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The convergence between rapid cortical plasticity and self-supervized backpropagation here faces several limits. First, voxels and artificial neurons are here averaged over time and summarize sound-evoked response to a single value because of the limited temporal resolution of fUS. Future work will thus be necessary to track the precise dynamics underlying our findings.
Second, the architecture and loss of Wav2Vec 2.0 is unlikely to identical to the brain's, whose loss is believed to be hierarchical and whose optimization is largely done locally. The use of a specific architecture, optimizer, self-supervised loss is only justified by the relative good performance this network achieved on speech recognition tasks. Further explorations and theoretical development are thus needed to identify which architecture, optimizer, training dataset and loss would provide the best predictor to the brain fluctuations here observed.
Third, the brain scores achieved in the present study are small. This issue is pervasive across studies {{cite:f243949806155a6b4fbb77149cbdc1147d4a3288}}, {{cite:460ea25e9133c1b6176b1f24176373055f580388}}, {{cite:3fa3e64e0d506531ab6510c4dd4270cf20ad520d}}, and results from the notoriously small signal-to-noise ratio of brain recordings. Interestingly, it is common to compute a "noise ceiling" analysis to evaluate the impact of noise. Noise-ceiling is a brain score computed, not from the activations of a model, but from the brain recordings of a repetition. By showing that the brain constantly changes as a function of previous stimuli, our study thus challenges noise-ceiling analyses based on repeated exposures.
| d | c79463f0cf58881b739dd41f01d9ec5b |
We have also searched for faint H{{formula:38005275-3e8b-4e83-9936-d103332f5e82}} emission in several of the recently discovered ultradiffuse galaxies in the M101 Group. Five of these dwarfs are seen in our broadband images: DF1, DF2, DF3, DwA, and Dw9 {{cite:45c11040efd70f776300d4fab5845f6a8fdd1433}}, {{cite:84f65bbcda017462de6c8b573b598c84dd4672f9}}, {{cite:000916ac3094e41e6d3b8affc249b5df896890dc}}, {{cite:522fd35781f8ffbc5ae33f19fc8e102161511095}}, {{cite:f013f2476ed546b5b67b89e91826aa908dd1cc7a}}. {{cite:ab4aa8aee1124df7a8ebc0baed3615fd12ee4889}} used a large NUV footprint and determined that all five of the dwarfs in our images lack a NUV excess indicating an upper limit SFR of 1.7 0.5 e-3. We detect no compact emission line sources associated with these objects, and aperature photometry over the ;;10 sizes (typical half-light radii as reported by {{cite:ab4aa8aee1124df7a8ebc0baed3615fd12ee4889}}) reveals no diffuse emission down to a level of 4.5e35. This places a strong upper limit on any star formation in these objects of {{formula:b89a4629-1002-4973-b683-385d113787e9}} .
| d | a4d6ee651bc9c758683cbe394ba312ce |
RV modulations of a few m/s on the stellar rotational timescale originate from active regions co-rotating with the stellar surface {{cite:2e92fe5e76941c8462e75e4733fda836bcc0c16c}}, {{cite:3a66e24a4d838f73a59cc5ae1d454b790b571aab}}, {{cite:782082797abb1592c1c91d8e9630d8cc99ff7e87}}, {{cite:dc4fff51e0bdf9325951e9d41663945380508ebf}}, {{cite:be2977dd3a188041dbaf546f0c103fb02bf1f0e1}}. These regions appear in conjunction with strong magnetic surface fields. They manifest themselves as either spots, which are regions of strong magnetic field and low temperature contrast compared to the quiet star and thus appear dark, or as faculae, which are regions of weaker magnetic field and of slightly higher temperature than the quiet star, usually surrounding spots but also found in isolation across the disk, as seen on disk-resolved images of the Sun. Spots and faculae introduce RV variations that can be decomposed to first order into two separated effects {{cite:3a66e24a4d838f73a59cc5ae1d454b790b571aab}}, {{cite:b8d4b99af0c5c7b998ed80096a22dd07791aff19}}. The first effect arises from the flux imbalance caused by active regions as they cross the oppositely Doppler shifted hemispheres of a star (unless it is seen pole on). The second effect arises from the suppression of convective blueshift by the strong magnetic fields of the active regions. Both of these effects cause quasi-periodic RV variations as the regions form at various latitudes (potentially depending on the phase of the magnetic cycle as in the case of the Sun) and the variations can sometimes have stronger peaks at half the stellar period if the regions rotate in and out of view from the visible hemisphere (for stars with an inclination close to {{formula:b0792e8f-7b7c-424e-a2f5-76cdc95e4f11}} ).
| i | be850edcb5332394d6d9c48c88c877db |
The CT, MR, transducer element positions, and focus position were exported from Kranion (Figure 5A) and imported to MATLAB to run full-wave acoustic simulations using the open-source toolbox k-Wave{{cite:fcff51faadaa26f232ff03ec6f42619a36865e8a}}. The skull was incorporated in simulations using a linear approximation to map HU to bone porosity, and then calculated speed of sound, density and attenuation from porosity ({{formula:82189844-39b6-43d1-9b55-ee7608d66c37}} ){{cite:e9ad2711903284f285c1bb1d2e2da8cc995f623f}}. Images were padded to match a simulation grid of [625, 625, 405] with isotropic grid spacing of 0.5 mm. Simulations were performed at 650 kHz, maintaining a spatial discretization greater than 7 points per wavelength in water. To minimize simulation time, a 100 cycle waveform was used. All simulations were run on a Quadro P6000 GPU (NVIDIA Corporation, Santa Clara, CA). The root-mean-squared (RMS) pressure was recorded for each voxel location. To evaluate the simulated results, the maximum intracranial RMS pressure and focal shift were calculated. The left or right thalamus target was compared with the location of the maximum pressure for each simulation to calculate the focal shift.
{{figure:6e1aa8a2-5197-4c82-8965-7fd1fc3caeea}} | m | d14e05e2b080db6277e057416179c8a2 |
We are aware that the rules of the game are with their rigour difficult to
accept. However, maybe it does not suffice to speculate that the world might
be a hologram As t'Hooft suggested {{cite:9d8b2e8094063feecfcd5ef675adbd6ae7838626}} and Leonard
Susskind sketched in his celebrated paper, Ref. {{cite:bca0b6d0dd581c463255e4ff11bf47aa3b749b06}}. - we really should play modeling games that might help to decide, if and
how it could be like that.
| d | e2af378a03d4d2915f8828bba64179a1 |
Such cosmological effects modify the neutrino capture rates mainly through the neutrino number densities.
