text stringlengths 54 548k | label stringclasses 4 values | id_ stringlengths 32 32 |
|---|---|---|
where we used Leibniz's formula twice as in (REF ) in the second but last line. Again Faà di Bruno's formula, cf. (REF ), is applied in order to estimate the derivatives in (REF ). We use a special case of the multiplier result from {{cite:b8b272d24ded7c4a18c4383c9ac877e65038026e}},
which tells us that for {{formula:e2da5892-852b-4d88-af4b-92c6bb70798e}} (by our assumptions {{formula:9274e674-cc47-4076-a472-d160e68afca9}} ) we have
{{formula:027c9ded-c52f-437c-8bec-42827e33abe6}}
| r | 09ab302664a47181866cb87edc4d6188 |
There are several state-of-the-art methods similar to the one proposed in this article as they use multiple attributes describing the aesthetic or artistic aspect of a photo for aesthetic evaluation {{cite:efceff584ef7ad5f85a95bf3b95d62f7d445364e}}, {{cite:60311c86d504423871da0024cdab611d90bd8b97}}, {{cite:a73b1e01b822dee88aaebaf46bf754ca9dc6776b}}, {{cite:21b7679cc03b0f74f7d7775b87dc60451428ed2b}}, {{cite:5456e6d879fc425bb034129250a11122a1e3af28}}.
| m | 2751637f99866254aebe11919e912651 |
so that Equation REF holds under appropriate assumptions on the set {{formula:2e092fa0-6582-48cf-a130-530269fd4272}} . Equation REF is analogous to the propensity score from the causal inference literature ({{cite:1e27d6d9cfbc9a533ce6d85eae3a9851f6c1ba50}}) and Equation REF would correspond to the weights for the average treatment effect on the control (ATC) ({{cite:7471f220e464531eead14d3668ef6c03078920fc}}). Thus, we refer to the estimated weights {{formula:8c61c9c0-5ae2-478c-9c18-b8e13d394b8c}} as propensity for self-selection into the convenience sample weights, or propensity weights for short.
| m | f321aaee0a1241592f76df7b5268b679 |
On the other hand, to fully explore the ultimate capacity of MIMO systems, the concept of continuous-aperture MIMO (CAP-MIMO) was proposed {{cite:5f7cef794db898aba05ee35d3a743398571884be}}, which is also called holographic MIMO {{cite:fc35677172b11fc60f84c264f9e830df2641288a}}, {{cite:452a682a3eded4f2da2bd996066ed1254f6eab4f}}, large intelligent surface {{cite:7eaff80c4137d4d8150867c5213e4c9fdf354f79}}, or holographic surface {{cite:84b52b8ae2645648262d753c4b191a2145a55a69}}, {{cite:da788af53a650f2d49f1cf7cb49a1f8b76719295}}. By deploying a large number of sub-wavelength radiation elements in a compact surface, CAP-MIMO can be regarded as a quasi-continuous electromagnetic surface with controllable current density. Although the recent
advance in highly-flexible programmable meta-materials has made the CAP-MIMO more practically implementable, some critical challenges still remain unsolved {{cite:fc35677172b11fc60f84c264f9e830df2641288a}}, such as the efficient estimation of ultra high-dimensional CAP-MIMO channels and the efficient current density optimization subject to a large number of practical constraints.
| i | 2c789326bdb77d0611bf94e511742bba |
For the di-{{formula:483bd815-59ac-409b-a771-0774a5af93c6}} spectrum in its production at LHC, the experimental information is still lacking. We thus adopt the similar production relation as in the di-{{formula:b6f5b9ea-e85d-4e1b-80cb-54c12300aee5}} production {{cite:d2e3c7bc43c3ea744cdad6e258af77f89c9c3489}} to investigate the di-{{formula:23719464-1272-4671-9969-48fbe63920ac}} spectrum. Considering that {{formula:12fc9761-11df-4f14-8652-915054e8bc12}} and {{formula:71785a3e-6c6c-4628-9c66-a3e8e2b86132}} both are the ground state charmonia, and both {{formula:32288ec6-4b54-4eff-a5a3-3db886697d1f}} and {{formula:7eb23b5a-4540-4d4d-a681-e720e26a10e9}} are well defined first radial excitation states, we fix the relative production rate but leave the phase angle as a free parameter in the numerical calculations, i.e. {{formula:369f4aad-3492-4938-8b17-84e76b87cb82}} . The dependence of the coupled-channel spectrum on the phase angle {{formula:84f2ed4b-2f46-4fc3-90f8-e2748bb439a5}} will be illustrated by the line shapes at different values for {{formula:b5c2d2fa-7610-49ab-9732-c1fbf55104c7}} .
| r | fa68e79347d88091e95088b3c0d953a2 |
However, several approaches have been proposed to investigate how to improve the accuracy of approximation. For example, in
the framework of numerical integration, we refer the reader to {{cite:15c80d3beb24ea93e4265fe2d6dfd330723d6872}}, {{cite:11e705648f4ed93ccf3eda71a1dd8c45e1187c83}} or {{cite:e6162e1a97919c6de67d6605704b996db9021611}}, and references therein. From another point of view, due to the lack of information, heuristic methods were considered, basically based on a probabilistic approach, see for instance {{cite:4e109b5ae8f5222cbf1c54c12cf1420fe8e1c13e}}, {{cite:8c5cabd00ded066725c295a2236b80eed0f8d67d}}, {{cite:92fd3d4bca963fe86bc05c9712be4df539cde15c}} or {{cite:49a5890178c00e90ae673d5966f03084b5862385}} and {{cite:d39ce0c0c72496c75edf9ea29723074cb1f8d723}}. This allows to compare different numerical methods, and more precisely finite element, for a given fixed mesh size, see {{cite:503e846edd41581857450367fb39bcb1ff53cd7b}}-{{cite:dda8e43fd586a3ccb9b7715b0fada969fe4cfbe0}}.
| i | 7bd0983dcb7e105f19446ae411692ff0 |
According to {{cite:a735dcaabc7f3c545d3ee3f7512728436e8f3681}} (see also {{cite:9cad80144b7973c1132d9ba3d7648e3d303983ad}}), if {{formula:b83fe47f-5ef5-4385-8b56-2c537b50df7b}} is an {{formula:2df7c822-3e2a-40dd-97d9-f302b22d1793}} -valued Calderón–Zygmund operator on {{formula:b6f55547-6e33-42eb-8f9a-7dd9aafe06e5}} , {{formula:c303cb17-e5c7-4c83-a451-80b32cba8671}} can be extended, for every {{formula:7e9f6ecb-18a8-46cb-b77f-fff847fbb3f8}} , from {{formula:fa19509c-82b0-4576-a55c-5ed23ad9682d}} to {{formula:0d76fc9b-ed61-44ce-a93c-fd46e656bf0a}} as a bounded operator from {{formula:d4927035-c11c-464b-8fe7-be867eb5ef1c}} into {{formula:5f56cf61-c7c7-4be3-a2ff-cba7bcf12f38}} when {{formula:c953c178-b459-407e-b5a1-1f1f003a8e99}} , and from {{formula:db94f257-e1cf-4f1c-81b5-5c2e335212b4}} into {{formula:3934e325-2ccf-456e-9826-3a33ce7f27cb}} when {{formula:98d93f8a-64e9-4bbb-b527-64c94116062d}} .
| m | a0e408177518e7d57058d11c8f535a8c |
Meta-learning. Meta-learning is also known as learning-to-learn, which trains a meta-model over a wide variety of learning tasks {{cite:aa7dfa2d5fe4f38c9e6ef1598fd1a819f238d484}}, {{cite:d5c44e6c54d6ffd892cd79d879941ab43a97f097}}, {{cite:a5495ef890a55fb2a48c270e375d6de30e436627}}, {{cite:5c0d7c6238cbc8e3916c7805fd9404fd4d759a03}}. In meta-learning, we often assume that data share the same high-level features, which ensures that meta-learning can be theoretically addressed {{cite:99cb5d0852f0b2027611c91e551840e56e9ec167}}. According to {{cite:6493312e005a206fa8fa3625737d1060d91006de}}, there are three common approaches to meta-learning: optimization-based {{cite:ca63cc7edc70333cd4f16f4207bf95486a3f44f9}}, model-based {{cite:fa85c66200e5360a90a20a865021b9d45ae07688}}, and metric-based {{cite:d1075eaa06011fd8f67d40c1ce9e8fbf9a6833c9}}.
Review of meta learning methods is in Appendix .
| d | 1283039c19c517b853031d5a4df30365 |
Next, motivated by island models, we considered two regular graphs with the same total number of edges
and we showed that the corresponding fixation probabilities are asymptotically different.
In particular, as the population size {{formula:461710e4-cbf1-4a2e-aad0-5be426a0adaf}} increases, the fixation probabilities decay at different rates.
Thus, in the asymptotic sense, the Isothermal Theorem of {{cite:872018d62f635f149088029777d5023c779c2621}} is strongly violated.
| d | 64758466e0a312783971fa68bbf0a98c |
In this experiment, we study the performance of spectral clustering for image segmentation when using different numbers of eigenvectors.
We consider the Berkeley Segmentation Data Set (BSDS) {{cite:d4914d28c215b2d2d6c4bc729f885ed48188fce0}}, which consists of 500 images along with their ground-truth segmentations.
For each image, we construct a similarity graph on the pixels and take {{formula:d21d41d8-5319-49df-80cb-b0dfc15f2dd9}} to be the number of clusters in the ground-truth segmentationThe BSDS dataset provides several human-generated ground truth segmentations for each image.
Since there are different numbers of ground truth clusterings associated with each image, in our experiments we take the target number of clusters for a given image to be the
one closest to the median..
Given a particular image in the dataset, we first downsample the image to have at most {{formula:fe8fa56f-0256-4b5b-bf06-611fb13bdfe8}} pixels.
Then, we represent each pixel by the point {{formula:9201b1f8-98e0-4331-98c4-56fbcad8b3a0}} where {{formula:7cdfa516-ad0c-4ec9-93c2-00264739a881}} are the RGB values of the pixel and {{formula:10b57dc2-535b-4c22-ac63-760ee902aa4a}} and {{formula:14a6a7d4-1655-4fa3-820b-af245a0bf580}} are the coordinates of the pixel in the downsampled image.
We construct the similarity graph by taking each pixel to be a vertex in the graph, and for every pair of pixels {{formula:ec1d4a5e-8abf-4418-8952-0dd13fafc799}} , we add an edge with weight {{formula:ab4f77a6-c58e-4dae-8639-5ef9b5dfd773}} where {{formula:397ffda4-5fc5-4d5d-af15-c1df07a98093}} .
Then we apply spectral clustering, varying the number of eigenvectors used.
We evaluate each segmentation produced with spectral clustering using the Rand Index {{cite:a161189102a8c19288c7a9a1b42fa88770888828}} as implemented in the benchmarking code provided along with the BSDS dataset.
For each image, this computes the average Rand Index across all of the provided ground-truth segmentations for the image.
Figure REF shows two images from the dataset along with the segmentations produced with spectral clustering, and
Appendix includes some additional examples.
These examples illustrate that
spectral clustering with fewer eigenvectors performs better.
| r | 74483e38054d4b98c18d4de80ac0c8db |
The purpose of this work is to present some progress on these two questions for various choices of potentials in (REF ). In the first part of this work, we focus on potentials with a finite number of terms, and for various examples we find a closed expression for the planar free energy {{formula:84a99e0c-9d00-4b17-8618-0dfdd9fc5727}} as an all-order perturbative series in the 't Hooft coupling. In some of these cases, we either determine or bound the radius of convergence of the resulting series. In the second part of the paper we consider very particular instances of potentials with infinitely many single- and double-trace terms. The potentials we consider are relevant for the study of four dimensional {{formula:8c681b99-eb6b-434c-b9e8-55be44dc8c2f}} SYM theories via supersymmetric localization {{cite:abc0ecf7186ff65554de96a38ac5234d60847b95}}. In particular, we provide the first examples of evaluation of {{formula:38bbbc1a-2169-4f2a-b15f-9a98a1edae98}} for four dimensional non-conformal {{formula:80bf7167-a1d5-4343-a3ca-19e91a8c12f1}} SYM theories. In this second part we don't present any new result on the convergence of the resulting series; nevertheless, we suggest that the results in the first part of the paper may play an important role in deriving analytically the radius of convergence found numerically in the literature.
| i | a9363054cf19d33a2effc6f73010502c |
Patch attacks against road signs are less effective than previously believed. From tab:synvsrealavgmain, a 10”{{formula:cc675e3b-1dd7-411c-9121-3cdf3d7c2258}} 10” adversarial patch only succeeds for about 18% and 25% of the signs (between the two models) on our realistic benchmark.
Even though we use undefended models that were trained with standard procedure and without any special data augmentation that would enhance the robustness, attacks are not very effective.
For comparison, a universal adversarial perturbation under imperceptible {{formula:f31765b3-74e2-4249-9951-372de0f96efc}} and {{formula:010ad2bd-e44e-456b-b59e-80d47a8a6901}} norms achieves above 80% success rate {{cite:cba30fa7bb8524119c4ca6ad246125342e3363d6}}.
| r | 7180eef9c6a243688bdd99500776b6a0 |
Let {{formula:af4b568a-a87b-407e-9e94-9267f4d8333c}} be a maximal normal subgroup of {{formula:52cfb29b-9a34-4f77-a932-745e6f29dbab}} . Since {{formula:984f43ce-9cf1-49ab-ab6c-4e3dea50a638}} , we see that {{formula:36da9918-68a7-484d-a9bc-4e94b1a12872}} is perfect by our earlier remark. Then {{formula:aef5f7dc-fd59-40c5-a9c6-8df93c5687b9}} is a non-abelian simple group. Since {{formula:6fa118a9-e4c1-400f-8967-03a1dd03d1e1}} , we see that {{formula:6f303c87-affa-4538-a17e-18be86855440}} is either {{formula:5e575c9f-c9b5-4e32-8fc8-eb780e731e6f}} , or 12 by Lemma REF and we analyze the cases separately.
(a)
{{formula:2661b21a-8985-4806-8100-c45a004d1aaa}} . By Lemma REF , {{formula:379e2dda-ede5-4749-aaa2-39c07ec886c0}} where {{formula:b2bd13fd-86d1-4715-be5c-52aee0d2bff0}} Then {{formula:0b7b28bb-a984-4a65-851d-5c5438f7d020}} .
Since {{formula:8f9a4af2-ca0c-47d9-9c5e-87f38dbe4cba}} is the only nontrivial odd codegree in {{formula:d0fb2dc3-5212-48d7-a1a5-554a0ca79231}} , we have that {{formula:56fb7191-8519-45fe-ae88-2be0fab2e26f}} .This is a contradiction since {{formula:36936aa0-e330-4dd5-8649-333f23a2976a}} .
(b)
{{formula:1d235213-4041-4a4e-a7f2-96248298a818}} . By Lemma REF , {{formula:adfaedb4-2d0d-4e31-b6dc-41eb987e8a43}} where {{formula:6245051b-0dd4-4a56-b783-caab2f7fea75}} is an odd prime power and {{formula:017d97b7-8a6c-459c-9c76-4a9c18faa513}} where {{formula:0b47d17f-f58a-4c5c-94b2-be9d039466db}} .
Then {{formula:62ad834f-43db-4d7d-b315-e414171ecf05}} , {{formula:5d6f8abe-9689-4a49-9894-9a98eb624291}} , a contradiction for we can't find a codegree which is the half of another codegree in {{formula:abe0ff35-8043-41d1-af6b-be79e7446db9}} .
(c)
{{formula:d17936b8-677b-4c9a-9724-55b9557fa650}} . By observation, {{formula:6d62e809-6702-4b4c-bd56-13388b0301dd}} is the only nontrivial odd codegree in {{formula:5f2dbc72-ea5d-4bc8-86f9-70cca436892c}} . However {{formula:077eb47e-f190-4031-89b4-55b6b5c739a2}} , so {{formula:6ca98191-31da-4d66-8442-c23909a3d05b}} .
Suppose {{formula:c8a62e4d-bf8a-4faf-8d9b-eee242ac78d3}} . {{formula:7c4aed09-f1f3-4395-8c87-27c1aac98f02}} since these are the only nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki groups, it is a contradiction.
(d)
{{formula:3c7015ed-7f92-41fb-a8ea-754b481b090d}} . We note that the nontrivial odd codegree in codegree sets of {{formula:a1c3211f-4e43-4fd8-b8f2-b7e4212c0636}} , {{formula:b20fb5d9-33a9-4ac9-a204-c80a64e63028}} , {{formula:787b592c-a08d-47b0-a219-3f1732f7bbf4}} and {{formula:fe45c6e0-7e6c-40a7-a2f4-2192a8ce78fc}} can't be {{formula:20dbe197-c0df-442b-af7a-c5d8a20df3c1}} .
(e)
{{formula:a70de8d5-448a-4e9e-8502-2d394427d468}} .
If {{formula:c5868893-ca8f-41b1-a8b0-3ef6f0600623}} with {{formula:39db0b64-112c-43bd-ab20-d1ad7af80627}} . Then if {{formula:fa57df0b-eb57-48f2-8b92-ac38ec6293d0}} is odd, we have {{formula:ba1e5084-e755-4259-9b83-015ed9fd02db}} . Since no cube of a prime divides {{formula:d23d809a-9f7d-43be-ac1d-6f55796577d3}} , we have a contradiction. If {{formula:90ff9b3e-1100-47c8-8664-3c824ce5d75e}} is even, we have {{formula:867f632f-c968-4589-aed0-3f8fc9ae3a91}} . Then this does not yield an integer root for {{formula:cf6f64ec-612c-4ef7-9564-81bee1a1a099}} , a contradiction.
If {{formula:7a57e368-66ec-45f5-b957-1a92350bbe81}} with {{formula:c76da8a4-233f-4ae7-8b94-b221c22bc92d}} and if {{formula:fa0af28e-efc0-463d-9170-a752c03b8dcf}} is odd, then {{formula:a3162447-b792-41df-81d9-b8864694e103}} . We have the same contradiction with {{formula:cc254d76-1d80-4d33-849e-406a5ce33992}} . If {{formula:74ab13f1-498e-4a33-b21e-08a4a848365e}} is even, then {{formula:47dd901e-18ab-48cc-b8d2-335769656162}} . This is a contradiction since {{formula:4aab15aa-3941-4e8a-98ff-0454ac3a4609}} is an integer.
If {{formula:c50430ef-d3c9-41e1-85fd-3eaba2512f25}} , we see {{formula:2ab92b68-c379-406e-a5e7-6470d9f3e9ca}} , so {{formula:d3349293-55e6-48bd-8399-340734c3cb1b}} . It is a contradiction.
(f)
{{formula:37ec04a3-6ac6-4955-90ca-baa4a05d3ff7}} .
Suppose {{formula:f099abb5-0f58-4b29-b68f-e59a0c51b823}} with {{formula:68cfa4cd-5909-4eaf-9370-e13b323db2cb}} . If {{formula:70b44693-635f-42a3-8dd6-daa3ed0a8987}} is odd, then {{formula:e7c5caaa-9b83-4653-b140-50a3b92fbed4}} . This is a contradiction since {{formula:0e6f204a-f3e3-41d6-b37c-a099fa1d17c0}} . If {{formula:60949919-42c7-4d3c-a680-65f8a50a7478}} is even, then {{formula:193990f9-7f0f-445d-bf2d-1a9040c411ce}} . Then This also does not yield an integer root for {{formula:50e68d38-369c-49b1-8bf5-a81eaabe4ee3}} . This is a contradiction.
{{formula:fa14b7f5-e6c5-4330-9cfe-543cc2032518}} with {{formula:735a2615-75ba-46ea-ae2f-b09670482dc8}} . If {{formula:9f5335b3-249d-4bc8-89bd-fd1456c9c714}} is odd, then {{formula:8ce3a90d-9fc8-42bf-ada8-ff174561c18b}} . This does not yield an integer root for {{formula:06c7d7a3-048e-4357-a40a-784e24d7db3e}} , a contradiction. If {{formula:4a2f1df2-431a-45b7-87d0-824f30fcb3e4}} is even, then {{formula:2b04c5ab-444b-4815-bb1e-0a827a802044}} . This also does not yield an integer root for {{formula:ef561b3f-911e-407b-9e33-ec92f1659735}} , a contradiction.
(g)
{{formula:612b71a4-55de-42ca-84fb-8f51826f3c08}} . Then {{formula:83dec1ed-40ec-43f1-af42-967cc1e1d97c}} . We note that {{formula:1b44e249-24fb-459e-a035-d888a961525c}} , we have that {{formula:edd56a3a-c3a9-4156-84b9-af1c197b808a}} . It is a contradiction.
(h)
{{formula:19380510-32fa-4f71-933e-cb961b02f6a2}} . By observation, the codegree sets of {{formula:c91e80c3-dd7d-4421-904f-07a5dc2fa32f}} and {{formula:15f9f6df-c216-4a89-80dd-fe2203754e5c}} contain {{formula:0f49d704-5202-4725-97e3-b4ded453904d}} , {{formula:d858748d-5220-4d9e-8afa-267443e43362}} and {{formula:8f07fe3e-0c10-4537-9e80-1ce0fbf2c3bb}} respectively, none of which are elements of the codegree set of {{formula:bae0b87c-e0ef-404d-a882-06487cc960fa}} .
If {{formula:b223a420-03e7-4187-ad4e-307e5ed9642e}} , {{formula:c1cde9a6-089c-4aa8-a26f-f46f9b03fe38}} . Thus {{formula:8ac4c515-4769-45d6-9656-739c5c6b7f59}} could equal to either {{formula:96a1d91a-c604-44cd-8271-cd2f48eec976}} , {{formula:58e4cbdf-0554-4829-8c07-d94ef30fd18c}} , or {{formula:587d8f04-d9f8-450d-b925-c487ec4eb6f5}} . By considering the power of 3 or 5 in those numbers, one see that this cannot happen.
(i)
{{formula:b0017f98-0895-4d83-a9ee-19e972b22657}} . Then {{formula:361c9b6f-0d56-441d-ad3f-f5779a59357c}} . It is easily checked that {{formula:40aee3ab-e613-4a50-a0b7-1521c77d8df0}} . Thus {{formula:aa50d299-deee-4aed-82c4-efbb46546a8c}} .
Lemma 3.2
Let {{formula:28663035-a77a-41b2-bf1a-c0253011b434}} be a finite group with {{formula:c49cc4c9-1b37-4fa0-b5b0-7bedf32b32b0}} . If {{formula:6f21cd02-88b7-47be-97e5-39ff5ea105de}} is a maximal normal subgroup of {{formula:90df73df-9f12-467d-800a-3525ce455e8a}} , then {{formula:df493458-7f1f-42ed-8bda-9654ea08c338}} .
Let {{formula:9d2b04c2-2ecb-4561-b34b-6a75fca0c826}} be a maximal normal subgroup of {{formula:33816f58-40b4-405f-92ac-a292dc119b5d}} .
Similarly with above lemma, we have that {{formula:fe5a88d5-1d4a-47ca-a50c-e34c8caa8f89}} is a non-abelian simple group and {{formula:8477f843-59fc-444c-ac50-e320b5e759e5}} is {{formula:d2fb9f02-abdb-4220-88bd-0203dc747b5b}} , or 13. Next we analyze the cases separately.
(a)
{{formula:a1d7e28b-f1da-4bb2-af53-7ad3ca21e8b9}} . Then {{formula:a96593e7-555c-449b-9180-b2ec79f76cc5}} where {{formula:49e6a1d9-d16e-434b-9a31-516ed6df2d5c}} and {{formula:efe7ebe0-f984-40f3-b3e5-b80477297549}} .
Hence {{formula:65aabb82-dd5a-4b41-b784-4b5f93ad4209}} , {{formula:1ba55c54-14c1-46bf-a806-16cf2eb3937f}} which is a contradiction.
(b)
{{formula:ab1fc8bd-010e-443a-a84e-2951b6a1b829}} . Then {{formula:3a7e6151-9f04-4dd2-baa3-6cdead82992a}} where {{formula:7fff989c-6e69-4890-8945-8347c1aa8088}} is an odd prime power and {{formula:963e17ba-7e04-4c37-9109-4c16ca000eb4}} where {{formula:88cff059-602f-4a1c-9e3b-ec9f58032f65}} . Then {{formula:d9fe7836-43d2-4124-84b7-bd1c0fbb3c83}} , {{formula:4c5d1ba4-1a01-4e91-8ab4-7f1d4fee6daf}} , and this implies that {{formula:1dbeab62-6469-4fe3-baf0-b2b0a4e0377e}} since {{formula:f79d2bfe-20ce-464b-b5fb-47903f4c860f}} is the only element of {{formula:b1dec261-c4c1-47fd-a3e7-444a706ffe43}} that is half of another codegree. In either case, {{formula:ed25aad5-9fee-40af-940b-cf2c22146d06}} which is a contradiction.
(c)
{{formula:aee49d19-991a-4faf-9e53-130887ea93e4}} . If {{formula:f6622c56-406f-4fb5-9d01-81bbd6999f39}} , then {{formula:4215bb71-f8f0-4b04-967d-14216c0cfb5b}} since these are the only nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki group, a contradiction.
If {{formula:87092554-744b-4537-bcad-2443d806923b}} , then {{formula:f95fa192-825b-416b-a36a-5a2afd908928}} and we clearly see that {{formula:65ff5c98-6096-450f-b262-a9c3253d404c}} .
(d)
{{formula:67134c2a-72e7-4cd2-ab1c-b0d2191375cd}} . Then {{formula:b73b6b26-fb1f-41aa-a5ab-fc7baf6cf5c9}} , {{formula:ee722817-1f28-44b6-a864-667986444759}} , {{formula:396bdc9e-b8d9-44bf-8529-7eda77824ddf}} or {{formula:b200d895-ae1e-482d-8252-6d4313bf0eaf}} . We need only to observe that none of the codegree sets for this case are the same.
(e)
{{formula:39d31175-60b2-4414-ba45-a31d9e213341}} .
If {{formula:36e5aa38-2d5e-48dd-bd22-d2aca0d8c3e1}} with {{formula:5547f7ba-727b-4435-b2c9-648a8c64ad41}} . Then if {{formula:9205d913-4a24-455d-bca0-db9ef8c8e2fb}} is odd, we have {{formula:c39712a2-311c-4a3e-bf6d-67488b18927f}} . Since no cube of a prime divides {{formula:77070906-6879-4c76-abad-7f8fd3fa8fce}} , we have a contradiction. If {{formula:e6d68674-245e-4a02-852a-a9a3d53b1e2f}} is even, we have {{formula:1fd2608c-35a0-4748-9da0-fb9b39a70acc}} . Then we can get a contradiction which means {{formula:020dba0d-bbe1-4087-b779-1a575dbfd73c}} is not an integer.
If {{formula:7790b9a7-e0db-489a-8a66-0256158ca899}} with {{formula:94e10de8-a993-41e4-8c03-5dba3cfd4045}} and if {{formula:14f0b8f0-3f21-41dd-8c10-531c9a5d4ce0}} is odd, then {{formula:10476ad4-4e9c-4b5b-b5be-8d5089bb77cd}} . We have the same contradiction with {{formula:97bce1b5-4ae6-480c-ad20-cd2b5b546e84}} . If {{formula:c41c6fe7-bd35-4e60-af38-77b85762b042}} is even, then {{formula:3c9e8340-360b-418e-933e-3c305813e37b}} . This does not yield an integer root for {{formula:519d197e-72b7-415e-a5ff-38af8b4916b6}} , it is a contradiction.
If {{formula:a6a5cd02-1353-443e-a152-de7ffc730845}} we see {{formula:01c06dea-7831-4402-a16c-7b0dd2c206e7}} , so {{formula:b07e03c8-0418-4c0e-b626-d1da87d8f9ba}} . It is a contradiction.
(f)
{{formula:6c7c3a04-fed4-4958-93bf-a03f7d312b3c}} .
Suppose {{formula:c1d4b46c-1cd8-4852-aede-e681eb125c1d}} with {{formula:d1db6484-300d-4dd5-a9af-ae3b0cb83905}} . If {{formula:9f48994f-e01d-457b-bf83-f9ad4e31a34a}} is odd, then {{formula:38f6f82a-77a9-47da-b0a5-907100172a3d}} . Then {{formula:299b5351-1cd7-4df0-a1c7-c07b04bb4217}} is not an integer, it is a contradiction. If {{formula:a1d4d974-f7f1-4ce3-a75b-9b7ce889d817}} is even, then {{formula:b6eb7778-6a71-45a1-ab65-e4298fda5af3}} . This equation means {{formula:c67aa0bc-9e1d-438c-b952-53058aa219c9}} is not an integer, a contradiction.
Suppose {{formula:7f09592c-2659-49cd-a615-4582b606d157}} with {{formula:5b7f4079-af7f-4c30-8b29-863f66e8e8ac}} . If {{formula:5ac7e54f-c972-4d9b-9d46-8fa9daf2d215}} is odd, then {{formula:8963f1fc-4294-4f19-bed5-fffb25006d3c}} . We have the same contradiction for {{formula:f490d89f-c6f7-4540-a59f-3c960a549847}} is an integer. If {{formula:7270df40-63a5-489f-a2dc-1b1c5d91a96d}} is even, then {{formula:54096f4e-73f3-46ac-a843-fbfecdbe3557}} . {{formula:f53690c7-977d-449f-be32-ab4c2a73a37c}} is not an integer, it is a contradiction.
(g)
{{formula:50b74ae9-13c2-4bb8-9d00-7431beb43e7b}} . Then {{formula:279d1824-afa5-4a66-b819-66a6da42286d}} . We observe that {{formula:9d892dde-050f-4109-8dfd-af2c9cb5fd6a}} . Thus we have {{formula:82eeb12f-6f88-4bf0-b561-4ae849219a19}} .
(h)
{{formula:879f098e-ed18-47d9-8d1c-173ce1f8f06f}} . If {{formula:226c9cc0-242e-4232-a7b5-8fbe38a97253}} , {{formula:594f0104-d59c-4aee-a818-f8326e29549f}} or {{formula:937aee4a-b65a-4537-a854-262d45947718}} . By observation, the codegree sets of {{formula:89b04e28-c72e-474a-8ad5-d0792f4cfd0b}} and {{formula:c10cc462-6aad-43c6-bf40-03058bdde184}} contain {{formula:13bac8b4-c730-410b-9293-dee6a77da0de}} , {{formula:201a7423-fa09-443c-bd96-f3b2f59689dd}} and {{formula:93b5e1cf-4252-4a15-865e-44ff53f4dea7}} respectively, none of which are elements of the codegree set of {{formula:d6f6d8b3-f0a1-42c5-9071-b3656493fbd9}} . It is easily checked that {{formula:f1498961-e863-4978-b4bd-968db290b04c}} . It is a contradiction.
If {{formula:d28a2c61-981b-4a13-ab88-fee93c499ecf}} , ({{formula:f3c64bbd-f016-4a18-b7aa-c5ea22ce00df}} ). We have that {{formula:8762d87c-3740-4f94-a5cc-80c8100069ec}} equals to either {{formula:41e0ac72-4734-4568-8beb-ec1f4e3ed561}} , {{formula:a940357f-e684-420e-96d2-fa45d24491ab}} , or {{formula:f5f6b6f7-3fb6-4c38-91ee-d8c343d16f38}} . This is a contradiction since {{formula:3986d9ab-c70a-4821-aefa-2b01382f5725}} is not an integer.
(i)
{{formula:b5197198-f9d4-46f2-993b-be22de9333d4}} . Then {{formula:fc4e70cb-aaa5-4634-a26b-05231aa560f7}} . By observation, {{formula:c2f1f736-a937-4904-bfdc-2c59fc2a71e3}} . Thus {{formula:3410e3f8-f172-4f65-966f-8e1efce22e8d}} , a contradiction.
(j)
{{formula:e8e801a5-cdf7-4255-b479-7484bbfb7340}} . Then {{formula:5712f287-9c42-4a66-a2bc-edd52b9d2a9e}} or {{formula:712e27e4-7cd6-4f54-a0a2-cd3a4cee7a4a}} .
{{formula:d154df64-2ef6-4398-86ee-50f814cfe6a8}} , for {{formula:c82d5e8a-c9d5-4e46-83a5-c31684dd018c}} . We can easily checked that {{formula:91f37150-f7af-4a11-9aef-dd4cb674f8ac}} , as the are the only nontrivial codegrees in each set. If {{formula:b03087ef-a6ae-4c43-b563-033257d94294}} , then {{formula:87d31717-6a1b-40e6-8ae7-2bdcb18b5157}} , this implies {{formula:1b546e36-ad34-4f68-93ac-0baa4c83ba48}} , which is a contradiction for {{formula:69106be2-f1c9-4bc2-90c4-fc0feebc3e4a}} . If {{formula:bd055993-e2cb-49cb-ac58-fbf8854ad656}} , then {{formula:d40bb6c3-8e67-4a37-9bbd-61ebbabe8870}} , this implies {{formula:517764fb-075c-4a15-85af-620b7b3bf27f}} , which is a contradiction.
{{formula:72b8cfd2-238f-4a22-a10f-6b300c9e65c0}} . We need only observe that the smallest codegrees for this case are the same. Thus {{formula:b9cf7100-071a-4f4b-a406-14aa2bdafeca}} .
Lemma 3.3
Let {{formula:492aa6cc-242b-4e8b-a2e3-6715cdfbb890}} be a finite group with {{formula:021cc1f0-626e-440d-9222-4adf94ae7212}} , {{formula:69941338-9ccf-429b-8431-683016ce3c13}} is even. If {{formula:3f27a01d-39ff-49e4-95e1-a919ab8b1fc1}} is a maximal normal subgroup of {{formula:8b2171fa-db03-4162-a4ef-b1d40089b5ac}} , then {{formula:7115b65c-8003-4faf-b2c9-c4229b3ba3e8}} .
Let {{formula:e2ee010d-b037-4258-8032-fd9fd79b0e31}} be a maximal normal subgroup of {{formula:a04ccabd-c683-4bf0-9649-879dc3cf4a7d}} .
Similarly with above lemma, we have that {{formula:04e741d7-1071-4af2-9014-04715d067106}} is a non-abelian simple group and {{formula:df890099-abdd-41fd-93cd-83ba8cfd34cf}} is {{formula:4ed68335-1c64-458d-9eaa-7ccf71fc4785}} , or 13. Next we analyze the cases separately.
(a)
{{formula:4aadd840-b4e1-4662-9089-5dbf9d77db14}} . By Lemma {{formula:96e29edb-f638-4c4b-a79d-974eab35df38}} Then {{formula:7f33484e-2906-4a13-8f60-0b221e3eac2b}} where {{formula:0f0091dd-6299-4230-ac60-0b269bc7d25f}} and {{formula:ea7a93c4-d172-4b4e-b477-57bb6fa1e239}} . If {{formula:b71423bd-90a3-4559-8b3c-9570f0b040f5}} is even, then {{formula:32be679f-c041-48c4-8b02-f2ee325297d3}} , as they are the only nontrivial odd codegree of each set. If {{formula:d6944404-9b17-4b29-9cb7-35e1a19a251e}} , then {{formula:7e2be4a3-2dd9-434a-a845-905e3070e042}} . Clearly {{formula:62999c8a-5b51-44ee-a1a0-000052e9ed8e}} , this implies that {{formula:90ff1692-514b-4d02-85ab-7756c9c68b0d}} , it is a contradiction. If {{formula:9b5bd738-b6ec-4334-b258-999932c1364d}} , then {{formula:f7bc3ac4-a14e-4ed2-b4f7-ba768e5a714b}} . Obviously {{formula:ec09e043-2e72-4689-b731-9da806bfde5d}} , this implies {{formula:a4fae149-668a-430c-823f-16d91f7ac0d1}} , a contradiction.
(b)
{{formula:7448e2a2-5a11-472b-ad04-1688a1899d59}} . Then {{formula:f0f1f1d6-c013-4709-8442-5d903b3df946}} where {{formula:8c5dd9f8-04ad-4d8f-9505-72b2c1003388}} is an odd prime power and {{formula:07800e83-c9c3-4671-8870-268dde877513}} where {{formula:667f9f17-8f94-40f8-bef5-21c0cfca4b10}} . Then {{formula:935db35a-dbf9-4c1a-b895-9e3452592d7e}} , {{formula:c2705599-3afa-43c6-a9da-34b4bd7d70d2}} , a contradiction for we can't find a codegree which is the half of another codegree in {{formula:5b9aa866-4f9f-48ea-945d-0e313d2345e3}} .
(c)
{{formula:8edf7741-e5f4-473f-b469-1a4bac28ea44}} . If {{formula:795711db-3f2d-41ef-a7bc-1e9492b58c49}} , {{formula:10ee46c8-9430-4b3b-bff9-3eff8fec4812}} is even, then {{formula:4e4d6570-1321-499a-b44b-df43d6e22cac}} since these are the only nontrivial odd codegrees. If {{formula:45222a09-3c67-42fa-b050-85b67bf62bb7}} , then {{formula:40b6a37b-5982-45ee-90e6-b58ae0d526c8}} . This implies {{formula:50260a12-f48c-4106-85f8-3d91933b9900}} is not an integer, a contradiction. If {{formula:470e4fda-cdca-49cc-994f-4eecb41b87f9}} , then {{formula:57d1bc65-a629-4a85-bcb4-52ffcebf761a}} , we have that {{formula:cbcd3e8e-e83f-41c0-84ce-48b766d45378}} is not an integer, a contradiction.
If {{formula:e094b55d-efef-4543-bdaf-ff937e9d5bac}} , then {{formula:fc312951-781f-4b2e-a7b4-74d0245caf7e}} . If {{formula:b20bbc7f-65de-4efc-b7fd-9a5bd46250ba}} is even, it can be easily checked that {{formula:792832ba-bfb6-40b5-a567-34369ad5b00a}} . It is a contradiction for {{formula:6cc08ae1-c85c-49a8-a07c-c01ea80b8cc9}} .
(d)
{{formula:c951c092-1d44-4a83-b30d-5b8075059cef}} . Then {{formula:ad15824b-e96f-4301-8e8f-82c9baab15eb}} , {{formula:967aa3a2-f61e-4e2a-8ff9-09fa3a428f81}} , {{formula:5abb05b0-0490-4fbc-9ce5-e773eefc1749}} or {{formula:f0721f83-9383-4647-ad6a-071922c2b9d5}} . We need only to observe that none of the codegree sets for this case are the same.
Suppose {{formula:ddce1bf4-5d8d-48e5-a715-724547ab4a19}} . Note that {{formula:a37307aa-2f94-427f-9f11-bac79217c38a}} is the only nontrivial odd codegree. If {{formula:5558c37d-8168-49a8-8e27-f0d839bb4d6d}} is even, {{formula:8e4dd65f-81bc-4a55-b9af-0ed7c32e6164}} is a contradiction for {{formula:e3e146aa-eb0d-40d4-b67a-3f77b5b78556}} .
Suppose {{formula:ff76418e-cd6c-4afa-b1b3-9734cf24ace3}} . Since every nontrivial codegree in {{formula:04a1a132-e615-463a-bbd5-f1d4dcc1428c}} is even, then {{formula:6e1fc04e-21e1-4087-a83c-eebd25d0aec3}} is the subset of {{formula:cdf2e36e-76ce-4e84-9ca6-073b8c398c01}} after deleting the unique nontrivial odd codegree. If {{formula:c2e3f823-aec3-4c6a-b374-5300deda9c20}} is even, then by comparing the 2-parts of the codegrees of {{formula:30f73e14-efb3-4b49-a15a-aeafd6e99cd8}} and {{formula:5507e5b1-41a1-47a2-aef5-8a58751af53a}} , we have a contradiction since {{formula:795d8ce2-3262-48bf-8ecc-a061960edcb4}} .
Suppose {{formula:b6fa08a7-19fb-4796-8883-925b013a7580}} . Then {{formula:3eebca31-da69-4992-83a9-ffc8af687cec}} .
If {{formula:e3fcd0ee-82e8-43d2-9219-a5fc365d76ed}} is even, then {{formula:15f1ce11-2cf7-42f0-9c68-c6de69b3e612}} , as they are the only nontrivial odd codegree of each set. Note that {{formula:180646a0-3fdc-4790-9302-0456f5db3587}} . If {{formula:4695e397-6d07-4fe2-b36e-6742ae96ddef}} , then {{formula:f3c4f323-0208-47ba-92eb-9d955469652d}} . It is a contradiction since {{formula:7196a12c-517d-4794-be09-ea7522bf849e}} . If {{formula:03883338-eab3-4590-ace6-a13d4f9f8cd3}} , then {{formula:a0d86802-a356-431b-94f6-6e32d82f7853}} . Obviously {{formula:733e1b61-2741-4478-a16f-670ca9d3f1a7}} , this implies {{formula:00e21281-ae49-4b66-91a7-b400ee886615}} , a contradiction.
Suppose {{formula:eb20f742-edce-48a0-8c0b-4df5982363bf}} . Then {{formula:3d35fceb-d8f5-4fe3-b425-53f459194bdd}} . Since the only nontrivial odd codegree in {{formula:ccd5395d-44a7-4756-b5b8-21ddcf8b059e}} is {{formula:55f491fe-fc18-43ed-9ade-4eed772c7255}} . We need only note that {{formula:cd3686b8-f0c1-4a68-bca9-73f14019c617}} has two nontrivial odd codegrees, it is a contradiction.
(e)
{{formula:ac4613c7-9123-4c07-b007-e016e31a1ee0}} .
If {{formula:da82210e-4a90-4699-a0a7-482b667ba5f9}} with {{formula:c2e7c614-39d7-4857-a57c-77473af2eead}} . We see that {{formula:fe01d7fb-b20c-4851-b9f8-b35fdb8be4a2}} the only nontrivial codegree which is divided by other three codegrees in {{formula:0786dc62-70f7-4389-bb1c-a2b0afe4043d}} . In {{formula:5f06ba23-7cef-47a8-8d76-f6d54ca3fbc0}} , there are two nontrivial codegrees in this case, which are {{formula:64517f72-e7d5-4adc-ad1e-2021e076ac2f}} and {{formula:6c72ba6e-ed32-476c-bed1-64fb5cd0cd35}} . Then {{formula:6c1e5ad9-ed69-43e9-8804-e1edaa295ead}} may be equal to {{formula:ec0ad077-bb20-4092-867e-fa23649dffa7}} or {{formula:899adef2-822a-472d-8ff7-fda6c1790fd7}} . Furthermore {{formula:89983ee2-05ad-4de6-8813-fee8efd9fd0b}} or {{formula:fa236662-c686-4a73-9df8-9dc23452bec2}} , as {{formula:ecd148b0-8177-4d26-9294-d4b6a50bfd55}} is the smallest codegree among three codegrees which divide same codegree in {{formula:0ff57868-fb49-4c6d-9c0e-aed2818fdb02}} and {{formula:6a6b8c28-cd1c-490c-8627-74213e8c02de}} or {{formula:9624216d-087f-49d2-bfaa-7fd814bf978b}} satisfies similar condition in {{formula:eae23806-1d23-4700-a37b-43d19f0d8427}} . This means {{formula:d35ea414-4100-4939-bab1-a1a9356bedc0}} , a contradiction.
If {{formula:3ac1fb98-45ba-4b88-83c9-2feb19c9e1ec}} with {{formula:b4235a0a-88c7-4c85-815d-53323ec50da0}} . Similar with the analysis above paragraph, we see that {{formula:b93aadf8-e24a-4c15-81e1-1bd0b40eebd4}} or {{formula:bf67b5ab-2a18-46a8-981d-30586a46a7c4}} . This means {{formula:d50def21-0596-4409-b36c-5b4479f538c2}} , a contradiction.
If {{formula:00b17291-f4dd-402e-acd7-89425864d846}} we see {{formula:a6bfd8dc-805e-4d3f-9c1b-74967048e36b}} , as they are the only nontrivial odd codegrees in each set. By setting this equality, we obtain no integer solution for {{formula:060038c2-becd-4fff-9909-8f07bc609f76}} , a contradiction.
(f)
{{formula:d97de4bd-adf7-4e39-a83b-ed06dd92ec58}} .
Suppose {{formula:b9fccecf-e245-4431-9305-09eea3a0d521}} with {{formula:d3171273-e0a9-4721-9d47-3be1e54ce7d1}} . We have {{formula:2bf4637f-c372-4e79-afac-e7d593890432}} or {{formula:7b137669-e0bc-4e12-8d5b-85790e728788}} . This means {{formula:df0eee81-6b3b-47c3-8473-39653e6a39bf}} , a contradiction.
Suppose {{formula:74a03e82-4ef8-47e9-97c2-50420c74cd3c}} with {{formula:ff923139-6ca9-4445-ad26-975022b3c7f0}} . We have {{formula:ffce2051-f4b0-471a-8c22-169d498e484f}} or {{formula:ca69bece-443b-441e-8fc9-5672949a8774}} . This means {{formula:52afbeb5-fc56-4f28-893d-518167f85e8f}} , a contradiction.
(g)
{{formula:6467be48-193a-44a8-ae14-7933b15df8c9}} . Then {{formula:16b1d004-4354-43bd-b18e-0c4cb1fa7d4f}} . We compare the largest 2-part of each codegrees set and obtain that {{formula:e5ae1ca9-4fd3-4ada-b41c-dfe85abac994}} is not an integer, a contradiction.
(h)
{{formula:7981b9dc-1214-4915-bc1f-57ae4513ab06}} . If {{formula:1e432bf7-bae8-471c-bc32-81411cc108ac}} , {{formula:a17d35ca-73f4-4e5b-b1fe-8cddf3643cb3}} or {{formula:2f9044eb-25e5-41b3-bad2-35d0bd34b099}} .
Suppose {{formula:2c6e192e-77df-4220-b722-7f7c8844f0b6}} . It is easily checked that {{formula:3536709a-edc0-495b-9795-756f89aae254}} is the only nontrivial odd codegree in {{formula:d1164681-5e1b-43e8-b319-b6b34de1b0b9}} . Hence {{formula:20e7d641-97b8-4e5b-b75a-4a4aa93fdc95}} . Note that {{formula:c53ca896-d714-4be4-8ce2-389847f50ff3}} is not divisible by a square of prime, a contradiction.
If {{formula:46262d6b-bde9-4036-b420-17d4dc6f026f}} , {{formula:4ea896a8-98b9-488d-865a-6fe861bd2c6b}} . We have that {{formula:aee90043-3e5f-4b34-8ffe-126932ddf5fb}} equals to either {{formula:24a3cac8-8451-4132-996b-30732e516c5b}} , {{formula:baba400a-8e5b-4aaf-b567-314968701ea3}} , or {{formula:1a4b182c-cbdd-4587-896e-4180f8660477}} . This is a contradiction since {{formula:be3a261b-e0b9-42d8-ae7c-4a3d859230a5}} is not an integer.
(i)
{{formula:8d22cb18-a817-4d8b-afa1-531390896ce6}} . Then {{formula:c0eada36-c66b-4bad-9256-d2a410cd4645}} . By observation, {{formula:ef5c8e44-3163-4f71-b0d6-b0dfe892c58e}} is the only nontrivial codegree which is divided by other three codegrees in {{formula:d783a9c6-c209-43e6-bdbc-0dea9378d763}} . In {{formula:d1ef7819-c57f-472d-b993-5010d0fe24bd}} , there are two nontrivial codegrees in this case, which are {{formula:83853f44-e217-4228-8084-a56ba87cbb77}} and {{formula:b51b7377-c0ae-4957-9f56-2ab902755020}} . Then {{formula:0cb295c1-bf40-48f3-bb70-79b259549dee}} may be equal to {{formula:ab405c14-c79d-4a58-a762-d2970617890b}} or {{formula:0398dab2-b2d6-4b15-8e3d-4f3dfe944c38}} . Furthermore {{formula:dbbff3e6-b54e-450c-b86e-8a59ab89b67f}} or {{formula:f1fdec1b-9ef6-4574-bd64-debb805083d6}} , as {{formula:1814b4e1-5792-4451-bfa0-0c3e1589e5fd}} is the smallest codegree among three codegrees which divide same codegree in {{formula:61b3c8e2-e0a7-4ca2-8826-1cec4d4f8d58}} and {{formula:f38407de-52f9-40c6-92f7-812635aac7bb}} or {{formula:ed2c28be-490c-471e-9518-14ac31622cc3}} satisfies similar condition in {{formula:646681b8-ca47-48dd-93cf-115f16d640d2}} . This means a contradiction for {{formula:488523da-f973-4637-8b5a-d7662b202fc4}} .
(j)
{{formula:244b1aa3-430a-478c-b984-f1627feac9cb}} . We need only observe that {{formula:8493033e-d3fd-4670-b2bb-11bd12e38dd0}} , as they are the only nontrivial odd codegrees of each set. If {{formula:b38b9d28-7543-4b92-933f-e398c60fb302}} , then {{formula:1ec01201-bb5a-41e9-a541-89bd9cc086c3}} , this implies {{formula:3e7727aa-b3a6-431c-a4e6-ad2edc9b6706}} is not an integer, which is a contradiction. If {{formula:5e0fc8c0-c261-4dce-8327-73b0784d7014}} , then {{formula:a53b911e-5af9-4fe7-847b-8f833d77a6f7}} , this implies {{formula:47077ac2-e239-4fa8-9dee-3a9a728cb51f}} , which is a contradiction.
Thus {{formula:aa83d11b-537a-4ef9-ac0c-f9ee6a71c645}} for {{formula:ec5126b8-cf4c-415c-967d-df682fcd3224}} . Comparing the smallest codegrees, we see {{formula:6b36b5de-8d4b-42c3-a85b-fcb66044db74}} . Thus, {{formula:cddef4e8-a6a0-4c1f-a6e7-a89f6574fc24}} .
Lemma 3.4
Let {{formula:5b133e7d-9817-4192-8774-c47ecd75d8e0}} be a finite group with {{formula:35f4b0a6-a865-4f4f-b183-bef08ffd4501}} . If {{formula:90161d99-1ecd-4da9-bfae-02e237fb5a84}} is a maximal normal subgroup of {{formula:59fa4660-9964-4f49-8dcd-ceede32abdbf}} , then {{formula:4e3ddd8f-ab03-4897-bbab-095e63ef221c}} .
Let {{formula:c743baac-ced9-4785-9c1d-63213d5dd55c}} be a maximal normal subgroup of {{formula:4bf75a08-ff02-4765-91e1-fa16a6d003ed}} . Then {{formula:8bc1c927-3de8-48ef-9c67-4d616370826b}} is a non-abelian simple group and {{formula:2ed7ab32-8bf6-49cf-93ba-86148c1ff6ed}} .
(a)
{{formula:11007a35-3a69-4e36-bb20-405ac03d9862}} . Then {{formula:137d2617-0f7c-42b1-ae22-82d8e202095f}} where {{formula:add2d5c6-b4a8-4696-95da-49c459f0e094}} , and {{formula:9ebd3ae6-d397-4496-9ef8-ca95a2859277}} . Thus {{formula:72ff0f10-dc3b-4dfc-bb1e-fde25b80a41d}} or {{formula:edb8014b-2c38-4cd4-9edb-a3fbbb18c36d}} , since these are nontrivial odd codegrees. It is a contradiction since {{formula:6ed75747-7b1d-4184-ad4f-40d41fc8c934}} .
(b)
{{formula:dfe7fa32-320e-4e6d-878f-26f828ea90ac}} . Then {{formula:32546d6e-ffa7-4002-b31a-79b910e68779}} where {{formula:d71c93c7-e694-4891-9158-5e71fe04168b}} is an odd prime power and {{formula:b32e9bf5-0336-4598-ba84-1d36cedf21b0}} where {{formula:e065a47c-1881-4a29-bda2-0e7941e885e7}} . So {{formula:251d6a18-2e23-49ac-84b5-2639bca9e33a}} , {{formula:e52e3065-6f40-493f-8954-1c79cd0697d5}} . This implies that {{formula:09d4aef2-3756-4995-9d50-20f9f3170547}} or {{formula:d00ea08d-0b09-459b-931e-14167ac1d3d8}} since {{formula:f0125f9d-4b0d-4aa2-af64-9c2b2506e671}} is half of {{formula:f217f915-b138-416c-993b-2747d4bdfda0}} , and {{formula:48035566-9efd-4545-bf07-9ee027147136}} is half of {{formula:bc9b813c-76a8-41d6-be75-ce653fbd025e}} in {{formula:ee833c5a-2c45-458e-a1e7-3178449d1b6d}} . In either case, {{formula:923a70b0-4ab1-41f0-af02-b4b86f0753bd}} which is a contradiction.
(c)
{{formula:007c204b-48bc-40a5-bae2-011a38313d48}} . By observation, {{formula:74653a69-dd97-4f7a-b7c6-c599d61b96a9}} since {{formula:bfd4af8f-8981-4fe0-a89f-c7ca74cbceeb}} .
