text stringlengths 54 548k | label stringclasses 4
values | id_ stringlengths 32 32 |
|---|---|---|
Retrospective simulation {{cite:dc7ce3ec80fc5a2f19f14284b7f8664f770bcbab}}
is an attempt to take advantage of the redundancy inherent in modern simulation algorithms
(particularly MCMC, rejection sampling) by subverting the traditional order of algorithm steps.
It is connected to demarginalisation and pseudo-marginal {{cite:11987fedefdfb063755f8cfbe0a7ef7279b0d315}} techniques
in that it replaces a probability of acceptance with an unbiased estimation of the said probability,
hence creating an auxiliary variable in the process. In the case of the Metropolis-Hastings algorithm,
this means substituting the ratio
{{formula:40f94a73-0f0d-4616-8f52-11b836712960}}
| m | a0301468d28ebb672d39254bdfce79a9 |
Despite the loss of asymptotic flatness for a magnetized spacetime containing a black hole, this solution has been considered to have some astrophysical uses especially in modeling the spacetime near rotating supermassive black hole surrounded by hot moving plasma {{cite:7047d371c50cc1dc347ee88d3ee04f19b8563df8}}. Indeed, a full comprehension for the interaction of a black hole and its surrounding magnetic field due to the accretion disc requires a sophisticated general relativistic treatment, at least by employing a costly numerical approach. If the full general relativistic or the comprehensive numerical treatment is not necessary, in the sense that we just seek for some approximate qualitative explanations, the picture of magnetized black hole by Wald or even the non-perturbative model by Ernst can be the alternatives. For example, these models can explain the charge induction by a black hole immersed in the magnetic field, or the Meissner-like effect that may occur for this type of black hole {{cite:a832e986093a308c0f34f4fdaadc10923f672850}}. In particular, the superradiant instability of a magnetized rotating black hole is studied in {{cite:c055e6c08ac98c013ad07721f6e6976f5d65f8cd}}. Another aspects had been investigated as well, such as the Kerr/CFT correspondence for some magnetized black holes in {{cite:ffafcde28934e812a7acd8fabef7de92630a015b}}, {{cite:6f3ce22b9f362442ee0575b9fab19f7c0ad0bc28}}, {{cite:bb539db1c2bb46814d27fcf22765540278af0496}}, and some conserved quantities calculations are proposed in {{cite:ddaed5e5780ed7e7df173a35bb7f3804c240bb9a}}.
| i | eae8b866f4f519c1377d8872a0ede18e |
In the SME EFT formalism {{cite:0cb2b40ccca94df10434fc60552e8cf3fcd05ee1}}, {{cite:c8df0547d54a6689a46dc915a8994433454fd273}}, {{cite:ae9f4f1891eee3d1f3814317dce6909887d62f86}}, if the apparent cutoff above {{formula:88dc9454-8df1-492d-8347-9b1e22ae2a4b}} PeV in the neutrino spectrum shown in Figure REF is caused by LIV, this would result from an EFT with either a dominant {{formula:c99a4774-39dc-4058-aa2e-b3329cd15f47}} term with {{formula:33f3a364-e655-4939-ab30-c6a72c5b44c2}} , or by a dominant {{formula:c77e560d-727b-4285-9d0f-d9f60c3ae0a5}} term with {{formula:79d820d1-544f-416a-8a00-8b8e12d9fbac}} GeV{{formula:c8395204-0a44-44d1-bddc-b811d4444ead}} {{cite:3bd03180cd4fe65f96be8d6d347f94dc679f888c}}. Such a cutoff would not occur if the dominant LIV term is a {{formula:742c2973-d895-47aa-85a1-15195b000d9b}} -violating the {{formula:997b8bdd-bcd5-4753-a9df-c591f225d888}} operator.
| r | bd148abc8d67b6bb58a92d006c0e8208 |
On the other hand, for a few known classes of games, {{formula:1712e650-7de7-4d94-a06d-239527c87280}} is computable, and sometimes computing {{formula:69494ed8-5796-485f-96e6-231d5a775457}} is even easier than computing {{formula:e3670a6d-372c-409f-a1d6-72fd57462c15}} . Cleve et al. in {{cite:d63ecefc2e452f494cd49c33f4fa5f63d2136c71}} gave a polynomial-time algorithm to exactly compute {{formula:c558a706-9bb9-497d-9d23-ec5cd24ac448}} for XOR games {{formula:8cf58bc5-2b88-44cc-9fc7-4af513b827ef}} building on the work of Tsirelson {{cite:0828ca4defeffad7675c6cb5747136353e9c8665}}. Kempe, Regev and Toner later presented a polynomial-time algorithm for {{formula:26e4435b-9692-4f48-92ad-ba2bf6888206}} for unique games with a factor 6 approximation to {{formula:3dd093ac-4949-4521-b1f4-ccacf57eec06}} {{cite:82d6ac83830d9b6145a3d364d6685ee9275ed903}}. However, it is {{formula:89e32b9b-172c-4597-b39a-dbda63916ff3}} -hard to approximate {{formula:121d7882-d8a0-4487-9c12-3603d4cb16eb}} for XOR games within a factor of {{formula:2abe0ca3-be84-4689-8540-b8e278f606c2}} and also {{formula:bbfdb30a-e16c-4e85-8f35-b1565c4280f8}} -hard to approximate {{formula:567b22f5-ba29-4890-9840-cc1fd04b83bf}} for {{formula:ab54b6b5-2dce-488a-9f6e-eb70d1356cff}} -XOR games within a factor of {{formula:fcc50b09-c86f-412c-9817-d55a129dfc51}} for {{formula:f6f46fd7-ab22-4387-99ad-16c9f6303a85}} {{cite:54abca549d2acfed1913e40eff667db2f05071ec}}. Classical unique games are conjectured to be NP-hard {{cite:051f573d5d09b77040ccc807b1eeb49a137768ed}} as well.
Both of the algorithms in {{cite:d63ecefc2e452f494cd49c33f4fa5f63d2136c71}}, {{cite:82d6ac83830d9b6145a3d364d6685ee9275ed903}} are based on convex optimization. In particular, a hierarchy of semidefinite programs was proposed in {{cite:18e197b0a302483917d3b78be6d1c4560b3604a8}}. The optimal values converge to {{formula:6088d5fe-804a-40fb-8758-b58ff8872082}} , which are the values of games when the players employ commutative strategies, and thus {{formula:68ef4f92-e338-4cc5-b34d-5149a6d85e9c}} is an upper bound on {{formula:acb9daed-9847-45ae-968f-0b770bdc2466}} . However, the speed of convergence is unknown. There are some other classes of nonlocal games whose entangled values are known to be computable {{cite:e31a8b4ba191949b47ab4613bb62dce73067e9b4}}. Readers may refer to the survey {{cite:43cdce6859c28584e41ae564761a899888543b9b}} for more details.
| i | d7fb702d3e62db0d0f91421eddf8c16b |
Along with the proof, we have extended standard initialization schemes to non-zero biases and formally shown that our proposal defines well trainable neural networks, while they support the existence of strong lottery tickets.
These initialization schemes include the 'looks-linear' approach {{cite:33998fd05626cb23267eb2af16430b88916b0a21}}, {{cite:749701bc9ee613bee89a14a33296a9c68567ab06}} that ensures initial dynamical isometry of ReLU networks, which often leads to favorable training properties.
| d | 3fe3eda0acea5d696c5d6b1110123d49 |
The above formulation has been shown to be effective in traditional offline hashing methods {{cite:acb4a83705605ca81a4ccce4a42b66789425ed75}}, {{cite:337d8f213576d79addb1e3a11f528c81ad079469}}, {{cite:e01d0bee83539aaf16ae9e1115e81d4cb51ec035}}, {{cite:c7639ff15a45a451c8ab0637744cb573611a7ee6}}.
However, directly applying this equation to online learning is infeasible due to the “data imbalance" problem, as identified in {{cite:01c2ba739f78a8619c152689ddb02ab9767e9111}}.
Specifically, at the {{formula:699520d5-d29f-478e-9051-080b40d0d90b}} -th stage, Eq. (REF ) can be re-written as:
{{formula:68eb709d-b46d-4ac1-a35c-941690366d34}}
| m | 1f7dad5c527c0688926f025be2f75e2f |
The memristive reservoir of the network type is an attractive option for hardware implementation, because the number of possible network structures, capable of producing different dynamic responses to input signals, can be drastically increased by scaling up the network size. This high scalability is the major advantage of the network-type reservoir over the array-type and single-node-type reservoirs {{cite:66e49f45fd74a6f27d72d01121f560078881c459}}. Since only a part of system components are controllable in real memristive devices and materials, it is a significant issue to find better system settings under practical constraints {{cite:f9e34a0defe772b1a57482a7c77c5e1f5dc1abe0}}. From a scientific perspective, our final goal is to comprehensively understand the relationship between dynamical properties and information processing capacity of memristor networks. The dynamical properties can be investigated through spectral analysis {{cite:93718d94f436fe29bb7814d14c99d7fbc582ecf5}} and nonlinear time series analysis {{cite:12c162797347d9b6d4e7f948656ec5ce853c1984}}. The features related to information processing can be evaluated by computing relevant measures such as memory capacity {{cite:d957c87ff8bb7c12ac00c61c711e4a7d5039d6c0}}, kernel quality {{cite:be72752ecbd2cc0fb405071b8a43892377dad6eb}}, and other capacity scores {{cite:1a3987d3630aa148b0063a819e0f0e81cff7c88e}}. Our simulation platform can contribute to both these purposes. A target for the future is to integrate numerical and experimental approaches for establishing a design principle of memristive reservoirs, thereby accelerating the development of energy-efficient RC-based ML hardware.
| d | b700f7cbea901f87300561a7c1b286d2 |
Hyper-parameter optimization (HO) is the process of finding a best set of hyper-parameter values that cannot be learned using the training data alone {{cite:62cd3d349117805c40614360e33824b64466088a}}. An HO problem can be formulated as a bilevel optimization problem; the outer objective {{formula:e4cd39a7-baae-48b1-9015-426b77e36666}} aims to minimize the validation loss with respect to the hyper-parameters {{formula:67717f68-44ed-4a9c-8c39-0f1270776634}} , and the inner objective {{formula:d59a2e50-17ff-4a70-aa6e-4483671d35f3}} gives a learning algorithm by minimizing the training loss with respect to the model parameters {{formula:f763b242-8724-44dc-8999-5f19f3060830}} .
| r | cf68395f2cc4706fbcd62635a79e9f5d |
In this paper, we separately described the travel mode, destination, and route choice behaviors of travelers using the free utility model. The free utility model has two basic assumptions: the traveler pursues maximum utility and needs to pay information-processing cost to master more knowledge about the utility of alternative {{cite:fbc64b28faa178a6c24c913587e229138eddca1f}}. Compared with the most used models in the four-step travel forecasting procedure, the free utility model can better describe traveler choice behavior. For example, the Logit model for predicting the probability of travel mode choice {{cite:c7394bd8818000662b1b70782347dd7b463dfdcb}}, the maximum entropy model {{cite:c052ed67243bf12627f3b74994b24f52ae6a52f9}}, and the free cost model {{cite:51a429210652e712a9612e73a68af8ca38a6f80e}} for predicting trip distribution all do not reflect the interaction between travelers. While the traditional SUE {{cite:b78fc3461b367da57888695ffef11cd2133ef4d0}} and UE {{cite:1275d842d83628f044a805e396e8bbb901149fba}} models are mathematically consistent with the free utility model in describing travel route choice, the integral term of the link cost function in their objective functions still lacks a reasonable explanation. However, the free utility model can simultaneously reflect the trade-off between the expected utility and the information-processing cost as well as the interaction between travelers. Not only does this model allow us to better understand the root causes behind the aforementioned classic travel demand forecasting models from the perspective of human choice behavior, but also allows us to study the travel decision-making process of travelers in a more interpretable theoretical framework.
| d | d80038e5f032eb75b0bf3e5049a064b5 |
While in this specific implementation we used random sampling and nonlinear least square optimization, our framework is flexible enough to support alternate sampling and optimization strategies with the underlying factor graph acting as the core data structure tying them together. As such it should be possible to use goal directed sampling strategies {{cite:410895b7cb97b906aa9a179136ba519d6e411255}} that can further mitigate local minima or bias sampling toward other desired outcomes. Recent methods in learning heuristics {{cite:418fa7eb8fa473cb6e17b99d4e9f19423676c020}} and deep learning based sampling {{cite:6542ae60cfd80fe4c20ef07c1d7acb626ca93d60}} can also be leveraged to the same effect. On a similar note, optimization can be further upgraded to use learned cost functions and weights {{cite:e71c6bc6db92f32561f1eb9b1478f0cc5879c0fa}}.
| d | fe3a16123564190824f87660533d7466 |
As a remark, in this paper we write down fields in MUR of the Lorentz group. This approach turns out to be more natural, compared to the approach that employs Lorentz indices, towards the goal of classifying all possible vertices {{cite:1b2bad1260a3402cd3f57909ccef65a7f9f18254}}, {{cite:c3977b024b273365d98883fc429e44a0d23172f0}} and writing down a consistent off-shell action with higher-spin interactions, see e.g. {{cite:e61d8ba413e0b49e2f8e5f4e86ae2c6d0479ef79}}, {{cite:eef1d861c91062cbb815d439d4dac4b630f4be5a}}. Moreover, this approach is closely related to what is known as the spinor-helicity formalism that is used extensively in computing scattering amplitudes due to its efficiency. Therefore, having a covariant action written in terms of spinorial indices to start with may be beneficial since one can employ amplitudes techniques to investigate the theory. This is indeed the case for one- and two-derivative chiral theories {{cite:eef1d861c91062cbb815d439d4dac4b630f4be5a}}. Thanks to writing down Lorentz covariant descriptions for these two examples, some of the puzzles in higher spin have been resolved.
| d | 1af1fbeb46c427214c55555e4483a2cc |
[leftmargin=*]
GRU4Rec {{cite:eb9221914d45d2476af355d4022861082a0c1bba}} is a GRU-based session-based recommendation model which also utilizes a session-parallel mini-batch training process and adopts ranking-based loss functions to model user sequences.
NARM {{cite:d5c0eab63de9958404e16ff30d6582a9e799af70}} is an RNN-based model which employs an attention mechanism to capture users' main purpose and combines it with the temporal information to generate recommendations.
STAMP {{cite:7959659c1530a2cc14ea6f6bf4b9f7d931717c87}} adopts attention layers to replace all RNN encoders and employs the self-attention mechanism {{cite:a13b1f1754c198355cc7ace0421c4ccbde1433b1}} to model long-term and short-term user interests.
SR-GNN {{cite:458b3a05522fbdeed0203427afab93b93e89e5c1}} proposes a gated graph neural network to refine item embeddings and also employs a soft-attention mechanism to compute the session embeddings.
OD-Rec {{cite:6bce3ea39b704f19557b77db128ea6b0b5d7fff0}} is our previous work that proposes a session-based recommendation model operating on resource-constrained devices where a tensor-train decomposition method is used to compress the deep model and a self-supervised knowledge distillation framework is proposed to retain model capacity.
| m | 1bea8f47e628777c4da7132da247d31a |
Intuitively, the idea of using the concept-feature dictionary to the ViT training could be similar to the memory bank mechanism in MoCo {{cite:3d2e6a8659d7a0fd33733567efa18e20ee3f4709}}, where the features are stored in a queue for replaying later. However, the difference is also clear: we have multiple queues that are indexed by concept codes while MoCo only has a single queue. Similar use of memory bank can also be found in {{cite:71b71d5245b116b85b64c1b052f7b365883e081e}}, {{cite:f3e13e8b8d708fb4b8559e13d7d77d8c3efaf714}} but they follow MoCo, and therefore it is used for providing negative samples when computing the self-supervised contrastive learning loss. Rather, our concept-feature dictionary is designed to make better use of the concept supervision via our concept-guided global and local losses to improve the performance on visual relational reasoning.
| d | de6ae8ed4fdd436f41cda024c5133c80 |
To illustrate the benefits of leveraging multi-step information, we compare {{formula:38bb7941-8b12-4584-b431-6b663ae18c85}} to the one-step estimator {{formula:14809fc5-e0a5-4999-8eb0-9447edad2b06}} and also with entropy production rate
estimates from the thermodynamic uncertainty relation (TUR) {{cite:2630c1ed678deec4117189601f3b3d6a45a22375}}, {{cite:137914edd40108a4a7b8e10e1be594b3834e45ee}}, {{cite:88ccd06b2595b0bb577935a407975b7cf49b3aef}} (SI {{cite:f8e89a983d4092716164d91555505a3372376e71}}). We find that in the strong-flux regime, when {{formula:5ab4a055-18f8-4b5c-8570-06418bb3c2a0}} and {{formula:f1f6470a-4c98-48ce-8e01-fa674ba08720}} are sufficiently different, all estimators reasonably bound the true entropy production rate {{formula:7be56b7d-eb4e-4f6f-830b-324114d9b531}} from Eq. (REF ) (Fig. REF C,D).
However, as {{formula:24712a7b-a52d-4582-b183-ace7a6ec6f9c}} and {{formula:768d2929-8a4a-45b2-9de2-b122b2447c35}} approach each other and the net flux becomes weaker, only {{formula:d4aabb3b-34f8-49d0-aa4b-465c033e2220}} gives an accurate bound (Fig. REF D). In particular, when {{formula:ef7c34c5-393a-4dd2-91f6-dd625d64c0b7}} , the forward and reverse observables are time symmetric, so the relative entropy between them is zero {{cite:d06270ac5c15670b1d1cc66117afad9376fcb0e7}}. Therefore, neither {{formula:966b19a8-7c5e-49fa-9757-ea6db35d9341}} , which here coincides with the estimator, {{formula:8b741fb9-08f8-4b9c-bd32-496365f43f29}} , in Ref. {{cite:d06270ac5c15670b1d1cc66117afad9376fcb0e7}}, nor TUR can yield a non-trivial (non-zero) bound, whereas {{formula:444fd317-1453-44e7-99a9-8e4d4c30b477}} can be computed analytically in this case (SI {{cite:f8e89a983d4092716164d91555505a3372376e71}}) and approximates the exact rate {{formula:64958489-e370-44e2-af99-059d2bc8d52c}} well for all values of {{formula:51f7e6bb-adac-4c82-be50-a8c96438d088}} (Fig. REF E). We next apply the two-step estimator {{formula:7342c74c-dd71-4150-ab3f-3b0c773efccc}} to data from recent experiments.
{{figure:784d5c2d-14bb-472f-b9ec-ce5748ffccc4}} | r | 2b930a698856f29e085f02c7bd54c61f |
In order to observe phase separation between the two components it is necessary to include fluid-fluid interactions {{cite:3e0f6760ab1c5e048cd78a0d16388cde2db504d8}}. We employ the model proposed by Shan and Chen (SC) {{cite:63ef8ceb060c03e00800cf3e9c13263858b36cb3}}, {{cite:d0f942f9cbe3ff8645bd72babe34498872e05001}}, where a force {{formula:b7337821-2079-4024-9f0d-aa30ec046474}} acts on component {{formula:dce509ff-3f3d-4e90-adaf-0accf609f720}} , entering implicitly in Eq. (REF ) through a shift in the definition of the momentum:
{{formula:aae8b8ff-4117-4ce1-af8d-561270350029}}
| m | 23865722445de0864091a5fefdb9b956 |
Table REF summarizes quantitative results on PASCAL-5{{formula:dfc61c12-c1ad-47da-a515-fb8adc9b32d5}} {{cite:cda1c6342ace693c6f0fd5c31d649f901d26ccac}}. The tests were conducted on two backbone networks, ResNet50 and ResNet101 {{cite:c9f019a819f14b07227539af0c3d5d1ade57b7df}}.
The proposed method outperforms the others on almost all the folds in terms of both mIoU and FB-IoU. It surpasses the others, including HSNet {{cite:7e6da91f386b86fdf162be6cbf2c39b5aae810e7}}, in mBA as well, since our ATD helps to improve up-sampling quality by providing higher-level spatial structure for reference. Consistent with this, VAT also attains state-of-the-art performance on COCO-20{{formula:2958530d-6d39-4aca-83c3-80acf95e0b6d}} {{cite:d83b2d9f336dbb20f2195028286234fb9374c243}}, as shown in Table REF . Interestingly, for the most recent dataset specifically created for few-shot segmentation, FSS-1000 {{cite:86c8cb69eb30012e9356a864d5937d1128b5b19e}}, VAT outperforms HSNet {{cite:7e6da91f386b86fdf162be6cbf2c39b5aae810e7}} and FSOT {{cite:0a4b34629a8bce84c790932cb9a285bca29d1ebc}} by a large margin, almost a 4.6{{formula:d11fd691-47f6-448a-b2bd-ba1913f69a70}} increase in mIoU compared to HSNet with ResNet50 as shown in Table REF . VAT sets a new state-of-the-art for all of these benchmarks. We note that our method outperforms HSNet {{cite:7e6da91f386b86fdf162be6cbf2c39b5aae810e7}} despite having more learnable parameters, which is known to have an inverse relation to generalization power {{cite:ef906988949124f84c90b397359353091c89f05d}}, a trend seen in Table REF . With the proposed method, i.e., 4D convolutional Swin Transformer, that is designed to address the issues like lack of inductive bias, VAT can have a larger number of learnable parameters than that of HSNet {{cite:7e6da91f386b86fdf162be6cbf2c39b5aae810e7}}, yet VAT has greater generalization power as well.
| r | 3c2764191dff0710e6443d50d7052f93 |
A critical task to evaluate the quality of embeddings is link prediction {{cite:3ea4dd4d0188672b9657ff62a2c1f258c4eb9b98}}, {{cite:545d220656ef0ed8d7d50a8591d558b5d2877d5d}}, {{cite:4b6cacb55b7209d9324fc588ee9c615f6b6ce1e0}}, {{cite:41b7fbbd39741d122b6f17c782180e2eb3677388}}, which tests the capability of embeddings to reconstruct the original network's neighbourhood structure, i.e., the first-order proximity.
On the other hand, it has been widely acknowledged that higher-order proximity has significant impact on the quality of embeddings {{cite:b346dbdbf5e19070abd6b38777c15bffb6d8e777}}, {{cite:061e05294811f820794d1c9da427a4a71fa684c5}}, {{cite:7f2d6cccdea460d48617ef0270ae08e98461851e}}, {{cite:8e00021c3d46643223b287e827054ae59ea0bcff}}.
While some existing methods {{cite:061e05294811f820794d1c9da427a4a71fa684c5}}, {{cite:4b6cacb55b7209d9324fc588ee9c615f6b6ce1e0}}, {{cite:2af2a57ef42255cef0aae4696e04ff932ef8f28e}} have claimed their strength in preserving high-order proximity, this capability is typically evaluated by indirect, alternative metrics, such as link prediction or node classification. Moreover, from our empirical observations, these tasks do not consistently reflect the preservation of high-order proximity.
Moreover, Zhang et. al {{cite:84b2ea7610754e149b4844793cdf34a160650a31}} reported values of the objective function of high-order proximities preservation to measure it. However, those measurements are indirect too, as they do not empirically and explicitly demonstrate their ability to well embed proximity of higher orders. Therefore, a natural question arises, that
| i | c00046c67c78fce43f2df44d8037aef4 |
We expect this nontrivial coupling of soft and hard sectors to
generalize
to nonabelian gauge theories and to general relativity, and to the
quantum
as well as classical regimesAs discussed in Sec. REF , the statement
that the quantum theory does not admit
a decomposition into two decoupled sectors is not in conflict with the
factorization theorems that constrain the soft behavior of {{formula:f62a5b51-3a7e-451f-860e-c88bb6fae416}} matrix
elements {{cite:f16a85394829bee73f00c650b939c1c4f42bcbba}}, {{cite:f0fd343d7ea74810455959fc590e82f37075f06d}}..
As a result,
the infinitely many conserved soft hair charges should yield non-trivial
constraints on gravitational scattering and on black
hole formation and evaporation, as argued in
Refs. {{cite:acb407dc970b4b0859ae227a4349eb59127e26be}}, {{cite:674bae988cd0f99249dd548a58c7c871e99d5424}}.
| d | 6f5f005208e9993c2f9e2b61eb6cecf7 |
The main result of the present work is a refinement of Theorem REF into a LDP both for a Nash equilibrium and the state processes at equilibrium.
It is well-known by the celebrated Varadhan-Bryc equivalence, given in {{cite:0e6e8120221290889d2a101fe36ca0581611f578}}, {{cite:d6eac622b64fbb997b5ce883f6845603d9f046d5}} (see also {{cite:04fd33ed6f334717dc54bfc9aaa4252e488c156d}}) that the LDP is equivalent to the so-called Laplace principle which can be stated as follows:
Given a function {{formula:a065f6d4-524b-4c36-8c78-786cd11bfb69}} with (weakly) compact sublevel sets {{formula:91959072-9147-4bd0-b48e-2e26ea72bf19}} called a (good) rate function, a sequence of measures {{formula:61640be3-5fe9-4bf0-a086-34cff8d21296}} on the Polish space {{formula:6d6463bf-ac29-4731-aad2-c04d60e68350}} satisfies the Laplace principle (in the weak topology) if for every bounded continuous function {{formula:731d1d2d-4387-49f4-972b-222e32c43d08}} it holds
{{formula:c648619c-895a-486c-91c9-c49271bf5b46}}
| r | 8f2e071545a8e2d49260e323099e817a |
To train the RL agent, we employ a deep Q-learning approach with a replay buffer {{cite:c74ecbf02def13a5c41a620250463fa5e4f34019}}, {{cite:8f482aaa2be03be3838116bf7f05723bf2477d7c}}, {{cite:85f383ce3fdda825dcb9789d39c0d5bf6d118b39}}. The deep Q-network is fully connected and contains 4 hidden layers, each with 64 nodes. The input and output spaces are 3 and 10, respectively; each of the 5 possible angles and 2 speeds are paired to form 10 possible actions. A concern of performance divergence for the UAV pursuit-evasion problem with RL was discussed in {{cite:9fe83ee84e831624955568dc2f3fe459f38064f0}}. We apply regularization through L2-norm weight decay in attempt to prevent this issue. Other notable hyperparameters are presented in Table REF . Our algorithm adopts the deep Q-learning code architecture from {{cite:c74ecbf02def13a5c41a620250463fa5e4f34019}}, and our own code is on GitHubhttps://github.com/LorenJAnderson/uav-2d-greedyshooter-rl.
| m | 925ab556eced4bdcc62a95de1a820ca1 |
Thus, regardless of the sign of {{formula:9f3e73b0-e18a-4de6-9a68-9a70ad192c8a}} , the band gap opens at the {{formula:6e69f243-a097-4e3a-9050-61d4b21ab044}} point and this gap becomes
topological. This is related with the fact that the irreducible representations (irreps) at the {{formula:0b130a4a-3092-44e5-a898-2e5788a16ae8}} points do not affect the values of the
topological invariant; namely, the gap at {{formula:1d274dd7-7173-4d16-af25-623df4295661}} is inverted between {{formula:51fe6016-6b7f-4daa-a7ab-e28861ec8842}} and {{formula:1ca32563-ed81-496f-8b5f-d0bd5cb32d8a}} ,
but it does not affect the topological invariant.
This situation is similar to the Kane-Mele model, which is a well-known model for the two-dimensional topological insulator (TI) phase with
TRS {{cite:1aa0ddbcd9850eb46b35e6fb38e3eadce78e6f5d}}, {{cite:0a8698321fe209f5dfe3e319426591a13645994e}}. The Kane-Mele model is topological,
regardless of the sign of the spin-orbit coupling {{formula:55c5e511-a3b6-46df-b275-aaca8c05b8fc}} , which opens the gap at the {{formula:5066f94f-e6cb-4742-8cc6-89a1adb021d1}} point.
Thus both in the BPI photonic crystal and in the Kane-Mele model, the topological invariant is independent of the irreps at {{formula:7020a962-506b-47c4-905f-7dbf3ebbcb12}} or {{formula:09ac1378-1fb9-4543-a65a-854b1360a0aa}} points, and regardless of the sign of the parameter ({{formula:63392768-9a17-42fb-8fe4-8fa6b5a53c24}} or {{formula:12b6f360-c4dd-474b-ba90-fc1dc6db5acb}} ) to open a gap, the system becomes topological as long as the parameter is nonzero. More details are presented in Sec. S1 in the Supplemental Document.
| d | 46b1d1bb57b9422a45ad3356e237f5da |
In Table REF , we show the reproduced results of the original
QDE method following the exact experimental setting as implemented in
{{cite:7028ee0e7c61fb02c7118b5586aac7697c87f3d0}} pearce:2018. These results are included to
offer a comparison between the 1-layer and 2-layer models (Table
REF and Table REF , respectively). It also
serves as a verification of our re-implementation of the method by
{{cite:7028ee0e7c61fb02c7118b5586aac7697c87f3d0}}.
{{table:68094d1c-1512-4aee-92dc-c8c63e608fc3}} | r | a424ef8fed0f7fe79abf4e6576796e8b |
Efficiency. From Table REF , we observe a significant improvement in inference efficiency between our TCSA-SNN and vanilla models. On Gesture, three-layer TCSA-SNN (following {{cite:1a61a6712cc7deaefd6465a2b8837490c9c525f3}}) and five-layer TCSA-SNN (following {{cite:2e532b9ac6e987aebe9a626e635c3f7c63b760d1}}) increase the energy efficiency by up to 3.4{{formula:82064040-7263-4d02-a74c-9407340ac886}} and 1.8{{formula:78e37ff4-49a2-4377-ae3b-64d0e5c00f29}} , respectively. We see a similar trend with regard to the effect of TCSA on Gait, finding that the TCSA-SNN outperforms our re-implemented counterpart baselines with both effectiveness and energy efficiency. Moreover, we observe that the NASAR is associated with the model size. Networks with more spiking neurons usually have sparser spiking activity, such as the NASAR of the three-layer and five-layer baseline on Gesture are 0.214 and 0.074, respectively. Incredibly, on Gesture, the NASAR of five-layer SNN drops to 0.011 with the help of TCA, which means that only 1.1% spiking neurons are activated at each time step.
| r | 451aa233d8ac38a658986eaddbd12521 |
Information novelty might help explain why politically-opposite feeds increase engagement. The fact that respondents indicate politically-opposite feeds as different from what they are used to seeing in their own feeds implies these opposing feeds contained novel content. Prior studies have demonstrated increased user engagement with content that is novel, for example, in the context of online news consumption and sharing {{cite:9dfd42c85fce76354e3a1921dcb0881e03c52a97}}. Furthermore, as novel information is often surprising, it also attracts more attention {{cite:f2754495964cdc886f3162d267efdd73316f58fa}}, {{cite:a2fceced7688d1e72262eeb3742db673485af66a}}. In the context of social media platforms, this enhanced engagement is encouraging as it demonstrates a consumer appetite for accessing alternative viewpoints and may actually dovetail with key business objectives such as increasing user engagement or time spent on the platform.
| d | 8ff8815e99344bba6710d4e9a37c4cc5 |
Meaningful Perturbation (MP) {{cite:8c89231cf4a24cd2e480a1448ace646300d56b01}} finds the smallest real-valued, Gaussian-blurred mask such that when applied to the input image it minimizes the target confidence score.
MP is sensitive to changes in hyperparameter values {{cite:eb208579d820ae88903ca5e3d5e8eb5c87985c14}}.
To ameliorate the sensitivity to hyperparameters, {{cite:3142ea881778ab09d8406724be592319d3b4595d}} propose to average over four MP-like heatmaps of varying controlled sizes of the high-attribution region.
Extremal Perturbation (EP) {{cite:3142ea881778ab09d8406724be592319d3b4595d}} is less sensitive to hyperparameter values than MP but requires 4 separate optimization runs to generate one AM.
| m | 6fc05635db582296f757a4d3ec3f3012 |
The {{formula:7220bf3d-0b7a-4d76-aa2f-24590dd694e4}} is used as an input to a highway network in order to reduce the interferences from the rest sources. The encoder, the decoder, and the highway network are optimized using {{formula:d86e1890-a685-4f5d-b8a0-b455a8616357}} between the true magnitude spectrogram of the target source and the output of the highway network, plus an {{formula:e360d2c9-ab85-43fe-9993-f6daff52ac88}} regularization term. The output of the highway network with the complex-valued mixture representation, {{formula:7a9fa6c6-230f-4ecf-9b02-375b39e9e5c4}} , are given as an input to a generalized Wiener filtering process. The latter produces the complex-valued time-frequency representation of the target source and further reduces the interferences from the rest sources. The final output, {{formula:d38b1c92-433f-4473-9c9e-df94cf304000}} , is calculated through an overlap and add synthesis procedure {{cite:8f9c01ebff41f1206c5fa7175b4230997264e262}}.
| m | a567c6e9946dbc8050775e775114682f |
Diffusion is a common phenomenon in nature and associated with a variety of processes, resulting in inaccuracy and misleading of the underlying physical phenomena {{cite:00a32aafbc2552522c63aa09806c86db829a2b68}}, {{cite:8daeb7f7c895dee57011a1322560d16727bbd7dd}}. In this work, the term “diffusion" will be referred to as the self-diffusion performed by nuclear spin-carrying molecules,
and can be described as follows: in liquids, under equilibrium conditions with a thermal bath, spin-bearing molecules perform Brownian motion (BM) due to thermal energy, which means that they follow random trajectories, changing their positions, without necessarily the presence of a concentration gradient {{cite:873f40755f3fc08c5e4f29283bfc0a91d811163b}}, {{cite:bdf9029067137c588ed6c188860d21d63721841b}}. So it can be thought of as a BM in the absence of an applied external force, so that, on average, no displacement is observed; however, molecules that were together, in the same neighborhood initially, will be dispersed as a result of translational motions. On a macroscopic level, this collective behavior, in contrast to microscopic individual movement, exhibits great regularity and follows well-defined dynamic laws. The formulation of this process can be done in a similar way to other diffusion processes, as long as it is possible to establish some distinction in the molecules that perform the self-diffusion {{cite:e300ba1b09395b93c1b1318242cd0f8d9e1c7900}}, {{cite:8daeb7f7c895dee57011a1322560d16727bbd7dd}}, {{cite:1f55cf929b37ed6ae6149810e6f4b819eff043c2}}. As will be seen, diffusion-based NMR experiments provide through magnetic fields gradients a noninvasive way to encode the spin random trajectory by its position.
| i | ec958584fe59a68cbebe33a04e7e394d |
where {{formula:f41d43bd-bedc-4347-8bd5-dc983b6831cd}} for Max-cut and {{formula:d59ac25f-31e8-4ec6-921e-edf93ebc94ad}} is randomly chosen to be {{formula:80adabc5-463c-4377-81e7-a34c4ff96c95}} for S-K model. For graph coloring problem, we first model it with quadratic unconstrained binary optimization (QUBO) form and then transformed it into Isng cost form {{formula:0bffa32e-746b-44d6-b9fc-cdc260325d7d}} {{cite:a42a068af52753793b5dcb9e7898225a39744eed}}. The QCover has an instance library to automatically generate the weighted graph {{formula:457d99da-2abe-48d1-92a4-0509329d7340}} corresponding to {{formula:abc5d076-a858-40ea-8a6c-6968ffad328e}} for above three problems.
| r | 1d3f034f3506412592a0a724ec092150 |
Embeddings from all layers.