In order to estimate the impact on the capture rates, we have computed the precise number densities of neutrinos in the current universe.
The neutrino spectral distortions from the neutrino decoupling change the number densities by {{formula:2febb6cd-6016-4ade-930e-fb5f603b59cb}} for {{formula:436f96c3-b186-4dfb-85c1-69b8581f5684}} , {{formula:6bfe9555-1c73-41c1-9316-39fffd680acd}} for {{formula:15293a5b-de7a-49ed-a4ba-0f2a8217073c}} and {{formula:8ba8ccb7-bb9f-43c9-aa40-83b178b9a8df}} for {{formula:84844d5b-9891-444d-92e8-7ec9836cdc46}} whereas the gravitational clustering effects modify those by {{formula:9f7e8a1f-a91f-452b-8cb9-21efd617607f}} for {{formula:ae60f90b-095a-4ce9-9360-af9c8480abe5}} and {{formula:1ddb17f3-bbe2-457d-98e5-1fe5ca81b174}} for {{formula:6d76ac08-d549-4c92-926b-733f9101f7f1}} {{cite:1b636353a3013acfedd291f3aa5ac0c0a650b954}}. The estimated errors of the neutrino capture rates mainly come from the uncertainties of parameters of neutrino mixing, {{formula:9648e326-32b2-4df8-a11f-ba98946b7898}} , and the reduced matrix element of the GT operator of tritium, {{formula:8d2aa509-60e5-44fe-af65-861dc08358c6}} . The current errors of the PMNS matrix are about {{formula:ccaa4e1f-7a10-4ff6-ab74-c4316d01ff24}} at {{formula:12d5e23a-07e7-41be-88b1-2d4cdc6b6a7b}} level {{cite:c35dcf6483e2a6c1746456204639871dbea394d1}}. The theoretical calculation of {{formula:2683444b-94a8-4c83-8933-16e1ad67552e}} still includes the uncertainty of a few {{formula:ed9fc319-eaaf-4359-a200-5ab838ec0ead}} , although the estimation of {{formula:c7c1d845-2764-403d-b25e-342f92e9abfc}} through the observation of the tritium half-life and the value of the Fermi operator, {{formula:7d457271-d79f-47e5-95a0-07e165507375}} , only involves an uncertainty of {{formula:cb187be5-f5f8-4fa8-af25-d067064949ba}} {{cite:e5bedc7e350152e0f3dd908a42f7443817b25195}}.
In order to observe such cosmological effects through cosmic neutrino capture on tritium, one needs to have about {{formula:725a1416-e687-439f-be13-190ec3bad768}} events since one needs to measure the signal with {{formula:36559591-e312-44f2-8c65-c27e949c8eb4}} precision.
To achieve this goal, one would need about 10 kg of tritium due to the half-life of tritium of 12.32 years, and the uncertainties of the PMNS matrix and the reduced matrix element of GT operator must be improved to within {{formula:5101865b-d399-4995-b432-746282bdb263}} level in future.
Planned neutrino oscillation experiments are expected to improve the precision of the PMNS matrix in the near future (see e.g. {{cite:e75beb3aac73fa668224c295afe9b7462fcc9f10}}, {{cite:677f962e72b1bc9fbf9f5ba06f170ee70006477b}}, {{cite:2d71fdc0d0a2b4a1dfa2f0753056d907e443b092}}).
In addition, to distinguish between the gravitational clustering effect and the spectral distortions from the neutrino decoupling for massive neutrinos, we will need to improve the computation of gravitational clustering and spectral distortions from the neutrino decoupling on neutrino number densities.
Since the lightest cosmic neutrinos in the Standard Model are expected not to cluster significantly in our Galaxy while massive neutrinos are, the lightest ones can contain a wealth of clean information about the physics in the early universe. To this end, much better energy resolution is required than is currently attainable.
| d | e5d64819c277ebee5573803e90be28c7 |
The former scenario, i.e. a dense absorbing gas in the vicinity of the host galaxy is less plausible owing to the expectation that this can
be quickly cleared by the powerful jet along the observer's line of sight in blazars {{cite:3d070b7489831653b92b0ffa90a5e4c6bee27ac7}}, , . A significant fraction of baryons in the observable Universe may be in the form of the photoionized and shock heated circum-galactic medium and inter-galactic-medium (IGM) {{cite:fdf7e097791084e51301777e2057d41572d426de}}, {{cite:4d8fe286a1667a4774f5aab78ba11d8268dd3a12}}. The excess absorption and spectral flattening may then be attributable to an intervening warm IGM (at a temperature of {{formula:1daf02b1-62b0-4379-8841-6eb24619aae8}} K) along the line of sight . This may not however present a complete picture as a fraction (possibly substantive) of the contribution can arise from the galaxy cluster environment {{cite:d5e3781cdb1822618208e1b423fb24095168c84e}}, {{cite:acb955c46359616fc53ce47ba90d49d59e683dec}}, the intra-cluster medium (ICM). The ICM composition and thermodynamical state is shaped by complex mergers, AGN activity (outflows and radiation pressure) and winds from quiescent and star-burst galaxies that compose the galaxy cluster {{cite:49ee1fd57e7f4079cdbd3bf4f08c3e36f421b476}}, {{cite:020c0daf2949a380426af659e3b0ae25559f833f}}, {{cite:68828ff9ff691fcbbfff5e997c3240a3258d260f}}.