Suppose {{formula:75a6ce01-36e2-44f9-8465-0d5eb86f6b5b}} . We note that {{formula:2c63c5b3-2bf2-4c9d-a047-d04ed406c202}} or {{formula:c5ad8774-e5df-46c3-a69e-f06c17711c2a}} , since these are nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki groups, a contradiction.
(d)
Suppose {{formula:a2287941-4b52-4f9f-8f9d-303d6a99166e}} . We note that the nontrivial odd codegree in codegree sets of {{formula:a0ea65df-5a59-4aa7-aa24-60d115dfc6bb}} , and {{formula:6bf60df0-ef0c-49dd-a615-77aafdbf3e39}} are {{formula:3a802cc6-fc52-4841-b73a-21293c45affc}} , {{formula:6fbcb164-46b4-41af-98de-a07e201ca6b2}} and {{formula:28b5fabb-e90c-4b06-bf24-c0769ea881dd}} , respectively. Also, {{formula:038c592c-8004-4df7-961d-a33dc88f6a30}} by an easy observation.
(e)
{{formula:9c613a85-c5d0-44d6-a2d3-89d9463db8e7}} .
If {{formula:af7262e3-9072-41b6-a9cc-162f7c7d6317}} with {{formula:8795f0c0-d18f-407d-bd4b-a26512c1eb5b}} then if {{formula:89377872-f148-481a-8002-31640e99b569}} is odd we have {{formula:0a5ed326-7a2e-4e05-92ba-2f2e096151ad}} or {{formula:3fdd5418-9bf3-4cad-8186-4632282e52a1}} .This equation means a contradiction since {{formula:3e68ab11-ff9d-41be-a6ce-76b86719ed80}} . If {{formula:d5f66570-8836-4acb-9487-c86de64c73c3}} is even we have {{formula:e0017b18-9eeb-4d8d-9e8a-3d5db3963c5e}} or {{formula:0eaf740f-103c-4a1a-882f-4b9e20543b41}} . Then we have a same contradiction since {{formula:0e7b9c7c-4024-4d00-9550-bae11d0f3423}} .
If {{formula:cbf8c7af-2eab-43bf-b0ad-b4d8d98fc875}} with {{formula:a64ee255-e60e-4cdb-bf0a-cabee8bd73d0}} and {{formula:321bd310-13fe-4d21-9d78-40f617b8a499}} is odd, then {{formula:552544a7-b867-4ac7-8d1f-e166973e2a3e}} or {{formula:469421d2-d624-4603-b9fb-8305fab25815}} . We have the same contradiction with {{formula:126d8b79-b169-4b9e-8e52-6a7a8244b07d}} . If {{formula:4093726b-0d84-4cc9-a323-90f0863f532c}} is even, then {{formula:b7ac47db-1aab-426b-aea9-a0b3ad139b9b}} or {{formula:62ff62d1-9e0f-4d4f-b5f9-d7d4f5f143e8}} . Then it is a contradiction since Then we have a same contradiction since {{formula:cfe82d4e-d427-46f9-b1dc-9b9cd043d031}} .
If {{formula:e08c112c-d957-4c7c-ab96-4d2b05e4fedd}} we see {{formula:efc85168-3534-47ef-b7b6-416119994289}} since {{formula:98f13e58-0039-449f-958e-daef098c44db}} .
(f)
{{formula:d97f7d42-6a92-44c7-b955-d71de399c1c2}} .
Suppose {{formula:75f9332e-6658-41b8-80c1-229fef5689e0}} with {{formula:5f0f5fe6-e9af-45cc-8e73-750a7af173f9}} . If {{formula:cc74836e-28b1-4cc2-bd86-a9942f79721d}} is odd, then {{formula:51af1957-324a-4e1b-96b6-02d977c71d49}} or {{formula:2ed4305a-20df-4775-b6cb-3360140451ee}} . It is a contradiction since {{formula:2ebee989-4134-463b-8dc4-866c106f1bca}} . If {{formula:fd5ae1a6-0c75-4101-87bf-e814e1e94809}} is even, then {{formula:dce31bd7-bc04-4d7b-9b33-07bb9deb4097}} or {{formula:7be40a0e-b637-443e-8b5b-6308bfe231cc}} . This means {{formula:d22d1bfc-43cb-43be-824e-233de6de96b2}} , a contradiction.
Suppose {{formula:d8434ce4-87fa-482c-8f92-e92b40ac6e7e}} with {{formula:ce8a6a4c-38f3-45eb-86ca-a96f95e44b4b}} .
If {{formula:fb1c7c7d-3d27-43b9-a08a-6f65a1a3f88b}} is odd, then {{formula:aed60fd7-d334-494f-a15e-e2ed0850e121}} , it is a contradiction since {{formula:500bcd63-ed52-4850-8627-417cf2712dbf}} . If {{formula:bee80476-dc86-4f29-a6c7-cb6610d278ab}} is even, then {{formula:1974c30b-283b-4690-b39c-019fac8d17bd}} or {{formula:f0bb2975-1083-4eb3-b3b2-575d8b021736}} , which means {{formula:21237dd3-7504-4c0d-9874-8c4b87be6db2}} is not an integer, it is a contradiction.
(g)
{{formula:4b886a6d-df65-4607-bb7f-42a111c7c6d9}} . We note that {{formula:be5c05e5-753d-4689-8eff-7e33356fb12f}} since {{formula:2149910e-0461-4795-88b5-e609b36ef711}} .
(h)
{{formula:cddd5248-811b-4a2a-a045-18b7dc4f0168}} . We can checked that the codegree sets of {{formula:0ff7c9f8-08df-43ad-9264-50ee7bf33ff6}} and {{formula:78a0cabb-27bc-4705-800f-a77d8807c144}} contain {{formula:5534edee-0ba0-43dd-ad0b-2f0245f847e8}} , {{formula:14a1536f-a530-4bfe-ab2b-2cbaa50d22b9}} and {{formula:dd9e5c89-3687-4dc7-a914-1c657f52abf5}} respectively, none of which are elements of the codegree set of {{formula:34ed7fb2-3fd1-4e9a-bae8-98d05fe05e03}} . We note that {{formula:9022e90b-6219-49d8-b3b6-6b27ef88e35e}} cannot be isomorphic to {{formula:5a68ba44-f6c7-4d96-ad11-dc391a4e0aa9}} , {{formula:5401ebf8-fed8-41a4-886f-dc5e386e3966}} or {{formula:19cb399d-e3a0-4616-b52d-1eb13f15790d}} . It is a contradiction.
If {{formula:5160cbcd-7605-4483-94c2-2b733a1f97c8}} , then by comparing the codegree sets, we have that {{formula:dc4ef6a2-2a6d-4f75-83d6-63cddba6910d}} or {{formula:cc4e8439-7190-4920-a76c-b34728c42257}} can equal to {{formula:7e10ae65-0834-4528-b788-2273e9b32ede}} , {{formula:5cbdd651-fdee-44f3-850f-10e3d83d7897}} or {{formula:ef8ce936-e351-4282-98ef-a28d2c900e8b}} , since they are the nontrivial odd codegrees in each set, it is a contradiction since {{formula:62bc33f6-d14f-4fb5-bccb-6a4d154ee71e}} .
(i)
{{formula:57ae09fb-fd8a-48e1-a0a3-065b9c67cb4b}} . Then {{formula:d7f54de3-da91-4c13-a621-0c345eda403a}} . We note that {{formula:07a265f4-8498-4753-aae4-fd0ca5649c58}} . Hence {{formula:09d0331a-12fa-4c2e-aa57-cd71591aa7a0}} , it is a contradiction.
(j)
{{formula:fc2ba1cb-0187-4483-836b-ca48e96933e6}} . Then {{formula:dd713908-4cd1-49f0-bed1-79bd276db7be}} or {{formula:719ebe70-30e3-4344-b4e5-72641fc2f717}} , for {{formula:63404828-478b-4460-a51e-096c48a781df}} .
Suppose {{formula:0a686d4a-1284-434b-9918-f142c1454202}} . We need only observe that {{formula:3bf17f4c-4a34-4edb-9c7e-594ca251a7e6}} which means {{formula:75210345-fa9f-4bcf-b035-8a7e808b55ed}} . It is a contradiction.
Suppose {{formula:09ef9709-1d64-4949-a445-dd82822b315d}} , for {{formula:1cdfb01f-f25d-4009-90b2-951ce36d6109}} . Note that {{formula:1111cf51-376d-4048-914c-db8a86e2328b}} or {{formula:5d71c7c4-7283-4693-81ca-d6170420bbc2}} , since they are nontrivial odd codegrees in each set. This has no integer solution. A contradiction.
(k)
{{formula:7d641205-81c8-4212-9e17-3dd0d4c77ae0}} . It is easily checked that neither {{formula:b3f97d34-6308-4f54-a51d-8dd10c2455fd}} nor {{formula:138f4899-0092-4947-ac68-152cf8e02eee}} is in {{formula:2e91e54c-cce9-48de-ab52-5c4dcd01ec13}} . Thus {{formula:45be4fc1-4574-4020-81cd-da2adfb30880}} and {{formula:81bc5bac-f7c2-4212-a684-c6fde3809014}} .
We have that {{formula:ccef13b5-a965-421b-a6e1-93eb78af6350}} .
Lemma 3.5
Let {{formula:62089aec-4a56-4e67-a73d-4c7d77f778cd}} be a finite group with {{formula:91b67162-e34c-4649-8cff-475388e32fe5}} . If {{formula:4f8008ff-58a0-4ee0-b4ef-31912969a48e}} is a maximal normal subgroup, then {{formula:1ce840aa-d85e-4312-8682-90156bb28ae0}} .
Let {{formula:e2715ac5-865b-4519-9040-3797e521245c}} be a maximal normal subgroup of {{formula:9102a4c7-679c-4240-92e9-3d916f1bb4b6}} . Then {{formula:b6fc543d-888c-409a-94af-30f94863c9af}} is a non-abelian simple group and {{formula:9648d20b-0219-45d3-9644-8bc4eccc9f2c}} is either {{formula:654b5eee-3d27-42d1-82b9-fcf8d13f15b4}} or 14.
(a)
{{formula:5648972e-9c76-4f9b-83d6-e6bb23f69327}} . Then {{formula:4fb68e0f-5960-43da-876b-d03ec220994e}} where {{formula:77dc508c-2848-4a0a-9f53-2e126a2134f3}} , and {{formula:e3e7e401-bedb-4b09-bf13-9010b12a56c4}} . Thus {{formula:70deb3f9-13d5-4381-b866-ec80d478b4a4}} since these are the only nontrivial odd codegrees. It is a contradiction since {{formula:a790a6f3-06c8-4c42-8836-6bc05bffa99a}} .
(b)
{{formula:fd2035d9-cf60-45a8-948e-8e3c84df413f}} . Then {{formula:d6fa1a20-96ca-4534-af45-00d3dc1dccb7}} where {{formula:af49ad7e-a1c8-488b-a3fb-e1ce4a19ecec}} is an odd prime power and {{formula:2827cda9-e122-454b-a607-35be594a51e9}} where {{formula:1aa45018-98cd-4cd1-802c-c11dae2eb5f2}} . So {{formula:a9ecba9e-f241-4268-92b7-308c026efac7}} , {{formula:4c178236-4b7a-4d24-9796-cae3abb4dbb1}} . This implies that {{formula:d6578967-d884-4171-9a33-a4477dcff56d}} , {{formula:801fda94-f454-4f39-9cd3-46cd0f55fbd0}} , or {{formula:50a56d16-b56a-4125-8a34-28f44d2e9712}} since they are half of another codegree respectively. In either case, {{formula:65a45fd9-0be1-4831-8962-9d51db2c19a3}} which is a contradiction.
(c)
{{formula:7127aa62-c812-42be-918e-9e72de8f6568}} . By observation, {{formula:6e20fb2c-2201-4624-8e80-a467e95328e0}} since {{formula:26e29e26-79af-4505-be95-df9429edd31e}} .
Suppose {{formula:acdc8ece-ec10-4330-8ad3-a7f9524496b5}} . We note that {{formula:c0e343c6-a048-454d-96e3-542fd4260f59}} since these are the only nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki groups, a contradiction.
(d)
{{formula:bedee429-c699-4eeb-ae1d-3894b4519df8}} . Then {{formula:252b6622-303d-46a3-b90f-563a9ab82067}} , {{formula:c3434d9d-0b5c-4252-a37e-528269e16564}} , {{formula:2e5f6276-b5f2-41dc-8ae0-6593ab5f1346}} or {{formula:1584ed45-9e8a-445e-952d-935a2a7ef1eb}} . We note that the nontrivial odd codegree in codegree sets of {{formula:4ae75413-28cf-4613-823b-f3c29a1aef90}} , {{formula:9be298d1-3eed-49de-ab92-b5fd8c1badc9}} and {{formula:001199d2-031f-4d6d-a7db-a9fb34ec41bb}} are {{formula:776cfc85-a3db-4b61-af2c-1ca678c3aff2}} , {{formula:f344c549-135d-4192-a69f-2ff835dd3234}} , {{formula:c6e14fe4-efec-4c7f-92ec-96881c3b1e49}} and {{formula:adca4435-59c0-400c-8136-01ea750246a0}} , respectively. It is a contradiction.
(e)
{{formula:10ea1a6c-0839-493a-94ea-2254f86b9fa0}} .
If {{formula:89572555-4f67-4cc2-8f33-4a492c573b2f}} with {{formula:c75fecf8-f3d0-4b9f-b37f-8afc81b04cc6}} then if {{formula:d87def8f-4ca4-4d53-a461-514c81ff8568}} is odd we have {{formula:0a3a6dce-d2cb-4529-84ac-76a5037a9dc1}} . Thus it does not yield an integer root for {{formula:30e5c851-1591-4d91-965e-dd50ee47ffa7}} , a contradiction. If {{formula:fc0de87f-a4f5-4ff3-8c59-b860c9f3a5b6}} is even we have {{formula:d8aacebd-a4f9-4e2c-bc58-3b394b9615aa}} . This also does not yield an integer root for {{formula:5d0a14c3-f719-4316-8271-ef4b21813ef9}} . This is a contradiction.
If {{formula:bc304c6c-926d-400c-93e6-04d8b541435f}} with {{formula:f02bb1a5-fddb-4439-9ba9-43672c43fc36}} and {{formula:e4802d58-d524-4c08-8dda-2167b03d449c}} is odd, then {{formula:d591dc18-95f9-4e26-967d-a6a896d687b1}} . We have the same contradiction with {{formula:f9eed257-1608-450e-9b7e-db62e83da033}} . If {{formula:054f08a3-e643-4d63-bb8f-aceeff95aef3}} is even, then {{formula:7d322a38-fcb1-4626-aaf1-594970e88377}} . It is a contradiction since {{formula:d18d694f-6043-4961-ab67-fb4c80b0f0ce}} .
If {{formula:3c72c56e-f132-4d92-8995-ae06550829ba}} we see {{formula:9efd4779-a34f-42df-a380-ddfe08bf36d1}} since {{formula:862989e5-ec48-4454-9e12-224a4bc33812}} .
(f)
{{formula:05406cff-b4b0-4779-a53d-7439e3e7b385}} . Suppose {{formula:4c0168c4-7775-4e05-8399-8957d6cc7c81}} with {{formula:7a1a01ce-c1be-46ff-a68f-c1668ca0b1e5}} . If {{formula:ab67ed37-3618-4037-9635-bc72979c06af}} is odd, then {{formula:f356c8e9-18b4-47c4-b4f0-1fb8695f3935}} . It is a contradiction since {{formula:d5392b57-4d54-4df6-8678-1be0165ec6cf}} is not an integer. If {{formula:886b6615-4f30-477c-b1f3-9cca935477c0}} is even, then {{formula:237f312a-9d0a-44c7-bb52-e5a093d69558}} . It is a same contradiction since {{formula:7934e179-e493-4a83-a0b4-090ceb138b8f}} is an integer.
Suppose {{formula:f2873769-f2e1-4439-ad81-9befa5745a52}} with {{formula:3ca72eca-9419-4fcc-9adf-d0bd96dc8b9a}} . If {{formula:6b5486be-afd7-4a9a-a3b4-efd62c2108ee}} is odd, then {{formula:ff967250-5740-48fc-9565-d4a6554bda3b}} , it is a contradiction since {{formula:821f19a5-88b3-459d-a9f8-03ae30ed17dd}} . If {{formula:523f18d0-80a4-4a0e-becb-6aac1eb2cc1e}} is even, then {{formula:45349601-d551-47ad-b515-ede50f156c94}} . It is a contradiction since {{formula:085d1b5c-d0c6-47cf-8760-22232f5a16a8}} in not an integer.
(g)
{{formula:0226ba7a-9bef-48d3-af2e-757351173700}} . We note that {{formula:b8035878-1dee-4a46-8486-6be69f7820cf}} for {{formula:60ba6010-c415-488e-a372-061f636f7414}} . A contradiction.
(h)
{{formula:b6658aec-fb9b-4695-8a70-bdbb1322db8f}} . By observation, the codegree sets of {{formula:e8c45142-10ab-4a6d-9aeb-6fba157bb4bf}} and {{formula:ef5cc574-f6ac-4a7d-8d94-86b47d257773}} contain {{formula:110af5b0-410a-4ec6-81de-773b9abd70ba}} , {{formula:bbe9134a-0c1b-4747-95ed-0c8a6d47e598}} and {{formula:66ca93ab-6c8c-422b-8ce8-c2c0fa2d7e14}} respectively, none of which are elements of the codegree set of {{formula:89a5899d-235f-4b3d-8075-0f39ae168517}} . It is easily checked that {{formula:56ff0a87-bbdc-43b7-9594-a33bbd51d831}} . We note by observation that {{formula:b724ee18-159f-4480-a92b-4d611c6d3278}} cannot be isomorphic to {{formula:a67c3f87-8ca2-4415-a5be-13408c2c6532}} , {{formula:b5415818-351f-42b9-95a1-b09466b29330}} or {{formula:22130c86-b230-46a0-aed6-6fbf44d32b84}} .
If {{formula:f8f5cd4a-3007-42af-92d0-9470ce09ad82}} , then by comparing the nontrivial odd codegree in each set, we have a contradiction since {{formula:71d6fc6b-2f34-44d9-a9ea-916855da2c36}} .
(i)
{{formula:3f7df187-9567-4a68-9bca-d43a024f9535}} . Then {{formula:69a6ba9d-59bf-45c0-8113-c40fff577f14}} . We note that {{formula:de253be6-6a20-472c-ba29-ef9d58e7a552}} . Hence {{formula:202dfa28-8c8f-4824-8fb0-7187e3b18b79}} , it is a contradiction.
(j)
{{formula:5fd4f445-766c-4d69-abde-3c459b844be1}} . Then {{formula:2c8a2a7a-ca5b-41bc-ab7c-ed23b0935e1f}} or {{formula:8aebea37-4515-46bb-917a-84ed1c3a8490}} , for {{formula:0fc0bc98-aff1-4550-a21f-b48ea34cbebf}} .
Suppose {{formula:692c62ce-93cc-4818-90bf-89eef3f8c8e9}} . We need only observe that {{formula:24f3406b-c5f3-441c-9a7a-cf226517de67}} which means {{formula:e1a18928-aa4a-4683-985f-47e2355011ab}} . It is a contradiction.
Suppose {{formula:f489c21b-65d6-4b9d-aa9d-bbb2a2061ede}} , for {{formula:b2eedab7-8787-41bc-8e8d-dd9d6a147ccb}} . We have that {{formula:0566ba4c-10a8-44d4-99f4-a1826a787d1a}} , as they are the only nontrivial odd codegrees in each set. It implies {{formula:ef98f19f-72b8-4dc3-b36c-95ae111121d7}} , a contradiction.
(k)
{{formula:42213c23-91ea-4431-9934-21b9b05ce360}} . It is easily checked that neither {{formula:3dc399d2-01ed-4c3e-961d-b3fbbad287a7}} nor {{formula:d62411ec-a0de-404d-92cb-978209dc2171}} is in {{formula:f341ea55-44cd-4124-8592-65dc7524acef}} . Thus {{formula:fad2fbfc-04a1-493a-84a6-d55a1f96c7f8}} nor {{formula:8d9e4873-8b45-454e-8682-0377b8378a30}} . We have that {{formula:22615625-157e-4c8a-8e55-0de501edd31c}} .
Lemma 3.6
Let {{formula:7336f3f2-6be3-4567-9110-037063f3ac2f}} be a finite group with {{formula:255da97b-83e4-4c9f-9037-80f3996e056b}} . If {{formula:04c8a216-901e-415a-a17d-ac071fd08196}} is a maximal normal subgroup of {{formula:41f1869f-b463-411b-bc07-60f69d9bc19d}} , then {{formula:afce0575-5b13-4fa5-be4a-0a9f5c2eeb01}} .
Let {{formula:c4c9ea28-0a74-4c6e-9358-bc4dd488ebef}} be a maximal normal subgroup of {{formula:8eb560f8-9f9c-4e47-8b84-e3e6a66b2ded}} . Then {{formula:9bcd53b6-35f1-4606-a146-7c4e317684c9}} is a non-abelian simple group and {{formula:9d378a48-5893-4d8d-9f81-8ac053a7d2de}} is either {{formula:a6a25055-cf2a-4ae6-8ac5-3ec0554e475c}} or 14.
(a)
{{formula:8a5e388b-72e9-42d1-9cfd-51e6bfc256a8}} . Then, {{formula:d584c87c-2ff9-4418-9256-99517bb56021}} where {{formula:204a5949-03ec-4717-aecb-55640ce0bace}} and {{formula:4012024a-db96-49c9-bc76-073e8377b85c}} . Hence {{formula:ded0d23c-21c7-4f68-9436-6cdaf154976c}} or {{formula:9a9338e9-75c0-4c57-88d4-bbb97ae28478}} since these are the nontrivial odd codegrees. In either case {{formula:05dad1e8-28ae-402c-ad07-c5372bca4a60}} is not an integer which is a contradiction.
(b)
{{formula:004b9cdc-3dd9-4795-9b0b-29ce7b68637c}} . Then {{formula:81499b80-cddd-4215-a444-c8a9c308ffc7}} where {{formula:c4dfd366-8263-430f-a3ea-c3ef4c585481}} is an odd prime power and {{formula:92c9f5a0-27a0-408c-b571-b9928b9c6ccd}} where {{formula:a2715172-24c1-4a3a-88bb-32e48195e19e}} . Then {{formula:e3a5c3cf-025e-4ebf-b7b7-19546983871a}} , {{formula:b5c1f6a1-57f2-4dbd-a311-b70bf50c6b60}} , and this implies that {{formula:c84fd277-3c8d-46bc-9099-320d1a741528}} since {{formula:f9b87385-e304-449b-a60c-2e48bfc3eaac}} is the only element of {{formula:53c945c6-df71-49dc-ae16-d82a0e4eb808}} that is half of {{formula:e82df418-e68b-46e2-a371-797b5ee9af84}} . In either case, {{formula:8125d183-df87-48ed-a057-ab58e33b016d}} which is a contradiction.
(c)
{{formula:6d39d655-d1f4-4a78-b2fd-a805aa9462dc}} . By observation, {{formula:eb70f52d-310d-45d0-b209-d0d3e4c29684}} for {{formula:8f621efc-af9c-4429-b3dc-312496fa2223}} .
Suppose {{formula:423f17ca-8546-4d9a-96b0-e0707ac95e6d}} . We note that {{formula:2d23adf6-95ea-4dc9-9559-aa613f0607a3}} or {{formula:7f12d66b-ecb4-4f55-8555-31171861f5a6}} since these are the nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki groups, a contradiction.
(d)
{{formula:2f805daa-5ac7-46fb-a2b4-49c34c87626f}} . We note that {{formula:e4d3e006-9fc0-421f-b185-2b791976c3b0}} , and {{formula:8c9397b5-8673-40a1-a448-327eba090a60}} can't be {{formula:54c2aced-4f49-4bb6-b6d2-b8f32ee501a1}} .
That the nontrivial odd codegree {{formula:b889b254-f896-47dc-94e5-9717c82deee4}} , and {{formula:686db0d5-3499-4dc9-9f8d-80dd7ddce86b}} in the codegree set {{formula:fb479b0e-3dac-41c4-8b95-050da9b3ada0}} and {{formula:6679c5e6-ed6d-4949-8671-ca83eb1f0ad3}} is not in {{formula:7d5e2056-ff83-4510-a9a4-9b9c3d0d47a8}} . Also, {{formula:28431e38-2a21-4db9-8f75-da41702d6a0a}} and {{formula:8cddaad1-cee6-431d-829b-d4f58976c7e7}} by an easy observation since {{formula:649f6cb2-e8e2-407f-97a1-9a71d03a7ca5}} nor {{formula:ee90093e-8bfa-4b14-88a7-e79c6ec2a89e}} .
(e)
{{formula:31be767e-5ba1-492c-b483-528bfe5229b1}} .
If {{formula:66fc27c4-1e10-4cec-ba10-9a1336bd2ed7}} with {{formula:51e7f50b-adb1-488e-bee9-c57dcc15339f}} then if {{formula:5865a82e-7932-4bc2-b13f-4b9f080fdb47}} is odd we have {{formula:1cc1eb57-5936-48f2-a02c-5a196be92d01}} or {{formula:431f00e5-ee10-47fb-9bbb-5c34163e16c6}} . Since no cube of a prime divides {{formula:300ce0e0-a021-400f-97ae-c4aa95ae9e50}} or {{formula:30757fec-68df-4338-a8c3-fbed9b177bd8}} , we have a contradiction. If {{formula:1bb8d965-214d-4bd4-9769-5e7cfd5b247c}} is even we have {{formula:edfafe01-96b5-4109-b767-83127a31373c}} , then {{formula:080b0072-2c2c-42e4-b38c-41ce851622b3}} . This is a contradiction since {{formula:34f026ea-c112-46e1-af36-224a7a11af0b}} . Or {{formula:4b7c4639-d345-473d-a413-f841a4a77c85}} . It is a contradiction since {{formula:95155f0f-d786-47a1-8e6a-72566e220773}} is not an integer.
If {{formula:f43aa648-110c-4261-bf31-74dbcb5ff41c}} with {{formula:b47d4d1f-29a2-44d7-99c9-3b271f926070}} and {{formula:8caba181-5878-4d81-875d-ea47ea534fe5}} is odd, then {{formula:d3d82ff4-b936-4ad4-ba9b-4bf669f8bd5a}} or {{formula:37bc9996-616b-4080-83ea-d8e0dff4c84e}} . We have the same contradiction with {{formula:f450cfca-00f3-4c48-9c93-82257d7ee718}} . If {{formula:57eeb6d2-1da3-43cf-a809-2577cb8d5d81}} is even, then {{formula:ebdf9d87-52ce-4d79-9ab7-acf0af25b113}} , which means {{formula:ea6b5e4a-9674-4914-abaa-c9968b7fbdce}} , a contradiction. Or {{formula:96d6d63f-6703-435a-b185-827fa9cf3451}} . We can get the same contradiction as above.
If {{formula:9cf1f483-70e4-4273-a476-2c6c0c1a3657}} we see {{formula:b8a2b02f-300a-4d0a-bfe0-bbacc97e0e7c}} since {{formula:a1d6d10a-1f28-4dc8-8866-5cd1e144d2e1}} . This is a contradiction.
(f)
{{formula:1708f6b1-98b5-410f-960d-c21716911fc2}} .
Suppose {{formula:ca063180-ce3e-432b-acdb-edf1c44f9d1c}} with {{formula:72dad8b9-d80f-44bc-9ef1-c8e295a6b39f}} . If {{formula:c99092db-a941-4bc7-89b8-c5f63149581e}} is odd, then {{formula:58c1d998-cb2e-450e-9d47-dba8bbd996a5}} . This does not an integer solution for {{formula:0c017903-3456-4f0e-bd0d-4f5505c65886}} . It is a contradiction since. Or {{formula:b062cb86-279c-46fc-9e83-9f133952064d}} . It is a same contradiction since {{formula:2103ea8a-940e-4b85-8fe4-26dc9be9d309}} . If {{formula:3c906286-7e03-4dd1-acd2-827cd071c596}} is even. Then {{formula:ffa73a17-c02c-4b3a-b533-8f8951d60037}} . It is a contradiction since {{formula:beb9b12d-c27f-4b94-a6c6-d99446520e69}} . Or {{formula:92788c4d-08a2-4ba9-a64c-afc4e0e12347}} , it is a contradiction since {{formula:1d301a61-20d9-4936-a8e1-e62e713f59b2}} .
Suppose {{formula:dd81f524-cc76-46d7-a52c-0779deda39b8}} with {{formula:6ca7a335-dc53-42dd-967f-bc22830d7cb6}} . We have the same contradiction with {{formula:4c3b0caa-453f-40fd-807b-f141f45425d9}} as above. If {{formula:00f4e7a7-c1b6-4c34-ae6a-cc0f5544952e}} is even, then {{formula:3e8a3f84-a79f-42e4-9924-8ad6c7dc0bfd}} . It is the same contradiction since {{formula:fec6c0fc-a368-41f3-88a7-3c5556aec0bd}} . Or {{formula:97adc931-df07-42ae-b3db-2408213f6524}} , it is a same contradiction as above.
(g)
{{formula:b077c836-3acb-4eab-b81d-cb51ee128c8e}} . We note that {{formula:84175b27-2719-4ce4-9001-e7da706e23d9}} since {{formula:0b22bada-5723-41f8-abd8-9e87244354e8}} .
(h)
{{formula:56e2f283-c1cc-4b31-a147-d240f75ffec4}} .
We note by observation that {{formula:52bb738c-1349-4902-9465-2588edee92c5}} cannot be isomorphic to {{formula:bb1f4212-ff9a-4210-8b32-7d604a42744b}} , {{formula:891b5534-ca13-4f54-90e2-37ea9a444239}} or {{formula:49e14367-f5b7-46cc-8e6a-cfef0147f3e4}} . we have {{formula:f87cb657-c3c0-4f25-bcfb-944413a8e325}} {{formula:93d4b038-42ab-495e-9477-c868eac1dc47}} {{formula:ce2e4ac8-aeb7-4fcf-aee3-7fb13349950d}} or {{formula:f5cc1374-9c50-4f9c-bce5-c898c44a9c9e}} for {{formula:708cf8a4-f860-47b8-bdcb-737e3f3a0236}} The codegree sets of {{formula:a4715788-271f-4900-8e14-dda585c411be}} and {{formula:42da2673-ac30-4d1b-8f2e-4a1e44898874}} contain {{formula:c20736e1-114d-4bbc-9b8a-893a8963bbe4}} , {{formula:243ed89a-f4b8-40a6-afee-a55f1625beb0}} and {{formula:ac876174-793b-4222-877c-4ba9e01279cb}} respectively, none of which are elements of the codegree set of {{formula:4173dd53-5b9f-4e07-b3b6-c5e86a8340f0}}
If {{formula:0e706b92-2293-4a44-9a27-db8d2df4746e}} , then by comparing the nontrivial odd codegree in each set, we can't get an integer root for {{formula:71ef7437-fb41-4fc9-85c6-08e88e2e0081}} which is a contradiction.
(i)
{{formula:0e77c954-90ce-499d-87a2-f693caeca3fd}} . Then {{formula:e08a4e10-859d-4ef2-8cf2-e1ddcd08fc27}} . We note that {{formula:fddae96f-171e-4c7d-bd94-0d79affc4036}} . Thus {{formula:2eb22a13-55ae-496e-9380-d936508baba6}} .
(j)
{{formula:ed369639-19d9-48f3-887e-76a48dbd0c87}} . Then {{formula:818145c8-ccd9-448f-8e0d-15123256e3eb}} or {{formula:595b44df-61c1-4f3b-9bcf-6056ec5cc25f}} , for {{formula:01b7ad59-aa0a-492c-bb11-98f98b459dcb}} .
Suppose {{formula:418a5283-72dc-438a-a28a-ca9fdaec2273}} . We have that {{formula:dae5e72d-e79e-470b-9b82-19a98f28c609}} by an easy observation since {{formula:787a234b-9cfa-47af-b40b-a7e272426a10}} .
Suppose {{formula:e0fd5c97-8c2c-4468-af32-d4a1e6d09a06}} , for {{formula:3f9ca8cf-65ad-4a42-857d-68ef1ec0f21f}} . We have that {{formula:95e91e06-ad23-4b12-af35-db4fdc4e8782}} can be equal to {{formula:0608392f-40a2-4a76-aa3b-6c9104750872}} or {{formula:7bcdd83e-0821-4c60-adf3-e8a5983db3f5}} . If {{formula:0654e49e-b335-4fe1-8d75-25fd24b0a12f}} , it is a contradiction since {{formula:95d67076-41a7-4474-8e75-f95938a5f241}} is not divisible by a square of prime. If {{formula:6150f91c-38ac-485b-b414-a900bae4145d}} . We can't get an integer root for {{formula:813c773b-af77-4e5d-a2f7-3b794f96c1e1}} . This implies a contradiction.
(k)
Suppose {{formula:e53ddd19-5004-4f9f-b86c-b850cd2edfe6}} . Neither {{formula:623d84a4-7f08-4774-8a1d-f391839bd533}} nor {{formula:8e7cf9be-d5d1-4667-8332-b344aefe1402}} . We have that {{formula:b348f427-0ae2-47a3-8fcb-2cc374116ad5}} nor {{formula:08e8abb1-c73d-4ad7-8fcd-8bce640c3594}} .
Thus {{formula:ef162694-f5e9-48d9-a6ef-54d155615860}} .
Lemma 3.7
Let {{formula:0ab41679-1183-40d0-bdb6-df531be9af94}} be a finite group with {{formula:dc1d4168-65db-468c-873a-339be7146cac}} . If {{formula:7307eb01-c105-4610-8eac-a784d433c981}} is a maximal normal subgroup of {{formula:526ffe52-a3f6-4b86-933f-4f4b2121ba22}} , then {{formula:2c688ea1-9f00-40bc-bcc0-d511d93dfc45}} .
Let {{formula:c691180d-5a31-42a6-98d4-99a34bb82f9e}} be a maximal normal subgroup of {{formula:570f87f2-0e15-4c61-ad34-706edf5ad901}} . Then {{formula:79d5aa9f-f407-4bc1-8685-c0520396b32d}} is a non-abelian simple group and {{formula:2ed7114e-b154-4e41-9cca-c87a8454de36}} is either {{formula:167d2e5a-0c89-492b-87e4-a3b3e14d0c90}} or 15.
(a)
{{formula:cacbaa1e-0041-4df7-97e3-9cf7bf7c0e54}} . Then, {{formula:03631a90-ca4c-4318-ab29-bc6802afc37f}} where {{formula:0f259a52-8db4-43f5-b5b6-e9c01fe2fe47}} and {{formula:eec8bb07-7a7c-4f0f-bf3e-28ddcf59b264}} . Hence {{formula:ab2e9cd1-15e7-4354-bc27-08addea070e4}} , {{formula:98dfeb39-ca9c-423e-8572-f86a4fe8de1a}} or {{formula:5522c969-23c5-4c03-a2fe-aa3c417b8580}} since these are the nontrivial odd codegrees. In either case {{formula:f9109e8e-e1a4-43b7-9263-094a223b6ca6}} is not an integer which is a contradiction.
(b)
{{formula:439ca7fc-7150-4a27-baf0-e5c243c0876f}} . Then {{formula:1b0f84d0-f7b5-47d3-9d5c-b0440888df2a}} where {{formula:28ca9782-5181-478d-976c-c11330bafb38}} is an odd prime power and {{formula:af89a000-dcc1-4cbe-adba-c9e081f47319}} where {{formula:34d7836a-fecb-4168-a29d-225e051a3b76}} . Then {{formula:7a50184d-51dd-4cf5-8200-e47e4198d4b2}} , {{formula:092afba9-df0d-4d69-a6d2-6c58b8da0d93}} , and this implies that {{formula:a1252404-b8e7-4b5f-af66-8a8a1b01c189}} or {{formula:aead08b3-4d96-4a17-a00b-f86df8b394d9}} since these are half of the other two elements. In either case, {{formula:b03c3f32-93a2-4100-be50-ad3fb13da877}} which is a contradiction.
(c)
{{formula:9175ba6d-0838-47f0-86fd-7c14ef03ae31}} .
By observation, {{formula:235b0f4c-8241-43ba-bc14-3ad8d110e54a}} since {{formula:e4a66755-e30f-4d4c-ab0a-113461d801f2}} , a contradiction.
Suppose {{formula:07c5fecf-9a7a-4a5e-8083-0f77a64f8580}} . We note that {{formula:97788dd2-23f4-4950-aca7-6ad2d0b5833d}} , {{formula:a55f8037-9935-48b3-980e-0b2a38468bd0}} or {{formula:756199a7-ee0b-445f-bbc5-b31f82c83efc}} since these are the nontrivial odd codegrees. We note that 3 does not divide the order of the Suzuki groups, a contradiction.
(d)
{{formula:2c0d2ce2-b979-4e02-bbc1-377dba9ddcf0}} . We note that {{formula:26a9020e-d4bc-46d0-acc8-01d05d0bcfcc}} , {{formula:55b0faf8-5c94-4f7b-9d6a-c98b9baf633f}} , {{formula:8322b3e1-08c7-43e3-baca-abc60ea7bdd5}} and {{formula:2c4ba1cb-9746-4c11-9aa0-0d27f9084535}} . None of these codegree sets are included in {{formula:d25c4773-4924-4e1e-ac3d-821229b55bfd}} .
(e)
{{formula:e598b61e-d942-4c2f-8e2d-c39bbc1bb581}} .
If {{formula:1d278d13-5e73-4874-81b6-94d8da72140f}} with {{formula:b34ed290-a6f4-401b-96ea-7b48aacd8301}} , then if {{formula:3a0a431f-5de6-48f6-b7e1-5317dae39230}} is odd we have {{formula:a14b89bd-8220-42e2-b446-a7e44d49c209}} , {{formula:47f87036-b210-4034-9cc0-4eab905a2f48}} or {{formula:bf9704f9-6b16-4e2c-9a07-66643390e417}} . These equations can't yield an integer root for {{formula:8a486e03-f1fd-4bf1-a4d9-40c1f8d6da09}} . We have a contradiction. If {{formula:00a19608-a8bb-4fbc-91f9-b217ed75c1bc}} is even, we have a contradiction for the same reason.
If {{formula:13c5d030-b19f-410e-8027-86532bb49d98}} with {{formula:24590625-b0d0-462a-840f-24c2a33564fc}} and {{formula:190b552a-436a-4df9-b0af-681c7792e03f}} is odd, then {{formula:0f9bf50f-3fb7-4c4c-a59b-2288800c2ae9}} , {{formula:d6b7de0e-673e-4158-99aa-f6d74845cb88}} or {{formula:51cb1726-38ce-4e1d-9dc5-0db1d4b39764}} . We have that these equations don't hold as above. If {{formula:fba078da-23b8-4f45-b8be-05405200df8b}} is even, then {{formula:b6d8e0d5-c085-4c53-b46f-b825ff67d8d7}} , {{formula:c842af58-2b7f-4c99-9f18-3b02edf54a0b}} or {{formula:e6e05fd0-c9b7-474c-8e6a-9db05410496e}} . We have the same contradiction.
If {{formula:ce07bda3-5a49-4567-96bb-f71ad69018ac}} we see {{formula:9c4af308-d51e-45a0-82b9-401a255e6fc6}} for {{formula:a53552c7-e8d6-4389-8bfe-5956466d14ce}} . This is a contradiction.
(f)
{{formula:adb028dd-15e8-4293-93f2-917c20bf0750}} .
Suppose {{formula:d3aeaf23-6112-4e26-8a3c-afe7b6294fc1}} with {{formula:50b0c5c3-7f25-43c9-a934-a33a42c9b69a}} . If {{formula:6c828e7e-5263-46cf-bfbd-fee168acd361}} is odd, then {{formula:345814f4-0fbb-4d8c-a269-6a06bd488ea5}} , {{formula:f31ad77e-17d5-464c-8272-fc4a0fe286c5}} or {{formula:441df4d4-82a1-4a92-9a78-7d26093c3208}} . This does not yield an integer root for {{formula:fa55c8f3-5cac-4bd7-baf9-895d765675be}} . It is a contradiction. If {{formula:8ad58d1a-bd16-4ba9-b728-796d579710a0}} is even. Then {{formula:2fe0b6ed-c2d8-440e-b1fa-8e08b89af109}} , {{formula:35f05f8c-df9d-4a4a-b575-e4a78fa4ed3c}} or {{formula:bdb28fc6-403e-4b77-bec3-f5e39041d265}} . It is a same contradiction as above.
Suppose {{formula:85b8c625-c5b6-48e9-9011-d31a86cf5df5}} with {{formula:cbd83d33-a091-4c5a-87d9-701fa4bd58a7}} . If {{formula:2f42d423-22cc-499d-ab30-c862fb850b0c}} is odd, then {{formula:b40e4669-874a-44f7-8b6a-4646e7028039}} , {{formula:48c820bd-ad90-4b22-aad7-ad64998b5c20}} or {{formula:ad07396d-5374-48bd-b861-91105c134b8d}} . This does not yield an integer root for {{formula:f6308349-d88d-46b5-9254-99a803f292af}} , it is a contradiction. If {{formula:543807cb-9d8a-4aff-a065-dc14d2cacb32}} is even, then {{formula:ca14dd2f-5e47-4bf6-88ea-bb5b2ffaf010}} , {{formula:b4571b25-29bd-4584-a96f-1e2c9c00798d}} or {{formula:9e3c8285-fa41-4f18-9531-ad244ac8b5ed}} . It is a same contradiction as above.
(g)
{{formula:efd3b06f-1c3a-4a09-8820-0fe3c1bc4b5b}} . We note that {{formula:98b31322-5997-4796-959f-1c775c4249f1}} since {{formula:61b12991-b05c-4b8e-b08d-f80693d03986}} .
(h)
{{formula:94c4d194-150d-4b50-bb39-b3c258e9d183}} . We have {{formula:ec867163-8cf5-4704-9b1c-a8fb1fc6bf90}} {{formula:0841f5b0-bd5e-4908-b398-3ac4da89f73a}} {{formula:5b9abcfe-3b4b-4ecc-b219-c920910247b4}} or {{formula:f675f5e8-e8ab-45d1-9b5b-535932e9f033}} for {{formula:1c5d30c5-1401-484f-bd23-4c4480ec0bba}} .
The codegree sets of {{formula:ac2bc9d8-eff0-4496-9f7c-bb3a167bf566}} and {{formula:3e36e5b8-c0b4-4f2a-9da2-127dd367381f}} contain {{formula:3d19f00b-05c0-440c-8bfb-04c8bb256220}} , {{formula:21a8ed8a-87c5-49d9-8f8a-1d9d51ff97aa}} and {{formula:2a39bb4f-e0e4-4fb9-bb91-def1e980a1d3}} respectively, none of which are elements of the codegree set of {{formula:2b2ffc7d-0aec-4ca8-8cb2-b26e9056011f}} .
If {{formula:805d034c-6c6c-4f46-9692-775e48c2a585}} , then by comparing the largest 3-parts, this does not yield an integer root for {{formula:1e26ccd0-f26d-48ca-be9f-ebca5595cd95}} . It is a contradiction.
(i)
{{formula:50ef0f1e-e225-4fe0-b776-5e1a750088c2}} . Then {{formula:349f6c89-145b-4f84-893d-ee137c8afdd5}} . We note that {{formula:9e429cfc-0318-4fa0-9c43-e5f25e1adc81}} . Thus {{formula:180c3cb0-fc36-494b-b3cb-b5e97fc4599e}} .
(j)
{{formula:b31fb4fd-1755-4f0a-baef-1c368e5abb16}} . Then {{formula:4e3c913a-bdf1-447d-8bfb-d452fc491115}} or {{formula:3405b0c7-99ed-4ffe-8cc2-866d454e6022}} , for {{formula:841dc21f-73b0-46f4-9212-25bfbf298794}} .
{{formula:54861a2c-bf97-421c-855f-128d01dad1c0}} , we have that {{formula:e928a3c1-053a-4724-9144-745a74d85936}} since {{formula:5007ecc9-605b-4642-8db0-cc803be20fd7}} .
{{formula:41445fa2-f8b5-4234-aa5d-b833f11636de}} , for {{formula:1c484d36-84a5-4b3d-9ed9-d8af703c1d91}} . Again by setting an equality between the nontrivial odd codegrees of each set, we obtain no integer solution in these three cases, a contradiction.
(k)
{{formula:46af4874-c7fc-4c3b-987e-80ac1ab6713f}} . Then {{formula:7c5a8086-2144-437d-9a7e-ede11b608b52}} , {{formula:04d81c2c-c804-453d-9a81-66826a3082ba}} or {{formula:c767ee88-9893-4644-980b-85983da03498}} .
It is easily checked that neither {{formula:ee856cd9-36b0-4649-bbd9-2e37f1bda4a4}} , {{formula:5be99558-e8ee-45b1-b9ba-45f83a5c9e24}} nor {{formula:94359520-65dc-45f8-a939-f825af24936f}} is in {{formula:11241583-494d-4dd3-a040-a29e0517ee68}} . Thus {{formula:647704d8-bf5a-4bfb-8a1a-722d7b89f395}} , {{formula:a267847c-4a7b-47d1-8448-1b40ce6db96d}} nor {{formula:1e2f5873-bd33-4a8d-b108-3a5497f75966}} . This is a contradiction.
(l)
{{formula:2c478257-ba62-45fc-8150-694bc9c6374f}} . We have that {{formula:8a890eb4-f534-4e4a-aa4f-430f4db0c13c}} by an easy observation.
Lemma 3.8
Let {{formula:d18a509b-8ee1-4649-b8d6-e19eadcaa73c}} be a finite group with {{formula:318bdd7a-c489-47d7-ab1e-446f252084bd}} If {{formula:f1eb199b-7ba5-4df4-ac2d-285343b4640f}} is the maximal normal subgroup of {{formula:a6677d1e-159b-4c10-aaa7-b8f7d8b70076}} it we have {{formula:267bc660-285b-4702-9913-7d9784c6beb1}}
As {{formula:d8dcbab6-bbec-44b6-b0ab-967ce08f77cd}} is a non-abelian simple group, {{formula:72be5095-406c-4312-a232-6ea69c72b46b}} , so we have {{formula:ce32194f-f18f-4be6-9c98-b1411ca89b08}} or 16.
We will show if {{formula:42f1a6e2-e915-40a6-b4a6-7ab6488648a2}} and if {{formula:d4b674de-74de-4c26-a0ea-9a2fb3637498}} , then {{formula:25ec1f9e-fc9c-4e84-87d7-ffc17f806133}} . Assume on the contrary that there exists a maximal normal subgroup {{formula:ed72a506-18da-43e2-93d8-b9bba6c38ba2}} such that {{formula:47f36cc3-0466-4bf9-aa3c-3f3438f0a277}} , and {{formula:f9a46ffc-69f3-4e72-abf3-e0841e874597}} and {{formula:220262ae-7d81-4ea6-83d6-19ca46ec1f41}} . We analyze each case separately.
(a)
{{formula:b3983080-6f7c-4c49-8faf-07fef0969163}} . By Lemma REF , {{formula:f48c575f-8a66-43a4-9b79-0b604f50f165}} where {{formula:b7810bd7-8841-4db1-b4b4-147147b9d57d}} so {{formula:bd578a62-9e23-43a3-a097-29f736765fbe}} . For all even {{formula:bdcd4269-deac-4dc9-a31c-d017fdbf9b3d}} , we have {{formula:7742f1b2-799d-4c31-a0f5-ddd5a454dc8f}} . However, there is no nontrivial odd codegree in {{formula:ad27036e-136a-49ed-b537-d9099e1c8eb2}} , contradicting {{formula:6a5735a5-6019-493b-9681-413faad7ef28}} .
(b)
{{formula:e107e59f-8980-4225-9e38-334a181b4ae3}} . By Lemma REF , {{formula:7102389a-1dab-43c0-8360-efc6a6cc3a1c}} where {{formula:38dd17ad-0772-4fb9-a149-36fdc324893d}} is the power of an odd prime number. This implies that {{formula:e8d2ce82-41e7-4088-8663-77aeb62d966f}} where {{formula:3d134dd3-f140-4f15-af4c-6f5b300d5cd9}} We have that {{formula:a0957dc8-dc03-45ed-9a25-185ee18d941d}} or {{formula:8f073b3f-3b19-4eee-a996-bf6779dc4114}} , since these are half of the other two elements. Contradiction since {{formula:818733e9-b125-4f1b-ad4d-155dd0cc33ee}} .
(c)
{{formula:add62900-a99e-48c9-b908-8f47308b2f5b}} . By Lemma REF , {{formula:802f9f00-b95b-414f-b715-182b5badfbb4}} or {{formula:f29e4bec-bad0-4337-adbd-67833c4ed592}} with {{formula:6c04f7c3-e163-4990-a5f5-3ff0ea192e66}} .
In the first case, notice that {{formula:33d092ad-3805-4d56-943f-613bb1a45dd9}} is the only nontrivial odd element, but there are no nontrivial odd elements in {{formula:b4d81b44-b637-4c97-8a4a-b7440c7c1bb7}} , contradicting {{formula:fc401ba0-52af-4d51-9716-9013a60fe3e5}} .
In the second case, notice that {{formula:17213333-73a8-4cfc-8389-48df0716c594}} . As {{formula:94c7047f-97b8-467d-88d3-6cfe7a29273e}} is even, {{formula:a9dc6f08-46cd-41cd-9a38-4d9a5e4351d0}} . However, there are no nontrivial odd elements in {{formula:0f6d6f58-3f43-48fd-84c4-1423dff2c044}} , contradicting {{formula:6e668456-6f7b-4c59-a5c6-975e709bddb8}} .
(d)
{{formula:14688f29-e98b-4a94-a4aa-eb3d3b25930c}} . By Lemma REF , we have {{formula:6ac8ebf2-3860-413d-bcfb-7b71b4a24bfe}} or {{formula:c8ee3b47-7c82-4146-999a-82a30e5ea36b}} . Notice that {{formula:4f5709e2-c918-442e-979c-42b9bea991f5}} , {{formula:77a840c5-07a8-44ee-a579-bb1f99ffd5b3}} , {{formula:d7f9648c-a211-4b96-812f-27c4b3a4dc2d}} and {{formula:b8dc7414-7484-40a0-9646-438020ae282e}} . By inspection, we can see that none of these elements are contained in {{formula:277c0027-7af2-4605-bf07-35ac6ab24281}} . Thus, {{formula:eeb481fa-4625-44e6-8f38-74b4172003e4}} , a contradiction.
(e)
{{formula:8b3ba914-71a5-4229-952b-b01628127aef}} . By Lemma REF , we have {{formula:64831427-b09d-46de-a048-adc227b18207}} with {{formula:2a0307c1-9c3f-4d24-94f5-844d5602a49d}} , {{formula:e8705236-70ba-413e-98fe-3db18a962465}} with {{formula:71e17961-5f44-492c-a415-be6122c3ad39}} or {{formula:3f4815c9-0be4-4e03-a215-d93fd82f1e0e}} . Observe that the only nontrivial element not divisible by 3 in {{formula:807261cd-e737-4919-9dbe-b4a8a6f69db3}} is {{formula:10d880ef-8ca2-4927-a653-bb9a3c09c32c}} .
In the first case, notice that {{formula:2d3b7b15-0aff-421e-8543-beba915db1fd}} . If {{formula:e52ad2bd-2b20-4bfb-abcc-a10407724917}} , then {{formula:0df4d615-c036-4de6-805e-fe6b98780346}} , so it must hold that {{formula:da63fe2a-1aba-44c6-92d2-64ab3216252c}} . However, this does not yield an integer root for {{formula:546a3d6b-0b7d-4241-8ea4-dda311688d16}} . Alternatively, if {{formula:653bef09-e596-467a-9114-62d2e8502665}} , then {{formula:1ee9b417-a35f-4bb8-b2cb-349018bd557c}} , so it must hold that {{formula:b8d6da33-aebf-4b23-8d59-a65acd4378c8}} . This also does not yield an integer root for {{formula:bac2fa5e-1f0e-45f4-b8ce-ddab0f61f714}} . Thus, {{formula:9d285bae-2abc-405c-b692-f5e1e1867f09}} , a contradiction.
In the second case, notice that {{formula:5a95117e-1158-4adb-89e1-f8d9b346b212}} As {{formula:fddeb751-d226-4b67-8223-741380984dc4}} , so we must have {{formula:1bb424ab-c493-436c-9d63-d72fdb98a656}} . However, this does not an integer solution for {{formula:d205a2ac-169e-496e-b538-440d63388fd1}} . Thus, {{formula:493e0d6b-79a7-466e-9678-19f567fbbab3}} , a contradiction.
In the final case, {{formula:2294289f-a2a2-42be-9f7a-85fe6ec27aab}} but {{formula:a9037c43-8d7e-4d37-9331-6ce724aa3248}} , contradicting {{formula:d379d297-6d63-41c1-ab3e-a45defeb8f92}} .