An alternative approach, inspired by {{cite:2be745d0a6918b0952b9ae893d81a06c56ee02c4}}, is to use pooled embeddings from each layer (not only the last) of the Transformer. We explored two pooling strategies: averaging over the timescale (common for speech Transformers) and taking only the first embedding as suggested in {{cite:2be745d0a6918b0952b9ae893d81a06c56ee02c4}}. This yields 9216 features in each case.
| m | 10aff5b49bd54643d200772d8a7d0b76 |
Complementary information to the above analysis comes from the
knowledge of the electromagnetic form factors, assuming that one
of the massive constituents of the bound state is charged. Their
asymptotic behavior qualitatively probes the compositeness of the
states: a rapid decrease would be the signature of a many-body
structure {{cite:4aeb1785b5351bbe3e45c968ffe2ff7937c063bf}}, {{cite:4bd34fe1807b2e83bec51b1552b278f9bec124a5}}, {{cite:f9ab01867c4afe78fdc07d8762839018fa3f497d}}.
| i | 4c69d46b2d91b8990a7e3cc6d64aba95 |
The network of gravitational wave (GW) detectors composed by the two LIGO {{cite:e6a267030d1f8e607254ecd5ea21fa704f7a0108}} and Virgo {{cite:4715525a70922b88fd334ec1936bf5054235f7ba}} observatories
has already completed three successful observation runs
{{cite:5beda308cbd3407ea67aa34a11c4e390936bcb2b}}, with the detection of over 90 coalescences of compact binary systems.
To maximize the possibility of detections and their (astro-)physics output,
collected data are analyzed via matched-filtering techniques
{{cite:fb41319fc64546198a22f423c8ce3dff5ddc7d98}}, {{cite:84381bb0dc2fb711d94e54cbddbfeaa35bf39769}}
by correlating them with pre-computed waveform templates, whose development
is the object of intense investigation {{cite:42f0d51fc61eb61fb66385d469ca16de4cc762a1}}, {{cite:beb9c498c809c6275af5c4a23cdc8102d305a00c}}, {{cite:3a24bae6fbad0dd90d4bb4c85ba47f71c251ef66}}, {{cite:af7ad7a3f7257116f66516675d01449d824313f8}}, {{cite:49ce927b134e480e43cb816157eecc67177cc097}}, {{cite:fea2a472bc655bf2884d14c16a835b886a7e5ee0}}, {{cite:beb3e93a51c5f73585a60c4a2b71274b09d2ae6c}}.
Improving the accuracy of GW templates straightforwardly enhances the quality
of astrophysical information obtained from these sources.
| i | 195dc297fd59d2d4efe4cc8cfe6290f1 |
Throughout this analysis, we have focused purely on linear methods.
In recent years, nonlinear methods for dimensionality reduction, such as autoencoders and diffusion maps, have gained popularity {{cite:ee4e697cb444791abe553f034699fbc4029da962}}, {{cite:132446e66b9ee02f916af965d18e81b3a149bf58}}, {{cite:a0314dc52c187a31e8620fbb070838719b370abb}}.
Nonlinear models similarly benefit from promoting sparsity and interpretability.
By understanding the structures of linear models, we hope to generalize these methods to create more accurate and robust methods that can accurately model a greater class of functions.
| d | e32947ef344ae03dafbb015c0a756172 |
On the positive side, we show that, when {{formula:bdef11ef-ce7d-4163-ba0d-65c60dec858d}} , the optimization function is submodular and thus can be greedily approximated within a factor {{formula:a57848c3-03a1-4e78-bb65-2d8d0029cb6f}} .
Preliminaries
Structured populations.
Following the literature standard, we represent a population structure by a (generally, directed and weighted) graph {{formula:3d5d2a6e-a2c9-4bd6-b267-11447c5e4af5}} of {{formula:69486aa0-96d2-4add-b10d-97ebe6dc76db}} nodes.
Each node stands for a location in some space, while a population of {{formula:c96aa269-f169-4a36-b9c4-2be12dd672ad}} agents is spread on {{formula:01974a0d-0414-4c3d-8430-370dbbada80f}} with one agent per node.
Given a node {{formula:5f5c4dec-4625-40f7-ac45-867c0f15ef26}} , we denote by {{formula:8b83155b-59d6-4380-ba51-483c3b2db5ff}} the neighbors of {{formula:6ad7ad96-d7a2-4ae1-8e82-70904847085a}} , and by {{formula:9525db05-61d0-4582-995b-41a9e5f2e083}} its degree.
The weight function {{formula:895c6a70-2fa3-44c6-953c-cdf92db0c82e}} maps every edge to a non-negative number.
As a convention, we write {{formula:bf4ecb4a-008e-4598-a278-4e6858d8e7aa}} if {{formula:0f5204b3-7656-45d2-9f8b-651a206492f9}} .
We consider graphs for which the sub-graph induced by the support of {{formula:e7b0e78e-ba34-4ea2-ae69-d3589d791d48}} is strongly connected — which is necessary for the voter process to be well-defined.
We say that {{formula:4c8ba755-cf6e-40b1-b5a3-66c30ce861b0}} is undirected if {{formula:7c320c16-f777-437b-b156-f741b13656b6}} is symmetric and irreflexive, and {{formula:6ca9fc6a-5a9c-49a3-ab57-9fd2ec2f5f1e}} for all {{formula:0e954111-f647-4bdc-82bc-4be5669948e4}} .
For simplicity, when {{formula:138a2ac6-2df3-483b-ae2e-1451421b1431}} is undirected we denote it by {{formula:5c29b430-76dc-47c4-9a82-f593283a4ab8}} .
The voter invasion process.
The classic voter process models the invasion of a mutant trait {{formula:cfb5ffd9-00e0-4533-b4e1-4c8157675ee0}} into a homogeneous population {{cite:36932f47bca93d2c172d55fd13650ab6aa622e1c}}, {{cite:9f063b7fc25c62b7fc4250bc191f71781c43a32a}}. The process distinguishes between two types of agents, residents, who carry a resident trait {{formula:05c1f81f-48bf-4896-b887-5d3a9133182e}} , and mutants, who carry the mutant trait {{formula:b033fc9b-2acb-4f9f-a39b-42c55c7960ef}} . A seed set {{formula:186e5c23-4533-476f-83d8-4f4f98991dbb}} defines the agents that are initially mutants. The invasion bias {{formula:3d6d4e49-5faa-4b12-8b16-938f1760b4a6}} is a real-valued parameter that defines the relative competitive advantage of mutants in this invasion. When {{formula:2f925f8d-2aba-476a-be0f-3b8cc5a06351}} (resp., {{formula:12283489-315f-41b2-a117-61d03bbfcfab}} ), the mutants have a competitive advantage (resp., competitive disadvantage), and the process is biased towards (resp., against) invasion, while {{formula:e443cae7-c5c4-41f9-8677-a3812d96d26f}} yields the neutral case.
A configuration {{formula:4875575d-824f-4f54-8f04-4d89ffb25425}} defines the set of mutants at a given time.
The fitness of the agent occupying a node {{formula:99a6cbc6-1d51-4263-b204-b3a89ed2f621}} is defined as
{{formula:91003bc1-4320-46d9-807f-d167dd9c9959}}
The discrete-time voter invasion process {{cite:fffb105feda6e222015e0d548970e7c203e01a30}}, {{cite:c66c2fc0e4ca25fef9c404846cca1d169c0dfb2e}}, a.k.a. Moran death-birth process {{cite:811a285878d13672cba919e5140ede02b94733be}}, {{cite:214f597a6e0dae9a9f77666d7170c9de6a504f7a}}, with seed set {{formula:8b22fe6b-4796-4499-88a6-88b9f235529d}} is a stochastic process {{formula:6f7ae2e9-ad7e-401e-bf75-81b32d973d2d}} , where {{formula:d39d887b-0713-454c-87a1-2ae3c2aea3f9}} is a random configuration.
Initially, {{formula:f2b1fade-b8c7-4c32-a0c1-2f4fbd021554}} , and,
given the current configuration {{formula:3b338739-c3f7-4d80-b616-d829d1b26aab}} , we obtain
{{formula:43d1f0e3-0fab-4fd4-a4e9-8db2968b3a3d}} in two stochastic steps.
(Death event): a focal agent on some node {{formula:117edc9f-aac7-44de-971c-44e2328a2809}} is chosen with uniform probability {{formula:be9c9278-38ab-46af-9451-adb744d2b536}} .
(Birth event): the agent on {{formula:d486fbc2-d6cd-440e-8f65-d1cdf14c903d}} adopts the trait of one of its neighbors, on {{formula:634ff604-40f2-4653-8030-abf6465c3f53}} , with probability:
{{formula:6a9d9e48-be06-44e6-a05d-d35d1dd2c382}}
Note that the focal agent may assume its pre-existing trait, and generally, the mutant set may grow or shrink in each step.
We often identify agents with the nodes they occupy and use expressions such as “node {{formula:51191f22-8582-46e5-8d3b-01d18886cd7f}} reproduces onto {{formula:64e5d6a5-9408-4e89-b0c6-2619fbfb2e96}} ” to indicate that the agent occupying node {{formula:4dfdab9e-adb7-44b6-952c-a8091473c898}} adopts the trait of the agent at node {{formula:d1bf1ddb-e8c6-4a83-ab5d-e7a07b9b690f}} . See fig:voter for an illustration.
{{figure:f139a98e-c39c-4231-b446-51ee448b5f44}}Fixation probability.
As {{formula:c37a767e-e1da-4886-93a1-9aa2e3b0e647}} is strongly connected, one type will spread to all agents in finitely many steps with probability 1. The mutant invasion succeeds if the mutant trait spreads, an event called fixation. Given a graph {{formula:fcef4cfe-f359-459d-b704-9cf680fcb556}} , a bias {{formula:70d98a5a-2abd-43c5-a162-62072eadeb86}} , and an initial seed set {{formula:a85100f6-226d-49da-9b36-907a3c1f6e66}} , the fixation probability, denoted as {{formula:b2f0ab0f-8046-418b-a018-4e535c26931e}} , is the probability that the mutants starting from {{formula:246f4d96-f8bc-41b1-ae29-834d025fa1ef}} eventually fixate on {{formula:4f5874f9-2447-4a25-b483-cd437e44e438}} . See fig:twostar for an illustration.
{{figure:dae6a37e-ec56-4b07-ac9b-b09a373d981a}}Whereas the computational complexity of determining {{formula:ba02448a-e960-49e8-9b63-7c59185bee45}} is unknown, the function is approximately computable by a simulation of the invasion process. In the next section we will show that when {{formula:eefeeca5-020e-45c1-b283-e0c9a4c78ba1}} is undirected, such a simulation yields an efficient approximation scheme.
Fixation maximization.
We formulate and study the optimization problem of selecting a seed set that leads to a most successful invasion, i.e., maximizes the fixation probability, subject to cardinality constraints.
Formally, given a graph {{formula:8d7b2b41-5255-4664-b3f4-b91c3626775c}} , a bias {{formula:76a41b09-1bf6-468b-9025-75ab2916d644}} and a budget {{formula:3ec8a6bf-ecc6-410e-bcf0-0faafbe62f6e}} , the task is to compute the set
{{formula:a03bb420-6924-4bc4-9672-753da5c8a8d1}}
The corresponding decision question is: Given a budget {{formula:7eb26adf-3dc7-4cc5-9638-f4cb9909b7d1}} and a threshold {{formula:136729e5-2bae-46fb-83ed-0b53f089b2e8}} determine whether there exists a seed set {{formula:8407bb8d-7566-49f0-bc07-d81c1e24677a}} with {{formula:1e18cec9-37f6-4eb9-a945-732f83ace354}} such that {{formula:fe506e49-e63c-40ae-8fd0-e7d5b89bce00}} . As fig:twostar illustrates, different seed sets yield considerably different fixation probabilities, while the optimal choice varies depending on {{formula:85a5f9d1-830c-4a88-9489-f0d1f365bd6c}} , hence the intricacy of the problem.
Computing the Fixation Probability
For {{formula:aa1eb4f8-2418-4429-89b5-450b3cbe97f2}} , the fixation probability is additive {{cite:c3f409b50cb72c67e9090a143265e8a9b12c2d2f}}. Thus, given a graph {{formula:1c41f3fb-090c-4971-9de6-8f34ca7c892b}} and a budget {{formula:30ecf967-5bf7-4363-b1ae-6c1f6304854d}} , the set {{formula:33d21e4d-0333-4832-9f8c-ce7afe50f2cd}} that maximizes {{formula:aecd7079-041f-4117-b577-3bde4c2d9a69}} is obtained by selecting the {{formula:0d39c5db-8399-4d36-a0c7-b55a7a18e307}} largest values from the vector {{formula:d2723d13-ca90-48b9-9ab8-fdb18a441f4e}} , determined by solving a system of {{formula:c04985e6-3818-489f-8b81-c39cb215382a}} linear equations {{cite:33cb99039e84ad604ae82334e4df71a32d8868ab}}; thus the problem takes polynomial time.
When {{formula:19b20635-6aa2-433b-a783-5860de2ebcee}} is undirected, that system admits a closed-form solution {{cite:c3f409b50cb72c67e9090a143265e8a9b12c2d2f}}, {{cite:ccf4c13656933ba16203c7e5de56bbaa95154033}}:
{{formula:0cef9a4c-05ba-43a5-a862-8af1f1003b5c}} .
For {{formula:9bf9f113-0cfc-46fd-8471-67d2b20908f3}} , the complexity of determining {{formula:7d9184cd-8c31-42ee-9677-58f1c8458f52}} is open. Here we show that, for undirected graphs, it can be efficiently approximated by a fully-polynomial randomized approximation scheme (FPRAS).
The key component of our result is a polynomial upper bound on the expected number of steps {{formula:3447a5d9-70c4-47ec-a810-d35d1a42246d}} until one of the types fixates;
an analogous approach has been used for the so-called Moran Birth-death process {{cite:6bdd8465bd0529ad1a84035ba90170f76298c51d}}.
theoremthmubtime
Let {{formula:3b3cc053-9e35-4e54-869e-65f10443de37}} be an undirected graph with {{formula:aa2bbf08-6e1c-45e7-8ebe-0b5183718bb3}} nodes, {{formula:49846f96-3b7e-4def-87e8-48f232cb95c3}} edges, and maximum degree {{formula:b730546a-f65d-4ee6-ad59-a7e5470d54ae}} . Let {{formula:6ff9efa9-c64e-41d8-97b9-dbd32705abb0}} be the initial seed set and {{formula:f137e6dc-4d03-4d9e-b09e-efa058d3260a}} . Then {{formula:366fe8f2-f946-4684-a0a1-b192b7240b66}} .
{{formula:3224d1d8-20d5-48a8-b0dc-d09c2c2ff6fa}}
.
Without loss of generality, suppose {{formula:db9ff96e-cc73-4ca4-acd6-4dda1253d732}} (otherwise we swap the role of mutants and residents).
Given a current configuration {{formula:9f1be051-d2e6-491f-a3d5-86adfba93406}} , we define a potential {{formula:5da03e05-1820-49fa-b2c7-7ced7cdfe023}} as {{formula:01249a48-14d5-4388-8b27-8e8cdfcf66f8}} .
We show that, as we run the process, {{formula:8ddfd326-1379-44f6-a245-785534923880}} increases in a controlled manner.
To that end, we distinguish two types of configurations: we say that a configuration is bad if every edge {{formula:7ae9bf2d-3c8d-41b9-8f20-25aa29fb2902}} connects nodes of different types, and good otherwise. Note that bad configurations exist iff {{formula:2d5b25b4-a5b1-41cf-9ccc-8a2ba29bd539}} is bipartite, in which case there are exactly two of them. In sec:appendix, we show that:
When at a bad configuration, the next configuration is good and the potential does not change in expectation.
When at a good configuration, in one step the potential increases by at least {{formula:59760f2c-9e5c-4cfb-9c46-25415ae0d6d4}} in expectation.
All in all, this implies that, in expectation, the potential increases by at least {{formula:c5899adb-5f22-4034-ab28-656e6a3ccc7b}} over any two consecutive steps (until fixation occurs). Since at all times the potential is non-negative and upper-bounded by {{formula:abf6b530-6989-4ffc-9441-c11fc2586fe1}} , by a standard drift theorem for submartingales {{cite:4a03632ccd12ba120cce9e6f5f6e9d5eda257911}}, {{cite:6b8ba825cbd137d5839bac0a0ff9cf06bdbfa3a1}} the expected number of steps is at most:
{{formula:0c8d6404-4dd7-490e-a469-c0e61a4d8802}}
thm:ub-time yields the following corollary, based on simulating the process a sufficient number of times and reporting the empirical average.
Corollary 1
When {{formula:3db49235-8490-4321-a46c-f574460ee960}} is undirected, {{formula:d62e2a5c-b82f-43d6-968c-241dc95f0485}} , the function {{formula:acde8c19-2fb5-49e8-af4c-309fd9c08612}} admits a FPRAS for any {{formula:c71fa3cb-a9bf-4c4f-a1eb-1350af960f0f}} .
Invasion with Favorable Bias
In this section we study the voter process for {{formula:891e42ca-bf75-4095-95e3-26d03276aa2e}} . We show that the problem is {{formula:c9661d4d-f4b1-4a8a-9267-fcf848a95bc1}} -hard (subsec:opt-hardness) but the fixation probability is monotone and submodular and can thus be efficiently approximated (subsec:submodularity).
Hardness of Optimization
Here we consider the weakly biased process, i.e., the case {{formula:369ceff6-90bf-40f6-9aa5-ae4a21cbb88f}} for small {{formula:c9a54828-dff0-4175-b799-ec528979bc32}} . Since {{formula:02b56bb4-c04e-401e-8a0b-6a50e098c7c6}} is a continuous and smooth function of {{formula:8fc9bb6c-d749-40a2-8ec0-7976bdcb09ee}} , its Taylor expansion around {{formula:bf8b4023-9409-4230-928e-6587b5f8184b}} is:
{{formula:5b9ba7f3-ea52-4c65-857b-bb955e69ed3e}}
where {{formula:2ee45c57-6e02-44ea-8494-d6b520403029}} is a constant independent of {{formula:3c84b9df-d228-45fa-adb4-798a0ffc0e72}} . Recall that {{formula:867b571d-bdb1-4548-b0d7-bfbdfed558d7}} {{cite:c3f409b50cb72c67e9090a143265e8a9b12c2d2f}}, {{cite:ccf4c13656933ba16203c7e5de56bbaa95154033}}.
For {{formula:ec453a96-2dd7-4fed-b89b-d7990b4a609f}} -regular graphs, this yields {{formula:5d60dede-00b8-4039-af71-1b5befb1b2c3}} , hence the first term in eq:taylor is constant across all equal-sized seed sets {{formula:a8886c45-36e9-4423-a473-2fe6c92f7d89}} . We thus have the following lemma.
lemmalemmaxder
For any undirected regular graph {{formula:f6a80b48-b89c-42d5-b7f0-39dec05bbc19}} , there exists a small constant {{formula:6234cd5f-c6a6-49a5-a845-7ba9663fd8e8}} such that:
{{formula:55367c98-d760-41e4-8c0b-0d911bf35b26}}
In light of lem:maxder, we establish the hardness of fixation maximization by computing {{formula:609d07a3-173a-43b4-8ab0-9edf8eaf4605}} analytically on regular graphs, and showing that maximizing it is {{formula:24e7dd4d-bc82-40e0-8b57-104072590b66}} -hard. McAvoy and Allen Mcavoy2021 recently showed that, given {{formula:0bfb4726-9254-45b5-8ec7-674347f0b608}} and {{formula:4e2dcafd-9dce-4b74-9715-b7c0f666f21f}} , the constant {{formula:448259fa-9047-466e-9a4e-6d903145e095}} can be defined through a system of {{formula:e4e2459d-4850-47c2-8cd5-1b85f0b4293a}} linear equations. Here we determine {{formula:a5f97bc0-1633-497f-8ced-451c67ba9d6d}} explicitly for regular graphs.
For each node {{formula:95b07acd-fb8f-4bfa-b33d-8ae46c099597}} , let {{formula:821da20b-9a33-446b-8d25-68de32a93156}} be the variable indicating whether {{formula:18f3bd23-6915-4f12-8f88-b1059a709d01}} . First, we restate {{cite:3b53f246851cf02cec4a8c21fedee065a3fe80a6}} in the special case of {{formula:34e258a2-7932-46c2-bed6-5785b7ce9ac7}} -regular graphs and adapt it to our notation.
Lemma 1
Fix an undirected {{formula:271e99f5-032d-4ca8-a747-084cf7cdea01}} -regular graph {{formula:9f4c497a-06a9-4d6f-aeaa-927e01ee19b1}} on {{formula:848032b4-078c-42c1-831a-75f1461603ef}} nodes and an initial configuration {{formula:f2704c25-7bad-4c61-a896-8aa8b60696f8}} . Then
{{formula:0922c3a0-53ce-4318-8012-3f28853e3da3}}
where {{formula:a9638c3a-c943-4305-bdd1-5a1212d73d44}} comprise the unique solution to the linear system
{{formula:771fce90-7313-478a-b35d-3b99f526d71e}}
Intuitively, each {{formula:6ba0fba9-3958-4b32-be31-4cb5485b0be8}} expresses the expected number of steps in which nodes {{formula:066e4562-a321-4ca1-9685-960dfe63f229}} and {{formula:b693d075-df7f-4c78-9583-ef184a245269}} are occupied by heterotypic agents (i.e., agents of different types). We say that an edge {{formula:33807459-fc02-4663-a80a-b0b2988f2811}} is active if its endpoints are occupied by heterotypic agents. The following lemma establishes that, on regular graphs, {{formula:bb0e9d33-e408-419e-9527-3340bb031b23}} is decreasing in the number of active edges induced by {{formula:08523c52-0eca-4805-ab7c-60151136f6dc}} .
lemmalemfpweakregular
Let {{formula:71c780f6-c059-4c98-a5c7-32fc2c7812f4}} be a {{formula:57692290-ad56-46ec-96aa-65ecaefc2ca6}} -regular graph on {{formula:1459c247-51f4-48a6-87da-22a527a6a6fa}} nodes and {{formula:7cdac99f-3808-409e-84cb-9788b3a026a9}} an initial configuration of size {{formula:6eed7338-b1f0-4070-8dec-5098fae54dec}} . Let {{formula:d3bc6262-2a91-4a7b-8314-54a42b82ca46}} be the number of active edges induced by {{formula:c9e5ce28-27cc-44f2-be8f-410d0b0f8afa}} . Then
{{formula:c69ab7dc-7d5b-4d2e-ae1e-dcfd03a16e28}}
{{formula:2ffb78aa-9819-4e12-aff2-8c46cb590e72}}
.
Let {{formula:e23b7fa2-67ee-4869-b9bf-b4540deaf115}} be the shortest-path distance between {{formula:0182afb1-c31d-4a8d-832b-77586a931ddf}} and {{formula:8e1fc8bc-d934-484c-a81a-d0f5b8df90cf}} in {{formula:e411946d-2183-409a-aef5-a5b0ba14a850}} . We proceed in two steps. First, we claim that summing all the {{formula:34bd34c2-7348-4a14-abab-a2c29fd0bf2e}} equations in the system of eq:mcavoysystem, we obtain
{{formula:bca8abb8-ff56-4530-ae61-54c1151a1c68}}
Indeed:
We have {{formula:3311fdf5-5354-4eeb-aae7-b339a4065980}} , since each ordered pair of
heterotypic agents
is counted once.
For a fixed pair {{formula:52da9557-8a95-42f0-a09f-22404a6cfab6}} with {{formula:b129feac-3ff6-4feb-98a4-4f5276ccf118}} , term {{formula:5c0c9136-0e79-46d8-8e66-881a8ddacbbf}} appears on the right-hand side of {{formula:e536a84f-6b1a-4ee3-a4f5-c1ab6a46d3d2}} equations, namely {{formula:423a3053-a327-4466-8eb8-311958dc0e10}} equations with {{formula:24a58e28-6c0b-4bdc-bcf6-b43df4d8ebef}} on the left-hand side and {{formula:2ea4c3b1-e694-4bc0-8747-e786dfd97243}} and {{formula:ee6190c6-676f-4651-a89c-16ba83508882}} equations with {{formula:38a72f26-add1-4925-b615-2aa7f472ab2c}} on the left-hand side and {{formula:548b7045-f83c-4b53-b443-fca4fca0a8cd}} .
For a fixed pair {{formula:16420d1a-87ac-464f-98ca-2bb9c5ed2513}} with {{formula:8cbce0b4-c370-41ea-a06d-689e2bc771a0}} , term {{formula:7f65e01e-c7df-4bf7-b5ce-a3d0f4b246c0}} appears on the right-hand side of {{formula:6d4e597b-940d-4a62-8b0b-95e5cf3d02f1}} equations, namely the above {{formula:7d40ad9f-e003-40e5-8b91-30d8482b2027}} equations, except for 2 cases where {{formula:d929d1eb-4c19-451c-b4b0-0e8749e55862}} .
Rearranging eq:sum-all1, we obtain
{{formula:7e5ab0e2-c892-49a8-ae46-e29755cb1b72}}
Second, we claim that summing the {{formula:b0e02ca3-590e-4d01-8e4e-2c93ab7fb450}} equations in the system of eq:mcavoysystem where {{formula:afdf1568-9fb6-4d21-9ef2-0246e3e81463}} , we obtain
{{formula:cbb82832-fc17-4b0f-a0a0-904907194875}}
ergo, using eq:sum-all2
{{formula:6003d12d-8916-4325-b4e0-d0949f22f678}}
Indeed:
We have {{formula:5ceeae2c-3389-479d-b308-188c05e2e215}} , since each active edge is counted exactly twice, once in each direction.
For a fixed pair {{formula:f03b575f-ce4e-4da6-824c-51f11b17918d}} with {{formula:73b69b5c-4dd4-41ea-a96d-382432fbdaec}} , let {{formula:59ddeab4-437a-43cb-b23d-064374d2964e}} be the set of common neighbors of {{formula:11f1ecd4-1513-4be9-a652-8943e40d4921}} and {{formula:02411951-67fa-4b2e-9148-96b8ca38fc45}} . The term {{formula:ce8bc2b6-8e16-4ed4-8d8e-2bec2e9b7a11}} then appears on the right-hand side of precisely {{formula:5d6e517c-4eb4-46e4-af87-296b3388f51f}} equations, namely {{formula:a47cf0fc-4f00-478c-a036-749de4ea90d1}} equations with {{formula:2c2c00f4-9ff3-44bd-8476-698da47a0a5c}} on the left-hand side and {{formula:afdd6e49-a0ed-40ba-aa1b-d057fa9da7c2}} equations with {{formula:e0385e66-cbdb-4345-a9e5-aa9999e2f137}} on the left-hand side, where {{formula:51ef2e90-4420-4736-9f8a-d68a7925d050}} .
From lem:mcavoy and eq:sum-dist1, we obtain the desired
{{formula:e3aa2450-5b2c-44aa-8ceb-3e08ed0ea3d9}}
Thus, given a budget {{formula:ac5a9b47-41d1-4b1f-b3df-51b74ee8e368}} of initial mutants, in order to maximize the fixation probability at the limit {{formula:7e92631f-b0eb-4c8b-a3a3-3d1ef71e197a}} , we need to minimize the number of active edges. We are now ready to establish the main result in this section.
theoremthmpositivebiasnphard
Fixation maximization in the biased voter process with {{formula:4dcaf899-a0dd-44f4-8447-f2755573f91c}} is {{formula:17d83a1a-c061-47f6-ad65-e03155e1e953}} -hard, even on regular undirected graphs.
{{formula:d62049d1-06c6-4806-814f-ddc3e2e500f6}}
.
lem:maxder and lem:fp-weak-regular imply that, for any regular undirected graph {{formula:f028bd39-2148-427c-8a11-b3f3e5995fd5}} , there exists a small {{formula:56361bb0-6859-44d9-a8fa-2025c80ef1d6}} for which
{{formula:3ce1f95b-11ab-4c07-b5cf-10c303d753fd}}
i.e., the fixation probability is maximized by a seed set that minimizes the number of active edges. For {{formula:15625a3f-f134-4426-9574-3c819971c94a}} , this is known as the minimum bisection problem, which is {{formula:1c6a09ac-28f6-429d-bf47-3aa698348208}} -hard even on 3-regular graphs {{cite:ef3d78d3aa03060c51038735bcae89731a5581ba}}.
Monotonicity and Submodularity
A real-valued set function {{formula:43d5f119-d301-452d-8607-7e63fb5c9600}} is called monotone if for any two sets {{formula:14b180d1-f47b-4934-ad16-f037c7085740}} and {{formula:b339d8a8-330d-4559-875e-4a005f50e1fd}} with {{formula:88deb7e8-621f-4ddf-be54-2874db9ece51}} , we have {{formula:e06bf523-e95a-4f8a-8d44-2da86b4a1c41}} . Further, {{formula:57be55de-52fd-4013-af88-83b6ebbfc2ca}} is called submodular if for all sets {{formula:aa0a1c9f-26a7-449f-a98c-a7aece6dd583}} and {{formula:0cc30730-204e-4239-b8f2-59ae15b93546}} we have
{{formula:1b776840-5279-4a64-934f-2d78907e9545}}
Monotone submodular functions, even if hard to maximize, admit efficient approximations. Here we complement the hardness of thm:positivebiasnphard
by showing that {{formula:82ce80f9-8a8c-4d7a-ac01-d868eaa1413d}} is monotone and submodular.
Although submodularity is well-known for some unidirectional invasion processes {{cite:187050db6d41080e9eb5b3171bd98e114e926ac5}}, recall that our setting is bidirectional, i.e., the mutant set may grow or shrink in any step.
This bidirectionality renders submodularity a non-trivial property.
Node duplication.
Towards establishing monotonicity and submodularity, it is convenient to view the process in a slightly different but equivalent way, via node duplication. Consider the voter process {{formula:1e1543a8-e9fd-46d4-9c84-67e998846312}} on a graph {{formula:2388197f-0190-47e2-8454-cb553a471d26}} with bias {{formula:6b30db57-98c7-40fa-baeb-04517fa0447a}} . The node duplication view of {{formula:3fa4320d-7b41-4a0d-b87b-6e9e66601d30}} is another stochastic process {{formula:732ad04c-ecbc-4614-9264-94c9c2055693}} on {{formula:ad7ccfda-d9f6-46fd-b266-578eb19a79bc}} , defined as follows. Let {{formula:8d44a6b0-9d1b-478e-8f00-f21cd024d7ff}} be the current configuration.
A node {{formula:853f06bb-3ab7-47b4-8095-25d4da73dfc0}} is chosen for death uniformly at random (i.e., this step is identical to {{formula:ff969c35-45cd-467d-afab-050e8ae77d7f}} ).
We modify {{formula:cad108a7-2e40-49d8-8c36-4e7e40753aea}} to {{formula:efd0bea0-a18f-4846-81b1-18a673023fc5}} as follows. For every node {{formula:ecc94cb3-bb7b-4803-9024-d447307dd279}} , we create a duplicate {{formula:a2312da2-025b-428f-a8c1-a2cef8dcd16d}} of {{formula:86924df7-835b-44bb-b657-caf3a5e014e0}} , and insert an edge {{formula:8d1adbfd-e464-4d45-bf1b-b4e4e91e2e95}} in {{formula:1edb24e5-e8e5-4248-9cd2-c5aadc78c5e2}} with weight {{formula:0418aa1b-74b4-429d-9f39-83ccf7d9e2cb}} . We associate {{formula:b41760eb-f078-4894-b1fd-d2256c94b98b}} with fitness {{formula:db25420b-0843-4dcb-9d3a-480425eda54f}} , while every other node {{formula:98f72d23-12a3-465c-9c6c-fe3280f3454e}} gets fitness {{formula:653e00c9-4059-4c37-89a0-f283152f12ae}} . Finally, we execute a stochastic birth step as in the standard voter process, i.e., we choose a node {{formula:97ce5dbf-02bf-4572-938b-162c9fe0fdfb}} to propagate to {{formula:c2e2c4ea-8b6a-4d08-92be-5ac48631be7f}} with probability
{{formula:7f7b154f-16d1-4deb-a065-37ff354aff18}}
The new configuration {{formula:5be39338-5f4c-413a-a026-196afdb5175a}} contains {{formula:a0fc2e45-1df3-475d-bf96-82337c91d810}} if either some mutant node {{formula:bc333524-ff29-4a67-9d39-7c4795882aa8}} or its duplicate {{formula:b51c9e67-0277-4d98-ba17-3c47e32b3290}} was chosen for reproduction. It is straightforward to see that this modified birth step preserves the probability distribution of random configurations, and thus {{formula:3472fa4f-a287-4417-b7f7-729cbffd22e9}} and {{formula:d6973653-c078-4cab-9dad-48dad2612f29}} are equivalent.
Monotonicity.
Using the equivalence between the voter process and its variant with node duplication, we now establish the monotonicity of the fixation probability.
lemmalemmonotonicity
For any graph {{formula:378bf48b-b4fe-45e4-9ca8-8c9c4eb43d60}} and bias {{formula:cf27d97b-82ce-4e85-8f94-20ebf1938b32}} , for any two seed sets {{formula:16dc5d4d-5d4a-45a5-ac8f-2e8ab4b1c992}} and {{formula:54749171-c132-427e-9051-5756d32c1f49}} with {{formula:cfb7b0d9-4037-42bd-a96c-24627ad24f9e}} , we have {{formula:92c1464e-1399-42a4-be6e-0ce05701eece}} .
{{formula:f016da54-6387-481f-9b47-e36c7a0ce633}}
.
First assume {{formula:234ccd8a-e07b-41df-93e8-249cbd71d16e}} . Consider two processes {{formula:f4c53189-1ab1-4ba0-8b9e-9b06bc3befac}} and {{formula:8a264f25-d278-4100-a219-419e3de39a89}} starting from {{formula:2c88567a-00dc-4071-93e9-1971fec96be3}} and {{formula:2095ad7e-016b-4c27-92e2-cc7416fdd888}} , respectively. We establish a coupling between {{formula:d3212dc6-e122-4c65-b407-a04290f9ff5f}} and {{formula:65152a67-4faa-4de1-8a02-0e18b36fb190}} that satisfies {{formula:0264c7d3-0066-4cd8-ba9b-070ef42454c4}} for all {{formula:07d8ba3d-21e3-4792-8ba8-ccfa85999110}} , from which the lemma follows.