| i | f033d8e425d7b647770bb37e7d003812 |
The rest of this paper is organized as follows. In Section two, we recall basic definitions in the
Hom-superalgebras theory and useful
results about Hom-associative superalgebras and Hom-Lie superalgebras. In {{cite:b20a3677e5dcb48fa20cd3af58d47151d86796d0}}, the authors show that the supercommutator bracket
defined using the multiplication in a Hom-associative superalgebra leads naturally to Hom-Lie superalgebra. In particular, we recall the
notion of Hom-Leibniz, Hom-Akivis and Hom-Lie-Yamaguti superalgebras. In fact, in Proposition REF we show that The
supercommutator-Hom-associator superalgebra of a multiplicative non-Hom-associative superalgebra is a Hom-Akivis superalgebra. In the third Section, super versions of some well-known properties of (left) Hom-Leibniz algebras {{cite:f29aedd71d89dd26f20e038737e8c55f13fff526}} are
displayed. Consi-
| i | cb557cafc6a8dc5a677678134ecee9ad |
Future work may expand the current project by addressing other healthcare domains outside the AIs area that use RL for either offline or online learning problems. A non-exhaustive list of examples is mentioned in the recent works of {{cite:bbf3c2c13cd561d6030612414823ed74c71a7122}}, {{cite:c6d4504dc01409ce46fc5c19f11144ee6a315d5a}}, {{cite:f8079e6ff4db6cccb8b0cc778575e5c7efb9adc3}}, {{cite:3b974799e7a0c0db8210bfac9f50085110b7bee7}}, and include among others the problem of automated medical
diagnosis {{cite:7904659a452de20f8dd0aa649f14a06f0cc273c3}} and the design of adaptive clinical trials {{cite:abd38786b7f824d760563071dd006e95387948ab}}. We aim to pursue some such direction as a separate work in the near future, starting from the growing area of adaptive clinical trial designs that is capturing an increasing attention from regulatory bodies {{cite:c7aeec3149f09878eef67ee45679912e4e26b471}}. In such settings, by utilizing and processing accumulating data in an online fashion, RL and MAB methods could contribute to make clinical trials more flexible, efficient, informative and ethical {{cite:33f0b3a4e3c26d9438eefb3eaf7904f91c05afa6}}.
| d | 3458e5b8658172559514e1c780563ef1 |
Photo-realistic and interpretable manipulation of face images is a challenging task.
We address this problem by employing 3DMM as an intermediate representation, which is a powerful 3D statistical model of human faces {{cite:5703b4a9113ebdfb611e364958b78f8fb3450b10}}, {{cite:5b69b01b47accdf2c4bbf8d5126b116fe8ecc7cb}}, {{cite:d07109730b10fffb8123c0aab873dcc1d2a225ab}}.
Fig. REF shows the overview of our MegaFR architecture, where the 3D face reconstruction network yields 3DMM parameters from face images. In this section, we first present our 3D face reconstruction network, which is designed to capture facial expressions faithfully, and explain the overall architecture and the proposed loss function.
| m | ab44342a664809ca61c3bc66705eaf6e |
In a regression setting, we obtain probabilistic forecasts using one of the algorithms described above through the estimation of a Gaussian distribution {{formula:50a651e4-5952-4b74-a25c-a09a5fb5040f}} , where {{formula:cd768899-33af-4ec5-b435-ecf52a8f4ec7}} is the mean estimated capacity and {{formula:b1e25ab2-2936-45b3-b822-120012f9cb01}} is the associated uncertainty quantified as variance. To evaluate the usefulness of predictive uncertainty for decision making, we create reliability diagnostics curves analogous to the work in {{cite:4fdbad7cbd70961e7784d403da8d5f9113e51206}}. To plot calibration curves, we divide each predicted confidence interval in {{formula:cc2d2c36-e40e-4507-85a7-c8cbc6e6f748}} confidence levels that are monotonically increasing on the interval {{formula:3e46b9c7-90b2-4020-8a7e-eddd33e05e13}} i.e. {{formula:deffa52f-8ec4-4839-8309-f221188e8baa}} . We then compute the empirical probability for each threshold by counting the frequency of true labels in each confidence level {{formula:f2c9ca79-0883-40ec-b27d-dffa4bfe72d2}} . Mathematically this can be summarised as:
{{formula:17848da8-1d01-4aaa-a549-b31d28adf4e5}}
| m | bfaee0fcae70c52b3bd6f3493c2e9b06 |
Long mean free paths are crucial for observing the intrinsic {{formula:52a4767a-24d9-4ef9-94c4-d488df3f498f}} of Sr{{formula:fbc49f85-2e2c-4eae-8199-a554e69b1288}} RuO{{formula:f18f0719-c956-49dd-9b8b-250debbe6f48}} : {{cite:f8eaf159014264431461e7f73e31c9715ca1c618}} found that 90 nm is the critical transport mean free path for superconductivity in Sr{{formula:d1b09c16-e05a-4fd0-bd58-0b27d75ea236}} RuO{{formula:f71a22d0-e62e-4678-91a0-e8f7c74ca35e}} —any shorter and the material does not superconduct; any longer and the {{formula:1e066080-450f-4fef-8718-5951a1b4d37f}} rises rapidly to saturate at {{formula:891a1b72-27cc-448f-9b7f-da86659bf682}} K. We find mean free paths more than twice this length on the {{formula:5790a12c-af2e-493d-a4d0-4ce81dbab812}} and {{formula:8f3bed8c-5176-4f14-a73d-225eebdf952d}} bands, which are thought to dominate the superconductivity in Sr{{formula:dee3bf18-7bb5-4a7e-9b13-01de1d1b71c6}} RuO{{formula:e12894d2-afc5-4e30-999a-a98705543c6f}} {{cite:5cf7d9252000e14259cd294c0cca7003347ab78d}}, and which is consistent with a {{formula:a415ebbe-f111-4f91-a738-088f4175b3bc}} of 1.05 K for single-crystal Sr{{formula:130b5847-d17b-433b-9fb8-6e4cf085395f}} RuO{{formula:5978f584-ed26-4e23-8668-e86f18ca39bd}} {{cite:f8eaf159014264431461e7f73e31c9715ca1c618}}.
| d | 1bec05e54243a5408e4b79620e9fc29d |
For any triangle-free graph {{formula:ded22b12-08f1-4fa0-b445-8f30c5aac83e}} without induced matchings of size 2,
{{formula:7654eac6-10db-4d25-b717-c0a3ff4673ef}}
(the bound is tight for {{formula:31a6c4fc-077c-4d72-9919-8e6038d30178}} ).
For any triangle-free graph {{formula:178f5485-e024-473e-8c90-e0267c203318}} , {{formula:9ba6b3ea-affa-4a3c-ab73-2a900d51842e}} .
Conjecture is true for any triangle-free graph without
induced matchings of size 2.