(f)
{{formula:b6589869-0da5-42d9-8c5c-cd9edee33871}} . By Lemma REF , we have {{formula:a1949394-7f6d-486e-bab2-cb19def6d8f5}} with {{formula:12ef538f-a0df-46e9-8ff3-fe5bc7ec5d62}} or {{formula:8dec8d9f-6b71-430f-a26c-b4235436362d}} with {{formula:09bb0508-06a5-468c-a8be-106a674b7d54}} .
In the first case, notice that {{formula:40448386-d9ec-4d5e-ba30-e2d0f2359313}} . It can be verified through a complete search that {{formula:1ed03d2d-ee61-48c3-a1e8-717ae43f3d64}} . Thus, {{formula:25efaeae-0011-44e6-8e6a-14fa711d31c7}} , so we must have {{formula:621ad94d-8ff6-4715-8b3b-3eeec650ed68}} . However, this yields no integer roots for {{formula:e4fefebf-aaf2-4acb-a4e9-d466a3307586}} , contradicting {{formula:04e8c99c-b621-4659-849a-3351739c6f25}} .
In the second case, notice that {{formula:0c79d4f1-031d-42f4-8fda-e9128596cc71}} . It can be verified through a complete search that {{formula:e1785b43-cd07-45ca-a0d2-f75a072c2279}} , so {{formula:4ce028cf-95be-4773-b6a3-784f2ce45329}} , so we must have {{formula:29bbb5d4-3ce7-4728-9991-cb2fd35dc9d7}} . However, there are no integer roots for {{formula:6418b5ce-a2ca-4bc5-9e8e-95068cc79520}} , contradicting {{formula:8969d7fa-2956-4a15-be16-3a0b9693a375}} .
(g)
{{formula:6be7bfdf-8a77-4e0b-99ab-669304f65772}} . By Lemma REF , we have {{formula:aac80e39-977b-4a09-a310-4f141b4b6967}} . However, {{formula:2b26dec4-f785-4bd9-a423-cdc0e3df7b24}} is only present in {{formula:0127046e-4829-4bc4-b965-e132d3d29ea1}} and not in {{formula:7e9015df-72c5-4eec-bfbb-ca1a2706e89c}} , contradicting {{formula:abb9b52f-1b8b-456e-a2c5-32d89196519a}} .
(h)
{{formula:b65d9335-042a-4493-aa70-85e50a2ccf59}} . By Lemma REF , we have {{formula:202515f2-18b0-44c8-a714-a6217d187c39}} , {{formula:e9508667-fd50-4612-9e8b-56ac8ed5e9da}} , {{formula:2af17ade-b470-49e7-a5c8-5fd30b3b3c50}} or {{formula:fe4c780b-f606-4c5f-b74a-7ff19e22f742}} for {{formula:79565945-6e42-410b-99ef-5ee1e1d01a97}} . The codegree sets of {{formula:53857d5f-7f2f-45b1-b8b1-e2c80e0da97a}} and {{formula:a8c8f8b8-3647-48dc-9d48-eeac605f851b}} contain {{formula:d6d6b3ae-911e-4713-9744-b3d1f0232509}} , {{formula:fb441643-186b-467c-9e90-8917bc554ee2}} and {{formula:5fbe8bc4-7839-479a-9573-ed9f86ee44e7}} respectively, none of which are elements of the codegree set of {{formula:38d9df79-b634-4fdd-aeff-9fa315a8c260}} .
Notice additionally that {{formula:f31b53a9-bb7d-40a1-bb20-410f2ef8f211}} . However, {{formula:08bd2864-1f05-40c4-897b-1ed6b97df5f9}} when {{formula:05f270c5-6264-4bff-b300-263d119e8e36}} is a multiple of three. This implies that we must have {{formula:4f9cb267-ead1-4ed9-b060-3d68d088d04c}} , yet this does not yield an integer root for {{formula:e57bc07f-81bf-4fe7-8d01-4b870a99b01e}} . All of these cases contradicts {{formula:cf90d794-05ac-488e-93f9-e988faa8cb7b}} .
(i)
{{formula:4245bedb-7c12-4316-af95-cc034dd09c06}} . By Lemma REF , we have {{formula:683ea218-9f92-4db3-9b1b-f34627ac0d34}} . However, the element {{formula:974b8ca9-1d99-49ba-b8de-7e5c331408cd}} is the only nontrivial odd codegree in the codegree set of {{formula:9f716f10-07ec-4b97-a312-f0f5d5639423}} , it is not in the codegree set of {{formula:1be3e508-cb17-4273-ac6d-b995fa40645b}} , contradicting {{formula:22e78f45-f4b2-48c4-a6b9-d6c2a8334d45}} .
(j)
{{formula:4b9c88be-f767-4bd5-aca4-a6402328cbc8}} . By Lemma REF , we have {{formula:99a9a295-3f75-428c-a948-a89a94550bfe}} or {{formula:23a2dd74-e6e0-4d6f-a719-2de8125031a3}} for {{formula:8142073f-65a0-49a8-bb11-1ba2899944d7}} .
In the first case, the element {{formula:3459efb6-34c2-4474-bb38-26a7216209c1}} is only in the codegree set of {{formula:2cf09465-60b3-456b-83d1-48f765e1c540}} and not the codegree set of {{formula:d7db11b3-b890-4b2d-887d-af08e2ae3037}} .
In the second case, {{formula:615bb8a9-1a50-41b1-a88c-b1f2b25b2520}} . However, if {{formula:89ad8584-b4be-4134-9083-cc4664e7616d}} , then {{formula:872ce621-9f7b-4a8a-b810-9af721a7ebc5}} , yet there does not exist an element in {{formula:f1062918-be82-4c6d-be10-a13bacd891c1}} that is divisible by {{formula:989c3d54-f7c8-42c3-b7e7-503d4e594176}} . Both these cases contradict {{formula:c7e98414-713d-4966-a8de-0492421a7f19}} .
(k)
{{formula:c4e480ef-c0e8-47a5-86f2-c1638c7e7ebe}} . By Lemma REF . If {{formula:f988dbe8-79bb-4b51-b7f7-17f1c3fcc858}} . We have that {{formula:550c8d4f-98db-45d2-9b8c-d0f0f79656c8}} and {{formula:20b7904b-688d-495b-9b3d-34853fbea914}} are both the nontrivial odd elements in {{formula:60a2ea18-7664-4709-9c3c-7862d00e6cfd}} , there are all the nontrivial even elements in {{formula:e39ab09c-bd36-4a00-9e00-1e42a54a5281}} , a contradiction. If {{formula:4158edf5-9437-41ae-91b0-b4d5cb1a421a}} or {{formula:fd9628b5-429f-4b79-920e-27263ad304da}} . the codegree sets of {{formula:2be62208-c47a-4ecd-932c-a6cdc307a651}} and {{formula:070614dd-ecba-418f-84ea-5a7d59280926}} contain {{formula:4579d51f-f31c-4449-b36d-4358df219335}} , {{formula:7c3c5518-56e2-4067-ac5d-fc9dd94b2db1}} respectively, none of which are elements of {{formula:b2d39435-3b1c-44c4-8851-5b4843bfaf1c}} . This contradicts {{formula:138c3cda-1db8-4d4d-9156-9841c46c3095}} .
(l)
{{formula:ff834327-635b-4374-809e-d4eaa7f67266}} . By Lemma REF , we have {{formula:2c961e8d-5ecf-45a7-ad09-dec4c545a4e9}} . However, there are three nontrivial odd elements in the codegree set of {{formula:c86a879c-b2ac-40da-9848-604e6ddb539e}} and not in the codegree set of {{formula:18703b7b-8948-4733-ba03-9922d7a5c33d}} . This contradicts {{formula:4e823e14-3283-4e0f-86f6-0760d2e83bf2}} .
(m)
{{formula:e67d2f3c-f71f-4fbf-ab4b-ebb9f64f305e}} . By Lemma REF , {{formula:fcd511f4-ddb3-4d52-80ca-6f777d56ca94}} or {{formula:a65cb753-ab58-44f2-bd43-53c506329c51}} .
Notice that {{formula:58bfb0b9-ef9c-4d96-97a5-81200546c4bd}} but this element is not in {{formula:3dfbe433-0ed2-4729-b045-66efa01eba27}} . This contradicts {{formula:c34f62f9-d466-4da1-80c1-558ff2f3d874}} .
Having ruled out all other cases for {{formula:70bb2bf5-2e30-427f-b5ce-f056e60f0141}} with {{formula:169989fe-1ab0-48e7-b0c2-7b16ce5ef877}} , we conclude that {{formula:76cac9a2-bd12-44e8-bc71-59e59fc525c2}} .
Lemma 3.9
Let {{formula:1791f321-ccae-4614-a9f5-5f9450ac85dc}} be a finite group with {{formula:127b29f3-b620-4c95-9942-3ae886852551}} . If {{formula:95c7a17d-6e8a-48a8-b26e-3be8779ef2f3}} is the maximal normal subgroup of {{formula:805ab5cb-743f-4306-9228-77ce4c67df66}} , it we have {{formula:24e67439-0935-41cd-88bb-90e7e18a5fba}} .
As {{formula:091fc950-a6ee-4bf1-8621-d4c8ac72239d}} is a non-abelian simple group, {{formula:f4fcb93a-db01-4ddf-baef-8c370c960617}} , so we have that {{formula:6a05aaab-77e4-4c96-b7bc-de8f543d3350}} or 16.
Assume on the contrary that there exist a maximal normal subgroup {{formula:c0dd9ca8-b2c4-42af-98aa-8bda37f3a941}} such that {{formula:2f2d916d-95b3-40f3-8504-1993464ad81b}} and {{formula:cbca5ca4-c149-4fb9-aa52-b17e989bd028}} . We exhaust all possibilities to show the previous statement is false.
(a)
{{formula:ddf1d264-b972-48eb-b385-1cae84d8f215}} . By Lemma REF , {{formula:fbbd8245-af05-4556-a82c-f962c746ea93}} where {{formula:b0c71f70-70b1-4385-8ea8-5f4582fad240}} so {{formula:71967bfd-6f47-48f4-9bfe-184a026a0106}} . For all even {{formula:e19d52f8-3c31-4427-9be4-83d6d4f7759d}} , {{formula:f0383c86-727e-4614-b74b-b74ea03ff3b0}} is odd, yet there are no nontrivial odd elements in the codegree set of {{formula:b8947d1c-4594-496e-b27c-7c19374a988b}} . This contradicts {{formula:f5e02a1f-edb8-4d5a-b490-1e308a024b34}} .
(b)
Suppose {{formula:8d067a9c-3e11-4d12-aa30-2f8df657a5e9}} . By Lemma REF , {{formula:32d174ec-ad36-45ff-8120-87e2e139e94a}} where {{formula:e34742b4-0775-463a-8f35-a8ad1d35c201}} is a power of an odd prime number. This implies that {{formula:bc3f586a-3640-49e0-bf63-91200a71b46c}} , where {{formula:8a0fa010-7d67-4bbe-9ba8-783bc7f2a5b1}} .
Then {{formula:14f7f74d-042a-4ab3-a091-c349fcfce455}} , {{formula:8ec3bff2-e58c-4fa2-8d6a-26cd75f8bc1b}} , and we have that {{formula:3b904379-116c-44c7-a299-e36d1c53f72d}} since it is half of the other element. This is a contradiction since {{formula:628e15b4-1ec0-44ec-9f2e-74ce8910da08}} .
(c)
Suppose {{formula:fb19bc84-6685-4426-86f5-ff5e919372ca}} . By Lemma REF , {{formula:257d59e8-1d90-4a10-875f-8e26fc2fe174}} or {{formula:45cb3c51-0415-4d52-8200-29df52652613}} , with {{formula:d043acb5-b6b4-4f51-a57a-371a066d85b4}} .
If {{formula:420416d7-8954-4e38-a433-806f6abab13c}} then {{formula:510c5921-57c6-4349-b2b9-a55063ae09e0}} is odd, but {{formula:d9b0e7c1-073c-4f42-8b86-9a69e84f3cbd}} , contradicting {{formula:3bc4af93-77ac-4872-a685-bf914acb92b5}} .
In the second case, notice that {{formula:efe42673-47f9-4094-ab0d-e360bccb8a7d}} . As {{formula:68fdffa7-04b8-42a2-a72a-36123bf70556}} is even, {{formula:81fd9e0f-52b6-477a-a005-95fe69cbee29}} . However, there are no nontrivial odd elements in {{formula:4616ce19-dffb-455f-9514-c49b3f1dbe80}} , contradicting {{formula:05c92999-cd1f-4576-8db4-a88d5037be7b}} .
(d)
Suppose {{formula:9808f2dd-e6e6-43f9-810e-df34f6a070e6}} . By Lemma REF , we have {{formula:5ddc2481-30bb-4533-b28f-04670ca50f3c}} or {{formula:afec337c-031e-472a-835c-1d23bbd9535b}} . We may see that {{formula:6d2b9820-3cbd-4ab2-94ef-09386dfbc244}} , {{formula:b45c19fc-4fc2-4361-b877-077cd31bce3d}} , {{formula:33c8437a-d4c2-4e2f-8322-ab42878acdb1}} and {{formula:7fbe6c59-a4bf-464b-9e5f-dcd3b38d7524}} . However, none of these elements are elements of {{formula:5b1e2baa-6126-4e40-8950-285bd28bcce3}} . Thus {{formula:13b2a8bd-ce20-4247-8f9c-f30b83b5618c}} , a contradiction.
(e)
Suppose {{formula:838e193f-16d7-45d1-98c0-dec11aeeba7c}} . By Lemma REF , we have {{formula:c4259635-38d1-48e8-833d-678b4d94c95e}} with {{formula:ee0a1971-a402-4591-b521-5fbce4a5299d}} , {{formula:2334bdf6-1fd9-4e9d-8278-834c5b950812}} with {{formula:e7722f26-d0a0-42f7-94b5-28371ce7b60a}} or {{formula:41456f29-aa78-463f-baaf-348cf8fd6c95}} . Observe that the only nontrivial element not divisible by 3 in {{formula:6871ff5e-f826-4570-83a5-0f9c91122177}} is {{formula:f4a9d978-3b13-4ac8-bbc4-9ddca3bf7bf5}} .
If {{formula:19b56f3d-03de-4793-9d60-564200b13c79}} , notice that {{formula:a0b26ec2-46e2-48e2-8bcf-88d53c8a3795}} . If {{formula:cb53766f-d0ba-4f30-877c-237cfaf5063d}} then {{formula:f62574ae-0261-4134-9e5e-9ced5da09ee2}} , so it must hold that {{formula:d39490e2-b622-43a8-ae13-f759725ef9e9}} . However, this does not yield an integer root for {{formula:115e4220-9aca-4f25-9f80-91d268d9b296}} . Alternatively, if {{formula:388052a3-1c0b-42cf-9402-d6e163700bda}} , then {{formula:d38ac624-ca9f-4d81-984d-2ef646dd86ae}} so it must hold that {{formula:ad767b2a-b52c-4e4e-b19b-137aa85e3132}} . This also does not yield an integer root for {{formula:472c0a6c-ad90-4826-a641-c37b65535a99}} . Thus {{formula:a0431e41-9866-4741-960f-807f4b555f6d}} , a contradiction.
If {{formula:9a9163d8-d12c-44b9-ae3e-5ac924f108a2}} , we have {{formula:43909fb0-b560-495d-9ff6-9be1e67cf81d}} . As {{formula:415f18c7-f265-4e2a-a3f9-50bce714211f}} so it must hold that {{formula:a69b7847-ba85-44f8-be71-5cf6e1378881}} . However, this does not lead to an integer solution for {{formula:4bf9c8fa-703f-48e4-98c1-b648ec9b81fc}} . Thus {{formula:7223af01-dad5-42a6-97df-881ae3b183f6}} , a contradiction.
In the final case, {{formula:2f7f86a7-b0b7-4d77-a005-a30466143a4f}} but {{formula:12496b64-3cb4-48a2-bbfd-82e5e789c05b}} , contradicting {{formula:352012f6-4e91-468c-a032-f12641887f2b}} .
(f)
{{formula:0d2a4e19-f3c2-49a8-9ffd-7f3cecb0752d}} . By Lemma REF , we have {{formula:fcdcd5e5-98c9-4653-9fbb-a8f35ae3c7a2}} with {{formula:f83827cf-6936-4ae1-8915-859f07b5f1ca}} or {{formula:1ba999ec-1585-416c-979b-ae1fd6c07500}} with {{formula:cb28a482-a25f-4b4b-b656-b50fd8f58169}} .
Observe that by a complete search of remainders modulo 9 that both {{formula:8c7e2430-af7b-462e-b347-f864ca7c6d9d}} and {{formula:d833483a-c41d-4567-b66a-a5b5b665a76b}} cannot be divisible by 9.
In the first scenario, we have {{formula:c5ec9004-b517-4f48-97d6-4a0a0a5f8ee3}} . From the above remark, notice that {{formula:18b4d886-c2f8-4d88-9259-951926f786b6}} . Therefore, we must have {{formula:38c73423-5500-4bdc-aa46-bd4fac14e495}} , which does not yield an integer root for {{formula:848f15fd-78cb-41b7-96f7-44b0b83cf402}} , a contradiction.
In the second scenario, we have {{formula:d239f8d6-1f45-48db-8d07-cd73a7d07802}} . As with the first scenario, this element is not divisible by 3, so we must have {{formula:7b9c70ff-acee-4f35-93c7-0df47ccc7690}} . However, this does not yield an integer root for {{formula:b577d85c-f164-4cf1-9fc8-228826da77fd}} , a contradiction.
(g)
{{formula:749440a8-d683-47cd-aa4a-05bfbaba1a3c}} . By Lemma REF , we have {{formula:9ed9a03a-a15e-42c3-bd3c-f817e778f10d}} . However, {{formula:84895fd5-23fb-48a6-8d38-1b7f96aae7f1}} is only present in {{formula:ad9fbf53-0a14-436e-8651-bcc53580e53f}} and not in {{formula:9431a622-1d93-4c05-b0da-210840365d3c}} , contradicting {{formula:8cf4885c-c5b5-4b32-a0e6-a17bd693ed78}} .
(h)
{{formula:869a34f7-5a4c-418c-9a65-da28d88c69d7}} . By Lemma REF , we have {{formula:d3a37982-1e97-4275-9ac8-1c0238168f6d}} , {{formula:75dd120e-08a8-4617-aea4-a4a905687198}} , {{formula:ccce63c6-a6d9-4ff4-8bd7-c410df36c391}} or {{formula:6ffd1c0f-9a36-4f89-93fd-060cb0018043}} for {{formula:a7f05356-2942-4b99-9443-25eb51772071}} .
The codegree sets of {{formula:1d8df113-41e5-4c3b-a1c0-fb68701f7ff0}} , {{formula:d3e605fc-54f2-4fb7-91a0-41c4e3e34123}} and {{formula:453667c1-7369-4666-8bc8-0ce891643a48}} contain {{formula:c4f2e252-db25-44cd-b8ec-984880b5944d}} , {{formula:6b0b65bd-1393-47b0-aad0-a93c79526586}} and {{formula:926d3ceb-2a82-49c2-8ec5-14d69d0df11e}} respectively, none of which are elements of the codegree set of {{formula:002a005c-053f-4c08-a60e-3967f11c11c9}} .
If {{formula:02fefddb-b7da-4b87-94f6-f05de56e8259}} , {{formula:0cea6607-aa81-4c63-aeac-acb3b518a33b}} . However, {{formula:5338e21c-a276-4333-bef1-6e1bf8e9f978}} in this case, so we must have {{formula:27c2ecee-a4f5-4e56-a077-978d7837face}} , yet this does not yield an integer root for {{formula:b8ba1e73-f640-4e1e-8054-488abbd465f2}} . This is a contradiction to {{formula:ff666dae-9559-4290-96b1-1e88c0d3d1e4}} .
(i)
{{formula:651921ba-2815-4231-991a-80aed3c901e5}} . By Lemma REF , we have {{formula:90451d00-eaec-47b6-a1b1-b52d366bcb53}} . However, the element {{formula:6d5a9fad-1b82-4df5-b563-d93d7363d9e1}} is only in the codegree set of {{formula:89a403e4-3146-41ec-a4bc-06480f0779a2}} and not in the codegree set of {{formula:457503fb-bbc0-4786-bad5-76e34d6e9fe1}} ,contradicting {{formula:38a32e17-2e3a-4cce-bf37-32dcd4b70147}} .
(j)
{{formula:6252c7a9-5268-48db-9748-5161cb03be35}} . By Lemma REF , we have {{formula:65682b3d-09fe-4a1d-9c7b-7d5343508874}} or {{formula:893b2f5f-c598-4b6d-9a05-8f70b54a00c6}} for {{formula:1330a3d3-6fa2-48e4-a510-d0578acfbe7d}} .
In the first case, the element {{formula:611cb5b5-1089-40cb-8379-fb9228738a4a}} but it is not in {{formula:858d6920-5351-4e08-a2ca-356e5681050d}} .
In the second case, {{formula:21aded3a-9733-4a06-b93e-19991e57f91d}} . However, as {{formula:e918aaba-a4f3-45cb-b1e6-c31bbd467543}} , then {{formula:631ff668-a65c-462f-8723-61dc0c0162ba}} yet there does not exist an element in {{formula:be27f9e9-aaed-41f6-8f5a-344eaf8e523f}} that is divisible by {{formula:46baa76b-c76b-4a07-b65f-b8844dab0563}} .
In both of these cases, {{formula:67ac330c-5450-4a62-a2a5-c7b52b21d0f1}} .
(k)
{{formula:756e6c23-3a51-4c56-83ac-42a69f4b0ee3}} . By Lemma REF , we have {{formula:b01ff8f9-e783-4f3c-b180-674d02a7d990}} , {{formula:53251678-f62b-47e2-a924-0c8c6f5f489e}} or {{formula:231ceb5b-f057-4970-ad59-21f921bbcace}} . However, the codegree sets of {{formula:e498dbdc-a00b-422a-acb1-7dbcf15f675e}} and {{formula:596c8973-73d2-427a-b5c0-e63a4a5a79c4}} contain {{formula:dc92bd9f-2179-4022-b850-104f460a2489}} , {{formula:1f744216-4ab9-4c8d-892f-7002673fc681}} and {{formula:77e09a5c-12f1-45d8-a134-8dd1b35b3b54}} respectively, none of which are elements of {{formula:0164832b-360e-4ba6-94dd-a87bea1bbb70}} . This contradicts {{formula:0e98d475-4874-4e2c-8ff8-2228bce7e48c}} .
(l)
{{formula:7417e049-86f0-49dc-9961-22833e266068}} . By Lemma REF , we have {{formula:6e497444-cc89-4f08-8215-5d90aa1eabfa}} . However, the element {{formula:79432045-5bd2-4805-91b7-a21a8af790d6}} is only present in the codegree set of {{formula:b47e7dce-e29e-4956-84fb-6c26a61d65ce}} and not in the codegree set of {{formula:964f93e4-7274-4800-a219-a940a9d8851e}} , which contradicts {{formula:dd1928cd-fcb5-4a97-a389-ee531a9978eb}} .
(m)
{{formula:c184961a-3433-4972-94a2-0f809b3d0e80}} . By Lemma REF , we have {{formula:e12de7f6-d4d4-48f6-b530-44f03ebb6051}} or {{formula:1964996a-454f-465b-8880-48550100d5b3}} . {{formula:6300f8c5-c1dc-48a1-8e87-f7f5f42ec8ae}} but is not in {{formula:635841a6-4f26-4a1d-8c1f-4cf2199e31d2}} , contradicting the assumption.
Having ruled out all other cases, we conclude {{formula:9add574c-73db-4f76-8208-c96a3a3cd68c}} .
Lemma 3.10
Let {{formula:1f201a80-abdf-4529-a916-45dbb543964d}} be a finite group with {{formula:0f5a0def-e79d-43e3-864f-e4d3f1324d99}} . If {{formula:56a99c1e-790d-4241-9525-200d8195b1cc}} is the maximal normal subgroup of {{formula:da84ce24-cb62-4475-b5f5-5f28c102c72c}} , it we have {{formula:6c1d0372-db9d-439b-92d1-d6acba3026fc}} .
As {{formula:11a95175-936d-4af6-bcda-1f996fd603ec}} is a non-abelian simple group, {{formula:0366eb5f-de44-452e-9e02-59ac1812b7fe}} , so we have that {{formula:bcb9f672-7feb-4abb-ae0b-5a2df1e4558c}} , or 16.
Assume on the contrary that there exist a maximal normal subgroup {{formula:18ca6632-c426-4ae2-b009-8b78e6b9f947}} such that {{formula:35eac01a-ca3e-402a-9485-274603ed3c3a}} and {{formula:be82f4cd-41c4-48f9-9e01-00bc90313db1}} . We exhaust all possibilities to show the previous statement is false.
(a)
{{formula:6acfc683-c11f-41f9-a9a3-4a4fa8e609f4}} . By Lemma REF , {{formula:2a3f7a1d-5e28-4a66-80a6-763ae20dea77}} where {{formula:ae9d2c5a-4603-4526-bab9-4d109e62ba13}} , so {{formula:98588649-fefc-4d12-91b9-532d758392b9}} . For all even {{formula:00a59013-e12c-45fd-ae4c-b0fe2834faf8}} , {{formula:63aaf758-72c6-4495-95ca-52706526480c}} is odd. Therefore, {{formula:fb209622-0127-4554-8305-c30021495114}} must equal {{formula:648fbbcc-b737-451e-8327-0e1d819ee6a9}} or {{formula:161acfbf-3d78-456d-bbad-80453c54ea15}} but this yields no integer roots for {{formula:ef4f2b4d-90f6-4a04-8bb3-43a601631817}} , contradicting {{formula:49ae4d7d-9dbf-4a23-986d-5233371cba1b}} .
(b)
{{formula:f2b06b3a-92db-4a51-b8f9-69ffb284cde6}} . By Lemma REF , {{formula:ce7a1cce-950d-4db5-ac42-6cdf6f565a22}} where {{formula:c3b8440b-4c47-4bc3-a06c-6d6457c952e9}} is a power of an odd prime number. This implies that {{formula:a65cebbd-7aed-4c45-afe5-06329f852832}} , where {{formula:1685d5a3-c717-4b36-8fcf-85a8fe430b58}}
However, there does not exist two elements {{formula:faf9a405-b458-44c5-941f-6ab49825a7d4}} with {{formula:6ae02577-61f4-4f76-a109-14362191bb81}} contradicting {{formula:c84c918d-02e1-47e1-9e59-61110b9f7e61}} .
(c)
{{formula:7f65cf9e-30e0-4850-a19a-b95f3666df11}} . By Lemma REF , {{formula:e68246c9-d888-4a02-9430-951318658d89}} or {{formula:f0ee5c23-08f6-40e3-b75e-cec9b95548e3}} with {{formula:5fb34508-448b-4002-b4af-ef0b81ec9ee0}} .
If {{formula:24053d61-30d8-46b7-a656-1c88bd8a246b}} , then {{formula:c1270300-32d0-4b09-9a25-f75f9ae029d7}} but {{formula:71078e4b-3a76-4e49-bc1e-cc4211d83aff}} , contradicting {{formula:2d04d7d7-5db1-40ce-8639-49f2711c6aa1}} .
Furthermore, notice that {{formula:09e6e0fb-2e74-4bdb-8933-0715c1756f34}} . As {{formula:85e00c01-7d52-4fbc-b242-d4836630c65d}} is even, {{formula:fc0be1fa-734d-4eb7-ab6c-49938205def1}} . This means that {{formula:f56e44a9-8dee-478b-b352-1a3fb0148bdc}} or {{formula:82b2aba1-d489-4cfc-a312-4908a7074396}} , which does not yield any roots. This contradicts {{formula:1925520a-1bf9-4d59-bb9d-952145a7a8a3}} .
(d)
{{formula:169ad03c-c443-4c47-ac62-2a3c904ae06a}} . This implies that {{formula:95383fc7-e4a5-4ede-a71c-3882baee8c55}} or {{formula:c2c301e2-df7b-45b4-b97d-36b18ccc31da}} . By inspection, the elements {{formula:4cca8dd6-1dc3-47a9-a2c8-c17a1f9bec41}} , {{formula:a01fb1ed-905c-4508-9751-1a7e09989f2f}} , {{formula:3b2a11f9-56ea-480a-967a-4aadbdf338f1}} , {{formula:ea16fde4-b2a1-42a3-922e-e9802f691488}} are in the codegrees of {{formula:8b4481db-8db6-424f-b62b-02f1424c3245}} , {{formula:f691b78b-ca50-4587-affa-e8165189e0e2}} , {{formula:884c508a-4f56-41c8-bbe2-beff56cd533c}} or {{formula:ffb49099-d910-4197-a53e-a9566b9a7a35}} , respectively. However, none of these elements are in the codegree set of {{formula:e73f9a8c-58fe-47dd-83e6-f0f019e9abc2}} , a contradiction.
(e)
{{formula:be06ff36-74b6-47b0-873d-896182d8bb7f}} . By Lemma REF , we have {{formula:eb4879a3-ce03-49f7-8bb6-6496587257ca}} with {{formula:3af60874-ae75-4a31-9efc-f3eca73aad19}} , {{formula:5063608d-e877-4523-b2a8-dd3ea17bbeb6}} with {{formula:c9a4aa42-cf34-4e2e-a359-ad07a9701954}} or {{formula:7203e649-6451-4c20-873c-21cea80a12c2}} . Observe that the only nontrivial element not divisible by 3 in {{formula:17459b32-0076-4930-98c8-134087954d48}} is {{formula:b7a64a72-3199-4390-93ba-bdd55afaf03b}} , {{formula:14806575-2770-4e58-aace-09b6c2cee584}} .
If {{formula:9b68cc35-e8db-4d88-9ff2-2b96ad10bdce}} , notice that {{formula:7b876db9-59df-4a25-861a-556f66434e8c}} . If {{formula:ab3d5e98-9b3e-4a73-99e1-ac14924166bf}} , then {{formula:847990a8-81cc-4868-b572-61beb62cb0a1}} , so it must hold that {{formula:a96c68a5-d1b9-4909-81ce-0adda0cba2f9}} or {{formula:b995cf7e-26c4-4d91-b548-d7b1afa9556f}} . However, this does not yield an integer root for {{formula:69e914ea-07b0-46d2-ba12-0443ef0bbc6a}} . Alternatively, if {{formula:69d026bb-4642-4dde-a7d9-b5238b71e311}} , then {{formula:882fab52-6ac4-4d21-a4db-cabbcd443784}} , so it must hold that {{formula:ab2da8b0-ec2e-42ad-8f7d-ff624b6eac76}} or {{formula:a65e7b0c-4d4f-4765-a3cf-068d043ca792}} . This also does not yield an integer root for {{formula:1c70d639-2e02-45e5-8d85-c27daebf2617}} . Thus {{formula:00924543-ea6d-4103-804b-da09a0530058}} , a contradiction.
If {{formula:04355ad7-8226-4f44-8375-002602f97b96}} , we have {{formula:cd68626d-763a-4488-b6a2-e87125f36a03}} . As {{formula:707d5488-5434-48d5-9ea0-f42dd3fa8707}} , so it must hold that {{formula:d088d9e7-d01e-410a-a0bb-b822e80047b8}} or {{formula:3f814f30-f41d-4228-88d7-8b114e315e08}} . However, this does not lead to an integer solution for {{formula:ec92033e-8789-4744-b6e5-d506b6a5c97a}} . Thus {{formula:4d7578ed-388e-4bf7-ac70-7bdde58431d6}} , a contradiction.
In the final case, {{formula:1f429ab6-c6f5-47cf-82a8-979fb546e53b}} but {{formula:1503cf3e-6c66-4547-adea-2da77d3ec23d}} , contradicting {{formula:c495b882-2a48-4f1b-98cb-6d34560a7e44}} .
(f)
{{formula:5a086e6b-7026-4abb-8a05-f0b3f8badd11}} . By Lemma REF , we have {{formula:4e449e8a-4167-4f24-b544-b3dab1822a63}} with {{formula:dba74430-194a-48db-92b4-42a357e8e1b5}} or {{formula:88272b71-a56e-435e-8f2d-ef2b419af429}} with {{formula:d2628c94-dfc7-4778-ada8-5f2058f1b564}} . Both {{formula:dcf1a497-fbba-437f-90b6-e44606521be6}} and {{formula:cb40d71c-cb24-4aa7-8963-298a380e2ae5}} are not divisible by 9.
In the first case, we have {{formula:6709003c-e1c6-4459-96c4-fb22a026bd03}} . Thus, {{formula:c9020350-80ef-4c98-8946-b99b681be977}} , so we must have {{formula:a841eaea-eae8-4099-b76b-357d2519a352}} or {{formula:5cbc8011-5d14-4711-8009-cefedb868b51}} , which does not yield an integer root for {{formula:dc5ed20e-8188-4cad-9844-40e4ab6503dd}} , a contradiction.
In the second scenario, we have {{formula:1b1b5969-130f-43b3-86bb-58d725403de2}} . As with the first scenario, this element is not divisible by 3, so we must have {{formula:03451444-b9bc-40fd-bc62-0f62c7a3b1f6}} or {{formula:5b97867f-b9ac-4e69-99e7-b745c563d740}} . However, this does not yield an integer root for {{formula:407c3d8a-2757-4844-a2a8-75943cbfed16}} , a contradiction.
(g)
{{formula:9f5cedee-dce4-4c50-8b03-b2fac8982dd0}} . By Lemma REF , we have {{formula:c0720e42-5bcd-450a-9198-04f717da206c}} . However, {{formula:115f7edc-5583-4eb5-9037-652caaa339db}} is only present in {{formula:7bba2236-f187-4589-9295-b6268274e1eb}} and not in {{formula:ed06c554-eb63-4315-a517-1504bec717f5}} , contradicting {{formula:44713a50-9452-476b-98b0-a85f90a0c65a}} .
(h)
{{formula:6a8281bb-49c1-4272-af19-68db8a6df236}} . By Lemma REF , we have {{formula:01d86bfe-a1cf-4cdc-b314-b96321eb9ed2}} , {{formula:37916db6-ba52-42f6-98cd-930b632dadd8}} , {{formula:3e3e4073-b06d-449a-b569-dd0a0a3cf4b5}} or {{formula:57073828-eb0e-48fd-9797-f2e26ba1d0b9}} for {{formula:2f22cd85-9987-484d-b989-0660d79371bc}} . The codegree sets of {{formula:e1170a91-2f3b-47a0-8919-b04b738a76fc}} and {{formula:d1eecd26-829d-491e-82ec-b5bc3ba0ea47}} contain {{formula:306928d2-5502-4e1a-8cdb-8fbd6e405681}} , {{formula:fc128503-691d-4f72-a0b5-a2c8393813ba}} and {{formula:87310dc9-a9d2-42c9-b12b-1d3a1b0aa2e9}} respectively, none of which are elements of the codegree set of {{formula:d1fa7009-cd3d-47fd-8d35-79ffca5d33bb}} .
In the other case, note that {{formula:cf202660-cfd9-4c25-a769-e65ee00ea309}} . However, {{formula:267544f1-c95c-4b0f-83f6-4d2b414694fb}} in this case, so we must have {{formula:0f7dd747-6c0d-4e2b-9b16-6529167cdf4f}} , {{formula:a9e23229-888e-4be8-ad34-3f99599e7847}} yet this does not yield an integer root for {{formula:1fe9ce3a-1fd6-46f7-963a-fe52f5126d2d}} . This is a contradiction to {{formula:f2477e47-283b-43d5-ae3a-4e3fda228fff}} .
(i)
{{formula:8bea7942-bf2b-4629-bb90-4b9f3c2bd0ea}} . By Lemma REF , we have {{formula:51988d04-6ebc-494b-acea-a5c855bb2cce}} . However, the element {{formula:ea3c276e-f712-4eb4-aaa6-5c8669fa369f}} is in the codegree set of {{formula:bc2d628e-6fe7-4eb9-8d25-27b21b6acda0}} but not in the codegree set of {{formula:d4dc823a-3078-45de-a7e5-2de5ec277fdb}} , contradicting {{formula:2dcb3658-e7d4-4b61-8413-6830be453d55}} .
(j)
{{formula:4f6ee3ed-fd31-4494-bcdf-d144133700cc}} . By Lemma REF , we have {{formula:9a7436ef-43b9-4804-8788-97c003afae88}} or {{formula:1cc6416d-bbc0-49e2-bafe-9cc5369946f5}} for {{formula:335ad48c-7014-45bf-b51d-70a5e39df8cc}} .
In the first case, the element {{formula:a9c13b83-c646-4c66-9ec1-b0f09e6ce6e9}} but it is not in {{formula:5e5c455d-f262-4690-ace5-47718bcf7bf1}} .
In the second case, {{formula:f1b36fa2-f274-4b4e-a06a-775ebe1d639a}} . However, as {{formula:554b94b6-f611-4ee3-81c2-371ce74642cc}} , then {{formula:ed0277d5-02bc-4342-b773-ec2943a2fa6f}} yet there does not exist an element in {{formula:6147c970-f096-48e9-aefa-71bdf82dc558}} that is divisible by {{formula:1102cf84-3211-43e8-82ac-ff4b5ed0cc9b}} , a contradiction.
(k)
{{formula:0248d524-72c4-4a93-a08f-3db68bd2d3f3}} . By Lemma REF , we have {{formula:d9a2bd44-356f-40ac-a4d3-ea7a8c50fb0d}} , {{formula:ecfb2179-8468-4cda-b9aa-31fa8b56e339}} or {{formula:afd2e9eb-8118-42e4-a78f-8f22cc451520}} . However, the codegree sets of {{formula:8a52fd7d-4f4d-4800-9c6d-0ce086611b12}} , {{formula:4c0d6d66-508d-4fdf-96a1-3857159d9ae9}} and {{formula:bf36419b-8d7f-4b5c-96cf-57b59d0a3470}} contain {{formula:83fc427b-972c-4fb6-8142-7de9f01972ec}} , {{formula:d5aade99-4e90-4a02-8a00-baf488cb9e93}} and {{formula:e9fe8822-a1d3-48e7-b739-68b1ac2c7e51}} respectively, but these are not in {{formula:90e1f196-ac6e-4d71-bb5e-3dbec5dfbc57}} . This contradicts {{formula:1e775e22-5f06-46f9-be12-379c0ad9c80c}} .
(l)
{{formula:3fcd44a4-727a-4c88-9d5a-ea1cee95acbd}} . By Lemma REF , we have {{formula:b818b7ba-53fa-4cc3-a332-2cc4974646e5}} . However, the element {{formula:3d623244-e903-46de-9be6-a96a24c904a6}} is only present in the codegree set of {{formula:4feb31ca-170b-4ce5-be86-6829df81b5d1}} and not in the codegree set of {{formula:d480ba45-b6d3-4665-b84e-b1b14edf4788}} , which contradicts {{formula:7a3d77fb-c544-4692-85ad-7f523f7283e6}} .
(m)
{{formula:64fb98c3-d076-40c1-8450-54a13d2a0b04}} . By Lemma REF , we have {{formula:e1887da7-cd40-4255-9bb5-38d0619e12d2}} or {{formula:ec65f375-eb9f-40c4-b1ba-b84519a222c0}} . In the first scenario, {{formula:3e3392b3-d663-4944-95f9-2b5c62abdfd4}} but is not in {{formula:bd12f854-426a-4b15-8858-66c9ba070094}} , while in the second {{formula:ae426a3b-6161-4aa0-958a-911bf7345c4b}} , but this element is not in {{formula:3836b4ba-04f2-42fc-9992-38bad5dcff6d}} . This contradicts {{formula:1e8bbbed-f4db-454f-a0b9-fb2dfbbf419e}} .
(n)
{{formula:b4da53a7-4512-44e9-8fa7-17eaf39288b6}} . Then {{formula:991959b5-4ebc-4105-bee4-89a398bfa3b0}} . However, the element {{formula:4aa63c91-635b-4e51-8b06-dba63318a4a7}} is only in {{formula:6d301fae-4631-40c9-bd06-a11525186004}} and not the codegree of {{formula:873eb5eb-a649-4b9b-b73f-71e397109609}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:49726aa9-9fef-4733-b641-ca2514b516ef}} .
Lemma 3.11
Let {{formula:d908283d-d9f1-4652-8408-c4a1584ea78f}} be a finite group with {{formula:a847b8b2-99ee-4651-9a89-2303644a9a41}} . If {{formula:a73c4907-4984-45f5-baab-85355722ebe3}} is the maximal normal subgroup of {{formula:2956e7bb-009a-4e78-8b73-a8f8ef251e58}} , it we have {{formula:a991f828-1a78-42a4-8579-f54fb0be69d1}} .
As {{formula:232b90b3-af92-48ff-86dd-42c3d17ef55c}} is a non-abelian simple group, {{formula:68d56cf9-0c47-4100-be70-f5e725b7fdcb}} , so we have that {{formula:2b0f940a-69c8-4a3a-b322-8f1b559d0915}} , or 17.
Assume on the contrary that there exist a maximal normal subgroup {{formula:c4a73e13-4857-4625-b025-8f04cf4a4626}} such that {{formula:07a0fd89-8a85-4a1b-8012-603e9ff6f1be}} and {{formula:f2698864-ba7e-4674-b105-cac120a3bb47}} . We exhaust all possibilities to show the previous statement is false.
(a)
{{formula:83d33e9a-90ec-434c-b136-a289606e1c25}} . By Lemma REF , {{formula:64316930-08b8-4620-b2a9-5ff6b3f6790f}} where {{formula:cff5c4a6-dd26-4312-9803-98726abcdbbf}} , so {{formula:7dc5359f-5011-48b5-b6cf-0c390715da47}} . For all even {{formula:e0812c63-337a-4a46-9ece-c5ec42eba0f5}} , {{formula:95a83687-5a8a-4179-ae48-bba4d4a5f864}} is odd. Therefore, {{formula:dbc542a7-18d5-4973-b313-250346670632}} must equal {{formula:3ea2f6d7-e426-4058-865c-c7ed28348624}} but this yields no integer roots for {{formula:09420d72-c522-4ac9-96ed-f342242884d2}} , contradicting {{formula:aed35d09-3e92-4482-aa97-86df86e0b7ec}} .
(b)
{{formula:42f9a949-2557-41fb-a1e3-2357f3f0033b}} . By Lemma REF , {{formula:27b75ccd-bbc4-48db-a914-1e08ba8e90d2}} where {{formula:239668ef-ce2f-4c3d-9494-1529e8007189}} is a power of an odd prime number. This implies that {{formula:37a7fc14-4601-4849-9f9d-bb4bee354d40}} , where {{formula:2dfbaf89-230b-4c8a-9bca-77673cc50073}} .
Then {{formula:5ee38e8c-3bec-46b3-9f9f-a2bea145ba8b}} , {{formula:c4c0f948-3223-4836-b9a7-946282de3e3f}} . By a complete search, and we have that {{formula:61da460f-ab37-4c75-8e33-3341242e554e}} , {{formula:4fada5a8-a340-4346-ad14-a1df89fba9eb}} or {{formula:35e06361-2ffc-4a70-b611-8b033030e5f6}} since they are half of the other three elements. This is a contradiction since {{formula:67fe1fe2-b892-41e9-8834-855a0b5f7393}} .
(c)
{{formula:09693f57-fa83-4b0f-83f7-751fdd4e5959}} . By Lemma REF , {{formula:c880a5ca-4397-4704-aebc-52d306f66bfc}} or {{formula:7aeacc65-f9f9-4da4-bbe1-ec66aebfd51d}} , with {{formula:44faca9a-f255-4431-aec4-5822399d470c}} .
If {{formula:e18080f7-1c26-4af7-8bed-2043ae2e4c2d}} then {{formula:6f8351d8-ec8e-430e-9288-23cf8ca176fb}} but {{formula:0d27558b-1bce-4269-89be-08a4b16ab0b8}} , contradicting {{formula:086441cf-42a9-47de-8e34-09cc32ca0c33}} .
Furthermore, notice that {{formula:6604f2e4-cc69-4136-94d1-331c9a003b0f}} . As {{formula:82cfecf8-ed2e-4c53-84c4-227d5155059f}} is even, {{formula:5b929d34-bf9e-4d9a-8774-767866467683}} . This means that {{formula:8290f436-40f3-493c-b189-17dc9354d7f0}} which does not yield any roots. This contradicts {{formula:5aa843d3-1cbf-4bd8-b5e7-528140250e95}} .
(d)
{{formula:8659c501-d414-43cc-9499-ce0dc21d6369}} . By Lemma REF , we have {{formula:6a95648d-ea0b-4d3d-a55e-1eb5b1328f29}} , {{formula:07afb997-e555-4d2e-80c7-9012fd20f1c2}} , {{formula:7f2c7fe5-8676-4ef7-8e9e-a402bb14dec9}} or {{formula:549c8985-2e12-4caa-bc26-6f6f1dc5728b}} . By inspection, the elements {{formula:27a51808-b4cb-4859-a9ab-0f0655a6aaac}} , {{formula:ec3960a8-217c-4cff-aab5-f39ee1044798}} , {{formula:5e089dac-9bdc-4833-9e4f-9857471b3548}} , {{formula:eeb96035-b994-4c61-9d8a-19c0ab90b2d6}} are in the codegrees of {{formula:d0e427a9-fe28-45f3-a5fe-09181a3278d9}} , {{formula:267ad168-7845-456f-a3f9-43dd645b5a67}} , {{formula:fffb218a-8850-4385-9562-845cf6362020}} or {{formula:5f266e47-9d45-4bb8-993c-a868c2f4170c}} respectively. However, none of these elements are in the codegree set of {{formula:0c3d2715-1858-4e9e-a6ee-d1506e9b6879}} , a contradiction.
(e)
{{formula:83394095-f40d-4922-93e0-a82583989c5e}} . By Lemma REF , we have {{formula:1a622f0a-934d-4167-8b69-67cc39bd56c2}} with {{formula:c197c8a8-0e7d-4398-b66e-562ca549b7ee}} , {{formula:7237ee4a-d526-44ea-8820-619c566f2b1e}} with {{formula:2cfe49e6-1f55-4df3-87b0-39310cabb1e9}} or {{formula:d566cfb6-f7dc-4299-94a5-a83145b0debb}} . There are no nontrivial elements in the codegree set of {{formula:590600d3-ae97-40bd-a6b9-59add032c923}} that are not divisible by 3 except {{formula:7cf0f42c-e5da-4416-99f7-5bebe122e2dd}} .
If {{formula:374e7e68-d990-4376-9559-d3a801474c3c}} , notice that {{formula:c4868328-d949-4e11-b761-1f1cbb8049db}} . If {{formula:ae2553f8-62d5-43b3-aa63-7efc1d6eff0d}} , in this case, we must have {{formula:b2a6e324-d96c-40ab-a050-42b51820bc06}} yet this is the nontrivial odd codegree, this does not yield an integer root for {{formula:a60249b3-f95f-4b5e-9ecd-88d5aa832c44}} . This is a contradiction to {{formula:e32e81f9-e71a-4356-8802-abe458c6755f}} . Alternatively, if {{formula:08451203-5adc-4edd-960d-e967db0a37ec}} , then {{formula:75563370-4b29-46e3-aa29-27e76f4508be}} . We have that {{formula:ef210e7c-8d07-411e-9a1c-222515b0130d}} , this does not yield an integer root for {{formula:2d6cc7be-0c5a-4a72-a26b-1522e9236316}} . Thus {{formula:02e96fb8-ff4c-4118-b1e7-1e1335fb384a}} , a contradiction.
If {{formula:e2a5d17f-cf93-4b25-a8d2-7af59bae08fb}} , we have {{formula:774680a2-b4e9-43f2-9d2a-93395d64fbca}} . As {{formula:bcc06b7d-ffcb-4c0a-9cf1-ac95ccf02a38}} . We can also get the same contradiction as above. Thus {{formula:36e53a54-6802-45fb-9d6f-473ce3621833}} .
In the final case, {{formula:0b167cf0-aaf1-4fdf-91f4-7659c2649204}} but {{formula:7755e84f-ba5f-45e4-beeb-840b2983588d}} , contradicting {{formula:183b94bf-a53e-428f-9a55-c903228bfca7}} .
(f)
{{formula:6bc64897-a065-48c2-8735-9d6eb21a0917}} . By Lemma REF , we have {{formula:2e2298ef-8a46-47f6-a34c-625124ae86cb}} with {{formula:33b3a11e-3060-49aa-bbb2-d16cfd31f5cd}} or {{formula:740095dd-ab36-4cae-98d8-7e5a4c4d5029}} with {{formula:24f8e45a-6cb5-456a-ab42-3de94646e00d}} . Both {{formula:101f5925-a241-482c-99ba-765dadab7a5f}} and {{formula:e7175528-175c-4ca8-8ea4-4bddc78951f1}} are not divisible by 9.
In the first case, we have {{formula:d5283047-ac9d-4c63-a86c-470a63c99e3a}} . Thus {{formula:4ae0c33e-7990-4e2e-a12e-65f3386d67bd}} , we must have {{formula:8c34c7dd-3060-4b35-ad17-30aa7a69f047}} is the odd codegree, yet this does not yield an integer root for {{formula:325b200f-fc3e-4a28-9d26-b613984a6233}} , contradiction.
In the second scenario, we have {{formula:b27ba0c2-0759-4a0f-a494-44aa8a3466da}} . This is not divisible by 3, we can get the same contradiction as above.
(g)
{{formula:13b2b115-fcaa-4652-8503-aad5d7d50a56}} . By Lemma REF , we have {{formula:f34c9af0-08f6-443d-9eed-25cee91c472d}} However, {{formula:509ce110-ea01-46e7-9f1b-5c266f3f298d}} is only present in {{formula:77732c61-e0d7-49e1-a12e-07207d4b0b45}} and not in {{formula:d1778a5e-afa1-4b8f-a70b-68120dea2da0}} , contradicting {{formula:dfeb0b4e-b293-4296-bf82-3546413aadcb}} .
(h)
{{formula:db7f59bc-61ca-4c05-960d-48c8f76008ca}} . By Lemma REF , we have {{formula:3146d256-48dd-412d-8745-75afba22e500}} , {{formula:43548a3c-0981-486b-a4e9-82a95e5dd97b}} , {{formula:439f9334-2d56-4d3c-9823-fe2b1f24ba3e}} or {{formula:6dece884-1568-4732-ae77-6ee22d7c3fdd}} for {{formula:200222d0-0523-46c7-a9f5-d3800f6d02d4}} . The codegree sets of {{formula:679ddc7f-b1b8-4d38-96d3-847190cc6157}} , {{formula:a6cf0dc5-a850-4a96-8c47-089620239280}} and {{formula:db000615-d753-4311-8cc0-8f86116c3f63}} contain {{formula:fc61523b-2fe4-4965-9196-fb533740c258}} , {{formula:db086f31-dcc4-456a-bc62-2e63169da839}} and {{formula:35967d95-050a-4562-88df-2bd499beec26}} respectively, none of which are elements of the codegree set of {{formula:8c0761fd-f57e-4fdd-9121-a39b3836edbb}} .
If {{formula:bdaeb907-c729-46ec-906d-e1485193c4a0}} , {{formula:b4c32f01-437c-4929-bbbc-b3b9898cb811}} . However, {{formula:9e9583e7-ec56-4cec-aa00-41d087744bd5}} in this case, so we must have {{formula:9f157de7-490f-45c9-bf64-d785e1c9ad93}} yet this does not yield an integer root for {{formula:62ddc81d-e47c-494a-ac80-b677293c5955}} . This is a contradiction to {{formula:67c64ec7-2735-4633-8fd7-a87f5d9c8aff}} .
(i)
{{formula:714fadb7-508a-4e2c-b903-d3ba0d60b082}} . By Lemma REF , we have {{formula:0b39545c-dd8b-4d15-95c8-504a47d4f37b}} . However, 17 is a factor of {{formula:4d789792-d3a8-4116-8ef4-e31d7730a9fa}} and not {{formula:70a72632-ae74-4df0-a46a-0eccc099d786}} , contradicting {{formula:2f9f43f1-ce23-47d0-acc1-1a2f4234d30c}} .
(j)
{{formula:893a3ca5-ed05-461a-826e-9c407baacccf}} . By Lemma REF , we have {{formula:6d44c7b4-f70e-4acd-86da-ac53d00ad9ed}} or {{formula:7bd8b452-c40f-4254-b742-2582bbab5e03}} for {{formula:eadcce56-eb7e-4df5-9c2f-08ac5e989556}} .
In the first case, the element {{formula:b2e5415e-c53b-4c8a-af65-cf6297685713}} but it is not in {{formula:b8355d38-07b0-4e4b-a192-8aee16a8eff6}} .