In each step, we first choose the same node {{formula:202cda85-adff-43ae-9d3e-9bfca98729c1}} for death in {{formula:c02053c7-fc6a-4584-abab-52f0ba83d78d}} and {{formula:d3146924-6ee8-4e91-bfb8-de53b80db0e0}} with uniform probability.
Then, we execute a birth step in {{formula:c3fee564-1eaf-4a23-8652-47588e7d98aa}} .
If the reproducing node {{formula:20559b19-922c-4c48-a67d-0dedcf64dcac}} is present in {{formula:711583a4-2d5b-4f75-8d00-778769513165}} , we perform the same update in {{formula:184d2564-13aa-4acb-af55-81a35bcb3339}} .
Otherwise, if {{formula:6c09ed9f-1a4e-47b4-b1f8-ac2c6d242319}} is a mutant duplicate absent from {{formula:ea38912c-1a87-4163-b128-cc285087c3b7}} , we perform an independent birth step in {{formula:c30fa561-3cc7-45fe-b961-948d54dec656}} .
This coupling maintains the invariant {{formula:82788833-0682-465d-bf8e-9a4134f616e4}} , as whenever {{formula:29e8895f-e07b-4c6a-bf7b-83b0150efe64}} takes an independent step, a mutant has reproduced in {{formula:a7b380ce-ebd2-4a08-a8d6-9e55cc9cc8f4}} . Moreover, the coupling is indeed transparent to {{formula:fa7b203b-0b73-4873-aa48-674a45f2fcf2}} and {{formula:09d7d68d-cedf-4e65-bdbd-cf12d50a14c6}} , as every birth event occurs with probability proportional to its agent's fitness.
The monotonicity for all {{formula:d6b031bf-456c-4c13-970f-9baaefb5c0cc}} follows by symmetry, as when {{formula:42c8164a-dd18-40ba-a8f6-f6cc7412c6a2}} we can exchange the roles of mutants and residents. Using the above argument, it follows that the fixation probability of the residents grows monotonically in {{formula:11bf874c-5ba2-4c9e-9c44-b255744ab81c}} and, in reverse, decreases monotonically as {{formula:f72fb80c-e93c-42cc-a502-876b655b4d75}} shrinks, hence the fixation probability of the mutants grows monotonically in {{formula:3cfdc304-033c-4587-b116-4c04ef8a65f8}} .
Submodularity.
We now show submodularity, i.e., we have
{{formula:89bff765-38a1-4dd7-a68e-7fd31bbe1eb2}}
To this end, it is convenient to consider the following more refined view of the invasion process. Given an initial seed set {{formula:70c96e00-da81-47fc-89b4-fc215313d44e}} , we keep track not only of the mutant set, but also the subset of mutants that are copies of the agents initially in {{formula:a0205c32-2710-44b8-abed-62bc8c4beb43}} . Consider this refined view, and execute the invasion beyond the point where mutants fixate. With probability 1, we eventually reach a configuration in which all mutants are copies of either {{formula:0d7db202-fc3d-4481-876a-f061636d15d5}} or {{formula:c155abbd-cde5-4d96-85b7-0bf8b8c28324}} . In this case, we say that {{formula:6fc510ac-0ddc-4f56-a80b-efc7273ca45a}} (resp., {{formula:1fb58cc4-6a04-4b01-9f60-1bcf4ded3511}} ) has fixated; with a small abuse of notation, we write {{formula:79342500-06b4-45fe-8083-bff1c8ebe31c}} (resp., {{formula:959473e6-727d-4957-a699-8c3bf98b7f04}} ) for the probability of that event. Thus {{formula:d6459e7d-6c88-4b72-989b-91811c523991}} , and it follows that
{{formula:1c573789-6027-4ad5-ae00-44692059f2bf}}
as a fixation of either {{formula:873c0514-a457-4523-8e56-3b72ac849e09}} or {{formula:c201e96e-fdc3-415d-b77e-904d1068ed43}} implies that of {{formula:cf9f5445-81c3-4456-9b9b-d5ba9e7e532d}} and, in reverse, a fixation of {{formula:59546142-49aa-4e9a-b6b6-ffc687f2f7c9}} eventually leads to that of either {{formula:9f0ec15d-3b88-4dc9-b21a-c798fdb1b4b5}} or {{formula:7dbd5ae3-29f2-4a7f-97ef-721f6e4a6eca}} . eq:additive is instrumental in proving submodularity.
lemmalemsubmodular
For any graph {{formula:dae744ed-738d-483e-a05e-245b265c8f97}} and bias {{formula:83da722b-eda0-47fe-91fe-994f4f044e34}} , the fixation probability {{formula:dd633709-39db-4cf6-9121-7b2cb8f95eba}} is submodular.
{{formula:08b65da8-e6de-478b-8bd8-e3111aee7ba8}}
.
Consider any two sets {{formula:46e428e5-4870-408a-8b67-d9feab002a3d}} . Let {{formula:1d185e83-d118-4b27-a3c5-4fe96e15d48f}} , {{formula:9388be8f-3632-4ff1-a2da-ed372c93ce7a}} , {{formula:dbaf6e24-ae35-4334-8186-9bed715cd587}} and {{formula:cbcfcd9d-0455-4ed0-adc3-ca2d9d452c4e}} be the four invasion processes, under node duplication, that correspond to initial seeds {{formula:df2ec152-9b93-4e04-bc23-fe5e2edd2038}} , {{formula:29f69d3b-d8d6-49e1-9e11-3cc63cbab01c}} , {{formula:6d8104e1-d9c9-48d6-a853-6d7980c2c9a2}} and {{formula:56b17a98-2d1d-4f95-af48-459df827be7f}} , respectively. For {{formula:30405fd4-6675-4505-86d3-2de7f47d8498}} , let {{formula:9c20e05c-39d2-4cb4-817d-29a2fd9e6f0f}} be the {{formula:5184af2a-f5d8-4f28-b40e-9ca59741b9bf}} -th random configuration of {{formula:1f08d12a-6588-4938-8c8f-6ccd88c2a64c}} . Moreover, let {{formula:fb0f2c7c-96b4-41ae-8d2a-be7cfcfd7b0f}} (resp., {{formula:6a99e58f-1f23-43d0-9c97-9d000e1830e6}} ) be the set of nodes of {{formula:d65d529e-7134-418f-96ac-cc21a17ba8d2}} in step {{formula:b25b82fb-6ab0-4b12-aeb4-1b6e341433f7}} that are occupied by mutant agents who are descendants of some agent initially in {{formula:f5b9cb96-2823-468c-8423-98504a8ba6ec}} (resp., {{formula:c638fd2a-9992-4461-93da-296ee609f6bb}} ). We establish a four-way coupling between all {{formula:515dad2d-d6f8-477f-8bc0-aec4816099c7}} that preserves the following invariants:
{{formula:98dea3c7-7dc6-42a9-a4e8-cf409f82e198}}
Due to eq:additive, invariants (i) and (ii) guarantee that any fixating run in {{formula:74c99e53-1328-45a0-becf-4c7038487db4}} is also fixating in at least one of {{formula:601bf275-7efa-4d32-8834-9f01ecab35e4}} and {{formula:d095140e-56e1-43b2-abd6-3d1343daf09a}} with at least as large probability. Invariant (iii) guarantees that any fixating run in {{formula:b1728e6f-ada0-4d3f-9e87-3697923be1f5}} is fixating in both {{formula:eda57d1c-be62-4ef1-a468-54c6ba88b210}} and {{formula:fd8a3c29-2bc8-4a1d-a7af-d07e3e4cfec0}} , each of the latter having at least as large probability. Hence, the three invariants imply eq:submodularityfp, i.e., submodularity.
We now establish the coupling. In each step {{formula:38bb315e-c041-443b-8d7e-a2b150af435a}} , we choose the same node {{formula:df1604cf-be38-48d8-9ab9-a6a94befce44}} in all {{formula:1955c27d-076f-4352-aa1c-63e3018952ed}} to die uniformly at random. Then, as long as {{formula:2e79476e-331b-4edb-9973-536d80225939}} we perform the following steps.
Choose a node {{formula:916abeea-bfa3-4570-9d3f-0f54fc16bf59}} to reproduce in {{formula:7f0b1ecf-bad7-4418-bda4-2cbeb4791d8b}} using node duplication (hence {{formula:8a2a5d2e-e605-4876-b27e-b2d04f15ad45}} is either an original or a duplicate node). For each process {{formula:b6ad4404-50d5-4cc7-91fa-7ee5bc77304a}} in which {{formula:3c06ed6f-acc7-45ff-891f-916c6451e1a5}} is a node (either original or duplicate), propagate the agent from {{formula:fe40b77c-f2c7-469e-8bfb-7366eab7abbc}} to {{formula:2293d7e7-b7a8-411f-b3e4-2c515806383f}} . Note that this step updates {{formula:757110fc-5deb-4b1a-a039-f475a84c80c3}} and at least one of {{formula:be3347bd-9e4f-4b01-88d3-80f964c0478c}} and {{formula:4115a14d-0e54-4f93-8baa-0c9f8b701be6}} , while if it updates {{formula:ecbcf076-9c40-4eba-84de-919dcfb259e1}} , it also updates both {{formula:c43fb59f-91e7-4e50-85be-f0febf1b9504}} and {{formula:faa251bd-bc82-4e0e-8054-fbc195eba9b1}} .
If {{formula:c0838874-ffea-424d-ab1f-496f88101cd3}} is not a node in one of {{formula:145e6913-8951-4319-aecd-7a8b861bf30b}} or {{formula:55476179-551f-4b7c-a886-bb83127a7366}} , then perform an independent birth step in that process, choosing some node {{formula:2fcbee5d-60a8-4754-a599-aeb3780b7f3a}} (either original or duplicate). If {{formula:cac7b4ba-c022-455d-bc32-731c228757e6}} is present in {{formula:595a10de-f0b9-4b1b-a365-dab00085efdc}} , apply the birth event in {{formula:778cbb33-012b-450f-b63f-a88e106e2eb8}} as well, otherwise apply an independent birth step in {{formula:2cb22c27-305e-430a-8d32-be1ecdbaf920}} .
If {{formula:19751980-ff68-4d48-a296-98f58c7adb26}} or {{formula:950b8be6-d5e4-4e23-ac27-acb1df5a8658}} , we perform step 2 only in the process not yet terminated.
As in lem:monotonicity, the coupling is indeed transparent to each {{formula:e8f276cb-c1d6-4a26-9933-ae5494c3d835}} and maintains invariants (i)–(iii).
Conclusively, monotonicity and submodularity lead to the following approximation guarantee {{cite:a711ec03b871ac396baa1b082f4db138ca66f176}}.
theoremthmpositivebiasapprox
Given a graph {{formula:ebd6729d-f598-4a7b-9ce3-8e310955822a}} and integer {{formula:f355b6ba-3597-4dc1-b95e-a8df9af07bc6}} , let {{formula:e528bd8a-663a-460a-9e0b-0acd7c777940}} be the seed set that maximizes {{formula:c66d34b8-87a4-4730-88d0-8955a13def2f}} ,
and {{formula:90c4d87d-042c-4551-bee8-c3e615455afd}} the seed set constructed by a greedy algorithm opting for maximal returns in each step. Then {{formula:f22315a2-1b0d-4da4-a894-c4db21e3af49}} .
{{figure:53a35362-f322-4396-a7e9-5eb30ed831f9}}Invasion with Infavorable Bias
We now turn our attention to disadvantageous invasions, i.e., {{formula:782bbaf5-c85b-4f4a-b687-e159c8548ce1}} , and show that the problem is intractable.
To establish our result we focus on the limit {{formula:1e28702c-3bc4-4d55-aee3-91efbd6363f6}} , i.e., the process is “infinitely” biased against invasion,
and write {{formula:adba17f5-06cb-44f4-ab38-83e8d3ced613}} .
Our key insight is the following lemma.
lemmalemnegativebiasvc
Let {{formula:a391110f-49d9-4685-a07b-b82c6c702f90}} be an undirected graph on {{formula:ca61bc66-94d0-4c30-a638-33711b76583e}} nodes and {{formula:44bea85c-c0d3-4ed9-8c90-09f1df67ad6b}} a seed set.
If {{formula:1a76c70f-71ea-40e9-bffe-77b987094f2c}} is a vertex cover on {{formula:8a0abfac-f05e-4e68-b5ae-7b203b47e604}} then {{formula:69568eae-eab3-4c94-a03f-e211861fd0e1}} , otherwise {{formula:3a5b7be8-17fb-4ae1-8802-624bd9430f44}} .
{{formula:173fb01c-2ff2-4aad-9a69-dcfc90d9e324}}
.
First, suppose {{formula:61cb9d87-26af-4f52-b595-6b13aa2b80a7}} is a vertex cover.
Denote the nodes in {{formula:332be3de-fad4-4c02-bfb7-788aec55e210}} by {{formula:dca04fbc-b1f1-46bb-94b3-d72b1483ebe0}} .
With probability {{formula:59922a74-f17a-4ecb-9088-aa70fe1e93eb}} ,
in the next {{formula:ace29046-73ad-4a2d-a089-de140ef4af77}} steps the agents at nodes {{formula:d2d1cbb1-1a74-41f2-a528-415f820c45d2}}
are selected for death (in this order).
Since {{formula:234ef72e-d807-419c-9996-8b283b81c31e}} is a vertex cover,
each of them adopts the mutant trait, and mutants fixate.
Second, suppose {{formula:cb70a3b5-ddb2-48c7-ae31-d60813298f3d}} is not a vertex cover, thus we have an edge {{formula:d3ebb76d-5bb5-4b14-8db3-52e2a709421e}} with {{formula:642f7910-658c-4ef6-bb83-400b0664ca99}} .
The idea is that, in the limit {{formula:7447d125-bc1e-43b8-a611-a8b01ebe1294}} , the agents on {{formula:3f10ced8-e012-4c0f-8cd8-874685d5175b}} and {{formula:5f0ca017-e21e-466a-8bcf-1430e3626688}} are
overwhelmingly more likely to spread to the rest of the graph than to ever become mutants.
Consider a node with {{formula:b9b69f97-4adf-4a1e-a2b8-de0297caf4c8}} mutant and {{formula:0c7b0ae6-fc3e-4e07-b9f1-171888de08ce}} resident agents among its neighbors.
Note that if an agent at such a node dies,
the node gets occupied by a resident agent with probability
{{formula:bfb15745-e7b9-462b-9167-a362550b112d}}
and by a mutant agent with probability
{{formula:cca5e438-e297-4008-8add-f262d127c477}} .
Since {{formula:bbfc5533-6eed-4c54-9e9b-32dc326dd47b}} is connected, there exists an ordering {{formula:f672ac40-c822-43b2-b049-cda39909d497}} of the nodes in {{formula:b1601e9b-2e28-4ad2-80ba-1b0c6e0a2f96}}
such that node {{formula:275d6877-8bd6-4601-8b1a-b65a2659ce01}} has a neighbor among {{formula:4a680bf2-3c62-4201-aed3-340ed7e7e482}} , for each {{formula:8fda5e8b-62eb-43f6-9ce2-1dd1790b9b4d}} .
Consider the first {{formula:80b5fe0d-4337-4ab8-b8a9-cbaede76ca99}} steps.
With probability {{formula:a310d6ef-316c-4b02-9a1e-f4c98894c9bf}} ,
agents at nodes {{formula:68582f61-b27f-4d1f-9b83-e7eec6767d8f}} die (in this order) and all become residents,
thus mutant extinction occurs.
On the other hand, by Union Bound, within this time-frame, one of {{formula:38ca1703-f5d5-4cc1-ab49-d85d94b3598d}} , {{formula:b0630650-9267-40cc-ac5b-80ad22cea812}} becomes mutant with probability
{{formula:61492b59-078b-455c-9cb7-5a02c0c6cc9b}} .
If neither event occurs, in the next {{formula:29cecfdd-efc0-41a0-8d2e-0adc8d33c455}} steps the situation repeats.
Thus, mutants fixate with probability at most {{formula:8225adfc-a53d-444c-9e24-eb3c01a7a60c}} .
Since vertex cover is known to be hard even on regular graphs, lem:negativebiasvc implies the following theorem.
theoremthmnegativebiasnphard
Fixation maximization in the biased voter process with {{formula:3442321a-bca8-4fb1-8e7e-dd0d45abc5c5}} is {{formula:4e988849-e94f-40c6-8d3c-f1937c4d13d8}} -hard, even on regular undirected graphs.
Submodularity considerations.
lem:negativebiasvc implies that {{formula:ebc9cdb6-3a98-4bab-9712-6a40f3a3f110}} cannot be approximated within any multiplicative factor in polynomial time,
unless {{formula:cc405a86-2fc2-4f9b-a41b-9449512e1e2b}} , since such an algorithm would decide whether {{formula:6b246225-f66d-42aa-9440-53cc82e05553}} is a vertex cover on {{formula:3decd5a2-0cbb-4a15-a766-31840eed8e8a}} .
That is so even though lem:submodular implies, by symmetrically exchanging mutant and resident roles, that the extinction probability {{formula:c4ef607d-d50e-4243-aeef-0aa8c08a343c}} for {{formula:4e790b41-0b77-403d-8cef-019f6005c80e}} is submodular.
Fixation maximization with {{formula:752188cc-2b0f-42a9-ab62-3a56b7418910}} is thus equivalent to minimizing a submodular function subject to the constraint {{formula:5ce26209-df90-4c5f-86b8-728f85b1c9a2}} . Although the minimization of submodular functions without constraints is polynomially solvable, simple cardinality constraints render it intractable {{cite:20bcbb2914ba54ed5c8c92942527b3d9ff8813aa}}.
Experiments
Here we present some selective case studies on the performance of various heuristics solving the fixation maximization problem for the biased voter process.
Our studies do not aim to be exhaustive, but rather offer some insights that may drive future work. We consider six heuristics for the choice of the seed set, namely
(i) high degree,
(ii) high temperature, defined as {{formula:44d941cb-0df6-4b90-b1f9-1c44c04c0139}} ,
(iii) high pagerank,
(iv) high eigenvalue centrality,
(v) high betweeness centrality,
(vi) high closeness centrality,
and the Greedy algorithm (thm:positivebiasapprox) that provides an approximation guarantee for {{formula:c2f9d716-a968-4379-ade6-d7b09e4f1e7f}} . We evaluate all methods on four networks from the Netzschleuder database {{cite:6ffdc8a698c57cf075e324db5e2867d2d25a46cc}}, chosen arbitrarily.
In each case, we choose a budget {{formula:1f8d3859-24fb-4917-98cf-f8e620f1d3ae}} equal to {{formula:6e945c0f-d782-49ea-aec7-ce87f4ef5cf8}} of the nodes of the graph.
fig:experiments shows the performance for various biases {{formula:a526c8ae-371a-4c29-9c45-eb2bd2b64040}} .
The networks Wiki and Polblogs are well-optimizable by almost all heuristics, possibly except high closeness centrality in the second case that performs visibly worse (though not by much).
The Twitter graph is more challenging, while the Elegans graph is the most difficult to optimize, with high variability on heuristic performance.
Unsurprisingly, the Greedy algorithm dominates the performance on this benchmark,
which is expected given its theoretical guarantees (thm:positivebiasapprox).
Among the heuristics, high pagerank appears to be the most consistently performant, and the first one to match the performance of Greedy on the Elegans network.
Conclusion
We have studied the optimization problem of selecting a cardinality-constrained seed set of mutants that maximizes the fixation probability in the classic biased voter process. We have shown that, in contrast to the neutral case {{formula:dc003a34-85d0-4761-bf48-69dfa3bee7ab}} which admits a polynomial-time algorithm {{cite:c66c2fc0e4ca25fef9c404846cca1d169c0dfb2e}}, the problem exhibits intricate complexity properties in the presence of biases {{formula:f9f49e30-fb4f-4d69-a44a-a4e4fcea4314}} . In particular, the problem is {{formula:cc3b127d-5585-4d24-b241-c883bfa6ac2f}} -hard in both regimes {{formula:4f9a3a11-0c69-420e-96d3-b9ae278da99d}} (bias in favor of the invasion) and {{formula:962a5bbd-1728-48bb-8838-684bb7bf3908}} (bias against the invasion), while the latter case is also hard to approximate. On the positive side, we have shown that the optimization function is monotone and submodular, and can thus be approximated efficiently within a factor {{formula:3e310fec-da69-4812-a693-0e43ba29f716}} .
Interestingly, our results on optimization hardness imply that even
just computing the fixation probability over randomly chosen seed sets is also {{formula:333c7727-d25e-4c21-8a72-bac42e3a87cc}} -hard.
Random seed sets are a frequent assumption in genetic/biological settings {{cite:017947946a8226869e05d8bf38a83defe506063b}}, {{cite:214f597a6e0dae9a9f77666d7170c9de6a504f7a}}, {{cite:811a285878d13672cba919e5140ede02b94733be}},
where novel mutations arise randomly (though we often start with {{formula:5c11b1f1-bd2f-4bad-977a-2ee535ce27c1}} random mutant).
Interesting future work includes investigating better approximation factors than {{formula:b2317681-ccbd-429d-b874-26f7a24e3a23}} for {{formula:e2a545bc-c631-4d69-a2c0-1fbf50850535}} , and an in-depth search for good heuristics.
Appendix
*
It remains to prove that:
When at a bad configuration, the next configuration is good
and the quantity does not change in expectation.
When at a good configuration, in one step the quantity increases by at least {{formula:c0bb176b-84d6-453b-ba93-736b25c57726}} in expectation.
Let {{formula:069dcce2-2b3f-4235-a285-c4e48fb43062}} be a current configuration.
For any edge {{formula:cbd98359-fcb1-447d-a3ea-a7fbda427fb7}} , let
{{formula:a96885ee-71b9-40cd-b0ef-e192d2de1238}}
be the probability that, in a single step, node {{formula:2c1bb69e-f8c6-4aec-b7a8-8ebbaa54cbcb}} reproduces onto node {{formula:dbabe621-aa4e-4cdb-a915-435a8485bd44}} .
We say that an edge {{formula:3a234c76-980b-4842-ad4f-0513a58ec0e2}} is nice if {{formula:5dbea4c9-2887-4368-b91a-5d225edf1729}} is occupied by a mutant and {{formula:940d6c32-4ed6-44bc-96a8-5e43306caec8}} by a resident.
Let {{formula:3ac10d48-11b5-4f71-9fc4-ae9d4b10e928}} be a (random) configuration after one step from {{formula:5a9589f3-c4ec-4963-846f-9bc5e44c71a6}} .
Then
{{formula:1e90865c-ef30-4ceb-8634-7c3b800f7cf0}}
where {{formula:4216c25f-18ec-4234-821f-5a8031e6a78d}} is the contribution of a single nice edge (summed over both directions).
Consider a fixed nice edge {{formula:f702ccc2-c7c1-4c06-b64f-4a9476c90893}} .
To prove the claims in points 1. and 2., it suffices to show that:
If all edges incident to {{formula:3e7f9758-aa3b-4123-93c0-3bea3de88025}} and {{formula:f942bdd7-a402-4161-a543-cb13d956c89b}} connect nodes of different types, then {{formula:c9db7ca5-0419-4ce9-833c-302d99931e86}} .
Otherwise, {{formula:f92c4118-1c6f-4828-a43f-654537089145}} , where {{formula:8930b9ee-45a2-4ac1-adaf-59f82ec2ca5b}} .
To prove (i), we simply rewrite
{{formula:34a2c30d-1090-479b-befc-a43a8e237371}}
To prove (ii), we distinguish 2 cases:
First, suppose that {{formula:54d96d6c-2ea0-46c3-91ce-2fc186d88511}} has at least one resident neighbor.
Then
{{formula:a8bf31c2-2305-4fe5-88ed-b726e8bdd25a}}
Similarly, if {{formula:46b6df91-affc-4cd7-9c40-02b167a93521}} has at least one mutant neighbor then
{{formula:65078238-ba66-4e80-a676-30345cc816b9}}
| i | 2fac653b720e38f94be501860611a156 |
After the trial runs, the results show that the overall mAP for all of the models and settings is between 68% and 81%. Table REF shows the complete results of the experimentation process. The overall best classification was the ResNet 50 architecture for all of the settings of the superpixel masks. The three variations of the residual networks (ResNet) used in the experimentation were among the four top performing models in the four different settings. These results are a
good indication that the optimisation achieved by the inclusion of residual methods learning with shortcut connections {{cite:b93f0c7d73d11d1c2f6aa16867cd122301248b8a}} has a positive effect in the overall task of weather classification, and it can benefit slightly from the use of superpixel masks as data augmentation. The results of this contribution are consistent with {{cite:b93f0c7d73d11d1c2f6aa16867cd122301248b8a}}, paper in which Residual Networks are described as the winners of the 2015 ILSVRC competition, having a smaller error than GoogLeNet and the VGG networks. Residual Networks are thus not only successful in arbitrary image recognition, but can also work well as a solution to weather classification problem. Another notable fact that is evident with the results shown in Table REF is increase in performance between the experimental setting involving raw images and the ones involving superpixel masks for all models except for PlacesCNN model and CaffeNet model.
{{figure:fd331e6e-4bfa-4f1e-89d5-2e02181b4dd2}}{{figure:a7a19cf2-8192-40b5-81ff-a9e7d21929e2}}{{figure:a81a8769-aaab-4756-84dc-aca649481e0b}}{{figure:1c56776c-08e3-4955-a79b-518c3fc68ee3}}{{figure:3a2b9ac2-084a-4121-a8a0-f997108ce029}} | r | 3f911bce4f9aff73e734cb2fc5772d75 |
The architecture of our model that maps the fingerprint to its location consists of a CNN inspired by AlexNet {{cite:4ed3e16c2c93e15274c30234411aa72a6671fde1}} and ConFi {{cite:a9511dff4b03362b9ea8a2cc7aeccb2a9e641dc8}} (see Fig. REF and Tab. REF ). The fingerprint – a CSI-image – is fed to the model which performs a regression on the (x,y) coordinates of the RP where the fingerprint was recorded. Note that as opposed to {{cite:a9511dff4b03362b9ea8a2cc7aeccb2a9e641dc8}}, we do not exploit multiple subsequent measurements to construct an image. Instead, each CSI measurement consists of 4 complex vectors (one for each port) of 39 subcarriers each. Hence, the CSI-image consists of a 39x4 image with 2 channels. The first channel is the amplitude, and the second channel is the phase. Each of the four columns represents a port-specific CSI-measurement.
{{figure:186a428b-0b23-4596-81ed-6c257dab98c9}}{{table:aeb2961b-834a-46da-a21c-faf3b04cd28b}} | m | 5c60130fefc45daca25a317dd503c0f8 |
In the context of gradient flows for diffusive systems, recent progress has been made in developing energy-stable methods through a variable transformation representing the energy, generally a nonlinear function of the state, as a quadratic form—such methods are referred to as invariant energy quadratisation (IEQ) approaches {{cite:29f5bf80c7725faf6daed47e168a6f1e800227fc}}, {{cite:e05f426d5c1b28bc113f98244e8819ec59a729d4}}. The main result is that it is possible to arrive at updating equations for a time-stepping method that are linearly implicit—so that the update depends only on the solution of a linear system, rather than the solution of a system of nonlinear algebraic equations. This allows for the sidestepping of the many difficulties associated with iterative solvers, and reduced computational cost. Alongside the more recently introduced scalar auxiliary variable (SAV) approaches {{cite:64e7bed0e6627516fdea28224a8fdefcaa5b9306}}, such techniques have been applied to a wide variety of problems {{cite:0682b3289d929b9567e41d8284e7df5420092d79}}. More recently, IEQ/SAV approaches have been applied to Hamiltonian systems, as in the case of diagonally-implicit Runge Kutta methods {{cite:a198c0a8df882ef923a7467795af6a4e77df187f}}. For an interesting overview of the relationship between quadratisation techniques and linearly implicit schemes, see the recent article by Sato et al. {{cite:4549abeac0968f541820ff3385f161912d27b253}}.
| i | 8978131cf0a939eac5f0f76ff9dbce4f |
Similarly, we train and evaluate a CNN based network TIRG {{cite:71f82e9b8979d9b959e495c7f3f0c3aa38de0873}} on GCD. It consists of resnet and LSTM backbone for feature extraction and {{formula:c5eb2827-5329-4fde-a411-1f31156dabce}} layer for feature aggregation. Herer {{formula:618e1aeb-f107-4765-b5a1-e75ab530fa01}} is computed by resnet and {{formula:3ae9cad7-f6c8-4ab0-bdd8-bf49c341fcc8}} is computed by LSTM. The results are listed in Table REF (first column). For CNN based network, we finally trained and evaluated integrate the CLIP and USE features with {{formula:bb13946d-0799-4537-9440-89e125940814}} layer for feature aggregation {{cite:1f3099b8b67f9819991572c33f1d7a429d22257b}}, {{cite:71f82e9b8979d9b959e495c7f3f0c3aa38de0873}} as shown in Table REF (third column). It improves the classification accuracy from 76.04% to 79.14% as shown in Table REF .
Hence, on average the proposed {{formula:9e9804f8-ff05-449f-9040-c0b5fca4bfc1}} shows the best accuracy amongst models trained on GCD.
{{figure:fe773c13-a67f-4551-84f1-6ca455577436}} | r | b76b8cd30717f2adc41ac12dfdd524de |
In this section, we present two technical lemmas which capture the
essence of the proof technique in {{cite:6c56b3cbfcc525a05930529d41f49ec667b78f21}}.
These lemmas are used in the proofs of Theorems REF
and REF .
Though the present author discovered these two lemmas independently,
it appears that Lemma REF is to be found in {{cite:f1b45fe0c72aa7c67c4a9633d9f2ed517cce7226}}.
In turn those authors were apparently unaware that their proof technique
resembled that in {{cite:6c56b3cbfcc525a05930529d41f49ec667b78f21}}.
The reference {{cite:f1b45fe0c72aa7c67c4a9633d9f2ed517cce7226}} is difficult to locate,
but the relevant result, analogous
to the present Lemma REF , is
restated as {{cite:e4799f0d77dd5f9eeb4b26b1c99a2ef30e37e4ee}}.The author thanks
Prof. Boris Polyak for the reference {{cite:e4799f0d77dd5f9eeb4b26b1c99a2ef30e37e4ee}}, and
Prof. Barbara Franci for supplying a copy of {{cite:f1b45fe0c72aa7c67c4a9633d9f2ed517cce7226}}.
| m | f7e4bcb09c06d96a0c4eac90ce291b71 |
While DL models are able to make predictions with high accuracy, their deterministic nature prevents them from providing uncertainty estimates for their predictions. Wildfire occurrence, however, presents a high degree of stochasticity, therefore uncertainty assessment is critical in decision making. Hence, Bayesian ML, DL {{cite:2108806e1de1180a0fa13269c85bd346ed086c7a}} approaches offer a promising route in that direction.
| d | 7b54f75307e9868edf0cb03435d55c79 |
Next, Fig. REF compares the rate performance of the proposed scheme to other counterparts under imperfect CSI, where we have {{formula:2b9c1634-dcae-4ba7-94ed-10ca7f05e7cb}} . In our simulation, we consider two benchmark schemes: 1) the AO method characterizing the near-optimal performance {{cite:3ab4c0b7d071f18c76dde4a8b6b70d31b966efd3}}, and 2) the random phase shift. In addition, three values of average pilot power are adopted for uplink training, i.e., {{formula:63913eec-2cdd-4336-aabf-ab506e6e5792}} dBm, {{formula:47af7251-e672-48ca-bad6-fb48e4e01666}} dBm, {{formula:7ce8d390-8019-4a7b-9b35-0c0008fe4f7d}} dBm. The DFT-based reflection coefficient configuration is invoked for improving the channel estimation performance {{cite:8479f2c45d48299465a127accbbee0892b5d12ea}}, {{cite:f4c483f522cf6f2ab62e42426b25f6f8cc4d955c}}. As can be seen from Fig. REF , the proposed scheme degenerates into the random phase shift method when {{formula:0efb7b69-d300-47f8-9ae0-79cf656a81ce}} . Furthermore, compared to the AO method, the proposed scheme has a moderate performance penalty under high pilot power. For example, about {{formula:ffec4b4b-fb79-4550-9d9e-7c8501ad98ec}} b/s/Hz rate erosion is observed when we adopt a fair training symbol of {{formula:dd2c403a-0134-4380-88ae-6c32b03af307}} and {{formula:7864afda-c052-4e43-a4ab-7bd5e2212a1e}} dBm. Nevertheless, for a moderate and low pilot power, the proposed scheme might outperform the AO algorithm. For example, given {{formula:e44fdf8c-9c98-46cf-8124-033aa43a60ee}} dBm, the proposed scheme outperforms the AO algorithm for {{formula:e797773e-1f80-4be8-9841-6b4e1435daaf}} . Note that the AO method still requires {{formula:29f78ee3-9c66-4563-bfb0-a7d3b8b782d2}} pilots to obtain such inaccurate CSI. Specifically, the estimation errors of the direct channel and all reflected channels will severely deteriorate the performance of the AO algorithm. By contrast, the estimation error of the composite channel has less impact on the performance of the proposed scheme, owing to the fact that the proposed scheme at least operates near its optimal performance even with selection bias. Therefore, the proposed scheme is more robust to channel estimation errors, as implicitly shown in Fig. REF .
{{figure:5f463f79-0512-4c34-9e9b-40b1db32b547}}{{figure:3cbb424e-d8d1-43ee-93bb-e683e38dd82b}} | r | fe9d901d959734dd93233bb5d477ff56 |
Full Fact's approach centred around using sentence embeddings as a feature engineering step, followed by a simple classifier such as logistic regression, which is what we used. They used Facebook's sentence embeddings, InferSent {{cite:efe832ed8d456afa7a7184cc49c6e860d6deaaec}}, which was a recent breakthrough at the time. Such is the speed of new development in the field that since then, several papers describing textual embeddings have been published. Due to the fact that we had already evaluated embeddings for clustering, and therefore knew our system would rely on Google USE Large {{cite:73bd7aac04d6d1d2c09f2ab8ef1ce818e9c74754}}, we decided to use this instead. We compared this to TFIDF and Full Fact's results as baselines. The results are displayed in Table REF .
| m | 24a7d7f31c9cd0b29e03f1a17a2962ac |
Spin dynamics {{cite:3c02bc7f1edc29dae8ad4585ade3d25f889fa23c}} are also regularly used in network clustering. The first step is to define a spin model on the network, consisting of
a set of spin variables {{formula:9df86946-9983-4b5f-9c80-a46f6692fdd8}} , assigned to the vertices and a Hamiltonian {{formula:d01a6d04-09a8-421d-8bb5-22f5429b77cb}} , expressing the energy of the
spin configuration {{formula:b46f0eaa-cf79-45eb-8320-0d3b194257dd}} . For community detection, spins are usually integers: {{formula:a69c779b-1dd0-493c-bbc0-860efcdb3bae}} .