Conjecture is true for any triangle-free graph with
{{formula:097ebe56-050e-4234-88e7-74b023472abf}} .
Recall that a regular triangle-free graph {{formula:6755959a-ed42-4d29-bb9c-00f9cc31856f}} is strongly regular if
{{formula:7425e1e7-db14-4dd2-80a5-055433aaa817}} takes the same value {{formula:ed3385de-1d53-47bf-9761-c5071b66d4b3}} for all pairs {{formula:2043d9a8-c039-4d01-98ee-7a3f8ee07bf2}} of
non-adjacent vertices.
Conjecture is true for any triangle-free strongly
regular graph.
For any triangle-free graph {{formula:8a8dd87b-02d0-4532-9fed-2b000149629a}} with {{formula:11cac47d-3ce1-4c28-865c-480f6c296562}} we have
{{formula:c2bdf847-f2d3-450f-9183-ce309f61d729}}
Conjecture is true for any triangle-free graph with
{{formula:b496860a-b296-4b31-ac9e-6c29add4b4e3}} .
Conjecture is true for any triangle-free graph of
girth {{formula:2b9cbd8c-302b-4fb7-9623-361fb9549b00}} .
Proofs
In this section we prove all our results. Some of the proofs, particularly in
Sections REF and REF , heavily rely on symbolic Maple
computations. The corresponding worksheet, along with some supporting
material, can be found at
http://people.cs.uchicago.edu/razborov/files/halves.zip.
Flag-algebraic calculations
In this section we prove Theorem . As we remarked in
Section , our notation for finite graphs is consistent with
flag algebras hence it is sufficient to prove the inequalities
{{formula:15e4103f-c718-40b1-819d-e14a983988ce}}
({{formula:c40db0c9-2281-4dd9-9a4d-b3145ec177bc}} is the matching with two edges) in the theory {{formula:01f959d9-ef52-4022-b47c-a419d7ed0689}} of
triangle-free graphs and then apply them to the infinite (balanced) blow-up
of {{formula:50ad227c-e28a-4f1a-b08d-977a6792fcc1}} .
We do it by a straightforward Cauchy-Schwartz computation in flag algebras.
Since quite a number of those have already appeared in the literature, with
varying degree of informal explanation, we do ours matter-of-factly strictly
adhering to the notation of {{cite:e885daab9c458a7f75771fe86770e7d43f33f9d7}}.
Let us start with (REF ); for that we need to consider
triangle-free graphs on 8 vertices. We have {{formula:8385940f-9814-4f3c-81bf-ecf231242c22}} and
{{formula:46c5a5a4-3fce-4cb0-aed0-a7cba247792e}} , where {{formula:76e8e692-4b2e-4496-90ae-f6c7f2b18c5a}} and the types {{formula:e21a4225-53ef-4e86-bda5-405ccdbcbd22}} are shown on Figure REF (with
the exception of {{formula:fa1ef608-4c58-4c4d-8795-245cd02965c5}} , these are the same types employed in
{{cite:534d96cd3a6951973107e5169559f2b13feb8d14}}).
{{figure:31367468-cdcf-44ef-bf3a-c0823b73aa2b}}We enumerate flags in {{formula:6dd72a65-cd9c-4ecf-a891-c5b2b21e8264}} in a rather arbitrary order
as {{formula:eb1969e7-7f61-4ee8-baa3-9bd33b3695b3}}
and exhibit PSD matrices {{formula:418a3ae0-bb2b-470c-b063-fbc32e0115c7}} of size {{formula:8719bada-3b11-4d32-9260-0f99ac4f1ee2}} with rational
coefficients such that
{{formula:d82457ba-6a2c-4c0a-94fc-2ca437ab47d1}}
where {{formula:6b7fec52-4a5e-47cf-b29c-689097119808}} means coefficient-wise comparison after expressing both sides
of this inequality as linear combinations of the elements of {{formula:97b674c9-c745-4d7c-afa3-aa4acd63d7ae}} .
The only further remark we want to make here is that the matrices {{formula:a75c3f3d-d706-46b5-95a6-b41198bad19f}} are
degenerate and their co-ranks {{formula:56b5d686-5b4d-42bc-8042-fa3428505b10}} are equal to 2,2,5,4,
respectively. This reflects the fact (and makes an excellent sanity check for
our calculations) that the Clebsch graph {{formula:f423f8c7-1beb-4cf6-b33a-1983ab754b60}} is an extremal
configuration for the inequality (REF ). Hence every strict
homomorphism {{formula:cdc5c362-4571-473b-82f1-eadd051b2f00}} gives rise to an element in the
kernel of {{formula:aa424c30-fb04-475c-93c6-a7a7177f94d1}} . The actual computation is deferred to
http://people.cs.uchicago.edu/razborov/files/halves.zip.
The inequality () is proved similarly, but this
time we need only graphs on 6 vertices; on the other hand, instead of
{{formula:84e4e4e6-87b9-4839-9e4f-8086ad46a843}} we need the type {{formula:23a665a8-f487-460c-855e-b7db0d1ad377}} . We have {{formula:0779d83d-9455-413c-aa51-69bf2ce7e0fc}} ,
{{formula:4e5e8ec1-6bd4-4b83-a2c8-2ad9a0dc0d2a}} , where {{formula:f2890507-6985-4a03-8c0a-b9f6f4c67763}} , and also
{{formula:6b35cd5c-336e-4ea3-805b-767210eea660}} . The computation has the form
{{formula:1f0f5b18-76ca-46c7-adff-8d638bbe9fd4}}
The coefficient 2 in front of {{formula:9e5f3eab-be61-4822-878e-da9f95109b64}} is rather arbitrary, we did no attempt to
optimize on it. As this inequality is tight on {{formula:8cacec37-cc6e-4d34-981d-8113659e277e}} , matrices
{{formula:8899c976-791a-4210-a6be-abab625bf6be}} also must be degenerate and indeed they have co-ranks
1,1,1,3, respectively.
Absolute lower bounds on {{formula:55801da6-575f-4995-a100-9adb8bc202b9}}
In this section we establish Theorems REF and
REF . As was already mentioned, they immediately follow
from Theorem and Proposition
so it only remains to prove the latter. This is simply a part of
Krivilevich's argument {{cite:67ef2f952c482544eea466dd4185e68f6cd9327c}}, slightly re-phrased, but we include it
here for the sake of completeness.