In the second case, {{formula:94ba4f76-0e26-4cc5-b138-3232a74c6412}} . However, as {{formula:cfc1fbd3-e03e-427b-8077-c76acd0f2045}} , then {{formula:6a84a116-f288-4229-9b73-4f0c715bdb69}} yet there does not exist an element in {{formula:c5187f57-65f1-4df1-8d08-f6d4e46f7461}} that is divisible by {{formula:ab7c4b95-3c82-4e2f-bbb5-ae2337c7bb0d}} , a contradiction.
(k)
{{formula:9ed031d1-03a7-4f29-b0a3-417ae11cd95a}} . By Lemma REF , we have {{formula:b8d296ac-3806-4b94-81a7-f18f0c30e85f}} , {{formula:fe95a89d-0613-4d2b-bfd9-d98d9b49dfbf}} or {{formula:7046bd8f-f48c-467d-a2f1-3294ec61d7fd}} . However, the codegree sets of {{formula:cab34597-7cd7-4478-a97a-e6a532278baf}} , {{formula:d00415b1-e980-4ebf-86ec-21d506c74318}} and {{formula:f79e6906-7222-4e5c-929e-16a0ce87a9b8}} have {{formula:b9f16d41-c7c6-439b-8ff0-167a4ff670a3}} , {{formula:dee7bb5d-af7d-4bc2-95b8-e38d7410fdd7}} and {{formula:47e82cc9-e740-4bbc-bde1-54e7006c4918}} respectively, but these are not in {{formula:8efd7bb0-c1d4-467e-b342-8a69f1b5af81}} . This contradicts {{formula:2f54e891-d2ec-4e7d-993a-ca966984138f}} .
(l)
{{formula:8217c348-60be-4a83-a2dd-3b1af3497ffa}} . By Lemma REF , we have {{formula:1ef46d89-aa76-4537-97e2-b90fc9e31ea6}} . However, the element {{formula:ba1a744d-1c6f-4f0f-ab63-8b92ce85fafb}} is only present in the codegree set of {{formula:8bd1d2fa-a862-465e-b7ab-7893ca9f6a2b}} and not in the codegree set of {{formula:11bc4ada-051d-4122-a100-6b25fa4d115a}} , which contradicts {{formula:758fd259-25e4-4013-9fbb-a39a4de78cee}} .
(m)
{{formula:cb6109ce-766c-46e0-9fe8-48c4f6d9d3f0}} . By Lemma REF , we have {{formula:c9f54603-9ab5-4f59-90bf-35c1fb00af11}} or {{formula:44148279-2fb0-4f28-bc87-434627d1c0da}} . In the {{formula:b9f0f98b-921e-44ad-9821-ea66cca4a8d2}} case, {{formula:a101b106-4afc-4127-9c8f-99c8a339677b}} but is not in {{formula:a43d30e6-718a-48f8-9fe6-89c984bacb55}} , while in the {{formula:65b26062-65c1-47a2-97f1-896414f70c6d}} case {{formula:12e2b32c-c192-42ed-b7ad-9b3df541935a}} but this element is not in {{formula:de07ddd3-31be-405c-b435-6489dd941722}} . This contradicts {{formula:cf55e506-a8a3-4bf3-a3e8-05a4850d9b01}} .
(n)
{{formula:3a656076-0915-4560-86f1-3904e3107b47}} . Then {{formula:0047d833-aeb5-448b-8320-3ce3b82b6918}} . However, the element {{formula:a42054ff-3010-436d-8cfe-5a79f52d75bf}} is only in {{formula:f7af077e-72ed-4526-b5ac-f690f64990fa}} and not the codegree of {{formula:b2dd2ec0-737e-4b4f-9e10-ab2c33da1a44}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:7037ad6c-9f09-4f60-8d39-fb09c414b6ad}} .
Lemma 3.12
Let {{formula:41fa1649-c1f4-4579-99b7-34685a14e606}} be a finite group with {{formula:a8b652fa-01c0-41d6-b316-c295b6c2c033}} . If {{formula:c7d5fcf0-e6d7-4876-af06-1d7f0f4888d5}} is the maximal normal subgroup of {{formula:a0dc5288-9c37-4b91-8db3-32a23d757fe1}} , it we have {{formula:0fc17eba-becd-4696-8655-9790c6d3d851}} .
As {{formula:c3c7a9cd-dca1-41d8-8f0a-9f9e1d3ce7f2}} is a non-abelian simple group, {{formula:39b81d8f-c3cd-40e1-af31-9feeb3cdc594}} , so we have that {{formula:c63c001d-0b70-4b3c-a3e4-62a04d6a9e5c}} , or 18.
Assume on the contrary that there exist a maximal normal subgroup {{formula:c189b3b5-0b50-470c-85eb-a4ed5c350379}} such that {{formula:e8dc6694-0263-400d-adcb-2d0412299129}} and {{formula:a74277fd-67cc-448c-a55c-1b8ce525e647}} . We exhaust all possibilities.
(a)
Suppose {{formula:f86b3ada-4119-4747-a03a-2ba09a907fe8}} . By Lemma REF , {{formula:188a6f93-b9ca-4bce-9e05-482934064437}} where {{formula:2ffbf9d7-1d89-4e79-8f7b-e883acfb477f}} , so {{formula:c42dcf56-c42f-46c8-a6b5-0d2eeea2cc12}} . For all even {{formula:413b235d-b963-4acc-a5a7-07b698d4f472}} , {{formula:d529d3e3-3916-4b06-9ebe-d1c667088b36}} is odd. {{formula:656696fa-3225-4b26-b63f-648424aaee91}} must equal {{formula:885a7ed4-460a-4792-ba4e-0576cf2c4980}} but this yields no integer roots for {{formula:5c86b196-2f77-48dc-bf83-4fe58c473602}} , contradicting {{formula:51a56198-1d52-43ff-9b47-092e8900b718}} .
(b)
Suppose {{formula:9709309d-d5e1-4a3f-9e41-a60c8ddb8642}} . By Lemma REF , {{formula:c87f5fc7-b176-4575-9d1f-836637b5d469}} where {{formula:03887f00-cd31-41ac-8c16-82eafe26e74e}} is a power of an odd prime number. This implies that {{formula:b6a7d591-6938-4897-8ead-df093c2c51c0}} , where {{formula:03ce3a18-42e8-46c1-a405-ab3d715d471c}} . By a complete search, and we have that {{formula:3dc21c14-e6ab-4341-8dcc-5c8fc77b8706}} , {{formula:08cc269d-bed6-408f-aef7-28047773975b}} or {{formula:378f6b39-f343-4c4d-843d-0a0afbc0c814}} since they are half of the other three elements. This is a contradiction since {{formula:720afb39-0ff4-4828-951d-d5b281417fb1}} .
(c)
{{formula:af56c942-5d11-4195-aaaa-461faca8fd63}} . By Lemma REF , {{formula:f0d9b44a-ea6f-4c88-8e3f-34a68c772f3f}} or {{formula:96d20d13-5fbd-4a02-801e-5ecbba8cd9d8}} , with {{formula:954e49fd-7f0b-491a-b9f1-5c7bf8803089}} .
If {{formula:d148cd45-5522-4fda-86b4-d796203c36d1}} then {{formula:1d07021e-8b8e-470b-baf9-26114f528425}} , but {{formula:952fc677-46c4-44a6-aa68-5806cdc938a4}} , contradicting {{formula:24202021-a5c4-4dec-83ef-88e53e7593e2}} .
Furthermore, notice that {{formula:42349af2-a2e7-4cd3-90d8-f32c9e35e55a}} . As {{formula:e86271f7-ece8-4b90-8bb1-dc6d061ef553}} is even, {{formula:2113dc04-c54d-42b9-8e33-7ae13dee16b9}} . This means that {{formula:75928e31-0347-4b8c-bcec-44c259ece1b8}} which does not yield any roots, contradicting {{formula:13c4568c-e20c-40d7-b007-8e8966738bcd}} .
(d)
{{formula:2166cd5c-c165-4d57-babf-1c5de8652adb}} . By Lemma REF , we have {{formula:7d1067db-63d1-4961-8995-a4c73f6ac42b}} , {{formula:1cfaecb8-09fb-456a-b842-1e378fc97e79}} , {{formula:783c6f84-a7bd-4327-a5f1-a99ba5dc8341}} or {{formula:a6d92fde-c376-4aa3-a49f-91236e12d94b}} . The elements {{formula:2c032215-f76e-4745-bc9b-de3b305fda5d}} , {{formula:1a5df010-c68b-448e-8772-d2337a40b568}} , {{formula:69ae7585-072b-49cb-bc7f-36aa10ce09b4}} , {{formula:01d65e52-95be-4245-bb98-b859876a4999}} are in the codegrees of {{formula:2fc13949-6075-41f9-82e5-5efc2fd59e35}} , {{formula:42560681-9f33-47cb-9bd3-f40d945a626e}} , {{formula:0de4087e-5a4f-4b9e-9f76-76b512c686b0}} or {{formula:27c7a31c-4479-4ea5-9e9f-f21440d7e5ac}} respectively. However, none of these elements are in the codegree set of {{formula:9f6788c5-894c-4ba2-9b07-7073e0d4525e}} , a contradiction.
(e)
{{formula:128d0da1-8335-447e-b878-0036f71ad892}} . By Lemma REF , we have {{formula:3e3f63d1-d479-41ed-9be2-1b7f1af25612}} with {{formula:1fe9d30c-2c8f-4162-b816-91f6b2a71c8e}} , {{formula:86a9a584-a173-4912-b027-2b6e008a3001}} with {{formula:ed50cd35-913d-475c-a835-ad3178670de9}} or {{formula:9e80f80c-a34b-4d73-9610-8a37bb755b34}} .
The only nontrivial element of {{formula:6a29be3a-6503-4688-90e4-5b854af2476c}} not divisible by 3 is {{formula:5293b153-4a58-43f0-9436-9cdffae96210}} .
If {{formula:9f3ed288-e494-4e9e-9ba2-0b2a0ea391cf}} , notice that {{formula:9ea6a4cf-eaf5-42fb-a912-55799c004efb}} . If {{formula:b0fa82ba-fb69-4b3f-bd03-e5293f0252d0}} , then {{formula:12b22a4d-49e2-4a38-8404-abd4279d4d87}} . Alternatively, if {{formula:39aa0e14-563a-4003-8f5c-c511127eb060}} , then {{formula:e64ab1d4-c0ab-46c2-817e-8debe8fef767}} . However,{{formula:2fe54cc9-3dee-459d-a992-c3af1dc72cb8}} yield no integer roots for {{formula:814022b1-21e8-4350-9399-8473ef263bbb}} . Thus, {{formula:51fd9974-3b2a-4504-b35f-4d761e20026c}} , a contradiction.
If {{formula:deb44b60-65fd-4f46-98ab-8d4014ace82a}} , we have {{formula:a517f87a-2535-4b74-a982-3928b6fb5f3c}} . As {{formula:01d96c4f-64aa-4d11-9c7a-fb436bcf7733}} . However, {{formula:c5098fb0-90fc-4ffe-9fdc-6a060932da16}} has no integer roots. Thus, {{formula:c9d01491-fbc0-4c6c-99b3-a2f6ff341337}} , a contradiction.
In the final case, {{formula:a83eed55-f038-4498-be24-a6c81881aa8e}} but {{formula:5d8739aa-f67b-4855-9581-bcf749887b2a}} , contradicting {{formula:6e149dad-c03e-4bd6-9899-d4de8960dc68}} .
(f)
{{formula:b30a341f-bda6-417a-a4b2-ceac990b9193}} . By Lemma REF , we have {{formula:56c3dfc8-ea60-409f-b229-e62e881db4f0}} with {{formula:b6a335fe-b0f5-4a40-b429-f47713f00229}} or {{formula:131e2ca7-e9bc-4151-b493-0b0b3219f299}} with {{formula:59ff1f0f-d976-4767-ae32-5e3868c77196}} .
Both {{formula:48bf0e55-7eaa-40d3-8a9c-3417b5cfbaa9}} and {{formula:a47aea10-7ee6-4e86-912e-4a1f5b709b7c}} are not divisible by 9.
In the first case, we have {{formula:50316e2a-9b8e-4fe4-bef4-ed105bfba241}} . Thus, {{formula:60817cf4-52f5-463e-ad6c-0bf30791f51d}} . Furthermore, we have {{formula:3548b899-97ee-4cb8-b0de-4933e625cabf}} . This is not divisible by 3.
The equation {{formula:bdf4ddc3-ab86-47fe-8294-aa72a4fa151f}} has no integer roots, contradicting {{formula:18dc9020-5605-4ad2-b2a7-90e1c0055d18}} .
(g)
{{formula:5dbcdf1b-8b67-4399-b0fe-8cb5ac0247d0}} . By Lemma REF , we have {{formula:9d9112b9-49b9-4a9c-8d7f-f71f8e987f9d}} . However, {{formula:c758abf3-34a9-41bb-8ec9-1440154d6641}} is only present in {{formula:9e869f6b-8cbd-42e9-8228-a1e2b080aa1c}} and not in {{formula:cfe2d697-6570-49e3-953b-7a8fd57eced8}} , contradicting {{formula:029c2286-b0d0-4046-9b19-d6887ef1b303}} .
(h)
{{formula:b0055893-15db-4fb7-99b0-f6c9e25abc43}} . By Lemma REF , we have {{formula:c251dfe9-8aba-47e0-abfb-771dd2e1377d}} , {{formula:cfcfe743-1a51-4e99-8f15-861afd2235d4}} , {{formula:fd48c189-6310-465e-addb-6d0a360b2a09}} or {{formula:906a05bf-53ae-4cc1-ab53-66de8fa154ca}} for {{formula:aba4f667-6fad-4384-828f-58e4eeb31c37}} . The codegree sets of {{formula:7dc8fe11-5349-4ee9-a420-65bbe8a86d93}} , {{formula:c67d1df9-7591-43c3-8507-a017cc849f7a}} and {{formula:a92ab1bc-e0a5-4ccb-a4aa-2d014e5d32e9}} contain {{formula:f35becb5-e2b0-4451-afa7-610c26b8094e}} , {{formula:9f236feb-8b32-47a3-b617-5a3b2fec499e}} and {{formula:10c43478-49c2-4175-8b29-3c653ebba64e}} , respectively, none of which are elements of the codegree set of {{formula:b3bd9918-e175-4095-869f-5e96644694b3}} .
If {{formula:5b854d43-5588-49cf-bca5-7665bef75a7d}} , {{formula:25d5cc8b-6511-4397-b4af-8db4a75e4b98}} . However, {{formula:1b0fec01-f7d3-45f9-a1e4-b74ad964fc84}} . In this case, so we must have {{formula:6d2a47cd-5832-475e-a2b9-6b03002d8b7e}} yet this does not yield an integer root for {{formula:55fd8f7c-3ffe-4999-906c-0ded03385f4f}} . This is a contradiction to {{formula:04ead800-c7f6-4457-8bd5-ef04b1cfb1b0}} .
(i)
{{formula:7bde699f-03c3-466c-b810-29ad5ba786e9}} . By Lemma REF , we have {{formula:3d314190-9f18-44fe-8c7f-7556f0eea1df}} . However, 17 is a factor of {{formula:65336f19-0dd0-448d-a809-6de1edfd4be9}} and not {{formula:ee85c624-12d4-4889-a9ec-e0f1a6ecb92a}} , contradicting {{formula:505ecdf7-3963-47ee-89b0-5fd0ca7da641}} .
(j)
{{formula:6e5b6eef-e918-464e-ad9a-93faff7a9817}} . By Lemma REF , we have {{formula:e99786f3-102a-4b4f-8693-c8f748f034a6}} or {{formula:df2ccf80-c146-48d7-bd55-5f2f5cce455a}} for {{formula:7dad0342-7262-4c59-b415-5818c0a20040}} .
In the first case, the element {{formula:77fc7cbb-6b18-486a-b9f7-963a430ac0c3}} but it is not in {{formula:6e24216e-14a6-4b86-b27e-05f3fa23156a}} .
In the second case, {{formula:cf8ac8bb-e76a-4a33-bd6a-bef1c023f0b4}} . However, as {{formula:08b068a6-ca88-48f5-9add-5452f9af71f3}} , then {{formula:07966c33-69e4-4a81-b2d8-7a18b9ea965e}} yet there does not exist an element in {{formula:fb78607f-fde6-4469-ac86-e36b0c5c48ad}} that is divisible by {{formula:4dd7051d-549e-4a2f-9537-6c5e227ce4c9}} , a contradiction.
(k)
{{formula:6373fadb-2b2e-4f87-a829-7e778326371c}} . By Lemma REF , we have {{formula:ef5fe549-8bdf-4fe6-a4dd-581a51bc56e2}} , {{formula:6282018e-b8a4-4157-b82e-48b003314d2c}} or {{formula:cc77321d-3cfe-43a9-90d6-690971f31a58}} . However, the codegree sets of {{formula:b0260def-dccc-4ec9-a034-f01f391f5a7d}} , {{formula:6cb414b3-2a13-467d-b7e5-8a6863d5c282}} and {{formula:d775cfba-c74f-4621-83ee-3164483c8f2e}} have {{formula:45389480-f30d-4f14-a350-94371ec4467e}} , {{formula:7e22aa58-11e6-44e4-9888-9f3984d3d2db}} and {{formula:ef59fd55-54f9-4f80-9e88-fc5082be02ba}} , respectively, but these are not in {{formula:214a011c-0e24-4e4f-8e5f-32ee4c0b29bc}} . This contradicts {{formula:7fb9e03a-67ac-4153-9c6b-e488547e47ee}} .
(l)
{{formula:b3cb3055-b2f5-4f4c-941c-2eeedb03ebd4}} . By Lemma REF , we have {{formula:4a4c111d-d445-4276-a92c-2cadcceb6123}} . However, the element {{formula:404972c7-b8ad-4e8b-b3ca-6bd2b4649e31}} is only present in the codegree set of {{formula:01e6c635-56ed-4141-af15-311c91cb684c}} and not in the codegree set of {{formula:06b8a63a-b6ab-448d-978f-ef6a2e5550ab}} , which contradicts {{formula:ab921b00-093b-42ae-ba1a-892829ee7749}} .
(m)
{{formula:1aef82a4-7cb7-47fa-9d9f-1e7f7e48fbab}} . By Lemma REF , we have {{formula:f7100f0a-3e00-479f-bdae-c31846346b81}} or {{formula:7f5edf6e-52c0-4201-8af8-445f821459ed}} .
In the {{formula:19ea3278-bf96-42b9-89eb-88dd1a045161}} case, 7 is a factor of {{formula:73670605-62b0-440c-8bbc-2f09ef737a17}} and not {{formula:ef1f04bf-e416-4a2b-99c5-5189de6f690d}} , contradicting {{formula:17c6ec6c-ce4f-4f9c-9632-1c7665fd2512}} .
while in the {{formula:b6902bd0-9338-4146-8352-047e2fb2c676}} case, this situation is the same as the contradiction above. 7 is not the factor of {{formula:1ce65486-572a-4dc2-888c-43045343d5b3}} .
(n)
{{formula:64c194e5-0f2d-4e00-8cb4-b3ca9b1a21bd}} . Then {{formula:9dd59c91-47aa-4f99-9b8b-257626212e44}} or {{formula:47514862-36e2-4d71-89f9-b42a5e177d5a}} However, there are three nontrivial odd codegrees in {{formula:b6c0a147-fd12-4305-9c09-3893a92831af}} , there is only a nontrivial odd element in the set of {{formula:fa6dcdd8-5043-45f8-adea-003f467a5f84}} , a contradiction. Additionally, the element {{formula:05381e9e-72c0-4643-992b-10e7e69088c5}} is only in {{formula:1a00af3b-c5a2-4e03-9764-3a2a6887de6a}} and not {{formula:0e50623d-60a0-447a-ba5c-da94dfd2b458}} , a contradiction.
(o)
{{formula:edd35092-78a5-4dd4-ab06-c8a299a84c59}} . We have {{formula:9b6e4a51-77be-4739-9dbb-c5053e2a9e86}} or {{formula:e3d87f9e-1bbe-425a-8f96-c75825f74e2b}} . If {{formula:b4b30776-7a7a-4ed8-ac6e-76f21cfceebf}} , then {{formula:b04634fb-6eb6-4d2d-9cda-7c3b3f617487}} . If {{formula:85d5871c-2e04-4ec7-a1cc-47e3281361e8}} , then 11 is not the factor of {{formula:91a91f88-bfef-4f91-a867-28ff42d2d624}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:c0b8db03-7f99-47be-861d-9191356e9919}} .
Lemma 3.13
Let {{formula:08fdd949-a6f7-4c8f-8f88-a2d70cbd63fa}} be a finite group with {{formula:1ea81d92-a57f-4a06-9ddf-fc9d5ffcd01f}} . If {{formula:b03f6805-0689-4c33-9da4-53d5fec641fb}} is the maximal normal subgroup of {{formula:2b5d89c9-e388-4f59-b17a-652ebe8d736f}} , it we have {{formula:37d2597c-aeb1-4e19-b049-4d3b13114649}} .
As {{formula:7bdd3434-b3d4-4ca3-8a53-9f6aacfe44af}} is a non-abelian simple group, {{formula:33004f83-6e7a-40ec-8853-0b82c3df6d68}} , so we have that {{formula:aa91f82e-cbec-4384-ab7d-11a79f43dc1a}} or 18.
Assume on the contrary that there exist a maximal normal subgroup {{formula:59b8d45c-2bb3-47e3-a459-a104c04eb72e}} such that {{formula:01b9bc38-ab4c-4f7f-bb69-c4c1c65f6e8d}} and {{formula:5a120843-5d3c-450c-9de2-1cccccdc49d1}} . We exhaust all possibilities.
(a)
{{formula:4515c3e2-6db0-4338-95c0-d64248e61fcd}} . By Lemma REF , {{formula:b6ecb816-fe35-42bf-bc0c-351cbaa0094d}} where {{formula:20a2cea5-4e9b-4843-86e9-89677034f58e}} , so {{formula:584d0cc7-b027-4e15-8ead-67e8016e9b12}} . For all even {{formula:3ad3b02e-5282-418d-a467-249eb7882a50}} , {{formula:05e99650-4ec3-477f-8d17-d9129003de1b}} is odd. {{formula:a68496f3-eb69-45a0-b69a-b8278f2c128d}} must equal {{formula:e2a10343-00ab-4962-9d83-f31d985211d2}} but this yields no integer roots for {{formula:f5370716-e0f7-4ce6-a1b8-b09c88c7911b}} , contradicting {{formula:d6250b29-1083-4baf-a0f8-72fb07043254}} .
(b)
{{formula:7094ebee-df7b-4509-a963-0b5d116ebd3a}} . By Lemma REF , {{formula:b238a8f5-46c9-40bb-8d29-2c5dd9d2b189}} where {{formula:1f5826b6-dadb-4e6f-8b2f-341ee1d66410}} is a power of an odd prime number. This implies that {{formula:93fd22bc-db75-4e78-a813-0970fcc74803}} ,where {{formula:03dc2b45-2b16-4a91-9cbb-71e95d31936a}} . we have that {{formula:189afd79-0dcd-4fe5-bce2-af24e74da773}} , since it is half of {{formula:18e4fce4-653a-4979-aa30-69693cbae0f9}} . This is a contradiction since {{formula:b3364b69-308d-4709-8ef6-d2f7e0635aea}} .
(c)
{{formula:c77aea0d-c8ed-43df-ac1b-687311fbec67}} . By Lemma REF , {{formula:bae63d03-0727-4309-b3d6-458fb73035a4}} or {{formula:bc3014dc-54dc-40a4-99e6-be8059bddbfd}} , with {{formula:ae2448f8-d303-410d-a0ae-714c0b8bcf86}} .
If {{formula:5d5b9452-5476-46c0-a9c2-7bf4f31d617e}} then {{formula:336cec4f-acb8-4280-8491-fa3242097fe5}} but {{formula:bf77db90-9cd9-4dcf-9bed-81b978dee0b4}} , contradicting {{formula:8e2ba339-fccb-4f77-b6ca-40cfd90e2c3a}} .
If {{formula:08bb5a23-f792-49a0-be2d-207b61fccbe1}} , notice that {{formula:2b9b9353-9bc1-4f3f-99bd-e349bb4bfff3}} . As {{formula:08049d7a-0967-4c6f-80b4-3abefd4132a9}} is even, {{formula:8f70d207-74e2-4fdc-a81e-b2f19939154c}} . This means that {{formula:6b1d93f6-9ad2-44ae-a517-1304c52ed38b}} which does not yield any roots, contradicting {{formula:903878b8-eaa5-43cd-a719-f8742332809e}} .
(c)
{{formula:772fe76c-8b30-45cb-a069-13ecb8ce561a}} . By Lemma REF , we have {{formula:c8dbcd43-89e5-4fad-9500-c4532e286744}} , {{formula:4c99318c-1992-483d-b023-84c5839ea8fe}} , {{formula:ae8d3124-e753-4cfe-ae20-a67c13b08885}} or {{formula:eed42436-a7ea-4eb9-bd9a-be443720e33f}} . The elements {{formula:0fd0d36e-f691-4212-af21-89f449f5f0cf}} , {{formula:6f0ca3cf-420e-407b-8bb9-e1e7019f62e8}} , {{formula:cd7a7253-6f3f-4601-afde-4764c761ffc7}} , {{formula:d36ed673-12da-4438-8689-df171f93764b}} are in the codegrees of {{formula:a310d983-bdb7-4d28-8df8-d612ca9c19ba}} or {{formula:bb1f11aa-8928-46ef-9ce3-3b40d57ccbd2}} , respectively. However, none of these elements are in {{formula:23fd36aa-0165-44c4-b379-ffe0466e46fc}} , a contradiction.
(d)
{{formula:0f2e352b-b056-428e-8272-d188039010bc}} . By Lemma REF , we have {{formula:d0fa1fc7-71b7-4991-8e2a-fe29d5d390ef}} with {{formula:b2aebe20-1900-45de-b9ee-15246b5b2b41}} , {{formula:3d3783ce-420b-4d04-8701-fe1a88d5b5c3}} with {{formula:b71fc012-2016-414e-bbfb-68fd7f1ca589}} or {{formula:a9cc0f78-4f56-4e38-b105-cb19413b15cf}} .
The only nontrivial elements of {{formula:c969dc31-4438-4bf2-9133-c1d34a9642f3}} not divisible by 3 are {{formula:c982e3f6-5671-4718-8fbe-edf5eb531639}} and {{formula:0b48723d-62b1-4fcc-ba51-64ae0313acf5}} .
If {{formula:a6f49e80-432f-4031-bb30-b16600fd9510}} , then {{formula:16fbdb11-b57c-4535-b0ab-125f5a2bfbb0}} . If {{formula:0d4b1ce8-6d01-4711-8d6a-b896aaf8d972}} then {{formula:1d0fd140-4bd3-4e86-857d-054407775da2}} . Alternatively, if {{formula:65e889e2-7b6d-48a4-9f5f-87e86fd66d86}} , then {{formula:10a79aeb-052a-41af-9541-35c75414b3a1}} . However, setting either to be equivalent to {{formula:5818534a-afad-48a0-967b-c2f1ca623897}} or {{formula:97529dee-b4af-41ed-8558-d27581464b87}} yield no integer roots for {{formula:c7ed47a6-83ab-466f-95f4-7eb5b9d9f7b9}} . Thus, {{formula:d7fbfbd8-d02e-47ea-a6ab-ce3d178df5f1}} , a contradiction.
If {{formula:c1287f51-cef0-4d5d-920e-a7dfab59dd86}} , we have {{formula:df77cb3b-10d8-4574-9583-5bb3d3aedcb9}} . As {{formula:ce87ab31-b4b9-464d-9d24-92b6435a05ea}} . However, {{formula:6d29dc8b-8e6e-4b2b-997e-3059c4fdf20a}} or {{formula:13082e21-f46e-4956-bbad-ad5a26289901}} has no integer roots. Thus, {{formula:6364eacc-dce6-4278-b999-4727ad827d84}} , a contradiction.
In the final case, {{formula:1cb49a01-c97f-4614-b376-9a638cedc88a}} but {{formula:b191116f-0f1e-40ba-810b-2fa3c9d332de}} , contradicting {{formula:bc82b261-1fab-4e2e-8e2b-a0edb811fdb9}} .
(d)
{{formula:6d09ac0a-cb45-48eb-99a3-cbfe78d5f55e}} . By Lemma REF , we have {{formula:e927b0bf-949e-4284-b612-40a7cc79ff3c}} with {{formula:209e4ccd-1a10-434c-9009-0147819ff43d}} or {{formula:11311c72-f5fc-4557-ab74-dbbe85063173}} with {{formula:b5acf6c9-76ed-4009-94b3-78ac4358c632}} .
Both {{formula:b6b81450-ea79-4fa5-bdcf-7cda2e73d565}} and {{formula:c897b5c9-8608-41b3-bcb1-50fcbea5dcf7}} are not divisible by 9.
In the first case, we have {{formula:84d6b81f-a35f-484c-b1ca-283e442f7785}} . Thus, {{formula:f9e7634e-f90e-4ea2-b5c4-67b868c354f7}} . Furthermore, we have {{formula:06af3b7d-d017-4e0d-8324-0a33c24eed8e}} . This is not divisible by 3. The equations {{formula:d23723d5-38e5-4036-80f9-ae93bbdcf0ca}} and {{formula:ea93b069-90d6-48bc-80c3-8b612fe55f9d}} have no integer roots, contradicting {{formula:b0628282-d898-4dd2-8b92-43c72cb6ae7f}} .
(g)
{{formula:f67b9922-84ee-40b2-8d2e-7ff387aaa3a0}} . By Lemma REF , we have {{formula:e63c20ab-176e-4ae6-be84-f6ee629b0ae2}} . However, {{formula:508bf38e-4ef1-419f-bfdf-ce744a9d355b}} is only present in {{formula:186707a8-a54f-4975-be72-d450a97d784e}} and not in {{formula:87ddaaab-849b-48c0-bd55-82de41059ba9}} , contradicting {{formula:77e75f55-d1fb-46cf-bcf2-d90352729e56}} .
(h)
{{formula:76369862-cb91-42e8-a767-45a828a1d518}} . By Lemma REF , we have {{formula:2cd2e38a-8604-434a-b41e-0a1317185262}} , {{formula:81ef947e-29cd-4be8-b256-a94b67273b6e}} , {{formula:959b07a6-f310-4aa8-ab53-fdca5b96363d}} or {{formula:68cdd532-4d5c-4c1c-9a45-1a2a08a0ede6}} for {{formula:67583d24-37b2-4468-bd01-40830391ba0e}} .
The codegree sets of {{formula:b74db6a5-c425-4302-9e92-f929f2ccbf0d}} , {{formula:a2780acf-85b2-492b-9cd1-6d48d27cba5a}} and {{formula:12f8994e-9f89-4f15-86ff-d0367b109261}} contain {{formula:2def60ad-a95d-49f2-9b84-debb64b5f596}} , {{formula:f1d1997f-bb56-47f6-8caf-b35dea85b2f3}} and {{formula:d12ab68b-1f67-4ccd-9c36-9cf55585f3b5}} , respectively, none of which are elements of the codegree set of {{formula:0f778753-88d5-4849-bf02-0eafb1001ca1}} .
Furthermore, {{formula:c499a503-88f3-4431-9656-106dadc64c33}} . However, {{formula:d9badffe-0344-48a2-934b-6ebd5d20ddb0}} in this case, so we must have {{formula:c92ba11e-4f04-4aeb-9a9d-129bcf5c870b}} or {{formula:35b25212-1c04-459d-a03e-e9d343cdccff}} , yet this does not yield an integer root for {{formula:9c88753e-d70a-45a4-8073-d8bc8a55bb08}} , contradicting {{formula:b667ce7c-962c-431e-bf7f-d396bc58eb1f}} .
(i)
{{formula:4ab8e1c6-ba74-494d-ba64-105a225bdb3d}} . By Lemma REF , we have {{formula:6f35ec63-1aa0-44a2-96c3-85a8161abb22}} . However, 17 is a factor of {{formula:1b65f1c0-2a61-47ba-87a4-d65eec5d441e}} and not {{formula:884a0d60-240d-4c18-99ac-fe4a05788f7d}} , contradicting {{formula:f3a7e8c2-edaf-4815-b63f-5f9bf005e07e}} .
(j)
{{formula:52d85e14-3548-47fb-ac9d-e6e265175259}} . By Lemma REF , we have {{formula:e0e013d9-45ca-458d-80b7-ce897caaf865}} or {{formula:eb08c67b-db42-44e2-a40f-6d428ee0434e}} for {{formula:8cb3ce13-79c4-4973-af9c-2bcd6daef963}} .
In the first case, the element {{formula:3c0ad41a-9252-4a0e-905a-fb0bd97057ba}} but it is not in {{formula:74d0290a-4e52-4bd1-bf57-ecda4217c6cf}} .
In the second case, {{formula:8abede3f-ed97-40c2-8688-ae931af0b6b8}} . However, by a complete search there is no integer {{formula:0e86c497-40de-41a2-9be8-f9b705fa711d}} such that this element becomes equal to an element in {{formula:79e3496d-e1a9-490e-aed3-8dc991b34071}} , a contradiction.
(k)
{{formula:3d90ee5d-fb80-4ac3-90f0-688322c559d9}} . By Lemma REF , we have {{formula:8b0cf49b-89c7-4ab7-8d48-fe3b63f1d2db}} , {{formula:c05107de-b001-4c0a-b0eb-f9d84927eced}} or {{formula:f9fb3069-84ef-4804-8b2f-1d53a7ba9ceb}} .
However, the codegree sets of {{formula:8f5e147b-0b68-4cdf-9d7e-441b68f26873}} , {{formula:7e224e99-7a16-467f-8734-6c5ccb636e62}} and {{formula:b4e85030-bde2-4081-91ec-a86a4357f8c8}} have {{formula:1ad1bd5c-cc90-4c7a-b075-1bedd3969266}} , {{formula:4a46bdf7-051c-49b5-a3ed-fb2bb49b25c6}} and {{formula:06d17618-abf2-4ffc-b82e-fa76cfbca129}} , respectively, but these are not in {{formula:cd4ecea8-d91d-40d3-85df-3e9b6c2b48b4}} . This contradicts {{formula:18594493-8955-4d44-8b72-b9ff01048775}} .
(l)
{{formula:7c15ada2-8185-4cce-9f03-d4ba8e9c5877}} . By Lemma REF , we have {{formula:6f84d8af-1fb5-4bed-8ff3-10ee5ff4ba85}} . However, the element {{formula:9aee98d9-dd55-452e-89ee-a6e6c5d3e40f}} is only present in the codegree set of {{formula:6ccada1e-2eec-4300-95ff-a5e80aa250c0}} and not in the codegree set of {{formula:cef95214-989b-4ebf-b53a-911a843953e4}} , which contradicts {{formula:c753d9f1-c300-4bdd-a781-eabb8aa634b5}} .
(m)
{{formula:985cc23c-fbfb-43a5-9634-5920af348f93}} . By Lemma REF , we have {{formula:5098d944-060d-441b-bd6b-61754cbcb97c}} or {{formula:bcb4d4c8-261a-495b-9dc9-906b3885c1a2}} . In the {{formula:ba74a45f-4ff0-48d2-85ee-df3b686469a2}} case, {{formula:192c1885-7162-4fad-bf96-fe3ccb60e085}} but is not in {{formula:0366d072-6b4e-4f9e-817c-96b2c21eb5ec}} , while in the {{formula:cd16a188-a0a0-472c-932d-c0f806aaa046}} case {{formula:52a7a1e6-a10d-4d68-a6bd-392bd8c5efb4}} but this element is not in {{formula:03175d7b-3163-4f8d-afe5-650d8a9ee6fc}} . This contradicts {{formula:3db09a2f-ce6c-4ba9-8142-4c28d78586d0}} .
(n)
{{formula:ea60e202-e4fc-4b75-a729-ce1892ef54f1}} . Then {{formula:72e78575-75a2-42b2-8ecd-9887e5ea9332}} or {{formula:64633947-ce9a-48e1-8649-5890cdd3a8b3}} .
However, there are three nontrivial odd elements in {{formula:d07d5a7f-e389-47b3-8118-8f93e5e5b633}} and only a nontrivial odd codegree of {{formula:237366e0-41ca-4b42-aa7d-bd1e712d3dc9}} , a contradiction.
If {{formula:68debedf-90d7-4b5c-8744-c2c2800baaea}} , the element {{formula:64cd7a7c-53c0-49a8-931f-1bc64d483212}} but it is not in {{formula:ddd8728f-babd-459d-a96c-5f546047b276}} , a contradiction.
(o)
{{formula:5eb1709a-7e33-4b72-8cd8-4823d6dab06c}} . We have {{formula:f8af44a5-59ad-455a-880a-b80a268bbea9}} , {{formula:2c84a7b3-bd8e-41a9-9891-ed02f4eb521f}} or {{formula:946a7c26-8780-4038-8cbe-90afb086987a}} . If {{formula:c032a14d-1355-43f1-9fc4-d1abe56c4e90}} , then {{formula:4bbc6342-926b-422c-a736-bbfc4b8d8e47}} . If {{formula:caff87a2-5c37-4ffd-8e5d-8baffb0283d0}} , then {{formula:e5c0cb4e-ad18-449b-859c-103e3a9533f2}} . Neither is in {{formula:3624293c-7800-4618-b743-530ca0bb3b4c}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:91041ee8-6312-4fd5-a991-f2bdacfebe23}} .
Lemma 3.14
Let {{formula:3c375ffa-112c-4300-8838-847f313151d1}} be a finite group with {{formula:e53c04fd-4a35-4a34-8241-3397552b37de}} . If {{formula:3ce988e6-9a77-4093-be39-ee4c64812928}} is the maximal normal subgroup of {{formula:886564a0-7d43-4415-b2ea-429e9db6c17e}} , we have {{formula:3bd5cd6f-bcf2-4196-ba08-d71adae83c92}} .
As {{formula:9912e585-dbb7-49c0-9f03-8562e1ddecac}} is a non-abelian simple group, {{formula:226a3083-1a6f-400a-86fa-3beecf436136}} , so we have that {{formula:f93b758c-936a-41c3-aa9c-f0219beda21e}} , or 19.
(a)
{{formula:4ef2aada-47c4-42b0-831f-cf97988be9b7}} . By Lemma REF , {{formula:f719ab06-89cb-4b2e-b2eb-fe0d4de3664e}} where {{formula:84b790e3-708a-49b9-9600-458eeb767143}} , so {{formula:de28304d-0bf0-4bd5-85eb-eda4222308fe}} . For all even {{formula:b0e54e52-5fdf-47f4-9419-cd6e1ce15722}} , {{formula:f619f76c-031d-4ca8-ac2c-9427d08bb240}} is odd. {{formula:2bf577dc-eb1f-4ef5-9359-4adb83ee3316}} is the only nontrivial odd codegree in {{formula:17ab6929-0678-4ed2-b21d-e6ff3b79920c}} , but None of the elements in the {{formula:c73f345b-f8a8-45d5-9705-cbb755fa524f}} is odd. Contradicting {{formula:45648a7b-63fa-4629-a197-80d3e1f63337}} .
(b)
{{formula:888c9bb0-6b3f-4b47-80fe-0dda3d74474c}} . By Lemma REF , {{formula:96bc6e9a-8dcf-47ab-a304-89c913aa427c}} where {{formula:ca0cfcf2-ff3d-4ae9-a235-e5f268bfce34}} is a power of an odd prime number. This implies that {{formula:29de3c33-4625-43ee-b0ab-065169f93625}} , where {{formula:0141a887-20f7-4b29-8d97-16d636956945}} . By a complete search, {{formula:9d55e846-612f-4cec-aa26-a6d5b6aaafc3}} cannot be equal to {{formula:8a78565d-8cc0-4ec8-b74a-c3363f2895a3}} or {{formula:7c1b4608-9b42-4598-9435-abd1d906d3c0}} in {{formula:aabd8051-26dd-4f6a-b642-e9a52e7ccd2f}} since it is half of another nontrivial codegrees, a contradiction since {{formula:41339b71-b62d-4cd1-9955-191aee7bebef}} .
(c)
{{formula:93f6b4bc-a7d4-4c2b-bf7f-214b36b75fbc}} . By Lemma REF , {{formula:07d820f5-7d40-48de-b5d1-53efb04a41b1}} or {{formula:f5797d82-207c-4b50-b5a8-734a945e41c2}} , with {{formula:38b7230c-b057-487e-97b2-ae21a9e50e2d}} .
If {{formula:ce53aa33-1bb9-41f8-8deb-4b3aa4e313e0}} , then {{formula:45fbe38b-bea4-47e1-b5e6-6505c80b88c6}} but {{formula:4df108e2-3b3c-4fe4-9a30-65f470ec4386}} , contradicting {{formula:5b34da02-92f2-42b4-b0ab-907520c36065}} .
If {{formula:184bf7ee-94af-42a4-8c58-b2e6570ddf46}} , notice that {{formula:fac2f4df-a22e-4724-a2a7-696ed7c35ed0}} . As {{formula:22d22c97-3492-48e9-a581-b495e2bd91b7}} is even, {{formula:e8f2bda4-1b1d-46f6-aaf5-034771c70193}} . This means that {{formula:df91ce3c-6cd7-4de7-8eba-1c802ea742fd}} since none of these are nontrivial codegrees. Contradicting {{formula:4e6dbe84-2c51-493d-b6f4-9f29574ea784}} .
(d)
{{formula:53d53a1c-81b3-476d-a160-6ed85060f9f7}} . By Lemma REF , we have {{formula:4150f501-7b09-4d3f-a48d-55785fc64e5b}} , {{formula:158f7b49-f269-4953-b589-8d5a2e37bfc4}} , {{formula:071b623d-4e68-4294-8af9-11312cab4930}} or {{formula:952e1a81-c6f8-479d-ada7-5ef5a03a78aa}} . The elements {{formula:18c77bbc-0bb8-44a8-b451-961c04b42e65}} , {{formula:6ada0256-72c7-47bb-8c74-994645f3f898}} , {{formula:1f836f66-b891-4d5a-84fc-5c8ba8db7af4}} , {{formula:d725b608-dc9e-475a-a3a1-79b90e5355a7}} are in the codegrees of {{formula:5d347a43-5873-4445-b99d-b85abe87e1fe}} , {{formula:b882982f-ef4b-4f4d-824c-2db942536c77}} , {{formula:5b33591a-a76a-481c-82ec-d24a6d5ede6e}} or {{formula:31f36f76-e74d-45dd-acd2-48fb877dfe90}} , respectively. However, none of these elements are in {{formula:d52fb00a-124b-4626-b742-f8cc91f47f6b}} , a contradiction.
(e)
{{formula:59bdb95d-cd7c-420a-9535-1c1aa97b15e0}} . By Lemma REF , we have {{formula:ab331615-941c-43e9-8d88-26f066757a96}} with {{formula:78cc4522-00f4-42db-b457-a031df71fe90}} , {{formula:c6c67693-4ddf-4966-b1e0-04c10cc9c9de}} with {{formula:1f152a4a-1b49-4ce1-9edf-f9c25bd99f90}} or {{formula:a907becc-0740-4629-83f7-9e3da691b45f}} . There are three nontrivial elements of {{formula:a4366d90-2238-4ca3-9469-cd6adef0213d}} not divisible by 3 are {{formula:e3186edf-5f5e-4f70-bc4c-b93569a3d976}} , {{formula:31c9d0da-4864-495d-a5ed-9a7d14bf2d49}} and {{formula:da561ecf-0ac4-4cf2-8323-82ea91ec590f}} .
If {{formula:2e46b64a-c9cd-4788-94f7-1a0e795aa991}} , then {{formula:6b9d54e1-a6ab-470a-97b0-4f972660de62}} . If {{formula:910298a4-221a-4131-af48-53bd0d0d1ad6}} then {{formula:5dca62e2-1df4-4b1d-8025-93c5fa0b0f69}} . Alternatively, if {{formula:318990cb-85e7-4a0d-853b-e67f53568a61}} , then {{formula:22d9ed78-c5d7-4f7c-9f94-d166f627295e}} . However, setting either to be equivalent to {{formula:ff884502-a3ba-4f5f-83ee-f18b9f68d798}} , {{formula:633d2921-4e45-422b-b54b-6dd1a5cec039}} or {{formula:b35ff148-2171-4b1a-b029-9f41bcfdb5e7}} yield no integer roots for {{formula:b08afc98-eee3-427e-a7d4-003b57357e04}} . Thus, {{formula:cfcfc235-9894-46d7-8c9e-ce39fbee0cb8}} , a contradiction.
If {{formula:056ed85c-2cbd-48f5-81ba-42d3abb1ab3a}} , we have {{formula:32dd2df3-05ed-4835-9f8f-005dba314aaf}} . As {{formula:e3ee5ffa-371b-4ae8-a66d-93b24e854b7c}} . However, {{formula:197bdef0-510a-4634-99fe-16fdadc1473a}} , {{formula:82e80729-be08-4caa-8a77-c57cdddfe027}} or {{formula:56eb7cc2-2e59-4625-a6ff-e4207cb1f234}} has no integer roots. Thus, {{formula:862b2c45-d9f5-413e-a73f-e15dc1876787}} , a contradiction.
In the final case, {{formula:b9a07af7-fe9e-475d-9159-4b04c2f31164}} but {{formula:c6309c3f-9711-4782-8de0-487bdc97c493}} , contradicting {{formula:891b560e-7fc8-4909-86cb-7207397d76f0}} .
(f)
{{formula:440c0d0d-aae7-4a6d-98da-d374be010a89}} . By Lemma REF , we have {{formula:de4e9e42-c998-4287-8693-0efd793babd7}} with {{formula:ccaabdac-05b5-4610-9bf6-af76fc47bc89}} or {{formula:407b92fe-1e05-4a57-992a-b92344020741}} with {{formula:c38a6ac7-943a-4345-87a9-b225ac2cd2ab}} . Both {{formula:0a29a4f6-714b-4765-b254-fa66750704c9}} and {{formula:66a5d133-5360-42c7-befb-a7a92a11c148}} are not divisible by 9.
In the first case, we have {{formula:62cce4e7-9462-4a84-806a-d671eae0bd2a}} . Thus, {{formula:ce415bfd-2b62-44b2-9cf1-3a1f954bcbc2}} . Furthermore, we have {{formula:ddb0e893-2e10-412c-aee9-810bd8827b32}} . This is not divisible by 3.
The equations {{formula:0237bc4e-465b-44b0-8ed0-ab20ae7ab005}} , {{formula:1d133590-18a8-4f8d-a5e7-6afb865f1fa4}} or {{formula:3b61a929-999c-4659-b850-f252f0b77d95}} , and {{formula:b5ae03c4-a073-45a8-82e5-f5f493945db0}} , {{formula:54d94d37-a72f-4604-9522-3d5fa21bcf58}} or {{formula:f98e480c-7aa0-474d-9c0a-d8fd58ef1528}} have no integer roots, contradicting {{formula:aec59a02-39ec-4bc0-9514-29d0d28d88b0}} .
(g)
{{formula:d4e1fe23-628c-4768-bdc9-f08d665c6d5d}} . By Lemma REF , we have {{formula:7e080aa9-1bff-44ea-acb0-d17ed560e8a8}} . However, {{formula:85ef712c-db73-4051-8838-78e7b289be5d}} is only present in {{formula:70b23f2c-e3ec-4bb8-b962-4dbaca1fe390}} and not in {{formula:f7c74688-c850-4d79-beaf-54e5f946c988}} , contradicting {{formula:02763fae-fc55-473c-b991-99c725310b04}} .
(h)
{{formula:2e6d3cfb-a3d1-4860-beb4-6388b79cad50}} . Lemma REF , we have {{formula:a8c7e095-f5a7-4a24-a125-566907a7691b}} , {{formula:371261f0-3621-486b-96fc-43e446ac5ddf}} , {{formula:fc00fc65-f720-488a-84eb-46e546a6d6d1}} or {{formula:4d1b08c1-8871-40f6-b0af-e17cde143018}} for {{formula:99861847-086a-48f7-98d3-5f74b3798949}} .
{{formula:a5f41e4e-a432-40f8-805d-c8afdb857242}} , {{formula:6365d118-7139-4a51-95cc-482b48536ea5}} and {{formula:c9d3233c-9943-458b-9f32-37938f7678e8}} contain {{formula:9a48f673-fa1a-478e-969c-68e7c4104ab2}} , {{formula:5959ec08-0cf7-40cd-9077-2466889ac05c}} and {{formula:0d9c771b-165d-437f-8d34-ce5fc4f5a9dd}} respectively, none of which are elements of the codegree set of {{formula:0299b7d6-bc88-4538-aa82-dbc836b3409c}} .
Furthermore, {{formula:29418458-a08c-4a5a-a0af-aa5af6ee6060}} . However, {{formula:42b7898d-44e2-4539-8c72-8681a74f011e}} in this case, so we must have {{formula:aef1bedd-9f5f-45f1-928c-7e92a4eab69f}} , {{formula:58e36aa1-a60c-4358-b707-803a44d94c61}} or {{formula:89bd6ed0-7484-4643-9df0-73ef858e3989}} , yet this does not yield an integer root for {{formula:f0f683e5-6e1c-4816-84b9-c83afe006867}} , contradicting {{formula:2fb43d4e-c57d-4e80-a320-bfa093de2519}} .
(i)
{{formula:0b6321ee-a2ff-40e5-8212-0b052899fc42}} . By Lemma REF , we have {{formula:af3ee003-91dd-48f7-b0f1-b290db02ac76}} . However, 17 is a factor of {{formula:eaeaaa34-29aa-4a18-aaab-bd2231f1071f}} and not {{formula:5e0356f6-c6bb-4f4e-b0dc-90fdc9797010}} , contradicting {{formula:7f75b942-9b30-492f-bbca-6bde23feb3b5}} .
(j)
{{formula:5a98c77c-7c34-4208-a331-7b9b40f6e53b}} . By Lemma REF , we have {{formula:6c855592-0d22-4f79-9459-4c1f1d9b8d2d}} or {{formula:bfc5e87c-826d-425d-8e2e-34e58269af9c}} for {{formula:1c7cac64-7136-4c6f-a3af-9d9ae7cd1936}} .
In the first case, the element {{formula:ceef623c-89f1-4143-903e-e0482be71475}} but it is not in {{formula:7dada14c-67da-4533-977c-f99ac0807d76}}
In the second case, {{formula:7001aae7-ddec-4c4e-aab9-c38226827a67}} . However, by a complete search there is no integer {{formula:9a3e080f-4df6-4dbe-ac2a-f99b40123337}} such that {{formula:612124c3-fb42-4edf-b281-330bcc91c7c5}} equals an element in {{formula:1a57bd48-bd50-48d8-88a3-aa731b148ad3}} , a contradiction.
(k)
{{formula:09e1f41c-934b-474a-b666-b96f46106b7d}} . By Lemma REF , we have {{formula:d7904dd7-925e-49a7-8755-30fddb77f8b7}} , {{formula:a2d4486c-2fca-4ad6-94da-2477e08cf5ca}} or {{formula:9b6c8156-f8ee-47f3-a28a-36c66e3e2f69}} . However, the codegree sets of {{formula:5b4a809b-474f-4e9e-8a19-c8a2a2b0022b}} , {{formula:ba8a2590-cfc8-4f06-8c2c-a25381574928}} and {{formula:cc3b759a-de20-40aa-a3e6-444d3ceb57d8}} have {{formula:34546372-fad9-4718-a4f6-128657c9f8c3}} , {{formula:af4715e1-6fd1-463f-b50b-8c8b7f23601c}} and {{formula:d17710fd-1a42-44c2-b9e8-a538f02d83e0}} , respectively, but these are not in {{formula:1e6be901-7151-4987-bd84-6bb93efbb00c}} . This contradicts {{formula:a5cdb8e1-d1e2-43bd-aad0-1c38ed2168e6}} .
(l)
{{formula:4cfa5886-03a1-435e-8485-b5b9c6ec178e}} . By Lemma REF , we have {{formula:047fbf85-e297-402f-9c2c-84d9f8f9aab6}} . However, the element {{formula:957707b5-e19a-4dbd-bd39-0f4ef90e064d}} is only present in the codegree set of {{formula:7c4a731f-a25b-47d2-a467-dec41740f478}} and not in the codegree set of {{formula:57bd2c0b-ba8d-4d4f-b228-db0f21720f87}} , which contradicts {{formula:d5156020-0e85-416b-bfdb-0d19a714492a}} .
(m)
{{formula:46333045-93ff-4656-abc5-d3d15ac0df01}} . By Lemma REF , we have {{formula:34c5d836-0d27-4cfe-966a-b3d1f5d8c23b}} or {{formula:b6360d3a-ac90-4adf-933e-90fb707e62b1}} . In the {{formula:cfa0685b-a066-477f-9978-073840678eb7}} case, {{formula:b3e3a66f-8562-4da0-bc4a-0ba7572c59ad}} but is not in {{formula:1c05c865-a31b-4475-93f2-c63c4df0a84e}} , while in the {{formula:604c26c7-89ce-4a72-9572-2b091a871c71}} case {{formula:c0f1975f-06ef-4469-abcf-f49fb89c066f}} but {{formula:d10c0d5e-3795-40e6-a773-f6ce7621f41e}} . This contradicts {{formula:92fe916b-4a57-4ca4-92ea-8b8b503bbfbb}} .