Contributions to the energy are usually given by spin-spin interactions. The coupling of a spin-spin interaction can be ferromagnetic (negative)
or antiferromagnetic (positive), if the energy is lower when the spins are equal or not, respectively. The goal is to find those spin configurations
that minimise the Hamiltonian {{formula:e1e99a91-fbad-481f-b4d7-4311823c30c2}} . If couplings are all ferromagnetic, the minimum energy would be trivially obtained for the configurations where all vertices have identical spin
values. Instead, one would like to have identical spins for vertices of the same cluster, and different spins for vertices in different clusters, to identify the community structure.
Therefore, Hamiltonians feature both ferromagnetic and antiferromagnetic interactions [spin glass dynamics {{cite:be815a950ca46e5e95b823acdac7250100dab34a}}]. A popular model consists in rewarding
edges between vertices in the same cluster, as well as non-edges between
vertices in different clusters, and penalising edges between
vertices of different clusters, along with non-edges between
vertices in the same cluster. This way, if the edge density within communities is appreciably larger than the edge density between communities, as it often happens,
having equal spin values for vertices in the same cluster would considerably lower the energy of the configuration. On the other hand, to bring the energy further down
the spins of vertices in different clusters should be different, as many such vertices would be disjoint from each other, and such non-edges would increase the energy
of the system if the corresponding spin variables were equal.
A general expression for the Hamiltonian along these lines is {{cite:9e7d353e7c6b32ba1d4c43b569dcaedb4a09cc6e}}
{{formula:b247912a-849b-4925-8be7-fef6b3052390}}
| m | bb76b847446496190f955164fd00a63a |
Furthermore, in practice, the phase-shifting of each PS can only take finite discrete value from the discrete phase-shifting set {{cite:9e501e56cc169722289ed477d3cabeb2440301d4}}, {{cite:c671941b61e3bbacdf94b97bf3df55a60f7bfec5}}. Let {{formula:ba5f5703-192b-4f92-95cf-d18267c7fda4}} denote the number of bits to represent the resolution levels of IRS, the discrete phase-shifting set is represented by
{{formula:fed7198a-8052-4f2f-85e6-a2f0c8262dff}}
| d | 327144bca50c84b670ee0af9d4c2c369 |
Many deep learning methods have been proposed to generate time series, including Recurrent Neural Network (RNN) based methods {{cite:7d1025ee22be1458bc3c6445f9103bd1287f1d12}}, {{cite:887e9403060ddce12018470bba919fb2f2d6c1e7}} and block-style methods {{cite:5e45808bbf420d68597c7538d6b51293cd597158}}, {{cite:9c87481e407283496d385da45ce54029e5715d12}}.
For example,
Cao et al. {{cite:7d1025ee22be1458bc3c6445f9103bd1287f1d12}} introduce BRITS which leverages the recurrent dynamics for both correlated and uncorrelated multivariate time series.
Fortuin et al. {{cite:f536b82c07362a67fbac04a064b67d5054e26866}} combines the Gaussian process to capture the temporal dynamics and reconstruct missing values by VAE {{cite:c407629d450a0a38e0864833cc0a38f64b39ddb5}}.
Luo et al. {{cite:0a3099f36220e6f99b432d05af0a76474c97f046}} introduce GRUI and propose a two-stage GAN {{cite:b5a7491c739b254c7729b9c861097ac8df87c9e9}} model, the generator and discriminator of which are based on GRUI.
E2GAN {{cite:4d030714f91a41086e707c075b81b8ae6ea1927e}} further simplifies the generator by combining GRUI and denoising autoencoder such that the GAN-based imputation can be trained in an end-to-end manner.
Gamboa et al. {{cite:c65202b00da64d34686cbeaf1201e5b8c1340333}} explore different neural networks for time series analysis.
Oreshkin et al. {{cite:5e45808bbf420d68597c7538d6b51293cd597158}} introduce N-BEATS for explainable time series forecasting.
Li et al. {{cite:faa9c1f37ea11674ce195a5e4887c209bb7da7d9}} propose an enhanced version of Transformer {{cite:5de4cc2f6b9c5a2030c9ac9d7192c38b8e854dfa}} for forecasting.
Zhou et al. {{cite:9c87481e407283496d385da45ce54029e5715d12}} propose Informer for long time series forecasting.
Wu et al. {{cite:b51e387183351b67621d6a399ddf0738946d75c0}} introduce AutoFormer, which reduces the complexity of Transformer.
However, when the missing rate grows to a high level, the performance of these methods drops significantly.
ReTime address this issue by retrieving relevant reference time series as an augmentation.
| m | 458c4a193a3fd8c83ef047978d9b51ea |
During outburst, the RG is engulfed by the ejecta and the near-side is exposed to the radiation from the hot WD. The immediate effect is to increase the distance at which dust condensation is possible. {{cite:fd70b2492aa147451f8fe4049ecefef208547e15}} showed that the sublimation radius reaches a maximum on about day 70 of the outburst and then steadily declines until about day 250 (about the time of the 2006 observations presented here). Although we can envisage a region behind the RG where the dust is shielded from the outburst (the shadow cone), the orbital motion of the star eventually leaves this material exposed to the hard radiation of the source. This would occur in just a few weeks (assuming the RG parameters in {{formula:ef127a71-dea5-45b2-8cf4-68532585657a}} and the orbital parameters in {{cite:f8752f21328c47a1a9a5f9ed0de88398c91de1e6}}) and well before the X-ray emission has subsided.
| d | 08856ee5f1af102ab8542a0063f66c33 |
On the side of adiabatic quantum computing {{cite:fa5a012c7dd93418ec0443c46f8aca4154d8c597}}, despite the fact that it has been commercially available for more than a decade in the form of quantum annealers (QA) {{cite:8ff3d6b50bd7442f9d0d3495d8ce14b546bc0d3e}}, this approach has only recently been used for LGT calculations. These include the pioneering studies on the annealer for the case of {{formula:6cb2906f-249a-4bbd-8fa0-44ba37024cbc}} {{cite:01b47543768b0a493f0c699df23ce077d188aad7}} and {{formula:85935326-0f45-460a-b50a-54edcefdb6b2}} {{cite:6a7d2590c1adb3952c58147785f8bceb3852e053}}. In these formulations, the number of qubits necessary to digitize the theory under study scales with the size of the Hilbert space of the problem rather than the spatial volume of the system. Thus, this formulation does not show the expected quantum advantage present in universal quantum computing. On the other hand, systems in a QA architecture s.a. D-Wave's Advantage_system5.1 already comprise several thousand physical qubits {{cite:6d46707b7ca323603120f906ec1b6874d424b601}}. Even at this stage of hardware development, proof-of-principle quantum computations in LGT {{cite:01b47543768b0a493f0c699df23ce077d188aad7}}, {{cite:6a7d2590c1adb3952c58147785f8bceb3852e053}} or other field theories {{cite:e9dd17e2af05c2c01941b73e5b3530a86f5e8ef9}} are feasible. The intrinsic nature of the formulation of problems on the annealer simply requires the mapping of the lattice field theory onto an optimization problem represented by the Hamiltonian
{{formula:368ab2db-f02b-4223-ba56-297080233c67}}
| i | c72b0585f2c2a85cdc3e41e518db995e |
Radford et al., 2015 has shown a way to build high quality representations by training a GAN model and reusing parts of the generator and discriminator networks as feature extractors for other supervised tasks {{cite:c30331ee3c6c125ace295bd7819e7af5d584f76a}}. This can be potentially a promising future direction for this work.
| d | c40f02dc6ea4b553eabec949049ed77d |
The IRL model framework is illustrated in Fig. REF . Model training can be conceptualized as a Policy Generator (G) and a Discriminator (D) locked in an adversarial process {{cite:765e68a281e9adf415a90a60e7860a5be1d0f81e}}. The Generator generates fake eye movements (Actions) with the goal of fooling the Discriminator into believing that these actions were made by a person, while the Discriminator's goal is to discriminate the real eye movements from the fake. More specifically, the Generator consists of an Actor-Critic model {{cite:0a77dd05065a2ce095ea8d5dcd714abfd2d00b0b}}
that learns a policy for maximizing total expected reward over all possible sequences of fixations, with greater reward given to the Generator when it produces person-like actions that the Discriminator miss-classifies as real (the logarithm of the Discriminator output). This reward-driven adversarial process plays out during training using Proximal Policy Optimization (PPO) {{cite:3d5dd9af3eb0b3765e0a2335a159fe09ed209402}}, with the result being a Generator that becomes highly adept at imitating the behavioral fixations made during categorical search. At testing, this learned Policy for mimicking people's categorical search fixations is used to predict the fixation behavior of new people searching for the same target categories in new images. These fixation predictions are quantified by what we call a saccade map, which is a priority map reflecting the total reward expected if saccades were to land at all the different locations in an image input.
{{figure:d224f99b-2378-4abf-8da6-8664f25f448a}} | m | 6402f7a2061eda391d06e77a26f123b7 |
From Lemma REF , we know if we get an {{formula:4cb007f2-cefb-4158-958c-4125e6b68d0f}} -biased set of size {{formula:6c319d83-c18f-4e5d-86a2-38f424e7c93b}} , we can use it to recover {{formula:1021b376-77f0-47b4-9b14-cc57d4d32922}} , by making {{formula:c2b1c0c9-3406-4ad9-b4a6-14530a8446e5}} queries. Thus, the key technique for designing an efficient algorithm is to obtain such {{formula:6314d528-ade5-44f9-afe3-34db68a4d6bb}} -biased sets by making a small number of queries.
To address this problem,
we start with a simple disagreement counting method, which has been heavily used under the fully-random model {{cite:11f6344f36cf933284b78df27192558f90641b63}}, {{cite:99cbc22738d09cf0cacf69783314462229708d70}}, {{cite:baada7f01898960ea7ec0222cb9c343a88741b64}}. We consider the following simple procedure. Lemma REF gives the theoretical guarantee for this simple procedure.
| m | 5df17f6ccf5b9d41212f0f2658a1c10f |
Let {{formula:7db5527a-14ce-4450-bb33-114b566059bf}} be the vector bundle on {{formula:f1148b8b-938c-4c95-8700-829b72fa2697}} arising from the {{formula:90767d07-5337-46de-a405-c078ca9d35e2}} th step
of the Hodge filtration for the {{formula:dd3afc52-a7ba-4b76-9468-74cba11d46be}} th relative cohomology of {{formula:2108f620-d3c2-4d0d-96f5-9824cede3337}} . We take
{{formula:1cf8994e-4b24-48db-b9f1-ecddb9c7691b}}
and {{formula:597c5ea5-7c51-4636-818f-e80472376636}} , and the map {{formula:32f2c0ab-3223-4b91-aadd-8f9b5b1c71d1}} then
arises from the Gauss-Manin connection.
(For the Gauss-Manin connection, see {{cite:52e579e839fdc4c90a0511bfbc3997921eea7f53}} and {{cite:d9eaa7784585e89e4c4ce8b5faf643c4392e5196}}.)
{{formula:57722dda-0e18-4f5d-a09f-b8a1d5076150}}
Lemma 2.2
Suppose that {{formula:f046eedb-ef78-445f-b286-a398cdf8f408}} is a proper {{formula:d1b6a82d-1d9f-4443-a273-8b4574361f0b}} -variety equipped with ample line bundle {{formula:4a2e7f44-bc7a-4424-9cc1-8dbd39fdbb8a}} ,
and let {{formula:a252fed9-7fad-4c70-9109-39d815ca7796}} be finite, with {{formula:f4b91542-d7fd-4e0d-9e28-7f69520b33db}} .
Writing {{formula:b3338cb4-81f3-4594-9f2e-4f0d4781ca0b}} for the degree of {{formula:de59f3dc-65b5-4055-9d91-b8ee13d64702}} at the generic point, we have
{{formula:8e0ea7a1-2206-4a18-9b58-bd7195c35571}}
We will apply this only when {{formula:4f734912-02c2-45da-bd3d-737c5208f8ce}} is a variety, but the argument does not require that, taking
(REF ) as the definition of degree.
By the projection formula we have
{{formula:9e791b59-c2a4-4033-ae3b-025890fd159d}}
Now {{formula:42f087fa-adbd-40e4-acf0-687f38e1d417}} is isomorphic to {{formula:cb359162-3ce2-421f-a4a2-a345844bfef6}}
away from a set of positive codimension on {{formula:ed443184-6efa-4088-9d49-d6f433d8bab6}} . Therefore
{{formula:153bb702-bbef-48d0-8837-4f64c23bbff2}}
coincides with
{{formula:7fec420b-1a11-4060-9a70-9e2b2e3f00f0}}
for large {{formula:f37492c7-0437-41f0-979f-5ee00d3ed80b}} , up to terms {{formula:facdd9bc-47b7-4828-aeae-3ffaa5c1bd8b}} . We conclude using {{formula:75b43284-3879-41e4-94c3-c1534e318193}} .
{{formula:36b102f6-829d-4788-8e5c-7443d06df7fb}}
The next Lemma
is {{cite:6877a10c554980fc56a17306637f5a7f5924e173}}.
Lemma 2.3
Suppose that {{formula:e49067cc-b9b4-4614-9b35-595b98221a05}} is a irreducible normal noetherian scheme
and {{formula:cd30765f-04f8-4ae4-8190-6fdf91a88372}} a finite étale cover. Then the irreducible
components of {{formula:2d1de4b2-77ac-4b9b-9d6a-723c02ef269f}} are disjoint.
The following result asserts, essentially, “boundedness”
of the set of irreducible varieties in a fixed projective space and bounded degree.
Lemma 2.4
Let {{formula:d8f7104b-3661-40de-b00d-308de5380b92}} be a field of characteristic zero.
(a)
Suppose that {{formula:5d2abbf9-e4da-49c0-971e-d2106911d110}} is a closed subvariety (irreducible, reduced closed subscheme) of degree {{formula:42c34e67-a392-4a9b-8683-fa376956b8c8}} .
Then there are bounds, depending only on {{formula:a62521c6-c934-42f9-96f4-0e43499ed6e8}} (not on {{formula:ca73f3fb-6e25-428b-b67c-3e7ab1b09be5}} ) for each coefficient of the Hilbert polynomial of {{formula:c19344b0-31ce-4fd0-a8be-5d92fd955cb4}} ,
and in particular the homogeneous ideal {{formula:9646b0cf-1aef-43c4-83f9-af83a2020b0b}} of {{formula:8147cb3c-b9cb-43b5-8d13-5293de98269c}} is generated in degree {{formula:fbe6028e-fa93-4e4f-aed7-2edaf51bfcb6}} .
(b)
Suppose given integers {{formula:18103597-8fac-407e-91ee-abcfe7a15765}} . Then there exist bounds {{formula:a5559690-c557-4a72-990f-4f33ed613598}} and {{formula:97e38eb4-860d-45c2-a08b-ae7eda8a91fb}} with the following property:
Suppose {{formula:3f67512e-8a39-417c-a794-f5571353b5a0}} (with {{formula:53a727b4-e690-4af9-903a-924bc5022c65}} ) is a collection of closed subvarieties of {{formula:4c3de797-566d-4a47-b779-84564117c824}} ,
with each {{formula:0f0e4536-7b67-4c3f-b80e-4a1832c9b25e}} of dimension {{formula:370489bb-a4fa-4397-aaed-7128dc55890e}} and degree {{formula:526178cb-b019-44f8-a376-b12dc2330beb}} .
Let {{formula:35a4d7c1-32c8-4da7-9448-ce553c139f79}} be the intersection
{{formula:988e2790-1e2a-446c-9b0d-d26e4aa60044}}
Then the number of irreducible components of {{formula:d9d1beba-defd-43a2-8f9d-a6d57601d38e}} is at most {{formula:c30376e8-374e-43a3-a05e-ab56170d86ca}} ,
and the degree of each such component (endowed with the reduced scheme structure) is at most {{formula:3d395fff-ff53-4a78-b986-5b8ac8c0d92a}} .
Furthermore, the bounds {{formula:02c64e0e-f32b-4144-8bc8-856ab212b764}} and {{formula:26fc4197-fe2c-413f-9e2a-d2b1ab5e9e28}} are independent of the field {{formula:a9ca81df-3335-45ce-8b08-0a03010a45e9}} .
(c)
Suppose that {{formula:667731b7-defa-4aba-a5b4-e7f35a8a657e}} is a
closed variety of dimension {{formula:dc7ce3cc-64eb-46a8-8738-1e928b32733d}} and degree {{formula:0d690fd8-23ba-4a7f-bbcd-109aa9d07871}} which is not geometrically irreducible.
Then there exist finitely many subvarieties {{formula:ab993651-66b2-4eef-8d5f-72fb39731203}} , defined over {{formula:a4c26ecb-c8ec-4cd3-b1b9-d7ce4bf5fa23}} , such that:
Each {{formula:2a684451-0c1c-4e10-a919-d1de24f5f4d9}} is irreducible of dimension {{formula:177a7c3a-f5bf-433b-bab4-67381083a84d}} ,
{{formula:5dd73e26-092e-4905-8369-727a805b5b16}} , and
the number {{formula:4e7b1e7c-2eef-4c9c-a53c-ffe6913ee650}} of {{formula:d8a90f2f-6735-404c-b5e3-2c1aee122579}} 's, and degree of each {{formula:3f14e82e-65da-4573-8977-677f54507b49}} can be bounded in terms of {{formula:4132fd75-a39d-496e-9d4b-94bd7b399422}} (but independently of the field {{formula:aa8c60c0-3c1a-4505-aee7-7e8a181df6d6}} and the variety {{formula:79e38004-a3a5-4526-a94c-5ed601e21bde}} ).
(d)
Suppose that {{formula:5ba23850-59c9-417b-98f1-fc9e7052b269}} is a closed variety of dimension {{formula:826bc554-054b-4adb-81d1-1ae8a277ad7c}} and degree {{formula:3113ba97-5dfa-4572-84ae-934f3ed663db}} . Then
the set of points in {{formula:60fdb3f3-3df4-4b3c-b323-f38996d21c38}} which are singular on {{formula:bc9a90fb-8506-45c8-be00-cb5ad281aebc}} can again be covered by varieties {{formula:03e289f9-f84a-403a-96df-2b58d83d45f7}}
with the same properties as (c).
Example 2.5 As an example of “bad” examples for part (a): take
a {{formula:11e7fc44-3ee6-4378-8106-6aa2b7d490a0}} -dimensional variety, and adjoin to it a large set of disjoint points;
this modification does not affect the degree, and shows the need for irreducibility or at least
equidimensionality. Similarly, consideration of
embedded points shows that “reduced” is also important.
As an example of the situation in part (c), consider the plane curve defined by {{formula:d4f065eb-ad69-4340-896d-2de7d5f7d292}} .
This is irreducible over {{formula:ed8e01a5-3e7e-4bc5-a065-b871f03da0ac}} but not geometrically irreducible;
it only has one rational point {{formula:c2be6516-a1bf-4999-a5e4-c780b1e85f58}} , which is contained in a zero-dimensional, geometrically irreducible subvariety.
More generally, suppose {{formula:ff72ee43-0a20-4d31-b475-c9a1041111d7}} is a number field, and consider “the affine line over {{formula:50c70ed4-cfc3-4f1d-9323-d90af67d5dbc}} , with the origin reduced to a {{formula:a24b8f1b-76ab-4755-b2ac-8927d8c82ef1}} -point” – that is, {{formula:6889906f-c8bd-4fad-9646-bdf7d9d0c360}} .
If {{formula:c4122389-76cf-4a15-9939-20ebbe4bd93f}} , this scheme is irreducible, but geometrically reducible, and its only {{formula:a7d804ee-0934-429d-87a6-83e3f76cec54}} -point is the origin.
(Taking {{formula:6a0d63ee-3636-4ed8-94a8-47e5d3da025a}} recovers the original example.)
The first assertion of (a) follows from {{cite:49683f76af358a8813fd77c14800b4db779c5baf}}.
That assertion applies as formulated to “special positive cycles” over an algebraically closed field;
in our situation {{formula:19b1d025-0778-4dc7-ab7e-da896c8530df}}
is reduced equidimensional and its decomposition into irreducible components give the closed
subschemes appearing in loc. cit. Définition 6.9.
The consequence
on bounded generation of {{formula:96c3f18d-597c-4a39-b462-16ece2c26855}} follows, because such a bound on generation of
the defining ideal is valid in any finite type subscheme
of the Hilbert scheme.
For (b), we may as well consider one fixed {{formula:0fc1b687-81b0-458b-a82c-762b3ad60f45}} .
Let {{formula:7b432d43-ca00-46e4-8896-df3148ab4ca9}} be the finite set of polynomials arising from (a),
and let {{formula:3bb0fec4-2ebc-4657-be26-069c6f957fe2}} be the Hilbert scheme parametrizing closed subschemes of {{formula:4a9e80c6-fc91-4b45-9168-554934e0f415}}
with Hilbert polynomial in {{formula:a9f2e12f-1a6c-49ab-b413-5298809abf76}} . This is a finite-type {{formula:a8b9083e-34bc-4c38-a3c5-aee2c0d44ccc}} -scheme by (a).
Now tuples {{formula:a940804f-a498-4684-ad8c-20be279c0499}} are classified by suitable {{formula:bd80ed40-3320-4d1c-95af-903c8cc337c3}} -points of {{formula:e597d1fb-eabe-4351-abf1-809fca0404a0}} , which is again of finite type;
and the result now follows from standard results on families over a finite-type base.
Specifically, we have the universal schemes {{formula:68979625-4842-4c1f-b7af-13550141076f}} over {{formula:9a8d7d42-3a42-491f-b13c-106789bb400b}} ;
let {{formula:a7602944-a0b6-47dd-9569-58688927e190}} be their fiber product over {{formula:6873254e-a030-4e5a-85bc-a94ae8cf7f58}} ,
so fiberwise {{formula:5f93292f-3406-4a7e-ba19-3351f9ff5d01}} gives the intersection of the {{formula:e14f89f0-5042-41cd-acb7-60b011e8c4dc}} s (although
with a possibly non-reduced schematic structure). We will work by Noetherian induction.
Let {{formula:4d5c2337-be45-4574-bf78-3da40b36c047}} be the generic point of a closed irreducible subscheme {{formula:75319e6c-d03a-4a45-8041-67142d11a4bc}} .
We will show that there exists a relatively open subset {{formula:c79b9425-c2b3-48ee-86d7-629e8e3d8b2b}} (that is, open in {{formula:b7a6ee36-91c6-4bd5-af86-8bb789bcb5fe}} , but not necessarily in {{formula:f1ffa02f-3c4a-48b4-9a34-dac42c8a8d69}} )
such that the number, dimensions, and degrees of the irreducible components of fibers {{formula:cc7054ab-ccfb-446e-9b91-d4a5d36e0995}} , for {{formula:1f797f28-237e-409b-9870-1780b8084df4}} , are bounded.
Here, as in the statement, “degree” is taken with reference to the reduced scheme structure.
The number of irreducible components of any geometric fiber of {{formula:9634e384-eaef-4f4f-a927-f498e8c9461a}} is bounded by {{cite:f5f72ce3009bdfec9f5b167c329c67dd6b2011d3}}.
This bounds the number of geometric components, and so also the number of irreducible components,
of any {{formula:9e0b8c86-bafd-4eab-abdb-da21f18c3e0a}} as in (REF ).
Now we turn to the degree.
For each {{formula:6778a4b6-f9a5-45fc-a364-019e262aec23}} with {{formula:97287acd-b76c-41f7-90dd-f3719e4fb4cb}} ,
let {{formula:8a195e46-b6f7-4c5a-8433-a8c77db35708}} be the closure in {{formula:02cd6748-bd3a-42c2-b8a7-7455fc9b0cb0}} of the union of {{formula:de27a18b-9aaf-45d9-b175-9109b8a30bd8}} -dimensional components of {{formula:a9f97d0e-a414-4c42-a2a6-2371452fbd96}} (thought of, for now, merely as a closed subset of {{formula:3fa390fa-3530-43d9-aa44-1f3256109521}} in the Zariski topology; we will revisit the issue of scheme structure shortly.)
In particular, the fiber over {{formula:48a59137-8069-464f-b9cd-37227e512b22}} of each {{formula:ee60e20f-32a4-44b4-86ea-db3f184e1c69}} is equidimensional of dimension {{formula:a497bbc0-fcbc-474a-a605-10f41c6a5661}} .
By {{cite:f5f72ce3009bdfec9f5b167c329c67dd6b2011d3}}, we can restrict to an open {{formula:6eaa6d6b-0af4-4c4a-8caa-b7b2747c93dd}} , on which:
(i)
For each {{formula:a2fdd7a4-027f-4964-8c85-2b94b1d603f8}} , the fiber {{formula:8e5c4f85-921c-43ee-a1c7-100b5bf94af7}} is equidimensional of dimension {{formula:9b66d169-f9a5-47ae-ac07-2f3c7b9d5788}} ,
(ii)
For each {{formula:1ca2bd7e-2b53-48b3-8c39-975a4f9e039e}} , the fiber {{formula:25d916ac-028b-425a-a8b5-25e3b16d06ee}} is set-theoretically covered by the various {{formula:99d0ed29-27b1-47b6-8b5d-58d548716738}} , and
(iii)
For each {{formula:7484dbae-4e97-42ae-8cc9-bece8b765db4}} , and for all {{formula:ff737a1c-f893-4649-9866-59c6fd8c94f6}} , the intersection {{formula:e53c625b-d7ae-4ec4-8a9a-21217821a6ba}} has all components of dimension strictly less than {{formula:bda1aaff-8ce8-41d0-961b-6c2cc6fe78ce}} .
Take {{formula:d59fa124-3250-4f70-8dec-1125195d1571}} and let {{formula:63d9fb5b-406a-4640-b902-683b874e5ee9}} be any irreducible component of {{formula:f67999af-4f95-4f75-86cb-9544b7c0a49c}} , say of dimension {{formula:2ab34849-fdb5-4cea-88f4-f224318631c9}} ;
by (ii) it is contained in some irreducible component of some {{formula:cbd2e5ea-f986-4be9-a7f5-5c80cb4a8672}} .
Since {{formula:fa0209fc-7fa8-45a2-b304-f285607a90e4}} is maximal among irreducible subsets, we must have equality here, i.e. {{formula:57880441-7966-4d91-847a-974eeb0e722d}} coincides with an irreducible component of this {{formula:fe695fec-7f01-4ab5-9c81-d71e7e6d0332}} , and
since {{formula:d607e886-269a-4a0b-8d1b-8417621cad5e}} is equidimensional of dimension {{formula:5ecbc6a4-0288-468c-9a8b-01e4538dbb24}}
we must have {{formula:3c5e0dfa-152c-424f-916b-9b325e4ee11f}} .
Now let us endow {{formula:86128f40-fc45-488a-86ca-4ae2829fc45d}} (so far merely a closed set) with its reduced scheme structure.
But now by generic flatness ({{cite:da5684618c7d272362ad8d6ef9f01606f182b16e}}),
we further shrink {{formula:a07a1b89-c340-488b-8d55-e83a1c1d5413}} so that each {{formula:0c80a6b7-c00a-4d86-b104-1d12ecbce5ce}} is flat over {{formula:3491774e-2ddc-492b-b01f-a2142da15e89}} .
Then for each {{formula:f038408e-3022-4cad-a227-0fe4b12689bc}} , the degree of each {{formula:70e3f09d-2ba1-4376-ab29-761167beb3c8}} is independent of {{formula:3c1b1ae8-2c22-4667-9b59-eb9179bc2e35}} .
Now, the scheme structure on {{formula:760996b2-f2bd-4c10-9fb7-453d5d0093d6}} need not be reduced, but
nonetheless
there is an inequality of degrees {{formula:78194bfb-e562-42cf-92e1-41111c7cb88d}}
(where {{formula:92d75dc7-2b88-4b6c-bd3c-977f7676c811}} denotes the fiber taken with the reduced scheme structure)
and the degree of {{formula:5217fc20-3b4d-45bc-b147-a7a9da291978}} bounds from above
the degree of {{formula:ff0e1c00-969e-4182-80fe-a3e66badfbc5}} taken with its reduced scheme structure.
Now we turn to (c).
Consider the geometrically irreducible components {{formula:ca40f559-7d8e-46a5-9453-bf9cd87da337}} inside the base change {{formula:ae84f6a3-779a-4e6c-afbd-81da66e0ed42}} of {{formula:d5ec614a-fa45-47fe-b86e-876b06eba0c3}}
to an algebraic closure.
Note that {{formula:07f65197-61ec-412f-a957-036ed97668f1}} is bounded by the degree of {{formula:56cd5c5e-7614-4550-a35a-152690a4a8b0}} , which we have assumed bounded,
and similarly the degree of each {{formula:fac743a8-20de-41ba-a431-5eb81e892072}} is bounded by the degree of {{formula:5009ca84-ee9b-4259-8c92-305b3a5895c3}} .
The intersection {{formula:c9770776-b0d8-4fec-bfa3-efacd37f4a37}} (with its reduced structure)
is a Galois-stable closed subscheme of {{formula:8a6d30fd-7494-46c9-8bbb-84687407b7d1}} and thereby descends to a reduced {{formula:4e764afd-0788-40f9-9117-80b77d02968e}} -subscheme {{formula:041b94ae-07cd-4f1b-b7c4-3974a53a764f}} .
Since {{formula:ea4c5870-bfab-4cc3-b647-aae38fcaafde}} is not geometrically irreducible, we know that {{formula:c8c8910e-cb80-498c-a805-0ca680870571}} .
Also (b), applied with {{formula:52a7ae6a-8a26-4588-9876-32a9657b179f}} replaced by {{formula:105828d2-d484-4643-82ee-b415124de9f6}} , implies that {{formula:155db8a0-b8a9-4007-8e7d-c9f01ec84a29}} has a bounded number of irreducible components, each of bounded degree.
The same is then true for the {{formula:55e04449-f87a-44e9-8599-0b349e523740}} -scheme {{formula:f779e9ab-1591-4602-a93f-4b8878d1ccf3}} .
We claim, further, that {{formula:9c65f2e3-5921-4e80-9fbd-9abe333d8abb}} .
To see this, note that any {{formula:4ef0b0e5-48b5-4f8a-85b1-d998ec6ad44e}} -point of {{formula:1dffde6c-54bd-44fb-af20-cacddb03bd3b}} must be Galois-invariant;
since Galois permutes the geometric components of {{formula:03c66568-b045-4a29-be7e-f2e2f284b0c3}} transitively, the associated element of {{formula:b89658a3-866c-45ad-99cf-244296ae44c3}}
belongs to all {{formula:82436d78-3f23-43a4-b340-e581faaa567a}} .
(See {{cite:08f5e295d06f6c2351554a86a17cf4ec5dd16975}} for a similar argument.)
The argument for (d) is similar to that for (b). Again we can parameterize all such {{formula:b8422e5a-e524-447c-a8d8-d7f463cdae39}} by a suitable finite type Hilbert scheme {{formula:1cfc889f-5f6b-4339-8473-56570c39c941}}
and, writing {{formula:a4122d23-ea31-4295-8b3c-009de8911618}} for the universal subscheme, the smooth locus of {{formula:2e2104f6-71f8-478d-a196-6a5759df2cfe}} coincides
with the set of points of {{formula:61bbf024-402c-45b6-92fe-20e8ee502fa5}} that are smooth in their fiber over {{formula:5d3c4d74-25c3-4998-be80-dbe5a4046fa5}}
(see e.g. {{cite:b5606ec4302407aeb6f85630913a7508a124068d}}).
Let {{formula:19f9eb50-5505-4452-9c4f-2bcb7bfca1d2}}
be the complement of this smooth locus, endowed with the reduced structure. Then proceed as in (b).
{{formula:4d212ac5-1a53-49c7-8ab6-b45553e0163d}}
Finally, the following theorem of Broberg {{cite:fdc55cfbe56760e0e4aad4ad0ec42fd2ddeacf28}}
builds on fundamental ideas of Heath-Brown {{cite:b520a4eca33248ea375d4e02aabf7a6e7865decb}} and Bombieri-Pila {{cite:87d4ccb6a1149f92e322fc2b2d0542c77757c306}}:
Theorem 2.6 (Broberg, 2004)
Let {{formula:b98d3fdb-8a7a-4ef7-87c4-504984370832}} be an irreducible closed subvariety of dimension {{formula:ef1a8496-1ad5-40be-a5ec-8bd33be9851d}} and degree {{formula:a54f819c-4ac6-4b68-9adf-f5dd8d5bd469}} .
Then the points
of {{formula:71767ca8-471f-4f6d-af12-16c4d3c787d0}} whose naive height is at most {{formula:d9ecf9f5-49d3-4cac-a4c0-2acc8aaed2f4}} are
contained in a set of {{formula:604a59ed-14bd-4b3c-9a39-e5fc115577e5}} -rational divisors of cardinality {{formula:c2e878a5-8902-4d83-bb30-e2237e024f2f}} ,
and each of which has degree {{formula:960ecb1b-4321-4dc3-88d7-824cb430b804}} .
We note that, as stated in {{cite:fdc55cfbe56760e0e4aad4ad0ec42fd2ddeacf28}}, Broberg requires a bound on the generation of the ideal of {{formula:47cd2ec8-fb53-4af7-a70e-494d6e2ddf69}} . This bound is however automatic from Lemma REF part (a).
Also “divisor” in the statement means “effective Cartier divisor.”
Remark 2.7
Because Theorem REF is so crucial, particularly its dependence on degree,
we briefly outline
where it comes from, taking {{formula:562bd97c-682c-400f-9aea-265e1b29aafc}} to simplify notation; this is not needed in the remainder of the paper.
One chooses a large integer {{formula:c2893906-5b20-4581-a072-46a60ae46ce9}}
and embeds {{formula:b66146ee-e8dd-424d-94bb-1a4afbb1dbb0}}
via a basis of sections of {{formula:37cb76fc-42ca-47e7-96fe-93cde39f887e}} . In fact, we can and do choose from {{formula:1d89e430-0a3e-4282-a236-7ff1c9807aef}} a set of monomials {{formula:002881ce-d73e-46eb-988a-d96792c3c2fd}} of degree {{formula:8e101214-6832-4b7c-b208-b369a084f25b}} in the {{formula:36c37527-fea1-4ebb-a790-533d32710c0f}} -variables which freely span {{formula:ad712285-ae3e-4abc-91d2-7e743081d29f}} .