Let us start with considering an individual edge {{formula:6582ee8c-2daf-46fc-aac1-4bb82b28bdea}} . Denote
{{formula:2bfbbc3b-4641-43bd-8f70-31ff7437e3a6}} , and let {{formula:32c65bc3-de61-4ed8-abb7-dde3b3957c48}} ; recall that {{formula:0d6e6870-9ec4-4fc0-b270-f717a1f44720}} by (REF ). Let
{{formula:03223a91-0af2-4084-8e0e-e0595ccf098b}}
so that {{formula:45172d0e-85c2-4dbb-9256-dc52bdf43fa2}} , and let {{formula:12011517-907c-497f-93c5-4aa398743c8c}} . For {{formula:6ee9009d-db3f-4ee1-a8c7-7d9c0d47780b}}
define the half {{formula:32da890d-61d2-462e-9ed8-60f0d210986e}} by
{{formula:ef103e43-347c-44de-b9da-d4691a06bdd1}}
Then
{{formula:96ec0ed7-c6d4-47f0-a9f1-4cfaafa037fb}}
Multiplying the {{formula:1b414067-a2a5-4d3e-a8eb-12d9d3185d49}} th inequality here by {{formula:0e43e84b-80ee-4741-a3fc-cfb1d95e3100}} and adding them together,
we get
{{formula:52b60d0b-1a61-416b-b5e6-a1f4d5d54ff5}}
where we denoted {{formula:deed7f51-032d-4a40-a03b-154584572b15}} by {{formula:7a80d133-e3fb-4089-8c3b-99630bd30729}} to stress that this is
the contribution of {{formula:f35c81e5-8d3f-4dde-b985-343067c8ff03}} to {{formula:34fea2f5-20f1-482c-8985-c0fad0e15b79}} . Finally, averaging this over all
edges, we get
{{formula:6a643b66-608c-4ae0-8842-c336e2d708bf}}
that is precisely Proposition .
Sparse graphs
In this section we prove Theorem REF . As in the previous
work {{cite:2c94f1d9a0a343921d8e8abc2d2076e4859de677}}, the analysis splits into two cases: {{formula:b7e0e8bd-e28c-474f-947d-68e45ca5ddea}} and
{{formula:67044d58-7533-47fb-9b66-0b14c92a14bc}} .
The first case is taken care of by the following variant of Proposition
:
{{formula:a619c829-15ff-41fb-a3b5-ca9cd19dd3b7}}
Pick {{formula:0a5bccb4-0b9b-45cf-9103-8bc1216f0d8b}} with {{formula:adfb552e-d2e9-461d-9571-e06b1da69e11}} , and let {{formula:cd0ece6d-37c2-4a86-bd5d-2b79b0242c54}}
(so that {{formula:5aa65b1a-5b9a-4e33-82a9-ce8187b570c8}} ) and {{formula:574b2da3-42c1-4e44-9993-efe4f68d41ef}} . Construct the following
halves {{formula:f714dc39-d67f-4c51-b030-b867ffdd67ba}} and {{formula:fcac7ef5-e02d-4ac4-9ec1-8be6b7c6381f}} :
{{formula:b105badb-9b54-4ea6-8282-4b2c835c231a}}
Then
{{formula:df93ce87-997e-4586-8155-e882d603a4ca}}
Multiplying () by {{formula:7e3bcb20-226c-4621-85d1-0489c4ae37b3}} and adding it to (REF ), we
get
{{formula:24f31843-9327-4be2-b61c-7a4b855a47cc}}
Now, the function {{formula:a3fa2b67-2d63-43aa-b76b-40871a6ff8b6}} is decreasing for
{{formula:4d0481d6-7e07-4eb8-9907-7cdb763581ac}} , hence {{formula:77f8dd94-685e-4af8-b1df-3b5562c8af7b}} implies {{formula:513834df-8f41-49bf-af9c-47b605c2364d}} and then Theorem REF follows since
{{formula:32290ceb-ace8-4005-add4-3a2b561c2678}} .
The case {{formula:e42a545f-c90e-40bc-bac6-410d25e94fc1}} is more difficult. As in the proof of
Proposition , let us first consider an individual edge
{{formula:8609b7b6-ed97-4af4-a34d-678e297e01d5}} (but this time we will not randomize over this choice but
will pick it up in a way to be specified later). We will re-use the notation
{{formula:1bebd90a-6af9-4318-8526-84e6848ecb7d}} from that proof so that we still have the bounds
(REF ), (). But now the condition {{formula:d142566a-3345-4bc9-b2ff-0cc482cd6386}}
allows us to form one more half
{{formula:65287d17-a52d-465e-960e-a427b50ea6f1}}
where
{{formula:72f48966-9fba-40e1-b80f-f07d82f19837}}
This leads to the extra bound
{{formula:a0ad910d-f976-4260-88ef-81227f118587}}
We are now looking for non-negative coefficients {{formula:40589083-d761-4eed-af4c-9dc73560c1d2}}
such that multiplying by them (REF ), (REF ) and
(), respectively, and adding up the results, we will equalize
the coefficients in front of {{formula:af0e13e0-d237-4d49-bee0-fdd78a66583f}} , as well as
{{formula:1581a1f0-082d-4ec3-b925-7a05846a1831}} . For that purpose we set
{{formula:73ba8d67-0984-4d59-a6d8-88a6d19305d9}}
Then (see the Maple worksheet)
{{formula:708ffc7e-a954-4079-abdc-3211a41c6a46}}
where
{{formula:9a4c94a4-5fc1-4d30-b00c-0e280e02fa0c}}
Note for the record that
{{formula:a4156d0a-3f86-484b-ac7c-c136ce634aa4}}
since {{formula:fc0ecf78-26d7-43c8-bbae-d5ea52823d59}} . Hence we need an upper bound on
{{formula:261e51bb-1702-4a0c-992d-318161d07b5c}} .