(n)
{{formula:a13391fa-0caf-4f13-980c-260a5fe7c9a3}} . Then {{formula:0d684ef1-5dd5-4428-a1df-9f18fac41ce3}} or {{formula:428d6a30-aca3-4bcd-a54d-aaf5e2dcfca8}} .
Notice that the element {{formula:5bb3e4a6-ae12-46f4-ba7f-aa3f4f1a6fa5}} is only in {{formula:98efa410-3a3e-4949-b828-9371f4c0e5d6}} and not the codegree of {{formula:cad8684b-7850-4aef-8cbd-1ebbf30d1742}} , a contradiction.
If {{formula:ed1fa7d8-85ea-45b2-bb3a-02532f3c8940}} , the element {{formula:8540f29b-da5b-44dd-aea2-3e1bcd216d62}} but it is not in {{formula:80699ee0-4cc9-4b14-8f02-f51ed9a14858}} .
(o)
{{formula:924f23e8-aae5-434c-bfbc-3e7ecb7e40b3}} . We have {{formula:a0edfefc-94d2-48bd-b393-f595c6f10c59}} or {{formula:8aa5468f-ef81-40dc-8103-10fe30589ad0}} . If {{formula:89cf9fde-3801-4001-bc0c-e2af6612ea46}} , then {{formula:70ebfa29-6158-43a2-836d-45629611c70f}} . If {{formula:200bd483-6050-4adc-81a0-1d93b5acd407}} , then {{formula:abe3725d-46e3-4314-8a0a-6d2c680dbf8a}} . Neither is in {{formula:4d8b0bca-e416-4df2-bd71-fa046df0ee22}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:0d91f66f-9f5b-4e70-a1cf-e1cff1e3344f}} .
Lemma 3.15
Let {{formula:bf4c92d4-da6a-4de4-a7f7-e60cf464991e}} be a finite group with {{formula:5ace5290-a85b-4baf-ad4e-ca599f8ae889}} . If {{formula:dc1eadab-99d7-44e7-929b-a25fe0345c90}} is the maximal normal subgroup of {{formula:2d15dc82-1208-4f1b-927c-7d305c2b8d81}} , it we have {{formula:3d10856c-e2bb-4cfd-9885-dcd0f2fd198d}} .
As {{formula:0d7efdd9-c3fe-41fa-99cb-383a12c3c436}} is a non-abelian simple group, {{formula:20b62fd8-c0e6-4c22-90dd-5b1848c760b7}} , so we have that {{formula:3439f761-0f72-422e-a82a-2ec94ef72077}} or 18.
(a)
{{formula:49a7f5cb-221a-4f78-8e4b-f60f72e85daa}} . By Lemma REF , {{formula:6de15105-865f-432f-9eb8-f3051f6bcaf5}} where {{formula:8bcd960f-1157-4f4d-bf67-436ef5df5954}} . The element {{formula:35ed8b34-55d9-4319-947d-715498f687d4}} . For all even {{formula:75cd9f8f-ce28-4ea0-aabe-3b8406a1c7bb}} , {{formula:8cb3654f-ff1c-4d34-a007-69903f6d8bfc}} is odd. {{formula:a77abf38-d315-4b93-b31a-c4e9575fe213}} must equal {{formula:0e895003-915a-4471-85eb-a6b2b6be71bf}} or {{formula:14b380a2-eed0-46ef-a120-f2e839ada6f5}} but this yields no integer roots for {{formula:0bb17a01-c343-4477-a986-8b5c2404bd24}} , contradicting {{formula:735372f2-4902-4b80-a7d0-5dcada3066ae}} .
(b)
{{formula:c070845a-1ded-4d67-9e37-6890ffb74cb4}} . By Lemma REF , {{formula:4de7b66e-8e2f-412e-9b1e-af7bb401d61d}} where {{formula:3cb3baa0-e110-456b-a5ea-ee24006dcd73}} is a power of an odd prime number. We have {{formula:0c0a0047-0180-412a-9486-4feea3a240de}} , where {{formula:2dcd7ae1-742a-49da-8d37-088ebb594e22}} .
By a complete search, {{formula:3ecf619e-5090-4208-a91f-446dc6509b1b}} cannot be equal to any of the nontrivial elements in {{formula:a1f68c6d-5943-4909-b657-4514be65b935}} for any integer {{formula:9a347793-c765-479b-891b-af933e6c0a86}} , a contradiction.
(c)
{{formula:aff6c4c2-0a6b-4317-904e-f1e9450bfcdc}} . By Lemma REF , {{formula:50f6a312-e264-4e04-b0f5-8cb9e321f44c}} or {{formula:e5d75f2d-9eb1-491b-8031-6d3e2ee27b4c}} , with {{formula:3c27e388-10a6-49c8-8b9f-5f039b42e7c2}} .
If {{formula:9afa8d2a-5008-480a-9e4f-26ce7e98e836}} then {{formula:a3b99ff6-7dda-4148-93fa-2daf06d6a8cb}} but {{formula:c76abdae-88a5-4568-a067-8feb432e7f92}} , contradicting {{formula:50d91402-32c5-4307-aabd-4bde0575308b}} .
If {{formula:57796659-221d-4d93-bace-66a1e78636eb}} , notice that {{formula:a18a5521-8de6-4dab-a0a8-88b4910e6b6f}} . As {{formula:a230a236-c634-409f-9c41-3aa4f0f16297}} is even, {{formula:e0a05935-8b8e-4535-a440-037236bb4eb0}} . This means that {{formula:27b90de7-f9c3-4d65-8045-5e3fad466a9d}} or {{formula:432d46a8-7b93-4380-9b68-8624ebb2b3d9}} which does not yield any roots, contradicting {{formula:d081b2e3-bd35-4340-b4a1-f27bf66d347b}} .
(d)
{{formula:915c6d5c-3e5f-4eb8-bab0-12a131e1b8e5}} . By Lemma REF , we have {{formula:5ade10f8-38d5-4397-a359-b92357f50185}} , {{formula:5c616b66-b7e3-4c05-94e4-45362f43df34}} , {{formula:5af36f22-0629-4043-987c-1e77ae5aeb50}} or {{formula:a9d5d6a9-41d4-473b-9d1d-a295ed4230b4}} .
The elements {{formula:706b0d40-32f3-4d9a-9e6e-510fdd0876d2}} , {{formula:2d3ec52f-8b52-4f7c-9ee6-122a33cf80b5}} , {{formula:802161d3-d559-4dcb-875d-1750e402f2e1}} , {{formula:feaa48fc-d2f7-47d4-8e76-4ef6c10841f5}} are in the codegrees of {{formula:920a7f81-54d4-43ef-bd8e-f3efa8eee84f}} , {{formula:d3b3e415-38ed-496c-8e22-6ad0fc57d7a3}} , {{formula:9db7d64f-075b-4d00-bfdb-98ab21c8f9fd}} or {{formula:eaf4f693-09cd-41f9-bbc1-5d7898972c85}} , respectively. However, none of these are elements of {{formula:761356cd-b4e5-44d0-a83f-1e2df461b920}} , a contradiction.
(e)
{{formula:8dca048f-9537-40bb-8225-3987638b1214}} . By Lemma REF , we have {{formula:7d4275dd-240f-4228-8c3f-955bc607a4bf}} with {{formula:e885e8de-bfaa-4db4-a744-0e50b492bc0f}} , {{formula:b0b4cb6f-a05c-43cf-b0aa-44ca6ceecb6d}} with {{formula:bffa95bf-a731-4648-8ded-be6b5e828622}} or {{formula:675abf65-c45d-43a7-a8bb-ab3131421651}} .
If {{formula:64150ec9-78d0-4870-a899-46f9864aac9e}} , then {{formula:db8dae12-e045-47b4-abc8-f447e7f2d609}} . By a complete search, {{formula:e8ccb1fd-e986-4162-b170-902f60e6c9fa}} does not equal an element of {{formula:cde9ba77-b911-4bee-afed-15819c5c8a65}} for any integer {{formula:275aee06-bccb-44eb-9457-69fc159be57b}} .
If {{formula:5fdbf54b-0ec3-4bad-8bc3-d78e333dd2b3}} , we have {{formula:481e2ffe-2101-4569-ae04-8f3c041cb708}} . By a complete search, {{formula:1df523e6-d838-4d35-b5c3-860160e7e16b}} does not equal an element of {{formula:2ad0299c-eac4-4f91-b2b6-3647554aadcd}} for any integer {{formula:062359dc-531d-4771-a0c5-5cc960d96e07}} . Thus, {{formula:b2589e1c-5be4-4a78-a6d2-c52546ce5176}} , a contradiction.
In the final case, {{formula:00c9f159-09ce-4e79-97db-de72e62fd019}} but {{formula:0ce5ca04-8850-47ab-a05d-6575c794d764}} , contradicting {{formula:c4949d91-9222-495e-88e5-26c9c217e559}} .
(f)
{{formula:10ad9907-cf44-4dba-9842-72b716a2932a}} . By Lemma REF , we have {{formula:c30fa33a-1779-44b3-b7b6-d63dc13c58da}} with {{formula:96828511-b9ba-430c-8f30-c9808090d58b}} or {{formula:ae4b66e7-12bf-46fa-8ca5-68639e74fc64}} with {{formula:317bbfc6-54bd-44a9-8ed0-5758aa59ee36}} .
In the first case, we have {{formula:3c60b9fe-4377-4933-a2b5-41e07c9dc07a}} . Furthermore, we have {{formula:e9ec7d07-e89e-428b-ba04-dcac325ee882}} . Regardless, by a complete search, these expressions do not equal an element of {{formula:42a33fea-be4e-4f96-984a-6740504fc7e8}} for any integer {{formula:58abb2c2-94ee-438a-9d24-a0bd800a771b}} , contradicting {{formula:1f4f1a5d-6673-4a8e-a04c-003b41eae7e9}} .
(g)
{{formula:0f3bd979-9c06-4a45-9127-420b93f4e327}} . By Lemma REF , we have {{formula:b01459c5-f85c-4465-8a32-e211b69f0963}} . However, {{formula:fb696745-634f-461a-acb2-0a573b30debf}} is only present in {{formula:94503db2-7762-40e6-8455-1993952204b9}} and not in {{formula:642af475-ac9f-4cc2-9fde-5f1884ed285a}} , contradicting {{formula:baeb5f6e-48c3-49c4-9436-2fbf517af91d}} .
(h)
{{formula:1a8030f0-1ef3-4383-b5be-78c97cfc9388}} . Lemma REF , we have {{formula:0998f013-c9b5-4aa8-abbc-d73b81a663fe}} , {{formula:8e68754b-2a34-45c8-9823-ac252843a24e}} , {{formula:1da555ff-068f-4198-b532-91d08df94e30}} or {{formula:d4da835c-0875-4045-8faa-deaec76d7953}} for {{formula:7d279611-da46-4e4d-8510-7fd84f5b7b6d}} .
{{formula:ba9d734e-7dcf-423f-a762-d2f803051111}} , {{formula:0096ea5b-3a21-41b0-bac7-8fdb85127607}} and {{formula:c45d2cdf-db2e-4bb0-ada7-9c9c3a332a32}} contain {{formula:5c5e27b3-4350-4f7f-bb7b-e2cdedc05c88}} , {{formula:d7957a89-3b14-4501-a2e2-120773bd77e3}} and {{formula:a1251554-32ac-4044-8cf6-b7f640a83bf2}} , respectively, none of which are elements of the codegree set of {{formula:23c731a2-6801-434a-ab72-f13a4e93a175}} .
Furthermore, {{formula:9ac57b0f-cee3-4c78-abef-0c5af8606c8d}} . By a complete search, this element does not equal an element of {{formula:a0d8e845-d37e-4663-b344-d018b4d438e5}} for any integer root for {{formula:42cb445e-0ed0-4c7e-8eaa-dff6fcf72c34}} , contradicting {{formula:0ff57db4-2cf5-49fc-bd2a-c82b91048727}} .
(i)
{{formula:1570f3ba-f047-4cf2-bbaf-31da1f1ba0c2}} . By Lemma REF , we have {{formula:507cc6fb-0baf-4d49-ae2b-eb12434a376d}} . However, {{formula:7ca121b4-43fa-492e-8c83-6ea002b62067}} is in of {{formula:0291248a-630a-44e1-a1d6-5a8808303ee0}} and not in {{formula:2f086929-4da1-4b72-a6e3-e421f51c1ed9}} , contradicting {{formula:772c7e11-bc41-45a2-b40f-007d445aba99}} .
(j)
{{formula:1f70d473-5415-4e4d-ac55-d7e3dee24c44}} . By Lemma REF , we have {{formula:b2542c7f-ce13-4e62-9923-d278ab033e3c}} or {{formula:99c9687e-173d-49af-971e-0b0eb9dec83a}} for {{formula:2e6d7999-b350-4f72-a3b2-293c7fd071cd}} .
In the first case, the element {{formula:4f4bf65c-351e-4414-98a1-1af64dd68cb5}} but it is not in {{formula:603cdad0-8a3b-43e0-a2c4-5005bf0f1c94}} .
In the second case, {{formula:cd5535c9-75fa-4a8b-ba9c-5cde86b04130}} . However, by a complete search there is no integer {{formula:bf262eac-eef5-4c98-acf4-00944dfe66c1}} such that {{formula:f9134202-147a-4144-a5f4-2dbdb45d5987}} equals an element in {{formula:72ab3ac5-2771-48bc-b38a-62dc721c4efe}} , a contradiction.
(k)
{{formula:b17f6a9c-be3a-4b76-9ac6-0ae9ea974464}} . By Lemma REF , we have {{formula:b6aad315-37b4-4401-b972-3a2dd0b0d0f3}} , {{formula:bbcf5feb-9ffa-43e5-b34d-9e765f291a9f}} or {{formula:51cbd26f-0647-42fd-8796-b5ca35a31950}} . However, the codegree sets of {{formula:fbf8a0b8-c8dc-425c-937f-3d0835f71936}} , {{formula:2cfa61b0-8473-410d-8693-ec35f9adc57b}} and {{formula:74689e63-563d-492b-a0fb-5b9ffd45223a}} have {{formula:6d36e0b2-c8fd-4433-8ea9-e073a0870d91}} , {{formula:7b6c01b0-9538-49fd-8cdf-43b3cad84905}} and {{formula:88c5f0ae-9437-4561-8f3a-88e50be5ff21}} , respectively, but none of these numbers are in {{formula:a0ff3641-a158-436e-9a15-60bef6fa3d07}} . This contradicts {{formula:cd6e9c11-a6a5-4804-8a38-d080c11217a7}} .
(l)
{{formula:04af4fc4-fbb6-43e4-9fd9-b9c462d07c45}} . By Lemma REF , we have {{formula:e441fc89-25aa-4a10-b33a-f65340719a84}} . However, the element {{formula:6d8cd555-5f54-4a92-a462-44f12632c325}} is only present in the codegree set of {{formula:3cb7c4e1-8834-4118-9217-d03b166bd484}} and not in the codegree set of {{formula:9c024593-761a-41fc-8019-f1d99132b10d}} , which contradicts {{formula:f292ab89-14a2-4ac8-a18f-2f63adcd35e7}} .
(m)
{{formula:918f142c-c90d-4bf6-802f-7e138b7f45c4}} . By Lemma REF , we have {{formula:cb6aaa5a-bf91-494f-b226-ca1be92db509}} or {{formula:0ac2fbc2-f580-4740-a7ee-9963ca95f5d8}} . If {{formula:35410a25-c8fb-49c0-aeea-c7492c45fde2}} case, {{formula:80bd0514-dee4-460b-a71b-95ea316c379e}} but is not in {{formula:6c623d68-4f45-4d58-baa5-978c2a739dcb}} , while in the {{formula:3b69f225-c706-4dca-87bc-bb30b5855c14}} case {{formula:f7c4c6a8-59f5-46db-ba4c-49e6e68976de}} but {{formula:25041c64-0e9c-45f5-88e6-b0ba80a82a67}} . This contradicts {{formula:8cdf09b0-92b7-4b95-9879-eecac0b667c1}} .
(n)
{{formula:83748d2e-9cad-4cb9-9acd-54ba029567f8}} . Then {{formula:f4f1bde8-6ef3-4dbf-a396-2410a857927c}} or {{formula:e73f94c4-138b-427b-9402-5ae148b62189}} . However, {{formula:3067cb9e-d213-4b79-a76e-1dd8a24fe68e}} is only in {{formula:51620b61-7eaa-412c-b8d5-a67dec627075}} and not {{formula:998175f4-9475-4f7a-a671-2884b140412a}} , a contradiction. If {{formula:531ebeb8-20b9-42ae-a42c-c2ccb8ba67bb}} , the element {{formula:10b02b2a-8322-4cf2-bde9-79978481f491}} but it is not in {{formula:2336a616-7bc0-4243-831f-6a11efc179a7}} .
(o)
{{formula:d020352e-812f-4dd6-8716-7d3f388ed5a9}} . We have {{formula:4762d691-b169-4d77-9a11-ff44c36244eb}} or {{formula:9ba2a76e-7be7-4055-8ed2-28dae02ad3bb}} .
If {{formula:b9979e17-f3cc-43fd-8f47-b9cd379ccae0}} , then {{formula:c46a6dfe-9b0b-4207-8373-9634777d94c0}} . If {{formula:c12d6436-36a6-4467-a7a3-0483d21444b7}} , then {{formula:97278d1a-0c2b-42b0-b6be-4ae98ee6e567}} . If {{formula:fd2924b4-49ca-4e8c-843f-404e73a72144}} , {{formula:5077cea5-539f-441f-9036-0594cec7bc64}} . None of these elements are in {{formula:ba191b6a-a58e-4ca3-8bcd-5bdce570bb02}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:6977cf54-d7a0-4f0f-beea-fe78d9b48abf}} .
Lemma 3.16
Let {{formula:c656425f-2cb1-4b05-8f55-fbbc272861e0}} be a finite group with {{formula:78a64c1f-0688-45ae-b11e-45ff172bfe7d}} . If {{formula:8c03848e-d797-45a1-8557-8a93caa26e2a}} is the maximal normal subgroup of {{formula:838e5ab5-dfe6-40d6-9db1-e990a84ba682}} , it we have {{formula:e869a9de-1fd9-4f94-b528-fbaa39821659}} .
As {{formula:ff175b7b-c5d6-49aa-b9bf-bf9c2b143046}} is a non-abelian simple group, {{formula:870899f3-a3dd-4b3a-b554-2ff85ff09c69}} , so we have that {{formula:e82a0f90-31d1-438c-9791-271f7cda286d}} or {{formula:e9b12150-2802-4e39-9206-aaa4e18c04fc}} .
(a)
{{formula:0762c2b6-218f-4aa1-9ed6-2081b9fb54b0}} . By Lemma REF , {{formula:f945876f-c432-49b0-83a5-52632112280e}} where {{formula:5fd5a9bb-a603-4013-9e42-34242c82f778}} , with {{formula:b631facb-7c31-4d7f-816b-10affc5471c5}} . By a complete search {{formula:a84d3eb5-dcd7-45c3-beb7-ed4b43aafe0d}} does not equal an element of {{formula:045eae20-d4ae-4886-9975-6839b28ba593}} since none of the nontrivial codegrees are odd. This is a contradiction.
(b)
{{formula:46da25c3-baef-458d-b504-efa0a53174c2}} . By Lemma REF , {{formula:13ba6e6c-4a0b-49b9-8971-b8c66f15be76}} where {{formula:581df5a5-e25b-481a-a0d5-d0e48e0bb2d4}} is a power of an odd prime number. We have {{formula:b2aa70ea-49f5-44bb-82dc-7d0f79374e88}} , where {{formula:ec08eea1-6f86-40bf-b936-c16184b0bc8b}} . By a complete search, {{formula:d084940b-97d7-488d-886f-49fb81c581ae}} cannot be equal to any of the nontrivial elements in {{formula:e1ebaba0-6d90-4508-b678-fc374ead9274}} for any integer {{formula:01caf3b3-79b1-4517-9c5a-0cb31498fbb2}} , a contradiction.
(c)
{{formula:5d4f15ed-a7d8-4efd-b4ef-ecffe275b1d9}} . By Lemma REF , {{formula:82e779ff-cccc-43e6-81c6-88d9837dd5c4}} or {{formula:862fe4fd-0b57-47b8-a5c3-e68c6be12012}} , with {{formula:0f991829-e544-451d-95e7-db2ee5f02a10}} .
If {{formula:1f2a9d3f-8932-4d15-8a93-6d4c3a008c85}} then {{formula:a57fcc1c-4f5b-4f33-9ec3-1d3ab546b2a9}} but {{formula:de3af79b-c703-4262-b378-24b3da7641c8}} , contradicting {{formula:0f3aba9c-5786-45e4-8b67-9a84714bb82b}} .
If {{formula:93ede761-f45b-4a90-bfa7-9c7ba7e17c67}} , notice that {{formula:41d6e8bf-e869-4e73-87e4-3265fd3dd918}} . By a complete search {{formula:93773ebd-6607-49f2-ae87-724679b8763f}} does not equal an element of {{formula:30de5e80-e92b-4096-bcb8-d7c8adfd2cab}} for any integer {{formula:afb8f7bf-c65f-4049-a7f9-9ca24270f24f}} , a contradiction.
(d)
{{formula:c84c8aad-485e-4817-af7a-c9ca8c74e38d}} . By Lemma REF , we have {{formula:bda208c8-c34b-4f94-96d1-ed4723a1cd94}} , {{formula:5e65ae2a-4bfe-4341-9193-09121c158e38}} , {{formula:84e4cfb7-842b-4cb4-a733-c9f7210ebef4}} or {{formula:d1069c3d-ee43-4ba6-99a0-806ae9f460a4}} . The elements {{formula:e5747b1c-6247-43b0-92d0-58402535e817}} , {{formula:93d8b740-c5ba-489f-bd07-5902f7d831c5}} , {{formula:5c15bb87-ab6a-455d-9ed6-4f5bd2f6700e}} , {{formula:f56f51a8-b83f-40c8-82cb-f261eb4e2f31}} are in the codegrees of {{formula:72a4fff4-a0eb-4d0a-b92a-3a368466dffe}} , {{formula:a3882e9d-5456-4a57-8a25-38ff9b8b90bd}} , {{formula:f54da9f5-0759-4965-93de-085603a0604e}} or {{formula:3b601663-2df1-49f1-814d-b6135558c2b7}} , respectively. However, none of these are elements of {{formula:0be47488-0346-497c-8cd5-d95d17c83418}} , a contradiction.
(e)
{{formula:1371525a-efdd-47fe-bb97-8308428da3f2}} . By Lemma REF , we have {{formula:d00bb6d4-613e-485f-bfe1-97418471c2a3}} with {{formula:522b9313-32a1-47bd-97a0-b749f68bfd93}} , {{formula:452a2137-97c5-446d-b0ac-b0ad09186ed9}} with {{formula:1f99d553-1e2d-480f-852d-79f293cbc38e}} or {{formula:9607eca7-380c-47cb-9206-e05460b75bbd}} .
If {{formula:d7503008-6c2e-4308-b484-d5ba7b541729}} , then {{formula:309d6863-963e-497a-bed2-5cb239018a35}} . By a complete search, {{formula:d5c6108a-80f8-480a-a4f1-6656ddfbd021}} does not equal an element of {{formula:f6ad4e6d-40ee-4545-891b-73cf966f1253}} for any integer {{formula:aa1a2119-6e22-45f4-83b5-efa8297f1059}} , since none of these elements are odd in {{formula:b018827d-1ca8-4639-8143-8173429de74b}} .
If {{formula:8637e61b-fbf9-49f2-a53b-9f7d229c8de8}} we have {{formula:90da2bbc-6430-4f80-b01c-2b27f7ea67c8}} . By a complete search, {{formula:571ffe45-1858-48d3-828c-1b024db4ac33}} does not equal an element of {{formula:a04076b8-77e5-4c2d-b4cb-aea8a2c52494}} for any integer {{formula:2e98f33d-4070-4d01-bfec-9b3027e086f6}} , a contradiction.
Furthermore, {{formula:9ecf8f3e-826a-4aa7-a1a7-f3298d997f68}} but {{formula:563acb0d-b03b-40f5-b140-e11f014ccc5e}} , a contradiction.
(f)
{{formula:729af621-9517-46d1-88b4-d8f9d805b18a}} . By Lemma REF , we have {{formula:47fc4d2f-9abd-4067-a311-9ac76848ed2b}} with {{formula:03b148f3-c2ce-4c41-99c9-7e15fe4e8c2a}} or {{formula:1d4d17e4-11f7-40d8-90a0-86777503242e}} with {{formula:42ef2b2f-f622-47a3-9d54-b304ed763a43}} .
By a complete search, these expressions do not equal any element of {{formula:ba88738a-d242-4523-8c43-c2fe87ccd94d}} for any nontrivial codegree is even in {{formula:f5d7087f-3555-42d6-8d87-e2b3dbaa2a36}} .
(g)
{{formula:c09635cd-4a52-4754-9cc6-42e788a978d3}} . By Lemma REF , we have {{formula:54c5eab5-1dd2-4b28-8df2-bf30baa961b1}} . However, {{formula:59052b20-f3b2-4f00-a578-1ae6ba1ddc5c}} is only present in {{formula:6acd8c5e-2b25-413c-811c-eb65560176f8}} and not in {{formula:fae8be93-2e47-477d-a33e-7c3bbd6cde58}} , contradicting {{formula:00698807-75e1-4cd8-bc5d-4a5ee8f71f47}} .
(h)
{{formula:5ced1cfa-b6d6-476a-9ebb-539140dc7695}} . Lemma REF , we have {{formula:6c53a63b-e978-4cd8-b1c3-3f5cec8f06f9}} , {{formula:b945f98f-e7c6-4341-ae79-933c398cdcbe}} , {{formula:0d0cb0dd-6569-4b2b-ad5b-6e6a6281f966}} or {{formula:be87c851-d2a8-45e9-90ef-736ef8d3e145}} for {{formula:b144ebdd-5c67-4a61-bc4d-6821018a0f07}} . {{formula:aaddbcff-34eb-444c-9210-ad3a737725aa}} and {{formula:b719dd53-2988-47ad-859b-32a2b93a8f49}} contain {{formula:67f6657b-59e6-4f7f-a192-a47a15c0b76d}} and {{formula:41e658be-da7a-4940-87fb-f4b4e4bd2ae7}} , respectively, none of which are elements of {{formula:799f73a7-28e1-431d-9003-5c25d290f0b9}} .
Furthermore, {{formula:39fca523-07fe-4445-8187-e83f09768c35}} . By a complete search, this element does not equal an element of {{formula:07de3971-2021-40e8-bc4a-eb37dbb02bbb}} for any integer root for {{formula:b768452e-e6bd-462d-898f-ab51d117fda8}} , contradicting {{formula:35176e46-2321-4949-b523-81f652302845}} .
(i)
{{formula:e0fe14d7-473e-4c19-bb28-13c201fa7efe}} . By Lemma REF , we have {{formula:923eb3c4-c371-4fc5-a6a3-2b8931a8003a}} . However, {{formula:450146c7-db9e-428c-92dd-ccf040d84a76}} is in of {{formula:1e605fdd-f3db-4995-bdb8-b3de21d52709}} and not in {{formula:2b6ffed1-f433-417c-909a-4a60d25efc26}} , contradicting {{formula:cf170e89-d318-475f-b73b-3befefb0595c}} .
(j)
{{formula:9028e5c3-3e3d-4335-a6e8-fe01a2517b6c}} . By Lemma REF , we have {{formula:cc618db9-db2b-4d21-9b14-63d3d96cdf64}} or {{formula:bdcbe0f4-b392-448d-9cb7-3c167b220de2}} for {{formula:c92b4442-aac3-4354-9092-b9cf352693fa}} .
If {{formula:737201c9-9367-4beb-91bf-50865888796b}} , the element {{formula:8f3cc455-7c72-49ca-a944-99721d876ab2}} but it is not in {{formula:d9671ed4-67d9-49c0-a27b-e5406b339af2}} .
Furthermore, {{formula:b2794a5f-962e-4057-a5f3-795651d6c18a}} . However, by a complete search there is no integer {{formula:4c1181a9-e6fe-44db-b664-2c689fd58e7d}} such that {{formula:a458e02b-d430-4f09-805a-29a4734c3b7d}} equals an element in {{formula:e7b23d0e-9565-4d81-8a0f-766b63e0683a}} , a contradiction.
(k)
{{formula:fcfa00f9-5334-41d3-bf95-b60c8e8f326b}} . By Lemma REF , we have {{formula:92cc17d7-3807-43ec-98d9-6cc8a1e2bc3b}} , {{formula:159fed91-ed85-4893-8ca2-953989ce3b47}} or {{formula:fd9bb353-ed24-498d-8d52-2173d9fb699b}} . However, the codegree sets of {{formula:cdaf5241-3cfe-48b7-8e31-f34781915879}} , {{formula:19047a0a-cb1d-4ded-95e0-a6f48731ec67}} and {{formula:6291795e-d6b9-4d8f-b62e-feaba64adc53}} have {{formula:771f1881-606f-43be-9bb1-f0d2ef0f9a78}} , {{formula:667a1d32-85fe-4096-9b67-72703bb2c443}} and {{formula:c0e2f138-b641-4b9d-9a0a-0e4b34b80d1e}} , respectively, but none of these numbers are in {{formula:3fbee89e-b6c9-41e8-8b61-836002eaceb3}} , a contradiction.
(l)
{{formula:869e8627-2db3-4aaa-be5c-d7308b7dcd0d}} . By Lemma REF , we have {{formula:26de9fd1-b0bb-4782-a1a8-63e1121804bb}} . However, the element {{formula:50349780-ece0-464b-bae8-90d02476196c}} is only present in the codegree set of {{formula:efc8bea2-a859-4fa9-8769-a40c216eec20}} and not in the codegree set of {{formula:0491d9e2-74db-4f73-bd61-bd41180eab8c}} , a contradiction.
(m)
{{formula:d0876d86-77f7-41cb-930d-4a8283b6ec99}} . By Lemma REF , we have {{formula:8935667f-b85c-4e07-888e-c04569d0e7fc}} or {{formula:b4eb139f-d144-40ea-a8d1-8b5510910919}} . If {{formula:6763ee52-7359-4a89-b4f4-4ec7ee95df98}} , {{formula:629ba663-6571-4352-9f5e-02230a6a83d9}} but is not in {{formula:011515a9-8606-484d-82de-9c7ddeaf518f}} , while in the {{formula:375e4d9e-26b6-4a51-bf12-b04302b6d2d6}} case {{formula:f4d1f37f-d44a-40ed-8267-52661e305e05}} but {{formula:4fd9b578-fbfe-46d9-b1c5-3cd8376de067}} . This contradicts {{formula:7ad9e0dc-b0d4-4404-bcf7-20d9e1248141}} .
(n)
{{formula:163735eb-1f7f-4bde-834a-40c03e5ab299}} . Then {{formula:68b00c37-6813-4443-b586-a214af94433b}} or {{formula:0ba6ca52-432f-4c88-961d-5d7e0e57de32}} . However, {{formula:824b8820-03aa-4aba-9010-51358bff901b}} is only in {{formula:21617fb7-c5f6-4d00-83cd-c34a3890b2d5}} and not {{formula:466626e5-a053-4674-9043-f6e074ef46e3}} contradiction. If {{formula:a9f25755-044c-48d7-9075-2ac3a50cdf54}} , the element {{formula:0d340397-7dfe-493f-b04d-edff5f84126c}} but it is not in {{formula:158bb335-551b-4319-9553-da29e4d66513}} .
(o)
{{formula:f84cdb79-205e-464c-87d6-e6f81784cfc8}} . We have {{formula:4de7a8b7-5f18-4a6e-80d2-ba17c3147a04}} , {{formula:071079fe-be17-4034-bf70-95259f1050be}} or {{formula:9fa36be5-1a86-40fd-9627-2f27fb01687e}} . If {{formula:4405ee67-d2c0-499d-abe4-d1da84668c12}} , then {{formula:1f6acc95-7d33-4296-a984-60ed6dde1ff2}} . If {{formula:b1070894-746a-486c-ab97-a5cce764f43e}} . then {{formula:29302c50-cb13-489b-99f1-58212515d134}} . If {{formula:cf2a28bf-f090-464b-8988-f3b6c4dc2fcb}} , {{formula:424387d1-2bda-4de6-a419-0315f7f742e1}} . None of these elements are {{formula:56e213f6-cf6d-418c-94bf-b1443b42a453}} , a contradiction.
(p)
{{formula:a618089c-7629-4694-b295-942781e89fad}} . {{formula:4b6cd31a-d067-4389-80c8-41c7deb6cfd2}} must be isomorphic to {{formula:c62182e8-2b78-4984-9c3f-f6ab561ab49c}} . However, {{formula:ce8599ab-0d63-4437-8dcb-cdf41621f745}} but it is not an element of {{formula:a2b7fe13-9f5b-4c77-8a42-ec6610638b3a}} , a contradiction.
(q)
{{formula:298f1130-c468-46a9-bf75-74aeeb66513a}} . {{formula:e7d17e92-7f5b-4179-b317-f2a4bd5819a1}} must be isomorphic to {{formula:dc4e7a29-0272-46d4-8722-9989381eb84a}} . We may find from {{cite:2a961900de284170beb08505166b81764ed9ce18}} that {{formula:a2b1baf2-0559-4f43-9845-8277ad84ad48}} is a codegree of {{formula:ab751348-cafc-43d3-be32-57053fcdcf47}} . However, from a complete search, we may find that there does not exist an integer {{formula:3f5cc747-5acf-4a3f-a73a-e8c37bc51ce8}} such that {{formula:b169cfb2-f202-4fcb-a973-95f3eaae6cf1}} is equal to an element of {{formula:5155c919-2ed8-47cb-9aca-49110a4a965c}} , a contradiction.
Having ruled out all other cases, we conclude {{formula:d6d03259-c87d-4a9d-9647-23b535487df6}} .
Now we give the proof of Theorem REF .
Suppose that the theorem is not true and let {{formula:afbdb02f-a3f2-45bf-993d-1cde317222b2}} be a minimal counterexample. Then {{formula:d8eb6dde-eba8-4887-9371-dbb73e3f26b6}} is perfect by Remark REF . Let {{formula:1c03124c-d48f-467a-a7cc-9a79a7ba0110}} be a maximal normal subgroup of {{formula:4e3a955c-2342-4b0c-9171-cdf0ec212733}} .
We have {{formula:c3ee89cb-d6ae-4497-91e7-0131500c715a}} by Lemmas REF -REF and {{formula:b7f17862-d693-4899-97b6-1805ac53b02b}} . By the choice of {{formula:8ba4c73f-ff96-481e-9d11-97aad3c79d12}} , {{formula:a1e03733-aed5-4134-88cc-57eae85113ca}} is a minimal normal subgroup of {{formula:b278d5a8-5db3-471d-bebd-83fb7968caab}} . Otherwise there exists a nontrivial normal subgroup of {{formula:d133b214-4f05-4dff-b5ef-93e9c7d288d1}} of {{formula:b9417648-297d-42d2-b5a1-4604d416ba5c}} such that {{formula:e0fe816a-e7bb-47df-aa64-ee51cd9443ee}} is included in {{formula:e80bb62a-b4fc-420b-8c43-54327fbd2903}} . Then {{formula:0418e78c-de7b-47b7-8983-a262e60f8a35}} for {{formula:960c88b3-5c29-422f-b14f-a27ecee3bb47}} and {{formula:c05cb763-2229-4415-b16f-f76d1cb73413}} for {{formula:60906137-5de4-400a-8813-6f0c28804104}} is a minimal counterexample, a contradiction.
Step 1: {{formula:8904ea3d-4583-4b90-8a00-e90882e13e51}} is the unique minimal normal subgroup of {{formula:92cee7a7-93a2-44d5-be3c-fdf77abaf615}} .
Otherwise we assume {{formula:1970d466-3611-4ccb-bfeb-09480b0ac515}} is another minimal normal subgroup of {{formula:5a2763b3-e028-46cb-bb2f-df7448a21e98}} . Then {{formula:f6fcffac-34dc-4e19-b9a8-86a24db0da24}} for {{formula:8fdf0782-3594-4531-acfc-e89715ed3bce}} is simple and {{formula:a0864ede-2e09-4a29-be03-0322f3190803}} for {{formula:10fc51d0-a5d1-45ef-b69d-3e63e3af9e01}} is also a maximal normal subgroup of {{formula:e7e570c9-1c2b-4d15-858e-37773e3d0587}} . If {{formula:fc5479e0-8814-4992-b841-275c0d9bb920}} , choose {{formula:b792a9ad-6cfc-40d9-a813-eb3111ceee47}} and {{formula:accbe66a-72a2-4ddc-a5d6-c04107eb511f}} such that {{formula:a3189909-7cad-4ad8-821d-030d49d70ef3}} . Set {{formula:e5908afa-7b5e-4c83-8cf8-62cdb709d4cf}} . Then {{formula:4447e16b-0cf6-4386-bc36-142d99f9d83d}} , a contradiction. In other cases, we can also obtain a contradiction similarly.
Step 2: {{formula:871cede8-22fb-47c2-ab7e-d50a2a1e856c}} is faithful for each {{formula:65a1a9b6-d311-4b9c-a553-02a13092022b}} .
It can be easily checked that {{formula:f8d7f098-d2b9-4883-a884-b985afdafbab}} is not contained in the kernel of {{formula:83d5ef3b-6cb5-4125-b5e7-826e09f3c738}} for each {{formula:469bfcf5-bf94-48b8-aaa8-636fc82283b4}} . Then the kernel of {{formula:d7aa0d3b-f4ed-42df-8a3b-242caaa6b986}} is trivial by Step 1.
Step 3: {{formula:72519795-4e8b-491e-8b8a-88a5c546cf56}} is elementary abelian.
Assume to the contrary that {{formula:2b78d3bc-b4ec-44a5-b549-3eefdf4b993e}} is not abelian. Thus {{formula:15fe9817-ac56-48e4-966b-cc93a86ed054}} where {{formula:35d2aeff-95cf-4577-a2ee-4f980c61ca45}} is a non-abelian simple group.
By Lemmas REF and REF , we see that there exists a non-principal character {{formula:17afd9ef-713f-430b-8859-ee7d5a83618c}} that extends to some {{formula:a5775aa3-8ac7-43c6-aa49-746956bd7119}} . Then {{formula:2ebb951e-5a02-4e14-ac9d-5733b4e75d99}} by Step 2 and {{formula:80b2ca84-b055-4408-985c-653995fd3f44}} . This is a contradiction since {{formula:0ad5aab1-7290-4d92-9812-3f7e2cb863ee}} is divisible by {{formula:31f00b63-59ab-47f4-9558-1172976b35b7}} .
Step 4: We show that {{formula:114246ad-a180-4a6f-9fce-b46da97ff82e}} .
First, {{formula:e03f5421-2a9f-41a9-8359-1b8461cc41a1}} . Since {{formula:bb2aca98-d500-486c-8362-8884f8ff2bf0}} is abelian by Step 3, there are two cases: either {{formula:bc2d8472-9405-40ec-bc9e-4d9895286f95}} or {{formula:84d8761b-43ea-4c7a-8db4-44e6236d663c}} . If {{formula:5076c132-c75f-4fc5-b2fc-7ec42023c012}} , we are done.
Suppose {{formula:2266afb1-15d3-4368-a2f1-c2eb75891d0c}} . Therefore {{formula:96193497-28ee-4dfc-bc9b-48425357a7fd}} must be in the center of {{formula:bdc3e24a-5812-44cf-a8b7-71c0e5317cda}} and {{formula:76689c38-76bb-4ca9-8976-7f4b691ced70}} is a prime by Step 1. Since {{formula:981086d2-f9b4-4da4-9a2b-2cff9ad26e04}} is perfect, we must have {{formula:bb595dfd-b216-465f-80f3-0a499efdb1b1}} and {{formula:6f6e1262-873b-436e-90ae-e0cbd6f9f12b}} is isomorphic to a subgroup of the Schur multiplier of {{formula:e1270a48-71f8-4040-a3bd-a3dbb79ddc78}} {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}.
If {{formula:0bbfbb3d-3656-4638-aac9-e26f52dce33c}} is isomorphic to {{formula:53c42161-576d-40c8-b9a8-e6d65499c797}} , {{formula:840627af-72ee-4c28-bddc-2cbe6991b382}} , {{formula:6299a404-7ffe-4bc6-a4e2-68f16175fd11}} or {{formula:35b69270-fad5-4e6a-9718-149731751caf}} , then by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}} the Schur multiplier of {{formula:095906a8-bf83-4286-974e-8cb0f8362f8b}} is trivial which means {{formula:63ce12c5-c04b-471c-b08c-ca56ca083254}} , a contradiction.
If {{formula:5360146d-4b7e-41c8-8cbd-244e27ec45b6}} is isomorphic to {{formula:52ad69f3-fd17-4d88-b4f1-1940f2590c92}} , then {{formula:24ce6f29-1c3b-4ed1-89e9-b986e1054b7e}} is isomorphic to {{formula:5b67efc5-e54e-48c8-a15f-63d930e7cf6f}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}.
We note that {{formula:77b93b38-a965-4d33-9024-497026b0439b}} has a character of degree
16 which means {{formula:c67e85f8-606a-40b0-aae7-24c74c3f385e}} , and this is a contradiction.
If {{formula:67180cb9-c5f7-4a59-b6fa-68341d74ef17}} is isomorphic to {{formula:f629743f-9564-42da-92ed-f06834348cec}} , then {{formula:7af01f6e-cf27-48c7-b3d8-c5106dae40bc}} is isomorphic to {{formula:05f31620-6f9d-4533-92b2-f7ead0fa35e8}} or {{formula:1b4c6a47-5936-496d-9de4-c77fc7ca0c1a}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}.
For {{formula:3bb0333f-4548-41e0-bac7-1f5c218e2ffb}} we have a character of degree {{formula:3fee2841-d367-49fa-95f6-497e9483bfb4}} . This gives a codegree of {{formula:44f96e16-3c1e-464b-b5af-2f53879c8174}} , a contradiction. For {{formula:df80274d-dd04-4305-a868-0a58cdf994d3}} we have a character of degree 729. So {{formula:8d8e2eb3-e92a-408e-913d-d762d6257319}} , a contradiction.
If {{formula:e5f4496f-70d2-408d-9078-50e84c5858c2}} is isomorphic to {{formula:925d9840-a37e-47b6-9cc8-28cca25fb656}} , then {{formula:853199ff-82f2-41d9-a3dc-822847f2c20d}} is isomorphic to {{formula:bd2e1afa-8ed5-45ea-8602-35be57084ef4}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}.
We note that {{formula:033b74d9-f4ae-4879-9cd3-8aa7b0c263cd}} has a character of degree {{formula:cea02f21-e43e-4221-8953-786a5f11ba19}} which means {{formula:c49abbce-99d8-4315-9e2f-6a7ec48e3ef3}} , and this is a contradiction.
If {{formula:a2c96f3d-85f2-44b4-b286-532b260329ea}} is isomorphic to {{formula:0f8b60a0-0e07-4e7c-8f19-7a3a237f36ad}} , then {{formula:c6127a26-bc6c-460a-b738-65971084054d}} is isomorphic to {{formula:ca1c1092-efad-44c4-9de2-8d6d76b8b51e}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}.
We note that {{formula:600c0238-ec7b-4916-8e2c-e9ba7c49a69a}} has a character of degree
5832 which means {{formula:e9499d5d-8db2-498d-a946-b2c4d8a4be85}} , and this is a contradiction.
If {{formula:105f8bc3-c8ea-46a7-b33b-3a4df461e476}} is isomorphic to {{formula:1bd3b69c-ef4b-4bb1-b6d6-bf5052e8041a}} , then {{formula:8589568c-c395-4624-b6d8-642a7d658ea1}} is isomorphic to {{formula:aac74df8-fcc3-456c-86f3-0328d663cb7c}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that {{formula:ee1329ae-ac46-4369-aef8-4cfe5acc30e4}} has a character degree of 8 which means that {{formula:d219e5fd-baf1-4f91-995e-8ba841c82274}} , contradiction.
If {{formula:9a17d96e-d905-4b60-8364-8e80f2b2e326}} is isomorphic to {{formula:2c2eecfc-1fc0-4054-b366-3c57d396acfc}} , then {{formula:9542a8ef-2b2c-4d6d-ac3c-2d2faf98f9b5}} is isomorphic to {{formula:1977ea34-283d-4931-be2f-38be16283bef}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that {{formula:4480a49f-a869-4c08-92d1-01ccda18b128}} has a character degree of 6 which means that {{formula:2b208959-22de-417c-a833-3c56258fdf43}} , a contradiction.
If {{formula:885ea964-2fc5-42a8-a8b5-cce6aefa604b}} , then {{formula:6c260b80-8246-436b-a348-3536b9bc06d2}} is isomorphic to {{formula:e957c79b-4d67-473a-b598-43e1510fd9f3}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that 40 is a character degree of {{formula:6be07077-d2cf-4aae-a61a-86d695eb803c}} , so {{formula:d0131dd6-eee8-4e08-8fa9-4563a041f803}} , a contradiction.
If {{formula:421c526b-222f-4ac4-a12e-02de60e93c85}} then {{formula:0f11fc4a-5d83-4644-9941-12275a9a0ffe}} is isomorphic to {{formula:c4394f9f-a6b8-47e8-a4f6-66c333f49ece}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that 126 is a character degree of {{formula:80b2e64d-48af-4134-80fe-c32a2427eb44}} , so {{formula:4f8c3865-d3bb-4318-8158-7055796ff928}} , a contradiction.
If {{formula:9d56ef07-ac35-43f0-96c1-528f4744ffee}} , then {{formula:76f75932-0f85-44ce-901a-8ff2befd4da5}} is isomorphic to {{formula:8332f0f5-fcfc-4f29-b59a-df1a3c45637a}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that 12 is a character degree of {{formula:cd5c8255-dc89-4bd7-9afd-afc455d8f140}} so {{formula:1788a5db-95df-4ab4-ba4b-fc25f8643c31}} , a contradiction.
If {{formula:c51f644c-8fd5-45c3-95dd-8994f53f618e}} , then {{formula:311b739e-8284-4c3f-83f3-32b7564306bf}} is isomorphic to {{formula:25815235-cb7a-41ad-b281-e7fc2130baee}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note {{formula:7c0a284b-cc75-4942-804a-aeafa258ce1f}} has a character degree of 12 so {{formula:78861601-a355-42db-8960-e944c369405f}} , a contradiction.
If {{formula:52fb67e5-0bba-452d-8162-3db9e7b5e620}} , then {{formula:77cb9596-5386-4933-b7e1-6416bcf9f365}} is isomorphic to {{formula:6fb25de2-5721-43ad-af9a-ecff2499893d}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that 56 is a character degree of {{formula:4ea89088-8cb5-43ec-b024-ff47bd21bbdc}} which means that {{formula:c719b035-bae3-4292-8a0b-89e79f3514cb}} , a contradiction.
If {{formula:e2bb1f9d-2f8b-4d6d-ad7e-162fcc29b857}} , then {{formula:cdc3dd53-8c78-4ceb-bce2-793668607c2b}} is isomorphic to {{formula:a73869a3-1d18-4cb1-89d0-b4405ca6f864}} by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}. We note that 342 is a character degree of {{formula:5ab6c25b-d647-4fd1-adda-6e67d1bd1e71}} so 4042241280 is in {{formula:2e7b4e04-dfd8-4be5-936b-4e785296dc03}} , a contradiction.
If {{formula:910ee087-9c16-4230-a4de-fcf77c52489b}} , then the Schur multiplier of {{formula:cd6a9313-c6b9-49a9-bb19-fcbcba52fb56}} is trivial by {{cite:c6307deb5534db33cc25deee70c8f9410894a7d9}}, implying {{formula:71155ad4-3c4e-45ee-8bd5-7ca65fdd18db}} , a contradiction.
If {{formula:edc65826-0e9a-4e7f-947a-d7214104acb6}} , then the Schur multiplier of {{formula:05348614-8972-4bd3-b1e0-7b570d61951e}} is trivial by {{cite:2a961900de284170beb08505166b81764ed9ce18}}, implying {{formula:c2eafde4-fbbe-4417-a809-bfee868a5dff}} , a contradiction.
Thus {{formula:458168cf-d951-4184-948d-af8fdad7ea92}} .
Step 5: Let {{formula:618ca52c-6409-4133-9f98-cba4d685152d}} be a non-principal character in {{formula:94276833-0cf4-44f6-8241-fc38ab3cfb26}} and {{formula:4d92635d-362d-44b6-bbb0-8c6d274001dc}} . We show that
{{formula:02d46068-0638-421b-9c93-120bd5a85212}} . Also, {{formula:f1ad1ca4-61bf-4529-9462-95662243866f}} divides {{formula:3a9f56ba-3df2-40ae-a688-de7eb8e32674}} and {{formula:42028b4d-d6b4-4260-b31b-aaf268177d5f}} divides {{formula:f8f8ee09-efd3-4b87-b122-b47c39ef530b}} . Especially, {{formula:4a4ae1fc-dd0a-43e4-b47d-c72ba7c21caa}} , i.e. {{formula:f551eed9-46da-41ad-aad0-1ede3ec4667e}} is not {{formula:d0ef37a2-bf8a-412e-83f5-a723bc64eda2}} -invariant.
Let {{formula:c175702e-f51d-42e1-9a99-a0d8a91cbdfb}} be a non-principal character in {{formula:022e99b4-d94d-4fe0-928c-5732168996fb}} . Given {{formula:2d009b80-c6f6-4cdc-87b2-4b68f3b347f0}} .
Note that {{formula:a3d3d4bb-3330-45f8-8e06-714b564ed6b7}} and {{formula:19e1b856-b564-48b2-8e9d-f47bfaec6708}} by Clifford theory (see chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}). Then {{formula:1ddf0dea-2aa7-42d4-b43c-9bf126e509c6}} by Step 2 and {{formula:1d54e753-33ff-427e-9de5-bbf3ee632c6d}} . Especially, we have that {{formula:cdb3dd95-11bb-461d-a6f9-3c4b39570bda}} divides {{formula:69824f71-803a-4d5f-b539-0e1cb4f7f2c5}} , and then {{formula:da41e3a8-2ee9-4d6a-a6ff-e8b0db646e5d}} divides
{{formula:914e2192-bc42-49f2-890a-93ff1f75d448}} .
Since {{formula:93005b9c-7f65-473a-b28d-cc832807aa39}} and {{formula:d993182b-051b-40a8-a188-8bbdc576b41b}} is divisible by every element in {{formula:f50e410d-56a6-4427-97e6-a2b20df88493}} , we have that {{formula:39ffcef8-a7fe-4d48-8a91-6ae52940ec5e}} .
Next we show {{formula:78cc874b-856e-4793-b779-8be94c5bcbd9}} . Otherwise we may assume {{formula:b9f84e82-3dd0-410a-bf9c-24e468177098}} . Then {{formula:039e1505-da03-4523-8ef7-b76b128eb651}} . Furthermore {{formula:c93f36c8-c5ca-4052-a22e-8140a8c39d2c}} by Step 1 and {{formula:8a061c58-884f-45bb-8369-8f87e405c556}} is a cyclic subgroup with prime order by Step 3. Therefore {{formula:fc0097c2-e890-4774-9140-101ee80e0917}} is abelian for {{formula:f72fc468-a759-4755-8b0d-4c5db4ce245b}} by Normalizer-Centralizer Theorem, a contradiction.
Step 6: Final contradiction.
By Step 3, {{formula:7e7d07c3-2c87-4634-8d82-80f423ebdf46}} is an elementary abelian {{formula:e821638a-4ea4-4ad7-9924-ffea7db23913}} -subgroup for some prime {{formula:fab7d984-a065-4139-87f9-b25a9fc4a982}} and we assume {{formula:be3ad853-7e0e-4b3f-899b-84037b2901bf}} . By the Normalizer-Centralizer Theorem, {{formula:af97c084-fe69-4d91-a8ed-49697bc67fef}} Then we have that {{formula:c8108aff-f7c9-4bdc-bb2d-eb964b8fb789}} . Note that in general {{formula:aef34e2a-882b-49e8-abe2-40335df46167}} . By Step 5, {{formula:ae4eb28f-690d-49fb-ac47-990ff1aa9cf4}} so we need only to look at the prime power divisors of {{formula:f1534fa7-7862-42b3-8418-00f9de7674e2}} .