Choose a “good” prime {{formula:572ac408-9f56-40fb-9961-db8d32b8da7b}} and examine a collection of {{formula:75baa33f-149c-4c98-b162-41b59672025e}} points {{formula:9e665fce-e8fb-47ec-a44d-3ab067cbcaed}}
which all reduce to the same point modulo {{formula:89b1a28d-460f-451c-bcd8-56d61b92c358}} of {{formula:3f0ceecb-3608-4d4b-a2cc-413da81b88dc}} , and whose height is at most {{formula:39c1cf93-a6c2-4167-91f1-d9e42e731bdb}} . Expressing each {{formula:694fbbd6-116a-405d-a787-eeec82ce4aa2}} in coprime integer coordinates,
we can speak of the evaluation {{formula:ae0dff82-e204-4478-af53-f1482a90a49a}} .
Consider
{{formula:9ab7cf3a-e5d5-45b3-b480-2eb3d5ff3b49}}
which measures the volume of the {{formula:8c6cd119-bbfc-4402-b4be-30468d9c905c}} -simplex in {{formula:a75fdbf8-0619-49b6-a52e-fc5e875fd7bf}} spanned by the {{formula:6c257c9a-b440-4c14-b252-ef1d145f819d}} and the origin.
On the one hand,
{{formula:97a1eff6-9ce2-40ba-a474-2d57676d2346}} is bounded by a constant multiple of {{formula:037e4f06-823f-4f22-a4a1-5b1da017f240}} .
On the other hand, {{formula:b28f29cc-fcb9-4022-a706-6fdac1a1cf07}} is highly divisible by {{formula:38ad39f3-e698-4bc2-a9d5-9f669bcbef58}} ,
because the values of {{formula:a39ac5a8-3a4f-4079-af8b-ac33b0940b0c}} modulo power of {{formula:9815ef27-fcd5-49ca-96dc-33bee5967e45}} are highly constrained,
and therefore there are many relations (mod {{formula:78fc194b-5dde-4617-bd5a-9478e3807126}} ) between rows of {{formula:f42a30c4-d4cb-4a70-9915-f7cf498f17c8}} .
To see this more formally, fix {{formula:d8174534-caed-4534-8713-71c4e521c665}} and let
denote by {{formula:ff404c41-5b35-43d8-af4f-4935d0490add}} the subset of {{formula:4e480db5-e87a-4735-98c5-cc641ab04787}}
consisting of points with a given reduction modulo {{formula:7f94fd2f-f881-4f41-a861-c33fc764dff1}} . Set
{{formula:70accf95-5caf-4d44-9f84-c18d70627abb}}
Each {{formula:59c99af1-ef66-4aa8-82d5-e747175261b5}} gives an element of {{formula:49cca971-3114-4933-97b7-9afbbc7f78f3}} , by evaluation and reduction modulo {{formula:f4e790e2-75e0-4a8e-9504-2156b12143e4}} .
For {{formula:17565c00-d520-47de-b9c5-7c8314abb5ea}} , all these functions (for varying {{formula:0ed7ba5a-bd88-4adc-b2d8-97131aabc99c}} ) lie in a {{formula:618b2f49-7052-44ae-a567-b5af20458128}} -module of rank one: the constant functions.
For {{formula:5e7047f0-e901-4be8-941c-4e590df50f66}} , these functions
depend only on the “constant term” and “derivative” of {{formula:1ba45aba-d7de-430d-b4bc-885eace6c4af}} ,
and thereby
lie in a {{formula:82352a92-d92b-428c-9665-ea1292f44391}} -submodule of {{formula:b538445a-c903-454a-9f29-75c58cecd537}} of rank {{formula:6b3a791f-35f5-4fbf-89d9-ee112b937699}} .
For {{formula:343421fc-7e7e-4692-bccc-5f26ae04016e}} we get {{formula:18c8629c-dad0-4db5-aeed-f5b8a12e4756}} -submodule of {{formula:9446ce14-23d9-4509-ac03-03a9c6aea751}} of rank {{formula:5baa9590-c89d-4bd2-864b-6abfe069b385}} ,
where the number comes from counting possible
Taylor expansions of {{formula:278c00c1-1b35-41a3-90fd-d5c62e748c60}} up to degree two.
Each such statement gives linear constraints on the rows of {{formula:c4d8cf88-fa6b-4082-b7a6-dcb495cefefa}} ,
and therefore leads to divisibility for {{formula:203a158c-ec63-40d9-84e7-cf46952e93f8}} .
Computing with this we find
{{formula:f9cd0832-6012-4170-8f4a-31264c2377d2}}
where {{formula:dbb92d73-d1f5-45e4-9364-3330e3044261}} arises on the right-hand side
eventually through the asymptotic behaviour of {{formula:48f35590-a05e-4c0e-aeab-8b79bbe83ae1}} ,
cf. (REF ).
Choosing {{formula:e373642d-75a5-4f73-9819-536f432a4678}} so that
{{formula:054de5e1-2fbf-40c2-96f5-cadfa870ca3a}} is larger than the size bound {{formula:ffaeba6a-f7d1-4141-915d-c62516df46bb}} then forces {{formula:92ebf66b-27fb-4db6-b9e0-f80992a27794}} ;
so the points {{formula:5c0461ec-ab0c-4d6a-981d-fb98259dbfb6}} lie on a hyperplane of {{formula:77540221-0d2f-4003-b59f-b18840311ceb}} , i.e.
all the points of {{formula:cb7b4861-fae1-4d6f-8630-9d5bfeb058a8}} with a fixed mod {{formula:d695f18c-50a6-4137-805b-b7f0f04d5d01}} reduction lie on a divisor.
So we produce {{formula:0da7b4db-22cb-4011-9d75-a83ea72a4d82}}
divisors covering the points of height {{formula:91b0d8c4-525b-420e-b912-62722fc786c1}} . The argument above is so flexible – in particular, using freedom to choose {{formula:50d7ebfc-6fc5-4e73-9ebd-78a6808b1a84}} –
that one can achieve bounds that are uniform in {{formula:95a65be9-586f-44ed-8596-0cba243eb3d0}} .
Notice the crucial point: as the degree of {{formula:ca67b006-71be-4c15-8695-2a60b3aceba5}} on {{formula:d6600d6a-58dc-46a3-8938-9922b7f386b8}} increases, {{formula:4c661d87-69af-4a8d-b093-b9a1f310fdc0}} has more sections for a fixed {{formula:a8b05357-0d5c-4210-b692-781cdcbddfaf}} , giving
stronger divisibility for {{formula:a9ee61a3-78a4-4cd9-bef7-68d8a21acf7a}} and thus stronger bounds.
Reduction to Theorem REF
We describe how Corollary REF is reduced to Theorem REF .
Deduction of Corollary REF from Theorem REF
We will focus on the first statement of the Corollary, with {{formula:209ffa82-91e8-43d9-a527-aaafcb134c9c}} -integral points,
and remark at the end of the proof on the only modification needed to handle
the statement about fixed discriminant.
For {{formula:74053a40-b7b7-4b9d-9c07-565a12dba30c}} (binary forms of degree three and above) one in fact knows finiteness (Birch–Merriman {{cite:5ee7faa20b01861484504f572565533900d6b784}}).
The same is true for the case {{formula:7a8641af-b925-4c92-8a9e-009837700677}} of cubic surfaces (Scholl {{cite:623a3434d6befc343e525213204bad443850e70e}}).
We may now restrict to the remaining cases {{formula:30af2f21-ebdd-44e8-946b-5d644797bfcd}} and {{formula:9f15a20a-4de6-4668-bb09-485d6d3e617e}} .
We will apply Theorem REF taking {{formula:ebdb172f-e44e-4dc3-b12f-e8905a3f2303}} the projective space
parameterizing polynomials of degree {{formula:fd111400-6575-4956-bc1a-1d13b36ccbd3}} in {{formula:3610fb96-df92-4454-9378-8ea9c57dcbaf}} variables up to scaling,
with {{formula:eecdc030-efec-4a4c-84f1-12816e1f3b4f}} the zero-locus of the discriminant, and taking the geometric variation of Hodge structure to arise
from the middle cohomology of the universal family of hypersurfaces over {{formula:ba58c844-7818-4ced-bebb-8753b94e14bc}} .
Here the infinitesimal Torelli theorem is known, see {{cite:8a959cf92358465d89fabf79b38e83cc258dea14}}; that is, every fiber of the period map is locally contained in an orbit of {{formula:e91244e8-73d6-4ea8-ba4e-6d0a38733e82}} ; so,
noting that the Weil height of {{formula:cebb7517-6fdd-46d1-b06f-fd4f07f3456c}} is given by {{formula:9b273d81-526d-498f-842e-fa83ba22fa2d}} and is thereby bounded
by a power of {{formula:758f185d-9f47-4018-8b1f-df8ddb4ef9af}} in the situation of the Corollary,
Theorem REF shows that the integral points in question are covered by {{formula:f656af13-b796-4f40-9b8f-6cc2f650b738}} orbits of {{formula:6ca7fc3a-e08b-48df-850f-558f3621d651}}
or equivalently {{formula:63440d1c-743a-40d2-bf1f-bcace98bc650}} .
We must replace {{formula:6fb45cd2-b912-4c5d-b283-aaf06e18db6a}} by {{formula:864beb97-6b6b-416a-b7d5-4570cdb322bd}} .
This is not difficult
but one must take care because a hypersurface could have automorphisms in characteristic {{formula:109b6145-2afd-4b01-9a53-5840bbce62d0}} that do not lift to characteristic zero.
Fix {{formula:9af9dfea-b781-434c-9adc-d932633553e4}} as in the Corollary. We will show that the number of {{formula:c0515c5d-b28d-49ff-a1f8-1472cd46dc1e}} -orbits on
{{formula:9d48e9f7-1b85-4861-acf3-e1f78cbacc6e}} -integral polynomials {{formula:0278200d-95a5-4660-8b47-fdee82347df1}} with {{formula:5a966fbd-07dd-4891-a5a3-04955e29affa}} -integral discriminant is bounded in terms of {{formula:cee3710b-e606-48a0-b4fa-8d283bd0493b}} . Let {{formula:137faac5-1d26-4ba4-aa21-07cb00fac92d}} be the degree of the discriminant polynomial. For any {{formula:17447c10-2852-478a-97b8-429498e31a97}} -integral {{formula:5c3c1d7e-a180-4e33-b1c5-98627c04d0fb}} with {{formula:a57bb652-0271-43e4-9d8b-b7cd98f1eced}} -integral discriminant, there exists a rescaling of {{formula:ef664b9d-5313-48ae-bb58-b09ddac04a3e}} by an {{formula:6f064858-3db6-412c-a143-85c23acf3f54}} -unit
whose discriminant has {{formula:ba86a859-c094-4d9f-9a89-b3b54223f1cb}} -valuation between 0 and {{formula:a9ad30dd-4a50-41a6-a8e5-1f1575a2621f}} .
It suffices, then, to show that for any integer {{formula:6bc1d085-cad0-47a7-898c-281f4f3eaf92}} (with {{formula:dead7164-7a61-40ad-b77b-ed72ee1156d3}} )
the set of {{formula:1331bc5d-d310-4123-9b09-14ce80c783b9}} -integral polynomials in {{formula:cca943ef-1f4c-4b42-b2b6-7d0e7e4bb6b7}} with discriminant {{formula:44195d2a-8ba0-47ef-a512-8f6c09e871a5}}
lie in a union of {{formula:637660bb-13ea-4347-9984-463cc20c8603}} orbits of {{formula:209b89a3-4649-488f-a380-24218c5836c2}} .
Now, write {{formula:fb7ba1e4-c275-4623-8e33-19f1bae67fd4}} for the affine hypersurface defined by {{formula:3daa9014-12ba-4595-b248-95e5cceacbba}} ,
which we can regard as an affine scheme over {{formula:f926e98a-08af-4ad2-8fea-6e7d623c2817}} (and even over {{formula:63d2fc43-64aa-4a0d-85f0-ae5a3fddb17d}} ).
It is equipped
with an action of the {{formula:97b23f6c-949d-48e7-b9b1-f0ffaaec0ad0}} -algebraic group {{formula:8a0b34c9-f843-4528-a3fa-7568056fff97}} .
We must show that the intersection of {{formula:4898cf6c-b97c-4c7e-b58b-482ec3caff27}} with any {{formula:571067ff-96d0-48d6-9edd-8e63cf6eb8c2}} -orbit
is covered by {{formula:bd1396d0-9394-4a20-b328-c5633c2b290c}} orbits of {{formula:07eecef3-4435-448e-9948-335d5d00634d}} . We will need:
Claim 1: The action morphism
{{formula:8b5ce3a1-43f0-4ec2-bb96-7057a0e531b6}}
is a finite morphism of algebraic varieties over {{formula:ef8b6b54-d1ad-4a23-bff0-30bc0c7941ab}} .
This follows essentially from the theorem of Matsumura and Monsky {{cite:b8105744a25e3e09fd8f3405a5ceba2051e2305b}} that each stabilizer
{{formula:17a732fd-f58c-4a12-ab37-4b934ff8ce52}} for {{formula:fae433a0-da51-4e02-987a-ab8a78f552ba}} is finite.
The deduction can be carried out using results of geometric invariant theory
but we give a self-contained argument using similar ideas.
(of Claim 1.)
It is sufficient
to prove
this over {{formula:4acacece-b4eb-419f-ac53-5818b10c89a3}} Using the singular value decomposition
one reduces to checking that the action of the diagonal subgroup {{formula:0868416c-5d8e-4209-84b3-cfb21f03e1a1}} on {{formula:9a23ea01-f899-47de-8539-a307bde87300}} is proper, which can be checked for the analytic topology {{cite:6877a10c554980fc56a17306637f5a7f5924e173}}. In other words, one must show that for any compact regions {{formula:a043b29b-9ce3-47bd-ae87-8dd354af9461}} and {{formula:53228c7c-f0bf-48e7-ae0b-c2cb00796493}} in {{formula:61156aa5-cb34-43de-a93b-804e64168dd3}} , the set {{formula:f7e9f88e-8d7d-45cf-a890-1186b02db614}} is bounded. It suffices to show that, if {{formula:52af40f3-a4b8-4fe3-8778-0450be3f3453}} and {{formula:c5e543dc-e7a2-49cf-b5df-099187ddaa84}} and {{formula:3d8cc30e-7ed8-4a4c-9c4d-505e49e8e06f}} then the absolute values of the {{formula:8f19d629-3b7d-4f62-a9b8-0239c37fb1af}} can be bounded in terms of the coefficients of {{formula:0edad140-0aa5-4db2-9d4a-ff96b9bbda35}} and {{formula:b83fcb96-532b-4cde-b8fb-7fc7bd37df42}} .
Write {{formula:9f0eec41-0a53-4c8e-ac73-86cc8aba10c7}} for the set of {{formula:3ae0b718-d717-4350-878d-f3b4213627c5}} such that {{formula:716071fa-8dda-45f1-a87f-bf7267466b94}} . For each {{formula:be67cc9e-5552-45eb-933f-5ec493330cf7}} , write {{formula:8f637078-0440-473e-bd62-a750529daa21}} for {{formula:4920617a-2c86-46d8-950a-23cf665c6436}} . Then the maximal absolute value of a coefficient of {{formula:e6b31856-d11e-48a2-acdf-f9dd4e242fa0}} is {{formula:5811d085-79a2-4445-ba47-2a1423e39390}} , so the condition that {{formula:53ac498a-688d-4d73-ad37-8a12ac096343}} provides upper bounds on {{formula:abdee5da-93c5-4258-9c92-c529b4abd564}} for all {{formula:6935a3c8-a670-4b2f-8a7f-f2457cfab3e3}} .
An upper bound on {{formula:4b378082-6e12-4689-8bb4-5ec680ca02e0}} restricts {{formula:c7766925-78bd-4f92-9277-888259d0c315}} to a half-space; we are done once we show that the intersection of these half-spaces over all {{formula:658b5e33-8a1f-44e2-9090-0a5759fd3071}} is a compact region in {{formula:c86699fd-092d-4126-b19d-a0ecd6d2f4d6}} .
This is the case exactly when the region
{{formula:7e165b6c-56cd-44fc-adb8-fdee07cf832d}}
is empty. Suppose otherwise; then there exists {{formula:ceebc6ad-1b17-4343-8b5e-8b06e0854e52}} in this region
of the form {{formula:9a55103e-d108-40b5-b14e-35fee51999ec}} for some (whence any) {{formula:464aa19e-9aa2-4071-af92-d9fb060f791e}} and with {{formula:fb938a7f-20cd-44d9-a0f5-dd1ae9690732}} .
By assumption on {{formula:6749c98e-e311-4619-85c2-9fd585466b08}} , the limit in the analytic topology {{formula:c80a9e3f-a704-4b8f-90f5-47242f652737}} exists; explicitly,
{{formula:65511c47-c055-4ae2-8505-55147fca03a3}}
where
{{formula:2a436ebe-fb87-4863-8339-2a7e7b682ce6}} is that subset of {{formula:33d5ca0a-98ba-4283-a648-a71c7cac8544}} consisting of those {{formula:87ee3e6a-d11d-4c49-8d6e-cc7bbebb4a91}} with {{formula:fca443d3-034f-4a52-ab9f-8f2783f32500}} .
Then {{formula:4a1583ae-63ab-4e6c-9960-aa41f29ed791}} is nonzero, so {{formula:4cf158ba-449f-4415-9555-388b7ba8a32c}} cuts out a smooth affine hypersurface.
But the identity {{formula:bc2afaff-8b83-439d-a696-832b6673263c}}
means that the {{formula:7ba13c98-5203-481b-8869-4d347779dead}} conditions {{formula:d4c17e21-c43a-42c1-95c1-d0203b400bcb}} cutting out the nonsmooth locus are dependent,
contradicting smoothness of {{formula:b126fd05-8622-4ebd-b3d7-80daf9d9853a}}
(cf. proof in {{cite:b8105744a25e3e09fd8f3405a5ceba2051e2305b}}).
Thus region (REF ) is empty, so the action of {{formula:f504fc58-6fb1-4074-b9f9-1b97a93e25f8}} is proper, so the action of {{formula:c1867dc3-88a1-4765-a99b-57d142a39722}} is proper as well.
This concludes the proof of Claim 1. {{formula:b9746ee1-4b0e-49c1-ae3d-1b440c0bef3e}}
Take {{formula:d44d556f-f15f-4ff6-98ef-3d4d1486947d}} . We will now prove
Claim 2 : The action of {{formula:8049be30-e0a7-4061-a861-6af3bc1334d8}}
on the stabilizers {{formula:8733e78c-163c-4c25-8143-7b5569fdad56}} and also on the set {{formula:08801668-02b1-410b-99ab-434c896f468b}}
is unramified outside a set of primes {{formula:1b270def-d08d-43b5-ac21-ef7311e6193e}} depending only on {{formula:d5f73e1d-3b61-4dd8-850a-ad72da9db89f}} .
(of Claim 2).
It is enough to prove the claim for {{formula:1055f3a7-36f3-4894-ab5d-bcd325c759a4}} ; take {{formula:717e3c0c-f2e9-4d2b-9453-be60848c5b5b}} to get the claim about stabilizers.
From finiteness of the action map we see that the matrix entries of
{{formula:f80adcc9-a81e-4ba1-a472-2d040ab62637}} for {{formula:e8ab49f8-0d61-4591-a87d-59b4dfde9aa8}} satisfy monic polynomials whose coefficients are rational polynomials in the coordinates of {{formula:6b8aa4e6-4a7f-48f8-9121-aaf6b534cf1f}} . Take {{formula:ad16aa1e-9a9a-48ba-b915-74174a91b442}} to be the finite collection of all coefficients arising in this way.
Now, for any extension of the {{formula:88537990-0cd5-44a5-9e59-eff723514ae3}} -valuation on {{formula:7b147183-b826-4662-a6fa-2198d1c264ea}} to {{formula:aa24681e-d8f8-40bf-9ab3-9393df75fd2a}} , and for every matrix entry {{formula:fa7ff775-ae89-44b7-b845-c7f9b966d215}} of {{formula:33a20716-faac-421e-8b34-337300f6f930}} we have
{{formula:6facb8cc-03c6-46be-a5c0-2a13c9d08a09}}
Take {{formula:9cd49bd4-8aed-4e1f-b2d8-824ee64c6123}} , larger than any prime in {{formula:a1feb895-3a87-460c-872c-904c6108ed20}} , such that the coefficients of each {{formula:db50f2f6-7366-4e6e-95fb-52c595aea46c}} are {{formula:58fe89ec-d08e-4f78-8ab1-25db65f7f963}} -integral for all {{formula:7cbd57b4-d318-4e27-bf65-9b93b1fe7999}} ; it now follows from (REF ) that, for all {{formula:31a2bd84-c9d2-4b4a-88f6-802986365644}} , any element {{formula:e9d7c261-c827-450a-9a88-5ac55ce30c90}} which lies in {{formula:a88fa147-a274-4fc3-8151-d9b265c39d01}} for any {{formula:398f2110-087b-4794-bd11-3f5c68dfa8c7}} has for entries roots of monic polynomial with {{formula:9caa0097-8dc6-4c5a-9afd-65cbb3cbb3f8}} -integral coefficients; in other words, it is {{formula:8f6efcea-a70c-41fb-95fc-d97ae1b0ddce}} -integral.
Thus, for all such {{formula:997e63ec-8cd6-4238-b7f5-2e967fe18393}} there is an induced map
{{formula:6bb45eef-3420-46d1-baa1-e469d5603253}}
which is necessarily injective if we also require {{formula:071ae5f9-bec8-4c66-a1cb-156aa2254eff}} to be larger
than the order of {{formula:7425c757-0dc7-4604-851c-d9e208deb7ec}} , since any torsion element of the kernel of {{formula:887d21c2-b485-40a3-ad4b-e209dbdb8055}}
must have {{formula:43fc7ee9-680a-46fd-89ab-ce590dfd6215}} -power order. For general constructibility reasons (similar to Lemma REF )
the order of {{formula:492f1ad9-e212-4fc2-809c-529c61f543ba}} is bounded above; choose {{formula:b947010e-f1e7-4b9f-8d26-d35d043ec3ab}} to be larger than this bound.
Now for any {{formula:82928b26-3686-4855-8437-6380adb5fc7a}} belonging to the inertia group at {{formula:b0d67f8d-e2b6-45fc-8ca7-ea8339dd4176}} , and any {{formula:86246ed0-aa73-4f80-8e98-87f402ccb2a8}} ,
{{formula:0b7488d6-3a89-4880-9378-914e63f19667}} and {{formula:d9bf2054-5373-427c-8ecc-91d41635eee6}} have the same image in {{formula:55eb69d6-fe1f-40e0-8714-ba2128744d34}} , so must coincide. This is precisely to say that the Galois action on {{formula:b1a65231-e37c-48bc-816c-ba8c8060a4b3}} is unramified at {{formula:fa5a6af2-1c75-4d50-8a59-3ca80220f0d4}} , so we have proved Claim 2
with {{formula:47b808f3-6209-4757-acbb-33f34848c7df}} the set of primes less than {{formula:37679e1d-2a1b-4e57-aef7-580df3e69635}} .
{{formula:fc1a96ac-83d2-4d74-a8b1-42177deb319b}}
Now, with {{formula:06c329d8-12ee-4862-b11c-d8e5a44e1823}} , there is an injection
{{formula:0cb81216-7974-4a3a-83a0-ef2966c2ed67}}
sending {{formula:f30ff934-ade9-4882-9f6a-53b91f896dfc}} to the right torsor {{formula:2bfb0082-97bd-4de0-af91-1ae71a327e3b}} . The Claim means that,
under this map, elements of {{formula:9ff05525-9ff7-4acf-a836-6767b40169e3}} are sent to unramified-away-from-{{formula:7f729058-010d-42f8-8672-99917eb14209}}
torsors for the unramified-away-from-{{formula:f242578b-01e0-4834-892c-4608e0a46489}} -group {{formula:510f51e7-4a62-4e04-aa81-144da789d233}} , whose order is moreover bounded. Hermite-Minkowski gives an upper bound on the size of unramified Galois {{formula:8b3d30e5-5d55-4862-933d-ab9923af76fc}} in this setting, and we conclude that
the {{formula:1329a906-67b9-47da-895e-ecdad1745202}} -integral points lying in {{formula:634a51dd-e047-4de8-b9d0-e57fead530d8}} lie in a collection of {{formula:451f588f-4bb8-4d21-81c5-4c08a2cc11db}} -orbits
whose cardinality is bounded in terms of {{formula:524b68a3-0c21-4473-af4c-069039aa1e13}} .
Finally, we pass to {{formula:3fe7a37b-58b6-4d4d-9633-0637fb433fda}} -orbits using (REF ).
This proves the first statement of the Corollary.
In the last sentence of the argument we used (REF ) only at {{formula:1d50f3c5-33f0-4701-8135-950a9dd290d9}} ;
but using it also for {{formula:4a2d564f-cd4c-4694-8de2-e38b4031324f}} allows one to conclude similarly that
{{formula:0847b84e-8b2e-4770-b996-3b9677d0884f}}
is covered by {{formula:cf39058a-502c-4e61-b073-0051b5f18b18}} orbits
of {{formula:391d65d9-32d8-4974-aa9e-2c0888a89136}} . Here {{formula:a01aa856-3e78-4218-ad36-1483c7c169f3}} just refers to the largest coefficient of {{formula:b01c8c58-9683-42dd-ad6b-93959b67c1ba}} , rather than a Weil height. This is the second statement of the Corollary.
Reduction of Theorem REF to Theorem REF
Suppose that {{formula:98b74104-eadc-401f-865c-e26ac4184db9}} is quasi-projective and {{formula:709a943a-c26b-47b5-bb26-1b26ef0203bc}} : X X{{formula:5e1f7aca-9fc7-45c0-98b0-4858091e3608}} R K{{formula:d5906fc9-df93-4127-a11f-26c122f146aa}} , finitely generated over {{formula:9b808d62-9b6f-4eaf-a209-a56135d6f29e}} :
{{formula:1bb21483-b4a3-4e00-82c2-63b2a49a47c5}}
where {{formula:86e906b6-a699-4112-8c78-231a749b1075}} is the spectrum of {{formula:3d8cc4f7-3711-4b02-a569-5d0e4e379d64}} . This recovers {{formula:d69be900-7bb4-42f4-a5f1-1c0ca31c85ed}} s: SpecS{{formula:4927db7b-a903-4e97-8d2b-b5f5d7619f0d}} S{{formula:a510d852-e8fc-44e9-8eed-0ab07a8eaabe}} S{{formula:83d94ce2-2c96-476a-9558-585d10effc1c}} S{{formula:302110dd-d2ef-4170-8079-5c4a24342bca}} R{{formula:137ed29f-c57b-4789-96b6-e986e7ab818b}} S Spec K{{formula:1f5ae448-6f30-44dd-b34d-800abc651808}} S{{formula:f3b40505-ccb3-4beb-9e47-91a0e3b26bce}} S{{formula:bf6202a5-ca56-4170-92e0-e39428039f4b}} K{{formula:8de6d637-85b2-4cd2-92ca-1631537ca369}} Consider the morphism of vector bundles supplied by Lemma REF , which, applied to the morphism {{formula:f3c17471-abe4-4bb2-80f6-27ef0d2744a8}} , gives
a morphism of locally free sheaves
{{formula:29b26b0a-9472-4d06-a513-00268edf588c}}
over {{formula:b5d63009-4472-4058-a6f8-46e1c46cf6a0}} .
Let {{formula:20b0b088-cbb9-496e-80cd-8604d5d7b6fa}} be the sub-bundle defined by the pullback of the tangent bundle of {{formula:4b39a6fb-9689-47ed-b78c-c9dc7b369304}} .
Since {{formula:c5ff2300-4b7e-4c91-9a44-43e48fccb577}} s' S(K){{formula:1bbe9abc-64c5-4e16-af2e-00b9a10cab8a}} X(K) s'{{formula:7137f2d0-8ca9-4ad9-a009-f1bb88cf8c7b}} U(K){{formula:fe19b422-3d8d-42cd-b688-cd6a16d338ad}} XR XR{{formula:89cb06f5-6944-42bb-91ee-c3774e8513d6}} s'{{formula:cdc13b94-ebea-4727-8d7e-29231560fc50}} K{{formula:6d99cb5b-9fc5-44ed-a981-55fde748b643}} Xs' XK{{formula:5c46c12e-5856-4110-b0d0-9a6b469a556a}} E XK{{formula:787c5f9c-bd5d-4630-9c36-8188aae31529}} gx,s'|TX{{formula:c28ee9b1-e08b-4716-a81a-5cd11b5be424}} E{{formula:b0df79b8-9336-46ba-83ae-9e998f7f169b}} X{{formula:66c02004-34e3-4aea-9b71-90e2fffb3756}} K1 K{{formula:e94d96aa-b805-468d-aec5-f50921042e2c}} K1{{formula:d421b8cc-f83e-4567-89f8-ca1d21ec7d6f}} X{{formula:f2987439-38f3-458a-955b-c67fc36f5191}} E{{formula:4c66048a-0809-4c5b-8fd5-155c68e287b8}}
Proof of Theorem REF
The remainder of the paper is devoted to the proof of Theorem REF . We use notation as in the statement.
We will fix throughout a good integral model for both {{formula:cb5013f4-e352-4ede-9672-062794cc2c3b}} and the morphism {{formula:03624b9a-2e25-4fdf-bb9d-f35dcd871142}} ,
as in Definition REF .
As we have noted, the proof involves proving that integral points of {{formula:a5624402-20f5-4bd9-8285-36bfffd84467}} lie on
various collections of subvarieties, whose dimension will be steadily reduced until they
are either points or fibers of the period map.
The key inductive statement used to reduce the dimension of the subvarieties is
Lemma REF .
It may be helpful to note, in advance, that we will not need to keep track of any integral structure on our subvariety. The notion
of “integral point on a subvariety” will simply mean a {{formula:aa68c4df-f21e-4c7c-b6ff-bc310b5ee8a3}} -rational point of the subvariety that is integral
as a rational point on {{formula:78feff87-b812-4873-8381-cce3b15aacc0}} .
Large fundamental group.
Enlarging {{formula:cdbe6441-9399-47a2-89b4-9e82e796bd32}} if necessary, we fix a prime
{{formula:49f94fe3-36f7-4138-8f62-1271717fee68}} for which the integral cohomology of complex fibers of {{formula:0111942b-81da-40cb-be3c-0b14737519c9}} over {{formula:fc757d08-d44f-41c6-9314-3728bdafcc53}}
is {{formula:cc58c9dd-e158-4734-82c9-e2efbbc83190}} -torsion-free, say of rank {{formula:172db436-2704-47f7-8af3-0e286a7e554b}} over {{formula:30100986-0db9-48d8-b129-9a5fc62dce05}} .
Fixing {{formula:8009efaf-4ce6-4cd4-a67d-ed99ef82ca29}}
we get a monodromy representation
{{formula:f043a30d-1f22-4228-82ef-6d4066282031}}
In this setting, the global invariant cycle theorem implies the following statement: For any complex irreducible subvariety {{formula:05ba5fb4-e6df-48b0-b95e-245e9f2ee7f7}} V{{formula:3fd448b0-a934-4a87-9503-71e47019f8e4}} 1{{formula:db02b983-8cc3-45c3-8ce3-3fe5475f347e}} (V){{formula:d1a42e69-b3d8-42c8-a291-cef298902968}} n {{formula:c99a0aa6-ba24-4eb8-bcc5-c59f4502d242}}
Next, let {{formula:0d0dacfb-5560-46c2-8312-a5a45e88f14b}}
be an arbitrary algebraically closed field, with base change {{formula:b9e5d5ff-514c-4e34-bde3-58a075f6aa92}} ;
let {{formula:9478609b-3de9-4f9a-922f-637adbcf5c95}} be a closed {{formula:6cc6408c-0339-474e-b76b-467201edcfec}} -subvariety
not contained in {{formula:e54242c1-6a85-4f8c-b72b-2117d8caab16}} or a fiber of the period map.
(Note that we can make sense of the latter condition without reference to {{formula:5183b3de-df4d-4bdd-ad5d-d1a2750125ad}} V{{formula:24c2d2bf-72d6-45aa-a606-d949bdf6a7c7}} V{{formula:5e6e546c-ac5b-477e-b97f-5740a94f7997}}
We get by base change {{formula:75fcac3a-e586-414d-86a7-392f16cd72d1}} and an étale local
system {{formula:aabd5e7d-4b76-4734-bbe2-0e2f22fb8a1b}} on {{formula:01bacf2c-a839-49c2-b454-40e5e31574f3}} ; and
the same conclusion as above holds, i.e. the monodromy representation () for {{formula:695570bd-72f0-479b-a948-d09d3b6801c5}} on {{formula:78e05fc6-9e74-4faa-8506-574e4dd738ef}} has “large image”
in the sense specified above.
We will use this only in the case when {{formula:62d683c3-0b66-462f-940e-10a7bea94d2e}} is the algebraic closure
of a finitely generated field; we may then choose a {{formula:0d6ebc8f-43f8-4c37-98ad-5ec1e052f497}} -embedding {{formula:4ed3352d-023e-4bb3-b3ac-b2d5b3692e1a}} V X{{formula:ac05bb3e-7c0c-49b4-a5cd-a4a2e552e9aa}} V XE{{formula:20119eaf-56f1-4d67-8579-0dc6dcb69084}} Ri etE* (Z/pn Z){{formula:d6bf86ae-b824-4d43-b89c-eaebb3c8a536}} VE{{formula:d2af04f6-b500-484f-ba69-d4d4b872c321}} V{{formula:3c5df060-1af5-4e4d-ab17-78596dbb4731}} V{{formula:4a154c0a-e19c-4c1f-9ab3-a1a0e8706558}} V{{formula:b604165e-be6b-4cf8-9eec-f4bdf3de0e8f}}
Construction of a suitable cover of {{formula:77a37e64-32f7-4f24-a2fd-499989c56415}}
Our proof will involve an induction over higher and higher-codimension subvarieties of {{formula:97cabd8c-1d30-47b4-ac4a-3994a60598f2}} about which we know almost nothing apart from their degree.
It is thus crucial to have at hand covers of {{formula:dd2b555c-4944-4b2a-aab6-60050cddd886}} whose monodromy is uniformly bounded below on restriction to every subvariety of bounded dimension and degree.
Lemma 4.1 Fix {{formula:c43d006e-e3c1-48bb-9f29-4117efebac81}} ,
and let {{formula:f41efd2f-78a9-4a00-a0b1-80d769e66e0f}} be any (locally closed, finite type) complex subvariety of the Hilbert scheme of
subschemes of {{formula:6d10ca25-ad02-464c-8b1d-504a4e0acb0b}}
There is a finite group {{formula:798ceae0-41f8-416a-9115-f166f7db03b7}} and a
finite morphism {{formula:e286686c-2ac6-4a96-b778-23521bcfb28a}} of {{formula:16fe386d-1d06-40fd-b7d9-82eb067d9012}} -varieties,
equipped with an injection {{formula:64de4635-752f-463b-a778-85cb7ab42a5a}} , such that:
(a)
{{formula:56fdbbb2-13f0-4bf5-9d19-399db9b7ce51}} is finite étale Galois with deck group {{formula:a3e602aa-54f9-4e73-a086-5d3fac496d16}} ,
and, moreover, extends to a finite étale cover of the good integral model of {{formula:3cb4ee49-2465-4ee9-b3b5-ff7e282c1185}} .