For that purpose we now specify {{formula:07644f0f-ab71-4b00-85e5-8c5a10baea39}} . The vertex {{formula:cf854f8f-d676-4cc4-8177-5babff6707f0}} is chosen as the
vertex of the maximum degree so that {{formula:8751d2a0-2ccf-4a82-9dbc-6b6598cc0ba0}} . We choose {{formula:dc414e03-db7a-46ea-9403-3ff27f91b996}} to have
maximum degree among all vertices in {{formula:4437c4b8-01e0-4636-9185-7b19a46915a4}} . The latter choice gives us
the estimate {{formula:0141c3a1-f092-44ff-8191-59a66db7ef94}} since
{{formula:895aed43-fc67-4c7a-9150-dac5210d39d0}} is simply the overall density of edges incident
to {{formula:f7cdc61e-551d-4e2d-9837-f48a9cf99840}} . Putting all this together, we arrive at the estimate
{{formula:e2312dd8-b515-4292-ad28-006144e89d43}}
Let us also remind that we have the constraints
{{formula:c373cb82-3568-4669-a874-35b3503207d7}}
This optimization problem is a bit nasty to be fully analyzed, i.e. give an
analytical estimate on {{formula:13d2ecfa-24e9-42b7-b1fe-56ed77bddb7f}} in terms of {{formula:06130882-93ff-4f2a-b6ff-3689049da16a}} . Instead, we compute
{{formula:cd3f2ff6-d78d-499c-a2b8-57ab457f5ea1}}
where {{formula:798304fd-2941-419d-af11-791b88eee820}} is a polynomial. Our goal is to show that {{formula:82f0c963-b1f2-4b92-b85a-af10f1a6f49b}} implies
{{formula:ff16e23d-8f48-46c5-9db1-5e76db395d4f}} .
We note that {{formula:092503b8-71d3-4e6b-8c9e-fcdc321203d0}} and that individual degrees of {{formula:20c27215-4e9b-464f-90b8-84d0ae6f2f09}}
in {{formula:7a77c108-bf99-4d7b-98c7-0d229f4124c4}} are 1, 2 and 3, respectively.
We first compute
{{formula:d3be3bb4-23bc-44e6-963e-987632a37369}}
Hence it is sufficient to prove that
{{formula:8b800ef5-844f-4d6c-b0ab-6bd34dae685e}}
{{formula:e0645501-5bd7-4192-b974-6028b6d32d42}} is no longer smooth in {{formula:6f48a200-fc08-49ab-801e-0eee93d0a2a2}} but it is still a cubic polynomial in
{{formula:c195f5c3-2596-4d5a-9445-1b51d8012010}} . We consider two cases: {{formula:03306772-6bf8-43d8-b278-94a5488cd03e}} and {{formula:9fb01feb-16c3-4164-9bad-fb78ee0caa14}} .
If {{formula:0fed5abc-4220-46eb-baf5-63a07cdf2c98}} then we consider Taylor's coefficients at {{formula:fa47ab49-f707-4d75-bfb1-dda5a7e13c37}} :
{{formula:adf589ec-d02b-4399-9b96-6d5d1c315b9e}}
and it turns out (see the Maple worksheet)
that they are non-negative for even {{formula:3dca55f3-6d51-4bf2-a725-0b6a4fe43153}} and negative for odd {{formula:ffad4b10-e26f-4811-8aea-eb3b8f33eab0}} . The
required inequality {{formula:4cea3aaf-ac24-42f4-a5ed-6f0f9a0dc37c}} follows.
In the second case {{formula:f6d0f7a6-c695-41ce-af54-a1eb0ef544ce}} we also have {{formula:487794fc-33ff-4b36-a440-3fcef090feed}} and
hence {{formula:ca029574-dc6e-4dbf-948d-5e3c7ad706d2}} is a quadratic polynomial in
{{formula:cab7f83f-7cd6-4089-887b-7f94dcf3cc65}} . It should be noted that it can be either convex or
concave. But in either case the required inequality {{formula:6643974e-d344-4e77-ae0f-6527ebd95382}}
follows from {{formula:4e3057ac-8c5a-4d5d-b4fe-effd95995fae}} , {{formula:6bcfc630-71b8-48e4-8f45-f9101977e5e3}} and {{formula:6935191a-20e3-4047-83bf-6a2e63d40c13}} .
Strongly regular case
In this section we prove Theorem REF . For a brief
background on triangle-free strongly regular (TFSR in what follows) graphs we
follow {{cite:6d43a610fdf92dbb9802b15bb80d21b31f1ba09c}}.
Except for the complete bipartite graphs {{formula:fbe29410-efc9-49e4-8096-05adf0b85d2e}} (for which Erdős's
conjecture is vacuously true), there are seven known examples of TFSR graphs;
the cycle {{formula:6d984ea1-710e-4653-921b-cdd0a4e6c623}} , the Petersen graph and the Clebsch graph being the smallest.
The obvious parameters of a TFSR graph {{formula:df741bcf-c155-44ed-861e-fb06b739ff76}} are {{formula:eb074ed6-708e-4f9a-91b9-e4dc9d10e65e}} , {{formula:47d311f7-23f4-4343-a429-cc77edd99ff4}} (the degree of a
vertex) and {{formula:4f8c0e97-4990-4f3d-a695-25d69394c6a8}} (the number of common neighbours of a pair of non-adjacent
vertices). They are actually related as
{{formula:f1a24517-4368-4631-9a0c-99255a81cd54}}
From now on we assume that {{formula:0a8f8fe3-db15-4c80-bf3d-2077267e6b80}} is different from {{formula:b932ced2-6ef8-4d7f-8967-54b177250fe3}} and that it
is different from {{formula:6212b478-ca0e-4cf1-855f-c70982f87198}} . Then the quantity {{formula:2014695c-7349-4180-bd5c-265aad4cb38c}} is an
integer, and the only positive eigenvalue of the adjacency matrix different
from {{formula:f309d91e-2390-4f9d-8f3b-84c7d0855261}} is given by {{formula:f9c4acee-481a-465f-929f-281c567628db}} It is also an integer such that
{{formula:ca4cc20b-0c4c-40b1-9a77-a2abc5b139c2}}
Furthermore, we have
{{formula:1a237a61-7b59-4ed2-a4ad-bdef654143ca}}
hence {{formula:971372ed-c660-48b0-93c6-71fdb7e089f2}} and {{formula:2fcf2e39-4869-463d-850f-1a75eb3a1767}} are rational functions in {{formula:8a0a3551-4344-4441-8cd5-2778a8e2f148}} and {{formula:d4b0b24d-ca74-4bd5-9922-26d4b713ba3a}} . In particular, we
compute
{{formula:9efd6a64-b564-4b45-97d8-3d22a02c0e37}}
Let us now analyze this expression.