(Case 1) {{formula:b94ffff8-7c67-4bbf-a93c-bee1f07edc11}}
Now {{formula:63289255-c46e-4d63-ab4e-c0529f4c4433}} . It follows easily that {{formula:90efdbf0-a147-4552-b9f2-f4da66b29b1b}} does not divide {{formula:30c24d6d-d11f-47e7-9a81-b3735fe8c58d}} when {{formula:c019549c-df1f-41e4-98c0-92f04017245d}} , {{formula:c99772f0-9f1d-427f-aede-aa6f4ae55d85}} , or {{formula:bfb75c86-01ad-44ad-934b-8c27b5db9583}} .
Thus {{formula:1e5ced1a-2919-469f-a221-ce9f1b0c584e}} . By Step 5, we know that {{formula:e61aa8ab-03f9-4192-80c6-cf0f4a1902d7}} and we show that {{formula:e6d85a74-863e-480f-ac48-4fe2839e25e4}} divides {{formula:ae15128b-bfab-4667-b1f2-786efdec7d87}} .
Upon inspection of the codegrees of {{formula:23719708-086b-46af-a6a6-a9a446b983ec}} , we have that {{formula:b721f905-dc60-48d4-8abe-4f9ee410c4ec}} must equal to {{formula:e38cec41-1dc6-4b5e-9c7c-b37cecf1508f}} , {{formula:96e85bd8-19c4-4743-bdac-27a76dbeb2e7}} , {{formula:df2d1a25-e809-4a66-ad21-b46681035479}} , {{formula:6c239130-b328-4933-8c6f-c0ed6fc87773}} , or {{formula:af0293a8-49e0-4e96-92f5-a01ab384e2d7}} . Thus {{formula:02521f9c-fcf8-491a-ad6a-938f79a84e63}} , {{formula:af7f47c7-2f15-4888-ab64-334ce230d560}} , {{formula:2d588644-814e-4d9e-a580-e448f0f54431}} , {{formula:be7af667-0bc7-4ca1-ba75-666a00fb13ed}} , or 17. From this, we conclude that {{formula:f267d7ea-3c1f-4623-aac9-d986eba3e848}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, we note that {{formula:1f61e4dd-69aa-4a39-8081-ef4a08f1c128}} is a sum of squares which are of the same form as {{formula:6e38362e-e9c4-4087-b5c2-5f6c794ee1bd}} . Thus, we know that {{formula:1823687c-0eb9-4e0c-959b-3b1f24741235}} . But then, {{formula:d3354c9d-7159-44b7-b44a-63ebfd5e02d2}} , a contradiction.
(Case 2) {{formula:3809d065-4eb6-48b7-89f2-7f4b1bb0fa59}} .
Now {{formula:ce9b8696-2034-4b12-81f0-647c3dc4885a}} . If {{formula:1373fc1d-dcea-46fd-9b22-91cc5168eff5}} when {{formula:ad95d310-befe-4239-8ffa-2ce4597d6c79}} or {{formula:c3414d3c-fecd-4c1f-9ae0-1b0d468eb127}} when {{formula:d8bdbee1-3784-4a53-ae33-d1d170e47af7}} , it can be checked that {{formula:8c87a77c-5cc2-4d55-a093-fd2ebe1478e9}} doesn't divide the order {{formula:f09f5eab-1b71-49c8-9e81-359d67f60c2c}} .
If {{formula:46186a6a-be66-4c9d-9f77-6efc2b7da11e}} and {{formula:aa9b425f-992b-4cb1-9e40-99049bcda692}} , we have that {{formula:7ceec54c-28d1-47e0-b086-91f81708ee3b}} divides the order {{formula:4740c694-2e4e-4c6d-89d1-6c6c904c8b3b}} .
The first circumstance is that {{formula:9ae9c404-08c4-4ecc-ad85-d6251b083117}} . By Step 5, we know that {{formula:16278ad0-5f8a-445a-bcb6-d78a5c78c43c}} and we show that {{formula:7a8522b0-6075-4c94-99ce-82ac97f69722}} divides {{formula:a7af621a-d38e-4eb7-ab3b-213c85c3c2aa}} .
Upon inspection of the codegrees of {{formula:d7f10b19-5861-4448-9889-41a1f77f4ba7}} , we have that {{formula:439b4a5e-42c2-45e3-b60d-76c64b235ae7}} must equal to {{formula:c06a76b4-5528-405f-88d8-858b9a70cc62}} , {{formula:7b32de53-12db-4e88-9e7d-0181bc896644}} , {{formula:9592f6c6-a88f-421f-936a-c8508d1b7120}} or {{formula:f32c1c65-39ad-42e1-98b8-30cd80d8c830}} . Thus {{formula:127200a6-d979-44e9-b52e-a5453f89af93}} , {{formula:1f64ed07-d353-4d4c-a45c-2cff5c465e33}} , {{formula:1d31fdb8-90c6-45e1-946c-738fd9ab3567}} or 5. From this, we conclude that {{formula:5b1ffcbd-9e07-4a7e-8c7f-03a6f5def392}} . In any case, whenever we have the {{formula:8e2176f0-6535-4e6a-a1ed-50d782ca3628}} -parts of {{formula:af1d58f7-cf62-4518-832e-aaf2a27fa1c8}} and {{formula:abf2bd64-0c28-4be7-b691-cda82217dd80}} equal to each other, we know that they must both be 1. This is because {{formula:0897b3ab-eb01-4092-9d04-e94a43aeb23d}} is a sum of squares of the same form as {{formula:235e9136-481f-4cc8-aef7-702ad1c16d72}} by chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}. Hence, we have {{formula:f5f19022-7d0c-4878-a19c-47788bf43199}} . But then, {{formula:24a82c11-9286-4e33-8daf-c92a1e5cfd60}} , a contradiction.
Another situation is that {{formula:158f786f-0458-464d-a964-40d902fbdf72}} . By Step 5, we know that {{formula:6ef2cccf-f26e-4aa0-999a-2e9705c53f70}} and we show that {{formula:e37a6902-225c-4312-8c0a-6fae2cbad6a3}} divides {{formula:0ffa7fa3-1dc8-440c-ae38-29aa926b0f7f}} .
Upon inspection of the codegrees of {{formula:0f59055d-fc4c-44c6-9c19-9315be5f6960}} , we have that {{formula:18e6f4aa-477f-4232-b162-c23d52f7fd8b}} must equal to {{formula:854e8d33-af18-44a7-9c17-dd6e79827052}} , {{formula:1d52a6dd-6371-40ff-8f2e-816e594b5b2c}} , {{formula:70fd6d8f-72b6-4ac0-bfe0-501524268a0a}} , {{formula:6b1b9319-cab3-46b7-8335-b88c1e46c77f}} or {{formula:093a6c46-465a-4ff9-bdc6-2afa15838c1e}} . Then {{formula:f03df68f-0e45-498c-80d1-8e02423869c9}} , {{formula:bd5ba2d7-adc1-45a2-b10a-4bcb53d166c2}} , {{formula:c10c634a-f954-4315-a8d5-c1d24b95b46e}} , {{formula:3417c90d-be3a-4bbb-871e-4d217ef6ba61}} or 5. From this, we conclude that {{formula:258daf3a-ea93-48db-80d6-d1feae2d097d}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, we note that {{formula:d86529f1-0dc6-4531-a911-199079659a42}} is a sum of squares which are of the same form as {{formula:47f2ad2c-0212-4f5f-81de-d165822b322c}} . Thus, we know that {{formula:4be199f0-230c-4817-80eb-1798ba1cc184}} . But then, {{formula:4dc0ca2f-bc7b-404e-82db-e42e489973bd}} , a contradiction.
(Case 3) {{formula:b566b027-228e-458b-8bbb-19dd9a890025}} , for {{formula:b1980269-2b38-41fa-b44a-bf3cc837c521}} , {{formula:8abec70d-72b3-427d-8d4c-fb9e6395689c}} .
Note that {{formula:ba9e09e1-ff5b-446b-8c9e-5f27c46fe4e7}} and {{formula:9cf0ed48-171b-4525-a8e8-eceba15d90ab}}
and {{formula:807a2580-8498-4043-849c-e555ce8f12a5}} or 2. Thus, {{formula:3ccc984f-94ad-4510-824b-c5a8a1659657}} is the largest power of a prime that divides the order of {{formula:65c84459-c9a1-49c2-83ad-4115ac3b4141}} , where {{formula:a39512df-86f3-4164-923a-76d54a2d9d4e}} . Then, {{formula:7d24cfb5-560f-4242-bbc6-20f00ab01b7e}} . Let {{formula:0be8d08d-2c5f-414d-84b0-20ea94fc48ad}} be a maximal subgroup of {{formula:325df918-0ba0-4a21-8158-587fbe0dd02f}} such that {{formula:0bf2fe20-dd6e-42ab-9d0a-65dc0e100a25}} . Then is a maximal subgroup of {{formula:7552a73f-4e94-4aec-8c42-7b4251015f71}} .
If {{formula:ce97fef3-83a3-4609-985f-4e1534313b32}} is not the type in Lemma REF , then {{formula:a2dd72f5-4a1b-4568-94e5-37ff29d9b353}} , a contradiction.
Therefore {{formula:b6059755-4e05-4d09-b12b-39f491a188ab}} is of type in Lemma REF .
If {{formula:19ff85ed-b50c-4d33-ad69-457c92f37b45}} is of type {{formula:ed9ed1a5-17aa-49f7-bf67-790ec7e4d43c}} , then {{formula:e2bf1768-6ad0-4c0f-8239-bd19e3419a5b}} , and thus {{formula:8faf8419-f1c8-41cf-a495-105a34b29fc5}} , a contradiction.
If {{formula:41fc3cc7-103a-4a36-a632-eac18503710d}} is of type {{formula:587dfa91-ac2f-48ce-93da-bab40a5084ca}} , then {{formula:c9ea1ec1-0582-4f4e-aad1-83703f891c4f}} , and thus {{formula:d2173a01-277e-493a-a52a-4775409ca12e}} , a contradiction.
If {{formula:bb548c69-daaa-4819-922f-7fbf174d082e}} is of type {{formula:e34bf678-cd4d-4403-952e-9e324907dff5}} , then {{formula:c8807f78-6bd3-4dbd-a4a6-82b769ce064d}} , and thus {{formula:3cdce27d-c7ee-4cbb-b6d1-029d329dd823}} , a contradiction.
If {{formula:c9a616ca-361a-493f-88fe-259f3a05d374}} is of type {{formula:18e83b0e-f793-422a-8955-02d86b6d8760}} , then {{formula:9727e3fd-7dda-40ba-bf0e-bc0facf1ced5}} , and thus {{formula:130cce35-b282-4da5-8fd1-8b0d94404cd7}} , a contradiction.
If {{formula:79d16122-6e16-4b97-9710-2743dcb8bfe3}} is of type {{formula:e3057cc5-a34c-4f71-822d-295aef0e0749}} , then {{formula:ee2e5da7-db26-4bfb-83fd-1911d1d9a71c}} , and thus {{formula:26ad837d-fad7-4af0-923a-04fa1c78b368}} , a contradiction.
When the maximal subgroup is one of these cases, this means that it is a contradiction for {{formula:d95f386b-cfd0-43e2-b744-f29579f18760}} .
(Case 4) {{formula:6b291c89-ec49-491d-b862-760bcf72c94b}} .
Now {{formula:e72c03cd-bd4e-48b8-ab4f-0f3ec0c4cf85}} .
We note that {{formula:c53e5f7e-77fe-494d-a9fa-5ad6817516a8}} does not divide {{formula:a6e0d05f-f56d-47ef-b746-2411addb8050}} when {{formula:52a2ebbe-b607-4f76-94ff-e27a35580488}} or {{formula:e9a41697-a2c7-4ddf-ac0d-4507971ea12f}} when {{formula:41402c6e-620e-4906-a125-8917d33f3590}} .
Thus {{formula:7c1e89f0-cf64-443e-be6c-72a281376478}} . By Step 5, we know that {{formula:9fcbef14-b5b9-4a12-9114-d7a3c87df857}} and we show that {{formula:788d5210-df83-4e28-96c9-e90aff93d0c0}} divides {{formula:cb61c693-630d-431f-9b1a-927f4579a272}} .
Upon inspection of the codegrees of {{formula:d408266b-8121-4970-ac76-2bb026c229aa}} , we have that {{formula:a3034c99-303d-468e-8675-a92977c360f7}} must equal to {{formula:ed329590-0454-4b03-812a-d70481e1498a}} , {{formula:0b218bb7-e41e-473c-a7a3-4ccae31fba06}} , {{formula:10adaa8c-ada2-4eaa-806a-50c33016b654}} , {{formula:6f0ac02d-79f2-4dfa-aa60-9fd866845b44}} , {{formula:9cbcc16a-a521-4dac-98ec-33a485eabc45}} or {{formula:610197ba-b07a-4d82-b563-c12c01cbb0a0}} . Thus {{formula:232101a9-5394-4153-b897-a831ad9ac038}} , {{formula:a779c3e5-5021-45bd-92fd-80006732db27}} , {{formula:ee1cbc23-df0c-4ba8-b3f8-a7ad48955ad2}} , {{formula:9965d50d-07df-4253-9ae0-247deb037815}} , 7, or 5 and we conclude that {{formula:ddcf4f60-17f3-4d2e-9e5c-b6e2d0569c80}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, we note that {{formula:03f6d07b-78cd-47c8-a61e-5720d49a5bf8}} is a sum of squares which are of the same form as {{formula:ebdd6845-d498-4381-a27f-8e6fae526856}} . Thus, we know that {{formula:fa5c2b61-7dea-4193-a900-32030cbeeea1}} . But then, {{formula:9d3e3ff2-ddbf-49e9-964a-3285864c92a6}} , a contradiction.
(Case 5) {{formula:4a3b0632-2a70-467e-9f84-e266fdd8a5ac}} .
Note that {{formula:2d5ea159-e68b-4f30-bca7-a93f3e8a0e9e}} . We can also simply check that {{formula:8b95ae91-c652-4547-b1ab-da73df679ba1}} does not divide the order of any possible general linear group with these conditions.
(Case 6) {{formula:b5af129f-a809-4979-8b05-ba9797947c69}} .
Note that {{formula:56526108-031c-426c-8ab5-458863527929}} . Similarly, we can check that {{formula:ddd0141e-219b-4854-9cd4-7e7e28a152c1}} does not divide {{formula:c004fd38-a5e0-433f-adf6-4156d129dae5}} when {{formula:c388f5c0-0377-4291-bde9-166a73b48708}} or {{formula:6547d2bf-218e-4459-999f-9e927be63066}} when {{formula:1caead3a-011c-485c-aeac-d1202155883a}} .
(Case 7) {{formula:5b9f8021-3092-4024-914f-549d46594a4a}} .
Now {{formula:701d1aad-4922-4d75-8e87-0d83c4e37d4a}} . We can also simply check that {{formula:b9968313-9986-4fc1-926e-6de2666c1e60}} does not divide {{formula:5ce56610-4830-467a-ab67-8cc4bd43b416}} when {{formula:f20128d3-1a78-43e5-adf8-baa2dfe6a7e0}} or {{formula:489a15e5-315b-4786-98f6-2331287dda0e}} when {{formula:41b5d425-851b-419d-8908-1bddb5485085}} .
Thus {{formula:7a824535-3f94-432a-ba84-42603fcf94a4}} . By Step 5, we know that {{formula:35a42588-6c85-4bfc-ad46-def66d31a116}} and we show that {{formula:32585be5-a5ee-46a1-ad1c-8fcbddb4b28e}} divides {{formula:6966dabf-c398-4658-a245-e10193b8f46f}} .
Upon inspection of the codegrees of {{formula:dc2a45e0-9049-4fbf-88b8-061e38aea8f5}} , we have that {{formula:58c20699-84a8-4638-b9b9-fddc4b7d63db}} must equal to {{formula:5cf0cce1-31db-4b95-a44b-19beed268f06}} , {{formula:89b18e7a-4287-47fc-9aa8-2ad4f5155d95}} , {{formula:4be79b42-ae31-4949-93d0-e0d779b7b5a7}} , {{formula:f6acd6dc-20ff-42f1-b441-1e900f692e7d}} , {{formula:a58738ba-2bd4-412f-988b-e275d2aa5e2d}} , {{formula:5f847b98-b5da-4ff8-8592-7586ce235f53}} , {{formula:7a6d7908-7cf5-48e1-bb6f-d1d164c92877}} , or {{formula:1047abc3-358f-4753-8289-189b3c59eb9c}} . Thus {{formula:12b12b3c-e734-4e01-95ec-46523b181889}} , {{formula:11595684-89b5-487d-b1a0-80e8b2628f65}} , {{formula:be25a605-4fd1-492a-bcc4-6e49dc48932f}} , {{formula:49fed380-8fe7-47e4-8638-6b59ce2c8af4}} , {{formula:cf84a388-5e51-4ea6-85ef-c33a34e0f0f8}} , {{formula:407df09f-7d0a-464b-a59b-1288fcfa504c}} , 13 or 7 and we conclude that {{formula:86d37b6a-67a3-4c63-b3c8-6ca6b6b5f2e8}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, we note that {{formula:0ab38b14-ebfd-43d1-bef2-bbd30940071b}} is a sum of squares which are of the same form as {{formula:7f96fe29-009e-4449-8ee9-24e89500d16d}} . Thus, we know that {{formula:71d942df-bb2d-43b8-a7d6-4656b51e6bcc}} . But then, {{formula:3a7e26ea-6e85-4a65-a76f-c1faf1309178}} , a contradiction.
(Case 8) {{formula:7d549516-0e5e-43bd-89ea-c8ef5ec1eb23}} .
Now {{formula:2e012d95-00d1-4395-9682-7c99734f475d}} . By inspection, {{formula:6a3a5460-eae1-4d75-b5f3-3353b0524935}} only divides {{formula:ef67bc03-b0bc-4016-900e-33343d94be5a}} when {{formula:ea60ac5c-7329-4a7d-aca8-557179c02083}} .
By Step 5, we know that {{formula:cd700446-e9df-4189-81eb-b8a27a4370c8}} and that {{formula:dc7df329-8283-46be-b908-2437595a3604}} , so {{formula:da355ce5-cf3e-492f-b94b-6e9063bade50}} , {{formula:630e7f72-3e85-4109-8bc9-b6a755bd5005}} , {{formula:7a011e90-10d2-4999-9caf-875129281575}} , {{formula:c52108a9-3870-4bb8-92e8-38f9ac0c612f}} , or {{formula:8f8ec795-2267-4bf1-a2fb-8a647b151223}} . Then {{formula:9dc3aaf4-c920-4fc8-b9dd-5c816cb2544a}} , {{formula:156433a3-61a6-4299-84ab-b312ffad7063}} , {{formula:fdb8beea-939f-41b5-9d47-fa470fe5dde8}} , {{formula:549eaf8a-1c2b-4231-ba87-166d7c613e88}} , or {{formula:a502ef2b-da47-497e-84ac-e8e7d28f7009}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, we note that {{formula:5ae04d00-190e-420a-884c-0e53dbd2f3ef}} is a sum of squares of numbers with the same form as {{formula:aed521d6-1563-43be-a0bd-011d9620030b}} This implies {{formula:163298b3-3c98-4afe-beaa-93837b77d387}} . Thus {{formula:780df524-1f3d-49b4-9255-cacebc4ca065}} , a contradiction.
(Case 9) {{formula:26b8e279-c646-4db5-a1ae-9d7954843ca6}} .
Now {{formula:3eb24c95-a3f6-42e2-bc59-33959b16fb2f}} . By inspection, {{formula:5fa088cc-618d-49f1-ab7e-97f33a6394fa}} does not divide {{formula:713f08de-f5e6-4eea-ba8e-2ef355252575}} with {{formula:181b24e7-435a-4b62-91d6-dc34db86d990}} , {{formula:a60e1f74-588f-439c-a58c-55ad09535e24}} with {{formula:93770c2c-19a1-4c4a-9b2b-39592d529999}} , or {{formula:ae145c83-bd76-45a7-ae91-88f984e6b768}} .
(Case 10) {{formula:6f124ac3-7114-4565-b164-bfd5f1014e33}} .
Now {{formula:de35b0cc-b6a6-44a7-87fa-ae336da3e17b}} . By inspection, {{formula:ecd4e321-2365-4030-bf6a-79e6b7c65746}} does not divide {{formula:c264785e-67be-4c3b-bfaf-75952213f953}} with {{formula:0f9c1fb9-8cae-41cc-afcb-3309b21b5614}} or {{formula:9636290c-fa4b-4f94-aa14-700c543cae06}} with {{formula:c1899d00-0011-4967-b13b-2dde16abf0a5}} . Thus {{formula:59048be8-31df-4a2d-9bd3-8ecd438cc3c1}} , or {{formula:0bba1971-dcaf-44b2-b9dd-9193a7a12de9}} .
Suppose {{formula:4c55489f-bf1f-4f69-a265-e8847bb7e1f3}} .
Let {{formula:ea3dc972-9dac-4813-a0bf-1770e2b6725b}} be a maximal subgroup of {{formula:c5586f46-1ae0-434e-adde-7c8e7bc98f0f}} such that {{formula:0a7715ec-0d1a-4d2c-939b-c86a8504abc2}} . Then {{formula:2d68f0ae-bcef-4eba-bb70-0ca681506117}} is a maximal subgroup of {{formula:412a72ff-44dc-4088-ad04-496e38617da5}} .
If {{formula:21f1dc12-ee78-448e-b2c9-53a9dc40cfdc}} is not of the type {{formula:2009fd72-5c1a-42fc-8409-8482be590352}} , then {{formula:315155d0-9a51-4cb5-89a2-55116ba2555d}} by Lemma REF , a contradiction.
By Step 5, we can again deduce that {{formula:3fa96164-0759-47af-a427-33e7a2e4f0a3}} , or {{formula:95714c62-f3fa-4eec-b364-be6407826ba5}} . Thus, {{formula:fef00a87-a54a-45d5-b05e-87c2e49698e1}} , or {{formula:089b2762-d7d5-442c-b64b-76c9913a359a}} . We conclude that {{formula:ba1aa23d-ad67-42c1-ba8e-b12a03ce75fe}} , i.e. {{formula:e261b539-e097-4ad2-9b1c-092d56b727d2}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, {{formula:13ba8b28-9f15-4070-be07-d500681fb6ce}} is a sum of squares which are of the same form as {{formula:d4645ca4-12f6-4529-ba72-5b7a33cca049}} , we know that {{formula:5c13925b-e5ea-43f0-b766-f1e8bb96a48e}} . If {{formula:eb1a7d1f-9481-4a55-8477-2e6673e579e4}} , then {{formula:6bcfa173-0cf1-47d0-b592-c4320b7954fa}} , and so {{formula:acc91267-fafa-493b-925e-a56480dc1078}} , a contradiction. If {{formula:0e62fb80-a952-4dae-b4da-d1e07362e54d}} , then {{formula:32a292d8-947c-4624-989f-7e0cbaf67b6f}} , and so {{formula:e3e86494-cabc-4375-be1c-9b344a37952d}} , a contradiction.
Suppose {{formula:bf6cb382-7cb2-4683-ba7b-98605fce8e75}} .
Let {{formula:b35e902e-32b9-448a-a4c2-9d16bad404de}} be a maximal subgroup of {{formula:c9625c8d-ef56-4202-a232-6fcf8256700b}} such that {{formula:f37fc688-654e-4e8b-9fe5-01be1a686f4c}} . Then {{formula:3c9b31cf-3ef4-41d9-bebc-289081695d23}} is a maximal subgroup of {{formula:e9a717fd-a84f-4223-aaee-5d81e48be2ad}} .
If {{formula:1b942362-2478-46db-bcd7-6b72d3773fb1}} , {{formula:26fa5738-4232-45e4-b3c1-f2b94609bf85}} , or {{formula:c6bed78c-1bfc-444d-874d-89f4b80b06bc}} , then {{formula:c525dbd6-c4d0-4c7b-aaeb-c3c0c4894713}} by Lemma REF , a contradiction.
By Step 5, we can again deduce that {{formula:6a0b9661-e463-40d8-a558-29099602a9eb}} , {{formula:cf2664e1-30f3-42f5-9717-d9d680b78307}} , {{formula:7dbba88d-882b-4886-bd52-e54101fcf55f}} , {{formula:eb9302c7-ca0b-4b6b-bbeb-d5e1ef5376df}} , {{formula:0b5204bb-9030-4107-9b06-892647495d16}} , {{formula:2801d630-4a4c-4d72-9483-d66dcc93ee31}} , {{formula:0fb96515-02be-4316-be94-18c71aa30afd}} , {{formula:cfe26d73-56ea-416d-95b3-0c69d3263245}} , {{formula:b0dde262-1da9-40ea-aaa5-0e7db09c4494}} , or {{formula:b72c9678-b483-4455-a42f-9a27cdcc6df9}} . Thus, {{formula:7028f0ea-5de9-4357-a42e-2a3ddab9f59d}} , {{formula:507a3395-a31e-4cf1-9d6c-b072554ceb68}} , {{formula:6edddc40-e32e-46ff-9120-bec1ca076d4c}} , {{formula:6283f0c6-f937-4714-a1e6-fe3c11cc3823}} , {{formula:2f8b13fd-c461-4293-8a51-e486aefc32e1}} , {{formula:6692acf4-53c9-495a-8860-69080f4308de}} , {{formula:a98dde1f-3b65-4c07-8e54-6b7684a680e1}} , {{formula:b7c895b5-ca31-4032-8865-8f066a2e5fa1}} , {{formula:926e3742-62b8-45a5-a387-c72f684861f8}} , or {{formula:43a8fb65-3930-4429-96ab-f6e028d4efe4}} . We conclude that {{formula:9b80c076-503f-4cff-80f1-ac930ffcfd21}} , i.e. {{formula:cd4a3d8d-0c32-4668-a6aa-db7e99bfaf7a}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, {{formula:ca805266-6744-418d-921e-e05708db40fa}} is a sum of squares which are of the same form as {{formula:a5a59ff6-5354-4c60-8713-34f9c186c571}} , we know that {{formula:828d4d7f-d26d-439f-ac2e-8a891a98f9da}} . If {{formula:54e78b20-3582-4392-ae4a-d0f4c8ce6656}} , then {{formula:b43737c9-9ae0-4a96-a695-ea8ebfc589e7}} , and so {{formula:292e1f75-470b-4748-9382-79679c0ba594}} , a contradiction. If {{formula:4aaf893c-0831-4fa3-b1bd-84b772349915}} , then {{formula:86af082b-caaa-4e04-b359-524e13a3306b}} , and so {{formula:db1fb0f3-5476-4587-8766-e7682c625ab8}} , a contradiction. If {{formula:06b1fcfc-4ca9-41a0-a556-7abadb3a1457}} , then {{formula:d795e5d2-928c-424b-9842-b7a1d92abc78}} , and so {{formula:09e93f12-2873-41a8-a615-e8fb710d6846}} , a contradiction.
Suppose {{formula:0696c1cf-08cc-431c-8cb5-a91528c7c55c}} .
By Step 5, we can again deduce that {{formula:0ceaedbf-34a3-4563-847f-9a13c4784e26}} , or {{formula:07b507e9-6809-49be-ac6c-82bf2614fcb1}} . Thus, {{formula:cdc4d799-a647-4c1b-913c-2674b521fe02}} , or {{formula:9d2b98fe-0e23-477d-b8fc-dfdf70ffb968}} . We conclude that {{formula:8474657f-c8b7-465c-b774-ae78379fb602}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, {{formula:02930326-6327-48ef-b449-6b54d82e056f}} is a sum of squares which are of the same form as {{formula:e5d7fb1b-b650-4ea5-b897-0799eff1552b}} , we know that {{formula:83df914f-6b00-45b3-a165-af6298d36d9e}} and {{formula:01ad8b9c-15fa-4613-bdc0-afadcf93c04f}} , a contradiction.
(Case 11) {{formula:e280a0c2-9915-4b6c-a0f0-5504bc7b190b}} .
Now {{formula:8593ddd4-8c5b-482a-becf-5d6f30326a5b}} . By inspection, {{formula:259ab2a1-b96d-493e-8550-37968604c1ad}} does not divide {{formula:c394d451-393d-414e-b4fa-bade3ccce756}} with {{formula:2781f431-3e54-4227-98b5-5bf6c27036be}} , {{formula:8d2223ef-4a15-4d60-a6b8-484bc7500439}} with {{formula:ac9d8c4c-657c-4dd5-898c-bcf1922065e8}} , or {{formula:b1aec66a-3733-47c7-af8d-1b0d8bc0e401}} with {{formula:3f2e1c16-f119-4ec1-82a0-cf824fad70b8}} .
(Case 12) {{formula:ae405207-fd77-4e73-8a3d-063a6bffda31}} .
Now {{formula:d552142c-bd98-45af-9e6e-20791f82e7bf}} . By inspection, {{formula:60113852-c042-4315-8d1a-c4d08e5b3419}} does not divide {{formula:db603e3c-01f5-4a06-bf12-51f105170aa2}} with {{formula:3e04ec57-45f2-464f-8340-b354c1865328}} , {{formula:ec90a5c1-39b7-49c4-aab8-d191b0ec4e4e}} , or {{formula:b1cbfec4-82f4-405d-acfe-e4e7f85a7eb0}} with {{formula:7700965d-cf9e-4c89-b4a3-761d8a995fd9}} .
Thus {{formula:47ed69b7-1679-4a77-98e3-bd0ef3d6c602}} . By Step {{formula:907ad930-83a1-4811-ae1d-6763e92bb032}} we know that {{formula:6a622071-7c3b-4f27-bd5a-ab0c8a451af2}} and that {{formula:9c0618c2-8e54-497e-bede-23c6ce449104}} , so {{formula:5be74acc-4802-424e-982e-826b0ff7e1ed}} , or {{formula:d53a43bc-d49d-438b-90d8-769bdb7f75be}} This implies {{formula:8cde788d-5611-48ea-b006-5c093fd65f07}} , or {{formula:218c21cb-c1e5-4fa1-8630-c9b1e390c15d}} . We conclude that {{formula:76da289f-4d1d-4970-813b-960e4fdcf3b1}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, {{formula:c61c0094-9d36-4106-9222-53dfb64a8d4e}} is a sum of squares which are of the same form as {{formula:ae548b41-1c44-4742-bd52-a1b2d1eb1315}} , we know that {{formula:1f34e442-5cd3-422f-92b1-8084bc084a70}} and {{formula:8deb47a4-31dc-44fb-8922-aa28a331a1a7}} , a contradiction.
(Case 13) {{formula:ea6828a5-cbd6-4041-b3ce-184d8f9ee6d2}} .
Now {{formula:48d7c4c7-269b-4ece-bc1b-64010ec2a2f3}} . By inspection, {{formula:23c04fab-ec10-41be-9d89-8cba0ad861b1}} does not divide {{formula:75cef9d6-d4a1-4416-a34d-3a3139396bf6}} with {{formula:d26a4cba-11b9-4854-af89-96f2bb6dbf10}} , {{formula:26a69a91-46ff-4f69-8678-ce73458fa3d8}} with {{formula:02ff16f7-e86f-4fa5-b016-4c0d0bc12806}} , or {{formula:b33414c9-cc38-4e56-9e68-f3cbde71467c}} .
Thus {{formula:37763c87-e503-46b4-8e99-aa9a905e911e}} . By Step 5, we know that {{formula:57dd9136-62fd-4644-b57a-be2bc8bb17b0}} and that {{formula:587734c4-3f2c-4896-a33b-4d8a5460765e}} , so {{formula:fc129545-5ee0-4978-ba1b-e4cb5f0a4b88}} , {{formula:9cdd954e-ec58-4747-a9e3-b1e6d529441b}} , {{formula:1113fb06-a297-419b-8aed-d9d2ede37b89}} , {{formula:6425558b-8d86-4f35-b5f5-208f65eea16f}} , {{formula:f9997338-4ea1-460e-b6c6-2731a1d9b073}} , or {{formula:68b78b0f-45d7-4ddd-bf81-04747b44678b}} . This implies {{formula:0c8e76a3-c23a-4dee-8617-8bce1768c339}} , {{formula:afe0b7cc-fb37-4689-8cfe-2ea2ab27a0a5}} , {{formula:31f61a40-f123-466e-bac5-198de013914e}} , {{formula:1e515dbc-378d-4bc4-80da-3228a9d86a76}} , {{formula:2ca4fc10-ea3a-4941-a6bb-13e4277a8129}} , or {{formula:0f5a0414-cf2b-4a8f-9155-4b1a119d0f47}} . We conclude that {{formula:445189c4-9b6c-41ba-a185-2f36cdac338f}} . By chapter 6 of {{cite:551cd37413e769428b2f696243de3c4232e87dc8}}, {{formula:198b81b5-01b9-40ef-a4d6-887f4c9e5e3e}} is a sum of squares which are of the same form as {{formula:8aa8c820-f135-40f2-a9f8-61f65f314e35}} , we know that {{formula:4057f300-f750-49fd-ac9d-688f240e8d3e}} and {{formula:541fe6af-f1aa-4328-8632-0901afd6fd2a}} , a contradiction.
(Case 14) {{formula:46241638-1622-4cf7-9d40-826cfbfc3b19}} .
Now {{formula:1d2050fb-1f26-42e0-a0cb-37922fdc8c32}} . By inspection, {{formula:9d1750f0-ebe4-4642-956f-9e384ab6747e}} does not divide {{formula:48019683-c7f7-42d9-8a39-cacdc7178587}} with {{formula:46a997c8-10a7-468f-b3fe-2df4868cde32}} , {{formula:7294ae8d-625a-4fdb-a591-3d3a251afe22}} with {{formula:790689a1-6c51-4550-81b5-a489cb36399d}} , or {{formula:588e8001-34b5-48b8-842c-4238ce14e86e}} with {{formula:fb2c8be9-f241-4432-a14a-82d602de75dc}} .
(Case 15) {{formula:155b2b73-91d5-4ea8-bcf2-abe9b7e75b4b}} .
Now {{formula:3e933700-cf3e-4f31-925f-1b71137d5760}} . By inspection, {{formula:54ff29ab-b1ff-46f5-9cf0-95acf3e80a71}} does not divide {{formula:f87de9eb-8552-45ff-a0c1-1ed637bec364}} with {{formula:d3c31960-298d-4b6a-9df0-df8ea40a6014}} , {{formula:fff110f5-e6a7-4eaa-a474-679fae490485}} with {{formula:e49bf907-4e9d-4379-b53f-8a7620381601}} , or {{formula:f7d2da9f-1e33-4e23-bde9-eec91ea73eac}} with {{formula:95c7bfb8-10b1-431a-b2a7-f8b5cbf4f997}} .
(Case 16) {{formula:c876a81e-943e-4469-b07b-d66d275698ae}} .
Now {{formula:4e755529-cc56-4db5-8cd2-ee3ed1326eea}} . By inspection, {{formula:8dfd3ac6-8aa1-4cc3-a652-c959e3a459af}} does not divide {{formula:8570a250-7a48-4ba3-b731-c036132b27fe}} with {{formula:4a05c5b0-f46a-4d96-867e-c3f99a8f54db}} , or {{formula:24051154-3033-4a3a-920b-ebba38e2208b}} with {{formula:edca05b2-1e11-4c68-a5ff-c7793c3af1b9}} .
| r | c843458886ec876dade4c11eca54842c |
Currently, one of the scientific directions of nonlinear dynamics, fractional
dynamics, has received wide development {{cite:06ed8218f8e1dd6e47dc452d281245fca980f221}}. She studies the hereditarity
properties of dynamical systems. Hereditarity (memory) is a property of a dynamic system in which its current state depends on its previous states {{cite:4c4b9c3d28fbf496dff4281940baa28c2dbf7c32}}. As shown in {{cite:06ed8218f8e1dd6e47dc452d281245fca980f221}}, the property of memory can be described using the mathematical apparatus of fractional calculus or using fractional derivative operators. Operators of fractional derivatives have many definitions and unique properties, but all of them, to one degree or another, describe the memory effect that characterizes information about the previous states of the system. This effect predetermines additional degrees of freedom – orders of fractional derivatives {{cite:a45aea6eeec085beda5fb7d29b6c9d105a842259}}. Such multi-parameter dynamical systems have certain difficulties in description and require special research methods to detect chaotic regimes {{cite:e3b0d0367d02110413a11bc30015500b6123b814}}.
| i | 73fa63de19c2153a4d1713bdcf224383 |
To perform this work, we focus on the quasi2D case where high resolution
experimental imaging is possible {{cite:aa2c1182ec1e4d719a84b099e17e328229286fe9}}, {{cite:7c38dc24e7d56323c1b3b6856d141c1b9929c9d5}}.
For clarity, we will focus on two examples: one is miscible and
the other immiscible, both far away from the transition so the character of the
excitations is clear.
We consider {{formula:00519752-d7a2-450e-9561-e67cb0690f54}} 100 and {{formula:21047f03-907e-4eaa-86f7-0863eb3fbaa9}} =1.01g;
for the miscible example, we pick {{formula:64d84ace-d3a7-48c5-86c6-5dc51e2f6c5d}} =0.5g and an immiscible
example, we pick {{formula:2dfb6f8d-453e-4a4b-a1e2-44d1d7e41ea7}} =2g, where
{{formula:ca0ba3d4-a28c-49ba-af3b-29b4b9df1f62}} is the strength of the contact
interaction, {{formula:b2b96a5f-2b7f-4d3c-b7fb-9036ba06bce7}} is the 3D s-wave the scattering length
({{formula:cddd0510-3984-40b4-8c5f-9982d2923b92}} ), {{formula:69358344-cba5-433f-830e-8c56b530cf57}} is the axial harmonic
oscillator length and {{formula:63f62ad7-838a-436f-9375-38dfc32a1275}} is the trapping frequency in the
tightly confined direction. We only consider {{formula:15d132ae-02fb-4e28-b33a-b7735721a2aa}} and equal masses.
We rescale the equations into trap units, so the energy scale is
{{formula:97e4a9c5-f13f-484a-82ae-a33c144d2bbc}} and the length scale is {{formula:5d36ea6a-3eb3-4f6a-af55-ea42a5f43c23}}
where {{formula:b14683a2-6f9f-47e6-b73c-195fa709d505}} is the trapping frequency in the x-y plane.
We can loosely relate this to experiments, for g=100 if we pick {{formula:16529898-0c6a-485f-a415-39cd47bf7aa9}} =1000,
{{formula:553ac8dd-fa14-492d-a091-456d805fc3d6}} /{{formula:2dc30afe-1ebc-4e19-9956-9a4c664030aa}} =100, and {{formula:e5d321b9-c3b0-41a5-bfd5-8e50cf99b578}} =100 {{formula:47042732-98bd-4fd6-a97c-89c5dac6e809}} ,
then this example corresponds to radial trapping frequencies,
of {{formula:39f60e41-ddb1-4c6a-aeda-3133a863c1ac}} 38 Hz for K and {{formula:84cf1234-4ec9-4152-b36d-93657d1e81aa}} 11 Hz for Cs.
It is worth mentioning that the chemical potentials for each component are about equal
({{formula:8cee2827-f037-4711-92e4-496f61818a76}} ). More importantly, for the immiscible system, {{formula:b48e288f-3c4d-4917-90dc-db9641be3d04}} is
{{formula:3f018469-7550-4eeb-a7ab-fb80ecc654b4}} and this gives a healing length of {{formula:a76851f2-630d-4eaf-81bf-373f55aba66e}}
(for the miscible system {{formula:47411f03-9f5c-4a9b-baba-d2ddf9a62cea}} ).
Further details of how we solve these equations (REF ,REF )
appear in Ref. {{cite:e662a4b060c967dd98c9d73c8cc52438541e6c7d}}.
{{figure:e9fd67be-e58c-48b4-bbb3-65146f6de19b}} | m | cda2acbfe01c8b0cf73936a0986a1f14 |
In our spectral analysis, we can estimate the inner radius of the accretion disk (R{{formula:84406d0d-cfcc-4b62-b29c-c5ab2ff47ac2}} ) in two different ways. The relxilllpCp model directly provides R{{formula:55b0f24b-25d3-4b9f-a069-2707e2955a6c}} , which is found to be 14.78{{formula:2ae695a8-8654-498e-8b0c-97d6095d07f3}} (r{{formula:b68c6457-a8eb-4f34-98fb-01205e70bb76}} ) and 12.88{{formula:30734f5d-0046-4636-8434-c4dbbd5e8291}} (r{{formula:2d40affd-3f82-44ee-bc33-e3ebe127f078}} ), for epoch E1 and E2, respectively. R{{formula:2e44a96a-3d12-4230-b3ae-550c39035331}} can be calculated from the normalization of the diskbb model. Using the disc inclination obtained from our spectral analysis, taking the mass to be 4.9 M{{formula:37e8b19f-93bb-4f6e-af40-443a1ee18f24}} {{cite:97cd0b50ee0e2888b5350c9705abc9c13cdb6c8d}} and the distance to be 2.4 kpc {{cite:559039a88a58d334064679875bdcba7d86512240}}, R{{formula:ee26797b-ddaa-4ef1-8877-e62de003d696}} corresponding to epoch E1 and E2 found to be 0.66{{formula:7f1e0c49-22d7-49ea-8d90-8a1f90147476}} (r{{formula:9057e073-ab06-489e-a859-9fafc3e4b514}} ) and 1.12{{formula:82ed6095-48c5-4d8c-b3c9-5f47c118ccb6}} (r{{formula:00e80077-829d-457a-9dfa-4f56bcc28832}} ), respectively. We find that the R{{formula:67c17ee8-9af9-4a81-a113-7130ee66a9b5}} values calculated from the normalization of the diskbb model are smaller than the values estimated from the relxilllpCp model. One possible reason for the discrepancy in the R{{formula:cebdf99b-ecdb-4203-aea5-7c4240e77e09}} values is that the relxilllpCp model is not self-consistent as it uses the nthComp model with seed photon temperature (T{{formula:cfc43759-c71f-4e8a-bc84-f0f657767c45}} ) fixed at 0.05 keV. This value of T{{formula:540cd6e4-3e7d-4b67-8d84-6446c5e29268}} is uncharacteristically low for a stellar mass black hole accretion disk. As a result, a higher temperature component becomes necessary to represent a part of the existing cold disk. Secondly, {{cite:9f9ae1f0b709822036cb60043a9b86694d70f2fc}} carried out a critical analysis of the usual interpretation of the multicolor disc model parameters for black hole candidates in terms of the inner radius and temperature of the accretion disc. The authors have reported that the diskbb model underestimates the inner disk radius. {{cite:9f9ae1f0b709822036cb60043a9b86694d70f2fc}} suggested that when the disk contribution is very low and the spectrum is mostly dominated by non-thermal photons, the radius inferred by diskbb is inaccurate. They have also explained that it is very difficult to determine the exact shape of the accretion disk spectrum for the aforementioned case. {{cite:630e74d10a65d791432818c3940a1c06b18533f1}} suggested a correction factor to improve the value of R{{formula:e5de5ac5-7ecc-4bb1-bedd-2a4fb14cfeb3}} calculated from diskbb.
| d | 9b2320d16d1a479c912638684d537a3c |
One-shot approaches stand for communication-efficient schemes where workers and master only exchange information at the very beginning and the end of the sampling task; similarly to MapReduce schemes {{cite:3c9470b135f2aa732e9babb90443fafae85ce02a}}.
Most of these methods assume that the posterior density factorises into a product of local posteriors and launch independent Markov chains across workers to target them.
The local posterior samples are then combined through the master node using a single final aggregation step. This step turns to be the milestone of one-shot approaches and was the topic of multiple contributions {{cite:cb5a049253d01744be732742ed85c82ce0832f6a}}, {{cite:7dded4fc50b367f2e8f15ee5c71601ed05805129}}, {{cite:7364f2b3c65bf0a83211a04453963effe6362718}}, {{cite:46cd0d6bfec036d51c198a9a11f7d09d32857a97}}, {{cite:2da1e8b2b722a19bd6fc449b52a05616b5a57753}}, {{cite:8cf5b59ddf9a2c02ded50763267122c4efc84b79}}.
Unfortunately, the latter are either infeasible in high-dimensional settings or have been shown to yield inaccurate posterior representations empirically, if the posterior is not near-Gaussian, or if the local posteriors differ significantly {{cite:f8b03a150d70a0f30f1240cfe6f6c6fa5f44695d}}, {{cite:85eb2b870073c59314aa53ff5dca4e38d6cc1ecd}}, {{cite:24d5053d6c832a836a930ece1e59ada252ecd96b}}.
Alternative schemes have been recently proposed to tackle these issues but their theoretical scaling w.r.t. the dimension {{formula:764ce07b-b7cd-412b-8c41-eefada67c98f}} is currently unknown {{cite:885a84a7c514cdfc8adfc75f44f05508e6c2b4fe}}, {{cite:f9d76d1cfd9a3bb144716c840793cb549de1deba}}.
| m | 893b6a64fc8855a54bd3c75c99731955 |
In this section, we show the number of operators and quantitative results at each level, for objects shown in Fig. REF , which are samples of the ECSSD database {{cite:38a9d1b70818176630cee705bdfd17ac5f993621}}.
To show the efficiency of our method, in table REF we show the number of operators for a single color channel of objects, at each level. we define an operand including three steps of multiplication, shift, and addition as shown in Fig. REF .
The three first columns show the number of operators needed for three levels of wavelet which are cumulative with their previous levels. The first column indicates that 1024 operators are used to calculate all {{formula:bd2561d0-19d0-4f77-ad9e-b602c77f846c}} hologram blocks; results are shown in Fig. REF (1,2,3)(c). In the following columns, although operational numbers can vary in different objects based on their saliency map, our proposed method can highly reduce hologram computations at each level, e.g. Fig. REF (1,2,3)(c-e). Note that, the fourth column shows the number of operators from {{formula:ca7876f3-b6a5-43ba-85a1-0e59d8a623cd}} to the object level; results are shown in Fig. REF (1,2,3)(f). The last column also shows a calculated fully point-wise hologram with LUT blocks of {{formula:980ae96a-f8dd-4077-89b2-1c6b3bede092}} for the object without decomposition, where {{formula:b55a82d9-297a-4bf2-9132-91e4e0b31d43}} operators are needed which here is 65,536 for {{formula:6f56abd9-ac18-4711-b35f-841beaa94281}} resolution. This reconstructed object is the most accurate hologram for a scene parameter. Hence, we compare our results with this reconstructed result, qualitatively in Table REF . Additionally, comparison with this also makes the SSIM parameter free of scene parameters and scales.
In this table, we show the progress of SSIM for different levels of wavelet in our proposed method, in comparison to a fully point-wise reconstructed hologram. Clearly, {{formula:3cc4c436-9317-47ef-a85c-4c7bfd7a56d4}} has the lowest SSIM, and the object level has the highest one.
According to the two above tables, we can interpret that the number of operators for {{formula:fd61ac96-4de8-45ed-a4d2-4ab3b44fcd89}} for the object level is relatively high whereas its SSIM is relatively close to the level {{formula:cff8c6c6-ab09-41b2-88e4-d72e11728ad0}} . This closeness can be visually realized in Fig. REF . Hence, in such a case we prefer our method to end up to level {{formula:7285afd5-d5d3-483c-af53-ddf0b096c683}} for both high quality and efficiency. However, even if our proposed method proceeds to the object level, the cumulative number of operators is still less than half of the operator of the original object made for all object points (fourth column compared with the last one).
In fact, the advantage of our proposed method is that some points suffice with level {{formula:24d7aa07-67d3-4497-8ef5-2701bbd3b9e9}} , the remaining suffice on {{formula:d2df6b67-3651-4535-836e-563280442c7e}} and others are in need of {{formula:58e86f11-78ac-4203-b9eb-23b66af211c2}} level which leads to high qualitative results on salient objects.
In this paper, threshold parameters of {{formula:57b6317b-17d8-4064-a99a-a402ffba8479}} , {{formula:65a44cd0-e449-4066-933c-91e05a069465}} , {{formula:5095daec-23ed-4fcd-b841-6db7d9b4bcff}} are 0.9, 0.7, 0.5, 0 for low to high levels of wavelet on the saliency map.
{{table:77467a0c-cdf8-42a3-92eb-6e7c1b39da5c}}{{table:10101ce2-b40b-4fb3-ab97-e19b64c70c76}} | r | 2538ac32964e83fdc89d219fb6d53ead |
CycleGAN {{cite:9b3b28e543b40856a44695d27fe500120c313a1d}} and UNIT {{cite:ffc92e445281a5702f502a65d2cf2c75e3469bdd}} are the typical unsupervised I2I translation methods.
MUNIT {{cite:3ed8ca915945d716989d2e89571d2d2f280955d2}} and DRIT {{cite:9b3ddd5aa63a7f4eea0021c52ed9b2d2f977f1a7}} are the multi-modal unsupervised I2I translation methods that are extensions of CycleGAN {{cite:9b3b28e543b40856a44695d27fe500120c313a1d}} and UNIT {{cite:ffc92e445281a5702f502a65d2cf2c75e3469bdd}}. Especially, we exploit DRIT {{cite:9b3ddd5aa63a7f4eea0021c52ed9b2d2f977f1a7}} as our baseline model.
INIT {{cite:711f0651d67087c30b2272a3503a3840f3117c83}} and DUNIT {{cite:f6dba751d5f63109e4b74634233bc2a2fb7b7002}} are the existing instance-level unsupervised I2I methods. These methods are compared only for quantitative evaluation and not included in the qualitative comparison, since their code (parameters) is not publicly available.
| m | 1064a8e66b8b6a69583f91b2a59a3ef9 |
We again note that the certified and clean accuracies of our networks are comparable to other (norm-based) certified training works {{cite:558aaaf262f9913898c1a909e57803d6c21de133}}.
| r | 9f6a78976be65fc7f955d16199a2c0d5 |
Also this kind of massive theory are helpful to understand many planar condensed matter phenomena. As pure CS theory is anyonic in nature, it is very useful to describe phenomena like fractional quantum hall effect and high {{formula:9303cccc-d15f-413c-ac95-f90fc6a3938c}} superconductivity {{cite:18c2563f9ceb62c2643009b1838d968fa312831b}} , {{cite:ef2b13bf947ccf9d2155f31031ffb64e2e6400a6}}.
| i | e2575e53a8826497929e0e481a4bc863 |
The final product of OHN model is the spherically symmetric ultra-compact horizonless object composed of homogeneously
distributed matter with infinitesimally thin crust. The equation of state of the homogeneous matter is {{formula:0d8f29c9-0df8-4266-a2dc-0f3bfcaeff01}} , where {{formula:133efde2-8309-4352-aef8-8e35440a7b5b}} and {{formula:0fcfef15-b9e4-44fa-98bc-e138f7c6921a}}
are the pressure and the energy density, respectively. By contrast, in this paper, we consider an OHM type model, but the final product
is the gravastar. The gravastar is one of black hole mimickers which was proposed by Mazur and Mottola in order to solve problems related to
the black hole (e.g., the information loss problem){{cite:90e0e5749509b880ca388838b367fdc73077af11}}. Its inside is occupied by the dark energy of {{formula:303802ea-11c1-4562-b3d7-4098772cbfc0}} , or equivalently,
the cosmological constant and hence the geometry of its inside is equivalent to that of the de Sitter spacetime.
| i | 7622b7485db6595450eb6df6efd62673 |
In {{cite:59ba93b55074c172de3a27004d76431d6d4a2143}}, the binary vector has been adopted as the architecture parameter. However, state-of-the-art architecture search spaces, such as {{cite:b9b60e3405788fccbabd05da897748853da56570}}, {{cite:3f71d55fedf61832070c3d643f6ba81566a0b631}}, are defined using categorical variables. In addition, repeating the architecture search is required to obtain multiple architectures with different complexities. We first introduce the categorical distribution as {{formula:af665f4a-4e01-46bc-8124-82e93796782d}} in the framework considered in {{cite:59ba93b55074c172de3a27004d76431d6d4a2143}}. Subsequently, we propose simultaneously optimizing multiple categorical distributions, each corresponding to a different regularization coefficient, to obtain multiple architectures with varying complexities in a single search. Each categorical distribution is updated by exploiting samples from other distributions to realize an efficient search process.
| m | ddf50297e848b051067412addd7ca6fb |
So, we face with a choice what the symmetry we want to preserve on a quantum level. Whatever the case, the dependence of the effective action on the Killing vector cannot be completely removed. The imaginary part of the effective action, which is responsible for the Hawking particle production {{cite:8849c7572010dacd3d4117d99e5ddbfecb01263c}}, depends on the Killing vector.
| r | 841b545e2d45ab49c2b08ed2ae12bb6c |
Comparison with metric learning. Table III shows a comparison of the LOVO loss and other metric learning methods {{cite:c06da6a1e1541e3fded27f800394c7bcf42846cc}}, {{cite:43d81e1434a131825873b685ef7038cc181820b5}}, {{cite:8f37db8b66ba0e9873885fd70d40c8446058630b}}, {{cite:a2f045540474f608aed875277a7b7d543adf844c}}, {{cite:d66bd0363887aebb242031033400d8d4a430401d}} with the same KWS model (LDy-TENet 12). As the other metric learning-based methods are employed solely for the KWS embedding vector, we only implemented {{formula:60f5f49f-9649-4858-9903-df57b391fefe}} in our model for a fair comparison and excluded {{formula:861ffabd-2380-4372-90d9-4240853c4c48}} .