(b)
Let {{formula:55bdbfd9-de96-46fb-89f6-93fa5c4a4862}} be any {{formula:72fc0b23-708e-4b5a-9236-3cc5d22d210d}} -dimensional irreducible closed complex subvariety
of degree {{formula:a6c9b7e6-ae2b-4f4b-8808-d4e26ebd9c82}} .
Suppose that:
The point of the Hilbert scheme classifying {{formula:58632282-7037-415c-9994-4ab403725773}} lies in {{formula:de3694f0-a0de-4e1c-a2d2-9fb107e5c32a}} , and
{{formula:c485611e-4ef1-4e53-b703-6b5b8fb43e1b}} is not contained in {{formula:49ed6118-1eed-4ab6-8417-1a4b854bd6b6}} and {{formula:a910256d-776a-4d90-8311-412154b0fff5}} is not contained in a single fiber of the period map {{formula:376ef379-81b1-43ba-af5d-5f4df9de13ba}} .
Let {{formula:55a6ce81-2397-46a7-80fb-de7a614f81cc}} be any irreducible component of {{formula:1fe0f805-a795-441f-99b9-97694604689b}} , endowed with the reduced structure,
and such that the induced finite map {{formula:32dd7b30-b637-415e-aae5-44dada639a2e}} is dominant
(note that it is automatically étale over {{formula:e9206451-f1bd-4239-9274-71e54beea151}} ).
Then the degree of {{formula:c071aa12-613c-4d4f-8c07-15852d9df407}} at the generic point is {{formula:0b9cb510-3771-4fac-98bc-c4d818fa595d}} ,
i.e., the induced map of function fields has degree {{formula:74e2abd6-0de1-4138-802a-927bc58c29ed}} .
Through the rest of this proof, {{formula:4e706343-fda8-4783-8c8a-c9e5d36a9d73}} will represent a single subvariety of {{formula:2bdb1575-325d-4a0c-8c09-00c3e6f692b7}} , classified by a point of the Hilbert scheme,
and we will use {{formula:4697b646-5608-4f3b-9def-e4f7695b76f5}} for the universal family.
We are going to find a cover {{formula:670c25c4-8b92-4bf3-870d-036e98270248}}
and a proper Zariski-closed subset {{formula:4d32de5c-6451-4f69-9b5c-572ec054e480}}
such that the conclusion of (b) holds for any {{formula:187a0b24-8d8d-4d9a-8ae8-9afacbafa357}} ,
satisfying the assumptions of (b), and
with {{formula:ec98f984-3f79-4b9d-8148-4911a405c826}} . The result will follow by Noetherian induction.
In particular, removing the singular locus of {{formula:f3a0f2e4-8bdc-4f31-9fee-43e69d17d1df}} at the start,
we may suppose that {{formula:4f3970e9-22d0-49ef-b7c0-4c970dc05852}} is smooth.
Take a geometric generic point {{formula:3acf32f2-0319-40a6-b4cf-253f9588be02}} .
Let {{formula:76282c75-453b-499f-8d64-7363a3d0a3b2}} be the
corresponding generic subscheme.
We may assume without loss of generality that:
Situation:
(a)
Every geometric fiber of {{formula:6bdc13d5-ca8b-4e30-bc65-143b237b0280}} is integral;
(b)
Every fiber of {{formula:3a158911-4dbb-4514-9db1-6453528b2f23}} meets {{formula:0ef8cb2d-1719-459a-ac86-a11452f40eb7}} ;
(c)
On each fiber {{formula:0035fa54-5750-4de2-bf9c-c64d82b39ad6}}
for {{formula:dc235782-31b9-4a33-b31d-ce56c78d6e83}}
the period map is not locally constant.
For (a), note that
the locus of points with geometrically integral fiber by {{cite:f5f72ce3009bdfec9f5b167c329c67dd6b2011d3}}
is open on the base, so if there exists one {{formula:d83bddd3-8ece-478c-a68b-fad17fbc62c1}}
for which the fiber {{formula:812d1135-7da5-479a-8cb1-aeda9d8c13d3}} is integral, then
(after shrinking {{formula:80a7069b-7e21-4143-ad34-8bbbea6261df}} to a suitable nonempty open neighbourhood) we can suppose it is true for all {{formula:41a2503e-0f8b-45a1-a2e2-ff3c1725ba81}} . If there is no such {{formula:55d836dc-1d90-436c-a7cb-e97b0a6f5a06}} , then the conclusion of the theorem holds for {{formula:e200ddc2-5af9-4c00-8161-ba2386f4ca77}} vacuously.
For (b), note that the set of {{formula:2b198872-1a57-4eaf-934f-1634aade0ca5}} for which {{formula:1cc4d15d-2aa6-4911-91d1-c436c5442429}} meets {{formula:6b7697bf-7708-4db4-a769-7809e276affc}} is constructible.
To see this, first note that {{formula:5c2c8a6a-df5d-42c2-aec8-062e267835a0}} is reduced for every {{formula:805e1a85-3ec0-4dd3-8cd2-97483d5a479f}} .
Now note that {{formula:c9b3813e-c3ae-4e02-8bcb-da91e316f016}} meets {{formula:520a4c20-e84f-45bc-99e8-da0aa5493354}}
if and only if {{formula:19acc511-7127-4854-8500-54ae0457a2bb}} is not surjective, and apply {{cite:f5f72ce3009bdfec9f5b167c329c67dd6b2011d3}}.
Thus, restricting to an open subset of {{formula:316fe57f-3b2a-4801-ae4c-bc7a0d370786}} ,
we can assume that {{formula:2d446964-9977-41f5-b480-c63bddcab39b}} meets {{formula:5cc2fb27-ac41-4533-aa35-d9469807737b}} either for no {{formula:322e8a56-84ef-4183-8dbc-d79544339be4}} or for all {{formula:0b8ac4fc-6613-45e1-8816-4e96c3df82e3}} .
In the former case, the statement is vacuously true;
so we can assume that (b) holds for all {{formula:a315b60e-a414-4a8e-80e7-220667d959f9}} .
For (c) let {{formula:3ff57dc0-1aa3-465b-b452-994f839f0a08}} be the smooth locus of the morphism {{formula:b55c4fad-2968-40fd-b87d-a97b31358323}} ,
which, by flatness of the morphism, coincides with the locus of points which are smooth points of their fibers ({{cite:b5606ec4302407aeb6f85630913a7508a124068d}}).
Note that:
{{formula:c868455a-ad82-4641-97dc-fddb05ea5844}} is itself smooth over {{formula:e7a0ac4e-5db2-4ff5-99ed-f0b46eaadd1c}} , since it is smooth over {{formula:28e47ea9-4c09-4c7c-92b0-f9847ac3ba8f}} and {{formula:4388cf9d-a3bf-4e34-aa93-c6ab41917ca7}} was assumed smooth.
{{formula:3cdf07c2-a692-41ea-98bd-26bf578c021e}} contains an open dense subset of every fiber, since these fibers are all integral.
{{formula:503e2c2a-10e7-444f-bc3b-1037f6a1e5d7}} is a {{formula:0f38545b-f403-4d1d-8dfe-03de53dda700}} -variety: this follows from the previous conditions.
It is reduced by smoothness,
and since {{formula:0af2e136-3eb8-4544-8709-d27b82add68e}} is flat ({{cite:08f5e295d06f6c2351554a86a17cf4ec5dd16975}}), {{formula:587edca4-1c61-425c-92bd-0da001db18af}} is irreducible, and the fibers
are irreducible, it readily follows ({{cite:08f5e295d06f6c2351554a86a17cf4ec5dd16975}}) that {{formula:0e132d91-8691-4755-8817-bb193e146d78}} is itself irreducible.
The tangent bundle {{formula:d6c7946f-8dc1-4e5e-ac56-3b4c9aa9cfef}} has a sub-bundle {{formula:e3eb20e3-442b-4493-8632-7783dd5db242}}
made up of`vertical” vector fields. Restricting the morphism of Lemma REF to this sub-bundle we get
{{formula:761fbf4f-790d-4e62-916d-cb57b1acba1a}}
Now, we may certainly assume there is some {{formula:42333728-b0a4-439a-a4e9-d8ba44a4f7ce}}
such that {{formula:ee2c941b-d671-4f80-ae51-80b604c5677e}} satisfies the conditions of (b) in the statement of the Lemma, or else the Lemma once again holds vacuously.
In particular, there exists a point {{formula:5c19e532-5002-4094-90f3-5a2d59fba347}} , smooth in the fiber {{formula:eb9930ff-fd84-4f56-bf85-ffe07e91e1ca}} ,
such that {{formula:9736020c-e9bd-429e-8398-8060a1813fd2}} is nonzero. It follows that {{formula:b8d404a5-e077-484f-acc5-86c557bba47c}} is nonzero on a nonempty Zariski-open
subset of {{formula:5ac5b3b0-646d-432a-b74f-3e7c07f2e7b0}} ; the image of this Zariski-open
by the dominant morphism {{formula:7c14d823-c8f9-4d18-b4e9-8e6a576e9fcb}} contains an nonempty open subset of {{formula:89c7d757-0c31-45ef-83fa-98e4db7232c3}} ,
and we replace {{formula:484f36d9-97c2-4b5a-8a9b-0f6f3d5e0860}} by this open to obtain the second part of the Situation.
So we proceed assuming ourselves to be in the Situation above.
We continue to write {{formula:f1aaca25-a470-4c7c-84eb-c9b52ef7a00f}} for the smooth locus of {{formula:2b0202b4-bf2b-4a0f-a272-dc167447912f}}
and {{formula:9cdf243c-a159-4acd-b644-fed511707aba}} for the preimage of {{formula:3046f45c-feb6-4b8e-ae36-de639bb2eaf8}} in {{formula:9c7fedb9-e23f-444f-b1bd-fbb8b8fa1c79}} . Recall that our assumptions
guarantee that {{formula:5a84d780-bb89-4952-b871-c553d20f8e2e}} is fiberwise dense in {{formula:1c8d1ffa-afcc-41a8-a6af-875b4bd90566}} .
By §REF we can find
an {{formula:ad1f6ae5-0e12-4970-8570-0d7563335336}} for which the image of the geometric monodromy representation of {{formula:6d622492-fc6e-42bf-a63a-72ca29de3dda}}
in {{formula:d087a1bc-1b8c-46f7-955d-b706f5a6fdc8}} has size at least {{formula:b02b7de1-f4a1-416f-bd09-9ecd9752b18e}} (same notation as in §REF ).
This choice of {{formula:37e07806-f196-4f5e-8caf-a893d0cae3b5}} determines a finite étale Galois cover of {{formula:65bae4cd-6980-464b-b263-387d13ce4ea0}} with Galois group {{formula:891e38dd-30c6-459b-ae6e-06c4ce204201}} , which extends to a finite étale Galois
cover of the {{formula:6d85f83b-f849-4b6a-ba15-e568612f1f76}} -model.
Let {{formula:404dd5c0-b421-48c3-9ed3-081f19618c0a}} be the normalization of {{formula:f03bf12c-d248-4d4b-8506-f91bc5fc7f43}} in this {{formula:fe66f201-6ce3-4bb6-b05c-5ebd7759ecb0}} cover;
then {{formula:0a64091f-aac1-41a4-ba5c-8ae2405018d3}} is a normal {{formula:7c8ef206-3009-405a-bf7f-d11306dfebc5}} -variety and the morphism {{formula:da2662e7-96ea-4ee6-8cad-45b878737e82}} is finite (although not necessarily flat).
The action of {{formula:25aae6c9-f098-4d0f-80d7-2b12f6ca337a}} by deck transformations above {{formula:5d88d842-d8e0-4beb-b6d4-1c54bb615f24}} extends uniquely
to a {{formula:753d5c6c-8415-46ae-aae1-4fadd864a200}} -action on the morphism {{formula:aa151a8e-68fc-43bf-905b-ccf4836e9d25}} .
The morphism {{formula:6b16165a-a97a-45da-bf66-ac3fb4dc9b71}} gives of course a morphism {{formula:7553536a-60a3-45b7-a469-f01f6427790c}}
after base-change from {{formula:caa73483-7f7e-464c-941b-e0e997cae079}} to {{formula:64f80408-c613-417d-8133-95fc8e3843a9}} . The restricted map
{{formula:13c554a0-4d84-4ffb-84f3-0862652b702a}}
is finite étale and has degree {{formula:ca0af066-e40b-49ac-b025-2f36a7c4a0c2}} restricted to
each geometric component of the source
by choice of {{formula:eda1a42d-bbb5-4ef5-82e1-c13bce4db616}} .
Now this morphism is the geometric generic fiber of a
finite étale morphism of smooth {{formula:ead284ef-e321-45de-903f-b1a21d87adec}} -schemes:
{{formula:d7cf86f6-a989-494f-8a11-c7c5835842fc}}
and we want to draw the same conclusion about degrees
for the fibers of (REF ) over
a nonempty open subset of {{formula:66a4038b-f914-4ba1-b94b-f6562e8665bf}} .
This will imply the desired conclusion, for – with {{formula:2bf58512-374b-4460-96fc-295c5fb01cfb}} as in the statement –
the assumed dominance implies that {{formula:e6a15750-0f17-49db-854b-c86f6c354ee6}}
is an open nonempty subset of {{formula:671206f9-ff70-4f2f-863c-ef88da40c9d8}} .
We now use {{cite:08f5e295d06f6c2351554a86a17cf4ec5dd16975}}
(see also {{cite:f5f72ce3009bdfec9f5b167c329c67dd6b2011d3}} and references therein).
which guarantees the existence of a morphism {{formula:1e662801-6cba-488f-9c13-ded43deba0f2}} (which in fact factors as a finite étale
surjection followed by an open immersion) such that, after base change of
{{formula:445f3f68-e4da-420e-bc7b-1470f37fa1a0}}
by {{formula:dc951dfa-6735-45e4-8138-1d86628dcf27}} – i.e. replacing {{formula:bbaea216-a586-4c35-9649-ba15049db47d}} by {{formula:588d3c11-5db2-4559-a99c-a5a5cf0b5b2d}}
and similarly for {{formula:f97f3b2b-3c99-4c08-974f-06a6de6f105f}} –
the following assertions hold:
(a)
Each irreducible component of the generic fiber
{{formula:a9d82881-7ff3-46b7-b0d7-aace5fe9172b}} (with {{formula:56f3141d-9f2c-4b12-a933-bc27e5e50a43}} the generic point of {{formula:5f8c6019-86b7-4407-845a-72de2ccffe82}} – not a geometric generic point here) is in fact
a geometrically irreducible component of that generic fiber.
(b)
Let {{formula:ff6d1274-50a4-4216-8392-15d7a90363ba}} be the Zariski closures of these generic irreducible components {{formula:aad9cc57-deba-4c32-ba7a-a09e11634249}}
inside {{formula:dd8da791-e81c-497b-95a9-4352ba94d979}} .
These {{formula:051c30ef-92b1-4bdf-b6bc-0e8906a825b2}} give, upon intersection with the fiber {{formula:1820491c-751e-4b7e-97b6-9f7704d88f79}} above any {{formula:1da03c81-be1c-4b94-89e8-5b25cfed864b}} ,
the decomposition of that fiber
into irreducible components, and indeed each of these irreducible components are geometrically irreducible.
In the decomposition of (b)
{{formula:84032929-2394-4e6a-9fd9-46e7dc3fe112}}
the sets {{formula:25e3a2e7-d148-4249-a815-e0741f79d685}} are disjoint.
Indeed, upon restriction to each fiber, this decomposition recovers the decomposition of {{formula:a41bcd62-08c7-4303-9418-ee311218ba39}} ;
however, this is finite étale over {{formula:638becd6-1854-456f-b798-ef3d0a046d4e}} and Lemma REF implies the disjointness.
In particular, the {{formula:42c29d4c-82be-498e-b5c4-2241ed5d6e26}} are
both closed and open, and in particular inherit a scheme structure as open sets in {{formula:a5a4af02-6ffe-4773-a40f-00f285da7966}} .
The restriction of the map {{formula:e673f615-b72b-4376-ae11-05c0f57546c1}} to each {{formula:ca78ad6a-dba6-472e-92d0-df380db7f611}} is then a finite étale map
{{formula:6770caeb-fc13-4e91-9453-6db7f69bdaa5}} .
The degree of such a map is locally constant on the base, and here, by assumption,
that degree is {{formula:6f801f85-1705-454b-871e-16eeb8d36e93}}
everywhere on the generic fiber {{formula:7858f82d-f6d6-4f28-a449-dc129d00dc8b}} .
That generic fiber is dense because it contains every generic point of
of {{formula:ec341b78-6f7e-4441-b06e-b72c51ab2047}} , and, consequently the degree of {{formula:a2b5d5c0-69c8-439d-9fa0-2edefe808772}} is everywhere {{formula:8cfb496a-3076-4239-8b25-c12c62c20086}} .
Restricting to a single fiber {{formula:0f1538b7-f78c-41a9-b800-e7295c407ce2}}
for {{formula:9be5d630-f1ce-4b69-9379-7c0e9df31fbc}} gives the desired conclusion –
that is, the bound stated in (b) of the Theorem holds for all fibers {{formula:e256e76f-6813-458e-98af-3ab615d47f9e}}
for all {{formula:afa8643c-7c4f-46ec-acf2-883c37852581}} in a nonenmpty open subset of {{formula:331148a4-651a-427b-9cb7-1c0fa453e7fd}} , explicitly, the image of {{formula:e03d343f-21f2-43e1-b7a1-4088ac405a6e}} .
Finally, we conclude by Noetherian induction.
We have shown that, given {{formula:c6e449e1-cdf1-402f-9742-0a0abb5e06fc}} , there is a Zariski-closed {{formula:5018d5d4-28ff-4642-b3a5-e44212297748}}
and an {{formula:0493d684-001d-4a9d-aebe-942c6b256d3d}} , such that the image of geometric monodromy in {{formula:ad393b77-5d39-4e11-a64d-2ffa410d6c24}}
gives an étale cover for {{formula:7b5fbf5b-18b2-41b7-a47c-56dc3c3f0c86}} , that satisfies the required properties for
any {{formula:ef83f776-551d-404b-bf93-356c8ad4ce77}} .
There is no harm in replacing {{formula:1fe1ebca-29fc-4d91-af3f-3606fb7afaad}} by {{formula:67c60968-9844-42be-b5ca-53ec2706a22b}} , for {{formula:34ae419e-6763-4f8d-a523-f0803f52de1f}} .
Thus we can apply Noetherian induction to find one {{formula:d27b77f3-a84c-4a84-9357-10c19d8e6a2e}} that works for all {{formula:11f72419-502d-4eb6-8302-110836b46112}} .
{{formula:45627d25-44d7-4eb8-9393-2c1d24f6583e}}
Bounding rational points on a subvariety
The following result is the key inductive step.
We note that all constants appearing in this discussion are permitted to
depend on the variety {{formula:085e7391-2c69-45e1-a989-3406bb4dcfaa}} , and indeed on the integral model
chosen in §REF , without explicit mention.
Lemma 4.2
Let {{formula:de645971-80d4-49d7-aff9-ab91719e75d0}} be a geometrically irreducible closed subvariety of {{formula:48f22e8e-c4dd-49f7-ade8-0e956f33c781}} defined over {{formula:cfd2cf33-69c1-4708-b7a6-3ca1a111ba58}} ,
of dimension {{formula:c08c7278-b248-4ad5-b20f-1aa44abac8ec}} and degree {{formula:06413059-cb09-4cf7-a3dc-937377504d66}} , such that {{formula:00108f92-623a-460d-85be-81f64573ab64}} X{{formula:191c8760-c154-40e7-94be-ce9da98ebab9}} B{{formula:3b739927-1ba4-48d4-9acd-c720e6cbce60}} V{{formula:3cffc290-3313-4cf2-945b-a5f7f03b6879}} Od, (B){{formula:f3ff93b6-4b14-42cf-bddc-36fd4c31746a}} K{{formula:cf4f32de-8fe3-4897-9bf0-ddc6026ed9c6}} n-1{{formula:21ee160d-496b-4b7f-b3fc-ecee7f6de1ab}} Od, (1){{formula:39264ece-1e8c-4d9c-ac0b-8dce396f6b36}}
Choose
{{formula:25badfdc-200c-4a1a-a24a-34bd2e1e18c1}} so that {{formula:fc7610bd-376a-4c7d-8c58-7a6a3fcc428d}} .
Let {{formula:f231e9ab-4807-4fe5-aaad-4f91768ae4e3}} be the
(finite, by
Lemma REF ) set of Hilbert polynomials that arise
from irreducible subvarieties of {{formula:d5a273f9-e3c0-4e63-a8de-901651c31734}} of dimension {{formula:13f62056-6ed0-4019-b568-f1d40f4241f5}} and degree {{formula:9652ac9a-2337-4385-9fbd-3d9edc3348d1}} ,
and let {{formula:f7bd50af-b6cf-4346-91ee-167e17affadb}} be the associated Hilbert scheme. Write {{formula:07424f24-acf7-4b15-9c27-c0445e409107}} for the reduced induced closed subscheme of {{formula:c7d806aa-e096-467b-abcd-10ad261dfc41}} .
For this choice of {{formula:ec16fed5-dd8e-4932-802b-35084e6e91e0}}
take a finite group {{formula:a2e74154-efe0-4452-b645-5fa8dbe776c6}} and a finite morphism {{formula:c435adaa-513d-4fe8-a56e-76e8da2c969e}} as provided by Lemma REF . We note that {{formula:45759194-3934-49d1-8b1b-c7caa8e5cc16}} , so the passage to the reduced subscheme structure is irrelevant for the statements on complex subvarieties proved in Lemma REF .
For every {{formula:3e103dbc-7ec3-4adc-a6c2-af905d2987fc}} , the action of {{formula:78acbc29-28ff-42a2-9592-eb115923313c}} on the fiber of {{formula:aebc0e9c-410c-4dc2-b9d4-00c8207f4cd8}} over {{formula:76691b3b-c2a6-41aa-a9a9-374b15449697}}
defines a homomorphism {{formula:d655f4d2-3665-4802-aa9f-1df20ddea532}} , uniquely specified by the rule
{{formula:461a2214-a6ef-4b98-befe-d57e07f1e130}}
for {{formula:50a1f3df-735c-43d5-9e7e-b3fb88822737}} above {{formula:2ea8af64-9a87-44cf-b403-8dafc81e0b4d}} and {{formula:7f3f5460-dd8b-46a2-b21e-5d3141dd2ced}} in the Galois group.
If {{formula:5b79fca8-c8a9-4b7a-9378-051f6df3c00d}} is {{formula:ba227732-d5b9-4600-9d60-ae27148c7509}} -integral, this homomorphism
{{formula:a69e0375-6c38-46b4-829c-ff700ab2ae8b}} is in fact unramified outside {{formula:bee45b4f-6e9c-4bf3-9431-6a95290d38a3}} .
Such a {{formula:ef97ec01-6084-4df0-b625-366f26ba5b76}} can also be used to twist {{formula:cfe34ff8-3f16-4edc-8f75-51fd9567772d}} , namely,
one modifies the Galois action on {{formula:bdae5b80-5e71-427c-9971-a09ba0345887}} through {{formula:14c59025-bb33-4bc9-9773-72eff6f760a5}} ;
and then (REF ) means precisely that {{formula:41ec6a46-37b5-4da6-956d-f84c2d8799d6}} will lift to a {{formula:bedfb480-41ed-42c8-a7d4-08cb7ac24532}} -rational point on the
twist of {{formula:60cc4baa-a3bf-4e5e-b603-1b3d43675895}} indexed by {{formula:89121b48-45a5-445d-a982-cb88a998df75}} .
(See {{formula:93903310-6fd2-4096-8045-ee0e678162f0}} 4.5 and Thm. 8.4.1 of {{cite:650b0fc784a317e90717ad725a86ccc546685a32}} for further discussion.)
There are only finitely many homomorphisms {{formula:fbee293a-7e6c-4514-add8-87fa0fa53fd2}} , unramified outside {{formula:43044932-8936-41ce-a3db-b4f8aa631acd}} ;
call them {{formula:584628e7-e31c-4b7b-b30a-a261d796ad28}} . This list does not depend on {{formula:b7331a5d-5ec9-4f0d-bb49-5da270dcb884}} .
Each such {{formula:276cf1db-f7c7-4e77-ac4c-17d6cd8d8836}}
can be used to twist {{formula:0868d681-74c8-43bf-be35-c96aee803bc1}} to a map {{formula:bda99721-9ccd-4442-ab46-25959aabe5dd}} .
Our previous discussion now shows that any
integral point of {{formula:fe15cda7-2a0c-443a-89cb-00751837d92e}} lifts along some {{formula:bbb13c6a-76f2-4fba-8560-7e9a0ba831ae}} to a point of {{formula:15d24100-dc0e-4911-b8b4-be6c95789eae}} .
For a sufficiently large integer {{formula:a5fce021-3b9d-4294-ae82-8b3652d089fd}} the pullback {{formula:688327e7-7908-4c78-8aa4-55f75b6ab966}} is
very ample and defines, after fixing a basis of sections, a projective embedding {{formula:d13d1c55-e40d-4fef-a42f-7399af404292}} .
Now the data of the diagram of {{formula:b01dd45b-d5b1-4624-a57f-17d69eb18954}} -varieties and line bundles
{{formula:e2c0f548-b21f-42ab-9be4-5f39207cfe2a}}
depends on various choices,
but these choices can (and will) be made once and for all depending only on {{formula:28d33c58-e9d6-490c-93ec-3a5e82247c45}} .
Then for {{formula:ad9dca1d-040b-4271-99cc-3b6cfae5bafb}} we get
{{formula:397c9bcf-51b3-4ab8-9e4a-c12d05986a2c}}
where the symbol {{formula:ffdefb14-8400-4a8f-93f6-68fddc7dba40}} means that
the ratio is bounded above and below by constants that may depend on {{formula:0cde16da-075b-4666-84ca-5243249273f1}} .
Since there are only finitely many {{formula:ab7123db-20ca-4447-bb8d-4eb0f129158b}} , and their coefficients are bounded in terms of {{formula:6956a41a-74ee-4fa3-9af7-75e6e4f811af}} and {{formula:7c4c1446-c547-4e09-bb29-b59e75b1ffed}} (and, as always, {{formula:e55bdbf4-c6dc-475d-91b1-3a911aae2c4f}} and {{formula:57eefc71-b182-4614-ac2e-932080d09e04}} ) but don't depend on {{formula:39727b53-c266-4d23-9186-eaf093b29dc1}} ,
these constants depend only on {{formula:2d43ef0a-0288-4e69-a6b1-d6756eecd40a}} and {{formula:26918273-f778-4dcf-8cbc-ceaa96633910}} .
Therefore, we have shown that the integral points of {{formula:2ae35c9e-97b1-4451-957b-e096da14a897}} with height {{formula:bb1472ab-d2ed-4b5a-945c-8fb137137f4a}}
belonging to {{formula:007d9655-6345-4846-a0cd-d7dd6f889196}} all have the form {{formula:fc1d06c4-e833-4702-81d8-0168aa79d175}} , where {{formula:1754f43d-9e42-434b-8c0d-773b10af78c7}} is a {{formula:0cfcce1f-e5ec-4ea7-9dd9-54c3070e1792}} -rational point of {{formula:1590a5ba-1a32-44ec-a4de-def3c23b1e27}} with {{formula:0313a759-7180-4818-9014-4df5a4c2144b}} .
It will suffice to prove the conclusion for those {{formula:056148b9-ddd5-4963-9282-10957593adc5}} for which {{formula:5f0cd2ff-1b03-4295-a85d-3d3c86e5ed44}} is a smooth point of {{formula:f81c2565-0ade-4ef6-8ad1-07cff8a258c5}} ,
simply by including each irreducible component of the singular locus of {{formula:4923e84f-221c-42f8-92e3-1f3077b00366}} in the list of subvarieties (see Lemma REF part (d) for the necessary bounds).
Let {{formula:a55714b1-540f-4b54-a34a-84302a2b5c84}} be the (open) smooth locus.
Consider those geometric components {{formula:e519e14d-fa3f-4c2d-bec9-780cbca2399a}} that have a {{formula:dfd2b1e7-d354-42e6-8916-4e4d214f2e24}} -rational point.
Because {{formula:31f70ad2-23c2-44a1-93da-bd06eedd79a7}} is a finite étale cover
of the geometrically irreducible smooth {{formula:e94b56b8-a7ab-46d1-83d0-ebdbaf987480}} -variety {{formula:a2a7074a-0919-4d3f-833f-41f2636c6541}} , its
geometric components are pairwise disjoint (Lemma REF )
and permuted by the Galois group; so
any such {{formula:6f0a0f9c-a1ba-420e-9ebf-9a12106f6718}} is defined over {{formula:10f33821-9c28-4143-8fac-07fb92f7a93d}} and the number of such {{formula:208de535-af69-476b-9ee4-be72ce9c5c60}} is bounded in number
by the size of the group {{formula:29d87ce7-5119-4701-8cc8-9f2c78b38c5c}} .
The Zariski closure {{formula:b429555b-787b-478a-9b59-b62a92db1c39}} of any {{formula:0c875ba9-b702-4c69-94c7-34201b20bded}} is again geometrically irreducible and defined over {{formula:ecfc4454-1f50-41f6-8235-845e2d14f01d}} ;
we understand it to be endowed with its reduced scheme structure.
The map {{formula:d030097c-6a4b-45dc-a685-679e735f38c3}} induces a compatible map {{formula:7c935804-29ac-469b-940b-529e71d112df}} ,
which is dominant since, by construction of {{formula:726c80fd-1acf-45ba-b07a-f71d9e77b9b8}} , the image contains a nonempty open set of {{formula:dbe8caf5-0f54-46dc-8667-1d1ac61d53bb}} .
Indeed, {{formula:edec910c-f597-44bc-a285-8d38024a3d41}} is étale over {{formula:e63a465f-679e-482e-aa2e-96ed1d5fd5c2}} ,
with degree between {{formula:acea4c45-c0bf-4b55-89d3-43af7e5a9a58}} and {{formula:3a58a9e6-9895-49ca-85af-266d54c2e171}} ; the lower bound comes from (b) of Lemma REF ,
using also the fact that {{formula:eb3c7646-2909-402a-86ac-643a7b7bf6cb}} is a twist of {{formula:e3afc47b-cb6a-4e61-8d66-c5f9edaa9caa}} .
The degree of {{formula:3863e0b6-3fb9-46ba-80d4-6883dc3e1c4c}} with respect to {{formula:aa4223f3-7618-4173-bca8-be61a306dd4d}} is {{formula:07db6eaf-f4d9-4930-a179-98921fdb3192}} , and therefore,
by Lemma REF
the degree of {{formula:965f8211-a49a-4fc8-86d6-6135d3616ab9}} , considered as a closed subvariety of {{formula:9bc75e06-5175-47a5-85e4-3cf30e93cb74}} via (REF ), satisfies
{{formula:edff8862-2474-4bff-9731-148b8de63b70}}
We apply Theorem REF to
each {{formula:e9296ab2-3a2b-4611-83bc-c77407fa1518}} that arises in the above fashion, i.e. to the
Zariski closure of any irreducible geometric component of {{formula:7316079b-e920-484d-b6b7-d9bdcdb36d68}} that has a {{formula:76784eec-6a22-421e-b418-18883fcf5000}} -point.
Theorem REF and our choice of {{formula:6052761d-c522-4768-8d02-657c769abb1a}} implies that the
set of rational points of {{formula:c07ec6a0-c609-4ff4-98a8-08bbb003a9e6}} of height {{formula:f75ce63f-7d04-4f41-8d29-8a7d11f195e7}}
are supported on a set
of proper closed subvarieties of {{formula:3bcbbdf1-2824-4993-a7d4-e1b809ac846d}} of degree {{formula:29aa1905-7417-4760-83a7-4968ba3a4821}} with cardinality {{formula:8f05d7a1-c153-41a2-b33d-6994c3da7add}} . These subvarieties are defined over {{formula:7346173b-ad70-4b99-a165-e4fa9ec56a33}} and need not be geometrically irreducible.
For any such {{formula:92c82c87-5bcf-45b2-b52f-49fb5af0cc31}} and any such proper subvariety {{formula:cce61807-ab8f-4f2e-9864-5137e0f5fe2c}} ,
the scheme-theoretic image {{formula:fd078eba-0b16-4b24-ac8d-a147d27bf234}} under the finite map {{formula:8f3e9bfb-27e9-4ecf-9b6e-1372c1b07214}} is a proper subvariety {{formula:50263377-387b-47c5-a0b1-834884b2486d}} , in particular, of dimension {{formula:5276d92e-7b44-4293-8a3f-d8ec471c4c75}} .
Moreover, {{formula:d0a9d9c1-c09d-465c-a3a5-f00f432feea7}} restricts to a finite map {{formula:0d1cc871-2b44-455f-9053-bd827670e280}} .
By Lemma REF the {{formula:b934d4a7-6aaf-4676-b98f-ef3872473fc1}} -degree of {{formula:358079a4-d675-43a7-9dc3-b75fbbf8d7ce}}
is no larger than the {{formula:b0a924e9-5f25-4733-89ad-0dc01694a513}} -degree of {{formula:3a0d0726-2ddf-4550-9571-c1c4a5d10e97}} , in particular, {{formula:177cc4bd-eb20-43b6-90b3-b971764ca0f4}} .
The number of maps {{formula:0ea4edbd-5648-4691-b16b-c7b08e9a688b}} depends only on {{formula:24e632d8-ef2d-4276-a1e6-b0ad55371825}} ,
and the number of {{formula:5beaea8e-12c8-43bb-8e00-968bfa5d5e4d}} arising is then at most
the number of {{formula:d1207e90-4b12-41d7-a699-4b6440299785}}
multiplied by the order of {{formula:3a438548-9a83-48b1-aa5b-1f42ba13b8e6}} , which is again {{formula:129b502a-0b86-40d8-851a-d660e411d879}} .
Consequently,
the number of {{formula:57e1e835-84cc-4742-8d7c-812cf34994c8}} arising as in the prior paragraph
is {{formula:8f989aea-75f8-45e8-8713-a089b9e7b0c0}} , concluding the proof (after the obvious
scaling {{formula:1509e58d-2f13-44a2-a10b-2b26e3b06760}} .)