Firstly,
{{formula:03946673-4a2f-47a6-a6bb-2f8cc3adc710}}
hence {{formula:bb5b2625-ac0f-4351-b043-e6ed09c73f25}} is increasing in {{formula:6930f567-c23a-43a4-aa59-568de93bc718}} .
Next, let
{{formula:b6cc6d16-6647-4708-adfb-96c55ba0824f}}
(this corresponds to so-called Krein graphs). Then
{{formula:20e8d9eb-2385-4058-9694-e11ca94a9561}}
hence {{formula:b52b7d97-e551-4abe-b46a-0963beaaad50}} is decreasing. As {{formula:8c38edc6-4027-4726-9214-bf0260e0a4a6}} , the proof of
Theorem REF boils down to the three cases {{formula:dcbdc7c3-e9ab-4953-9578-becd3cc344ad}} :
all others are taken care of by Theorem REF .
When {{formula:5c893771-5fca-4997-a64a-442bd399023b}} , we have either the Petersen graph ({{formula:d41ad1a3-919e-4e58-92b8-5b4f77b7c559}} ) or the Clebsch graph
({{formula:f3f5f7fb-3b69-400f-a7fd-f4cf9d537d7e}} ). Conjecture for the Clebsch graph is verified by the
half {{formula:b07e47b1-2413-4a08-b70b-c6b0f0332eef}} , where {{formula:220db970-4a3b-424f-8702-ce08e0c490bf}} is an arbitrary
edge.
When {{formula:085a6551-07c4-4755-aac4-ad0a12270c01}} , we have {{formula:3477c1d3-3413-4968-b6cc-acc7f499777a}} (this is the
Hoffman-Singleton graph) hence it is sufficient to consider the cases {{formula:01b98255-f8ff-42d8-bb67-8406a2c3cfe8}} . Well-known “arithmetic conditions” rule out {{formula:c15027e4-861d-4a03-9b5a-0e5960b9b331}}
{{cite:6d43a610fdf92dbb9802b15bb80d21b31f1ba09c}}, and the three other cases correspond precisely to the
remaining known TFSR graphs: Gewirtz, M22 and Higman-Sims (they are unique
for their values of {{formula:4924dc0d-0055-4836-a9fa-83d83479c53b}} {{cite:233b72944827071588ad6ef07680cb005b896953}}, {{cite:41fa36b21389bc3b092cd38c0487a609c31d297f}}, {{cite:64607ba10df51e1c14a34d803f31365658bdeeca}}).
For the Gewirtz graph, we pick up four vertices {{formula:ad8dc3f6-27f7-474a-a26f-b3a06f2bafc1}} spanning an
induced matching with two edges and consider the half (see the Maple
worksheet) {{formula:1bfc9a94-73f2-45d0-b6fa-5deede4f1366}} . It spans
51 edges which proves {{formula:51ab548c-28eb-4378-9c05-d19215ecfc68}} .
When {{formula:e2ad02ed-682b-4c93-a33e-d85f7f177084}} is the {{formula:2cd741b4-70ec-4cb7-b533-8171bfd8c9bb}} graph, we similarly let {{formula:8e009381-e358-4b28-bec2-d5f2ef7740aa}} , where {{formula:8713ee8a-8eeb-4a38-bea2-15f543c4977a}} and {{formula:e950b1c5-8201-4fc8-8432-02bd35492342}} . Then {{formula:ecb7b987-1b47-4fd6-ada6-244c790243df}} and {{formula:6a8e9142-c9a6-493e-a730-5fe50e73047f}} . Moreover, there
exists a vertex {{formula:a9a2696c-c228-4bc6-80b4-13acd3d3dc12}} such that {{formula:58895d0f-d396-4a50-a615-cace6b1a60c0}} . Adding to {{formula:1a68a0d7-634b-4b2e-8de7-aa03d5afd0ec}} half
of the vertex {{formula:7b3c91c0-90a2-48cc-97ad-e552e5c2381d}} , we get a half witnessing {{formula:c9cab1c6-ddcd-48be-9e1d-40c78890be3c}} .
For the Higman-Sims graph we present an ad hoc half achieving {{formula:39e62316-bd3d-49a0-90d2-24f82de1eeac}} . It was found by a simple optimization program
remarkably suggesting that this bound is actually tight. If it is true (we
did not attempt to verify the claim with a rigorous argument) then the
Higman-Sims graph comes very close to the bound in Erdős's conjecture.
Finally, when {{formula:4a338872-8173-4649-8a61-e7c759fe2039}} we have {{formula:2aae165b-cfe4-40ce-ad34-635343289809}} . Hence the only
case to consider is {{formula:ae22cb4c-7073-4caf-a069-427b00ec4d32}} i.e. a hypothetical 57-regular Krein graph on
324 vertices. A simple solution is to note that such a graph is known not to
exist {{cite:94795675cca5b9e7ed1b5df497cf7de725d11080}}, {{cite:f59ad3a93c96038d401b8be295d5a4c60e617913}}. Let us, however, sketch another argument due to
Grzesik and Volec (unpublished) that in our opinion is more instructive and
may be of independent interest.
As we already noticed, in the bound (REF ) the quantity {{formula:a71e1347-6816-4ff2-b524-d7b793a0d978}} can
be eliminated via the identity
{{formula:74bb359a-4411-4b5d-b9c1-dd63917c96ce}} . If {{formula:ca31f289-c0fc-4ac6-abb7-2fb1e7dca8af}} is also known to
be regular, then {{formula:eee37e5e-892c-4cd1-bb04-3b393026d1d2}} and {{formula:b9b3a7ed-e67e-46a6-97a9-9d6b42e72e0f}} can be
also eliminated using {{formula:8a8a45d6-21d3-4f6f-80d8-fab96cd580c1}} . Plugging all this
into (REF ) and averaging over all choices of the edge {{formula:3d5bd87f-1825-431a-adf4-5063c710ef00}} ,
as in Section REF , we arrive at the bound
{{formula:1a173780-b384-4c87-9d0b-addf108c5f65}}
that holds for any regular (triangle-free) graph {{formula:e48a1f82-3fe4-4b45-9eb9-60268bb42efb}} .
