de1258}} and the last vector has nonzero {{formula:884ec9e5-4179-4842-84b5-879d8c6d03a6}} component, and this case is settled. If {{formula:b2d27a7c-1755-4af5-9a8e-06d5d0c1fb41}} , one of the vectors is {{formula:7d209b4a-a348-4e86-9f8f-78b4bcab375b}} only if {{formula:d7255353-884b-4634-9498-aa21a7855d73}} . In this case one of the other vectors has nonzero {{formula:6a13072d-a0d6-4e37-9a43-756a17230305}} component. In both cases, they cannot be {{formula:aeed6485-e60d-4d26-83a0-a8981156cfc0}} and {{formula:582d2689-0bb1-4fbc-8d13-19ca675740ba}} .
Applying the automorphism {{formula:86c8f817-86e1-4af9-87a8-efecf9655f80}} in {{formula:73dcaa4d-ddf5-4e34-8db6-a0180dc1c2bf}} with {{formula:3bb08fc5-3942-4595-b34e-f77d6465c533}} , we obtain all elements in {{formula:9dc64b76-cfb3-4b4e-8054-1c21feb1de6c}} with {{formula:b8b183ff-e99c-4363-853b-8e8084e56401}} , and {{formula:c49bc7e0-d56f-45d1-927f-d696a3f7bf06}} . Similar arguments as case (REF ) show that by applying an automorphism {{formula:0d9d476c-92ce-4eda-960b-83aa52686119}} to {{formula:b217b759-415b-4458-b247-d30289fb1ff4}} , so that we obtain vectors {{formula:1e44edbe-7ae8-4529-b4bd-57e8e7566984}} and {{formula:a448efe4-9cc5-4d6f-98b9-0ed1854a6be9}} , the other vector has nonzero {{formula:a5103707-9366-4a33-bf38-0f03107943b2}} and {{formula:5a57f142-4137-4b9d-ba22-b2ebfaa6ab09}} components. As a consequece, the orbits generated by {{formula:d594b12a-0788-4a65-80b6-549a3dcca2f0}} are pairwise distinct.
{{formula:ba1e2bc4-9231-4049-a8fc-874ba4c927da}} .
{{formula:1428c957-0ae8-4c28-9ec6-6365eb9efce2}} contains all elements in {{formula:87b9abcb-ab3e-497d-834f-1b2fbafaa9bc}} with {{formula:7df54fe3-5eea-4377-94d2-07fe8024c1e6}} , and no other elements of {{formula:78d17cab-3900-405d-9e19-b7932dc2b0be}} .
Applying an automorphism {{formula:b8e7fc46-11d4-4440-a434-0af961e6d532}} in {{formula:fba2cb0d-76bd-469e-80ee-fd171dc71fd1}} , we obtain a subspace with {{formula:67791861-4210-4062-95d1-32b13dd61d7a}} as one of the generating vectors only if {{formula:89523f22-abe1-4919-8e85-30f1bbda23d4}} and {{formula:63c6f858-e2dc-4d30-845f-cad6a5b96570}} . In this case, the resulting subspace is {{formula:e0d6511e-9606-4fed-ab38-fcb73f9faab5}} . This lies in {{formula:1ad3999c-c6a1-4b66-a7eb-2787d821c07e}} only if {{formula:02f1af47-761b-44df-9407-b6e91e85e356}} , i.e., when the resulting subspace is {{formula:fc0d9b38-dfce-45be-9f15-11ca2ef0f163}} itself. In partiular, the orbit generated by {{formula:e37117d5-27a8-4d03-b12e-dead7c7207fd}} does not contain {{formula:6fb3d50c-e03c-4de5-b948-362a463b05af}} .
Applying an automorphism {{formula:991ff5f5-7d10-4c63-9c9d-15076d339edb}} in {{formula:e3ade412-120a-466a-8b00-c494a18270f7}} , we obtain a subspace with {{formula:59ab4a4f-d8b0-422c-a1a6-38508e510939}} as one of the generating vectors only if {{formula:87a237af-aae0-45c7-939f-515bd87ec1ee}} and {{formula:ba3c3656-27eb-4930-abeb-2ce9a36d16e1}} . In this case, the resulting subspace is {{formula:f5175341-507d-4900-8602-09464dc23242}} , with {{formula:40f5455c-3c05-4988-97a5-305d07e8873b}} . In particular, this orbit contains all elements of {{formula:7ae49a6c-e0cc-4cc6-8b54-79391a4d1532}} with {{formula:6e224cb9-5fdf-4256-b25e-c1e707293ada}} , {{formula:6f5dfa91-0c7f-4493-82fa-40416bad1ff6}} , {{formula:7de7e144-e01a-4174-b0dd-4198e18319fc}} and no other element of {{formula:8ccc46c8-03c2-499c-91f1-d4594799b6f2}} .
Applying an automorphism {{formula:7f30266a-832e-429c-b4fd-3cb28eb18517}} in {{formula:3f423eb5-16e8-46bc-9780-1133632e6451}} with {{formula:3867f925-355b-482e-bc39-904db244c162}} , we obtain any element of {{formula:f684cf35-5458-442f-94e2-954c688e0692}} with {{formula:67c99107-7c23-4a25-8ba7-0f5c3c49ac94}} , {{formula:d2ca0ec8-25b3-41fa-9407-599f1c18f5b1}} .
Moreover, applying an automorphism {{formula:a28b2a4d-79c8-401c-bbdb-eae15a698a9e}} with {{formula:5193f993-345e-41e8-b2c8-f89c47e646d9}} , we obtain a subspace in {{formula:bee1c240-dae1-447d-83e4-185953f5f2ea}} only if {{formula:e30680e7-c9a7-44cc-a0a1-af50ebdf1d67}} . The resulting subspace lies in {{formula:949efd3a-706a-4789-8d06-c8e427e7069f}} only if {{formula:b07b67ce-35b5-4b3a-90dc-cfd59108eb83}} . And the resulting subspaces are of the form {{formula:c12973d8-7a63-43d9-85ab-c503e8dfb064}} , with {{formula:3e724f77-0f1c-4bbd-be70-58a781866be7}} . Similarly, if {{formula:a40fd7c8-1fdd-4e22-83b9-cfa06ad47869}} , we obtain the same subspaces (with the same arguments). If {{formula:0f174d83-51df-47aa-b4af-de6cccc8c57a}} and {{formula:993fdce4-4796-45bf-b5a9-4f2fb7dfdbf5}} , one of the generating vectors is {{formula:a75327ac-5664-46ac-8d67-c4a0b8aace26}} only if {{formula:982eaf6f-77fe-4eb1-b3c8-34b5a1c21a38}} . In this case, the resulting subspace does not lie in {{formula:2ce55b8f-7422-42f9-b22b-b675f2551f94}} .
Applying an automorphism {{formula:04fb5dc3-bd60-4fb5-9f3e-059c607f6cac}} in {{formula:77330e06-f5a8-4119-bc95-b98dece931e2}} with {{formula:6827afcc-c53b-4e4f-b9f1-007042f87d67}} , we obtain any element of {{formula:af49cd99-9c51-4b5a-a3ff-f58a8a3cdd55}} with {{formula:8f37fa26-18e4-43f3-b3af-cf93005f91b6}} , {{formula:0782f069-d2c1-4671-aab4-08e3cae72467}} . Similar arguments as the above case show that the orbit generated by {{formula:2cca9f02-c90c-4d64-a7b3-83933fc51876}} does not contain {{formula:53e54cdc-9107-4386-a0e8-1c6a96a90ab4}} . In particular, the subspaces {{formula:4d66931a-02ec-4265-a8d9-c8a202d884d0}} generate pairwise distinct orbits containing {{formula:0c45fae8-f8ce-4f3e-860b-9ce6fcdb3c51}} .
{{formula:d9c0ba6f-73d4-4fd6-bf8b-e52d92baf8fc}}
{{formula:ee7143fb-04c4-4462-9166-62726e7a88a0}} contains all elements in {{formula:9e7551b8-2715-404a-b70c-e7fc2ffe76b1}} with {{formula:7c6e29ad-9b07-49a2-8d2e-d963be6ecf1c}} , and no other elements of {{formula:60e04a92-e2ee-47f5-9a26-0b9fae92625e}} .
{{formula:61f659e2-4748-45f4-a715-5e589059d399}} contains all elements in {{formula:3f677426-9f34-45ad-9a79-98a144bf5344}} with {{formula:e490f8ad-bd27-4b87-b0bf-91b85e1caad9}} . In a similar way as proved in {{formula:1fcb4446-f2f7-4f21-bdb6-e31544b896d4}} , this orbit does not contain other elements of {{formula:f51638ae-0400-4d4f-ba8c-8655462c23a7}} .
Applying the automorphism {{formula:5779430f-906b-40fd-8ef1-26e05a3ab39b}} in {{formula:e76e29e9-be04-4455-951d-2416658825aa}} , with {{formula:f9620ab0-ce85-4dab-8810-28a7ddf926f7}} , we obtain all elements of {{formula:61b5ac3f-d1fc-4c22-97d4-825fcbffaec1}} with {{formula:fdfd3b2b-08df-47e0-985a-8cd3b2c661df}} , {{formula:aed5c5e5-6506-42ce-ab71-e578f2d45865}} , {{formula:79974dd3-4194-4c4b-8456-05aeb5ad442c}} .
Applying the automorphism {{formula:ff792307-df49-4790-910e-2650a90dfb07}} in {{formula:4cfdc983-f157-4bfe-9ed7-aefa0854d8fe}} , with {{formula:ee117fe5-506c-4fc6-b5df-50cd56350006}} , the resulting subspace does not lie in {{formula:36a2f177-5332-4fb9-9797-f80ecc193ea0}} . If {{formula:93ab71b6-60d1-4dd4-9c91-71fbe55b8c0b}} , the resulting subspace lies in {{formula:227dc0ae-2a80-479d-9cc9-dc20f2e08ffa}} only if {{formula:3a3dd4b9-a878-446d-90f3-447bb58b610b}} , and we obtain the subspace {{formula:f884a8cb-dd20-470f-95ff-e24efbd3e6ad}} , i.e., the element of {{formula:5b7b8078-a707-4cb3-a1f1-49dffef13992}} with {{formula:3238fe03-e46e-4d8e-9667-fd6e7e48184e}} , {{formula:a1918198-d831-40de-a28b-04559610db12}} .
Applying the automorphism {{formula:35e40a3c-6bd3-4fa7-a9a1-e808aba2638b}} in {{formula:b1c1f2a0-21bb-43ea-97b3-b15927b3cef9}} , with
{{formula:549a4b1a-36ed-4763-ba63-50b72fd60200}} , we obtain all elements in {{formula:09719859-d03a-47a7-8efc-f601bdbc843c}} with {{formula:52d13bef-80ba-4a85-9fa6-922003ddcf42}} , {{formula:bb250909-27b0-4b34-850c-ceb93ccd6590}} .
Moreover, by applying an arbitrary automorphism {{formula:43e33ed7-fe31-4740-b84a-7a13f52b67c2}} in {{formula:52f8bbae-6ce8-4c57-936a-b3a21ac9194b}} , if {{formula:bfcbc1e4-e29d-421d-947e-9a99cd1a51ba}} , the resulting subspace lies in {{formula:8bccffa8-21d3-47d1-9f24-fd3963b537b7}} only if {{formula:cfd7baaa-97c3-4042-9c4e-139045b9d519}} , and in this case it is equal to {{formula:8c8914ad-ba84-48eb-8485-8c6419f4aa62}} , with {{formula:f576287e-2511-47c3-99cb-25dc6b513029}} . If {{formula:58309832-d165-47a1-b2ca-9d85e9fb6d7f}} , the resulting subspace does not lie in {{formula:dfffd183-e5db-42bf-ad6c-a4657c4fb765}} . If {{formula:16c54529-8753-44a5-a3cf-e96a57c786ad}} and {{formula:a59464ac-5a9d-4e72-b31c-baa04e9cdd26}} , the resulting subspace lies in {{formula:9724a3e5-15dd-43ea-8966-7ae152640934}} only if {{formula:dfef0d71-2aa8-405c-bc7e-1a488084ca93}} , {{formula:6f5ec658-aa1f-4726-9f53-f2bb5acbb2c0}} and {{formula:e765f53c-106a-4f8f-9caf-39353dd34477}} and this yields all subspaces of {{formula:55162f40-0b7c-4d2a-8399-b61fe09aa903}} with {{formula:ce248771-fb71-41c3-b751-1a24a3d83c62}} , and {{formula:a0ea4ec2-85f5-4b66-884b-5968664a808f}} .
Applying the automorphism {{formula:88bd2308-be3d-45da-8bae-847fc5bdcdfe}} in {{formula:aebe1336-d550-415c-88cb-a4db9ac22731}} with {{formula:02062a3a-2b0a-4482-a858-1148de1d2ea1}} , we obtain any element in {{formula:41dc3e23-15f6-4cd8-bbf5-7780902b6fdd}} with {{formula:cb904ef2-d5a5-4057-a656-686cee318bee}} , {{formula:085a758b-2648-4dc4-9f6a-568b49f104bf}} and {{formula:328c15aa-08e0-41f7-9c4a-1ffa0b928b98}} . And all elements of {{formula:6615856a-5197-481b-95c8-10c8c1b64633}} are in the distinct orbits generated by the subspaces {{formula:5408960d-1c72-4190-8fdc-4472013d5891}} .
{{formula:5b6b2546-2765-4148-b605-c3dd356e9299}}
{{formula:27cbfe37-5cea-44e6-a8b9-ff1f22381f90}} is an element of {{formula:266c76f5-9997-429b-9a52-c2d7fbb4fbd1}} with {{formula:6018c844-3997-4bb7-9dd3-50a553236820}} .
By applying the automorphism {{formula:3cffa16b-ec97-4dc6-9e58-59ce016046f3}} in {{formula:bf6c729b-bb3c-4ace-8bac-0c5903e532ac}} with {{formula:a0d947e2-5a6c-40c6-b2dc-40b02887bce4}} we obtain any element in {{formula:60423b53-f490-425e-a0dc-43d65a065265}} with {{formula:a7367d29-1cc9-46a8-89e5-0989b33483dc}} , {{formula:380808f0-77bc-44e6-abb6-eccde849a215}} and {{formula:04e68798-6a6d-4591-80db-6a65538769d7}} .
By applying the automorphism {{formula:9be3e12d-bb22-4c12-b7f3-8000c1eda6e5}} in {{formula:b3b80fa4-27c4-485d-a2a8-9454bb9e500e}} with {{formula:62189d90-1c6a-4b29-b623-97c9d9bc2cb1}} , {{formula:12a5cd49-64a7-45f5-847c-c480e2ba5727}} , we obtain {{formula:e618deab-433f-4b1c-883a-8e8b916f007b}} , if {{formula:0ca975b9-9445-4a13-8518-e2cdaf443a35}} and {{formula:ec66ce4e-3366-444d-9fa1-12cea2cac778}} , if {{formula:7278bab7-b992-40a1-87f9-3e4c52f27fe8}} .
An element of {{formula:f1af243f-8f7f-495f-bdce-7babad2a0cd4}} with {{formula:8fd529dc-682b-48c9-a4fe-31da25db31e0}} lies in the orbit generated by {{formula:85afa86a-d80e-4dae-854c-7586a7c4d70c}} . Indeed, one just need to apply the automorphism {{formula:507d43ca-c098-4960-aea1-f513a5dc8f76}} with {{formula:ab709df4-32f2-4223-aeef-e3b8302e3237}} to the later to obtain the former.
By applying the automorphism {{formula:02f658bb-3d10-478f-a6ee-3ca75be67ef0}} in {{formula:7c81fef3-14cc-4321-be57-5ae0281b943d}} , with {{formula:b42d6848-fdf3-4aa4-95e6-964a4857a857}} , we obtain any element in {{formula:96c23649-e487-48a2-9fee-cabda1fedeaa}} with {{formula:98577563-2445-4864-b318-8d03a1953283}} , {{formula:0baa1488-ac44-4973-bbec-350d3f0abb70}} and {{formula:50abc26d-68e2-4d0e-a11e-3fa63794e237}} .
By considering {{formula:1b4738b2-701b-47dc-945b-ffc1126af7cb}} , we obtain any element in {{formula:e07c20a1-1b8c-4863-bef9-ec46085dd81b}} with {{formula:c954ccda-5249-48db-812a-fdd25d0de907}} , {{formula:b5a4b88a-ce64-48e6-9a80-07d3ca5bda59}} and {{formula:f3ba3441-9da5-4813-bea8-fd7e01058900}} .
Let us now show that this orbit also contains all elements of {{formula:93e82d97-fd27-4475-a4a0-e83628538913}} with {{formula:97726180-ea0a-4b14-aa57-a8ffa75a7b18}} and {{formula:d1781c07-a66d-4963-ae9e-df48cbc5e2f1}} {{formula:d27a1d5d-7c10-4a4b-99ef-50ea27e565ed}} , provided that {{formula:e4956bad-e053-4903-b41a-c180bc105ae6}} . Indeed by applying {{formula:96580283-f057-4f9e-afb4-48ab0a7607d5}} in {{formula:440047c9-6961-4758-8164-023370c2b033}} with variables defined inductively by {{formula:27488f95-a06b-4ab7-a8a4-6ee2106e9727}} , {{formula:f033ade9-047e-49b1-919d-ec3d7cb575c6}} , {{formula:acf09530-fa08-4612-8ffa-7baeea354d99}} , {{formula:bc7fc3cd-e4c8-4c36-ad11-8ae13e47b045}} , we obtain the subspaces {{formula:30dced50-5914-4f27-aadd-fccf50a9e64b}} , where
{{formula:4d8a8121-077a-40ca-bb25-aa496f01dc60}} and {{formula:9ebb912a-493c-4c11-b998-6958a535b77d}} . Now Lemma REF shows that if {{formula:3de0c326-54d3-45f2-877c-7f99fcaf34af}} , we can find {{formula:8394d6d3-3e5c-43b7-b347-b632567c4c93}} satisfying the above equations, and this case is settled.
Now we show that the orbit generated by {{formula:3b8f87ba-aa13-4f68-bcb8-7c0c92da6411}} does not contain {{formula:5c033792-88bd-4678-9961-6e254e47c306}} . To that it is enough to show that this orbit does not intersect {{formula:366ae005-cb8b-42bd-a351-8a7b008a4a33}} .
We consider different cases. By applying an automorphism {{formula:2abe23b8-ac32-455f-82df-979ac0e002dd}} in {{formula:30a9dac0-5494-486d-9811-61531cce295b}} with {{formula:9ca4f393-d5b1-41e5-bf4f-1a496b219cb3}} , the resulting subspace contains {{formula:baa4168d-29f9-4097-8755-1afbb69fb694}} only if {{formula:788ad9ed-10aa-4db3-87c2-d82e4ab0c305}} . In this case, the vectors {{formula:086d4998-cbbf-4173-bc5c-decf39ec6a7a}} , {{formula:879212e9-aa7f-43d2-a8b8-6d776e115e08}} and {{formula:a0da20cd-86aa-4382-b581-6f4992d750dd}} do not lie in the resulting subspace. As a consequence, the resulting subspace does not lie in {{formula:b781c146-e8d5-4b82-ac04-53612342f0b0}} . Similar arguments apply to the case {{formula:88d8674b-8968-4bc8-ab30-965936a531f1}} and to the case {{formula:6e4ef403-e63a-4677-b256-619e5bf489e1}} , {{formula:af97b036-c9ab-4ffb-8b40-823fffb1bc3e}} .
If {{formula:2cb8a1c2-6207-44d2-951f-616bdcd53eed}} , {{formula:c626b6b1-1478-44a2-9276-b0e21831f9ea}} , the resulting subspace does not contain {{formula:b140b5f8-6e26-4223-bee5-970f4226eab3}} . In particular, it does lie in {{formula:707a9e26-e79a-4b0a-a5c2-0852e2ae5782}} .
By applying the automorphism {{formula:4f56504c-30b0-4cf4-bb29-dca192ec2d1f}} in {{formula:7d788fc0-a6c5-41a6-8324-574f5c8c0410}} , with {{formula:0407c7a4-6f72-401b-821e-570a4c0df753}} , we obtain any element of {{formula:74eb61ea-84eb-4685-b97c-41efb9aa0226}} with {{formula:f77bae73-ca71-41bc-b14d-c6a6204b53a0}} and {{formula:2024249c-4e0e-4ad7-843c-dd0e4ab4c73e}} , and with {{formula:09efc0d2-075e-40a4-9471-74ced4aa4353}} , we obtain any element of {{formula:45bbd677-21cd-4a03-97d7-aff66ac2bc32}} with {{formula:027bd6e6-572f-4bc4-93fb-fd9347c3715a}} , and {{formula:f19300d4-fe88-4954-9864-b356a66fa46e}} .