The results indicate that the pairwise learning methods {{cite:43d81e1434a131825873b685ef7038cc181820b5}}, {{cite:a2f045540474f608aed875277a7b7d543adf844c}}, {{cite:d66bd0363887aebb242031033400d8d4a430401d}} perform better at high SNR levels, while other methods, including ours, which is not pairwise, show robust performance at low SNR levels. As the pairwise distance computes all possible pairs in the mini-batch unit, it shows improved performance in the clean and high SNR noisy environments which are similar to the clean data while they show degraded performance as the network is biased to the dominant classes. In contrast, non-pairwise methods show little performance degradation in the clean environment, and it takes robustness in the low SNR conditions.
Particularly, as our method would map the input data to the orthogonal class centroids, it shows robust performance over the other methods. Although the performance improvement between {{formula:76dab6b9-b91f-4e55-a6f3-1c6a6d5a5b56}} and {{cite:8f37db8b66ba0e9873885fd70d40c8446058630b}} is not significant, {{formula:c7517591-5364-48b2-b6cb-b0b80aeb199c}} which is our proposed method produces a {{formula:f4cd3d45-6bd2-474b-9b46-a311e4a6a03f}} improvement in the WHAM 0dB condition. By considering the computational power of the classifier and the performance in a clean environment (almost 97%), our method shows reasonable performance in noisy environment.
{{table:970442e7-8a3f-45e6-8838-7514a9bfc581}} | d | a57b1b42e598a1f1fb4e103c57cbd474 |
The position of the highest particle {{formula:8d9fe5df-e94f-40b2-882e-fddab4142d1f}} has been the subject of intense studies since McKean {{cite:9b64fbaaea1737dfa3f1e4ec518d48707ff14819}} who linked the distribution function of {{formula:7b1a419a-37a7-4e05-8d33-eccf5df0b599}} with the F-KPP partial differential equation. Then, Bramson {{cite:cac5d76f060e76298e9faec49da44a7d8beff741}} obtained the right centering term {{formula:d898bd3a-4638-457b-b1f4-8617dddce3e0}} and Lalley and Sellke {{cite:1845d036b0ee954b236263d78e8a0f54d0a19468}} obtained an integral representation of the limiting law using the limiting derivative martingale {{formula:f659a7bb-97de-4e0d-8bee-a891c3dc63b9}} .
A new step has been taken with the proof of the convergence of the extremal process
{{formula:7af746c6-f8a2-4825-80e6-ae46c40ef0bc}}
| r | a2da2702f2b72244329b69abb1bddc99 |
But surprisingly above 10 a.u. of heliocentric distance the
systematical deviation of experimental and theoretical data was
found {{cite:e8d47c91f6ca0f994ba1bf20606ed9f30bb3c35d}}. This deviation can be described simply as a constant
acceleration towards the Sun with magnitude of about
{{formula:9a0cca1e-f4dc-422c-9a87-34d84f1ee6dd}} m/s{{formula:bdedec2b-a415-4f1d-a94d-77198b6a9f99}} . This value is the same —
within error limits — for all the spaceships Pioneer 10 and 11,
Galileo and Ulysses and for all distances from the Sun {{cite:e8d9194112ed6c1e17c8ca0105be8d60528dda25}}.
| i | 23a66ab1838f46a55864a64ebf3d1d32 |
Twelve Regge trajectories are constructed by fitting three states and four Regge trajectories are obtained by fitting more than three states among sixteen possible convex Regge trajectories. Same as the example discussed in REF , twelve Regge trajectories obtained by fitting three states maybe arise from the insufficient experimental data or the inappropriate assignment of states. The convex radial Regge trajectory for {{formula:cb7a2105-e5f7-4923-86b4-d16aabf2a87e}} including five states will become concave if {{formula:1bf4df27-548a-4318-8e59-b7412a420866}} is excluded which is assigned to the unwell-established {{formula:9b651df0-4cd3-4789-b5b4-4f5894ee89b4}} state. The radial Regge trajectory for the {{formula:84f5fe40-8327-48bb-a0df-192c0e8ecae7}} {{formula:adfb92e8-515f-45b5-90f0-7cd15f3cc54a}} state is convex if {{formula:fe31c3de-ba40-40ce-bc57-08b56eff2aeb}} is included {{cite:70b7fd9d7209ad281345ee179fc6a1b1eb64e9f7}}, {{cite:ba34432207196e017deee86e2eaeea67ab04c927}} and becomes concave if {{formula:e7ca5585-805b-4944-9c96-a071a09cdcf4}} is excluded or {{formula:e4cb8b55-7ec4-43a4-a72a-59f8380fe027}} is assigned as the {{formula:88e78f3d-b583-4fdf-ba2e-e9d164bf5cbe}} state {{cite:ba34432207196e017deee86e2eaeea67ab04c927}}, {{cite:e6f40a5f924775d90c083ba18083b9f7415cc66c}}. The orbital Regge trajectory for the {{formula:4bc64e5a-2519-43d5-8424-87c4649ccaf6}} {{formula:ba51eff9-3879-48e3-80b8-5f62d8f4372f}} state will be convex if {{formula:6319a4bf-2834-4bfa-87f1-967e574ae836}} is assigned as the {{formula:ca52423e-8ef6-4e04-a250-64588277f5f6}} {{formula:c6e521c9-63fa-464e-8a86-5beb4313012d}} state {{cite:70b7fd9d7209ad281345ee179fc6a1b1eb64e9f7}}, {{cite:ba34432207196e017deee86e2eaeea67ab04c927}} while it comes to be concave if {{formula:bad073f4-1b56-48ed-a20a-039befc89327}} is assumed to be the {{formula:6ac3ec08-5096-4f08-90bc-2447077c1056}} {{formula:983d7191-2c39-4b59-8c26-2aa4aecc1847}} state {{cite:2516788c8523264e0b71b0c36f1ceb7e75c97a5d}}.
| d | 3ac0cc8efa1f8225d44526d73ad8df65 |
In fact, the perspective of inexact Newton procedure may provide potential inspirations for developing more efficient sub-sampled Newton methods. For example, {{cite:d7b54813e58503a7e49c7a870abec9e7fbb3ccc6}} proposed use to conjugate method to solve {{formula:7583d279-3e34-477d-b75f-5eef98e82d5a}} approximately, where {{formula:22520412-33e8-4e03-9dd4-1ce74c503693}} and {{formula:6763fe98-fb35-4c8f-b340-f0d02f10d980}} are sub-sampled Hessian and sub-sampled gradient respectively. This is a method combining inexact Subsampled Newton with sub-sampled gradient.
| d | 7de59fbd703eadb0d7cb0c0901eb8e8f |
While our results are encouraging, we believe that significant improvements in learning can be achieved through improved network architecture and increased depth. As was demonstrated in {{cite:477f6788d4fa9d49b1c2045737924eeea2822c7c}}, network architecture has a significant impact on the performance of deep networks. While the networks described in {{cite:9d7dff2ffa12af046159cfb5666408bbd5198923}} were able to learn complex features in just three layers, our results suggest that extremely large datasets such as the YFCC100M can support (and possibly benefit from) deeper networks with improved high-level concept learning.
| r | 237eabeaa80d50a9b84e879fd0b5e3bf |
An alternative determination of the LO-HVP contribution stems from an ab-initio calculation in lattice QCD {{cite:a364d93874f7bf308f81208df575caf4efdaf862}}.
The latest result from the Budapest-Marseille-Wuppertal (BMW) collaboration {{formula:8a60c11f-a3e6-4dac-9830-45ec11a60e4b}} {{cite:144b4543cfe3e80ebf7b678f7953425308163833}}, consistent with previous calculations {{cite:5a0b97d3d10005f7452ec1b226f86a04705611fd}}, {{cite:69d9660b2771338f075a270175806142bdf33436}}, {{cite:69161a2b8857de9e177bba823f3996aad51f1096}}, {{cite:0e24e50c7168107a1e663244176f245ef82a283c}}, {{cite:bff2afb42e6b96384b57a34bc2adf90988561ec6}} but with a three-fold reduced uncertainty, is about {{formula:8e578370-8680-45d8-abe7-4aebda485063}} ppm larger than the {{formula:5161f401-dac5-4d96-aac4-ee02508f5119}} -ratio value and agrees with the experimental average in Eq. (REF ) within two standard deviations.
| i | bb5c8e43d5aae77c1e3b9fc3c98a8650 |
The introduction of stochastic modeling principles for geophysical flows has been a focus of great attention in climate study, meteorology and oceanography since the seminal works of {{cite:a67484ad3d21cd1e4b417c05f467d31065c1aa92}} and {{cite:5b66dc49b375f04bda77838fe60c8c31fb523fa5}}. Most of the schemes proposed so far are built from pragmatic considerations or from homogenisation/averaging techniques coupled with linear or quasi-linear ansatz dynamics to encode the evolution of the small-scale random processes (see {{cite:e6aa4d58c58fb1f7bd60cd4ca9fba1bd6f31f326}}, {{cite:a06113f262306636019cc880fb63d433a737ab9d}}, {{cite:8ea5d75545eddeac0a156baaca34807e4f75f2a8}}, {{cite:d404300b6f3793eed4489788c3d76b95473c7623}}, {{cite:3b75ae2de8f234ff41f7927f99de3d297a6f21ec}} and reference therein for extensive reviews on the subject). In particular, pragmatic schemes built from parameters' perturbation have shown to bring some noticeable improvements in weather forecasting applications (see for instance {{cite:3b75ae2de8f234ff41f7927f99de3d297a6f21ec}} for some examples). However, these schemes remain ad-hoc in their conception and lack of general solid grounds to be extended to other models or configurations just as does classical physics. They also face some theoretical issues, for example the control the variance brought by the random terms, which may, beyond numerical stability, deeply change the asymptotic nature of the underlying dynamical system, even for low noise magnitude (see {{cite:d7ac54848eaa75ce1a5f292903540f0332a9ec03}} for an example of this phenomena on the classical Lorenz 63 model).
| i | 9bb8ae594cc36cd9e09f0e54902b5e87 |
Overview.
In this section, a novel multi-modal video summarization method is described in detail. The proposed method is composed of a contextualized video summary controller, a textual attention mechanism, a visual attention mechanism, an interactive attention network, and a video summary generator. The flowchart of the proposed method is presented in Figure REF . A pre-trained CNN, e.g., ResNet {{cite:1e2c2f0093daf1fb0ccc1f35c1e009b713a9d715}}, is used to extract features from the visual input and the “Visual Attention Mechanism' is exploited to generate the attentive visual representation indicated in dark green. An input text-based query, e.g., “Jumping with a skateboard”, is sent to “Token and positional embedding” for generating the input of the “Video Summary Controller” which is composed of a stack of decoder blocks and a “Textual Attention Mechanism”. Each decoder block consists of masked self-attention, layer normalization, and a feed-forward network, indicated as the red dashed line box, and the 768 color-coded brick-stacked vectors serving as the input of the summary controller. Note that the masked self-attention is considered as a function of {{formula:a0a796d7-f668-4f9c-b872-00c322ea79a0}} , {{formula:cd0a7cdd-ae1c-40de-98bf-89f4f933c29c}} and {{formula:e1b859d8-5c5d-4256-a2a7-8f1608ce42ac}} , i.e., {{formula:4cc9f551-1f34-4b02-9c72-81edb2ae56d0}} in Equation REF . We exploit the “Textual Attention Mechanism” with the output from the last decoder block to generate the attentive contextualized word representation colored by dark blue, i.e., the output of the summary controller. The proposed “Interactive Attention Network” takes the attentive visual representation and the attentive contextualized word representation as inputs to generate an informative feature vector shown in purple. The informative feature vector is the input of the “Video Summary Generator”. The job of the summary generator is to output the query-dependent video summary.
| m | 5cc8cd5054f6770ccb983dd73e8a227c |
Inferring the location and properties of underlying latent events is often of scientific interest.
However, this can be challenging from a statistical perspective, since neither the number of latent events nor their locations are known.
Traditionally, inference and estimation in NSP has been approached with RJMCMC algorithms, which use birth and death moves {{cite:4d6e1c11d9f87bca6dd228c18bbfaefa28158d1e}} to address this trans-dimensional inference problem {{cite:1a0b9d1981faad54a3c339e63a3e2b06fb01bab7}}.
However, in practice, RJMCMC can suffer from high rejection rates, sacrificing performance unless proposals are carefully crafted.
Alternatively, there a number of specialized algorithms based on minimum contrast estimation, which optimize the parameters of the NSP to match statistics of the data, such as Ripley's K-function {{cite:804d7dbbf84fbca83b5658d88af02817222d6178}} or the pair correlation function {{cite:aa76c1f661d3633cdd76dd0f5f05ce502f193ff8}}, {{cite:a57da55040251672755ba03a7e9d7bffbfb77da6}}, {{cite:bbd192a6edc2dbb038085d39d50c378a9ec971ff}}, {{cite:1a0b9d1981faad54a3c339e63a3e2b06fb01bab7}}, {{cite:23b3224ec96e3c435d8dbc467e30134aa0841db1}}, {{cite:5c175959a73f522eb46ab2cb0babe48d4a150b97}}, {{cite:00d1e12dc21ecd1602a6296d9598814e27ea4549}}.
Though these estimators may be asymptotically consistent, they may produce biased estimates with finite sample sizes, and their theoretical properties are still an area of study.
Further, obtaining closed-form expressions for these second-order statistics often requires strong assumptions about the parametric form of the NSP.
For example, several methods assume that clusters are spatially isotropic and of equal size in expectation {{cite:5f0e557738531dcde3507e6f67402e429b0376dd}}, and subsequent research has aimed to relax these constraints {{cite:0ea6635adc92f9726899ada36519002ab2f3b3a7}}, {{cite:e3eecef78e354ca15133f33251db3003000912ca}}.
| i | 0e78ee6dfc2349491c7780057f16ade7 |
This section describes
the proposed unified generative and discriminative approach to ACE
(Fig. REF ).
To tackle the frame-level ACE (Section REF ),
we formulate a probabilistic model {{formula:f5ca4927-007b-49a3-97bc-e05f8558add6}} representing
the generative process of chroma vectors from chord labels and latent features
(Section REF ),
and then introduce neural statistical estimators {{formula:acab5279-8432-4b9a-8cf6-d27eb03e18d9}} and {{formula:b02cb0b6-662c-44b1-b2f0-c930f36fe7ee}}
that respectively infers
chord labels and latent features from chroma vectors (Section REF ).
These three models are jointly trained
in the framework of amortized variational inference (AVI){{cite:a5561b53d51f05f2477ad63e229c7ecb2ade30ad}}.
In theory, {{formula:40a81651-a4bd-402e-941b-1c60366f34be}} can be trained in an unsupervised manner
only from chroma vectors without referring to their chord labels
(Section REF ).
In practice, {{formula:19aff56c-3d75-41c2-aab3-92a63c326020}} is trained
in a supervised or semi-supervised manner
by using paired data
(Sections REF ).
{{figure:6d258246-f9df-4c47-a830-ed92a096f05e}} | m | c083c93f215f840c315bd2792096ff1d |
Motivated by this example and returning to the abstract setting, we equipp {{formula:88940b99-cd9a-423d-ac39-cc83b22c9fef}} with an abstract Banach norm {{formula:9ae41164-f463-467e-ad2b-724b2c16990f}} , the space {{formula:8eb81bbd-d043-471b-b834-538c5a846e8c}} with the graph norm of {{formula:2f539177-cf2c-4adc-bcd4-fb496a583a3b}} , and assume that the embedding {{formula:f5ca635c-4e56-4614-8c5e-197c2b31bf30}} is dense and bounded. Drawing further parallels between the abstract and the PDE/ODE settings, throughout this work we distinguish between the strict inclusion {{formula:9888d940-4ea8-4a73-9914-1c731cdc1fff}} and the equality {{formula:761289ee-4c68-45b7-bb85-bd28a74dcf8a}} . The case when {{formula:914dbce4-a197-47e6-9c3d-dc171f520a2c}} is strictly contained in {{formula:f4dd0e22-e33b-4ab9-9a68-4db15aa8a160}} is closely related to the notion of quasi-boundary triplets extensively studied in the pioneering work by J. Behrndt and M. Langer {{cite:8eac6756ff6f8fcb44805530f036832e31415380}}, J. Behrndt and T. Micheler {{cite:cc6f8e8b6fbd2e147a19912199dbfff90a9d370b}}. In case when {{formula:f247cc80-0e05-41ca-ac8f-e0fcdc1aba6a}} the triplet {{formula:1dcd605a-cb5e-42e4-813f-13decf4d576a}} is called the
abstract boundary triplet. This case is understood much better and was developed, in particular, in the classical work by V. Gorbachuk and M. Gorbachuk {{cite:5aa06cef6121adb9dfbd262c46e41de8f2888af6}} and A. Kochubej, by V. Derkach and M. Malamud {{cite:ff0a7b59f88b3c95093941677031c555ca1cf919}}
and many others, see, e.g., {{cite:a66b9d677f9728e5a7dc3454f7e464e3554e8f96}}, {{cite:8fc8c645419b8faef79c47e71795a73406cfdba4}}, {{cite:aad7c5fd9e4c309305b9690984d84e56f2d4d9d7}} and the extensive bibliography therein. The main reason why we consider a non-surjective embedding {{formula:1e9725a8-68d6-4a7c-a2fe-e30fafb1980e}} is that, when applied to elliptic operators, it allows one to use the standard Dirichlet and Neumann trace operators as components of {{formula:d8cfe4f1-bfa5-46a0-9f98-85af823459cf}} and therefore discuss physically relevant boundary value problems (e.g., heat equation on bounded domains). The disadvantage of the condition {{formula:526cc23f-fcd1-492f-b9a1-e696142f76df}} , however, is that it restricts the class of admissible self-adjoint extensions of {{formula:170e6fea-1654-4451-855a-26ae8d33eda7}} to those with domains containing in {{formula:a37201d2-72e5-45d7-b11b-d84210a47d55}} . On the other hand, the case of ordinary boundary triplets {{formula:95e219a3-7ae6-4160-aa41-d4728e4b0db7}} covers all possible self-adjoint extensions at the expense of dealing with the trace map {{formula:a606d819-fde0-4161-966b-1f0617ed8106}} which, when considered in the context of second order elliptic partial differential operators, is a non-local first order operator on the boundary of the spatial domain. The trace maps of this type have been studied, in particular, by G. Grubb {{cite:7489938cec0dd44cefbfc531e7850ddfacc21169}}, H. Abels, G. Grubb, and I. Wood {{cite:38c48e4e6f9b6c4eddecdde03c723df107b425c5}}, F. Gesztesy and M. Mitrea {{cite:212d9b7bd39c6b737bcb37b69ce435d74e6af5d7}}, {{cite:31bfa65a75fcb1134366d08115a1b592afcc475b}}, {{cite:8cfce9de46466f2bfcfbb6bc51bb8fb018c60929}}. We stress that ordinary boundary triplets are particularly well suited for ordinary differential operators and quantum graphs; we will exploit this in Section .
| r | 254930d97493334e22bf89d29b1f1e93 |
Extensive experiments with models of up to 110B parameters demonstrate large performance gains over standard data and model parallel training strategies. Our approach also matches or exceeds the efficiency and performance of previous sparse expert approaches {{cite:a3aa38b83990b6a94489266d3208580aff9873a8}}, {{cite:1b22d8b5a7ccba654e6898ea1ce54ad8a33188df}}, when controlling for computation budget, despite its relative simplicity. Taken together, these results demonstrate the first drop-in conditional compute layer that can be easily added to any model with no new hyperparameters or training loss modifications.
| i | c8157f04391a6bb03cc9871762d8b920 |
To demonstrate the effectiveness of our model, we compare our segmentation results with the state-of-the-arts (SOTAs) methods on three referring segmentation benchmarks utilizing relevance filtering as the localization module, as shown in Tab. REF . It can be seen that our model achieves the best performances under IoU metric across different datasets even though we do not utilize the time-consuming post-processing, e.g., DenseCRF {{cite:c82735f0e8e1af369a3c3cd37e8371884c6b7fcc}} and ASNLS {{cite:a721d397d5217924cff062dd143dd1a1b89d4013}}. Note that our model can further improve the performance when adopting relevance filtering for two times as shown in Tab. REF . Specifically, compared with the best competitor CGAN {{cite:22bd2483996c108726e1207b5e1c0578b24c28ad}} which proposes a cascade grouped attention network to perform step-wise reasoning, our model significantly outperforms it by about 3% absolute IoU point on two challenging datasets (RefCOCO+ and RefCOCOg) performing only the simple relevance filtering once. The improved performances over the best competitor indicate that our model is very effective for this task.
{{figure:13397142-8986-4ea4-ad16-eb96420cba12}} | r | 94e87ec9f60b4b918ff7b7f528fd82f2 |
Table REF summarizes the results of our case study.
For the accuracy results, statistical significance is determined with the Wilcoxon rank-sum test {{cite:8002497cd78474155050e5cb8a75aaba74a083d2}} due to non-normality.
For the efficiency results, statistical significance is determined with an independent two-sample {{formula:568f4583-3c9b-43ec-809f-7860fafc39f3}} -test.
| r | 61849f0896750c423dae3e76ad0c69cc |
Another unavoidable error contributor is the noise appearing due to the low level of detected intensity. WVA is performed at near orthogonal pre and post-selection configuration which reduces the intensity of the post-selected beam significantly {{cite:175f9ee3b56bddc3e1771bcd4c8e0608e27060f8}}, {{cite:d6ff6dd1fdef7a3a998ab07dd8d1312431668537}}, {{cite:9175f7ee36c3c051d5267acbdeac51b6af21f665}}, {{cite:dd2c2cfb6efba3740585e8838d6dcde47bdea33a}}. For {{formula:7a329699-cab2-4b07-8564-cf21d782409e}} input polarization, at {{formula:ab51cef3-41cf-4d7d-bfcd-3774c5e50490}} , the variation of the detected intensity is plotted with changing {{formula:37430010-8f65-4b56-9fe2-8c878306877c}} which shows certain decrease in the intensity level around near orthogonal post-selection region {{formula:1277bfcf-660c-4e3a-8cd0-7351713d47a8}} (Fig. REF ). On the other hand, for the horizontal input state, when {{formula:41a764a3-d83e-4750-b1a2-c20ec4917580}} approaches the Brewster's angle, the air-glass interface itself reduces the intensity of the reflected beam rapidly {{cite:56098e8691aee53ee6f97dfc04da566b1e6afb06}}. Again, the near orthogonal post-selection makes the beam more dim. These low levels of intensities usually affect the estimation of any parameter from the intensity of the beam, such as, {{formula:5855574c-5cb1-4d1b-bd45-8f13cf161bb0}} etc. It is also to be noted that around Brewster's angle and subsequent near orthogonal post-selection, the beam structure gets distorted, which causes errors in calculating the centroids {{cite:1cb771fc38ded37c785ee3ba283084f0239c7275}}.
| r | e0a0f60bd49a54f08602cfb56acb653f |
{{formula:2f49cf9a-0579-4e8f-8a8f-0af7c363c3f0}} -LaYH{{formula:b0dbcd3e-2753-4430-af97-0bf3be2275c9}}
can be regarded as a
compound based on
the {{formula:0c878857-6f06-47f2-808b-ef2aff607b40}} -YH{{formula:90831901-8fae-4ca4-8f83-c3a72786872a}} dimer
with La substituting for one Y
(Fig. REF ).
The {{formula:5e63fddd-b82c-4ae2-ac27-14880f6307b1}} -YH{{formula:cc8b6fb8-0f16-4754-a322-b470ff7971b6}}
actually appears as a
stable structure in the
convex-hull analysis given in
Fig. REF .
Our prediction, {{formula:42156236-afcb-4bc5-9811-7d3e0a218d72}} -LaYH{{formula:4489eaf2-49a7-4c10-a729-bbb7eaedb0a5}} ,
is actually a distorted
structure of {{formula:fed8b29d-8089-4384-aa7b-83467bf36d32}} -LaYH{{formula:5bb7bd4a-54f3-4315-aaec-fbe18cdfff67}} ,
showing considerably similar
X-ray diffraction (XRD) peak patterns
as well as that of {{formula:125e2a5c-8fe0-4b52-9ce5-9f225a3efbd8}} ,
as shown in Fig. REF .
Furthermore, the possibility of the
structural transition
between {{formula:1b4bc465-f08b-4454-9a13-1cd677142d8d}} and {{formula:382fa575-244f-4189-8dad-fdee6539d617}} at
higher temperatures is pointed out.
{{cite:caabbe50879e8ca027897f453baf89ecc402d5ef}}
{{figure:37947ba4-d7fa-4ff1-9a4b-b214ebcc5070}} | d | fe3bce45f5fd2035bf8d0dabe35bb0c1 |
There is strong evidence for dark matter (DM) comprising approximately
85 % of the matter in the universe {{cite:b02d1857d355f75704bdac70853bc7f40dea2432}}, {{cite:a64b0bfc8bcec7358c38d68cae08d8ef7233a2ea}}.
One possibility is that the dark matter is a weakly interacting
massive particle (WIMP) {{cite:a985bd58bd7726596cbcf2120fbe8b42a72d7280}}-{{cite:49e83f2bd13f82dda8e3412cd3289428b4ea54eb}}, entailing
associated new physics beyond the Standard
Model (BSM) {{cite:d8dd4747fa00330e1773aa676ce58b3b0fd45d98}}. Here we discuss a model of this type involving extra
spatial dimensions with fermion wave functions that are localized in
the extra dimensions. The underlying gauge group is
{{formula:00a39256-ac2b-47ab-90fb-1743e9e65b45}}
| i | 649949de7f10fef5cd022af17fad5867 |
We shall argue that these features can be naturally unified within the framework of the Page curve and Page time {{cite:2f534c96cd1ca7e78fa81a1b2815500c4ab21ba6}}, {{cite:069ecd0891b9997c2f33f3fa1cc24edbd479d2db}} assuming the entanglement between the brain states and time series of neuron excitations.
| i | 0c4a2c3e41facc76739b94f1865c641a |
The neutrino is a fundamental particle and was first discovered in
1956 by Cowan and Reines {{cite:ee49b9c5510a2057bd8a3941cb57c93e433d0139}}{{formula:f6863ec3-2020-4e38-bd03-cf298ef8c068}}{{cite:f1f7bb65b26cfe941873d54c39e7a1508a747571}}. In the
last few decades, it has been proven that the observed neutrino
oscillations can be described in a 3-flavor neutrino framework. A
parameterization of the standard Pontecorvo-Maki-Nakagawa-Sakata
(PMNS) matrix describing the unitary transformation relating the
mass and flavor eigenstates, defines the three mixing angles
({{formula:123ceded-e887-4a4c-9ebf-4f74cf5a36bb}} , {{formula:2ba7d06e-d463-422b-94f9-bc0c7f9207a1}} , and {{formula:c0e9a57d-ebb9-4d41-8ce6-c245414c2726}} ) and one
charge-parity(CP)-violating phase {{cite:32ed9097426fe6bfc61b6ddf4c0fb710b2981b52}}{{formula:c289d36d-8299-418e-96b3-e8035081fdc5}}{{cite:9396a0ad3fa72559be3aa954a3f46e6b584e7132}}.
{{formula:30420f75-e2fb-49b8-b592-a16aa5d673e3}} is about 34{{formula:40071a82-e99d-45e5-bd21-f18f517287f9}} and determined by
solar and reactor neutrino experiments. {{formula:96d4cdb1-a541-4e7d-9f90-05e4492f9afe}} is about
45{{formula:42977b36-04c5-4321-9c6f-7684d72ff4b0}} and determined by atmospheric and
accelerator neutrino experiments. An upper limit of the last unknown
angle {{formula:259029d0-b6d1-4c91-ad15-24cb52012671}} was given by CHOOZ
{{formula:0d6b5f53-7125-4d83-907b-e2f8e7ef08c9}}{{formula:260ce36b-75a4-432a-93e8-4dabd573b4b1}} 0.15 at 90% confidence level
(C.L.){{cite:e99ac1355fe86041046a724ce48667b8d663a7e8}}. It was hinted to be non-zero by recent results
from the T2K {{cite:5da441a424eb0e87da47e635e817e9ffe32c4f5d}}, MINOS {{cite:1711c3a859233590ed15b3dbb428d380f1d7bbe7}} and Double Chooz
experiments{{cite:a6d6e786b1c41e7d60bba00d9b6233deaee590cf}}. The value of {{formula:303f3663-ba2a-434c-99f3-26bcbcbb8346}} will
guide the designs of future experiments for the measurements of the
mass hierarchy and CP-violation.
| i | edaefef4f1f886f3ed73b3d7b21670e6 |
Lighting estimation has been investigated by direct generation of illumination maps {{cite:6df60d6eae730edb8ce4407902029e089aca766e}}, {{cite:d6a4e4fb530f89bd3734c224da40ec57f6168a3d}}, {{cite:9e4c2cd1e10dd696a35a291977c3705a57379624}} or regression of representation parameters, such as spherical harmonic parameters {{cite:89fd618be9e8bc6af1a92c70a4d56a9fa1c20ec0}}, {{cite:bd9204081c40f7bb77d02281e8b5c6c4f39969e6}} and spherical Gaussian parameters {{cite:7941449f3d3a5bd2b0d8cf586348b9983ef55d10}}, {{cite:bafa7109ba26e4f12bda439a0791c14ca87eed95}}.
However, representation-based methods often struggle to accurately regress frequency information (especially high-frequency information), which often leads to inaccurate shading and shadow effects in object relighting {{cite:bd9204081c40f7bb77d02281e8b5c6c4f39969e6}}, {{cite:82e5e2304d9936e83ce1d9684abf3305af947623}}.
Meanwhile, directly generation methods enable certain high-frequency information to be regressed, but often leads to poor generalization capability {{cite:6df60d6eae730edb8ce4407902029e089aca766e}}, {{cite:9fb98bb3719bc2031af14fe0acd0839d8db7a966}}.
| i | 2ce2ca3178f9552cb73bf82265c837a4 |
If {{formula:41b20552-64e7-440e-9bed-9169af020102}} are linearly independent, then there exists {{formula:b3507527-f8e2-4268-be9c-13b8b03ab538}} such that
{{formula:93b12d9e-5471-42dd-8eac-20c57cb422d4}}
where each {{formula:a698f3f0-fe3f-4d96-8f08-72e42d568278}} is a linear combination of the polynomials {{formula:3430c54d-9cea-4738-b694-8752f022d1a5}} . The zero set of the {{formula:40b1e35d-0b00-4525-8084-59e071f8c809}} minors of {{formula:e693b8a5-dc8a-447f-a4b0-66da56343e4a}} is {{formula:3b2a271d-174c-4647-a350-ff91570cfb93}} , and by definition it is the eigenscheme of the partially symmetric tensor {{formula:d5261429-91a5-4d86-82d9-a0cbb4f4cdf7}} .
Conversely, assume that there exists a partially symmetric tensor {{formula:6f6f6d63-a8f4-41c5-9460-8df1abc45a19}} such that {{formula:2d3fa2a7-89d7-4dda-a2b4-70d0ba9a3fa1}} . As the {{formula:5d997da5-8187-4211-98ac-465c3875959c}} minors of {{formula:8f85ba31-4f7a-4d9b-8b50-cb6b7ab12a8a}} define a 0-dimensional eigenscheme, the Eagon-Northcott
complex associated with {{formula:e2df8fcb-f76c-4b91-b296-819c67d64d6d}} is exact and. By Remark REF , its first terms are given by
{{formula:ed6487e5-3b1a-4e59-903a-3a9ca5dbd2a5}}
By the definition of the first syzygy map {{formula:7ea77555-61b9-4428-a4d7-ab834ac7f381}} ,
it is possible to see that {{formula:51f933bb-3a5c-45a8-b774-78f50cb79b95}} is defined by a matrix which is linear in the entries of {{formula:558b3775-7f0e-4e41-9a86-ebda9fe43537}} . More precisely, by setting
{{formula:1ad0eba5-eab1-47c5-b276-6b8d75560288}}
the generating syzygies are given by
{{formula:c862a885-898f-4c16-a36c-69d3d7761aba}}
for every {{formula:2d784d98-8159-47d7-a4d3-aa451536ac80}} . In particular, the first {{formula:a097b6e1-e0cb-4e3f-b425-37f89d5bb33c}} columns of {{formula:197822c9-51cd-4a72-b2ae-6d7b15e821ff}} contain only the forms {{formula:6994e102-2319-4bd0-b7bc-519804dd6d4b}} and 0. Since {{formula:7b53a73a-fc1f-425f-bf65-9e5c38f42a9d}} is an eigenscheme by assumption, its ideal is generated also by the {{formula:ab65e978-70f1-49ae-9982-1668e04ed66f}} minors of a matrix {{formula:f7f0198e-5e8f-45cd-a12a-aad2e490b234}} as in (REF ), so by Theorem REF the Eagon-Northcott complex associated with {{formula:4b85ba46-75f0-4dee-ad28-c93f5863d2bc}} gives also a minimal free resolution of {{formula:eb5098eb-5f31-43b4-932d-43b7b7d1cd39}} .
By {{cite:4026d1827c6e40a5121f52a0890a58e59d53e704}}, the minimal free resolution of a projective module is unique up to isomorphism of complexes. This means that there is a matrix {{formula:fd9f49e9-48bd-487c-8345-2910dcdcd416}} with degree {{formula:ba7be6b8-6933-4324-b0bb-e026e79c8568}} forms as entries, and there are three invertible matrices {{formula:b92645f7-1068-45d2-bd2b-6a2944ea09aa}} and {{formula:ffd973ad-6576-4f36-8017-a41ac735426a}} such that if we call
{{formula:422dd6d4-5530-454c-9312-732972524c6c}}
then we have {{formula:4bfe13a0-adb5-409f-b355-dad25660453b}} . In particular, the linear entries {{formula:b3b0a001-d399-451a-bf7f-a72d947e3f24}} of {{formula:c7170d80-ed1e-45bf-a0cb-4749d9374eff}} are linear combinations of
{{formula:9c59b343-38a5-491e-a438-cc3e86d3d713}} . So these forms span the whole of {{formula:ebb8fed2-d36d-41ce-828f-2253ea8399a6}} , thus they are linearly independent.
| r | 2006ab174aec1e781f4dd16112b07055 |
In this section, we briefly discuss how to directly apply the results from {{cite:a6430efba53d4ba0f7b1f24a72f3e45c2339d493}}, {{cite:f5ca8d64fb9f964b4ae83c29d2c9de2198895ce8}} to obtain Theorem REF . First, we match our notation to those used in {{cite:f5ca8d64fb9f964b4ae83c29d2c9de2198895ce8}} and instantize the convergence results in {{cite:f5ca8d64fb9f964b4ae83c29d2c9de2198895ce8}} on (REF ). Recall that {{formula:8f91f0b4-e2fb-4a72-be95-7370cc6845c8}} and {{formula:a8a44e9f-7930-4283-8d18-74ae77210c89}} . Their dual norms are {{formula:d5a1ef9e-f4af-4ca7-8003-d3431acc81e9}} and {{formula:3330c06f-b20f-4b9b-bb24-6adb521351b3}} , respectively.
The complexity of SMD is known to depend on the diameters of {{formula:e4f5cda8-b7ef-4ca6-bd40-684e5a6aa49f}} and {{formula:091457c8-e570-4dcb-aaf4-73f0c170a83c}} measured by the corresponding distance generating functions, namely,
{{formula:37159081-6dab-4be6-8484-99385362d4a6}}
| d | e37a9d89d6540ea8e08529c54fb865c6 |
Experimental implementation: Rydberg wire gates introduced above can be implemented in optical-tweezer atomic systems, which have been previously demonstrated elsewhere {{cite:11770925f9bf0bbc292e9d456ce415244771066d}}, {{cite:20880537668044d0f2a17549274ab5701cff17b1}}, {{cite:c21a3247b41885c4684bb843a9a8255fbf041b9c}}. As an example, we consider three rubidium ({{formula:56627eda-3827-490c-9a48-c99c8cd9260a}} Rb) atoms arranged in the linear chain geometry. Once the single atoms are loaded to individual tweezers from magneto-optical-trap, the atoms are prepared to one of magnetic sublevels in hyperfine ground states as the ground state {{formula:0df572b9-6337-43ca-876d-6252f0e30ffc}} (for example, {{formula:51ba21dd-5fc0-4269-ac26-029faa6441a7}} ). The states {{formula:42743764-dbb4-458f-bb27-9a43e0d0d4d0}} and {{formula:cca79f1f-2b95-4a04-bfae-63f458b17f2e}} are coupled by Rydberg state excitation lasers, and in general two-photon excitation is used to transit to {{formula:50d2fce0-8308-4a98-909a-22a34756c1f6}} or {{formula:8169b01c-badf-4c0f-ada1-8228ae60343e}} Rydberg levels via {{formula:03baa2d5-eb83-45f6-8b61-2012d423e031}} with 780 nm and 480 nm lights. For {{formula:69ef5c60-692d-4751-999d-8d09d17967f8}} the atoms undergo van der Waals interaction, and the interaction strength when the interatomic distance {{formula:fb4f82be-61a3-40a3-925f-3b6b027006d8}} m becomes {{formula:dbeb232d-09f3-49cf-a35e-ac71a3cbf475}} MHz, where {{formula:d02a821e-a998-4aae-907b-7a2b7e99c89a}} GHz. Individual atom-addressings to couple between {{formula:22dec562-aec0-4435-a4f1-5c9011bd0439}} and {{formula:666c024b-3622-4d9e-ae22-3a8f8bc19265}} can be implemented by diffracting multiple laser beams from an acousto-optic modulator (AOM), then focusing to individual atoms. The switching of individual beams can be done by controlling amplitude and frequency of radio-frequency wave to AOM. The individual addressing lasers can be either ground-Rydberg resonant lasers {{cite:c21a3247b41885c4684bb843a9a8255fbf041b9c}} or far-detuned lasers {{cite:af6d7eff5752a0e66e4d4f6db59cdd9cf4f52783}}, in which the latter suppress the Rydberg state excitation with additional AC Stark shift combined with global resonant lasers.
| d | 7ca656aa26ae7e60ae2ce5786b05f677 |
Lemma 8 {{cite:f4dd03f18d9cce5b6dda77abff430e65309441b1}}
Let {{formula:50dd64bc-7101-4718-be34-57bf121ece7b}} and {{formula:f1be7a46-bb0c-40d9-8339-4138670eb9e3}} be two vertices of {{formula:0406ce03-ef86-4d25-81d2-630311eacfe6}} . Then
{{formula:e881780d-8446-49dc-899e-a448bec4f590}}
| r | f87eea36c5a995325a4289dcfbf9b6e7 |
Efficiency analysis. We also examine the computational complexity of GPR-GNNs compared to other baseline models. We report the empirical training time in Table REF . Compared to APPNP, we only need to learn {{formula:cf3669b7-0362-4e1a-a9d3-215b285a0081}} additional GPR weights for GPR-GNN, and usually {{formula:28abd59b-df7f-4256-9950-8eb35fc69f33}} (i.e. we choose {{formula:bc5763ca-7308-496e-a44e-5283e0fce447}} in our experiments). This in turn requires {{formula:59f1eaa8-0c3d-4985-b52e-570d8f846fba}} gradient computations which are dominated by the computations performed by the neural network module {{formula:9d62a258-a6e0-49ef-a1d1-aeb49e18dadc}} . We can observe from Table REF that indeed GPR-GNN has a running time similar to that of APPNP. It is nevertheless worth pointing out that the authors of {{cite:d06f722610b51134cc48b7b0481114a3ef219e1c}} successfully scaled APPNP to operate on large graphs. Whether the same techniques may be used to scale GPR-GNNs is an interesting open question.
| r | f8f7dce27cf8c16e093b56aafd9833e6 |
Limitations.
The strength of our analysis is based on real world data during a massive surge in our facility where ventilator capacity reached fullness. There are several limitations to this study. First, the results cannot be applied to other disease states such as novel viruses that may arise in the future. The model needs to be re-trained with new data for each specific patient population. Second, the observations occurred during the height of the pandemic in New York City when little was known about the specific management of COVID-19. Results may be different with better therapeutic options for the disease. However, this is also a strength of the study given that it matches the scenarios in which triage guidelines are meant to be deployed. Third, the results could be different under different surge conditions, e.g. if the rise in number of cases was sharper or more prolonged. Finally, the simulation cannot mimic real-world work flows that might have required some alterations of the movement of ventilators between patients. From a modeling standpoint, our algorithm may not return an optimal tree policy, since it only returns Markovian tree policies. Additionally, we use a nominal MDP model, and do not attempt to compute a robust MDP policy {{cite:6bf3bf5b9610a5b1d05509fbae43c745f560f732}}, {{cite:b65c2e88ed94e0770d0481612c4cb06f08d18175}}, {{cite:81903a5d5156053bc4c608532d73e793fe4d1172}}. This reason behind this is that we have a fairly small population of patients, so that the confidence intervals in the estimation of our transition rates may be quite large, leading to overly conservative policies. To mitigate this, we use our simulation model to estimate the performances of a policy, and not simply the cumulative rewards in the MDP model, which, of course, also depends of our parameter choices. Therefore, even though the triage guidelines computed with Algorithm REF are dependent of the (possibly miss-estimated) transition rates and our choices of costs, the estimation of their performances is not, and is entirely data- and expert-driven, relying solely on our data set of 807 patients hospitalizations and our collaborations with practitioners at Montefiore.
| d | fed8ae340c19d8e789fdc2a04b347d1e |
fig:overview-scenarios presents an overview of the papers. At the top, papers are represented as a UMAP {{cite:9177360a9f90ccad032c8daef9a41fa68dcaeaee}} 2D projection of their coding and colored by coded scenario. The UMAP projection reveals a clear split between two clusters. At the bottom, the average coding per scenario and per cluster is represented as a heatmap.
The cluster on the right resembles papers approaching scalability primarily from a visual angle, as characterized in the Visual scenario (blue), with Clutter/Readability as an effort variable. The papers from the other scenarios in this cluster are likely those that discuss aspects of scalability that are primarily associated with the blue scenario (i.e., Didactic/Argumentative, Case Study/Examples, Few Samples). For example, three of the red papers are related to tiled-displays (screen or projectors). The cluster on the left covers papers approaching scalability primarily from a compute angle, as characterized in the ARS (green) and PCS (red) scenarios, with Compute Time as an effort variable. The papers from the other scenarios in this cluster are those that discuss aspects of scalability that more often associated with the green scenarios (i.e., Plots/Table, Experimental Validation).
Overall, this overview shows that multiple aspects of scalability coexist in the community, and even within our four stereotypical scenarios. It also hints to categorizations of scalability papers finer than, or different from, our scenarios.
| r | c31fb8922629d4977e04d6ad78ed1924 |
Multi-orient Box Matching.
Compare to DETR {{cite:656f41e41409c119766a32707766d16d86279378}},
the difference is that we propose an angle prediction and corresponding loss while only horizontal boxes prediction for DETR. Let us denote the ground truth set of objects by {{formula:e7ae533b-a367-42b2-b01e-a8425f014a6f}} , and {{formula:f9f56a72-e661-4010-aef9-fe9408f07d49}} the set of {{formula:02db0aeb-84d6-4f73-9fff-69a2e62549ed}} predictions. {{formula:9ec86946-8364-4137-982d-43621a4bb13d}} is as a set of size {{formula:9c355aae-3b78-43bc-a4c1-f601f097b733}} padded with {{formula:43073fd2-a44d-48e2-8142-a00be62e55ac}} (no object).
To find a bipartite matching between these two sets we search for a permutation of {{formula:81d468bb-5910-4e41-b43d-e744c44a8eb0}} elements {{formula:4229794a-c912-4c27-a85f-3fb43c4af613}} with the lowest cost:
| m | 1442df2afd18b902dc0e7bbed06dff8e |
We also provide a comparison of our frame-level detection performance with existing methods in the literature. As shown in Table REF , our proposed method with the BiTraP trajectory predictor outperforms using stacked LSTM as predictor as well as both previous reconstruction-based approaches, the Conv-AE {{cite:e06101905c600fabd665fab43cca2d32b4d4bc7f}} and TSC sRNN {{cite:e94618754686d5ff808f28e81f5c81a2d9825d37}}. However, the simple bounding box coordinate-based representation ({{formula:5e391778-eddd-4114-b2be-f14da4e95594}} ) used in the current pipeline does not seem to be comprehensive enough compared with the more complicated features used in alternative prediction-based works, such as the intensity and gradient difference for all frame pixels in the U-Net {{cite:161773fcbaf1c13a15dec697081fe6f5269bac56}}, the full-body skeleton representations as used in MPED-RNN {{cite:3d781031671ee0aada67b0eaaa9cbc445eb9cf10}}, or the fusion of pose features ({{formula:852d3619-87fb-4623-a7a6-9cd687e2ebd8}} ) across multiple timescales {{cite:546f847934366eef72f0ca132bcb3cb1326ccc12}}. Nevertheless, our work shows that it is possible to perform anomaly detection using only a very simple and efficient bounding box trajectory representation and that our unsupervised, prediction-error-based pipeline outperforms prior reconstruction-based methods. Future work will include incorporating additional motion, pose, and appearance/texture features from the pedestrians as well as the environmental contexts to further improve the detection performance.
{{table:805684f1-8f6c-42bc-aefb-afc89244669c}} | r | 5721c742e37f792a98e44cec1639a664 |
In this paper, we presented efficient methods for Whittle–Matérn Gaussian priors in the context of Bayesian inverse problems. However, the techniques developed here may be of interest beyond inverse problems in fractional PDEs and Gaussian random fields. Several extensions of our work are possible. First, while we focused the derivation and numerical experiments on zero Neumann boundary conditions, this is not a limitation of the framework, and it is easy to extend the framework to zero Dirichlet and Robin boundary conditions. The latter may be particularly suitable to mitigate the effects of boundary conditions (see {{cite:1a61ddfb4e46150ae55c19d0e8a1436218aa8f5d}}). Second, the techniques in this paper may be extended to generalized Matérn fields on compact Riemannian fields, see {{cite:d9af5b54f5f63a564aceb5193a95cc65c6bf5388}}, {{cite:cf51e5594cb9b53fc2137f87e1f2709bd06c7f4f}}, {{cite:c91d99a7de971ed2cb208d061271b2a3a33e652e}}, {{cite:543571f42f9ef92e16ae23b7f6357d056ee85939}}. Finally, in Bayesian inverse problems, it is straightforward to extend the techniques to nonlinear forward problems within a Newton-based solver for the MAP estimate {{cite:2912d98a6494ebb543d170ec5334b16f6ade19ed}}. Another interesting line of research is to extend these techniques to dynamic inverse problems, in which the parameters of interest change in time and we need to use spatiotemporal priors.
| d | 4ba84a800135b30a1436bc9f18b80e4f |
To compare the results obtained by the original version of ML-Constructive {{cite:0aac7a0ad727f6983d34b9651b62ecbaa666787e}} and what it is proposed in this work, experiments were carried out on the same 54 instances selected by Mele et al.
{{cite:0aac7a0ad727f6983d34b9651b62ecbaa666787e}}
from the TSPLIB library {{cite:828ebade27db77463478d0d256c264fdc03ed516}}.
The size of the instances varies between 100 and 1748. A brief recap of the results – with the heuristic executed using several ML models in the first phase – is shown in tab:mlg-results.
A more detailed version of this table can be found in the online compendium, where the results of each instance are shown and discussed.
| r | bf35b9b646e53d6c76f255c933666a11 |
The forecasting of low-level representations like landmarks or facial action units has been recently tackled with deep learning methods such as recurrent neural networks, graph neural networks, and transformers. The usage of such deep and data-hungry models has been encouraged by the recent availability of large-scale multi-view datasets, often annotated in a semi-automatic way {{cite:3c82045cba91ccdb2d5006298d45224c80ce69aa}}, {{cite:8f65894bb9ab787ac83aa59715a0215133a69473}}. The increasing accuracy of monocular and multi-view automated methods for face, pose, and hands estimation has contributed in reducing the annotation effort. Still, the largest available datasets that provide thousands of hours of audiovisual material and feature the widest spectrum of behaviors do not provide such annotations {{cite:5d92ac53feefd84407df05634c6e2003c526debe}}, {{cite:b053bec018d35746dd36f34985def8f34f7434f0}}, {{cite:448f0878c08d079f198e6ee3764f923a3bd576a9}}, {{cite:a728be87b7a31e5a6152398f88bbf9a3f1f81c29}}. In contrast, the automated methods for high-level representations recognition such as feedback responses or atomic action labels are not accurate enough to significantly help in their annotation procedures. Consequently, such annotations are scarce, and are only available for small datasets, as shown in our survey. Accordingly, recent works have opted for classic methods such as SVM, AdaBoost, and simple recurrent neural networks, which have traditionally worked fairly well with small datasets. We expect future work on high-level behavior forecasting to also explore semi- and weakly-supervised approaches {{cite:2675280cdd69fcfd18d0486e4442aef7b64197db}}.
| d | 06cab15671a715174f4ef9d8ab093a1d |
There are situations where we need both an explicit density and a flexible implicit sampler.
For sample evaluation, it is not enough to merely distinguish samples from real to faked one, and one may also expect to provide fine-grained evaluation on generated samples, where the energy values given by the explicit models can be a good metric {{cite:3d3f438e5f7cc4ef3b8f093dbb37a3dd3ac26063}}.
Another situation is outlier detection. Implicit models often leverage all true samples (possibly mixed with corrupted samples) as true examples for training.
To make up for the issue, explicit models could help to detect out-of-distribution samples via the estimated densities {{cite:497be5237ae3a6020d09b2eafaf6f64a69700481}}.
Also, when given insufficient observed samples, explicit models may fail to capture an accurate distribution, in which case implicit model may help with data augmentation and facilitate training for density estimation.
These situations motivate us to combine both of the worlds in an effective way so as to make the two models compensate and reinforce each other.
| i | 310117ac9d65189e97ab2ee61e9e31d7 |
Imputation settings. To investigate the impact of different imputation methods on prediction, we selected a wide range of methods based on their popularity in previous research. List of the selected methods and their description is provided in Appendix , Table REF . Mean, Median and FeedForward are selected from a single imputation group. Also Group mean, based on severity scale of previous lactate (as shown in Table REF ), is investigated. PCA, Matrix Factorization (MF), SoftImpute, KNN, MICE, and MissForest are picked from traditional machine learning solutions and missing indicators, while Auto-Encoder (AE) are examples of recent imputation methods. Generally, there are two types of architectures available for autoencoders {{cite:2f712349ac0917a0af19946436d970184b3cac87}}: overcomplete, where there are more nodes in hidden layers rather than input layer and undercomplete, where hidden layers have fewer nodes than input layer. Our architecture for AE is similar to {{cite:861b3fe7e0cd91fa1fc063b31977bb9d2921ba42}}. As autoencoder needs a complete dataset for initialisation, missing values in training and test data are pre-imputed by mean and zero respectively. All the methods were applied using python based on fancyimpute, predictive_imputer, and sklearn open-source libraries. Also for all the machine learning methods, the default parameters proposed by their authors are used.
| r | 1caddeab62034100d5612c02ba7b4183 |
where {{formula:169af54e-f141-4d57-9c37-7d50da7556b0}} reads {{cite:c28ee1068f9c7c61e8d6ec0083e49a168a152df8}}
{{formula:f3b46d53-8672-4edb-9b43-0e4be8ebc74a}}
{{table:710d57b5-43d8-4288-8fce-0e3d050f3335}}{{table:702a5401-22dd-4d85-b1ad-1e103efe933f}}{{figure:aea1b531-cbd8-4e45-83f7-9373b4ac6cb5}} | r | 1b948cb45adfad9defada57582452ae0 |
Let us now consider the incorporation of a soluble and non-volatile cosolvent, i.e. glycerol, to the aqueous dispersion.
The drying process leads to a gel phase saturated by a binary mixture of water and glycerol.
This results in a combination of the pressure flow component (REF ) and a diffusion mechanism{{cite:a03e26fa853ed21b0f7fdcb5825c2b3d73017f4d}}, {{cite:8dfa62e9b0cd212871c4948430e8a8b2622249a4}}, {{cite:d3de4ddc3f6512128b435f2db9028d9cfc52f570}}.
In that way, the diffusive flux for each component, {{formula:e1d1674c-f412-46b2-a5f2-79054937cdd9}} , expresses in accordance with the Fick law:
{{formula:4bd0f443-8779-4d5f-bb69-ab91a9f1b416}}
| d | 0031ca7327044beb00012928a3d8376d |
To see the effectiveness of RényiCL, we conduct additional experiments for fair comparison with other self-supervised methods.