Conclusion of the proof of Theorem REF
Fix {{formula:8e0c1ba8-2854-4a4e-aca5-a0173b573646}} .
We use descending induction via Lemma REF . The inductive statement is the following:
{{formula:0581ae95-24e0-4ac0-9dc2-022d7088064e}} : For every {{formula:52047227-014f-49b9-8657-45e75c79e122}} with {{formula:066d69ba-216b-4e74-b87a-958de4b1f628}} , there exists an integer {{formula:079761b4-e6b4-412b-8e1d-a247f957526c}} with the following property: for all {{formula:21ea82c4-aba8-4c0e-aa97-4e1ee1b3d6e7}} ,
the {{formula:5591a990-fb02-4c37-9c65-f4a2266c8277}} -integral points of {{formula:46a4450f-ac5d-4887-9369-c10c9380ff12}} are covered by a collection of
{{formula:ff6b6d42-2eab-4df2-8890-d10c22ce1a87}}
irreducible subvarieties of {{formula:7af10b7f-e085-4b38-b98e-93451f3a15de}} , all defined over {{formula:c6f7bdf5-45b6-4db8-ba9e-81be0a1f9d28}} , each of which is either
–
{{formula:2e4d7461-6153-4030-bd6a-3907669caa94}} : a subvariety of dimension {{formula:ba9e1331-fe86-48d2-9ca3-d39e61cb98a8}} and degree {{formula:3673755e-3525-4a9a-9347-a891129796b7}} , or
–
{{formula:c173a476-517b-4ab8-a329-29c2386f06bc}} : a geometrically irreducible subvariety that is contained in a single fiber of the period map.
The base case is given by {{formula:0f514fb0-d2ae-4ce9-b78c-596c43298bdb}} , in which case, of course, the single subvariety {{formula:d53effda-f113-4a07-9713-a4f0e24f5158}} suffices.
The implication {{formula:e2d0249e-46e7-4892-97fb-89117f871477}} follows from Lemma REF :
Let {{formula:40296f38-cc29-40fa-bae3-bf45e2334489}} be the collection of {{formula:16dbcf2a-4406-45d2-af88-1ec9df5a5032}} -dimensional varieties in the statement of {{formula:51e0e61c-dfff-47f1-9644-a88facecf5cc}} .
For each {{formula:0dcb4ec1-6f95-4029-9f7d-03c6d87802c4}} , we will
construct a set of varieties covering all the integral points of {{formula:254abae6-8647-4e45-8b19-315acdd96a57}} lying on {{formula:655175b9-1271-4012-9cfc-ee3e006f6769}} . We subdivide into cases:
{{formula:154867b0-442c-4076-ad58-f00296efb805}} is not geometrically irreducible.
In this case,
we take the set {{formula:f3b29575-8ec6-431d-a366-29bb0b592a7c}} of subvarieties given by
part (c) of Lemma REF .
These varieties number at most
{{formula:df593ce4-e822-4506-bf31-3761dd042dc1}} and they have
dimension {{formula:920eb3cf-9049-4c77-abf0-a8a9b40b2a62}} and degree {{formula:bf88d0e6-0bfd-4315-a4ea-b58ac32afafe}} .
{{formula:2282f2f2-e88b-421b-8ffc-f4ef45474729}} is geometrically irreducible but {{formula:b7821e78-fe18-47a0-b7bf-a7e04fb8157a}} is contained in a fiber of {{formula:1fa6fe77-69a5-492d-b571-7a7c681a8dee}} : then
we take the singleton set {{formula:8d79f4c0-3057-41f0-a396-8954ae5b03e0}} .
{{formula:a311084c-4f3c-406c-b0b0-23df207e748d}} is contained in {{formula:b2258530-0472-46f2-836f-4fb8928503d0}} ; in this case
we can take the empty set {{formula:fec3b96a-756d-43df-8d7e-63f595b2cb08}} .
{{formula:d0b27e1d-4580-440a-92f1-5967f1f98dd8}} is geometrically irreducible and not contained in {{formula:95e92b5b-4931-4fb4-8bc4-e3a7f25f629e}} , and {{formula:29eb343e-dcc2-42ff-b946-95409c5c3d60}} is not contained in a fiber of {{formula:e54e85ff-0a9a-4814-8c8f-42e76b5aaec5}} ; then we may apply Lemma REF to show
that integral points of height {{formula:7ef4413c-4589-42ad-ace4-ded4d6c3ba96}} on {{formula:8e09728d-2bdb-4197-ab2b-4e5e6fdba76f}} are covered by {{formula:f6867853-c03d-4e1e-9b18-f8eef74e934f}} irreducible {{formula:055fa811-63dc-441c-b3a2-beb73664a300}} -varieties of dimension {{formula:70496bd1-73e7-47d0-9034-3d71ce3e032c}} and degree {{formula:51e801e3-355b-472d-95ba-9ef1d3071521}} .
We take {{formula:a9172471-51c5-4586-890d-16e78d0eb06a}} to be the largest of the implicit constants {{formula:c3ad8dfc-56d9-457c-8793-653456137303}} and {{formula:fd3583c7-30da-4010-b134-f3c0b080a1c1}}
appearing in the above proof. Then, to sum up,
by {{formula:b6ca142d-e008-4244-a5e6-173c7f9560e2}} we know that the {{formula:50d7940e-d47f-43b9-9d94-a10e11a1d362}} -integral points of {{formula:75792c49-c9c6-4a89-a479-02c8694b7919}} of height at most {{formula:8b3a11e2-7e82-4a8d-9894-a3b087a74e53}} are covered by {{formula:b6a77c56-245b-4163-8222-da5031a5e63f}} subvarieties {{formula:0dbb35eb-5b34-4a7a-b06b-0932778823a6}}
satisfying either {{formula:28145f23-fe91-4bfa-8f9d-11b9f1b34c1e}} or (b),
and we know that for each of those {{formula:b3e1fa6e-6bf1-4d5d-ba61-a8e01eeb1b62}} , the subset of those points lying on {{formula:d83445fe-f268-44e3-a6ac-9f904b65513c}} is covered by {{formula:d4dacea2-ac17-4943-8310-62fdf04b875d}} subvarieties
satisfying either {{formula:5f43de01-a740-488d-9e55-f2e79b509e69}} or (b);
together, these facts yield {{formula:e9aace5a-ca52-4f60-84d1-f339c9fac47a}} .
We emphasize that this is the point in the argument where the uniformity in Broberg's result is crucial. We have no control of the heights of the varieties making up the collection {{formula:2f9e1a98-041d-4f36-9210-ebd6b83b90c8}} , and indeed these heights will grow with {{formula:f783662b-bbe0-4427-ab1f-5961ff0c6c4e}} ; but since the implicit constants in Lemma REF depend only on {{formula:e80be482-03a4-4d24-9be3-a3a686f1b604}} and {{formula:f67b5537-bd61-401f-9e09-6446230291b8}} , not on {{formula:44db2e37-2963-4bd9-802a-1eaf6192f012}} , this lack of control does not present a problem.
The case {{formula:9b957059-6b81-4198-a486-26e15a92373d}} gives the Theorem.
| r | 1b31d2a0373d0be729a363e1b8634c39 |
The first part of our experiments (sec:Experiments-Quantitative) comprises a quantitative evaluation of the accuracy and performance of our optimized implementations on the KITTI 2015 stereo benchmark {{cite:8b108117aa9ef8297cdc90c00279e7799090bef4}}, as well as the Middlebury 2014 stereo benchmark {{cite:f9832be4cfb6bb11b18e38ac04ea895814da6744}}. In this, we have also evaluated and studied the effects of different configurations of the processing pipeline for real-time disparity estimation, e.g. the effect of reducing the number of *SGM paths or the improvement of the additional subpixel refinement. We compare the results of our implementations with state-of-the-art approaches and analyze their performance with respect to the power consumption of the embedded system.
| r | a2ef96cc519864e3a5015d25f711d1a6 |
Limitation. 1) Currently, our method does not applicable to multiple input voices. 2) Since we do not focus on fast inference, the high-resolution rendering process cannot meet the real-time requirement, but our model can benefit from some concurrent NeRF acceleration works {{cite:54e4c4af9cf5adb92b414007fcffbc126b4540ae}}, {{cite:0a28454eb729797aca25799f33e04395054265a3}}, {{cite:e6977fd8f5f82e2fecbb924b94f980481b659936}} to alleviate this problem.
| d | 1c38d4295613bff4ca7600ecc5835a47 |
There are millions of photos of natural landscapes on the Internet, capturing
breathtaking scenery across the world.
Recent advances in vision and graphics have led to the ability to turn such images into compelling 3D photos {{cite:449dc0d3d8238d40adb58aba785c27e1f6fbc7ac}}, {{cite:69f7e4adbc1b1614312b0db86650a4b7bd7f41a6}}, {{cite:accdfe26115465b4e140ee7fa910f8c526c14966}}.
However, most prior work can only extrapolate scene content within a limited range of views corresponding to a small head movement. What if, instead, we could step into the picture and fly through the scene like a bird and explore the world in 3D,
and see diverse elements like mountain, lakes, and forests appear naturally as we move through the landscape?
This challenging new task was recently proposed by Liu et al. {{cite:c67887ad5923500554db974984889d43201fab5a}}, who called it perpetual view generation: given a single RGB image, the goal is to synthesize a video depicting a scene captured from a moving camera with an arbitrary long camera trajectory. Methods that tackle this problem have applications in content creation and virtual reality.
| i | f13aed5d6f077e649f154dbcff95b1f4 |
The Tikhonov method {{cite:176d7cc00ea1bb184a3c29157f0f04e9da841c73}} is to solve the above question takes the form
{{formula:e3766b73-33a5-4bd4-b56e-fc5e55db0b55}}
| m | 19b6b0f8eb5e7a7ab21c909b76aa56f2 |
Actually, our counterexample shows more, namely that the following substitute of (REF ) for “long-range" potentials (i.e. {{formula:4217ea53-a871-4fe1-a413-05b070020b6d}} ), due to Frank {{cite:0fe269e938de11212bf6592ceb67b4fd630e70b0}},
{{formula:85049b10-2cd4-41a2-97a2-0c39f52d69da}}
| i | 60da32b666d24faad98b0d413f84f33f |
DBSCAN is an effective algorithm to discover clusters that present unusual shapes even in large spatial databases{{cite:c01b1387ed947963e265dfb6062f998147c9beca}}.
| m | c118bdd291ce262e54d78dc2c6ae832f |
The recent experimental discoveries of superconductivity in rhombohedral, or ABC-stacked, trilayer graphene{{cite:570c35396383065ecde6b9ce23eeaa1b10016913}} and in twisted graphene bilayer systems {{cite:925557e0ae3cfb13ebbc665b81ea2c88649e31c6}} have received a lot of attention. A vast range of theoretical proposals has already emerged exploring the mechanisms and symmetries of the superconducting (SC) state in these all-carbon systems {{cite:14a3268690ff081245edaefd7d795e0146eaa768}}, including the SC pairing mechanism being both phonon-mediated {{cite:bdd5dd94d3874aaffa1293dcb3bb288a93a68c46}}, {{cite:1ecb6bc6aa9ac238c7a2e433caac238699cd705a}}, {{cite:e30e3a84b6c865ece043cb9b2b0a42e327109692}} and electron-interaction mediated {{cite:f6523f92d8dd702ab5a72ee0cb4189296bad4ec8}} and various SC spin-singlet and spin-triplet order parameters, ranging from {{formula:37cb24c3-c2f6-473d-af10-ec99b6009c76}} -wave and {{formula:898612bf-e587-4504-8cef-0653712f226f}} -wave to {{formula:ea7a4446-a95b-4090-bbcd-f88ddfd49789}} -wave and {{formula:8e285a0c-2c55-4861-a570-a8e1619c1e4a}} -wave symmetries {{cite:cfeb8c91649a6e9d3c7cfbc34cc933c4d0875df6}}, {{cite:5c23a5816e155aefd202ca26e6dec433968673c0}}, {{cite:79bff7c6762196e7a2aaf163482824fc9d5c0327}}, {{cite:33c9a1d4d4d3840cdbe27c120323dbcee129ac97}}, {{cite:b23973a1594d3410b74a3013890ccb5a9faa09e6}}, {{cite:0f1c62dfe4beebf942b41c63a8e841c638a57231}}, {{cite:09ee4add9f3cb9e3a47e67e6fbb557040788d6e4}}. However, at present, there exist no experimental definite confirmation of a specific mechanism or pairing symmetry, nor an emerging consensus concerning these issues.
| i | 5a83482e76ba4d5188bb78e08aee9f5b |
up to subsequence, {{formula:24ce8d56-7365-4222-ba7a-88a2586993cf}} converges to some {{formula:eee07300-a8a3-4a5b-b54c-394179c4ae6c}} with {{formula:79f156d4-f35f-4455-9161-fe4649078ec1}} . From {{cite:3807fa58e8d777edcc7d3daf720563040d62245d}} we conclude that {{formula:73125aca-9eb6-4753-9160-50b4f233b91e}} is a bounded {{formula:0314ee8e-64e8-4f0c-ba82-596927ecfaae}} -periodic solution of {{formula:7938e412-3564-4e02-94ee-0b430bbd4e9d}} for {{formula:f1c6da30-adac-4925-9707-4e25925a6c74}} .
| r | 019a98fae9d43d42f0c3ab16e701ff71 |
where {{formula:5ff33d99-f7ac-4da6-90ee-9ebcddf1bf1d}} is the minimizer of {{formula:de7977e3-38b8-47be-90b5-9397c3b88a32}} obtained in
Theorem REF . The function {{formula:e48a3610-f4eb-49eb-97ad-15285fc65d4d}} is named as the
regularized solution to (REF ). Let {{formula:c705cffe-1197-47ec-a3e9-4f2d613bcb0a}} be the solution to (REF ) with {{formula:e28e3d6f-994b-4909-901a-955bdfbf8492}} and {{formula:4a316387-38ba-44a6-bb87-48fda959230c}} replaced by the corresponding
noiseless data {{formula:13ec9c5f-f316-49e8-8f62-730f975e5bd8}} and {{formula:a1d88a12-bb6b-45f8-91e5-d7d0a26592a9}} respectively. The following theorem
confirms that the minimizer of {{formula:703e860f-f3a8-4a51-84ae-263b9ec51495}} can be used to
approximate the solution to (REF ) via (REF ). It is a
generalization of Theorem 4.5 in {{cite:f10fd27e1336b030355d0d5144c7c89d374e2abb}} and
Theorem 5.4 in {{cite:f10fd27e1336b030355d0d5144c7c89d374e2abb}}. In fact, in those
theorems, the function {{formula:8a227f99-076f-4054-b9e2-f208a646e0da}} has some specific form and does not depend on the
first and the second variables {{formula:71e60e30-6132-4a42-826b-a65a2dcdd07e}} and {{formula:8e7e6798-1760-4bbe-9aa7-b6b7d79947b2}} .
| m | ea7a204b48535715641865e4a724d8dc |
We evaluate our method both qualitatively and quantitatively on video pairs of walking animals from the YouTube-VOS dataset {{cite:53e60f0b45cdd2a93d103366fed9a42aec68fec6}}. All pairs are challenging as they contain different shapes such as a cat/fox or a deer/horse. To demonstrate versatility, we also present results for humans and flowers, and for synchronization of short GIFs. For segmentation, we used ground-truth if available or extract it using a pretrained network, see more details in Appendix . In addition, we show that our learned keypoints have semantic meaning by performing simple editing. Lastly, an ablation analysis is performed to illustrate the effectiveness of the different components. For all experiments, videos can be found at our webpage: https://rmokady.github.io/JOKR/.
| r | 40c7f9bbd67442e15d3080cf9c35ce82 |
where the sub-kernel {{formula:d188f2e1-17d1-461b-a463-a269842a0ae6}} is given by (REF ) with {{formula:e5bf6ae3-fd27-4243-a27c-15812310a85e}} and {{formula:aa3f66cb-fad4-44a7-ba20-5c3d03397923}} . Thus, problem (REF ) is transformed into the problem of determining {{formula:ee60d8d1-faa1-432f-b4d8-6f5f22fd5399}} as well as {{formula:56fca8a2-e0ab-44d7-b6c3-d22f08bccb51}} . This problem is known as multiple kernel learning {{cite:e24b0d05ea8ab84e5c7c9f67722f58520ed27bd6}}, {{cite:dc71c195704ff90ae467b1fb677fef8241125a0f}} in the context of machine learning. We also denote the vector of {{formula:bbec597c-f1d3-4945-bcfe-46a5c11b73f0}} as {{formula:6d49b4a4-4662-4cc6-9305-5a5644a2c5cd}} .
| m | 817c96a9852e81a84ff07d7661ffd125 |
This theorem enables the estimation of the gradient of the value function analytically, whose evaluation in the context of sample complexity scales only logarithmic in the parameters {{formula:b16bdfd0-197c-4170-b0e7-eeadad274bcb}} of the policy {{cite:1ff7a82ffd1f176f0eda1f4f95bfec51038b4352}}.
| m | 792fc47f0a6534df0762e91ae1076a85 |
A novel method for cell detection and counting, called ACDC, was proposed, by exploiting a fully automatic pipeline based on the watershed transform {{cite:2546961f85cb10b8679d9d3fcec894d05e6d2fde}}, {{cite:0376060002e12440d31dff88a4b4972c44574849}} and morphological filtering operations {{cite:dd685c6fba4802c925a024220299e96a81e96fd3}}, {{cite:3f8cc5a143014aeacfc2837822b118bc68b206fc}}.
We benefited also from the edge-preserving smoothing achieved by the bilateral filtering {{cite:92770f8d240225f6b5e63dfdd1c2c4114fc18f11}}.
We tested our approach on two different cell imaging datasets characterized by significantly different acquisition and experimental conditions.
ACDC was shown to be accurate and reliable, thus representing a laboratory feasible solution also thanks to its computational efficiency.
The current ACDC implementation can also distribute the computation on multi-core architectures and computer clusters to further reduce the running time required to analyze large single image stacks.
To this end, an asynchronous job queue, based on a distributed message passing paradigm was developed exploiting Celery and RabbitMQ.
| d | 683c0e15593553920f042602d8568f6d |
While there are many definitions of robustness, the focus of our work is on distributionally robustness, that is, being robust to the distribution shift of the local data distributions. We consider a distributionally robust learning perspective by seeking the best solution for the worst-case distribution. Another key focus of this work is to investigate the fairness of the performance across the different devices participating in the learning. Fairness aims to reduce the difference in performance on the local datasets to ensure that the model performance is uniform across the devices participating in the learning process. In the FL context, achieving fair performance among devices is a critical challenge. In fact, existing techniques in FL, such as FedAvg {{cite:20fbe57ef252f4d26debb8ccac3f674948760df7}} lead to non-uniform performance across the network, especially for large networks, since they favour or hurt the model performance on certain devices. While the average performance is high, these techniques do not ensure a uniform performance across devices.
| i | e412750fc1f0d169a5d9ed1a8db5b02f |
In recent years, several studies{{cite:50ef947c1b78ab2a1c0c835025a7274dffb2c5be}}, {{cite:8f74bdb719dfa79269b5d8b76de383ccdd6804a1}}, {{cite:24eecb3cb9a6f5be051335640a01710af8b6f82f}}, {{cite:3a732765b2e10e1fa009fd907e2e41b783bb1033}}, {{cite:713680dd26582615fe8a53d4a57203394a648cc0}} has been proving the efficiency of the non-equilibrium (NE) techniques, make free energy calculations much more accessible{{cite:50ef947c1b78ab2a1c0c835025a7274dffb2c5be}}, {{cite:24eecb3cb9a6f5be051335640a01710af8b6f82f}}, inclusive available in the widely used LAMMPS molecular dynamics (MD) package{{cite:895b1be99f6d045432f3fe123d3a9ec8a6aeaf47}}. Contrary to equilibrium TI method, the NE approaches estimate the desired free-energy difference by traversing the thermodynamic path between the system of interest and the reference in an explicitly time-dependent process and have shown to give accurate results using only a few relatively short non-equilibrium simulations.
| i | 58128bffbf726d58b5535db7b8ba6c67 |
Theorem 1.1 {{cite:5ef56e749bfdb2d489f19fea6ac7252862e9d3a5}}
Let {{formula:196465b5-50cf-4ac9-bd25-65dc3de92602}} and {{formula:23aa2c97-c57e-44c1-ad67-1ef24791a7ed}} be separable Hilbert spaces and let {{formula:272b8214-5f82-437f-ac43-5d433387c744}} , {{formula:548cb672-2212-4629-8070-447efd35cf63}} . The following statements are equivalent:
| i | 5e08e1277ca50b7f807b2cabd4eb3f93 |
This subsection highlights the results attained by measuring three different metrics to judge noise reduction and the quality of the reconstructed low dose CT images. We use the following metrics for the evaluation - Peak Signal to Noise Ratio (PSNR), Structural Similarity (SSIM), and Root Mean Square Error (RMSE). PSNR is targeted at noise reduction and is a measure of the quality of reconstruction. SSIM is a perceptual metric that focuses on the visible structures in an image and is a measure of the visual quality. RMSE keeps track of the absolute pixel to pixel loss between the two images. We compare our results, examples shown in REF , with architectures that share similarities with our model in the sense they are based on a convolutional architecture. As seen in Table REF , CPCE {{cite:23330faebc2bf40b3e660eb774f170bfed1269e6}}, WGAN {{cite:e327e71d5e1e04e24f4456a7b8454bdb6a67c999}} and EDCNN {{cite:79ee919aa9b80887da06044ba6a54e3e44408eac}} like ours use a combination of commonly used losses to train their model while REDCNN {{cite:0547f2fbf15333244c095b8555487d2ddbf2e99a}} only uses MSE. Table REF shows that our proposed models, Eformer and Eformer-Residual, outperform the state-of-the-art methods in both the PSNR and MSE metrics, indicating efficient denoising and our comparable performance in SSIM also suggests that the visual quality of the image is high and important details are not lost in the reconstruction.
| r | 90c191848551a1f796d6f58fef787e0b |
Human intelligence is often understood as a process of turning experience into new behavior, knowledge, and skills. Humans achieve this by constructing small models of the world in their brains to explain the sensory experience and use them to infer new consequences. These models can be understood as turning experience into compact representations: functions of the experienced data which are useful for a given task.
Representation learning {{cite:fa693d55696002cb8b44465e0d7196afb492ee46}} aims to mimic this process with machines, via studying and formalizing what makes a good representation of high dimensional data, and how can we compute it in the form of an algorithm.
A central topic in this research field is disentangled representations, which advocates that a representation of an entity should capture the different latent factors of variation in the world where that entity is observed.
This abstract concept has led to several formalizations, but the community has not yet settled on a common definition.
| i | a6c7afb0a4f471011156811702f76cda |
The idea of variational inference is to optimize the parameters of a variational posterior in order to come as close as possible to the true posterior. Following {{cite:9300e75b0a997de748141a8de7d4e1a7d85529a4}} we choose a Gaussian variational posterior,
{{formula:d4a4f95d-d875-4306-b8b2-b962541f614b}}
| m | 87f82d4d8edf09bfa5ef6039b00694ca |
We primarily consider methods that utilize GNN models which adopted messaging passing operation as their main backbones. GCN {{cite:c2b2eeb70d28ea17a03f2347947179b6d089141e}} and GraphSage {{cite:cf392d566b0af58c3805b446783335a265557c23}} are methods where convolutions are strictly based on first order neighbour aggregation scheme for each layer. GCN-Cheby {{cite:650ba7a06ca45e4e87a3d12472725e7132e33840}} generalizes convolutions with the help of k-hop localized spectral filters. GAT {{cite:3cad9c251330940251c53cb62a74584d65cd0017}} adaptively aggregates immediate neighbour information using attention coefficients which are also derived from node features. MixHop {{cite:df5eca800ff45308c1778847dd516bfe41029b92}} and H{{formula:ea7e2ceb-1bce-49cf-84cb-fc12adf35060}} GCN {{cite:62e1aad046c57d0dabad4f3bbcec4010b646f646}} generalize the node aggregation beyond the first order neighbourhoods and dynamically considers node features {{formula:c37f4b8c-133b-4dc8-88b6-63da00481d38}} -hops away. It is important to note that all baseline methods operate on the original graph and thus have access to only proximity information albeit in different forms.
| m | b5b67e9a93e73902dfe05b5d2dd76151 |
In the last few decades, RMF formalism emerges as one of the most important and prominent theories capable of interpreting the finite nuclei results of heavy-ion collision experiments and the data obtained from the astrophysical observations adequately. A lot of RMF parameter sets has been developed in the last few years, which endue us with different types of equations of states, like, NL3 {{cite:75ae8d0b7c1be894171c9d1cf370bdfd2fb73bfc}}, the most familiar and fundamental RMF parameter set, provides the stiffest EoS and others like FSU-Gold {{cite:f3ceb18cb579022c06b77cf77920444754d804d4}}, IU-FSU {{cite:f0b1abc6adc9c7873b48776a83d7ea08f6e19f32}}, G3 {{cite:c8b3a490d1b21704979b377d5a338af8f537d209}} etc. dominates the softer region of EoS. Later various theoretical studies put some constraints on the RMF parameter sets' consistency using the experimental and observational data. Some of the RMF parameter sets failed to elucidate the experimental studies and have been considered incompatible for a more consistent study of astrophysical objects. In this work, we used G2{{formula:215c7ade-bd09-45bc-882c-ca692ad71385}} and IOPB-I parameter forces to explore the thermal properties of the nuclear matter and the neutron star. However, G2{{formula:8d34c48f-86da-432b-a2a0-72c5d68c153b}} is a well-informed and consistent parameter set that satisfy all the constraints set by the observational studies {{cite:002fb42f584adcf419d43868225fdde439cc4324}}, and IOPB-I is the recently developed parameter force by our group, which has also been recognized as a compatible RMF set in the theoretical studies {{cite:9290770e26878880842da760958f0f11872288f8}}. The numerical values for all the coupling constants of G2{{formula:acca9e48-9899-4c01-a5ab-31f0b6547778}} and IOPB-I parameter sets are provided in the upper portion of Table REF . The saturation nuclear matter properties at {{formula:79ef052a-2e70-4318-b114-9101f9ea5bf9}} have also been presented along with the available experimental data in the lower segment of the table.
| r | 8979962968ca59eb026508400d7ef94b |
The protocol setting is as follows. The employed DQN {{formula:0cd70c5b-ffab-4e16-a4a8-768e99e6f04a}} consists of two hidden layers, six residual blocks, and {{formula:5a1574be-2fbe-4174-9693-89c3ae255045}} output neurons, where {{formula:3338f1db-f497-431d-8465-c57acd9093a2}} for Clifford+T group, HRC gates, and Fibonacci anyons, respectively. We exploit the Adam {{cite:88fd0d1373c3517e74580bb29e9e152e7a37c17a}} optimizer to optimize DQN, and set learning rate as {{formula:ff66da18-0210-4fc4-ab23-cd0576e7335b}} without weight decay. We set the accuracy threshold in Eq. (REF ) as {{formula:069cf863-03b9-4c9d-b732-b0f457ea3d8e}} and the threshold of the mean square error loss in Eq. (REF ) as {{formula:d087d297-161e-4368-88bb-d457d29eb3ae}} . The gate sequence length {{formula:6b2e0ff9-eb3b-435d-8cd1-d7f9710e2091}} in Fig. REF (a) varies from 3 to 40. In the inference stage, the number of test samples is set as {{formula:6545bb11-0508-48e6-9ebc-fc35c98c88cd}} . See Appendix for the omitted details.
| r | fe5db9d63c75cc3be180f58d0d232afc |
However, without principled guidance, the underlying mechanism of transfer learning is not very well understood. First, when we are pre-training deep models on the upstream dataset, we lack good metrics for measuring the quality of the learned model or representation in terms of transferability. In the past, people tended to rely empirically on controversial metrics for predicting the transferred test performance, such as the validation accuracy on the pre-trained data (e.g., validation accuracy on ImageNet {{cite:2974fe805e4446327ae14e4dca794c2d0bd23a35}}). For example, some popular approaches (e.g., label smoothing {{cite:aa4205ebd167698a42e04440ff48c107245c4dc2}} and dropout {{cite:76dd2ab303d4f279064df1ade3b2c81fbc0a2931}}) for boosting ImageNet validation accuracy turn out to hurt transfer performance on downstream tasks {{cite:051f815f5dec1e8641ea669a0e57a4764f8a8d7d}}. Additionally, when pre-training deep models, many heuristic methods improving transferability, such as the design of loss functions, data augmentations, increased model size, and projection head layers {{cite:ee8f3b45c0d9d4b4746d12f87e5f0c2c7f5d9b13}}, {{cite:8a6b5af7864b57f5ee27f485ee3f5bdc546a41ec}}, are designed largely based upon trial-and-error without much insight of the underlying mechanism. Second, given the pre-trained models, how to efficiently fine-tune the model on downstream tasks remains an open question. Although fully fine-tuning all the parameters of the pre-trained model achieves the best performance, it becomes increasingly expensive as the model size grows (e.g., GPT-3 and transformer {{cite:74eebb4efafee7407c78d3f2a155da4867880c47}}, {{cite:074f986382b65b7d73ebc2ad58ba486965f4246e}}, {{cite:1cfdf95f94e6b4e2e9000bd7e528bcd5193d7f55}}, {{cite:ac2f31b8ae5cce8941172d32b93fe8448a9e9e2b}}). All these challenges call for a deeper understanding of what makes pre-trained deep models more transferable.
{{figure:a8807558-cc7a-4526-be0e-bbf188394ba4}} | i | 43228449212499695ce04814678629f9 |
Based on this claim, Cut&Paste {{cite:b2426ba7120ddce447b5c8844c2c2a50ee272027}} developed an SSL model in which the proxy task is adapted to detect irregular patterns. They created a data augmentation strategy in which a patch in an image is copied to another location after being randomly modified. This data-driven strategy outperforms the state of the art in terms of image-level classification. Nevertheless, at the time of inference, this method must use image patches to accurately locate anomalies. Our HaloAE also leverages this data augmentation strategy to regularize the proposed HaloNet-based AE.
| m | 31795b361a4b668040daa888b0fefae8 |
In order to determine the nuclear modification factor {{formula:55b30261-2ef8-4f82-a0f7-78d181d0c38b}} , the interpolated {{formula:b7d5189f-8990-4360-922a-ba84748d8da0}} -differential pp cross section is scaled by the average nuclear overlap function {{formula:fb9ec8fe-0ec2-4028-9e1a-f23e6b88df92}} .
The resulting nuclear modification factor as a function of transverse momentum is shown in Fig. REF for nine centrality classes and compared to results from Pb–Pb collisions {{cite:a5ece4442a6db926e27025436d155aebed948a6e}}.
The overall normalization uncertainties for {{formula:9316db1d-a564-465d-aa3f-3ffe079cd5ec}} are indicated by vertical bars around unity.
The uncertainties of the pp reference and the centrality determination are added in quadrature.
The latter is larger for Xe–Xe collisions than for Pb–Pb because of the less precisely known
nuclear-charge-density distribution of the deformed {{formula:7cbeedb5-bc8d-4254-a71b-3c36c5680daf}} Xe and the resulting larger relative uncertainty in {{formula:83a63262-af9d-4048-a9ac-091ce8bb8b5a}} {{cite:dbfeeb5d66f6e50a8692833c67b5eaa5e03ab094}}, {{cite:eea34f49f3154c838c7f67be3647f1bf7342e2c7}}.
The nuclear modification factor exhibits a strong centrality
dependence with a minimum around {{formula:58538a8a-2246-4db4-a115-fc7cd06442eb}} –7 {{formula:7bd78ed8-be67-47a5-8a14-81aa9eb673e9}} and an almost
linear rise above. In particular, in the 5% most central Xe–Xe collisions, at the minimum, the yield is suppressed by a
factor of about 6 with respect to the scaled pp reference. The nuclear modification factor reaches a value of 0.6 at the highest measured transverse-momentum interval of 30–50 {{formula:3f16334c-daa8-4b2e-83d5-7161e73a2ef5}} .
For comparison, the nuclear modification factor {{formula:e16d1381-95fc-4fb1-a6c8-b9d83daef069}} in Pb–Pb collisions at {{formula:06d7a31e-d4e8-4749-aee4-deb05703c0dd}} = 5.02 {{formula:f27ba1a6-1656-4882-a1fe-603c2e8f47ad}} is shown in Fig. REF as open circles for the same centrality classes as Xe–Xe.
In both collision systems, a similar characteristic {{formula:5d01cc86-3d10-4c39-9bcc-a13bb358127e}} dependence of {{formula:d25bb880-c5ad-425d-9256-25cd443a57df}} is observed. In Pb–Pb collisions, the suppression of high-momentum particles is apparently stronger for the same centrality class but still in agreement with Xe–Xe collisions within uncertainties.
| r | bbd5d529ce4241a5f1eac4733f5a73e3 |
In the Merton jump diffusion model (MJD), see {{cite:6165cd54d57fe1668bdc49543ff0da2e97ad34e7}},
which is a special case of the Bates model, see {{cite:2c326e0bc6998556d4b3f74c90512a6ad86af68e}},
the stock price is modeled by a jump-diffusion processThe number of jumps are modeled by a Poisson process with intensity
{{formula:968db7a6-1c03-417c-b296-51248d76d7cc}} , i.e. the expected number of jumps in the time interval
{{formula:096ad457-5e3e-4098-baa4-45924a02d62b}} are {{formula:4b0444de-5200-4961-9945-2a91a63fe9fc}} . The instantaneous variance of the returns,
conditional on no arrivals of jumps, is given by {{formula:eb5dee93-197b-4592-953d-47d5d7df6164}} .
The jumps are log-normal distributed. The expected percentage jump-size
is described by {{formula:5ecdf7c8-14e9-443c-bbc0-b319cd185214}} . The variance of the logarithm
of the jumps are described by {{formula:b60c7543-75ba-483f-90e2-94cd36edea28}} . In the example we choose
{{formula:ddf4c413-0889-48f7-9138-dc50486decd3}} , {{formula:6622fd81-2235-4d8a-bd96-b1a0272dd59b}} , {{formula:59d15c45-24b4-4d82-aed9-8db05b2443f5}} , {{formula:beee4118-cd89-490d-8dd0-18ca58afdaec}} , {{formula:2c5e1e63-7792-4f08-9a23-4b6afee9df6b}} ,
{{formula:88bb13c2-c8b3-430a-bb76-85687789bcf3}} and {{formula:8042dd67-3fc4-451f-899a-4202b01ead79}} . Under the Merton jump diffusion model
the characteristic function and the density of the log-returns and
pricing formulae for put and call option are given in closed-form.. Figure REF shows the price of a call option
using the closed-form solution of the MJD model and the prices using
the COS method with the interval {{formula:622e9405-7763-46c1-b49b-b4574f686587}} based on cumulants ({{formula:ae8a7313-bccc-4056-b79b-7ad3b6bd2a32}} )
and Markov's inequality ({{formula:36c0ff47-1e96-4411-abd7-fc729e475107}} ). An application of Corollary REF provides
a satisfactory result. However, we clearly see the approximation of the
price by the COS method does not converge properly using the cumulants
interval. The relative error is about two basis points (BPS), a significant
difference, independent how large we choose {{formula:2809bda6-3aa8-45e5-9533-88e9481fb064}} . The cumulants interval
is too short and does not fully capture the second mode (the jump)
of the MJD density.
| m | ced57388893d51211adee6d73e8b9e05 |
In this paper, we introduce anomaly clustering, a problem of grouping images into different types of anomalous patterns.