Up to here, we have that any element of {{formula:f79d1da6-dbfe-4ba6-895f-dfd150de030a}} is in one and only one of the distinct orbits {{formula:494be481-1125-401d-94dc-8f210d8d8f1d}} .
{{formula:9bf65551-7087-45a8-b2cb-8675f8e74354}}
By applying {{formula:0d5abfc5-bb0a-49ce-83ae-2b3adde01751}} in {{formula:ce48f53b-c649-4185-ad9f-4d1dc0565e9c}} with {{formula:10c7a887-61ef-4247-8e99-0fb21c686869}} , {{formula:6d182937-2f7b-4d61-9e2f-4480ce52dd58}} and {{formula:a6108440-d786-48a7-b19d-0cb84aba47d9}} , we obtain all elements of {{formula:c015d47b-42fd-4a6d-8a24-ad0b1054a530}} with {{formula:58e303f3-dfe3-4b34-9a4a-1120365274fc}} and {{formula:9a3ff7f2-12f8-4e77-8368-7759e240579a}} . The element with {{formula:20c69216-71b7-4df0-8e22-7be4be5ec5e8}} and {{formula:f93f8f1b-62be-4baa-9f35-bc092339cbc9}} lies (trivially) in the orbit generated by {{formula:31842fc3-86c9-4e08-ac9c-7eb1ac672c90}} .
By applying {{formula:77b65804-28ec-44fc-838c-f9029db852ec}} in {{formula:20ab99b5-64a5-4744-8e40-d6d2a2ba7d1a}} with {{formula:d8a86cf2-6510-4c22-bae0-39b402c4f180}} ,
we obtain all elements of {{formula:f138da58-f746-470c-8aea-1b7dd6d6e2cb}} with {{formula:8cc7f5de-2f3f-47f1-a44e-94722a9217da}} and {{formula:4b8da5c6-e310-46bd-b4a2-16ff6f14a7b0}} .
By applying {{formula:aad7074c-c752-4766-a331-cce9b1f50388}} in {{formula:271d7a7a-3fa0-4bc5-9700-e8880838f597}} with {{formula:598ebe01-9066-4a4d-8fbb-b32c0dd77d05}} , {{formula:16e1f224-55ec-46d6-b2b0-6a5985d08fb0}} , {{formula:7a0ee28c-d295-4e46-b8a2-465cc70a3d06}} , {{formula:98908ccb-184e-4edc-a2b5-1cde933d8433}} , we obtain all elements of {{formula:5291978f-bb6e-4a97-91e2-906d86b8c9e8}} with {{formula:b0cf9bc8-20ea-4647-8921-7dfec6f62ad1}} and {{formula:ce71236b-1857-4a60-9568-bf7a7bbdd30f}} .
And we have obtained all elements of {{formula:f466cb8f-4e84-4972-a64b-3a04aecdda9c}} .
{{formula:52cf08b3-6fef-422c-9d92-921b053b5ccc}}
By applying {{formula:a03efa4c-9a29-4e3b-986e-0ba405fed932}} in {{formula:388195e4-6082-4d55-9ce6-31b5b36dc0ef}} with {{formula:8a317c2c-e064-4580-be60-280cbfda17d1}} , {{formula:75e33461-f879-4e4b-b904-8244e7c67b36}} and {{formula:c97c94ff-4e57-40ee-a33c-da3a67ba83d8}} , we obtain all elements in {{formula:63895783-c431-404a-9be2-a3eebde2f86a}} with {{formula:024763bd-5d39-4afb-a73b-42e6d12bf0aa}} and {{formula:ad200342-9a9c-460a-8f0a-1ad6cb0eefd8}} . The element with {{formula:dfb2ece9-6983-4904-9017-47f65b2e8832}} and {{formula:281faca5-706b-4784-8080-d602c2f97ecf}} lies trivially in {{formula:bd51378a-5bee-4782-a2ce-f41ec86e1a0c}} .
By applying {{formula:42e5cfd4-b8a9-4d49-b422-e78e6688c33d}} in {{formula:03fe6758-5b0c-4ec4-8d41-70475d7f484f}} with {{formula:bb7b26b8-47a6-482f-a6c7-b743a80d99c3}} , {{formula:9c7359da-4dc4-4d1e-ac81-8d103f95f412}} , {{formula:6df303f4-c3a3-4502-a622-9cec54ac7086}} and {{formula:c8e90008-ed7d-4a45-b53d-11788c2ace58}} , we obtain all elements in {{formula:f377062d-9419-4156-af0b-536ff4ab9eb9}} with {{formula:28078958-65aa-4f0e-8f73-0ae240c81c28}} and {{formula:7f03196a-709c-4adb-aff0-7c3472ea419e}} .
By applying {{formula:2b6c9c0c-a531-4455-a300-bf469d174cdf}} in {{formula:30bbb502-3a21-4aa7-95cd-d67481f5735c}} with {{formula:863ec92b-3c01-4d5b-8048-0e7589ec3ceb}} , we obtain all elements of {{formula:84129e4f-063f-499f-a1b3-fd596dded30c}} with {{formula:9d89906a-67f9-4e92-a361-b9a9644f79f9}} and {{formula:33ec4679-d2b7-43ac-a10a-cc2241316d31}} .
{{formula:ee913a42-a71c-4eb9-890a-d9a1f04e607b}}
By applying {{formula:dc1e64dc-1498-4957-9c57-c662191e8070}} in {{formula:11145e39-f64f-44c4-90e4-25b69b079dcb}} with {{formula:615d2d75-1c67-4461-a538-913fb15609c9}} , we obtain any element in {{formula:23ba0ec9-488e-42c0-82ee-d3b21f5bad40}} with {{formula:ef971fac-c240-49e1-bae5-98aa9fb7bb71}} , {{formula:3a715913-210b-454c-9e83-294b7a996971}} and {{formula:0c8efc45-441a-4ba6-9de5-923fdbcff23a}} . The element of {{formula:1cba9d8f-cc46-4130-bb23-f0ad14b56ad2}} with {{formula:c8510a2b-9f13-44c5-9470-334bf6a831d9}} lies in {{formula:508bdcd9-e423-48bc-ab05-a4cb215197f5}} .
By applying the automorphism {{formula:30fa5ccb-1500-4432-93d3-a53f650c880c}} in {{formula:bfc04688-6852-407c-8b0c-a7ff3f563dc2}} with {{formula:b9dff478-24fb-4e48-ba92-382ea20dcd67}} , we obtain all elements of {{formula:6b6bf82a-6c36-4140-a849-3014d57613c2}} with {{formula:0c22fe22-3caf-4072-8957-f05bf1153a52}} , {{formula:ef352454-3e2e-4c37-9948-35f42760e270}} and {{formula:811a6360-3231-4ba9-9c7d-01c0099c0274}} .
We now apply an automorphism {{formula:108dbf2f-dbd3-4524-b543-674f7d4b3403}} in {{formula:28ad1020-0676-447e-9e16-a95875a11bc7}} with {{formula:54716161-22b7-4ed6-a6fc-27822acb1022}} and {{formula:1124434e-d484-4552-b3c1-20661702a8fd}} . The resulting subspace lies in {{formula:147e035a-a8dc-4e2c-91a7-43855621fa77}} only if {{formula:ae75177d-b800-4550-8162-a2d41901bbbf}} , {{formula:161ec704-b7f4-4c05-98aa-960f9060dabe}} , {{formula:cbd345f4-660b-4672-a665-c32999501407}} . Moreover, {{formula:f8f0447c-3b10-447a-9c5d-3d760081d253}} and {{formula:42da45c5-2ee7-490d-81d5-71c3bb303e8e}} must satisfy {{formula:dcf9a243-bd00-4881-8476-9cd190794a69}} . With this condition, the resulting subspace equals {{formula:8e43f6a4-62a3-4cf8-8456-d29f36740f4a}} . Here, {{formula:302f8011-ded8-4dbb-9c91-be9d2d63a495}} and {{formula:35532c99-007c-4569-8943-bd16ba411276}} . In particular, we obtain {{formula:19017299-5834-4f89-8ae0-15140c4fec64}} , and that {{formula:f648ec30-ff5f-4292-bbda-385e8d930cde}} and {{formula:38be394c-b5ed-4c2c-8f65-9e47417b9b14}} must satisfy the following system of equations:
{{formula:0e6e8b7a-7dae-477a-b21b-f8bd8315e688}}
Now, from Lemma REF , if {{formula:0cd315c4-c3f7-4f7c-941b-4b0043a8b4ef}} , the above system of equations has a solution if and only {{formula:76c9ec3c-9469-41ce-b2ad-a148c5838d75}} . This means we have obtained all elements from {{formula:b6d12704-09ff-428a-a5c1-fa9286810a52}} with {{formula:326c9a44-61f9-40d7-b5b6-9ebadcdd891e}} , {{formula:3217b1b9-b065-4f5d-91f3-f7a174e2b99e}} and {{formula:1c56f3cf-d68a-4b01-8401-8391552cf82b}} .
By applying the automorphism {{formula:339cb024-7a37-4b41-8f56-aa88c03dbbc8}} in {{formula:4b9d5265-a069-4e30-8561-4d9d2d64769f}} with {{formula:53dbcc33-9b1c-4c8d-bd8c-e1c1322053c4}} , {{formula:adb3bdac-1bf8-47b2-b7c5-a8caf044368b}} , {{formula:93a6ac8b-95a6-4855-9961-7bef7a31aee2}} , {{formula:92be1788-0081-4855-a1ec-01ae6a0d2c47}} , {{formula:b1171c9e-3940-466a-993c-4ce1684ed428}} , {{formula:26ca6e83-71c4-4e67-8eb6-c4343f43b425}} , we obtain all elements of {{formula:7ce04b15-c17e-42da-ba4a-cbc8a2dd4c8f}} with {{formula:369d27bc-dcae-43a5-ac88-1b88753a62ec}} , {{formula:5f8a74cc-dfc0-4e67-b970-123f016dad2f}} and {{formula:4a39d983-3e5e-4314-9a5d-5999c3f21a45}} .
By applying the automorphism {{formula:b8ecd393-cdf6-4da6-a297-bf5cd7f7dc5f}} in {{formula:91705e28-67bc-41f1-a9b1-ec62276a3f04}} with {{formula:cc4e360a-c087-4d0f-8ed1-f54db85680dd}} , {{formula:0c69690a-6f18-4fdc-9887-87bd93decba4}} , {{formula:a8f65951-95a6-4306-9391-b4a7edba989b}} , {{formula:1bdff8cf-9d3b-4ec8-aacf-d30155137aef}} , {{formula:b61ad995-2656-41b9-b03a-9eef89ae5b36}} , {{formula:f36e4009-ddbe-4a54-b0c1-b18421e4147b}} , we obtain any element of {{formula:d3a376d5-2ca5-48a9-9c5f-50798080451f}} with {{formula:b0d13f33-d0b3-4167-af7e-b48a0d377eb3}} , {{formula:8436a78c-86c4-498f-ba6c-534bd3b193dd}} and {{formula:e6889813-000e-4caa-8437-d92104e3f6f9}} . The element of {{formula:884a4940-f5a0-4c78-b4ee-38aed655c9bf}} with {{formula:9f8625e8-df66-4e37-8714-2e4a1524fa16}} and {{formula:b14fee71-cfec-4ba1-a01f-15ef4bdf1bec}} lies trivially in {{formula:6c180431-6bad-492c-82a5-6af48c1ecf29}} .
The element of {{formula:b5bd650f-b34c-4562-8fe7-fb4e096aab70}} with {{formula:515424ea-a2ca-43d0-8b40-61319c81e67a}} , {{formula:c468ebce-d303-4b6f-8758-15df960e42f3}} , lies in {{formula:78d4fb42-1af8-42e7-86ad-26a00a81b1af}} .
{{formula:5746d8e1-cc03-4329-a4ca-7017e1757543}}
The element of {{formula:4eb91eb0-c9ce-4cb3-9640-712b1a7709ad}} with {{formula:262b1a62-3708-4370-962d-4e5c15841074}} , {{formula:229263fb-9665-418f-ab66-e9bf64b231e8}} lies trivially in {{formula:fcfa5b64-fd8f-4e27-ac52-81e177358864}} .
By applying {{formula:5342c72f-8734-42c4-9034-e87377e86da3}} in {{formula:144a212c-d965-49fb-b2cc-dfb3443fa7fe}} with {{formula:17140dc8-8527-4ea6-8c82-58c6db1dbb70}} , we obtain any element of {{formula:3478b65d-aace-4264-80ed-179181877a1a}} with {{formula:846f09b6-7ad8-4090-a316-1535caf64a8b}} , {{formula:1cad3294-7940-40b6-bc4c-03605e7ad39d}} and {{formula:c8f4292e-1773-4e4e-9d0c-e2c8986179a8}} .
By applying {{formula:c565acd0-ca74-4f64-b15b-34577e1e4083}} in {{formula:d092fc2c-6b67-4d76-8d22-587232628503}} with {{formula:8f58f408-29d1-42a9-97db-92b7fd017d47}} , we obtain all elements of {{formula:bbc4be84-05c6-4580-a75f-a5f7ba64a8ce}} with {{formula:5a8826a3-1127-40d9-9a1d-8160aaab3af9}} , {{formula:4da0f4dc-4108-49c7-b2aa-d38ff0700d43}} and {{formula:2d6a0fee-2aae-4f2c-bc54-2cb5aeec69c0}} .
By applying {{formula:5d1d2518-de42-408d-a160-91926e90039e}} in {{formula:ef00b447-2c37-4eb2-a3bb-7ae117ba91c8}} with {{formula:2ae6facb-c46e-4e9d-841e-1739ba913a49}} , {{formula:8356a5b1-c246-45e8-9e32-c83949294788}} , {{formula:0f6fd8b2-fc6c-43a0-bd96-86264a887477}} , {{formula:70e5cc40-bb7e-458a-9e62-32e611d90ac2}} , {{formula:6ff86027-d40d-415b-bee3-d864b915b1fd}} , {{formula:ce41cea2-601c-4d8e-a747-f881376ebef3}} , we obtain any element of {{formula:fd47db15-a094-474f-8aa8-703a30090fdb}} with {{formula:924d2bbf-8ea5-4c59-8ab5-7c7da5d306b6}} , {{formula:7597f15d-432c-401c-8364-6d8db7a20e93}} and {{formula:26129699-933b-4db8-99e8-c07854d5f297}} .
By applying {{formula:be82570d-285a-42f9-943f-6246a58053f1}} in {{formula:11ee57a4-9b9b-4fad-adf2-015dd49c4636}} with {{formula:bf7caafa-4bd2-429f-86c5-a152d3e72cac}} , {{formula:d487f4d3-836b-480f-bb34-6b122cd28a77}} , {{formula:fb1e6f27-0a62-4e40-88b8-dba4d9c6118e}} , {{formula:6577bbda-6d3a-42b1-a876-12863f362ad4}} , {{formula:d1800191-6548-4f50-93e3-8fce70cf8288}} , {{formula:8f8a39e6-d900-41f4-be7c-cb6aa32bf575}} , we obtain all elements in {{formula:ca3286a5-a88f-472e-9900-2e3d3e30589d}} with {{formula:fdf02a2f-6494-4867-9cfb-cf4788fc430e}} , {{formula:811de8a9-8e2e-40ed-99f3-882334998112}} and {{formula:d7920bb9-652a-42f6-88b7-44202d1c60a8}} .
By applying {{formula:b0fbd07c-5355-4ca2-8127-71c9fb534477}} in {{formula:cb830042-ac43-49f1-a731-7849b9ac9dbb}} with {{formula:3e61c2d2-e867-4b6a-adfe-39a18757a96f}} , {{formula:7fd3b251-273b-45be-a4e8-d448642f6e7f}} , {{formula:e905ed6f-0aca-4784-8b09-4f2d4628fa17}} , we obtain all elements of {{formula:80fae0aa-642a-4f50-ac6c-71c804e36651}} with {{formula:f34f3c90-f6b0-4f27-a191-c3d5f7199050}} , {{formula:49d1838d-66ef-4149-a2c5-12b19685354f}} and {{formula:2f8f8a2f-d0c3-430e-a787-75296c2a497e}} . The case {{formula:e04f637f-c63c-4ffa-8ef3-808267949a86}} was treated in the previous case.
By applying {{formula:f9c9c762-64d9-4939-b8a3-3e11fa928960}} in {{formula:a1ccca29-1430-4e99-b8d9-d11a3d8355ca}} with {{formula:4d51288e-e521-47e1-bd43-b2041bf5d061}} , {{formula:e01a876f-df4f-47f7-a53e-2449108f97b8}} , {{formula:1d6db3b1-aa46-4215-bfe6-fc61b2055bdd}} , {{formula:59412088-bc65-42e3-a8ac-63444e5dc12e}} and {{formula:a3830b22-0714-402e-92d4-0d3cbb971283}} , we obtain the element of {{formula:ae67d793-80b6-49ac-95df-09076da68369}} with {{formula:5a2c7c8e-d549-4880-a5d8-7e70b0e860b9}} , {{formula:7ee6f50c-9337-450f-b68e-62229053eaea}} and {{formula:b81ce1c1-203f-45fc-907e-b48afcbcae2f}} .
By applying {{formula:6af41257-c54b-428b-a009-5e783b463ce4}} in {{formula:a15585ab-459f-487e-a4c4-68d41bf1c936}} with {{formula:93710d2d-51d7-4c77-8c4c-d26ab5485b40}} , we obtain all elements of {{formula:f9fb2cb6-823c-4a6e-92fa-aa4b7aff277c}} with {{formula:9fc10787-148d-4de5-b3c4-4c0ea39329b6}} , {{formula:f733a873-1196-451a-84d8-88bdb7ff1ebc}} and {{formula:9831c6f1-046c-4868-b303-befc070baace}} .
By applying {{formula:660c2800-6db8-4ab2-a132-4b7555a1c927}} in {{formula:2e3e2065-6616-4bed-a0ec-0dea5336bc96}} with {{formula:dc973709-dcff-4869-a915-7c4229e26ad5}} ,
{{formula:f98b7fb7-0b1c-42e6-bd7d-6c960bb5b7fc}} ,
{{formula:c9f415b9-5edd-4fb7-813d-c0e04afc425e}} ,
{{formula:be4ad634-22f2-4840-b753-682d37d5e50c}} ,
{{formula:ae1d4cad-1f32-43af-8efb-128a5bc96d82}} ,
{{formula:872d4237-5fda-4446-9115-d93a0ac8181b}} , we must have {{formula:07b96683-26b3-4bad-8689-8ce4d1b964d0}} . In this case, we obtain the subspaces of the form
{{formula:6c6a019a-99f4-4b71-9bdc-93a7474d00c6}}
where {{formula:129fc652-cddf-45bf-ab36-6baab914092a}} and {{formula:053f989e-89c1-41f3-b690-d6df035b66f1}} are rational functions in {{formula:21bff008-cf92-4350-9200-10e4859cab70}} satisfying {{formula:329a1ae2-4413-4be8-a48f-bf483aa74a5e}} .
Once we find {{formula:184fbc78-f05d-4123-97ef-6eb425fbe994}} , such that {{formula:754b894e-b856-4c97-968b-b0b86c14fabf}} and {{formula:d66e47ff-c9a7-4cce-87d2-3babdbcf5651}} , we obtain {{formula:336b3ae5-8974-4993-babd-72982ec4a07c}} and {{formula:c6a2512a-163a-4701-bd42-9dead064d9c2}} , i.e., the resulting subspace is
{{formula:dbe192a1-7848-4c9f-a83b-3fb36b436b4e}}
We will show now that it is possible to find such {{formula:4394e0f9-2d58-4985-a428-6bfd9a940d71}} if {{formula:749db20c-72a6-4e1c-bb3e-411a164cdd3a}} , {{formula:d00e81f3-785a-45a8-8854-924f680bd8c0}} , {{formula:b0c6fbdd-64be-4721-a2dc-f902f05e4f2e}} and {{formula:9f0fd2c1-fb11-482e-a93e-61595377cb77}} , i.e., we will obtain all elements of {{formula:b9d1a7a1-3d3f-4267-9b9d-c1169d5fcd89}} with the above conditions on {{formula:dcb8e2b1-0b50-463a-b1ad-8e8446b18b69}} and {{formula:29a46937-8ab3-4563-88f3-227b28a7cf42}} .
The numerator of the polynomial {{formula:247ef129-919a-438d-b43c-2f087c4d7d44}} is the nonzero polynomial
{{formula:f44dec84-639e-4fa1-b818-b53290fb867a}}
To finish this case, it is enough to show that {{formula:03c426fc-2383-4238-97d1-d1865187a19d}} are not roots of {{formula:4e1e9e58-4a97-45a6-9490-aa26b181d038}} .
First we observe that 0 is not a root of {{formula:3788a5f2-7f58-4f71-9138-28de859f7357}} since {{formula:02c2d816-6555-4972-bf54-dd1d371933d7}} which is nonzero by hypothesis.