MoCo v3 {{cite:28783f4fa348326aebd1cfcfebde1ee176fdc5d7}} is a state-of-the-art method in contrastive self-supervised learning. For fair comparison of RényiCL with MoCo v3, we apply harder data augmentations such as RandAugment {{cite:7d709ad0ce7de6c6b4501ac136b425ddd8b8d69f}} and multi-crops {{cite:44a04f986e277ba6c750893d62a9190491b84c5f}} on MoCo v3.
We first reimplemented MoCo v3 with smaller batch size (original 4096 to 1024) and in our setting, we achieved 69.6%, which is better than their original reports (68.9%).
Then we use harder data augmentations RandAugment (RA), RandomErasing (RE), and multi-crops (MC) when training MoCo v3.
Second, we compare RényiCL with DINO {{cite:508ae5d548d50d4b8d275326996ef4aaf0a56a1d}}. Note that DINO uses default multi-crops, therefore we further applied RandAugment (RA) and RandomErasing (RE) for a fair comparison.
For each model, we train for 100 epochs.
{{table:471ef125-33ba-41f9-b3cf-f9d8cd8fc356}} | m | 9cf10b599ce83d52018e373e42791804 |
The proposed model is based on Federated Averaging (FedAvg) {{cite:9b65f5f2c71a20a636d8fdbac868aef1238392cc}}, which is one of the most widely used aggregation algorithms in FL {{cite:269211f3b2d68d5ee95a6b413f33e2c18df2e9a6}}. In FedAvg, participants receive the latest global model from the server, update the model using their local data and the resulting models are transferred to the server. Then, the server updates the global model by averaging the weights received from the participants, i.e., the server and the participants collectively train a neural network without sharing local datasets.
| m | 0adefe8246de4b576a0e1d5e74e70ec0 |
Training loss:
Following previous works {{cite:66a7384e36487b3e68ada22d556b9f0d03986b75}}, {{cite:4d93a246d8b0b1eec16277242ba8c480d3ff53e2}}, we use a scaled version of scaled version of the Scale-Invariant loss (SILog) {{cite:dc4d317e72a5c1e0265e52136f6393c594e129bb}} to supervise our network. Given, the ground truth depth ({{formula:7da9d4ee-505e-442c-b4a7-d440076644fa}} ) and the predicted depth ({{formula:f45a222c-3047-48ad-9367-16ff97921a0d}} ) at a pixel location {{formula:9155d1a3-7f7c-4b81-bac1-ee7d953ec7cd}} , first the logarithmic distance between {{formula:0eef0b3a-d069-4da2-9191-9cd94b0d1131}} and {{formula:5f3d8aec-1585-46ef-b856-3e6b3aadada0}} is calculated as: {{formula:9e98fa57-e0e1-40f8-b9de-9c3e609b0b1d}} . The SIlog loss is then calculated as follows:
{{formula:8438e1ac-6f70-4a52-b431-fb8a876892b4}}
| m | 07c0dc4611dedab5ab0a6a39c3ed49ec |
Due to the intrinsic nature of the {{formula:27b28e53-8000-4a45-8af6-92c1bb5f6159}} -advection equation, mass conservation is not guaranteed {{cite:f4751d9c4e3456e2d9ff9ddd2166449b6f8e4fa1}} as the required numerical stability requirement of {{formula:f75becf0-c6eb-4664-b968-d81487963c81}} is not automatically fulfilled.
This implies that a re-initialisation technique is computed after Eq. REF {{cite:f4751d9c4e3456e2d9ff9ddd2166449b6f8e4fa1}}, which requires resolving another advection equation, Eq. REF , for the signed-distance function, {{formula:ae60320d-9df6-47ed-82ab-2b000674487f}} , solved with the TVD-RK3 scheme (Eq. REF – ) according to an "artificial time step" {{formula:440e5880-f227-482a-8aad-fe41f771e337}} = CFLLSM{{formula:fb6c997b-b5b0-4464-a720-d59d3571a0a5}} .
The smoothed signed function, {{formula:af35278d-6c53-4b5b-a225-28229593bdec}} , in Eq. REF is the calculated according to Eq. adopting the initial condition {{formula:3a85289f-c8a9-4351-a317-c28a98c35e5d}} = {{formula:bcb5d2a1-810b-476a-8e93-56a0042c2a19}} .
{{formula:8d93a20e-95f4-4d3d-8726-d81485a2cda0}}
| m | fc19e0c23b4f467de0d35f82500ba6cf |
Our argument will be extended furthermore in several directions. One direction is to use this kind of singular but invertible transformation to find new modified gravity theories such as scalar-tensor theories.
Some of degenerate higher-order scalar-tensor theories (DHOST) theories {{cite:8b8244fe16b4687ef049a8714f43ceaa530e5487}} can be obtained from the Horndeski construction {{cite:59aeb58ba4d774ca60e4d1d6b8b5ec3f7b5717e0}}, {{cite:fd71b1da79cf2caa9f79f30396af59d9372d032a}}, {{cite:d20b512e4ace881ecc4af186abb8b72f6810501d}} through an invertible regular disformal transformation {{cite:235168c3a96704a707fef47ba6169954f0301695}}. It might be interesting to apply this kind of construction to a singular but invertible transformation. In the standard regular case, both gravity theories are physically equivalent without essentially different matter couplings (that is, if we transform the matter couplings as well). But, in our singular case, a transformed theory could represent a completely new theory, even if the number of degrees of freedom is preserved.
| d | 65ba34eec0c42997c144534d23ab1b08 |
In the direction of Cygnus OB1, a gamma-ray source MGRO J2019{{formula:f70eb0a9-c44c-4a6d-b0a2-0687f48a0341}} 37 was discovered by Milagro at the median energy of 12 TeV with a source extension of {{formula:ad148445-4422-4af5-9f07-4c827d759d88}} {{cite:f79cd5e12e8403a08ecc11ce0c33be0a44ba6379}}, and this source was also recently observed by HAWC
{{cite:973521def49340373a1d1095fccd352a5760b1e9}}, {{cite:e5808fd5cd9352cbe2adae8fcf5426454194efc8}}.
VERITAS observed the same region above 0.6 TeV and separated the gamma-ray emissions into two sources: VER J2019{{formula:d1665c61-7e0b-4f61-8267-59d93637e37b}} 368 and
VER J2016{{formula:3548095c-0bf4-4d36-89b1-9cebc01c7114}} 371 {{cite:4b4fb6c3bb4407632d6e893eeccf7f1b7d0de1a9}}.
VERITAS also reported that the morphology of VER J2019{{formula:757e96be-1954-4ebb-be53-d8b5f19be919}} 368 is asymmetrical with {{formula:6f30e851-b6c2-4922-a51c-44fdf274aa9f}} .
The flux of VER J2019{{formula:d5063ffb-b9aa-4a25-8291-ced460d60432}} 368 measured by VERITAS in 2014 {{cite:4b4fb6c3bb4407632d6e893eeccf7f1b7d0de1a9}} is a few times higher than that in 2018 {{cite:0a51c47b5e6593a49166ad6771dfbc7fcc3d2489}} due to
different collection areas of photon integration windows.
| i | 9a64e240cf370644cb549a4a2f793181 |
The question is how to incorporate such semantic information in VQA. Traditionally in linguistics, semantic information about a verb has been captured via so-called thematic or semantic roles {{cite:d54a0cce4cdf36cfa99ad05a5f66bba14e31a145}}, which may include roles like agent or patient as encoded in a resource such as VerbNet {{cite:9b8d9a98e4d9f149e07aefa140190c724771b718}}.
Semantic role labeling has been shown to improve performance in challenging tasks such as dialog systems, machine reading, translation and question answering {{cite:71402ea6f4cc2109c4599ed0ee8800b21b3fd7e8}}, {{cite:4aaa91f166148be1b4c380be2565b2bd12b39be3}}. However, the difficulty of clearly defining such roles has given rise to other approaches, such as the abstract roles provided by PropBank {{cite:411656f15843bb17212076c1251a3decb4bb657d}}, or the specialized frame elements provided by FrameNet {{cite:de3017346a9e312aa152b01b9873e0ce2752cfc7}}. In FrameNet, verb semantics is described by frames or situations. Frame elements are defined for each frame and correspond to major entities present in the evoked situation. For example, the frame Cooking_creation has four core elements, namely Produced_food, Ingredients, Heating_Instrument, Container.
| i | e57019a304422a481e38adabbda04c05 |
A graph state vector
{{cite:62e11d0de1efc84e2a6cda54afe8f561d8ed5439}}
{{formula:b2aca097-fc74-4384-9a05-11650bd9e9d6}} is defined by {{formula:329dc01a-320a-4152-923d-ac25d2109b30}} qubits in
{{formula:092e4abf-7194-4b9c-b5af-75d372568c86}} entangled via {{formula:0a482b81-1a19-44b9-8474-78f63de8f69a}} gates for each edge,
{{formula:ef6062b0-480a-4af1-939a-89ced9e25570}}
| m | 6d16b2cf67a7ca72d229d8fb72cc2ca8 |
tocsectionReferences
Appendix
Asymptotic Variance
To state the asymptotic properties of {{formula:9345278d-4844-4693-b650-ef9593d7fa95}} , let
{{formula:c6518457-6274-4a44-b64e-9adfbbe6aa14}} be the individual's contribution to the estimating equations for {{formula:18a7ab5b-687b-4ea0-b5cf-3e155c2aaad8}} ,
{{formula:7f3e107f-265e-4970-a68d-b8a0c01c6cbf}} be the individual's contribution to the estimating equations for {{formula:3bfd6908-deba-4fdc-b352-276f01660fa4}} , and
{{formula:0edefb9d-c48a-471d-9848-2c1bbbb4d9f3}} be the individual's contribution to the estimating equations for {{formula:5bac1ea7-9da7-4459-a899-92068201891b}} .
Define
{{formula:2304585d-de97-493c-8844-223dbc140725}} ,
{{formula:b7d938d4-e927-4b22-bb2e-ae65fba1e056}} ,
{{formula:aa3eb275-d181-4249-99ad-bee564034645}} , {{formula:06e1bcc5-a3da-40da-998b-d7e48a78bc31}} ,
{{formula:7fbba396-cabf-4288-9b58-262786e07f74}} , and
{{formula:0496650b-5d79-478d-9d08-bc0d7a1033d2}} .
Theorem 1 If either the missing data model or the covariate model is correctly specified, then
{{formula:8fe08bf9-50aa-4668-8e53-fed9a747843f}}
where {{formula:e05a0470-c1bc-4d63-bcf5-83eea3029313}} is the true value of {{formula:d9c05e4d-5d92-4f87-9e25-d627cbb9233b}} , {{formula:14a0182d-7ebc-43ca-bef2-e5a2a8c7215e}} and {{formula:65aeab94-409c-4b54-9bad-8a61a90c8dd5}} are the probability limits of {{formula:80811e3c-30e9-455b-93e0-2f1eca148186}} and {{formula:c4b3c8be-08d0-4599-a5ec-b7f62ec8cd35}} , and
{{formula:e1352617-7031-439f-b9a8-5d90fa18fc2c}} .
Inferences for {{formula:0683bcd4-542d-476b-8e29-b95e47f5ff9b}} follows by replacing the unknown quantities in (REF ) by its consistent estimators. We make use of “generalized information equality” {{cite:65c70b6ea87872120c9851cc21bcba3c6f583377}} that
{{formula:c804cc1b-31fc-4798-9452-cae29fff364a}} , and
{{formula:d5607a0e-6b56-48fb-95c2-d8f4be19496c}} . Similarly {{cite:086dead9a546e8f19bac2d4f31fd1af675d40472}},
{{formula:a13a608f-7bdc-44f4-827a-035d8c1ee938}} , and
{{formula:2217b20f-1780-4241-8cc8-21482ec37850}} .
The matrix {{formula:acf64c40-711b-4492-b2b4-3c3e0d647510}} is replaced by {{formula:0492a92b-855f-48db-b5ef-5789d9be02a3}} , and
{{formula:9f9501c9-7617-4765-bad0-a5e95a4492df}} by {{formula:bc6a883e-c8d4-4ba4-a1b9-810cb44918f8}} ,
{{formula:b25d0289-70c4-4f8d-927b-bc6442b941ce}} ,
{{formula:02f23558-55e0-477b-b9de-1331f42eec0b}} ,
{{formula:1ca828f9-3c17-4a8c-b698-e0439b31f59b}} ,
{{formula:0426b984-7cb3-4dd7-9537-e46b4fe5f368}} ,
{{formula:00bed47c-a97b-4c75-ae77-b18e75efa636}} .
The proof is similar to {{cite:eba2da38e871468bc5c8054a2f3582e8ad59da93}} and is omitted here.
Additional Simulation Results
{{table:08ae8f36-2615-4244-aebf-dbf0be3441da}}{{table:5f9dcd11-1446-4c6c-9109-fbd0c958c6a4}}{{table:c0a527a5-a68a-4f1b-8baf-50abe4787609}}{{table:b274cf99-e7a4-4741-858c-f6ea23123f0a}}{{table:eafa6dd1-8154-47bd-a691-42bf7a0e3daf}}{{table:2955a4be-cf40-41cc-9844-a2bd565261b0}}{{table:66951c3a-d61f-431f-a3ff-2214cff5406b}}Table REF shows the convergence rate obtained from the simulation results for seven association structures and four sample sizes. The convergence rate for the working association structure (C) was calculated as
{{formula:024d3ede-19fc-426d-be30-adb3cd9d44d6}}
The local odds ratio parametrization presents a clear advantage over the correlation coefficient in terms of convergence issues, specially for unstructured association matrices and small sample sizes.
Additional Real Data Results
{{table:215bb97a-407a-4e53-ac96-6bba0876d597}} | d | bd0f4f3d6187c36978154c747904907b |
For the baseline model {{cite:09975148cc9a41e7a796f23dca85f3b175833ade}}, we obtain an accuracy figure of 44.76 which is below the reported figure of 46.88. Use of bidirectional LSTM for the baseline model doesn't make a significant difference. We report the results on validation set from both this paper's evaluation script and the standard evaluation scripts (released with the dataset). The results on the standard evaluation scripts are just slightly lower than the authors' evaluation script, but the difference is not significant.
| d | c6934a43f3bad600a95623708ce0fff7 |
L2 regularization {{cite:ebd8befa16e0503cf17a529f1cdc07d5a528fdb6}} is used to avoid over-fitting. The following objective functions {{formula:2e42e876-2f90-4133-8192-0859ece4de7a}} ({{formula:2da323d1-3bc6-468e-bdf5-c1ce8f458a81}} or {{formula:c37cb2ff-1150-4925-8c55-9b0f5f614d22}} ) is used by incorporating the L2 regularization on weights to Equation (REF ).
{{formula:51469cc9-9be5-49a0-8d77-b96899d79e0a}}
| m | 4664276e3db2bc01a49e7a94ddf8a880 |
Comparing speedups, our method obtains the greatest speedup of {{formula:34fad919-a7a6-41f7-9674-65e419213d14}} over the
autoregressive counterpart, with latency significantly reduced from {{formula:1209aeae-df39-4eb1-9e12-8afba9b70e31}} to only {{formula:0a3d8168-c21e-45f2-be3c-f6994e997e1c}} .
Previous works often rely on adding extra components, e.g. iterative refinement {{cite:ea422bcc6f96349502f16aba1a0cd8390b30d0b5}}, autoregressive submodule {{cite:2fef44f6fda96a6a259999ac8f814f9d9ad67d2e}}, or CTC decoder {{cite:a00648a75ad1b8a4589c5bace430b763f34eede4}}, on the Transformer to achieve better generation quality,
which, however, inevitably sacrifice the decoding speed.
Our method adds no extra components except the lightweight length predictor, and therefore is able to maximize the decoding speedup.
| r | 72fc19489d495e3dccd2c81f26f40f8b |
AdS/BCFT {{cite:df75e089d53382d52c6c001a4536230179e3fb59}}, {{cite:5420ccccbf14b9c3df6adeee7d131be1720f07ab}}, {{cite:5edcdd223e9883de3590719aca83483c338e658c}} studies the gravity dual of boundary conformal field theory {{cite:f6a722906081cfff1da5183269a9de263a8bb911}}, {{cite:7698e60fd90262f12be9f989c615cb1a8e673068}}.
The simplest, bottom-up model proposed for AdS/BCFT is a spacetime terminating at the End-of-the-World (EOW) brane {{cite:5420ccccbf14b9c3df6adeee7d131be1720f07ab}}, {{cite:5edcdd223e9883de3590719aca83483c338e658c}}, {{cite:88a75192110014aaf3ab8eaa0e3efc6ede7703ea}}.
The EOW brane is anchored to the BCFT boundary.
This bottom-up model captures some qualitative aspects of stringy models for AdS/BCFT {{cite:a44ad5889f62e8acc46a3d378bee129f1423006a}}, {{cite:d6672c735510b3f8707232d0957c1c872c05a857}}, {{cite:51a8c4d474c9b9519fae95a299b6ff6d558c715e}}, and has served a useful toy model for black hole evaporation {{cite:0693881a7b46d457081d24fd932a484bd1f6ae4f}}, {{cite:da40e2562d2a1ba40aef865982195aa75fb16287}}, {{cite:9e3fea707cda148f775649c92162d8fe0c77da63}}, {{cite:f334a17e74d59c3935b5604377d9f02d72a7ac69}}, {{cite:69c71bb6161960a3a91a456a1118f147e39e23eb}}, {{cite:387c1cce028bbbe7b743a0503c792ace3fa8b938}}, {{cite:af4e32d4019381f08d7785d7f4eb0b78f2a9b5f0}}, {{cite:5036885d4bc21cf21c8da12e936f6a225c3ca97e}}, {{cite:d1fc7c2a3d98806db510404276d71140dd3421b2}}, {{cite:28e1f0a1f03589db19b7b0e268494434513b3782}}, {{cite:4feeceb9ce63897f928b5acaf324047807ac3bea}}, interpreted as doubly-holographic brane-world models {{cite:9545a6a290dc83c3dd66386b2d8e08c09d144f94}}, {{cite:28611d1781d9966f6d310b97d6e148e819646a26}}, {{cite:bccce34cabaad12504cf224e7e915b77ebfcca8c}}, {{cite:327c7d3ba6e124608f225ce258d0f93e09bd57a5}}.
On the other hand, this simplest model of AdS/BCFT is known to have several atypical features in the boundary operator spectrum amongst all holographic BCFTs, such as fine-tuned boundary operator spectrum {{cite:760e4a7307217d85b7ed53e1c522da22d132a4b1}} and the absence of interactions between distinct EOW branes. The lattter, which is the main focus of this paper, results in a fixed gap between the lowest eigenvalues of the BCFT with two distinct boundaries and the BCFT with identical boundaries {{cite:872b8e241de100be8081569a79569ea462f4ea3e}}.
| i | b77f051acf2dc57c537e056b9c7bfb9f |
Factorization Machine (FM) {{cite:34359ebe949766210ae2849d8f496a15d0bc91f8}} is the state-of-the-art method in recommender systems. We compare with its higher-order extension {{cite:128582ddf8fbdf9ab7e824cd4313a218462bb558}} with up to second-order, and third-order feature interactions, and denote them as FM-2 and FM-3.
| m | 8b77cd0b970f50d2458b9f45198ef105 |
Definition 4.1
Let {{formula:019c1423-c869-4a38-8b76-2f2f6e3f9f5c}} , {{formula:d601fec3-cdf2-4ad1-a9c8-b437f9140390}} , and {{formula:93c2272d-0d83-4cb7-8cf3-438bf069711b}} denote {{formula:67b318e0-c36a-4509-bd81-1828c6b290a6}} real positive variables. We state that a real function {{formula:8536df4e-cedc-4c1b-8cd9-6553028187d2}} is a monomial if {{formula:51daee91-7efe-4827-a562-aa74a67e27cc}} and {{formula:b8157538-9c2f-4a91-be5d-2d99b2d0737d}} such that {{formula:39bdd54e-e4f7-4a46-a26d-72873cf4070b}} . We state that a real function {{formula:2327fabf-e742-4bed-b23a-b1a0c0d67e5d}} is a posynomial {{cite:6ee19527f5874404abe84f1d872fed678d9ab1ad}} if {{formula:080d5318-9245-47a7-b21e-9481a20694df}} is the sum of the monomials of {{formula:1a8b7eb7-39ea-4572-b4f8-d063bb6024b8}} .
| r | d35001961a755068e893e850efe889e5 |
General relativity (GR) theory of Einstein, which is assumed to be the most interesting and simplest gravity theory,
obeys the conservation law of energy-momentum tensor.
Although, since its establishment researchers are looking for different gravity theories and several modified gravity theories
have been developed. In this campaign, Rastall {{cite:98306dbc694f683d25611e534d8273e324a53ace}}, {{cite:35dfe807ac303656c8125df78e0a44d7a7585237}} introduced a very interesting potential modification of
GR theory, which does not obey the standard conservation law of energy-momentum tensor (i.e., {{formula:a493561c-d3f7-4f84-8bfd-ea674502e254}} ).
However, a non-minimal coupling of matter field via space-time geometry can be introduced in the form
{{formula:7a13d9e7-b050-4d44-bc17-32716b6915c7}}
| i | e6a5be6f98b29967717f2fe246a6916b |
Before we go to the detailed studies of the two schemes, we briefly summarize the present experimental information on the strange vector mesons. As listed by the Particle Data Group (PDG) {{cite:77af9d533c19501042036f238b0359ceff88eb61}}, two excited {{formula:0b574f31-2336-4b32-b5d5-aac739eeb1dc}} states are observed in experiment, i.e. {{formula:a6de1e1a-0f3d-4b44-b765-3e5498a719f1}} and {{formula:0d6fcce1-1d0b-4225-b09d-19edb734077a}} . While {{formula:48c4f0d0-35f8-4e16-acdd-c3b7cc850d1a}} can be well accommodated by the first radial excitations of the vector meson nonet, the property of {{formula:eb0d8c00-c1da-4e60-8428-29a9b56c5e9c}} is far from well explored. Note that the second radial excitations of the isoscalar pseudoscalar mesons can be occupied by {{formula:0a01f769-8591-47b3-9b6b-3c06d4879977}} and {{formula:1a9dc519-708a-41e0-ba1f-f3b4b79c321c}} in the Regge trajectory {{cite:5630514a766e689f151947173485e522689ac1fe}}, the mass of {{formula:57fe90cf-41f7-4518-98fb-4aa727aa40b5}} as the second radial excitation in the conventional {{formula:a84359b8-f676-47ac-a823-0774fff86fa1}} vector nonet seems to be too small. We also note that the strange pseudoscalar partner in the second radial excitation nonet has not yet been established in experiment though {{formula:77f719b7-eda0-45d4-b3a5-07b26135baae}} could be a candidate {{cite:77af9d533c19501042036f238b0359ceff88eb61}}. In the following analysis we first treat {{formula:986f86c6-5990-4fe6-a031-9b507351e831}} as the strange partner of the {{formula:de96b630-bd76-4972-b55d-6fb7b5c135ba}} nonet and examine whether it fits the constraint. If not, we then investigate the mass correlation of {{formula:0a2417ed-2664-403c-b4bd-de8446db23b4}} with the mixing angle and other multiplets as required by the Gell-Mann-Okubo relation.
| r | 0fb1561410710df549f819dd57da29f0 |
As the mitigation of the iism is an important consideration in pta gravitational-wave analysis, dm time series of pta pulsars are studied extensively {{cite:0bbcaa06d0d03521b1c70690852889c1b6fc8452}}, {{cite:cb86fe6739348d4661a8c0a39cdaa33b50c99930}}, {{cite:2bb7aaa038e270d227ab3d25fa6143b508576cbb}}, although these use much lower cadence and are obtained at much higher frequencies (typically {{formula:e15172bc-0374-4637-825a-afc7b65ee79c}} 300-2000 MHz).
The published pta dm time series present datasets that mostly end around the start of our dataset, making a direct comparison of the dm time series difficult.
| r | 7e88b31478db4fcae55d1f5527f85f23 |
This is related to a more general neural network problem known as the stability-plasticity dilemma {{cite:d1b9838df814c88662ba35b42dfb508711d4ba21}}, where plasticity and stability refer to integrating new knowledge and retaining previous knowledge respectively. Such dilemma is at the heart of the research area of Continual Learning (CL) also known as lifelong learning {{cite:9549d5176abbca87e47d168d9e068cf361e6a545}}.
| i | f3b9d497bffae34aa9313dd7092f4a3a |
Bilevel optimization is attracting more and more attention in the machine learning community thanks to its wide range of applications. Typical examples are hyperparameters selection {{cite:cbc952e6c5644d359a3ff81348e5bdfa346eda37}}, {{cite:9ef52c9dc6721db6bfd55838c3fd7597067d7614}}, {{cite:b844d367bf15995f09671663e2fb86ec0a4ada9c}}, {{cite:1416af5337b1a5af71c22f230e7d8ee4c5146f83}}, data augmentation {{cite:867e4a8a03ddcf8c23c2782c201b453719a35240}}, implicit deep learning {{cite:6aafbdb60d9062bb3fe4a89770f1ec4f2c582556}} or neural architecture search {{cite:a0b2e14a078394f2075064bbe376c022a177985d}}.
Bilevel optimization aims at minimizing a function whose value depends on the result of another optimization problem, that is:
{{formula:1ef18149-e10c-4a8f-9b19-4ea5b007717b}}
| i | 65dcec417872e7656e63965c0a8b178f |
Stochastic control problems with terminal constraints have been extensively studied in the literature. Optimal control problems under stochastic target constraints have been studied in Bouchard, Elie and Imbert {{cite:97cdf6ec8d97741927a9cc4aa69d06342daf2c54}} using the geometric dynamic programming principle proposed in Soner and Touzi {{cite:f2f7a161fae32e063eaaec8418a586448963c816}} . In Föllmer and Leukert {{cite:a6bf03e4144b8a715c36b3f16b737371556c4b06}}, the authors introduce the notion of quantile hedging to relax almost-sure constraints into probability constraints. In Yong and Zhou {{cite:b7ce8646cb1984efc29b13a80b73dd8bbf2e82db}} Chapter 3, necessary optimality conditions are proved in the form of a system of forward/backward stochastic differential equations. More recently the problem with constraints on the law of the process has been studied in Pfeiffer {{cite:310f7e4473c423153985fd0069e841dbe6f2ae47}} and in Pfeiffer, Tan and Zhou {{cite:a13daffb26c075aab57450e59a621b7c0e0e7ae1}}. In these works, the authors prove that the problem can be reduced to a "standard" problem (without terminal constraint) by adding a term involving {{formula:062cb450-2c1d-4e9a-b725-2cee34c0c100}} —in the case where the constraint has the form {{formula:cccff611-fef7-437b-a9b3-d6c3e00985b5}} — to the final cost for some optimal Lagrange multiplier {{formula:148aef08-a9f8-403b-a6c6-db9dd57cf378}} . A dual problem over the Lagrange multipliers associated to the constraints is exhibited using abstract duality results. In Pfeiffer, Tan and Zhou {{cite:a13daffb26c075aab57450e59a621b7c0e0e7ae1}}, the authors provide necessary and sufficient optimality conditions for problems with multiple equality and inequality expectation constraints with much less restrictions on the data than we do and in a path dependent framework. However {{cite:a13daffb26c075aab57450e59a621b7c0e0e7ae1}} needs to assume some controllability condition (Assumption {{formula:bf977241-72b8-4c64-aaab-cac24b03ce72}} ) and works with a compact control set. In our framework, the corresponding controllability condition would be to assume a priori that there exist some control {{formula:5955e79e-cb8a-4cdc-b528-3a4c5d0b9cf2}} such that {{formula:fb515beb-c68d-4013-a3a1-eb761bc5b047}} . In our analysis, this condition is a consequence of the requirements we have on the data.
| i | b5fd64a06a9ee46bed67d21b3e4baf97 |
With the advent of the LRW dataset, it became possible to train deep neural networks and utilize them for real-world applications effectively.
At first, a lot of research was focused on 2D fully convolutional networks {{cite:e3b6c837d77af30a307629e1a57c7156ca1f94b5}}. Nevertheless, thanks to hardware evolution, it soon became possible to use 3D convolution over 2D convolutions {{cite:477ad27f91d0bbd040e94d0f50c7e60dcf6a6927}} or recurrent neural networks {{cite:a0016f70480da6fdc6a1036b229da38bd7ab53b7}} for more efficient temporal information usage.
This idea has grown into a type of architecture where information about the local lip movement is extracted using 3D+2D convolutions backbone, while final temporal information is summarized using recurrent layers at the end.
This approach has been extremely successful, and many state-of-the-art lipreading solutions used it as a meta-architecture even nowadays {{cite:f0225ea37b77c759a8c325b31f07f5307957d145}}.
| m | 34f3876de80b0945623dcd5baa5bddc1 |
the difference between a multiple of cumulative mean rewards of choosing the best subsets {{formula:8c800b2d-df12-4da3-af54-1ac013cdd172}} of size {{formula:31d92051-9a07-4b07-86bb-94ac7a8cfa50}} (subject to matroid constraints omitted here for simplicity) and the mean utilities of the subsets chosen by the decision-maker, for a constant {{formula:ebbab08e-2851-4fd3-aef1-5b8fc6e35394}} . We assume that {{formula:c0bd7671-b864-48fc-9b8d-207a3f70bdd5}} is an arbitrary sequence of contexts that may be chosen by Nature adaptively. Furthermore, as in {{cite:265ade6d28329fdeff9b1a57da788d1f25b6bc9a}}, we assume access to an online regression oracle, as specified in the next section.
| i | 03ccae9e7e80c7ba24e19fafe52913ec |
While these approaches are very interesting and gave promising results, so far they have only been used to produce critical points for a fixed scalar potential, which is resulting from a single specific gauging within the large infinite family of possible deformations.
This leaves open the possibility that other vacua with the same residual symmetries appear in different gaugings.
The approach we are going to use in this work uses instead the power of the embedding tensor formalism in a way that allows for the search of critical points independently from the choice of gauging.
This approach was pioneered in a very different context in {{cite:db9e97f2e6c51e2c9e75e5984d438978f302d48c}} and used in the context of maximal 4-dimensional supergravity in {{cite:35c8087cdf5b2dbe3c579d40693d360b20920d31}}, {{cite:0af9d065f7bae8bcc4634f9f25628e9221ab519f}}, {{cite:693f756a8b6a5e7c779ee80cac501df0632efb10}}, {{cite:f7452e35a70d8e4e8a73859a05eb55932b028f94}}, {{cite:2c8a7d3d6486d1cdacaf1e5e6a3e19595f7d03bf}}, {{cite:d532a36df8d281111d54f060c46b831c74ac2fdf}}, {{cite:0e909615d9b058aa77f39dd8e1de9cd5949e0f01}},
as well as in half-maximal supergravity in four and three dimensions {{cite:08820f19f8787b8c1facabf8b74c0feee356d1dd}}, {{cite:a82570ce20ae7017d54215fa416a859b409b0289}}.
In addition to the power of investigating in a single sweep all deformations of maximal supergravity, this approach has so far produced analytic results for the critical points and their full spectrum, also providing information on the gauging, the residual gauge symmetry and supersymmetry of the vacua.
Moreover, for Minkowski vacua this led to understanding the moduli space of these theories {{cite:d532a36df8d281111d54f060c46b831c74ac2fdf}} as well as their uplift to string theory {{cite:bc027c9102143768ae54598e21b433024cb8f57b}}.
Finally, since the vacua are obtained without specifying first the gauging, this means that we can exhaustively classify vacua with a given residual symmetry for all possible consistent gaugings.
| i | 5431537754e8218c05dfe00728c3f755 |
The cross-Kerr nonlinearity is used for constructing the two-photon QND, which will mainly influence the feasibility of our schemes {{cite:781fc1757b1e57d0bd9aa5cf42ff9ffbff586de0}}. Although it has been widely studied in the past years, the effective nonlinearity strength is still challenging in the single-photon regime with the current technology {{cite:602ef32f06ee874bdfc5baad737b43b25ad8ce9a}}, {{cite:a2d45c3592060ba4d81803a24b37916b3e9d15ad}}, {{cite:c45d46ed6993d251d7b72bd65b1dd29c1408dbb8}}. Fortunately, we just need the weak cross-Kerr nonlinearity that could produce the small phase shift in the coherent state. When a sufficiently large amplitude of the coherent state satisfies {{formula:982a088f-5cdf-4fc0-a659-a5dc997f3765}} ({{formula:e2000b07-4aa1-4cb1-80a3-61d08be16c82}} is the cross-phase shift), it is possible to distinguish the small phase of coherent state from the zero phase shift. Actually, the theoretical and experimental researches have shown that it is promising for us to construct the QND required in our method, resorting to the weak cross-Kerr nonlinearity {{cite:b7de13cd4df99decaca080215e613bb8e8712bd0}}, {{cite:5b677f5a7019ee37ae4be14cda24a64623957cad}}, {{cite:06d7c6fdd60bd4bebd840588299f6769e739b4a4}}, {{cite:f5eb48f722702e67b001856078593af67c529b80}}, {{cite:d744fb0609c505498d24a6c2aee21af373969e3b}}, {{cite:4ad8d9d7e76d28ffa2377369e8da430451775557}}, {{cite:e7d8e54875615d647a9903314243f34b246049ff}}, {{cite:6d1cb5d9b03ff859f98131091789226ad8708394}}, {{cite:15bab3d6ea6492949aa88ebf2ee9ed416ebd1e81}}. In addition, some other kinds of interaction can also provide accessible ways to realize the photon number QND, such as cavity-assisted interactions {{cite:07c3182e8156330e8a7c76f554566b8bdbb48d95}} and quantum dot spin in optical microcavity {{cite:31d817162cf1fe797a771bad3253a9c12aa49f9d}}.
| d | c43d2b85281d10728436a3326fd6c5e6 |
In most applications, a network's community structure is unknown ahead of time and cannot be inferred based on visual inspection alone. Instead, we use algorithms and heuristics to partition nodes and edges into sub-networks, called communities, modules, or clusters. This enterprise is referred to as “community detection” and involves many steps {{cite:1d603f89970edf2bd64afb2295f3ae13dc31dd61}}, {{cite:8f2cc485c786655d7b021f58d2d57d007c802cf7}}. Because the space of community detection methods is paralyzingly massive, it is impossible to discuss every method and contingency as part of this article. In the following sections we break down some of the common steps for performing community detection and decisions that a user has to make, focused through the lens of the modularity heuristic.
| m | 0c53a942854c78370a887b082f239147 |
Finally, this paper has considered problems in which agents seek to ascertain the answer to a question of external empirical fact, such as whether it will rain tomorrow, or which of two teams will a sporting event. It should be noted that some attempts to leverage the predictive power of social information instead focus on questions where the answer is endogenous to the community from which that social information is drawn. An example is the use of social media to predict which movies will attract large box office returns {{cite:e7022ea9ec485617b44dc84e677850aa6ccc7767}}, since presumably the commenters on social media represent a sample of potential movie-goers. In these cases there is likely to be more value in simply measuring the aggregate opinion of individuals, since the expression of interest in a movie is itself a predictor of attendance, regardless of whether that interest is itself socially driven.
| d | 691a634faad4a4ff4534ddfa3930267f |
Policy gradient methods present an opportunity to avoid the pathologies discussed previously in temporal difference targets while still preserving other properties of the RL problem. While these mehtods tend to exhibit other pathologies, in particular suffering from high variance, there is no reason to expect a priori that this variance will discourage interference in the same way as in TD updates. We investigate H2 using two different algorithms on the ProcGen suite: PPO {{cite:ea734947948f658f579915a170d1dc4ccec4289c}}, which uses a shared representation network for both the actor and critic, and DAAC {{cite:9aa6a262f6aa0ffffec5c6569e474a6fa327ec28}}, where there are no shared parameters between the actor and the critic. This setup allows us to study both the effect of the TD loss on a network's update dimension, and long-term effect of TD gradients on the representation. We run our evaluations in the ProcGen environment {{cite:d8ec49a936938e7aad9f549ac7765e6348febf36}}, which consists of 16 games with procedurally generated levels. While the underlying mechanics of each game remain constant across the different levels, the layout of the environment may vary. The agent is given access to a limited subset of the levels during training, in this case 10, and then evaluated on the full distribution. We will discuss generalization to new levels in the following section; our initial analysis will focus on generalization between observations in the training environments.
| m | bd0df308bf51e273f4204d5f22b5714b |
Before discussing the multiphoton effect on the DCS, we show in figure REF -A the laser-free DCS as a function of the electron's scattering angle {{formula:4db5542c-5d85-4751-9241-e68b64e45d4c}} . Four curves are presented in figure REF -A: The first curve in blue represents the (e-p) scattering process in which the proton is regarded as a spinless particle, whereas the red curve represents the case where the proton is a point particle (Mott scattering). The green and orange curves show the Rosenbluth formula {{cite:288de74d54c09ad2c08ed5d352b6224d7340c6b7}}, which includes the electric GE and magnetic GM form factors of the proton.
The {{formula:ec69acb8-31d5-4276-9589-dcc003f67eba}} and {{formula:5ddf300e-89af-469a-a3da-c449269ec63e}} form factors are specified in the orange curve using the usual dipole parametrization, while they are defined as {{formula:bc6fa281-f693-4d9e-9cf1-b2c323497ed0}} and {{formula:87379ead-889b-4fbf-a071-b2ef710a9e1e}} in the green curve.
In this figure, we can observe that the scattering process represented with a linear electric form factor in {{formula:a78d71ac-5cda-4e09-91b9-3fce75db4b88}} (see equation (REF )) agrees with the other models for small values of {{formula:04d07207-2b5f-488f-adbf-be047b8b6f84}} , and it starts to differ from the Rosenbluth model (orange curve) at big scattering angles.
We now go to figure REF -B in which we investigated the (e-p) scattering process by using a pulsed laser field He{{formula:6481b32f-6e75-42c7-94cb-a4ad81d8a414}} Ne for various summations over {{formula:bf779aa1-c90e-4fd2-9be0-4dc0427f7e39}} with {{formula:0a7a228e-b47d-4789-a781-7f564efca3a1}} and {{formula:29870bcb-8817-4a8c-a3e7-61288a8a715c}} .
For a number of photons exchanged {{formula:51195c47-02ce-404d-bbd3-af5e9538c775}} , the DCS in the presence of the laser field is still not confused with the laser-free DCS.
This result is explained by the fact that the value of the cut-off that corresponds to this laser field strength has not yet been attained.
However, when the number of transferred photons reaches {{formula:3324e1bd-045a-413e-8a9a-4d8672224e97}} , the multiphoton cut-off of this process is exceeded and the laser-assisted DCS will be equal to the DCS in the absence of the laser field.
For the (e-n) scattering, we assume that the DCS of the laser field multiphoton interaction with the scattering system is similar to that of the (e-p) scattering, with a difference in the order of magnitude.
This is due to the fact that we have considered the nucleon as a free particle, and only the electron which is embedded in the laser field.
This characteristic of multiphoton interaction with the laser of circular polarization is frequently found not only in the scattering processes {{cite:046dfec363abd5e4683bbf90a64902cf30d517a0}}, {{cite:131bd3a2edb3672695739d6ae5894ef8388c6de6}}, {{cite:a2877e3278e6c891c5c34992e184a223827e6190}}, {{cite:6ad8772a9534c4cf23f824e58d418e8e755cdd46}}, {{cite:cb5e4974d5c74fe99375bdc6ae79c29937263e40}}, {{cite:4e9cb25c3dcf476b46511b4f097c7a9a256bd21e}}, {{cite:739230b31ea36cd8cf7303f5990800c39b3d6c7d}}, {{cite:b8e983f6fb7497306a1bd8d2b2eae69cb50a5af3}}, {{cite:db38e536d51871ad601a22e8d3f19555686dcecc}} but also in laser-assisted decays {{cite:f346d14c29c0d1691919ea93e6a5d988f98c8ec2}}, {{cite:5dbb7ba4368f01999da4b512248e2e46d50e3380}}, {{cite:94797ba92a346649a94071fd33d6ed49ea1db4af}}, {{cite:bbede91fbea6c00d308476006ba82a785d5c3109}}, {{cite:8ff32fbd9b3016d79d799486999a2823bdc95fe6}}, {{cite:ca9e061a6fe9db630ed59ecef875ae88b89e47fc}}, {{cite:a59471a471bc9fd19f8685c8500e2b78ca405c77}}.
To compare the DCS of the (e-p) scattering with that of (e-n) scattering, we exhibit several variations of the DCS in the following figure by varying the number of exchanged photons and the strength of the laser field.
{{figure:ac0ac91c-fca5-4b14-9d2b-321950bb233f}} | r | 61fb645d347a53bd55fb05cfa563d2d2 |
The spatial structure of a population affects its evolution in many ways, for example by promoting cooperative behaviors {{cite:8623a4fc8f11554628b40ed8896e04da9adfafaa}}, {{cite:83b7c7d659b531981973a6dfa983cf4fcab06a04}}, {{cite:2a5b2175aaa4d58d717d3b6ac2d3c69264d006f7}}, {{cite:9487f20bdbd817127a9c33ff5efc93e8a90d1d38}}, {{cite:39e52f0e99c363187b88b306821780047638a047}}, {{cite:c1a5958f42bf60cc88e29f69ec99037e68a279d7}}, {{cite:b90b396adb7020cfa761807d183492040a6da205}}, {{cite:e1d8fddb51e7046e50730b5cca722b4df9e1ea37}}, genetic variation {{cite:e2d581d20807f6b629b80df3551325da259a7059}}, {{cite:d6c1add1c2e37a106618dc8b9205300f82c92466}}, {{cite:3b07576342f2de42e6dad11d2be58e86d2452575}}, {{cite:ea302c4e710c4b4b9a8f53d9a8b600770256253b}}, and speciation {{cite:07693f870fa56d48104b9adc893ce8d798e09b25}}, {{cite:e696bb53675a5a211efa7ab4c73f9486d90f96b7}}, {{cite:e49ef5633c84903fc0855421b90f052584783f71}}. Asymmetric spatial structure in particular is known to have important consequences for adaptation {{cite:e4a7eaddd57ea641d164f76f76e6a49f14549670}}, {{cite:3bbbaa863cab640ad62eecbd5e42af8d9626a04f}}, {{cite:a9b090190e2b4c0520488ba31654e2d01e3bb8ba}}, {{cite:bfe043ead7043969c88701c4b30f071d7674374e}}, {{cite:9bba9efff68c889c9b06180398ded1a5d9436682}}, {{cite:8506b71c398a67f340e9b82ee5a809b10b7bc20e}}, {{cite:471bec226f7b87975a633addc63dee3405559f14}}, {{cite:9062265fb7daaad7be8322b40d32e3f048bc59a7}} and for genetic diversity {{cite:671bf199e7a4e37c50558cb407a4607b29baa545}}, {{cite:feb58d4c349abc9c30699e8232ab50917a09c1a2}}, {{cite:29312ac28045c0fb91ccbed0b1394aea0f37504b}}. Our work shows that asymmetric spatial structure also affects the rate of neutral substitution.
| d | 4119de899ba781e90dc20bc40ed99a26 |
The origin of the latter can be ascribed to accidental BIC formed due to a simultaneous destructive interference of leakage channels{{cite:b1d340652d18d138529400844674ec189dc8e858}}, as demonstrated at THz frequencies{{cite:0e9d3fe894e15d8c6b6d24eb35a4c956e932d50e}}.
| r | 71c2af5c7323fcdc26466c02f0af7dd7 |
We can also define the evolution of the discrete system on the Hamiltonian side, {{formula:12f02d2d-7aee-40a2-991d-dede203e84b2}} , by any of the formulas (see {{cite:b3b761af59e72b5cd71cbfbd6c27263af2969437}}):
{{formula:99055e26-fb8c-4490-8dd4-52223d46211b}}
| m | 6b29f2709ca6a885ba5679a627cd5452 |
In conclusion, we have demonstrated the applicability of MPLC as a building block for quantum information processing of entangled photons. We have certified high-dimensional entanglement with high fidelity by programming the MPLC to switch between different mutually unbiased bases{{cite:60f8eaf57d59899cf1dcd48c7b4cc4539764b732}}, {{cite:775212b54b126cac91655827ebaadf391d434c4f}}. By scanning the phases of the spots at the input plane and measuring in the DFT basis, we have observed high-visibility quantum interference, which is fundamental to many quantum information processing tasks. Furthermore, to demonstrate the universality of MPLC, we have experimentally programmed 400 Haar random unitary transformations on four modes using MPLC with five planes, observing high statistical fidelity{{cite:83062ebb80206d0e343818b7796c135c4b776d03}}. As indicated by our simulation, near-perfect Haar random unitaries on four modes can be implemented by as few as ten planes in the MPLC. Besides demonstrating the reconfigurability and universality of the MPLC, Haar random unitaries are important for boson sampling, QKD and other quantum information processing tasks{{cite:1b21f12ff0b9ec128743a53200c3c1638106027f}}, {{cite:83062ebb80206d0e343818b7796c135c4b776d03}}, {{cite:41a151f9143796f3a3225fb6810bb4886bf6c047}}, {{cite:b34d27c67672d06f4e29d97d29793556b31b67be}}. Finally, we have considered the use of MPLC as a mode converter for entanglement distribution, demonstrating correlations between the position of one photon and the measured LP mode of the other. Being a reconfigurable, scalable and universal approach for quantum information processing that is also compatible with free-space and fiber links, MPLC can serve as a leading technology for processing high-dimensional entangled photons for quantum communication and computation. We also anticipate that the wide bandwidth of MPLC{{cite:8bc78390ef7399b6289d4353fd8925155a3f28b2}} together with the ability to control all degrees of freedom of light{{cite:f5d943a89955692afcfeeb71dc44b7f4814da83f}}, will open new possibilities in controlling hyper entangled states{{cite:342ba71bd628d048756ecc92f673a934ce1b1b0d}}.
| d | 4489cdfe3d381d57f89b7d07375d6c4b |
The macroscopic state of an extended system is therefore best understood as a patchwork of domains that meet at various types of defects {{cite:050ef0ea92d3c73b66d57c4ad87bf47410afd9ca}}: disclinations, dislocations, monopoles, walls, etc. It is thus of
interest to
develop tools that allow us to understand, predict, control and manipulate energy driven pattern formation.
For energy driven systems, these defects are singular solutions of the order parameter equations that arise from averaging the free energy over all the microstructures consistent with the macroscopic order. Since these
equations depend (largely) only on the relevant broken symmetries, they are universal, i.e., the same equations arise in a variety of physical contexts {{cite:a0feb2d95d011512f0714ba231426067018a10f5}}. The topologies of allowed defects are also universal {{cite:feaafd9858489853190be64292723d7a0bcd07a4}}, {{cite:6fa9a505d34a04b10aa75b57af6dc7ef30816407}}, and they are captured by discontinuous and/or singular `solutions' of the averaged pdes that are initially derived to describe the domains {{cite:e00df09d89a0e0392794dfa8d7401dd665568130}}. This universality is one motivation for the work presented in this paper, namely, the idea that there is a common modeling and computational methodology which can be applied to study defects in systems with vastly different physics at widely separated scales.
| i | 77ba2f9d4aefc0fcd0083ec224225e03 |
We refer to fig:TRvisualization for a visualization of Algorithm REF .
This basic description does not cover all TR variants and does not detail assumptions that have to be made for the individual components of the algorithm.
A significant part of the algorithm is to choose the model in Step 1.
In general, only a few assumptions need to be fulfilled for the model function {{formula:523685e1-6b55-49c5-a1d2-efa166cf64d6}} , concerning regularity and approximation properties of {{formula:cd5bc27c-239e-4786-822a-e73569c96d02}} .
We refer to {{cite:83f72078e53f52650e77e4e76c6c680147d6f197}} for an elaborated discussion on suitable assumptions.
As a commonly used example, we introduce the following example of a model function:
{{formula:3b55bb98-8f7c-4c35-9dfb-15993b4b4ddf}}
| m | 1a27f26efb99a5ab4a5a36ed0d94468c |
Keeping the search global comes at computational price. Whereas typical shape recovery with RANSAC-based algorithms uses seconds-to-minutes on a standard computer, the genetic approach described here takes minutes-to-hours - up to one day for large, high-density point clouds.
This is on par with the computational cost required to obtain the point clouds from pictures using photogrammetry software.
In addition, improvements targeting optimization speed-ups are possible. Lowering the point density with an intelligent thinning operation can lead to important performance gains. The iterative initiation of optimization runs on increasingly denser point clouds could further relay optimal parameters already identified in prior runs to improve performance. Such a sequential fitting approach could be improved by adding a transferability objective relying on a surrogate model {{cite:dd37a8a32b1870d6ed8a7192279552dd563e3cd2}}, {{cite:1cc9661d12bce947122c2f103d4f7fead8fa67b3}}.
Alternatively, coupling the efficient but locally-based RANSAC search with our time-consuming but global genetic search is a promising idea. A simple hybrid scheme could involve the segmentation of the point cloud using the best-fitting cylinders of the genetic search, and the local search of cylinders with the RANSAC approach.
| d | be942ff9e23584abc0a123cc104a0500 |
As described in Section REF , our datasets are severely imbalanced. In such scenarios, evaluations of performance using only accuracy measurements and ROC curves may be misleading, since they are insensitive to changes in the rate of class distribution. We therefore perform our analysis in terms of precision-recall curves (PR) and the corresponding {{formula:c50d3072-a757-4b68-843d-aea89e7773c3}} score {{cite:0847238946a5aaec3d1f1e5c1009543bb70c1db1}}. Precision is normalized by the number of positives rather than the number of true negatives, so that false positive detections have the same relative weight as true positives. While the maximum {{formula:2b1e965c-fe88-4832-b5f8-4313d84613ce}} score indicates the optimal performance of a classifier, the area under the respective PR curve (AUC-PR) corresponds to its expected performance across a range of decision thresholds, such that a model with higher AUC-PR is more likely to generalize better.
| r | 7a7b8ce720143d5fa7eeeed6915003c4 |
Limitations for real-world applications.
Currently, our rendering speed (about 30s for each view) is bounded by the intensive network queries and nearest neighboring searching operations.
When deploying to real-world applications, we might consider accelerating the inference speed to fulfill the real-time rendering demand
with recently proposed coefficient caching techniques {{cite:ed5dfaf50f08731e5490b37e34ec27e4593f5eb8}}, {{cite:8251ae00272a395552736d1ac8940d215b88431d}}
, multiresolution hash encoding {{cite:107f31e1f5cfddb32342997d3882a83c6e1793a0}} or MVS priors {{cite:caa4895ff5b30533e90f66546ca66462e29efa88}}.
Besides, we rely on 3D modeling software to select vertices for the region of interest, which can be replaced with some semantic annotation approaches {{cite:5ac34ff917733b6984b448c382a1300d518d5a85}} to facilitate broaden users.
| d | e87e5e6eaea434c16099801c58b6c3fe |
Although many efforts have been made to perform robust tests of
CDDR, the lack of adequate observational samples and
model-independent methods should be taken into account. Specially,
it is difficult to obtain samples that satisfy both the luminosity
distance and the angular diameter distance in roughly the same
redshift range. This redshift-matching problem problem was
recognized a long time ago {{cite:bb464c45bdb3747bfb0bfc0f02387c35a192e3ad}}, with a heuristic
suggestion that the choice of redshfit difference {{formula:4acbf160-1de6-461e-a4a5-90ca65f397c5}} could
play an important role in model-independent tests of such relation.
More recently, many authors presented a new way to constrain the
CDDR with different machine learning algorithms {{cite:6231e7de64d4fabc7ad4bf5174c2e5c0fe733340}}, {{cite:7541ac5e33fd3234cfe92c0781f19bd92e80a1dd}}, {{cite:a7ccd4cb00e8c83258ae9ad259696f6a5526cf76}},
with the luminosity distance and angular diameter distance
reconstructed from complementary external probes (Type Ia supernovae
and gravitational wave (GW) standard sirens) {{cite:91fbcf6242c741841d24c9db758fe5535cf0e836}}. Their
results demonstrated the effectiveness of machine learning
approaches in the high-precision test of the electromagnetic and
gravitational distance duality relations. More importantly,
considering the fact that the purpose of modern cosmology is to
establish consistent and robust theories, all alternative methods of
testing the fundamental principles of cosmology are necessary. In
this paper, we will use two non-parameterized methods, Gaussian
Process (GP) and Artificial Neural Network (ANN) algorithm, to
reconstruct the newest observations of HII galaxy Hubble diagram and
ultra-compact structure of radio quasars, respectively. These two
approaches are data-driven and have no assumptions about the data,
suggesting that they are completely model-independent. The
luminosity distance is inferred from reconstructed HII galaxy Hubble
diagram and the angular diameter distance is obtained from the
angular-size relation of compact radio quasar. The advantage of
using these two data is that the redshift ranges of the two samples
are roughly consistent, and can reach a relatively high redshift
range {{formula:3bc7bda3-7718-430c-b871-677ddef8004e}} . Since no models were assumed in our analysis,
our method produced a clear measurement on the CDDR.
| i | cf5df24c760a2ab21f431e22f60f1f0a |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.