While there has been significant progress in image clustering research {{cite:5ad2db06160b71f88b58e853c2e5ee37e66609ec}}, {{cite:d73251348631ce73f77e639c87375fd425bc6df6}}, we argue that the problem poses unique challenges.
Firstly, unlike typical image clustering datasets, images for anomaly clustering may not be object-centered.
Rather, images are similar to each other in most parts but contrasts at local regions.
To the best of our knowledge, grouping images by capturing fine-grained details, as opposed to the coarse-grained object semantics, has not been studied in existing works.
Secondly, it is common that the data is limited in industrial applications, making state-of-the-art deep clustering methods, which are usually trained on large datasets, less applicable.
We highlight these challenges of anomaly clustering via empirical comparisons to deep clustering methods in Section REF .
| i | c827e2f85a71c48ceda289cb1b0572c7 |
Nevertheless, in recent times, a lot of research has been done to understand
the Hamilton–Jacobi equation from a more general geometric approach,
and some geometric descriptions to the theory were done in
{{cite:06cba4f87ec0e73f16e2bed2f4f5b1364d427f70}}, {{cite:77f64cc303faf84e1cb8b30a2a74b984a0a8a55b}}, {{cite:4d39a1b8d1bcbb62ba29a19543c8a2c98dfb9789}}, {{cite:138c5e105b8da8772fa2b9202b5d428f2eb67973}}, {{cite:1735cd6a2ac3ba5d4fe0ad3fbf33feb8ef177512}}.
From a geometric way, the above mentioned canonical transformation
is associated with a foliation in the the phase space of the system
which is represented by the cotangent bundle {{formula:b3ee8ed4-d1eb-4ac0-8210-cb56377e3b82}}
of a manifold (the configuration manifold {{formula:2331e4c7-7d99-4453-8ffd-06a135e2195f}} ).
This foliation has some characteristic geometric properties:
it is invariant by the dynamics, transversal to the fibers of the cotangent bundle,
and Lagrangian with respect to the
canonical symplectic structure of {{formula:1e9ba17e-4d65-46d3-9d0d-8d6169cb6574}}
(although this last property could be ignored in some particular situations).
The restriction of the dynamical vector field in {{formula:5c050388-fd71-4257-ac74-11b61ff31056}} to
each leaf {{formula:79fee2fe-6eed-4cb7-99b1-f2a0f99e6ef0}} of this foliation
projects onto another vector field {{formula:4abee1b9-5b9c-44db-8747-59081b1d2833}} on {{formula:638d8f7b-ff04-4813-8b2a-020ce33ae3d4}} ,
and the integral curves of these vector fields are one-to-one related.
Hence, the complete set of dynamical trajectories are recovered from the integral curves
of the complete family {{formula:8f5bcafe-e607-4e33-8895-086c01616a08}} of all these vector fields in the base.
These geometric considerations can be done in an analogous way
in the Lagrangian formalism and hence this geometrical picture
of the Hamilton–Jacobi theory can be also stated for this formalism.
The geometric Hamilton–Jacobi problem consists in finding
this foliation and these vector fields {{formula:21c09bce-fb94-4a58-8aec-c69c77899104}} .
| i | 0299d2e4cdba54436749342ae8c8041c |
In our simulations, magnetisation dynamics is governed by the Landau-Lifshitz-Gilbert (LLG) equation {{cite:65c63ee7e3ff2e1a375b6f4bfbb022dc9d73361e}}, {{cite:b84ca280a315a5c83267ebf4ffeb5b081a4c3bbe}}
{{formula:15e4ea1d-a1df-4ec8-9411-59ce099b6a5f}}
| m | 9ce0964a327de32f1a0a131b362b0838 |
Based on the complexity of the SQL, the examples in Spider are classified into four types: Easy, Medium, Hard, Extra Hard.
Here, we provide a breakdown analysis on the Spider test set, as shown in Table REF .
The BERT results are adopted from {{cite:c3287c6b9fd90b1ceeed1bcb445c6fb4ed783011}}, which is the state-of-the-art system on Spider dataset.
Comparing the RAT-SQL+BERT model and RAT-SQL+BART Encoder model, we can find that the performance of RAT-SQL+BART is comparable with the state of the art, but with fewer model parameters (12-layer transformers in BART encoder v.s. 24-layer transformers in BERT-large encoder).
We also find that the RAT-SQL+GAP Model Encoder can have significant improvement over its baseline RAT-SQL+BART Encoder on each hardness level.
{{table:bb860a5a-94c3-45f5-a484-3900dcf58275}} | r | adf480640acaac8ba58ab9afb2e40aa0 |
Strongly robust toric ideals appear
in geometry,
as defining ideals of non-pyramidal self-dual projective toric varieties, see {{cite:b73686519822d67b8965e22175881d7bd2c6f251}}, {{cite:dc8fb0879db6f02e53475490a736af5da8670b58}}; in combinatorial commutative algebra,
as toric ideals of Lawrence matrices helping in the computation of the Graver basis, see {{cite:834089d2bb6f2746f675c71c174910a433132ed3}};
in algebraic statistics, as toric ideals whose Markov bases have the distance-reducing property, see {{cite:bf129b0ee5f057b90acd975149bb0da0d88c7ce1}}, {{cite:443b9cee8f7838331e3472d32f651505d89a610d}}.
Note that for a toric ideal being
strongly robust implies being robust. In {{cite:bee588a1c65eaf46d73ee613346e150c8004ed41}}, Boocher et al. wonder whether robust toric ideals are always strongly robust. In fact, they proved this is the case for
toric ideals of graphs. On the other hand, based on {{cite:a48ecc05af08b5610b4f2f440e083f99d51d3bb5}}, using the theory of bouquets, {{cite:42462dd726f7a0804fa07c61ffa71ab336dbfaf3}}, it follows that if robust toric ideals of hypergraphs are strongly robust then this is true for all positively graded toric ideals.
| i | e9f3ae7c4232a8fd4dcef451cf8a10e0 |
First, we perform a linear evaluation by fine-tuning a fully connected classifier for 100 epochs on top of the frozen 2048-D feature vector from the ResNet-50 backbone. Table REF reports the results on Imagenet1K after 200 epochs of pre-training. With different batch sizes of 256, 1024 and {{formula:54bce379-db2f-4945-9335-c00622e3b8a6}} for network pre-training, CaCo even outperforms the state-of-the-art SWAV, BYOL and NNCLR with a much larger batch size of 4096 when we used small batch-size. It is well known that the performances of these SOTA models improve with larger batch sizes {{cite:9ecad5435323d6441bb1a523dbc3798b2bd5243f}}, {{cite:781a2ae925a72976236fcf2c4c3c9446664b6494}}, {{cite:2886a39d28f5fc014bf756b3e23c0dd12a1168ef}}. However, larger batch sizes require multiple GPU servers with at least 32 Nvidia V100 GPU cards to accommodate 4096 images. This makes the network pre-training unaffordable and inconvenient to set up across multiple servers for most of research teams. The proposed CaCo not only outperforms SOTA methods with much smaller batch sizes, but also allows affordable network pre-training on a single GPU server. Our training time is on par with AdCo {{cite:7c6022f79e38bd39e02a362b1830b66a49767b3a}}, MoCo {{cite:be91dfdb3f5e6a38ec0e3ceb521d401a77f57ebd}} as shown in Table REF
| r | ff1f90cb6c2f22a025876a47d95f2740 |
Dynamic models are based on two levels of specification: time-indexed observables {{formula:4f335289-36b4-40c4-af30-c9c87b65a219}} , {{formula:84da7fcf-2afa-492e-8dc6-11444392e722}} are assumed conditionally independent, given latent states {{formula:53d7e17f-65f7-416d-847a-ab4d3df04a93}} , governed by stochastic evolution rules which assign those non-observable quantities a Markovian temporal structure and formally address marginal autocorrelation among observables. The latent dynamic states guide the behavior of a dynamic predictor based on structural components which may relate to trends, seasonal patterns, covariate effects, propagation of intervention impacts, among others. Under normality of the response (conditional on states) and evolution errors, and a few additional assumptions {{cite:e2436bd310e8b814f66d8b98a30a724f0df2ebcd}}, the Bayesian update process is analytically conducted by well defined equations, which guide the evolution from the posterior distribution of non-observables at time {{formula:b5cd8eb9-1d72-44d8-8bb5-1255243a3e71}} to the prior and predictive distributions at time {{formula:7205ed69-4264-476e-9616-46c935dec82f}} ; finally leading to the updated posterior distribution for the states at time {{formula:0b524ea0-4a02-4cfa-8869-a8b64335efbb}} , when the cycle repeats. In that framework, retrospective analysis is also analytically viable {{cite:7f20ea7df5f5a23bc2ebfe6503109a2243c276cc}}. On the broader non-Gaussian DGLM context, there is no general analytical solution for the updating scheme and some sort of approximation is needed. Our aim is to produce on-line inference in the context of DGLMs defined in terms of uni or multivariate {{formula:af1b5716-5d5b-4e28-9244-c9932df90c96}} -parametric exponential families, preserving the tractability inherited from conjugacy properties.
| i | da1ccf2e1f6a3c19ef19a81f60ec720f |
It is noted that in this discussion only models employing leptons as primary particles are considered. The stellar
wind and the circumstellar disc provide reservoirs of target material with which relativistic protons could also
interact to produce gamma rays through {{formula:026c2ba6-ec2e-4cdf-8ef8-dc14c9c08182}} decay. However, hadronic interpretations proposed for this
source {{cite:815c528b8e26233fb6c1f10520d50cb753c2a027}}, {{cite:f6461e4e11e6b033785588e330a29901824bf41e}} were not able to explain the light-curve profiles adequately
unless complex disc morphologies were invoked {{cite:b7f1be910416eb6641ff1095e6fa35c823ed0698}}.
Furthermore, arguments based on the time scales of the observed variability favour leptonic
scenarios {{cite:969adecf03ff4a274d50f6c027cb711a0282eeac}}.
| d | 0015eb82506b550325761f2b11008298 |
In another context, it is often not straightforward to choose the most adapted statistical divergence for a given application. Indeed, there are many types of statistical divergences that are indexed by a scalar parameter, for some value of which the KL divergence is recovered {{cite:3663f250f219aa326e1990b1c2bd656c9085bcb6}}. There exist some comparative studies {{cite:34ef268f0d5be182cb69023995cdbfad479aa71d}}, but they are restricted to particular contexts. The notions of relative smoothness and relative strong convexity allow us to show that the KL divergence can be used to construct tangent majorizations or minorization of the Rényi divergences, which seems to be a new insight and may help guide the choice of a divergence.
| d | ec00173c45b936c87372aae1e6eb9094 |
The fast modes most likely originate as sound waves from the photosphere that mode-convert into fast modes at the equipartition
layer. However the fast mode wave amplitudes quoted above at the equipartition layer are significantly larger than those expected to derive
from waves associated with photospheric convection. In our models above, an acoustic wave energy flux of {{formula:38065f79-9b84-46b6-8df8-bfbd0904feaf}} ergs cm{{formula:e07477d0-44b7-4f13-a931-5678ff943d83}} s{{formula:9e3b8c67-3a8e-4758-8a32-0ccd8e36cf53}}
gives an acoustic wave amplitude of order 1 km s{{formula:176853b9-2cf8-4cac-8387-04325c066736}} at a density of {{formula:0940cba3-2140-4d7d-a294-c9d3f4829591}} cm{{formula:10181ca1-4d94-4008-bfd2-b60a1b5d545f}} at an altitude of 300 km, comparable to
amplitudes observed {{cite:f1f140f93e2ae18bc94019451f66c4ac806f5be0}}, whereas fast mode wave amplitudes of several times this value are required in Tables 2 and 3.
{{cite:0ef017a69c8bbccc675bcc8ae99f8b81aa832978}} identified the small patches of Inverse FIP plasma observed in the corona during a flare with sub-photospheric reconnection between
two interacting sunspots. {{cite:3518e31ee1ff9679d744e749a014c250bffd7b0d}} investigate further, and find regions of Inverse FIP effect plasma coinciding with strong light bridges within
sunspot umbrae. {{cite:d91c73f572e0c1221ffd7f392177169e612108c1}} have investigated MHD wave generation by reconnection, and following this work, the geometry of reconnection
at the light bridge suggests that mainly Alfvén waves, not sound waves, should have been produced. These would propagate upward along the
magnetic field through the equipartition layer without mode conversion here to fast modes. Such mode conversion can happen elsewhere
if the inclination of the
magnetic field to the vertical direction rotates, such that the Alfvén wave polarization is rotated to become fast mode.
| d | 917a4f30c29e381a41e169ec16eacd0c |
All models are trained with the Nadam optimizer {{cite:74c86ab19af713e3da41e6c7d7f399fbc04ceb5f}} on the training set and the model with the smallest loss {{formula:1da90bb7-c46c-4c8f-b77a-f9e789d73638}} on the validation set is chosen for evaluation. 5 training runs are performed per model with different random seeds for weight initialization to avoid initialization bias. The models are implemented in Tensorflow 2.1 in Python 3.6. Training is performed on an NVIDIA Quadro RTX 8000 GPU with 48 GB memory. As the focus of this work is on survival prediction, the results for the pretraining on ISUP scores are provided in apd:isupclass.
| r | 181d839978bd57fb5111ca7a3ae88ae0 |
Different auxiliary property-based tasks can also be integrated into a hybrid method. Hu et al. {{cite:ae57fe20e924793a20f4aa25eb7622527c9f2209}} present to pre-train GNNs with multiple tasks simultaneously to capture transferable generic graph structures, including Denoising Link Reconstruction, Centrality Score Ranking, and Cluster Preserving.
In GROVER {{cite:c324134fefb2d7a7fbc2f9dd6248830d03e97569}}, the authors pre-train the GNN Transformer model with auxiliary property classification tasks in node level (Contextual Property Prediction) and graph level (Motif Prediction) simultaneously.
Kou et al. {{cite:d59627cc603a31260c04e16f8a95fee22f4b421c}} mix structure generation, feature generation, and auxiliary property classification tasks into a clustering model.
| m | 4d8a93207a40d05c9b76e8f58857da83 |
Statistical physics offers a way to pass from a microscopic description of systems to a macroscopic one. Its numerical realization is known as “molecular dynamics”, which is a simulation method used and developed over the past 70 years; see {{cite:21fc44ac9282dbe2e4f2efef344690691e24645e}} for a historical perspective, and {{cite:98f7c67a0630e142aec29b07be6b3603ac62f4d5}}, {{cite:958543515c15ce68d67640fbee5f46232acfbabf}}, {{cite:b3a7f81f3d4bac8f501efc5c8990d7fbf37dc8d1}} for reference textbooks in the physics literature. Molecular dynamics has two major aims. First, it can be used as a numerical microscope, which allows to perform “computer” experiments. This was the initial motivation for simulations at the atomic scale: physical theories were tested on computers. Another major aim of molecular simulation is to compute macroscopic quantities or thermodynamic properties, typically through averages of some functionals of the system. It allows to obtain a quantitative information on a system, instead of resorting to approximate theories constructed for simplified models. Molecular dynamics can therefore be seen as a tool to explore the links between the microscopic and macroscopic properties of a material, allowing to address modeling questions such as “Which microscopic ingredients are necessary (and which are not) to observe a given macroscopic behavior?”
| i | 6803d0b336cd70e1ec0dd0642c6a7822 |
Label sparsity does affect all relation categories, apart from one-to-one category.
We further investigate the changes in system ranking by different relation categories when the label completeness is improved.
The relations in KG can be commonly categorized as one-to-one (1-1), one-to-many (1-N), many-to-one (N-1), and many-to-many (N-N) according to the average number of objects per subject and the average number of subjects per object {{cite:6ee69c87f7eec73da2288a61d4f1b188362ba6df}}.
Moreover, each question is either a Head question that predicts the subject {{formula:88d7c4f5-cc83-4dcc-b659-bf4b3d2c69fd}} or a Tail question that predicts the object {{formula:c4e508b4-0bfc-43e8-961a-791f27d2ad08}} .
As seen in Table REF , comparing the correlations of systems rankings from FB-Test-S-C to FB-Test-S, the system ranking is fixed on 1-1 category (correlation = 1), while changes drastically on N-1 (Head), 1-N (Tail) and N-N (Head and Tail) categories, where even negative correlation can be observed.
Herein, the above results are due to the fact that the 1-1 category has no sparsity issue, while N-1 (Head), 1-N (Tail) and N-N (Head and Tail) categories had many true answers which are added by our annotations. Those added labels are indeed shown to have important influence on the evaluation. As a result, non-negligible changes of system ranking are observed.
However, the situations on 1-N (Head) and N-1 (Tail) categories are a little different.
In principle, both should have little change in the correlation since there is only one answer as given by the original labels, but the correlations on 1-N (Head) in terms of Micro MRR and Micro Hits@1 drastically decrease to close to zero.
As our annotations only add more true tail entities for 1-N questions and true head entities for N-1 questions, the difference on 1-N (Head) or N-1 (Tail) categories between FB-Test-S-C and FB-Test-S sets is only the scale (i.e., the number of associated labeled questions like {{formula:00147a21-f878-40a7-bdf3-42fc288001b1}} where {{formula:d0f5b503-5052-4c37-be5a-24f814eeb732}} are newly labeled) of the triple questions.
Thereby, the changes on 1-N (Head) and N-1 (Tail) categories could attribute to that micro metrics may not stable on the size of test set, which is further analyzed in Section REF .
| r | a19851a774f7410f46221ab3c1376982 |
This second-kind superconducting phase transition {{cite:2efafe4dd1c4a006a6014907a48d7ebeb0b30453}} is known from low temperature solid state physics {{cite:e77db8b4ff0471c8d9691a66e1965de35091baf9}}, {{cite:c6e5e8e8662dad9c3fd08581db14c8fae1b68f29}}, evolving Meissner diamagnetism based on electron pairing and condensation that pushes the magnetic field locally out. In mirror modes the possibility of similar condensations has recently been demonstrated {{cite:4b912560e6eaa559da3a97e08d991164c10fd270}}. The transition is initiated by the mirror instability, which starts under the necessary condition of positive ion pressure anisotropy {{formula:0bb6659a-0d91-401d-99b6-925c075d0839}} . In addition, the sufficient condition for instability requires the magnetic field strength {{formula:46c1de0b-b6d9-4b3f-97c4-5c2b68758bcc}} to drop below a threshold {{formula:be457afe-2fd2-41d5-b34f-6e41320e1a23}}
{{formula:e76f44bb-b161-4910-aea5-a06b331ff67f}}
| i | 8c631d1f87ce70afa8ec5253ff113eb2 |
We find that the AHE in EuIn{{formula:f3177445-fc29-4775-9e67-2fe5287f7a56}} As{{formula:ce13b034-edcd-412b-a8bf-7333ad4af9ab}} fulfills these features of the intrinsic AHE.
We plot the anomalous Hall conductivity {{formula:ce90f02d-ee81-42c1-a897-3ac2b65c9a6f}} against the longitudinal conductivity {{formula:85e5a6da-7fa3-40b3-9011-6a8c648bf07b}} with other FM and AFM systems in Fig. {{formula:ef707f4e-1b4b-4c01-87a1-b1cf5517458e}} (a). First, we find that the {{formula:3e2cfa09-e400-4c23-ad34-0fdaa70ae818}} is in the range around 6{{formula:e108c5a1-9031-440f-8e13-46c2a3a9f851}} 10{{formula:d99367ed-77dc-4064-a08f-e9940cb4ccf3}} {{formula:8375fac7-f1c7-4fc1-850b-767578708677}} cm{{formula:5395e682-03b9-43b9-ac2e-3b15e1c13ad2}} and the {{formula:93251633-e97d-4e26-b745-2fde0706640b}} does not depend on {{formula:87043dcc-b2ea-4728-80c4-fbba229da193}} . Both features indicate that EuIn{{formula:402c99b5-a1cb-4e94-bf94-54a47415c463}} As{{formula:63ee5bd0-1cd3-4fb1-acc0-c3e1b239fb7a}} is located at the “intrinsic regime”, where the intrinsic mechanism is suggested to be dominant.
In addition, as shown in Fig. {{formula:c8d39bab-d06d-4974-b76b-963d9eec303b}} (d), we find that the temperature dependence of {{formula:1da48d4b-f574-41cf-a409-0bc628e4c846}} almost follows that of {{formula:b17a98f8-cc25-4232-8df1-705cb321758c}} (i.e. {{formula:a1ae190d-ff2b-4a37-af68-bbcbee09a37b}} ), supporting the dominant intrinsic AHE in {{formula:379dd966-3c70-425f-a99d-4d80f2f2e4ac}} . In fact, in the time-reversal-symmetry-broken systems, a band crossing near the Fermi energy {{formula:2dd8d2ce-cde4-42cf-94df-26861581665e}} with a strong spin-orbit coupling (SOC) can lift the spin degeneracy. The energy gap induced by the SOC produces a large Berry curvature, contributing to the intrinsic AHE. It has been verified that noncollinear antiferromagnets with zero net magnetization can produce a large AHE when their electronic structure exhibits a nonvanishing Berry curvature that acts like a large fictitious magnetic field {{cite:3579d96a732addb4c92c28a79e1e48a407aefdf5}}, {{cite:ab32552bc5f1ff9df5f290e6ee16bb8bd7e365c7}}, {{cite:e2b7eec09100bc4a17e30dbea525f49831ba92f2}}. According to the recent neutron diffraction experiment, EuIn{{formula:b3aad676-1c6c-41a9-9f74-6d8f9cf6ea34}} As{{formula:3b186ad1-0247-4976-a5b2-93e9727899f3}} has a helical magnetic structure {{cite:3f86a70e6847b991dd251bbf9a6f5c60ba94080d}}, in which the magnetic moment ferromagnetically aligning in the {{formula:ce6c1582-a050-4fc3-b6b7-0272d5df2a63}} plane rotates antiferromagnetically along the {{formula:4f985ba4-5055-4961-8251-ed7c23265d5a}} axis with the magnetic space group of {{formula:d1fbc9b8-c80d-48d4-98d9-9f95b64c610f}} 6{{formula:bc6d9fd7-e098-4c88-8052-9daa52e413cf}} 2'2'/{{formula:c8cf4f6e-0d83-4a82-af00-1294fd714dcf}} 2'2'21 (Fig. {{formula:be20c474-39e4-4a79-bd6c-6e0b30d8f901}} (b)). These symmetries break both the space and time reversal symmetries, giving rise to local nonzero Berry curvature. In fact, the unitary operators (space symmetry, such as the 6{{formula:438c1491-1190-4d95-9112-3ec2409747f0}} ) and the anti-unitary operators (such as {{formula:770f82ea-d142-41ff-95fa-233b5d5143ec}}{{formula:19b6e893-3160-4a2d-b8b6-f369453ddb76}} 2{{formula:38f109b3-52c0-4391-9464-a1b9f8c7bc99}} ) can own a finite Berry curvature in the reciprocal space, ensuring that AHE can emerge in the helical AFM of EuIn{{formula:ab7344f2-c29c-48a6-83ed-5109c24330df}} As{{formula:34267da5-1601-4668-b56f-381b46398743}} .
{{figure:7b9c37d7-db08-4093-97d8-80f97168843e}} | r | ef97738922eabaf2abd99af0cb730924 |
The formalism will be exploited in a companion paper to investigate the Hayden-Preskill scenario of throwing a diary into a black hole and asking when the information is returned in the radiation {{cite:75a709b93e132102eb1dd6231e4911fce1315ad5}} and the Harlow-Hayden process of distilling the purifier of a late portion of the Hawking radiation in the early radiation {{cite:c4bb481cea3a7bb29423b58646816a5d0d5d96c7}}. Another feature that can be investigated is the existence of obstructions to decoding the state of the radiation and black hole known as a python's lunch {{cite:ba9b0e80bae7829e5fb181e383865512a93cd3f6}}.
| d | 372020834d464474e8b196df51eff1d5 |
On the other hand, the idea of using the Morse index of a solution for a semilinear elliptic equation was first explored by Bahri and Lions {{cite:5055bb1087bf67e40a98dfdd7cf838df2b6a80c3}} to get further proved that when {{formula:dc6a5afa-f3f0-4d01-8ae1-98c17b996b56}} , no sign-changing solution exists for (REF ). To prove this result, they first deduced some integrable conditions on the solution based on finite Morse index; then they used the Pohozaev identity to prove the nonexistence result.
So, motivated by {{cite:c101d657644fd262f30eeca0884868f6f467937d}}, they used blow-up argument to obtain a relevant {{formula:1bf09cba-11ee-4f9a-a660-982fa7769f6e}} -bound for solutions of semilinear boundary value problems in bounded domain from the boundedness of Morse index (see also {{cite:5055bb1087bf67e40a98dfdd7cf838df2b6a80c3}}, {{cite:b95bc8e297021359de0aca33eb20e6161cf3b2e4}}, {{cite:4f5a605e7ade82fba5c3ec86d6bec80626d94edf}}, {{cite:df9764c78dacd6a96078158255b8ed61e2ce0f23}}, {{cite:1531f6533a176b5d9eb32a6098f96a2c789c2b87}}). We mention also that when the Palais-Smale; or the Cerami compactness conditions for the energy functional do not seem to follow readily, the proof of existence of solutions is essentially reduced to deriving {{formula:2e4f365b-4858-4fdf-a5a8-2687e4bf9392}} -estimate from Liouville-type theorems via Morse index (see for instance {{cite:ceb8aa575d66186903e536b95870009552ae826b}}, {{cite:c73c94f32d5e04dbfead1c4ee55c2a6e2b324cba}}, {{cite:e0e4512fac2949963773e0ceb83fcd2a8d2af24a}}). After these works, many authors investigated various Liouville type theorems for solutions with finite Morse indices in subcritical case such as problems with Neumann boundary condition, Dirichlet-Neumann mixed boundary and nonlinear boundary conditions (see {{cite:ede25249c8505f23796888f789ba6b8c53492eb4}}, {{cite:85be89f7af214ed90ae53fea2b90e4843ecf72e0}}, {{cite:4f5a605e7ade82fba5c3ec86d6bec80626d94edf}}, {{cite:5192fd787fe358997ceee78336c1ec266b3ce317}}, {{cite:d38cd906d915b89011c59bd10796455ad62e7d2e}}, {{cite:f26322ee74c879ca288140765d4f46ece07c1652}}, {{cite:59c1140cda99cca4926a7aa89add4380e78f1e11}}). In the supercritical case, the finite Morse index solutions to the corresponding nonlinear problem (REF ) have been completely classified by Farina {{cite:b95bc8e297021359de0aca33eb20e6161cf3b2e4}}.
| i | fed70707c048d72bfff6355ec2d5dfe1 |
The vector space {{formula:ad9c81a4-c257-4643-9b28-7c09f9be99bf}} was explicitly calculated by
Peterson {{cite:5f60a6e1fb0b28e301f18f703fd6271458edb23a}} for {{formula:1726cb63-140f-4f51-8123-c9197336dac2}} by Kameko {{cite:ede054b207f12e5caf75b0d16e9768c6129cbb4c}} for {{formula:95865e1b-869a-48bc-aa70-2c8f40ab0ea8}} and by Sum {{cite:f149c4db2e66c249c0f552a2c37a7c3d299c6f0d}}, {{cite:a5be9139c038c7a45dbbae6774669b5911eea5fb}} for {{formula:5fa4381b-ad72-4e8b-8e39-497570eb2f5f}} . However, for {{formula:4f5ea291-b77c-4313-88d4-aea38393c4e9}} , it is still open.
| i | c79b6518e8dc3fe7d3431e8f0d4ccab6 |
where {{formula:8166179d-ff12-4cc2-a979-90ec42530ea7}} , {{formula:8b18251b-e48c-480e-8394-63a18f6d1d19}} for {{formula:7fda118e-0dd9-4704-960a-96a42ba3b1ce}} and {{formula:55370c8d-e85e-47d6-a9ea-ccc9cbb5481f}} .
We remark that {{formula:451b158b-aa61-445a-96b9-dc8892ac936a}} denotes {{formula:6c4b8221-42a8-4b30-91b2-eb55382e3b0c}} and that
the angles {{formula:8d6d998c-c343-4035-8201-572cad3ea5e7}} are set according to the following scheme:
{{formula:f67746e5-80c3-44ea-ade6-59e253f8f5b2}} is the angle between the {{formula:78f43ff9-7625-4aff-b046-f2246e08d27c}} axis and the vector {{formula:3b4ab025-0499-498c-93b8-27751b108d55}} ;
{{formula:14396614-57f2-4e8c-aee4-f1461a1865cb}} is the angle between the projection of the {{formula:49e3be99-46ba-420b-be00-6c935cc1bbc9}} vector on the span generated by {{formula:891b17f2-b5e5-436c-b5b2-e929a74427ef}} , which we denote by {{formula:c3ef5c5a-7b19-4927-aca0-fcb4ed579b49}} , and the {{formula:325e58f6-abcc-4a25-927d-3037550d64f9}} axis;
{{formula:e6d71cf8-dac7-4494-8401-5f1b74f7dd38}} is the angle between the projection of {{formula:d0a505ab-b394-4764-af7b-dbca410c3485}} on the span generated by {{formula:4cbf44f9-aa2c-4e71-8b43-ce4aeb02508c}} , which we denote by {{formula:6c3cbf53-5ae1-4ca9-9e49-c58c8f2aa586}} , and the {{formula:cb60342d-8893-420b-8a12-2219bb6bcc47}} axis;
{{formula:92938185-ea30-47a7-9e3a-9653117979f8}} ;
{{formula:f35ad036-99bc-4bde-b40c-5ff11c9886fb}} is the angle between the projection of {{formula:7803ad2e-961c-426b-a410-9965e29259b7}} on the span generated by {{formula:10e98a2a-62a5-4fd6-a198-e6e386c66b32}} and the {{formula:835fb3c0-a7f8-4256-9f84-6e46d59ec4e5}} axis.
It then follows from {{cite:f6c4e76dfdd602d3092b45254a016e0979678e2a}} that
{{formula:0f94265c-96b6-4316-a26e-567c0aed155a}}
| r | b1affcb50a441a44eb1f6e10fb791441 |
Overall, HaloAE performed competitively on the MVTec dataset {{cite:233a75e69c85603ec9a5fe8dcd45178ced5ab283}}, with an average score of 91.4% for image-level detection and 91.2% for pixel-wise segmentation. The performances of our hybrid model between CNN and local Transformer suggests the importance of integrating global and local information at each step of the process. This study therefore implies that Transformer-based vision models could be improved by simultaneously applying the self-attention operation between and within patches {{cite:4ecd8789214d0882fda64657d34fff592bf1429e}} {{cite:9cd0cd081cea1651c53ac9b60a69e15891e00590}} {{cite:836fbcdffb8bd4a1b13c9d886aa53d310eada046}}.
| d | 416f8f074819a815936a9b040a9e8018 |
For MedMNIST3D, we implement ResNet-18 / ResNet-50{{cite:679155bf94eb6da7e246dc30d6c24375ea94cbe6}} with 2.5D / 3D / ACS {{cite:e3e74d06916267ba6cc57495b3640bc01f823fac}} convolutions with a simple early-stopping strategy on validation set as baseline methods, using the one-line 2D neural network converters provided in the official ACS code repositoryhttps://github.com/M3DV/ACSConv. When loading the datasets, we copy the single channel into 3 channels to make it compatible. For all model training, we use cross-entropy loss and set the batch size as 32. We utilize an Adam optimizer {{cite:66b6a459ebf3512afac5d950e427abc763801d47}} with an initial learning rate of {{formula:9f2da0b6-cc34-4382-8dc3-9144e040b99f}} and train the model for 100 epochs, delaying the learning rate by {{formula:e585b158-3679-4329-b738-4b0e72d8d495}} after 50 and 75 epochs. Additionally, as a regularization for the two datasets of shape modality (i.e., AdrenalMNIST3D / VesselMNIST3D), we multiply the training set by a random value in {{formula:cb8e617e-6881-4612-85af-d891b19cc9da}} during training and multiply the images by a fixed coefficient of {{formula:869861f9-2d0b-427b-8e60-a57e7ceac6b2}} during evaluation.
| m | 4974f866fe00da74ce5b856e15dca82f |
Furthermore,
{{formula:f3f17887-5704-43e6-8cd2-275c375b3537}} serves as a more direct signal to characterize locally how
vortical the quark gluon plasma is and we may extract the magnitude
of local fluid vorticity {{formula:9ec469ca-8691-4dbf-829f-75838fc4b835}} .
In current study, we find that {{formula:b5f8b379-6c80-4ad0-9ca1-32a43109a7ee}} is much larger than other components in the helicity polarization. Because the enhancement
of fluid vorticity in low-energy collisions due to the nuclear-stopping
effect {{cite:33b670e79b9e0a4d8b38b21fc60692847e1befde}}, {{cite:b4e0d35a8283914de4e44ca23c8a72e733c49e89}}, the {{formula:a9ddaab8-8fe0-4447-bffd-591a5b09d048}} is expected to dominate {{formula:805a7db2-28f7-45e2-8654-8564a9ff6f06}} in low-energy collisions.
Therefore, it is tentative to further
investigate the helicity polarization in both theory and experiment
to extract possibly strongest local fluid vorticity from the beam
energy scan.
| d | 09b77a3da919c5fa83b6cb8ce19acb5e |
Other time-sensitive protocols.
While the attacks we demonstrated were specifically targeting Lightning Network, we believe that a wide variety of Bitcoin second layer protocols {{cite:24d4fd00c68bfcdc5ee19b6f94094f3253ea6053}}, {{cite:f28f2eefafdcf502772adf9d404436d002a5f9d9}}, {{cite:11c4dd874aafbfe959d9b741cae37b51e16e3196}}, {{cite:1702d02d9bd9399a32edb18dc8b5c4a6ef898ea8}} may be susceptible to time-dilation attacks. This applies to any of them where timelocks are used to arbitrate parties willing to commit concurrent on-chain transactions. We believe that designers of those protocols should take time-dilation threats into account whilst arguing about their security.
| d | 99e386936fce8159da17607a7f5c846b |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.