The same with {{formula:d71bfa69-a74a-47d6-84a6-189b2a6fa092}} , since {{formula:be2959b9-10df-46ca-ae27-c77657ce41a0}} .
Also, {{formula:6c6ab72e-d21d-4349-9b6c-eda1e4f5ea80}} are not roots of {{formula:daab78fe-5438-4768-97ea-c76e8e7955e3}} , since
{{formula:5c11b63e-3287-4058-9009-f9583de19967}}
Finally since {{formula:af7f3e4d-cd2e-47a2-bc19-0a5ac79d981e}} , we obtain that {{formula:4594931b-d79e-4a8d-87eb-eb966641f74b}} are not roots of {{formula:ff7f8f9c-c476-4d32-be9a-e3959ca179b5}} , and this case is settled.
By applying {{formula:30f9550d-4bdf-4d63-8e8e-d1bf07199f4d}} in {{formula:615ac45b-c8bd-41a0-88f6-72e7e95164d1}} with {{formula:230028f9-8abb-4624-82e3-6228e0413a02}} we obtain all elements of {{formula:078565aa-edd3-4c48-baca-e37fa40832af}} with {{formula:eb97c434-7d4a-4a71-8070-42713df388af}} , {{formula:c4375454-fc42-45a6-b0ae-b0e31f686f2e}} , {{formula:1f7ad553-9bdf-4a3d-a6df-48bf4dab6af6}} , {{formula:02710f58-a40c-4b90-b87f-4bd81780d392}} and {{formula:5b282dca-b8f2-461a-8ed8-aadc1ce0e123}} , for {{formula:101e9e05-13e1-4f5c-ba8f-a4b35a81061f}} . If {{formula:49a7bfb6-6729-46c7-8f4a-2da388ff6793}} , we have {{formula:d3aca262-08e3-464d-bd2c-5e97393e5796}} , and this case have already been considered in (REF ). The case with {{formula:7450d2ad-1528-44e1-8c1e-cf5ca8043ecc}} was dealt with in (REF ).
{{formula:8babb6f4-626d-498d-96db-a232437f5f80}} , {{formula:59560dd4-e2e9-43aa-9e86-03bb505d404c}} .
Let {{formula:21641355-bd55-4870-8628-d23eb36fcdeb}} . This orbit contains the element of {{formula:58ced4e6-83d1-490f-98a7-a5c065409b9c}} with {{formula:310d1165-8383-42a3-a0d9-decd89f31333}} . Moreover, the orbit generated by {{formula:db316ec8-ea34-4445-bc63-275792a69c20}} does not contain any other element of {{formula:bf7ce6d6-943c-4fb8-ab6f-8174d1154b0f}} . Indeed, by applying an arbitrary automorphism {{formula:65fedc6c-d039-4427-a60e-eab9c531c548}} in {{formula:f293f637-a9df-4435-9397-d3337096d401}} , the resulting subspace does not contain the vector {{formula:768237ac-bc5c-48ea-80d9-9042c1c43640}} . As a consequence, the orbits generated by {{formula:cdd9f5f4-08fd-49ba-8a5b-250336e42307}} are pairwise distinct.
By applying {{formula:928684bc-9fd4-42c4-bf89-823dff2b35ce}} in {{formula:81b5e735-2ad1-47e6-ab7f-21eea863f344}} with {{formula:e1e5aefa-7ca2-4a50-bf5c-5553b748287a}} , we obtain all elements of {{formula:6d6031ca-931f-4c5f-bd1e-835f8e41660d}} with {{formula:571a98b9-a200-40cb-8757-b278814d2800}} and {{formula:bfb4b26c-19cb-430c-ab7c-3ff1bb6ee5cb}} .
By applying {{formula:09546e9b-592e-49e1-bfe6-959db77e64ff}} in {{formula:dc589160-fd45-42f1-a088-ede4fd87c1c0}} with {{formula:2ef47e9c-1aae-43c0-bba7-bb07320befa6}} , we obtain all elements of {{formula:aa5744e3-2170-490f-808d-0cd8d689316f}} with {{formula:af0b103c-7b2d-4b5d-81dd-5f21412ccdb7}} .
{{formula:e3b9313f-d9b7-4b17-9763-5fa46db513a7}} , {{formula:8b031766-7628-43e9-b6de-123f7c5e9108}} .
The element of {{formula:5a183487-d7ba-4e0a-8be5-4c122a28fded}} with {{formula:e4c11387-7a0c-4d58-bb3e-eea7efb6e707}} is {{formula:74adf802-c8d9-4197-ba96-4f82a5d93908}} .
The element of {{formula:1bf752f4-d335-424e-b101-0636830a40ee}} with {{formula:5fccc3c0-8d95-4e3e-9fc4-9a68155a0261}} is {{formula:991004dc-ea20-450e-b20c-ad46b85e7e81}} .
By applying {{formula:214d9865-6f63-4259-9189-d875ba41dc81}} in {{formula:e4bdd608-b1f0-481a-85e1-e4278ce6e083}} with {{formula:705a35ac-7142-4f5e-9f61-1c459d0b6fa8}} , {{formula:39d71321-e2f7-4c1b-a433-3c9c614cc03c}} , {{formula:5446e40a-59d4-4640-8ab7-d56a9eb2fc6d}} , {{formula:be4b464e-5402-4c90-956a-5999b14d91f0}} , {{formula:dae80f25-42cf-4e49-8232-89cd272f5134}} , we obtain all elements of {{formula:ae47a821-0b5a-4de7-95c3-24698eae436e}} with {{formula:0814a916-6552-4584-8e76-291251169678}} , {{formula:84b8af70-f756-4657-9946-65a15cdabd07}} and {{formula:69c04b45-43ef-4aa9-8079-06fb72c540bb}} .
By applying {{formula:dddd7314-2889-4adc-8d86-2b54acff2ab8}} in {{formula:b7570468-ef61-478b-aff3-5c4b52b96484}} with {{formula:a8eaa52e-dcca-4aca-bc8d-e470a079ea84}} , we obtain all elements of {{formula:e30f70b5-dc8d-4ace-bf11-7e9f64d0139b}} with {{formula:21af3a5c-bdec-4050-87b1-d8a55ce2630f}} and {{formula:deebf0b9-1373-4a25-9147-1898988bc3e0}} .
By applying {{formula:5ac0c95b-ebeb-4ad1-881a-c223212fef95}} in {{formula:e9b06e0e-4285-4277-b56b-921b8997b5a2}} with {{formula:afd5e5bb-51d4-4e22-8ec3-1e154c105315}} , where {{formula:cf9c18dc-7a68-4304-b301-93d3b310f831}} is a cubic root of {{formula:53db0141-f358-4109-88e3-f80041888c5a}} , we obtain all elements of {{formula:b6645430-5281-467f-81ed-451cbafc8275}} with {{formula:b0f94629-c6d6-4e9e-bf8c-dc20084cf8f7}} and {{formula:7901f2b7-e532-4e15-828a-1c55296a951a}} .
All cases have been considered and the lemma is proved.
4-dimensional central extensions of {{formula:c3bc69bb-d515-4407-867d-35c9d47de70b}}
We may assume that a 4-dimensional subspace is generated by
{{formula:84469b0f-6292-44f3-9a2b-adf77df6e994}} , {{formula:2f9d16c1-a8bd-435b-ab64-b480c275eb43}} and {{formula:b9b72df7-d9a5-467e-98d2-d803b90dbc01}} , where {{formula:ad1b0b5b-005c-4871-bda2-3f35598da4d8}} .
Lemma 12 The following subspaces generate all pairwise distinct orbits.
2
{{formula:f9aa8457-ff16-40f9-b01d-c4cdb783a731}}
{{formula:a8bd72aa-9526-4324-a87e-3583b199ac5c}}
{{formula:5acd2baf-d3e5-4c06-8b53-33f6857ee331}}
{{formula:38c22d39-d68c-425b-8a8a-aed4248190da}}
First, we observe that any orbit is generated by a subspace contained in one of the sets below:
{{formula:17d75003-ce92-4b05-945b-ce291dacdf28}}
{{formula:367472ac-01fc-49f2-847a-53091b7fc8a9}}
{{formula:05ea0106-9916-4622-8fea-7a82f5aa5642}}
{{formula:1092ef8a-c1c6-4eac-ade6-9714092be41b}}
{{formula:5154e5af-6992-4327-9488-7072249e728a}}
{{formula:f4e67077-3082-41c5-9d67-eb08440ce798}}
{{formula:ee43b6fe-d0dc-4ec9-a18e-44e820d3bef6}}
{{formula:c380fe1a-5deb-4917-931a-420ad10d4cb5}}
{{formula:c4c2e97f-ae5a-40f6-b4fa-a17f6ff812e6}}
{{formula:5ec1e9a0-ce2f-4f19-aaa9-bd8431e49223}}
{{formula:1cb2fd22-1c78-44f4-bed8-7b903ebd81c9}}
In order to prove the result, we need to show that all elements in {{formula:4ffd020b-b7df-42d2-948f-00018669515d}} lie in one of the orbits {{formula:a6465b00-0d44-44a7-89e6-9538ece7a859}} , and that these orbits are pairwise distinct.
{{formula:d4d9bcbc-38bb-4005-9118-7b0facd736eb}}
The element of {{formula:2ac17648-07fd-4e49-af17-c0d18445cdb4}} with {{formula:b2dca1c4-0088-40ce-8d08-fe65a3096ac1}} is {{formula:c730389d-c370-4221-b950-b1d7dc0bb795}} .
By applying {{formula:28040601-a0d0-43e0-94af-b8999187d479}} in {{formula:e34eabab-2fa5-4fd4-b8e5-eb15f9156abe}} with {{formula:2a395a91-c719-4c55-a49e-77c18c364aa9}} , we obtain all elements of {{formula:fafb4dbf-b964-433d-ad5d-89114b1f5673}} with {{formula:6c0b6a81-8629-4fd4-acdc-3a7d3920e132}} .
Moreover {{formula:926d6719-444c-4e0b-a88b-4bfb84626e09}} and {{formula:0dc87dc9-f3a8-4031-a1c4-40a06bf1d6c0}} generate distinct orbits. Indeed, by applying an arbitrary {{formula:411db5ac-4fef-4ef1-95b5-b07dd24c57a4}} in {{formula:0927b402-2ffc-4372-8fc6-efbfd9131f37}} , to obtain the vector {{formula:a9887084-90a5-4c12-938a-f73b734f2b48}} we must have {{formula:afce9c19-58f0-40c1-bea6-b471dd213154}} . In this case, the resulting subspace is {{formula:4a4b9761-0253-46fa-8f1f-2e6105336cec}} itself.
{{formula:0c276490-a66d-4074-8075-07dbf75ee7ac}}
The element of {{formula:cd343d31-9949-412d-8757-8a82f39b8152}} with {{formula:9d4fcd78-10fd-4e7f-950e-09a56cd6c452}} is {{formula:53fa6b56-b1bb-42fe-8ea3-e7250d646a19}} .
By applying {{formula:92b8ea74-dde9-485a-b49c-25e1a206d954}} in {{formula:a173c6f3-e11a-4105-bb61-3a0c33bcb702}} , with {{formula:b64dd31a-3bf9-4fbe-961d-576951d6635d}} , we obtain all elements of {{formula:ddee2baf-82bc-4ef6-bf9d-9a7977f50d7f}} with {{formula:47002008-21ad-42b4-bc25-76c262820467}} .
Moreover, the orbit generated by {{formula:1b1c9b9d-a0e0-4d6d-b625-0ff5f7056a09}} does not contain {{formula:730da06d-2a1a-426e-9185-5a7b7212ecec}} and {{formula:4b7b41ae-da30-4979-8129-e1c823699d49}} . Indeed, by applying {{formula:a8d9b623-0838-4a51-b8b3-8f583db716fa}} in {{formula:c9fc9027-f272-43b0-a418-df5360e2b034}} with {{formula:6e31ebde-e086-474e-9d47-818a1e9c9b4e}} or {{formula:419a735a-c794-418f-8519-2f0def6e4eff}} , the resulting subspace does not contain {{formula:73652344-529b-4326-af9a-e9792107e199}} , so it can be neither {{formula:e977e95e-e5a5-4a24-9024-b5fd8b986aa6}} nor {{formula:8d59153a-f381-4613-a9f1-6b8392df812f}} . Similarly, if {{formula:be21ec0c-8634-4fcc-b292-761bf322037e}} the resulting subspace does not contain {{formula:579df976-703d-41aa-9e44-90569b410445}} , and the same conclusion holds. As a consequence, {{formula:9970e622-4e00-4448-98c8-8d7266ad6120}} , {{formula:825d7e17-b911-47a9-add5-6af226d5775e}} and {{formula:ec50f308-cf17-4c1a-9832-3cbadb37adf7}} generate distinct orbits.
{{formula:cd8b1f31-26f5-4018-89cc-06310a5765c2}}
The element of {{formula:b3df469f-aa28-4dbc-a85f-c0c966a934e5}} with {{formula:9bc4494e-7a41-49ca-85bc-7b4cd5530aba}} is {{formula:cfe0a729-6a29-4f7b-86cd-b5a4d24b4a2d}} .
By applying {{formula:0cc94279-886d-4389-b452-95ec4648a0d5}} in {{formula:7b035817-f3f6-45aa-9293-5bfe5f2fba24}} with {{formula:bdf42b72-bff8-409a-a242-bce00ece0425}} , we obtain all elements of {{formula:43594871-3e75-4a89-9299-7a324275ce59}} with {{formula:21cffbd3-974b-48df-8a47-618454f699fd}} .
{{formula:dea664b8-7871-4592-affd-6aa32fe9a63a}}
By applying {{formula:609e7663-238f-4865-a643-55621faff189}} in {{formula:2a6b57e8-4165-432f-af8d-43443ead23d2}} with {{formula:139f3c82-727a-427b-b267-23b695d70177}} , we obtain the element of {{formula:005112dd-76c3-46e8-b7aa-4b061bbadb5c}} with {{formula:7227737b-4256-462f-99b6-c4d0541a581b}} .
By applying {{formula:db055ded-cd1f-4332-8944-47387f9ae2dd}} in {{formula:2414753c-3dc2-40f6-bf93-d6ba577de7d3}} with {{formula:e8c61264-ca8a-4243-a3a4-0bfaec56f6e3}} , we obtain all elements of {{formula:cc49c4d4-1e3e-48b8-aa18-9600970f4fdf}} with {{formula:b4e30adb-8511-4d63-99d4-2ebf03d0a216}} .
{{formula:e39d56b7-6a9f-4afd-a8b1-7ce03affdffb}}
The element of {{formula:9e3441a4-ee6e-4740-b9c8-1a78cda5379d}} with {{formula:b9326bc2-785c-470b-b74b-24e634a8b0dc}} is {{formula:96ad1082-79f6-4a0f-b4ac-7a5b0f004d9d}} .
By applying {{formula:40477ff4-86d9-4981-96f4-181eb18f6b57}} in {{formula:28a322d1-fc42-4b24-8ce0-de5288628251}} with {{formula:1234655f-1977-4563-8479-92e084571f98}} , we obtain the element of {{formula:ddb29563-0a07-4e90-be58-5647f40266f8}} with {{formula:e5ae4822-a1db-46cf-89c1-d8915740ba4c}} .
By applying {{formula:d74eaf9b-9d37-42d3-aae7-d73e91d057ca}} in {{formula:015a95e4-62ba-403f-a70e-ac79f63c0862}} with {{formula:3bb14dd0-7a9b-4179-b2c6-0a2363517633}} , we obtain all elements of {{formula:3791184b-7f80-4e3d-a7bc-4c8de5156bf7}} with {{formula:f83d4d4b-af5e-4e0a-878f-c06015d5b574}} and {{formula:ea3c1bf8-83bb-4435-abbe-b2f351b31ea8}} .
{{formula:caeb9833-f34b-4c4e-bf6c-3ee9404d6460}}
The element of {{formula:fcc5577c-407e-4f70-ba8e-eff97c840b92}} with {{formula:f58c92ce-f795-4c45-a277-5061d6a1312c}} is {{formula:1d091d82-8b6c-4d97-aa52-57bf921c4017}} .
By applying {{formula:691c13e3-3ca8-4eea-9082-b00ac88a6677}} in {{formula:99d8944e-4ac0-49af-b51c-743529c2bcf6}} with {{formula:b5cb42a6-4b09-4a89-b5ce-cb1950d5857f}} , we obtain all elements of {{formula:364da36a-a79b-451b-9ebf-34fd7a3db07f}} with {{formula:a0a7cb69-5932-43f1-9503-bba8eacad124}} .
{{formula:148503f3-0c90-497f-9d9f-141e5b22b10c}}
The element of {{formula:3f812f55-0c64-40c3-9992-772a99e2d27d}} with {{formula:4b866f27-ece7-404c-befe-6a48263986eb}} is {{formula:5efbec22-38f3-4e73-b11e-cbfbb24479df}} .
By applying {{formula:088e91f5-d0bc-492f-b158-387a13372e9e}} in {{formula:c30ecaaa-cfe0-4d69-ac6a-11351a39dba0}} with {{formula:48ddf197-cabb-430e-b5ef-d8f30292a8e2}} , we obtain the element of {{formula:23d9ee61-0294-4a5f-adf8-ee891caf862a}} with {{formula:990377f8-4971-4cac-93bc-780194742bb6}} .
By applying {{formula:98a27249-2113-4388-83ca-d11bfffdf7c9}} in {{formula:9789928f-8132-4b7e-97c0-4ae92d6176b6}} with {{formula:e9addfd3-5347-46ec-ae22-3127f248795c}} , we obtain all elements of {{formula:0f58389f-1e67-4325-98c3-e70cdc3d9924}} with {{formula:5a25727e-3df3-48be-ab94-d6c907c1b8ef}} and {{formula:4c39ac75-bde8-4fd6-9fc2-63739ab6f803}} .
{{formula:476eafcb-06a0-41a6-bf03-e3384b035075}}
The element of {{formula:24da40d3-a10f-4f84-878d-1100963466b2}} with {{formula:41a92d0e-647b-48e2-8011-06414521eab6}} is {{formula:de899898-fc43-4467-a592-1b5b5c1e4d1d}} .
The element of {{formula:a3b26a12-154a-4f2d-80f9-391deb51b34b}} with {{formula:1b1445ec-4f2b-4642-97fd-13aeee552970}} is {{formula:9182676a-3601-48bf-aca9-3b7646dfc8e0}} .
By applying {{formula:e12e50e5-5bc6-4b9d-b3c4-87f4e92f460a}} in {{formula:4dc87eac-0146-4912-a7a6-20a9800f1762}} with {{formula:fbc17432-0d3d-4312-8c90-0cdb1e1f06c5}} , we obtain all elements of {{formula:55d64c37-0d84-46e6-9030-5188c18919fe}} with {{formula:c34f8c94-fe87-491e-b297-b3efeadfe372}} and {{formula:2cf5b476-b548-4614-8095-96cbf0912cb9}} .
{{formula:f3f1b39e-f8ac-4b40-a0ef-cf4ce215545a}}
The element with {{formula:03f935ce-865d-4ec3-9226-43bfc1a5856d}} in {{formula:1f367789-fe74-4ff5-ad5d-1e72ace767be}} is {{formula:3e75cb6e-9868-4477-be30-464d4993f0ab}} .
The element with {{formula:5a3c2e37-5cb2-47bb-a7fc-e4e89a56b25f}} in {{formula:6f35b02d-943c-45d3-845d-535f903e225a}} is {{formula:099d72ba-083c-4893-9d80-84b7b0d88c34}} .
By applying {{formula:f53a5143-6318-42de-810f-12dc01a44d7c}} in {{formula:1e636f79-920a-472b-a5a2-f9be2dc9a9ee}} with {{formula:7bc1b054-7318-4e98-8ab3-c2cbae7e8912}} , we obtain any element of {{formula:e61755e6-d739-42ea-bed8-521d33517808}} with {{formula:968249ea-8806-4b80-9f24-9bbdcfe324ce}} and {{formula:912c3f3a-6406-4362-9e83-38af9460d501}} .
{{formula:b78fec10-7ebb-408b-9ee9-ca860fc20a32}}
The element of {{formula:bd497c11-a7b2-48ed-bab5-857402ae9e64}} with {{formula:41725e99-92f2-400e-8307-1427cd4d8768}} is {{formula:fce40a62-07b4-4b65-95e9-ba97f31fbf9a}} .
By applying {{formula:2ec367a0-d0af-49cf-864f-a5dd2f80f764}} in {{formula:1a3a39db-13e1-41b7-860b-55906b6efcbb}} with {{formula:bb307c60-06e8-455f-b9ba-587ac9d3324d}} , {{formula:656d9a7d-74a2-469a-aab8-3305927ed31f}} , {{formula:1a932646-2d12-44c7-9b33-cc7373da3a33}} , {{formula:7dfd2c38-0b52-497a-b884-dea4b61c2d9b}} and {{formula:59d49fb5-4257-4d53-b8e8-8208481a0805}} , we obtain all elements of {{formula:acd8af97-93e2-4b47-b2d0-72f73fc7b51f}} with {{formula:8622da90-634f-4685-beda-12a6e9b2dd70}} and {{formula:02d0b6fa-8cbe-4d42-95c0-e2e58f024163}} ;
By applying {{formula:6fa87e32-4dda-4345-9a0a-9aa9b1b599ee}} in {{formula:72fcaa88-93d1-4654-a102-d1d465f1abe1}} with {{formula:3c06332a-2744-4c1f-85a1-c8393ee3038a}} , we obtain the elements of {{formula:87c4aee7-8280-4e3c-a53d-b56ee6cce367}} with {{formula:6373acc5-ee34-41d9-bace-f23d3cf1f704}} and {{formula:fb8db0a1-9155-4002-894c-6e89f547f3e0}} ;
{{formula:5cd908f2-eda6-4b43-b74b-d0e012f4c59e}}
The element of {{formula:8925ec87-bbb8-4f50-9963-edabff33a31f}} with {{formula:9a1fb375-5eaf-4130-b052-2711b9747767}} is {{formula:1a819f9f-66b4-463b-bc07-1ae44f84753b}} .
By applying {{formula:a559e44a-5fe2-4bc6-ac4d-2f5ed71b97ff}} in {{formula:0fc0a088-3cfa-42ec-a068-0f350841c56f}} with {{formula:638d47b2-18ee-4531-a2ad-389a96cc71ea}} , we obtain all elements of {{formula:6af0b01c-5848-4768-804e-d3178318c883}} with {{formula:8ca4ce7d-34e2-4843-af43-7398922227c7}} .
All cases have been considered and the lemma is proved.
5-dimensional central extensions of {{formula:260e3d93-9c82-4ba1-b790-dc50834febbf}}
There is only one 5-dimensional central extension defined by
{{formula:01e02c8a-8ce0-4cbf-aae5-5467dadfe8d2}}
Classification theorem
Summarizing all results regarding to classification of distinct orbits,
we have the classification of all central extensions of the algebra {{formula:877eff27-17f4-4784-b70c-4d927ab34d26}}
Note that, we are interested only in non-trivial central extensions,
which are non-split and cannot be considered as a central extension of an algebra of smaller dimension than {{formula:26a03af7-fef5-4083-8dff-5b953abc6b9d}}
Theorem 13 Let {{formula:505dc520-4ad4-48f3-b5b0-cc71aa1a99bf}} be an {{formula:56815970-8b1a-43b7-9d71-07a9112aa0b3}} -dimensional non-trivial central extension of the Zinbiel algebra {{formula:2aabcbbb-df93-43ec-8a0d-331d4931bb58}} Then {{formula:1ea6e1d2-1582-4b24-a89e-defecb6fc490}} is isomorphic to one algebra from the following list:
if {{formula:2032794f-effb-4344-8707-472c3a17cf67}}
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{{formula:167f776a-e2be-4e27-ae8e-cd44aa2f1db1}} {{formula:e4afee4f-a2f1-4854-b22e-fa106bc00f4f}} {{formula:f9f41eb1-b142-41ef-875b-bec78ce781af}}{{formula:1ed0f024-dac0-4073-93cf-28bd395c326d}}
{{formula:21390973-0c58-4ad3-859b-a67fd43a9f7a}} {{formula:9ecdc793-38c6-4eaa-a369-c1cb93e787aa}}
{{formula:b8a8533a-75ca-48d3-a347-be7f9b82b68d}} {{formula:d9c631ec-3fe1-4128-b301-5d5ced1d5385}} {{formula:2e8dd255-2047-427f-9139-979e095394c6}}{{formula:f9a1ae8a-a4dc-47d5-b7aa-e67d67fbf880}}
{{formula:9e471571-075e-47f8-bdd7-ae71e7cbdd1a}}
if {{formula:f8a88485-9a3a-4538-b1a2-09871f175828}}
llllllllll
{{formula:06a09d7d-4397-4929-ac2b-6a4ada5339f4}} {{formula:1a790ad9-0421-46f5-9ebf-07e43f62dc71}} {{formula:c9f36fbc-3d17-4d0a-baae-64a2010df580}} {{formula:8d7355c5-5c5c-4418-a2ed-8a2b11ad715d}}{{formula:aad2ebf7-4c37-4874-8d4e-1c9ab52ac61d}} {{formula:62a68837-3f6d-4d0b-9415-a3868bfd8542}}
{{formula:eeb4d608-83d7-4100-93db-65b1be054210}} {{formula:8f1e5ccb-aa1d-4433-8ba0-cb2bd2b4c67b}} {{formula:0eea3e55-7638-4d87-99a0-c8ea6caf79ce}}{{formula:55d78df4-823a-4567-b6b5-e55509ed86af}} {{formula:de6ba6f0-0a28-4ac8-99f5-d0114c3235a5}}
{{formula:5329e1d3-3041-4733-9ff1-37696a34fb27}} {{formula:70334a99-d541-41a4-83b8-219294594486}} {{formula:9082f8b3-7ba1-400f-861f-78de90cf4bf9}} {{formula:c689c184-42b9-4f09-b2e3-2a47acf8a09b}} {{formula:b4c89ff1-6a85-40ab-b313-92f6969cc00d}}{{formula:c9be3ba5-7120-4811-9bc3-ba9c8c4f7782}}
{{formula:c402ea74-15ef-48aa-9d2f-7f846758c129}} {{formula:c5fd0d3e-dd36-457e-b4b4-c69087a75899}}{{formula:370796df-9677-43e6-995a-7a9837516c3b}} {{formula:d30051b5-c69f-4e81-abfd-998e0adafc7e}} {{formula:3d70161f-a233-405a-88de-395718bd6593}}{{formula:c9b57ba4-afa7-4bd7-937b-a4e34ef120d9}} {{formula:36bc515a-704a-46bf-be65-2f820becf666}}
{{formula:ac1188c8-ec6e-48b5-8ac7-51fb709ce58d}} {{formula:81cdcf8f-d83c-4a3e-a338-d8f566a8e989}} {{formula:f918d699-c3ec-40f9-b23e-8265a54249e7}} {{formula:e96b733c-b18d-44a4-a531-05edd33098ed}} {{formula:f406ff51-4057-4f6d-a65c-9d9dd496db56}} {{formula:81f91662-55db-4198-99ec-563b1ab9e573}}
{{formula:7dcb7704-af83-451e-a937-1775be872d36}} {{formula:a7b0d6ff-7b8b-4a44-87ee-480d91f4e255}} {{formula:762b6e65-22cc-4e79-ab4b-e723abae5094}}{{formula:6ccea61e-2780-4851-97ca-9d4417cd73a9}} {{formula:c7d16cde-cbbc-48e8-a324-e457d5f024b8}} {{formula:9dbc54c8-272a-49f3-a8ea-4aab1c89fd2d}} {{formula:bb4cfe58-11e8-484f-8e12-95d073783d6a}}
{{formula:a0e4443d-5c35-40f1-8fa2-c8e6908d506b}} {{formula:ee6ec208-eeea-4bfa-bd33-3c751727aef0}} {{formula:97b2e4bd-5245-47b2-b7b0-baa237216ccd}} {{formula:dbaa0c7f-9547-4df2-9144-6ad8183d45bb}} {{formula:285ca5cc-e479-47b9-9026-5285c48d4f00}}{{formula:537956c5-58db-4788-853b-7c8aba2f4cc2}}
{{formula:6dab3099-afb0-4fda-be5c-6c9c5c5dc4d5}} {{formula:e0828ce8-2499-4516-9b68-cceea7aa48db}}{{formula:580a79e1-f012-4212-8690-5760f04cb88f}} {{formula:78e6223c-d305-476b-a028-e9b68191953c}} {{formula:a50bbf98-df82-454e-a50a-a5ff390896de}} {{formula:41e28e9c-43fe-429d-8b36-6b2c44a56b1e}} {{formula:e3e5e3ca-3881-40b6-ac34-ddeab7eac04c}}{{formula:5859e0e9-ff0f-4a81-99a0-69849bf50540}}
{{formula:c6d83a69-80a0-4341-be8b-0b524793077b}} {{formula:6659ad6f-f480-4a71-81ad-7d7e3e02e178}}{{formula:ce0a22e4-f589-4f2c-b9f0-080169015a7f}} {{formula:98a47442-34ff-4285-9c85-b1316ce035ed}} {{formula:5d45b732-0cbf-48e5-8aa1-d0352712325c}} {{formula:f6debd3b-4613-4712-a322-fee059d73750}}{{formula:4b188201-1036-4895-94bb-eae56d54712a}}
{{formula:484a1bc1-4a3b-4434-9661-786f03a69b0d}} {{formula:4b4e0c97-452d-4864-8d98-00caaf2b0541}} {{formula:30278c5d-4f0b-44b5-8d38-8e0879f17168}} {{formula:79aa0167-e774-4737-99c3-2ad2cf6b9b51}} {{formula:b15103eb-59da-495f-96b3-38436ddc2802}}{{formula:a9979a3f-695d-4493-9d9f-d5d9f95f620f}}
if {{formula:11947d8a-c9f2-487b-b7b2-e18627445558}}
llllllllll
{{formula:32a71f86-6ce6-4a7c-8445-a8d80e921705}} {{formula:5c54a196-7538-41d6-b9ac-10141e33d054}}
{{formula:335bb7df-1771-4bf6-8125-b321dc830704}} {{formula:4f2e8332-6b59-46ad-b621-02be35f934b2}} {{formula:101476ae-0777-4dc3-af3e-144c98c751ad}}{{formula:0705ef0d-7aa5-4b52-9837-473d40ff2aaa}}
{{formula:ff91758f-5778-415b-8aca-77d79402cb27}} {{formula:596aa9b9-000d-4882-b812-8f1326cf806a}}
{{formula:f7f8ae25-65db-455b-8cba-a4745bb8762f}} {{formula:fa290b50-5cf5-4998-8258-58cacbf6af44}} {{formula:6ba089e7-8074-4a25-b89b-639abedd838b}}{{formula:0b8d03bc-5f0e-4f3c-b45c-85ea9b7d2852}} {{formula:981fde57-a6ac-4f2d-be80-cf0e7be93f3d}}
{{formula:96fb33df-72f9-4640-864f-0537b3615f77}} {{formula:d2c6d0b0-289e-447a-870d-b35c052b0978}}
{{formula:2363dd19-e145-445f-add4-8040a103b093}} {{formula:63264da5-1e05-4807-8cd9-5bf6113a97bd}} {{formula:cbb2d8fd-6f22-4463-817e-d78c688cf7ad}}{{formula:cf8d4ac0-45a5-4ce5-b253-437e52aace89}} {{formula:d43956b8-3fa4-41fc-9d81-0ed5f1ba1864}}
{{formula:e2f89278-6e2c-45b5-b3d0-7f285da13700}} {{formula:22ee574c-ec4c-4253-bd7e-0021f75b02e3}}
{{formula:311f5779-327b-4e98-9153-34a7660e903c}} {{formula:cb4def35-e887-40de-82b9-b96b9f7b9481}} {{formula:25af1ecc-cdea-40ee-873e-18e2c24aabfa}}{{formula:cb212028-31a4-43d1-a303-8514af61fd34}} {{formula:f67fa603-ead2-4284-90e9-b06f20e5d1c5}}{{formula:5bc89ad5-a43f-4588-882c-8566f41826b3}}
{{formula:82e99cea-2608-4d08-8588-b7914d30dce0}} {{formula:cd69f4f9-aa72-4cbe-980b-736c9fa46ff4}}
{{formula:fe5ecda7-3a88-496f-b0a2-86bd9c4569cf}}{{formula:55636b56-efdf-48e7-a5a5-9c4d57ccd6d4}} {{formula:05010364-89e9-44a8-a570-d2719e84e1fb}} {{formula:5b158e40-c8c3-4f87-9590-18c61647ba90}}
{{formula:eb26d000-7b18-4561-83ca-afa087ca93dc}} {{formula:90ef8e39-d7fd-41a7-9f3c-1a700c22df66}} {{formula:85ea533b-44d8-489f-ab52-b06b95d067df}}
{{formula:3eac0d45-95fd-4c42-8844-ec20dd05164e}}{{formula:40d04907-1732-42f9-bd27-e092c286f704}} {{formula:f6f96215-a8f9-4979-8fa9-613137f510db}} {{formula:de601de1-cd38-4add-ac8d-8412c2341573}}
{{formula:788ef225-4ef3-4c06-8096-b0516d6247dd}} {{formula:ec831d11-998a-4553-8ef0-bb9cf8076632}}
{{formula:f432bd5e-169b-4ce0-9eba-675e6c019b70}}{{formula:105acbac-dda7-434c-97f1-cb30c263b9c4}} {{formula:dfb62f3f-b521-466a-b84d-68736991f6dd}} {{formula:cebd45a3-7959-4678-be9c-977bd5edab56}}
{{formula:cb52dab6-af6b-4cb8-94fa-99b20b0928ec}} {{formula:f685ff0a-14a6-46ac-be09-3312f0cfdf21}} {{formula:8e92dba8-9564-44ed-b055-60e4b01e747c}}
{{formula:1f0d9075-6d43-458a-9aee-0e484b793d44}}{{formula:fcfcb7b0-03bd-487d-a733-3dba3337b22b}} {{formula:abe4a64f-74e3-4cb2-bca7-a8ebda35348b}} {{formula:b3d086ee-7bad-4237-bf5b-56e535e1fb09}}
{{formula:61603bb9-443c-4da8-b643-18cdcfe825aa}} {{formula:25a0fb02-f1aa-4999-abda-11441cd48d92}} {{formula:eb43b6e5-b0cf-453b-bd9b-2433faa7dfbd}}
{{formula:e9209fa3-4e00-4248-8610-9bdf5cc4da16}} {{formula:f62f2238-8876-4342-a7fb-7c6734972b4f}}{{formula:ead7af3d-f134-45e5-813b-d43e1fcd5df9}}{{formula:c9977333-94d9-4344-92c0-ff7dfe00d8b4}} {{formula:8ff5792f-abf2-43d2-883e-6d63badfc6e6}}
{{formula:8bfeab8f-9eb4-4c8c-97d6-f642ffdc0c8d}} {{formula:cc788d38-170b-4b6f-aa36-fbf24d00d663}} {{formula:75d2f43c-183e-40d1-930e-493d1cefc0c7}}
{{formula:0e83b322-4b5e-4250-b875-1c4f74d01e3c}} {{formula:82e1b8a9-1eb8-41cc-8ff8-c77decd26855}}{{formula:68bfedf1-649a-4caa-87b3-17dd99a6a403}}{{formula:ee85433f-c428-4c9f-b2a9-38d5f0447289}}{{formula:fd17a36d-6f63-4479-ba59-c8ead9d08f80}}
{{formula:c2f4778d-e4dd-4632-ad09-22d7cc47e2d5}} {{formula:a210548a-1465-4b9f-bee9-f564f3d8a71b}} {{formula:a628947a-6730-4cf5-b660-2368cdb0aa85}}
{{formula:a8df781f-da5c-4847-899d-820ac3a6a945}}{{formula:136e1eb4-0bc9-47ef-a9b3-fac699fabfcb}} {{formula:4f60681f-ed09-4ae8-8c76-1aef7bc8e558}}{{formula:ac278c3e-4e21-4032-9b6a-177ccf60dc1f}}
if {{formula:76da23d4-5e13-4658-a10b-d55931431301}}
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{{formula:da2eb7e3-8348-4eab-b7e5-c60ea453d9c9}} {{formula:119db311-9b69-4305-9d3c-808f501ae74b}}
{{formula:7b8f0d14-e77f-4902-b285-f6c3e0c942ef}} {{formula:fc006be8-9681-4773-942f-ec1639b258dd}}{{formula:181bf214-787f-46d0-bbff-1f2cd4ba72cd}} {{formula:77ebdff9-e766-428e-8269-c4c3b4789fee}} {{formula:fb028100-2a74-46ef-b6e0-81594d5ec9d6}}
{{formula:6ea26bb4-3d79-4747-8892-19ebda16c793}} {{formula:442e76bd-efd6-46de-81ac-4bdddfd70951}}
{{formula:ad7feba3-11ca-4ba5-a369-ce52a5073a52}} {{formula:522e2da6-eaa5-41b9-8c22-7a3ab3ff3b03}}{{formula:8404d83c-7c61-445e-8758-02097e9da832}} {{formula:8529be67-7e34-48e4-9473-7e9824be31ce}} {{formula:b60a7787-0f68-449e-8aca-7d3c50593a25}}
{{formula:da11ae1b-7483-4172-a35c-41326c1ff725}} {{formula:7db1acad-2d05-4773-8b6c-3bba01407095}}
{{formula:88c848a9-b7ea-44ba-8359-5b3ad8e8d986}} {{formula:1685ebd0-bd48-40d3-a2f3-6a0d985809fc}} {{formula:6cd23e41-6739-4970-8d88-4eabd2c7a494}} {{formula:f1b05f2e-c0eb-411d-8655-514e16091605}} {{formula:5ee78bde-1bb1-42ce-8c43-cbce5bd02be1}} {{formula:2da31577-cee4-44e4-813e-4a71da631721}}
{{formula:e4e219d6-ac08-4d46-b989-fa4effa6d947}} {{formula:5bd3e08b-1c53-40d0-ac42-5d249d35a76f}}
{{formula:73afb471-fcfe-43c6-ba3f-0800251084b3}} {{formula:cd19e481-ec11-4dc5-a7b4-a656734b6695}} {{formula:cc7ea2e3-cfa4-480a-ad36-129e9090229a}} {{formula:46836450-ebfe-4053-8e0b-0ba7cb174d49}}{{formula:bbd612b2-659f-4d5e-930c-dabe665cf620}} {{formula:b47ead4c-fc3a-414f-8a17-8f1f5eca0f3f}}
if {{formula:597ed9db-304d-4ff1-a106-2e311f669f2d}}
lllllllllll
{{formula:5a70d7ed-04c5-4bc7-b46a-cc2fb7235347}} {{formula:01c1d7bf-d3b2-41ff-bdb7-261f26a838e3}}{{formula:3b7d6cab-e7d9-495f-9593-e2d829fc4b63}}
{{formula:7e234a33-8981-4a85-8117-1928294e0811}} {{formula:12fa7301-e1dc-42f4-a36d-2a224c26f171}} {{formula:d74373ac-1dcd-4a73-a754-a683507e5750}} {{formula:cec9124e-79d4-4924-95e4-e8023a0cf8a4}}{{formula:22e4921f-6c17-44cf-aedc-72318051c3a9}}
| m | 7f3a4511b5b06dbfd39137a6d1f94e08 |
We evaluate and compare the following approaches: (1) pFedHN, Our proposed Federated HyperNetworks (2) pFedHN-PC, pFedHN with a personalized classifier per client ;
(3) Local, Local training on each client, with no collaboration between clients;
(4) FedAvg {{cite:9b6fed75d622faff8196aa3b5067f7aae060333c}}, one of the first and perhaps the most widely used FL algorithm;
(5) Per-FedAvg {{cite:d8d601e77190b0be466560b96e6f5f6427c32c64}} a meta-learning based PFL algorithm.
(6) pFedMe {{cite:b3d5730a6a210bd88dccee3aaf745968c7f62e12}}, a PFL approach which adds a Moreau-envelopes loss term; (7) LG-FedAvg {{cite:ba4092061e90e5c80ebf4ee59a1ed17ea418d652}} PFL method with local feature extractor and global output layers; (8) FedPer {{cite:74e54cfa8438d25a70cde1b6743819317782ffff}} a PFL approach that learns per-client personal classifier on top of a shared feature extractor.
| m | 67b4cf1070cb99ab09f49f32c2e17585 |
A review of the literature shows us a wide variety in supercluster masses and sizes. {{cite:468da3b2e6c3f0d7067cb62da3807d00b734ed2c}} find a massive ({{formula:e3f1dd04-e695-4c3c-9281-d9aed5e72117}} ) supercluster at redshift {{formula:f5ca71a6-de56-409b-8b60-cf7bca81f036}} , with a size of {{formula:8dfe667b-5612-4cc4-af10-31ac3c11cbfd}} Mpc. {{cite:e30f1baf6c5932e9a0df4bd01a93296c85ca68bc}} compiles more than 500 local superclusters at redshift {{formula:f07b7610-e491-4786-a690-21a28f761044}} with effective diameters in the {{formula:efeb11f7-40b5-4a01-a2e1-3c97bd334850}} Mpc range. Researchers have tried to look for superclusters in multiple ways. {{cite:6a1a2cd0070eb9e1be61c2115518fd627939e85c}} looked at a large population of red galaxies in the space separating two galaxy clusters at redshift {{formula:80027dff-c2ac-443b-ae6d-d87552ed3857}} and concluded the existence of a supercluster. {{cite:afb14ff5f4e95dc323feb105c5b10cb9b8eb79b9}} used color selection to isolate overdensities across 30 Mpc in the sky. {{cite:dcf0037e4f173660836f161f0689528a16d378d6}} optically selected overdensities and obtained mass estimates from X-ray temperatures of member clusters, finding a total mass of {{formula:2c54a3c9-2185-4244-89c4-1837ded908c6}} . {{cite:2d3cdfe517acf2de13304d663e5dc6841dba9370}} identified an extended structure at redshift of {{formula:15c17c1a-f49c-4a35-9342-be86c6e957e8}} with total mass of {{formula:a956426d-7151-45a0-b71b-a32e4d715e6f}} . However, the total survey volumes searched are missing from most of the literature, so these are not shown in Fig. REF .
| d | 4745b2ea602f1f119a6a46e1c03cdc9c |
In both the {{formula:33761396-a6ea-4825-836b-ba4a8b7b18eb}} integrable case and the chaotic case with random interactions in the large {{formula:53776806-3039-46a6-8109-0ccba65450c6}} , {{formula:92016b63-391d-48d5-ae5e-e324e20c77b6}} limit, we derived the two-point correlation functions of the Abelian anyons with the lowest scaling dimension. Intriguingly, the two-point functions of these two cases behave similarly, exhibiting a fractionalization into different characteristic velocities. However, their four-point functions are drastically different. We showed that the anyon OTOC four-point function of the large {{formula:25e18f26-aaaa-4b9c-a040-533ff4ab942d}} and {{formula:1cfb33fe-a704-4825-aa81-317958bd68ba}} limit with random interactions has a positive Lyapunov exponent similar to that of the chiral SYK model studied in Ref. {{cite:34c5d9ad899b5d0139f456b8843e746cc296e553}}, as expected for quantum chaos. The maximal VDLE approaches the maximal chaos bound {{formula:ee5c7978-047b-4dfd-b3f9-bb717979f6a0}} {{cite:c9b85b3305c29649f93b3923641914cddffd106d}}, {{cite:396d79465a19924b61d265cbaaafcbd18d41d30c}}, {{cite:26e3a820c402624fb40f23204e243e45728d8b20}} when the interaction strength {{formula:fb428a9d-525c-4927-bc97-53188611cd94}} reaches its physical upper bound {{formula:46d2d907-7d00-45d4-9d94-68ae08a7204c}} . In contrast, the integrable case with uniform interactions cannot have a positive Lyapunov exponent {{cite:34c5d9ad899b5d0139f456b8843e746cc296e553}}, due to the solvable eigen-spectrum of the model.
| d | bd2b53dad30643c9969cdc1a8d184fb9 |
Multifractal fluctuations lead to considerable theoretical challenges when we move from equal-time to time-dependent correlation functions {{cite:5801f42b068fd96304641d308d5b1389777b160f}}, {{cite:a3c86ad7ddc9c1d22d707885052400ca0a57fa8b}}, {{cite:2a324b4d8dc454b1833fd7c87ecd825a186eb514}}, {{cite:4b41ebf83d80302022835a5f2cbddaab1595be8d}}, {{cite:9651da4f4fd5fd5537d64aa558c8b4398be436bd}}, {{cite:449340fc571cabdae13e2a268d146eabae1913ae}}, {{cite:97a4285072d19753956d2deee5981624735db912}}. In simple critical phenomena,
power-law forms for time-dependent correlation functions come from the divergence of the correlation time {{formula:2cf3c96d-b7f0-4061-b670-455f09f41ecc}} that is related to the diverging correlation length {{formula:3ba6ad97-158e-4fb8-9e35-9239daf2de55}} by the dynamic scaling Ansatz {{cite:082305c959f9b8b14b095105eeff39ebc9722113}}, {{cite:abb8b76ab0864a3d16804f12150168099e10e200}}, {{cite:5cb5e145a1e7d9a5a4da34c8eebc35405bcbbe53}}, {{formula:e2b2d569-2e21-49db-8e3c-434c10020c62}} , with {{formula:6d0d5674-1419-4a09-bec1-55d72cd7ddad}} the dynamic scaling exponent. Multifractal velocity fluctuations in turbulence lead to dynamic multiscaling of time-dependent correlation functions for which we must use an infinity of dynamic scaling Ansätze. This has been discussed in detail in a variety of hydrodynamical partial differential equations (PDEs) {{cite:08259cbaf8a5b0714c30bdcc0628078d5d44ab4d}}, {{cite:5801f42b068fd96304641d308d5b1389777b160f}}, {{cite:a3c86ad7ddc9c1d22d707885052400ca0a57fa8b}}, {{cite:2a324b4d8dc454b1833fd7c87ecd825a186eb514}}, {{cite:4b41ebf83d80302022835a5f2cbddaab1595be8d}}, {{cite:9651da4f4fd5fd5537d64aa558c8b4398be436bd}}, {{cite:449340fc571cabdae13e2a268d146eabae1913ae}}, {{cite:97a4285072d19753956d2deee5981624735db912}}, {{cite:000f92519ca3ff5c3425ebd74bbeffd8f2a0e781}}.
| i | 792a22a443bab176600b3538c642c8f9 |
Here the results of using the algorithm on synthetic datasets and a real dataset are shown.
One synthetic network used for testing is a set of random Erdös-Rényi (ER) graphs produced and connected together to from a connected network by choosing randomly members to act as boundary nodes.
The other synthetic network is produced from connecting independent communities graphs produced using
preferential attachment.
The well known Zachary Karate club dataset {{cite:c3363323aaf14a0cf1dc1ed44d9dda839547e661}} is analyzed and presented. The Enron email dataset {{cite:0c9d0424618a0e3c404b86d6403ac742203a5f1c}} is
also analysed utilizing the valuable semantic data associated with the nodes to show the qualitative validity of the algorithm.
Lastly a new dataset collected from monitoring a Twitter hashtag is presented where the volume of
Tweets and the volume of boundary nodes are presented against a random set to show the importance that these nodes
have in spreading information throughout the network.
| r | fd2dfdac0ef1d327ffb54ec3a09e503b |
In this section, we outline policy gradient methods, which have proven empirical successes
in function approximation settings {{cite:105c7ea7c59209ded6e9c8d5c80aa806cebec2be}}, {{cite:d02fa1df527f44a22408e26ca72b09a2b1481ac3}}.
They requires explicit policy parameterization, which enables
not only learning appropriate levels of exploration
(either control- or parameter-based
{{cite:5175db8d805102aa5ead358b687aed6409270b53}}, {{cite:53037b4ab5770909caff95c573ede3b6dc014218}}),
but also injection of domain knowledge {{cite:3b306102e5842b82433a61191f8cf81d38b435a4}}.
Note that from a sensitivity-based point of view, policy gradient methods belong to
perturbation analysis {{cite:db7fe047bad99952b0bdaacf554db8de62c38d70}}.
| m | 47632e6f93a7cf8b6e21c9003a50f707 |
However, using metasurface optics for optical deep learning is not without challenges. Determining the metasurface parameters to produce a given PCE specification, known as the inverse design problem of a metasurface, is known to be extremely time-consuming, because it involves exploring billions of design points, each of which requires simulating the time evolution of the system by computing solutions to Maxwell's equations {{cite:5a2f2c7850d14457a74b8cf4345e1ae6efe94612}}, {{cite:b9984ff08aaaa650d41675d896c17040da4db370}}. Second, an optical layer when directly plugged into a DNN model will likely not match the exact results of the mathematical convolution due to optical design imperfections and sensor artifacts (e.g., noise).
| i | 369344b865af1cc71087367c4499c88c |
A coherently driven single two-level quantum emitter is well known to radiate non-classical light, which shows perfect anti-bunching {{cite:17bf60597694079b8af65540a3325b6fce35e8f9}}, {{cite:8124c87e9f2025a9334bc674c9c43150c19528aa}}, {{cite:40159c6ad471e5ed30fb29ccb92d420faa331fad}} as well as squeezing {{cite:5a00c6a3135037686291668dc08e10c2086981f8}}, {{cite:545d6a29d7aec17865e7661ffb513a6240c30aee}}. It is, however, very difficult to efficiently harness these properties directly. Usually it requires complex optical elements such as high-Q cavities, high aperture lenses {{cite:9958f5e5048bf776ea2d652c3b2d5bf697d30052}}, {{cite:06e4c366ea5abadf4035cc06798c999c8895ffa2}} or tailored optical structures {{cite:cc1e3b327f68a9d08818c50a8e8e837b30152911}}. Using larger ensembles of identical emitters increases the radiative output power but the nonclassical properties typically average out to create narrow band radiation but with close to thermal statistics {{cite:c66059c09fe476d890b2d15ba7e13810a022d102}}, {{cite:95d22b0c56fd58ac592eb5a0935ea11e27cf6a1a}}. For small interparticle dipole interactions in a dilute gas, the dynamics of the whole ensemble in the low excitation regime then can be mapped to an effective harmonic oscillator using the Holstein Primakoff transformation {{cite:b06861ac66319c0b296786b3ec2a52de4fe11c23}}, {{cite:acb8de808d26cb7d2d470b06220ccad1d7ec5c99}}.
| i | 5d5610d7ec746edb5079af9a24105ca0 |
We have considered the black hole mass dependence of the entanglement entropy of the Hawking radiation obtained by the island recipe for the asymptotically flat non-extremal Reissner-Nordström black hole
under the certain class of constraints (REF ).
The entanglement entropy of radiation of the eternal Reissner-Nordström
black hole was considered previously in {{cite:82e1f896241c5c7a48493b709da8e9348e71ab1b}}, {{cite:2514645a54e1ef7507e88dbbb65535648967fb05}}. The specific of our consideration is in the assumption of the special constraint (REF ). This constraint
permits to consider a near-extremal case and avoid the explosion of the temperature at the end of evaporation of the Reissner-Nordström black hole.
| d | 867392d23f2712de8c149903596e75e5 |
This equation can be solved by using spherical harmonics technique as was done in {{cite:1f9c95721602c11ae72913ab59add4b6b6665ce3}} for the scalar field. But here we shall follow the traditional WKB method for tunneling.
| m | 9db85c31c8faf8ed36a917d375e714a6 |
We use the following standard metrics {{cite:cf88b604a82ce1418e96b00d3a1791ffecdc2b0b}}, {{cite:dcd8046136a4b4e62ff43c85efc8467ae9134f2c}} to acquire quantitative results:
mean absolute depth error (abs), mean absolute relative depth error (abs-rel), mean absolute inverse depth error (abs-inv), and inlier ratio with threshold 1.25 ({{formula:12167f04-0def-473a-894e-f95d31bce27a}} ).
Since most of our competitors are limited to a minimum depth of 0.5 meters,
we do not consider the groundtruth measurements below this threshold for evaluation.
For a fair comparison on full field of views, we run the inference for our models at {{formula:6b7eacc1-b56b-44dc-bb60-6f65c10aeda5}} resolution without cropping.
We acquire the predictions of each method at their native input resolutions by following their suggested scaling,
then upsample the predictions with nearest neighbour interpolation to the original size ({{formula:45e94caf-5912-4dc3-8862-60c8de94ddf4}} ) before calculating the metrics.
| m | 92edc96291ca40ba0554ec053d58cb1b |
Applying {{cite:09713f46bfdc8da67226658474d840992accd7ad}} in our model amounts to using GMM to estimate solutions {{formula:1666fd9e-e219-447b-b107-af3a9e5e9d0b}} and {{formula:63085823-0d0f-42b1-a018-de7034308f4f}} to the following moment condition:{{cite:09713f46bfdc8da67226658474d840992accd7ad}} allow the instruments {{formula:e9417a93-aea4-4990-b817-c4428c08ce16}} to be replaced with any vector of transformations {{formula:7dce5720-0742-4c56-afa1-d32f8d030090}} with finite variance. However, if {{formula:0bbd29c3-5b18-41fb-a17c-0ea4c90617ab}} is nonlinear then the resulting moment conditions are valid only when {{formula:7a79bb36-6ad6-472a-8d2f-3e4364384f33}} and {{formula:eba19cdb-1d36-45be-97f4-680841038383}} are mean independent of {{formula:9f42136a-b100-4047-a6d4-50c8277cba1a}} rather than just uncorrelated with {{formula:f1beced7-e791-4c14-8f17-d712d1eca271}} .
{{formula:882d7adf-46f3-4f48-b98f-4fb540002be7}}
| d | 4dbb7d1ac7c8009c957aec15ddb4ab3e |
Our investigation focuses on the settings wherein the weights of the trained neural network stay close to the initialization.
Further, given the initialization scheme we consider, the {{formula:87e7389a-6873-470d-a686-906ddea04801}} deviation of the incoming weight vector at any neuron of the trained network from its initialization precisely captures the neural tangent kernel (NTK) regime studied in several prior works {{cite:44479d24216cfca42406941fa581daf032c33b7b}}, {{cite:a7bd6094a36857a2db72d181d01aa7711d58efe7}}. This {{formula:87af3e69-13dc-4fc3-bb6e-7f04d764b7cd}} deviation bound is also the largest radius that we can allow in our analysis; see Section for further details. In fact, our results would also hold for adversarial training given the same initialization and the bound on the deviation of weights is met. Curiously, our empirical results exhibit a phase transition around the perturbation budget of {{formula:7ea35711-e6ed-4d59-a580-6281a7674bd8}} – we see in Section that robust accuracy increases as we allow the network weights to deviate more.
| r | 618c4c32be07fe2088fe2d0343ba925c |
Several mechanisms are known to cause strong changes in the X-ray flux and spectral
states of AGN: The first mechanism is changes in ionization state or geometry of ionized or
neutral absorption that fully or partially covers the continuum source {{cite:ede66e086097c0e72c4173cf6c05fb4d173426b2}}, {{cite:5a2e66a2b3d666c0c10dbbd2d776a395ef51038e}}, {{cite:35657f611b9f5b9a27c1d7132607c7144e5c399c}}.
Varying absorption is frequently seen in intermediate-type Seyfert galaxies whose variability timescales can be as short as hours or days, but can also occur in
NLS1s {{cite:aeb898c8c327a965eb976bc56daba7ce9e55116e}} when the absorber is part of a clumpy accretion-disk wind or
resides within the BLR.
The second mechanism is changes in the X-ray reflection of photons off the (inner) accretion disk
{{cite:497106ba8991903c01cd515e68218d226d30b387}}.
This model has frequently been applied
to explain short-time spectral changes in type 1 Seyfert galaxies by changes in the
location and geometry of the corona.
Third, on longer timescales, changes in the accretion rate including disk instabilities
can cause high or low states in AGN
{{cite:f3428a67be5954e583905a255351c3e51558e7da}}, {{cite:d1f1ebd59cf0d5b426d81b845635e57ee75b5af7}}. This last mechanism has been favored to explain
some of the extreme changing-look AGN that also switch their Seyfert types.
{{figure:8db8be22-4981-4d9b-a4e4-e2ffbf4a7082}} | d | 745d538cc71ba2ac4ac613d8fd3eebce |
Methods such as SPN{{cite:50f289b2cf9a19839ba924885235d9be34d1a9d7}}, CSPN{{cite:fb53111f591c3fceb002080f7d4ca71b15926130}}, CSPN++{{cite:b64a1441d5a9cc06b5c3c46f97aee71c0de28148}}, DSPN{{cite:f5b462ea06bee447078edf182e81c15e50a9b100}} and NLSPN {{cite:6dccee170bf7aba6f6db04d83bd7d824db890f47}} focus on learning affinity matrix for high-level vision tasks including semantic segmentation and depth completion. SPN {{cite:50f289b2cf9a19839ba924885235d9be34d1a9d7}} serially propagates the affinity matrix, which is inefficient for real-time systems. CSPN {{cite:fb53111f591c3fceb002080f7d4ca71b15926130}} improved SPN {{cite:50f289b2cf9a19839ba924885235d9be34d1a9d7}} by predicting affinity values of local neighbors and updating pixel values simultaneously. Both of the methods suffer with the problem of fixed local neighborhoods, which often have irrelevant information. To solve this problem, CSPN++ {{cite:b64a1441d5a9cc06b5c3c46f97aee71c0de28148}}, DSPN {{cite:f5b462ea06bee447078edf182e81c15e50a9b100}} and NLSPN {{cite:6dccee170bf7aba6f6db04d83bd7d824db890f47}} methods are introduced. CSPN++ {{cite:b64a1441d5a9cc06b5c3c46f97aee71c0de28148}} proposed adaptive learning of kernel sizes and the number of iterations of propagation, which helped in reducing the computation time of CSPN {{cite:fb53111f591c3fceb002080f7d4ca71b15926130}}. DSPN {{cite:f5b462ea06bee447078edf182e81c15e50a9b100}} and NLSPN {{cite:6dccee170bf7aba6f6db04d83bd7d824db890f47}} learn deformable convolutional kernels to relax the fixed local neighborhood of pixels, which enabled them to focus only on relevant pixel neighbors for depth completion. All of the methods mentioned above utilizes a single branch AutoEncoder (AE) {{cite:ab7ec8473cd899f26372862baa3981a1f968c974}} network architecture. The input of the AE is the concatenation of RGB image and sparse depth map, which outputs a dense depth map.
| m | de69acbe317131236a18a9f0c1ebb00f |
For the 251 TeV photon from redshift {{formula:df512368-0194-4a79-be80-3656b396b32d}} , the relevant photon field for {{formula:c86e222f-ffb3-4ff9-a7bf-cb0f1c643391}} interactions is the CMB {{cite:f9409884572a96d6faab3a6e6546e2a142a9562c}}, {{cite:d05e6608fe1e91796cbaa32c3e7b4203f1976127}}, {{cite:2a8bdef7ab15cb26fa35429b6acbbe1582c5dcf2}}. Unlike the infrared EBL, the CMB is known to very high degree precision. We show the model calculation of {{formula:eb9830cd-f4a9-44c8-9cb8-326197c555ac}} (from Equation [REF ]) as a function of {{formula:4810898e-a974-47a0-902e-7a09d91f4e14}} in Figure REF for the Carpet 2 case. From this figure we can see that the 251 TeV photon from Carpet 2 gives the constraint
{{formula:52082e34-54cf-410b-8a17-9fb4946be45f}}
| r | 6c5b1139851dca0089ad7359ba7d0b67 |
It turns out that, in general, this measurement is so-called locally
optimal. This means that the optimal measurement depends on the actual
value of the parameter we want to estimate and the classical Fisher
information given by that measurement is a local maximum in the parameter
space that characterizes the measurement. Thus estimating the parameter
using a locally optimal measurement is rather impractical. Two
approaches have been developed to overcome this problem. The first one
is based on adaptive estimation schemes which updates a guess for a
locally optimal measurement {{cite:2fa0f2c52ab04f29304fe7a39b8a75c37a384db6}}, {{cite:9601e730a0b46c57f9b0fd9198386b0597689823}}. The second method seeks a set of initial
conditions that do not depend on the unknown parameter
{{cite:0a9d7cac23e2f56c3c5f837f6f5ba1a4fd204e4a}}, {{cite:f4b1e10855f86b56cce3aad8e2c00f3f51d1eab4}}. Both methods are based on the maximum
likelihood estimator (MLE) which implies that in order to obtain it, the aforementioned set of regularity conditions must be met
{{cite:79512b0e053dd6afd6a53b936e6f4b25825d3d00}}, {{cite:c1c8e93b7ecc4757f4e8f1cb18fb737b07a2d40b}}, {{cite:5114d8ff5eb015fcb7fd017d660a97857b5327d6}}. We will see that the MLE
derived from optimal measurements may fail to satisfy these conditions
and thus these two approaches fail to attain the QCRB.
| i | 1feb7ffe5b1070acd44594627d673212 |
In the OOD experiment, to prove the generalization of the method, we used three different backbones, VGG-16{{cite:c9e3e5b87429a246b81d652ab80d29ed4b4bbabc}}, ResNet-18{{cite:6ea937d1680055cc4aa2a846a44436b80a2f1785}}, and WRN-22-8{{cite:320a0b7f101a6fe1da4aae2753b193d2d9b91a7e}}. They are all very classical network frameworks in deep learning, representing three different sizes of networks: small, medium, and large. ResNet-18 is currently the most popular image classification network, so we use it as the basic feature extractor, and the two splits are validated on VGG-16 and WRN-22-8 respectively. Due to the large differences in the size of the three networks (i.e. the parameters), the selection of the hyperparameters during training is also different. In VGG-16, batch size is 150 per epoch, learning rate in CSE is 0.0001, and the learning rate in NICO-Aniaml is 0.01. In ResNet-18, 50 batch sizes per epoch, 0.001 in CSE and 0.01 in NICO-Aniaml. In WRN-22-8, 25 batch sizes per epoch, 0.01 for CSE and NICO-Aniaml, and 100 epochs for all three backbones. Images are uniformly resized to 224{{formula:7e6dbbdd-d65c-44b8-84d0-f4d038db540c}} 224 before they are entered into the network. The network optimizer is SGD and the momentum is 0.9. For a better fit of the network, the learning rate of every 30 epochs decreases by 0.1 times.
| m | f59d9a653b7c2bc74c93aad76c44b10b |
The extracted invariant mass agrees well with the values published
earlier for the {{formula:af9aa7de-1df2-4da1-84f0-ef0c5c6bcfda}} Be {{cite:de63e1d051e0b5a690ef57e85060e39f6844165c}} and the {{formula:5025dd23-51f9-45cc-837d-6ccb64051aa7}} He {{cite:e8e221d9f9abd98b71ed4a5fe8ecc769ccacd267}}
experiments, which provides a convincing kinematic verification of the
existence of the X17 particle. The branching ratio of the X17 decay
differs from the previous data, but, on the other hand, agrees well
with the theoretically predicted value {{cite:d937e2dd25ee84416aecb441ca889ea29784a8a2}}.
| r | 402ad05f0a9576aac190bff6da4e2669 |
In this work, by using OP and OPAL opacity tables reconstructed with AAG21 and Caffau's mixtures,
we constructed rotating solar models in which the effects of convection overshoot and enhanced
diffusion were included. We obtained a rotating model, Copal11r, that is better than the SSM GS98M
and the earlier rotating models of {{cite:dc3a0161e533e8baa668ffdadc72841945550eb7}}, {{cite:4862d49a0444c69341c18527ca44dcd05e4389f0}}. The surface heavy-element abundance of
Copal11r is {{formula:2f4c086b-be7b-46a9-92df-9abb387542b2}} , which is consistent with the value of {{formula:f9724581-0b0f-40b3-803e-215b390554be}} determined by {{cite:3b9f3ff44da24058385a97523923fa8b0098f1d6}}
and that inferred by {{cite:1d4d2e9f5b4abf852bb566e5f3e521c2d4ebed07}}. The surface helium abundance of {{formula:59f261e9-7c31-4555-bed0-b1a9096f184e}} and the radius of
the BCZ of {{formula:4c7a947f-7080-4b2e-81db-11ed7961a736}} {{formula:5d0f4e5a-146b-4654-9096-ba494666afcd}} are in agreement with the seismically inferred values at the level of
{{formula:2873c871-2465-4523-9646-33d9c065c028}} . The initial helium abundance is {{formula:660e11b3-4ef3-4b97-b2ca-0a92b6496ad0}} , which is consistent with the value of {{formula:1c90f948-f437-481e-bfc4-b5f413dd6092}}
inferred by {{cite:9043f4d68fa79f172feb362aeb28566ec1155860}}. The ratios {{formula:45dc116a-9e1d-42b9-bc32-defcb4c18331}} and {{formula:7f45c0e8-8ff7-47f5-9f91-e17781e5edb6}} of Copal11r agree with
those calculated from observed frequencies. The sound-speed and density profiles of Copal11r
are better than those of GS98M. Moreover, the fluxes of {{formula:e5fdd612-a716-411a-a798-8d8c83d71fec}} , {{formula:319b1958-b9da-47b2-a104-90ebed2572b4}} , {{formula:bfeb0e92-1a6a-4607-b97f-64ef46b8f53f}} , {{formula:a838871a-fb7b-4024-843d-72418929d4bc}} Be, and {{formula:0072d620-3da1-4cfa-8c6a-b41c1c189ad7}} B
neutrinos and the total fluxes of {{formula:1fd8efe8-8e2d-4e70-90ec-3d1b3d676108}} N, {{formula:fdbcccb2-55e4-4f80-bd41-cd9b2d57ce7e}} O, and {{formula:9928509f-455a-4345-a2eb-1bec1bec2952}} F neutrinos calculated from Copal11r
agree with those detected by {{cite:ef17ca3b87627212c18ee7308d04f6e2a39dbda0}}, {{cite:b7433a2b744a4aed226867657e2798ce1d12b02b}} at the level of {{formula:2b87fd24-4a5d-44f7-9c78-2a82a2f9511c}} . The fluxes of {{formula:3e0c2fea-86c1-49a6-8b5e-a3751da71acb}} Be
and {{formula:35203632-da59-451e-8542-39626b64db1a}} B neutrinos are also consistent with those determined by {{cite:8a9eb9bfcf121105d4947dcb39b0d595850efd52}} and {{cite:8ef92e0c2743a1af80b2a17727624d7c792bd037}}.
To recover the seismically inferred depth of the CZ, a convection overshoot of {{formula:e78ad480-299d-4303-9c5a-2e2e741edffe}}
is required in rotating models. Rotation or enhanced diffusion alone can improve sound-speed and
density profiles, but the combination of rotation and enhanced diffusion is required to bring the
rotating model into agreement with seismically inferred results and detected neutrino fluxes.
More details of Copal11r are given in Appendix .
| d | ac33365c24c98d5d2f3cdbf599422025 |
ODIN {{cite:22f69e423182444135c3c979f9a1a8a98e5339bf}} and
| r | e2809a8f93ad737bad05c28102201b86 |
One of the original motivations of the cosmological inflation scenario proposed by Guth {{cite:b40635f26c003b1c2675086e1fbd7a34cfdba189}} was to solve the monopole problem by diluting the monopole density.
If the value of the Hubble parameter during the inflation {{formula:1a51a73e-dbec-4638-948c-c68c6c6770d8}} , both the GUT and the PS monopole problem can be solved.
Simple inflation scenarios predicting {{formula:a5d51d67-16d9-451f-8632-e5fd62d76fe2}} GeV {{cite:abf37754f39eefd9534394fd9b1c31475598d6ae}} cannot solve the monopole problem arising from the PS symmetry breaking.The lighter monopoles produced during the symmetry breaking could survive inflation for {{formula:5528cf67-95af-43ee-964c-0036edbcb50d}} GeV and may exist at an observable level {{cite:5649536c9e814885fa5d420722ad2dc9fde39378}}, {{cite:34585493b6d7a781702f1fc06d26f0c71401188f}}.
However, if the inflaton is a Higgs field that breaks the PS symmetry, the monopole problem can be solved even if {{formula:4cd30b34-fd9f-417d-9735-62d70f87c6de}} .
This is because the PS symmetry is already broken during the inflation.
| i | 25534ca961dc27e0dd7cdd981b4d30a3 |
In principle, one could instantiate a hierarchy of CCNs, where higher level CCNs process the output of lower level CCNs, effectively modeling part-whole relationships. This is similar to Capsule Networks {{cite:6b37abda861177686b741d12d7f00fae31d0d3e1}} and GLOM {{cite:e2ad542b8fd2aa4326ba1dc91697b86ffae4ae9c}}, and corresponds better with the 1000 brains theory {{cite:81a0fdccd551d8b5e0ed35a46d29cc78720d005d}}. However, given the limited scalability of state of the art implementations of such hierarchical approaches {{cite:6b37abda861177686b741d12d7f00fae31d0d3e1}}, we adopted CCNs that operate on the level that is most important for a robot operating in a workspace, i.e. the discrete object level.
| d | 00a42cdc96afe35ceb07bce29a66c28c |
Though we propose a comprehensive framework to synthesize surface code on various SC devices, there is still much space left for potential improvements.
For example, besides heuristic data qubit allocation schemes, it is also promising to train a neural network to allocate data qubits.
Non-local bridge trees may also be explored for better parallelism of stabilizer measurements.
It is also possible to merge bridge trees to resolve bridge qubit conflict. However, this requires careful tuning since large bridge trees may be detrimental for error detection.
We do not include the error decoder design as our work mainly focus on surface code synthesis. Though we can reuse previous surface code decoders {{cite:9daabf3116de4f0cbbcf8647919906c295a61e49}}, {{cite:a2294935d8006adf4390976422ddc3e6673cb704}}, {{cite:3382c1c9275660b7c4c34701f729063ba1a935eb}}, {{cite:88e2ed8fe6f24fc1233565bb48cd9e7990e9ab93}}, there may be some opportunities to devise more accurate error decoders for the proposed surface code synthesis scheme.
| d | 90cf8daddf436f926c1e080845001950 |
We will model floating point computation in a precision {{formula:c3f6b170-499d-41cf-b631-96c559dc341f}} using the following standard conventions (see, e.g., {{cite:6207d346509ff2e2b0c831267be3cd46f85f25bd}}): for vectors {{formula:41415114-56d3-48cc-9b2d-390ea6f698e3}} , matrices {{formula:fecf29df-d439-490d-9df6-160be1d0c890}} and {{formula:27032a42-c7ee-460a-8f74-fe36a08cc18f}} , and scalar {{formula:b2724d12-b897-4df7-ab60-0076fe42a112}} ,
{{formula:6a396cb1-0f1d-4919-8654-9bc99faa173b}}
| m | a23231af6182cb6951fc4871bc518242 |
In our algorithm, as motivated by Section , we effectively consider the following joint distribution on the graphical model shown in Figure REF : the prior distribution of {{formula:7b1803fc-81b9-473d-acb2-af7920f2a663}} is {{formula:b8ad1c22-4dad-46c9-bde6-cd054c0ecebb}} , and the distribution of the rest of the graphical model is determined by an options with failure policy with parameters {{formula:962d22ed-b2ac-4acf-9040-2ea3a80829cb}} and {{formula:f3520a92-8b60-47aa-ad5a-1a67b4c87955}} . From the EM literature {{cite:b07ae1a129cc92a8f929be63652f93b43d1d9673}}, {{cite:53615a5e3442a22f7f3aaaa70f3cd5d578b3b036}}, the complete likelihood function is
{{formula:728cdee4-fc6d-43cf-9733-3b9f3a1f913d}}
| d | a193318f7ec937f665c23755ee0e2bd1 |
In this work, we implement a Susceptible-Infectious-Recovered (SIR) infection process {{cite:cff0c2de774c1d4460a9bc7e8107bb62b7d0df17}}, {{cite:6446063ad116f5b308b0bc4048dca9fd3934b76b}}
in a system of self-propelled agents with alignment and repulsion interactions to explore the impact of self-organized spatiotemporal structures on the contagion dynamics.
Our results show that the epidemic outcome depends strongly on the specific emergent properties of each collective state.
We observe that the outbreak thresholds are significantly lower in ordered states with large-scale clusters or band-like structures, and higher in ordered or disordered states with homogeneous density distributions.
By comparing the contagion dynamics of different states, we find that the timescales of the emerging spatial structures and of the infection processes play an essential role in the epidemic outcome.
| i | b81a68c7b79f37e3cb4eb6580d07921a |
[leftmargin=*]
Graphs usually contain rich saliency information and regularity structure (e.g., functional groups in molecule graphs, coalitions in social networks).
Thus, randomly corrupting their properties (e.g., topological structure, node properties, or edge attributes) could lose the discriminative semantics, making the augmented views far from the anchor graphs and misguiding the contrastive learning.
Figure REF showcases random augmentations are likely to pervert the cyano group (-C{{formula:814c6fcd-0460-4a48-89a3-db68bf35d756}} N), which is the key feature making the molecule hypertoxic, thus easily derailing the discriminative power of the augmented views.
A research line leverages external domain knowledge to identify the salient semantics of graphs and guide the augmentations.
For example, underscoring the functional groups, such as the cyano group for toxicity in Figure REF , is beneficial to molecule representation learning {{cite:93c21654a40f3d1e8ee4cfd3e4c3ff5f22f0d182}}.
Nonetheless, it is prescribed to expensive domain knowledge and easily undermines the generalization performance in unseen domains.
| i | 97155c317628171397ec9725294e6d7a |
Polytrope gas sphere was once used in the early study of stellar structure {{cite:9515134aee9bc4e8a32ac9272f4ecff4ebd362af}}. With the development of modern computers, researchers tend to solve much more complex stellar structure equations {{cite:97e4be40d338b00b8f357050e684833cfda98b45}} with sophisticated numerical codes, e.g. the MESA code {{cite:f5070e1388895e67f14d35b2c14095724be746c0}}, {{cite:d0ea65cf4305dd8c754a408780e8033d4a205ca2}}, {{cite:a5d88b368b619689e851f5ad53118daa1f3f6195}}, {{cite:63e391fa50ba0fbad2a97950833fcff09da6a05d}}, {{cite:6a758561584e0680511e073dafdd7054a22af378}}. However, polytrope model has its advantage, namely it has very simple equation of state but in the standard stellar model the complex tables for equation of state and opacity are used. In some stars, the equation of state is unknown and the polytropic relation is often used. Or when we calculate stellar oscillations with very fast rotation or strong magnetic field, because the problem becomes two dimensional (Coriolis or Lorentz force couples different degrees of spherical harmonics), we have to develop a new code other than the MESA code. In these situations, it is better to adopt a simple equation of state, namely the polytropic relation.
| i | 623ef86fd7e9eeda51894a3db30bb43f |
Galaxy clusters are the most massive structures in the Universe, and are therefore powerful cosmic telescopes for observing faint and distant sources via the gravitational lensing effect {{cite:8c12073a4264ca3bad9790b78f29c71f72d7b16e}}. When a background source galaxy lies behind a galaxy cluster, the light rays that we see from the source are deflected by the gravitational potential of the foreground galaxy cluster, i.e., the source is gravitationally lensed by the cluster. In the regime of strong lensing, we see multiple magnified and distorted images of the background source. The lensing magnifications produced by galaxy cluster can reach factors of {{formula:a6ef9008-594d-4a18-9413-2261b6177b9e}}{{formula:eeede4cd-2164-44d4-b799-fa375e699d5c}} , allowing us to detect the faintest sources that would be otherwise impossible {{cite:8710f2b179377f4a0d980a187127c081e1cdf60d}}, {{cite:f2ec75d15f93e565cf909d749fc9d010ecddc60b}}, {{cite:95f812a58f7ce1817bbe3a0f310937b1ed508027}}. The positions and morphologies of the multiple lensed images also allow us to probe the distribution of matter, particularly dark matter, in the galaxy cluster, which is crucial for understanding the nature of dark matter {{cite:3f0080b72f149e92bb706f0f33eb9bd79982d5b6}}, {{cite:7f696cc5890130f897f87d39676fc112ac56392a}}. Furthermore, an individual galaxy cluster often lenses several background sources into several corresponding “families” of multiple images that straddle at different locations around the galaxy cluster. Families of multiple images formed by sources of different redshifts provide measurements of angular diameter distance ratios between the cluster and sources, allowing us to probe the geometry of the Universe {{cite:35e1833456f1ffccf08eb8b4b39521829d2d26f2}}, {{cite:c83c747cd063fa1c0e8523a0c4be7a1813559566}}, {{cite:7cde8aafee0ac9cb93817f37a4d284ce973bc98b}}, {{cite:b27c084ae7ded67aa53a3127b2f09dbc16156426}}. In cases where a background source is time varying, such as a quasar or supernova, the time delay(s) between the multiple images provide measurements of the “time-delay distance” and the Hubble constant that sets the expansion rate of the Universe {{cite:5ddcf36f1ebe0c2afec0b9568e2ff2ee548049f1}}, {{cite:ef6a693be28c6a49c73edbb8dcd7c998e11089bb}}, {{cite:8bde53ce5ca81d5464db655e2543f9177647fbf4}}, {{cite:5be890efc705db5f57db14a4c3a4a275c9331420}}, {{cite:945f418c337749ec268a557839be08f1b0cc1d19}}. Strong lensing galaxy clusters are therefore excellent laboratories for astrophysical and cosmological studies.
| i | b35c71383ea21390e2403a2182fe5db6 |
where {{formula:efa182d6-fdc4-44df-8157-e7f26d03c952}} .
One can easily verify that the characteristic polynomial of {{formula:7067e305-b51e-4802-9c9b-9319688d9779}} is just in the same form shown in Eq. (REF ) {{cite:bcef20de85106bff03b995de1bdf375130fe0029}}, {{cite:cb73e92195b2878c52e79e82a421d8fd1ab26a35}}, and hence it indeed can be viewed an effective Hamiltonian. Obviously, due to its relationship with {{formula:0fee2278-d961-4d3c-ac0a-2b3a51a7faf8}} , {{formula:12098631-c972-4e99-ac89-f0c81d881038}} must have a convergence radius same as {{formula:9533ecc7-4abb-4a61-8e7e-fa1c0ba83322}} for {{formula:9a49e345-05d5-49bb-8bb4-6f24e6974ec4}} -expansion.
| d | 54d1fb7436320e3eb763673d868e5099 |
Imaginary parts {{formula:c1d9d61f-8faa-487f-b80c-3e683fb95036}} , {{formula:c4a6d5d8-9c84-45a6-83db-0bce59826afb}} can be calculated using the Mandelstam-Cutkosky rule {{cite:de52ad13076a125293abcd9e231c3f46d72bf530}}, {{cite:c32a678364b97f04ca69539b7fb51d7d1658c8a6}}.
For example, the imaginary part of the function {{formula:c1737b39-6abf-4e03-8126-351715a59f72}} is equal
{{formula:d038d342-e297-48ef-b94d-278f9665635b}}
{{formula:e033cab3-e460-4ab2-874a-e115061c215b}}
{{formula:38c3f025-5f94-4ddc-b617-f8a3d9a91b8b}}
| m | 23c033a3641012ffa99c2b2644df1fd2 |
for any subtree {{formula:f718b698-8eb6-45a5-94d4-cb75f6ced85b}} .
First, we use the fact that {{formula:43f5f9a1-dcda-4690-bd08-05858db6f56e}} is a {{formula:35163057-4d93-41ef-bef8-187a761576cc}} -exp-concave loss function, hence {{formula:d02c6e19-67c6-4f89-b920-4c8ffcc2d023}} -mixable, see Section 3.3 from {{cite:5b1e64ea926192065ecf57d5108b0061fd0282bc}}, which entails, since {{formula:5d8c95be-d397-48f0-bf46-e0944789cc4a}} is a probability measure over the set of all subtrees {{formula:da92cbeb-d0c2-4184-9e36-d0edc589c249}} , that
{{formula:73ec7f4c-f8a7-4d61-af5b-92b7d7429123}}
| r | f1a15485b6c434a694183cdcde8f9b0a |
The function {{formula:5acd726b-383b-44aa-9290-c2205ae42356}} can be interpreted as a probability to be at point
{{formula:a308b1d2-0110-48e9-b693-00b4958fdf39}} in the collective space. The GCM+GOA provides a general framework to
compute the collective inertia tensor and various zero-point energy corrections
associated with the collective motion {{cite:703953f9aa0b90894c63c097601a05c1a98341b5}}. Indeed, after using
the GCM ansatz (REF ) for the many-body wave function
{{formula:5131b4dd-acf2-405b-aa96-74c84d404223}} and inserting the GOA approximations
(REF )-(REF ), one can show (after some rather lengthy
derivations presented in {{cite:2e99fb930c7651bce7f10b6278faa27e7580259a}}, {{cite:fd25a8b0382b3b4136f7eafe0a2732286ea28187}}) that the expectation value
of the total energy {{formula:ea224a28-d6fa-4559-9672-db961283426b}} becomes
{{formula:4844034a-71e9-4f51-9e90-f1b43b3339a0}}
| m | a79101adbb88ee33dce51b5fd0a94589 |
As we noted previously, one of our goals is to make the greedy algorithm work in the distributed HPC environment.
Since this tiling solver would run for every program executed in that environment, it must allow running at such a speed that it does not significantly impact the actual user program's total run time.
With this in mind, although the exhaustive search would yield good results, in certain cases, the number of involved matrices and operations in a user program could result in a lengthy solve process, as typified by the Parallel PCA and Power Set Multiplication programs.
All of the programs we used were less than 18 matrix operations.
For comparison, SOTA deep neural networks like BERT{{cite:60402b2a61ee5f787c54879635ae6e83a0dead2e}} or GPT-3{{cite:354ccd097cbc1e6a63d7fb18905bc3aa98fbbf75}} often use a large number of layers (in large BERT's case 24 transformer layers, in GPT-3's 96 transformer layers), without taking into consideration any preprocessing steps for data, meaning user programs for investigation could substantially exceed the size of our experimental programs.
We also kept our search space small with the number of implementations of edges in these experiments.
With these facts in mind, in certain circumstances, it could become prohibitive, even with further parallelization of the search process (on server processors or across multiple nodes), to exhaustively search the entirety of that space.
| r | 563841887fdca8dc9c24d203dd19670c |
Note that assumption (1) is standard when analyzing the stability of optimization algorithms {{cite:05bc5f6e2f628ff5e2d8d49574c1b620e3f625fe}}, {{cite:5b673f30654d36fe57aad954fc9d78e7f90e27e8}}. Assumption (2) is easily achieved if we restrict our domain to a compact set.
Assume that we run the AOM on {{formula:6ae69591-cfa4-46c5-b431-d2c8d0d9c3f8}} and {{formula:bea7a137-5462-4d38-8e2b-70979bec2e97}} for {{formula:bb6ba5d4-98b9-482f-84a7-de9b6e1eeb69}} steps to get the outputs {{formula:7039e072-0ae8-45f2-a224-c505a37c4af2}} and {{formula:630225e9-fc05-41f1-8249-e91a094a5e6f}} . We first have the following lemma which controls the difference of the loss at {{formula:099e2aa2-708b-465d-bc2c-5b4c9bcc90ce}} and {{formula:51f8cfa8-7a36-4c83-8962-889b13243420}} . The proof of this lemma can be found in {{cite:5b673f30654d36fe57aad954fc9d78e7f90e27e8}}. For completeness, we also provide a proof of this lemma in the appendix.
| m | 0f1b406be18bffb6d1ff72535b3fc15e |
We first evaluate the performance of VoxSeT on the Waymo open dataset. The results are summarized in Table REF . VoxSeT outperforms most of CNN-based models, leading PV-RCNN {{cite:67c8fa8a4cde2566a80b20d44fe265c97b8396cb}} by 5% LEVEL_1 mAP and VoxelRCNN {{cite:a159e1e9d077c3544e03196f947e01d9b33992d6}} by 2.4% LEVEL_2 mAP. As one of the few transformer based models, our VoxSeT achieves better performance than its transformer-based competitor VoTR-TSD{{cite:c08b2f52a8f96ab6085bf301b3337c5a12a90833}}, which brings 0.9% and 1.4% improvements on LEVEL_1 and LEVEL_2 mAP, respectively. VoxSeT achieves comparable performance to the state-of-the-art method CT3D {{cite:376805256ab6e8d9499762c8b34a62ef95a0dd44}}. It should be noted that, however, CT3D actually employs a heavy transformer based RoI head, which has three self-attention encoding layers. By adopting this RoI head into VoxSeT, our model achieves better results, outperforming CT3D by 1.5 % LEVEL_1 mAP and 1.2% LEVEL_2 mAP. This demonstrates that as a new transformer based backbone network, VoxSeT surpasses Sparse CNN based networks. VoxSeT works especially well in the range of 30-50m, which indicates that transformer modules are better in capturing the context information in long-range areas.
{{table:cbae2af1-a5e7-43f9-b431-764e6fd5dc14}} | r | 48c1533fd816ae16c4dd308bc1ec5612 |
Pulsars can potentially emit electromagnetic radiation via many
mechanisms {{cite:f35b6898a509fcac99dc598431f2d3fb7f621282}}, {{cite:d20f41f30aaa6e73bec31bc8460c22168b6c9bd0}},
including thermal emission and other processes in their hot, plasma
atmospheres, and dipole radiation from the rotating magnetic field of the
neutron-star core; why then, might the pulsed radiation detected on Earth
be dominated by the small volume of superluminal polarization current?
The similarity of the “plateaux” in Fig. REF of the SI to Fig. REF (a)
provides an important clue. At the focus polar angle and over a short window
of {{formula:4c138a7b-7034-4733-be3c-d5a81ebe97d9}} , the frequency content of all of the
emission processes occurring within the rotating polarization-current element will reproduce
exactly, and result in a detected signal with greatly enhanced amplitude;
the result is similar to coherent emission {{cite:3c355d903004f08a8ac69d65394fb3c88fe9b67e}}, but via a completely
different mechanism.
At all other observation angles and observation times,
radiation from the emission processes will
superpose incoherently [c.f. Figs. REF (b), (c)],
leading to a greatly
reduced amplitude, and scrambled frequency content.
The sharp focusing in the time domain at the focus polar angle
is likely to allow the radiation produced by the superluminal
(outside the light cylinder) mechanisms to
dominate the pulses.
Note that this explanation of the brightness of pulsar
pulses does not depend on the incorrect proposal {{cite:2760118c163f8a025f9813a2a76b4320e9d0605a}}
of non-spherical decay advocated by Ardavan {{cite:c970da86a69ee342e91261fcc85c459e16b58dce}}, {{cite:eb8716c5e8f932c4e379962f81a40f6fcd4f123d}}, {{cite:db5952bb1af2a5208b18997a5f062332a1cbc36f}}.
| d | d207b0a86f283cfeeba5dfbea2ceaa9d |
We implement TTransE, and TA-TransE/DistMult based on the implementation provided in {{cite:897128f1fcf908d33749d6386a405f3c8bd441a2}}. We use the Adam optimizer to train the baseline models and optimize hyperparameters by early validation stopping according to MRR on the validation set. We set the iterations to 1000, the batch size to 1024, the margin to 1.0, and the negative sample ratio to 1. We implement LiTSEE based on the implementation of TransE {{cite:05717d78be7ec1edc1851ce01103400afc650d57}}. We implement KnowEvolve in PyTorch based on the sourcecodehttps://github.com/rstriv/Know-Evolve. We use the sourcecode for RE-Nethttps://github.com/INK-USC/RE-Net.
| m | 909cd2ceb1ec756ef5c14eddb747a55e |
To supplement the results of the experimental measurements, we have performed calculations of the electronic properties. The electronic heat coefficient {{formula:9e11d54a-e41f-454e-a2d6-e657eeed883b}} is directly dependent on effective mass {{formula:61f1f494-c503-4ade-96c3-aa14cdb6d0f9}} and carrier density {{formula:60d48962-c9ae-4d4b-8f2f-950ccaf2e4a3}} of quasi-particle via this expression {{cite:e545f2beac172e47fcde28c37a2a072057d25d2c}}.
{{formula:aa6fa3ee-39fd-467a-b0c9-823b5d2ca27b}}
| d | 622c574c0b52402186133f88bfabf707 |
Our method in this project is mainly based on the GE2E model proposed by {{cite:b590fc5069d124484f30022c30969c291d58237a}}. The main advantage of the generalized end-to-end training is that it enables us to process a large number of utterances at once, which greatly decreases the total training and convergence time. In this section, we first explain the proposed GE2E method.
Then the necessary pre-processing and data preparation, the training procedure, and configuration will be described.
| m | 5aba6f95e917f85f5c1b4ea0ab3708b3 |
We show our results on the VisDial v1.0 dataset, where 123,287 images are used for training, 2,000 images for validation, and 8,000 images for testing {{cite:03a3003ace31ba505d279fee901cd06d62abd2cf}}. Each image is associated with ten questions, and each question has 100 corresponding answer candidates. We use two MRR models (i.e., {{formula:e8315f52-b262-4121-88ed-8243ccc690db}} ), FGA {{cite:7820a70a21b1def837dbd41a975447ae639d4641}} and an ensemble of LS {{cite:91b1d4c7c710b62b57fec41b604c12a43565c871}} with FGA. We use LS(CE) as the NDCG model. We set {{formula:3e382370-6a97-43fe-8dba-414455773a35}} , {{formula:ee9864e4-5095-46d1-b386-6ed1362ae3fd}} , {{formula:3fc6ee86-8471-4a03-952b-d436bbf9dd9a}} , {{formula:15cc0371-f122-44ce-b4be-7ad74b8241ee}} , and {{formula:9ef63b83-8e75-4428-8ab3-7508f4b54c09}} . We tune these parameters using the
validation set.
| r | 275b5e823033e66ba25446fa62fc6125 |
The large-scale structure of the Universe, as traced by galaxies, provides fundamental physics information through its connection to both initial conditions in the primordial universe and general relativity through the gravitational formation of structures on the largest observable scales {{cite:063e54990c0a44f0e58a510fd6303fac440b53de}}, {{cite:efbed615b8de73ab0907c5645e1487a962bf2284}}. A well-established measure of this structure is the redshift-space two-point function as measured by galaxy redshift surveys {{cite:d246a520a9678486c44564ca38d432d6ce3b22e3}}, {{cite:c8e1752a42e4600f7949787c7d5a051f1364bf4d}}, which encodes both the power spectrum shape of fluctuations in the early universe and, through the quirk that line-of-sight distances in such surveys are inferred from their redshifts, cosmological velocities in the form of redshift-space distortions (RSD).
| i | 9f4390b79b0fca96d926ceafb3160954 |
We introduce the main model architecture in this section. Each subsection forms the foundation for the next one. Section REF describes an attention based network inspired by {{cite:0ee7e258ea9eb4ffc0be47cd5d65b2aa3b402f32}} that exploits the correlation between the current ranking task and the entire historical sequence of items for which the user has expressed interest. Section REF details an elaborate Recurrent Neural Net backbone that can efficiently handle a batch of uneven sized sequences, and how it is used to capture more recent user interactions with the search engine. The output embedding of the attention network is simply fed as an input into every timestamp of the recurrence. Finally we discuss how to build an actor-critic style reinforcement learning model on top of the RNN structure in section REF .
| m | 223273c2574add6a1f923b1bdfe3bdcb |
[leftmargin=*]
GRU4Rec {{cite:d3338399546a77924653393e25e2a141edb7b4d5}} is a GRU-based session-based recommendation model which also utilizes a session-parallel mini-batch training process and adopts ranking-based loss functions to model user sequences.
NARM {{cite:6c9ab2fe96388d01b7a7c2608417b2d5274ac5cc}}: is an RNN-based model which employs attention mechanism to capture user's main purpose and combines it with the temporal information to generate recommendations.
STAMP {{cite:4449d4b33de2fcb8a312cf360c763c6e3c4b6dbb}}: adopts attention layers to replace all RNN encoders and employs the self-attention mechanism {{cite:0ddece9ab521e47f895df4db2ce12da6b4608fc9}} to model long-term and short-term user interests.
SR-GNN {{cite:fc1cfe02e08123b9839bb11a8b20b57cd978d253}}:
proposes a gated graph neural network to refine item embeddings and also employs a soft-attention mechanism to compute the session embeddings.
{{table:8a882afa-c42e-41be-80ee-ad5d35c5d7f4}} | m | bb99ba4f368f8c3fe1ec8226631a6076 |
MS-COCO object detection.
As discussed in previous works, the performance of the object detection task greatly depends on the quality of deep features to locate interested objects. This rule also stands in transferring knowledge between detectors {{cite:12028055608680ed1550cd69a361b4102eeb6f7b}}, {{cite:a99679553bc205570d71585eaa6362bddac5825a}}, , feature mimicking is of vital importance since logits are not capable of providing knowledge for object localization. As shown in Table REF , singly applying DKD can hardly achieve outstanding performances, but expectedly surpasses the classical KD. Thus, we introduce the feature-based distillation method ReviewKD{{cite:fb1aa00dad27d60f36e63e5dc1632e8980f77365}} to obtain satisfactory results. It can be observed that our DKD can further boost AP metrics, even the distillation performance of ReviewKD is relatively high. Conclusively, new state-of-the-art results are obtained by combining our DKD with feature-based distillation methods on the object detection task.
{{table:90b00b46-8d17-44d6-baba-3f3be1bb36da}} | r | 3f8315222731169b345ccc4fd97f729d |
On reasonably regular graphs we expect the contact process to show a phase transition in the infection rate {{formula:fef3a590-6c8b-4a31-abe6-7d640f03416c}} . There is {{formula:04518b1a-197c-4188-9e9e-865645ce0705}} such that for {{formula:df748689-0cc9-4e20-a2f5-4e14c892a286}} the expected extinction time is bounded by the logarithm of the number of vertices of the graph. For {{formula:497bc643-376c-4430-ba35-e437c261a55b}} however if there is an outbreak of the infection, with high probability the extinction time is exponential in the graph size. See {{cite:dce0e4e20173953631a9723aa1ad468ca673b2e0}}, {{cite:046e8a3cb6dd0cd756cc143e8bd5f96e82f8accd}}, {{cite:ed8b588f510a111804136590567f94dd86612b61}}, {{cite:0f111f6336884776e528830fd2b18cb4dffcfaac}} and {{cite:3dad6157e8ce9f349c1ad5a01470c509da25b4da}} for results in this direction.
| i | 19f2b15c51a63dae0c99236060dc3650 |
Baseline: We compared our method with a standard VGG-16 network with Batch Normalisation {{cite:3a1768f49b1fe832bdb95af911c5e48934ec658f}}. We ran the following baseline experimentation:
| r | 18ba8aea944050f4e8e9da98b352b46f |
{{cite:ed2e65993210fe7b8f1492936a6ee609596e6dc3}} experiments using PCA and found that the majority of variance in weight updates lies in only a few dimensions. Perhaps these few very important dimensions are preserved with IMP. It could also be the case that larger learning rates could dislodge the weights from a gradient desert and find more amenable points in the weight space, allowing the recovery of useful training for even randomly masked networks.
| d | 54e4928a0d123b1a4c0bd8db2ea1ba8d |
The dashed blue line in Fig. REF b shows the {{formula:fe1dddeb-8a67-4074-8728-ad2c597cd8ad}} -factor obtained from Fano fit on the reflectivity curve obtained from the full vectorial simulations (calculated as {{formula:43603b40-a9c4-45df-9edd-55cfa0f5aa5f}} ).
This is in excellent agreement with the eigenmode analysis results, meaning that the two methods are consistent and reliable.
We observe that the Fano parameter, {{formula:d6dbe343-b10d-44fc-8421-3f8c7027edb7}} , is always negative except in proximity of {{formula:0a26a03c-1fe3-4a25-af82-f1927d121b4b}} , meaning that in the latter situation the quasi-BIC and the continuum phases are shifted by almost {{formula:dd736107-9e1a-4854-9319-75044c84c5ca}}{{cite:2bf9ec23567d69a0f70bc99c1ec0a47223d28b06}}.
| r | 4cb49a91dfba8e3a54690665ff5de19b |
It is possible to see a closer analogy - from a formal point of view - between the vector-like theories and chiral theories, if one considers
color superconductivity phase in the high-density region of QCD {{cite:e1aaa6975895c52d7130810d8ebd7236b077d1e6}}, {{cite:9c6e09c8c5396fc8b063db2a8b86f60428cd54a5}}. The dynamics of QCD in that phase is believed to be such that some colored di-quark condensates
{{formula:35d679a4-bc09-4c6d-80bf-e4cf3513ca9f}}
| d | 008f783d3162ce7eb2c1c59d74fabb2c |
Several real-world phenomena involve the emergence and spread of novel traits in populations of interacting individuals of various kinds. For example, such phenomena may concern the propagation of new information in a social network, the sweep of a novel mutation in a genetically homogeneous population, the rise and resolution of spatial conflict, and the diffusion of atomic properties in interacting particle systems. Network science collectively studies such phenomena as diffusion processes, whereby the system of interacting individuals is represented as a network and the corresponding process defines the (stochastic, in general) dynamics of local trait spread, from an individual to its neighbors. These dynamics can vary drastically from one application domain to another, and have been studied extensively, e.g., in the cases of influence and information cascades in social networks {{cite:cf871a9f35f1b680cbca67cfa391db96b0a273e3}}, {{cite:61cecaf5a36da2ee1a4655a8d4bbdaa5396fdd83}}, {{cite:f2a27ed71c039642b59c3a8ffd92f1ada8497e8f}}, {{cite:a4d514f28549352e660d590d452e4f91cf3ac6e3}}, epidemic spread {{cite:a1f59d815dd9ee042d05ac067d22861a64cc5c47}}, {{cite:33cd1569d4734fd1fa5412dba223ec143e49212c}}, genetic variation in structured populations {{cite:b0573e343bb893d3e543ebe527b7e210ac7c7495}}, {{cite:b0320ee860250ad5b7770525a4fc252b9aae062c}}, {{cite:7c7444e21387da232a3e7a3b01c1e5fd9846a3ed}}, {{cite:f5d772ce0a63a77fb8a284b48042ee4480fde430}}, and game-theoretic models of antagonistic interaction {{cite:d1c6a131a155dc2f11c087cfa603d666f7e80f89}}, {{cite:e66c4b34f23412c1b7ef295cd9654b372bc7363a}}.
| i | b03c851a7d48b01e61957259912bb3b0 |
The extension of these systems to more than one local moment gives rise to models of molecules {{cite:553cff4a9989d84a70350711d5fb18dd4e2886a1}}, {{cite:8065119c94e5a922e3d693b37f5bd61ed696d548}}, {{cite:154d6a497ea66e4d3925c773637684a229901d42}}, {{cite:128326f964f67e56ae45fd1262f6203155d404af}} and spin chains{{cite:406c4648982e1d9e1956035d543d4525084446ee}}, {{cite:d9fb8988927010d7214741e7a6ec3614668aff72}}. These systems have become particularly interesting both from the experimental and the theoretical side as they show properties of topological superconductivity and Majorana zero modes{{cite:f556ea7636982210e558878265219fcb8b380518}}, {{cite:a96e7742c9fee1bb03bd7c1afb51c1014fc8fe70}}. Here the influence of the quantum nature of the impurity spin is expected to be of particular importance. The theoretical treatment of such systems therefore requires true many-body approaches that properly take into account the quantum nature of the local spin.
| i | 70f12ee2049cbf79de1bed050960d319 |
Section REF compares GSAM to SAM and vanilla training. In this subsection, we further compare GSAM against Entropy-SGD {{cite:1ac760d10b486c14a7b6662f0f4a4bd019e102e5}} and Adaptive-SAM (ASAM) {{cite:6ddfd6171382d25fd97be56d767e528108132993}}, which are designed to improve generalization. Note that Entropy-SGD uses SGD in the inner Langevin iteration and can be combined with other base optimizers such as AdamW as the outer loop. For Entropy-SGD, we find the hyper-parameter “scope” from 0.0 and 0.9, and search for the inner-loop iteration number between 1 and 14. For ASAM, we search for {{formula:745127ff-e33d-4b6e-a632-bf74e398bfec}} between 1 and 7 ({{formula:ed442b6b-3c4e-40ba-9e60-789b0ebf165a}} larger than in SAM) as recommended by the ASAM authors. Note that the only difference between ASAM and SAM is the derivation of the perturbation, so both can be combined with the proposed ascent step. As shown in Fig. REF , the proposed ascent step increases test accuracy when combined with both SAM and ASAM and outperforms Entropy-SGD and vanilla training.
{{figure:0fc6d7c8-c6ec-40ea-bc2d-6aa178eb26c7}}{{figure:7351376f-03fe-4fc7-b9c6-3cd25951004e}} | m | cf04ef80455f89970f9c711b92273ed4 |
Replication of results. The numerical example of sec.br is tackled with the open-source finite element environment FreeFem {{cite:10b14c9050a572280ead013222dde2e8283dd0c3}},
and the precise source code used for the resolution is available on demand.
The treatment of the numerical example of sec.canti relies on minor adaptations to the open-source, educational implementation supplied with the article {{cite:4c43a0116b344a515112bc599e572a56a8726090}}.
| r | acfb5758d3185da65ad1e243104e6be3 |
Results on Real-world Datasets.
We also compare our AECR-Net with SOTA methods on Dense-Haze {{cite:25040b82b4d420b5e1f6e347d443c932815b22c2}} and NH-HAZE {{cite:78e60a93c3b403c96a3f26602a299b0f8a0d855c}} datasets.
As shown in Table REF , we can observe: (1) Our AECR-Net outperforms all SOTA methods with 19.88dB PSNR and 0.7173 SSIM on NH-HAZE dataset.
(2) Our AECR-Net also achieves the highest PSNR of 15.80dB, compared to SOTA methods. Note that MSBDN achieves only about 0.02 higher SSIM, but with 12{{formula:299eb879-6289-49a1-aa34-4af97a15bd7a}} parameters, compared to our AECR-Net. (3) Compared to RESIDE dataset, Dense-Haze and NH-HAZE dataset are more difficult to remove the haze, especially on Dense-Haze dataset. This is due to the real dense haze which leads to the severe degradation of information.
We also compare our AECR-Net with SOTA methods on the quality of restored images, which are presented in Fig. REF and Fig. REF .
Obviously, our AECR-Net generates the most natural images, compared to other methods. The restored images by DCP {{cite:eec8f7e825aa9dcb2e527c79f80f60aca98a5fb2}}, DehazeNet {{cite:552893d114e3d88669619bd8507982469207f0ef}}, AOD-Net {{cite:c40b7047058dbccb9434b0f423eaf4f81089c2a4}}, GridDehazeNet {{cite:803193603fde21d0aea38ffad74e2591497bf12e}}, FFA-Net {{cite:cab3d1bd3216448f8779c2765beb364a3f80b8a7}} and KDDN {{cite:07722c6df3a6c9034198f782785c63f200018999}} suffer from the serious color distortion and texture loss. Besides, there are still some thick haze existed in the restored images by MSBDN {{cite:6a7a1972da53184f6f962ec56f6325abe35e7710}} and KDDN {{cite:07722c6df3a6c9034198f782785c63f200018999}}. More examples can be found in the supplementary.
{{table:32945fed-3b9b-477a-9876-4e5cf2c53d40}} | m | ef0e96fec8d816c317cc41f09ef5a192 |
Another relevant aspect concerns the ratio {{formula:074d91f4-ad76-4e46-b3b1-6b6f84070e1b}} .
If {{formula:55c953fa-0184-4c15-aef2-c042e4d1d27b}} ,
the bad conditioning properties of {{formula:787fb10b-d4f7-4a04-b08d-d320598dace0}} inevitably affect the consistency
of principal component analysis (PCA) as a factor model estimation method.
In fact,
the sample eigenvalues follow the Marcenko-Pastur law {{cite:701944c79aeef86d96b07c317005137d4bc877f8}},
which crucially depends on the ratio {{formula:7238a0ee-c32e-455a-b57f-72e011a09358}} .
In particular, if {{formula:70d096cd-b9b4-497c-9d4b-55dbffda59e8}} , it is more likely to observe small sample eigenvalues,
thus making {{formula:03e6ce8c-23c5-4895-8225-d8d371037d52}} numerically unstable.
| i | 18256a9605a98091312cfd53ae7fe831 |
Inspired from Bell inequalities of two parties {{cite:bc7ce2e56510ed0c8c61f8cbefe8f5e7088f2c8d}}, the multilocality of correlations of a network follows from the GLCM {{cite:f92161ba09996e59d17310b541059463add7ad1b}}, {{cite:1b094d2577a656ab5acd02ecc4be3dec64822377}}, {{cite:a1b46687adba78c86a872af338fce897e6de1637}}, {{cite:c746274268c4784aa7bccd87307b5a77692ba69c}}. Formally, all systems measured in the experiment are considered to be in the hidden states of {{formula:24b302cb-630f-41ab-b3c1-d021db205a71}} , where {{formula:8b9fc8e0-3ac3-43b4-b444-ff4812274454}} are arbitrary and could exist prior to the measurement choices, and {{formula:b502662d-ccb6-495b-89a0-89dbbd5b8558}} is the number of hidden states. The dichotomic output {{formula:b27d089c-ca32-47ae-99f7-e6d7f61c543b}} of any particular system can arbitrarily depend on hidden states {{formula:5004895f-294d-4db7-9874-0542e38b2ad6}} and the type of measurement but not on the measurements performed on systems (here, one bit {{formula:442bce43-7fca-473d-bcfb-e4a6bc18f167}} denotes the type of measurement). Thus, the GLCM suggests a joint conditional probability distribution of the measurement outcomes as
{{formula:48855f69-c8d3-4f32-8802-b2e52caa9478}}
| r | 5bf1ec4c40836c7504c10f52e6d80419 |
Remark To show that {{formula:7334d7c9-d4a2-4a86-a4ca-1e9ebc3323ce}} is proper, we shall use Theorem REF . Hence, it is enough to assume that {{formula:ccf82937-7b84-4bb7-bd2c-af4ee1719dfe}} is a pole in the sense that there are not conjugate points along any geodesic emanating from {{formula:a7faa5eb-dd75-4019-b9f1-76b4c6c3826e}} , (see {{cite:e4359b2334d8503589379ea79d867de8ddf4e5c5}} and {{cite:b5b0c8fedfed4e0a1151d1e2a071f7f41d4385b6}}). Therefore our statement about the properness of the immersion includes ambient manifolds {{formula:58c32d35-502d-4fd6-b3a5-96304b9e05a2}} that admit non-negative sectional curvatures, unlike the ambient manifold in Theorem 1.2 in {{cite:082adbb1aa3a9bc7dd80eb52632b5de3075370d2}}. On the other hand, to prove the finiteness of the topology of {{formula:e9e56427-fe8d-4952-87d0-ea416bb73a85}} we need to assume that the ambient manifold {{formula:354f5a52-9901-4636-95be-e5edb81d0a20}} posses a pole as it is defined in {{cite:9767467e6fd3c0ba96d5f32048ed253cb52def57}}, namely, a point {{formula:1b395221-8004-420a-a41a-3f92d9a9e34b}} where {{formula:5f5945fe-610e-4ed9-92fc-11da06b235ad}} is a {{formula:763100bd-0de9-4588-a627-7280de24f25f}} diffeomorphism. However, although our ambient manifold must be diffeomorphic to {{formula:f121a021-3bc9-4d40-aee9-ccf274fce466}} in this case, (as in Theorem 1.2 in {{cite:082adbb1aa3a9bc7dd80eb52632b5de3075370d2}}, where the ambient space must be a Cartan-Hadamard manifold), also admits non-negative sectional curvatures.
| r | 186d5a27926313b626ef8e7e7a460c2b |
Given one observation of such {{formula:739ca279-c6e8-4a3e-9b12-90aaf4f8f3b3}} , our goal is to develop a simple and scalable algorithm that outputs the true partition, i.e., {{formula:33f4960f-180a-4b4a-90a1-970561484eae}} for some {{formula:8885d994-0634-4cb4-8686-b7e45b1d9602}} , with high probability. Since exact recovery requires the node degree to be at least logarithmic (see, e.g., {{cite:aea9b943fd819ce2fd19ba5d66d214020367acf8}}), we focus on the logarithmic sparsity regime of the symmetric SBM in this work, i.e.,
{{formula:d6c0ddaf-930d-447e-be33-6727f493417c}}
| r | 47f2695fcbc8dd26695ce045cf4febd6 |
We have studied here the mass spectra and decay properties of the {{formula:b6583d06-0bdd-4f20-b6b2-994ee61ba58c}} meson in the framework of relativistic independent quark model. Our computed {{formula:5208e41b-41d6-4ca8-a2f8-537f8f4c5c91}} meson spectral states are in good agreement with the reported PDG values of known states.The predicted masses of S-wave {{formula:49a7dc63-ce64-4013-826f-50248572dfc4}} meson state {{formula:d5def2f9-825b-4de9-afe8-57068e674bee}} (2605.86 MeV) and {{formula:925352d3-78f3-4931-8052-b69ae7e94d29}} (2521.72 MeV) are in very good agreement with the respective experimental results of {{formula:a758d06a-c668-47b5-bd00-5c66b5620383}} MeV {{cite:52377c436ef549deb27bb1b2a8d5271015cf8f34}} and {{formula:21e5a260-6695-4554-906a-e8eea4f2dc12}} MeV {{cite:52377c436ef549deb27bb1b2a8d5271015cf8f34}} by BABAR Collaboration. The expected results of other S-wave excited states of {{formula:186226e5-b90d-411d-890c-f54e7026c900}} meson are also in good agreement with other reported values {{cite:2def3464f3a1d7b7e43d2c747cf3acb9741528e0}}, {{cite:6a5cf8137907fc845b0559214ddc8e4a49c21fc3}}, {{cite:0a721d54823f2b353ee5e0eae8f0a142e68b9d85}}, {{cite:4ca61d8f42a76f8e86cec7784742fc1e837a7fcd}}. The predicted P-wave {{formula:b921b172-9ea2-49f6-a74a-6fc520e02ae0}} meson states, {{formula:52d6345f-66fe-4443-8061-78c7b44a6181}} (2468.22 MeV), {{formula:62bf5ba2-c470-45a0-a443-db63e509cd75}} (2404.94 MeV), {{formula:d294b2f2-542d-4bbc-8323-868feae2b203}} (2315.24 MeV) and {{formula:cd5a2d61-33d7-405f-9870-bb6e64eee086}} (2367.94 MeV) are in good agreement with experimental {{cite:0146d4298bbae2cdfb20d5c19fc70fa15cd74c76}} results of {{formula:8969cf39-c2cd-47e4-8d93-f88a973c746c}} MeV, {{formula:daae2200-ff1c-4597-8207-51ff2ec47b08}} MeV, {{formula:acf553af-acd6-47f7-8dd7-dca3ad33a4c0}} MeV and {{formula:ef4c3a4f-7b14-4fb1-81ec-1508ab83bfc5}} MeV respectively. The {{formula:c4e8c78e-dec5-4b55-864a-6e7baa1253c5}} (2760.15) are very close with the experimental result of {{formula:01796945-3d5e-46a2-88e3-f14dc8f1c239}} MeV {{cite:52377c436ef549deb27bb1b2a8d5271015cf8f34}} and we predict its {{formula:f2efa80c-ae8f-4109-8d75-fcd0ef91d01f}} value to be {{formula:24295380-8eec-475a-962e-dd2ab8e78fdc}} . We have also compared Lattice QCD and QCD sum rule results with our predicted results where the numerical values in Table REF for Lattice QCD results are extracted from the energy level diagram available in {{cite:eef855bc694270c6c719fca161ec6b09088f1a3e}}. With reference to the available experimental masses of {{formula:b7e04a6c-dc1a-4221-988a-fec1b07be720}} mesonic states, we observe that the LQCD predictions {{cite:eef855bc694270c6c719fca161ec6b09088f1a3e}} are off by standard deviation of {{formula:123af5e6-6163-4cc4-9758-3c8b907a15d7}} 58.52 and that by QCD sum rule {{cite:52e246133ac4ce465d1f50243572a9aa80318e1b}}, {{cite:6702736658ab82a9a1d431af81e469f7aabea5c5}} predictions are off by {{formula:638a02ba-bb8a-4306-a47d-8ce05962f922}} 59.22, while predicted calculations show the standard deviation of {{formula:54e9a989-6cd7-405a-be08-805f87f889fc}} 21.88.
| r | bb9419b9a3ca7e50028da5da31d75c72 |
OJ 287 is considered as one of the best candidates for harboring a binary black hole (BBH) system in its center. The updated BBH central engine description for OJ 287 allows us to track the changes in the orientation of the accretion disk and the primary BH spin {{cite:7d07916c13fc21528069bcf3178888825ee3dde4}}. However, additional assumptions are needed to explain its decades long radio jet observations. Interestingly, the assumption that the jet is launched perpendicular to the innermost disk axis leads to BBH orbital time scale observational implications {{cite:03b0b5d4b7ad50bf34f366cbcc744a29d590d68b}}, {{cite:f62e0a909a685856e783a3bcaeeb2ef09f71565c}}. Further, the time evolution of the innermost disk axis shows up as a bending of the radio jet. This is because the changes at the launch angle are propagated to the more distant parts of the jet with a time delay. The rather small variations of the launch angle are further amplified by projection effects since the viewing angle of the jet is small. This is in agreement with our observations that confirm a progressive jet bending with increasing angular resolution up to the smallest spatial scales probed by RadioAstron.
| d | 83be8c8008ed3686ffe808488f8163ac |
We start by reviewing the basic tools of the ambient space, developed thoroughly for EAdS{{formula:e426d5b7-04e9-4743-bc58-894862e57541}} symmetric traceless tensors by {{cite:99e18a93fb041c302955b3cfab71db2bd3524488}}, adapted to dS{{formula:af980fe0-4f53-4127-b58a-bd81f4683d30}} . Our aim is to set the logic and notation, and give a survey of the existing literature. The discussion of local coordinates and the scalar propagator is a classic topic, see for example {{cite:6403f5e345b51e49cfd950699701ce5853f6ddff}}, {{cite:0aca8589ad65d7d824bb36733eecd08243622acb}}. The analytic structure of the Wightman function and its link to the in-in formalism is explained in its full generality and details in {{cite:3b0c23124f7eb1a7b294b7edb7b7fde952f14a0b}}.
| m | 27da77da4516f9b87550cf5e1caa9950 |
We address this issue using Point-NeRF, a novel point-based radiance field representation that uses 3D neural points to model a continuous volumetric radiance field. Unlike NeRF that purely depends on per-scene fitting, Point-NeRF can be effectively initialized via the inference of a deep neural network, pre-trained across scenes, leading to efficient radiance field reconstruction. Moreover, Point-NeRF avoids ray sampling in the empty scene space by leveraging classical point clouds that approximate the actual scene geometry. This advantage of Point-NeRF leads to more efficient reconstruction and more accurate rendering than other neural radiance field models {{cite:614fffb45deb0a9be96004bc2d81861a5da34edd}}, {{cite:86868e35fe5df979031f1af8e48503fb7158e5ae}}, {{cite:f2029baf97dbf847d282c450b8e0c9e98fd3afb1}}, {{cite:f3d5a5810515c1bfb88b39138a50a4df2891f4fc}}.
| i | 3fc2b52d617a7d38c024df6199fc8532 |
Our choice of circuit width and design was motivated by Ref. {{cite:a94fb71c8ac3141610fde68cfd19962cb9184631}}: it was shown that to fit a polynomial of degree {{formula:16ff915f-69ad-4a66-ad03-465b78485a28}} , then a circuit must have at least {{formula:41e65d22-7bac-470c-a7d5-c1ac74ae9954}} qubits. We initially assumed that a 3-qubit circuit has sufficient capacity to fit our target function {{formula:81b60772-e3f0-4909-80dd-cf93fbad453a}} . On the other hand, Fig. REF shows that circuits with low test set MSE have difficulty fitting values close to {{formula:ae97d053-d99c-4c20-8147-ca71d86125bb}} or {{formula:64bd7d94-e361-4eae-a132-2ce5d3b2c3eb}} . A larger number of qubits (6) with deeper circuits ({{formula:beead4ec-dd30-4a7e-ac5e-40ea01a24359}} ) were shown to fit parabola {{formula:c24f02a4-59f1-465b-94d5-11d4b5583803}} when the data was constrained to the range {{formula:032a54f6-377d-4586-9b8b-e0ca6a557b68}} {{cite:a94fb71c8ac3141610fde68cfd19962cb9184631}}. Our results highlight the need for measures quantifying the expressibility and trainability of PQCs for different learning tasks. These concepts have been studied with respect to general circuit expressibility {{cite:7cb48ea7478a025d7c626fd77654ddf40f3b213f}}, quantum classifiers {{cite:6bf139bae1e4678ec2670772f0ffda260a8f88e8}}, and general variational quantum algorithms {{cite:e7a2adaba27cacddc62434f552ac1da420241cc0}}.
| d | cc70b73798030d35242a7b95c63fab19 |
Here we additionally report the quantitative comparisons to the concurrent works NeuLF {{cite:7cad239379557f51dd64c36566f953f992b824a2}} and Ray Space Embedding Networks (RSEN) {{cite:b9234447fcced468ba471a383babe6dec0f67cb0}}. Since the codes of these works have not been made public yet, we compare the quantitative results with those reported in their papers. Table REF provides the comparisons to NeuLF on the Shiny Object dataset {{cite:ccc0c34a06efdb9d3910e7fe1e48ac90c17674a0}}, where our method performs better rendering quality than NeuLF. The results of NeuLF on the LLFF dataset are not reported in their paper.
| r | ed54e89e535f446d8b2bc810fa346ad2 |
As a side point, we note that this method of carrying out arbitrary transformations of variables is standard in statistical physics {{cite:a556645ed6257932b7b5ae090ef553e72dcda5f2}}, but is apparently less common in statistics {{cite:ee9154110cf3e0c2299c6b3871263103fab7733b}}, {{cite:a2ac18ec05c3a40c47a5ab5104e8f882505695d5}}.
| m | e82109e703ea7285b6ca4704f52a72ea |
Wide field imaging has been a main tool for the survey of intracluster GCs in the nearby galaxy clusters {{cite:be818c9d80a43575ff993bbb68525adb6a3fe6ab}}, {{cite:73fcf8f65d9d7fe7e6dd1cb3bf8a774a6134b528}}, {{cite:bf5a847ece881783718d8f1a5553eb401bdcc47d}}, {{cite:ee52eb24c386c63c92c7c05b163bc1eddebc4e41}}, {{cite:6dae82ce8777b200bd990a2be72dddd203d2c935}}, {{cite:4df06667d4101a75501ccbf584fcfea6e8907a25}}, {{cite:a69e6c4c71fd44f22f04f953ffc69f83215a46bd}}, {{cite:2238071875a1e2c7572ada4009f0067688dc0147}}.
High resolution power of the HST enables us to probe the intracluster GCs in the more distant galaxy clusters, and study the relation between the intracluster GC and dark matter distributions {{cite:883e5198a3e6cc2ebef957381c6913327442d036}}, {{cite:df6f99d86845b8c193b21da84f64535dc8897ca2}}, {{cite:0f28ca0e96ac8dfd4fcef3e568983d73b91a5912}}.
| i | 8f693e5f3ed2456670f08b104bc7b286 |
Sentence-level Argument Role Mining. Our data's argument role annotation is at the sentence level, thus, we perform sentence-level classification experiments using the same data splits employed in summarization to detect argument roles.See Appendix for argument mining training details.
We experiment with several contextualized embedding-based techniques, namely BERT {{cite:5e4921b794282bf87ee8c0f0a828a1559c792c0b}}, RoBERTa {{cite:26563b9e70e02eed24dad91b71c46bf5d7c4a2e1}}, and legalBERT {{cite:d16f978f0a4b742fdf7e2618d7ddb41d1ac815ed}}. We employ these models to predict sentences' argument roles (issues, reasons, conclusions, or non-argumentative). Figure REF shows that legalBERT achieved the best classification results. We achieved a macro average F1 of {{formula:8fb5b8fb-466b-4fb6-bf1e-998db410a6e8}} in argument role classification and {{formula:54da2a71-6229-482c-ae74-8ac28554a055}} in binary classification using legalBERT. Thus, we rely on its predictions to integrate argument roles into summarization below.
| m | 70a0878f8e782b0a057907f53d6d5408 |
The prediction of flows through unsaturated porous media is of great importance in a wide variety of applications in science and engineering. There are plenty of relevant applications in many fields such as hydrology, agriculture, water resources
management, risk assessment of environmental contaminants, enhanced oil recovery, etc. The Richards equation {{cite:da3410c95d54dcd2a912d894e26ec565802dbbda}} describes fluid flows in unsaturated porous medium due to the actions of gravity and capillarity. This equation was derived based on a multiphase flow extension of Darcy’s law and the principle of mass conservation {{cite:07831c2804ee6dbb3d1caed3d3d3a15f62353d16}}, {{cite:da3410c95d54dcd2a912d894e26ec565802dbbda}}. Some analytical solutions of the Richards equation are available, but they are limited to very simple geometries and specific initial and boundary conditions. Most practical situations require the use of numerical techniques to produce approximate solutions. The design and analysis of appropriate numerical schemes for the Richards equation is a very challenging task due to the highly nonlinear nature of the equation. Despite intensive research, there is still a strong need for more robust numerical schemes for computing multiphase flows in unsaturated porous media.
| i | 411a6e1a87296942ae8770991474aeab |
Our goal here is to illuminate that evolution by considering a closely related yet simpler
physical setting, where a spherical particle levitates above a liquid bath owing to the Leidenfrost or inverse-Leidenfrost effect. To that end, we shall employ scaling arguments and the method of matched asymptotic expansions {{cite:508859be42e409742f9362106495646bc5474cf2}} to describe the double limit where evaporation is weak, in a manner to be defined, and the levitated sphere is small compared to the capillary length of the liquid bath; we will also assume that the densities of the bath and sphere are comparable, though not necessarily the same. Crucial to the formation of capillary waves in the vapour film, the small-sphere restriction does not imply that gravity effects are negligible there, unlike in the small-drop limit of the classical Leidenfrost scenario involving a flat substrate {{cite:03ccfe2cfc84b25c7194d907031b2fbb22382962}}.
| i | b0a15a5d065d8ff4db335e79f85606d5 |
where {{formula:65abd0a6-0c91-4988-ac08-0770e855a485}} is the excess flux in a specific wavelength interval {{formula:9d920743-8d23-4034-a75e-0620c2df2bd1}} (in {{cite:8f154969db94e5f0125cd095c9d188eee91e22b2}} intervals of {{formula:48e0a68b-f71d-413a-b7b6-023c103ce7c4}} Å are used) and {{formula:58bfefee-c995-4c7f-a5d3-6d72e4d76f10}} is the normalisation at this interval.
| m | 3b4dd51a477161d42541f5cce5aa71db |
Table REF shows the confusion matrix for the visual Turing test. Differently from our previous work on GAN-based {{formula:a63c426c-7cd7-467f-8102-f01300157a3a}} /{{formula:7e848d1c-c31f-4fa5-955e-7fd1c25151ec}} MR image generation, the expert physician easily recognizes {{formula:bb72d0d9-5207-4c9c-9658-7d35111dc374}} synthetic images {{cite:eae1ee3305275cb218df286ae8326cb58a99737d}}, while tending also to classify real images as synthetic; this can be attributed to high resolution associated with more difficult training and detailed appearance, making artifacts stand out, which is coherent to the ResNet-50's low tumor detection accuracy with only the PGGAN-based DA. Generally, the physician's tumor/non-tumor classification accuracy is high and the synthetic images successfully capture tumor/non-tumor features. However, unlike non-tumor images, the expert recognizes a considerable number of tumor images as non-tumor, especially on the synthetic images; this results from the remaining real images' ambiguous annotation, which is amplified in the synthetic images trained on them.
{{figure:a0befb56-b80d-47ee-b7a2-6e67a213820e}} | r | 75e0d9701a884b773eed3bfdcb20337d |
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