Now, if {{formula:c089903d-b15e-4384-98b9-5c8b12ecc805}} is also strongly regular then {{formula:c9d95c2b-749f-42bd-84bd-88e5669269a6}} can be easily calculated
as
{{formula:5173a9d9-00b5-40dd-b822-209950d6aa11}}
(recall from Section that {{formula:dfce38fc-0860-4488-85b8-a3a8f2ff9b8c}} counts degenerated cycles
as well!) Substituting this into (REF ), we get
{{formula:2402c03c-7035-4036-9bb0-ef90e0221e2a}}
In particular, when {{formula:43cc76cb-c6bb-4c0f-9cc4-d04c6d7e7619}} we have {{formula:56713377-a36d-43b4-81d4-a8e97cc67b7b}} .
Graphs with large independence number
In this section we prove Theorem REF ; as we noted in the
introduction, for {{formula:f4296ffa-8f25-46ab-9de5-df742677f0b1}} it generalizes several previously known
results.
It will be convenient to assume that {{formula:b7ecc2f9-b680-43b5-b508-08d10940fe9a}} is even: this can be always achieved
by replacing each vertex with two identical twins. Let {{formula:c59d95ed-c483-42c6-a83f-79db7c3430fd}} be
an independent set with {{formula:3c9ae0c2-8ecf-40a8-b4b4-87129d8afd78}} . We build a larger set
{{formula:ff8f32bd-ca59-46d3-9f90-5ebd5aedbadb}} by recursively adding to it vertices that bring with them only
a few edges. More exactly, apply the following simple algorithm:
{{figure:b2cfdc09-7294-4e56-b45f-212c78dbaf84}}If this algorithm terminates since {{formula:7faf1a3c-d1ba-472e-a672-fb5b20742a99}} reaches size {{formula:59d7c40f-c344-48db-b23d-5c8cd247a876}} then {{formula:6d806e31-1389-4efd-9a1a-41ef33f778b9}} which is {{formula:f2c1ca8e-c1f8-4d15-9558-bb800caa97ff}}
since {{formula:087a63f5-3675-4983-ab14-e90bca99a0ee}} and we are done. Hence we can assume
w.l.o.g. that the algorithm stops when the required vertex {{formula:613473c8-ba08-46d3-87b2-1cf25998bc30}} no longer exists. Thus,
we now have a set {{formula:4b48905b-a88c-4b08-af5f-243be4d46695}} such that:
{{formula:0ae12543-881a-450c-ba93-1fb763695db8}}
Let
{{formula:6cb7186b-e8a9-4ed7-84d2-d4bede7976c8}}
Then {{formula:b7141261-cd85-404c-8f1c-2dc46c4038d0}} is independent (as any two vertices of {{formula:71bb81d7-30de-4fd0-b9c0-0e3fa11971dc}} have a common neighbor in
{{formula:27ae373d-eefe-4010-94db-cf4928ac3dc0}} ). Hence {{formula:26131c20-278f-4758-9888-7dfd7d00e3f5}} from the definition of {{formula:8d4b2899-44b5-4e65-a723-0ec751515e03}} . This allows
us to choose a set of vertices {{formula:2b19d0c3-5d74-4b95-adad-92ef858e6cf4}} disjoint from both {{formula:c1ee1c1b-4455-48e5-ad18-25260a9c96a3}} and {{formula:e35a89b8-e19b-4f51-882d-7a592ffdb0f4}} and such that
{{formula:8940a39b-74a5-4c5a-bd40-83aa93a0ac82}} . We now consider two cases, depending on whether there
exists a vertex in {{formula:ff99cd86-6e80-42dc-a32f-f67ad65ea2da}} that has many neighbors in {{formula:68359d13-701f-4eac-8c59-0f493d7371be}} or not.
Case 1. There exists {{formula:cfc472a5-edd3-446c-a84c-e49cc1f94e52}} such that {{formula:496a364e-6636-4fe9-9e57-45bdf223776e}} .
Pick up arbitrarily {{formula:84610191-21f0-44d9-8132-f46f59053f5e}} with {{formula:bc6576fa-31a4-48a7-86b4-5acdf9c35ccc}} ; note that
{{formula:5e9a916a-87d7-43d7-b900-e0ffd31367ff}} is independent. Also, for every {{formula:9130df48-b9e7-4a71-a8a5-7557d23b719f}} we have {{formula:b53237f9-40e0-457a-b52e-4c9941f09f43}} since {{formula:b3f29b66-a9fd-40f1-8a24-7352be6b5bb8}} . Then we have (note the absence of the
coefficient 2 in the last term!)
{{formula:924d1ae1-c6f4-4232-9477-77b2754f86b6}}
The right-hand side is a concave quadratic function in {{formula:55e4660d-77b2-4e30-be3f-2ebc018d447b}} , with maximum at
{{formula:a614bc70-387e-4f4b-ae15-a5ddcf5c2ff0}} which is {{formula:63b4d080-0f72-416e-bc28-271147baf12f}} since we assumed {{formula:73cc7beb-def0-4296-951a-a34a07e9a3da}} .
Hence, since {{formula:a72746b8-5387-4602-90d8-c6150c0ea1a5}} we can plug in {{formula:ca7e6527-1974-42ba-9f58-b3b6db314b1d}} and this completes
the analysis of Case 1.
Case 2. For any {{formula:17c9bc94-56f2-4161-b6e6-a198dbc37dcb}} we have {{formula:d2ff647d-cb6e-49d4-bbb4-b28ad2f967f6}} .
This case is slightly more elaborate. Let us first fix an individual {{formula:03b80d4f-6296-4484-b587-cb3fe1f35f01}}
(we will later average over this choice). Let {{formula:e3d5e943-1ddd-4c42-ab83-35abaccc6558}} (so that
{{formula:76bfff12-05d3-4256-a82b-ee3ccfb26633}} ) and {{formula:a7ca38fb-fcc5-41c8-956e-fe355939c9aa}} ; thus, {{formula:df885b3c-5f54-4e1f-951d-886f4952b828}} with {{formula:16800f5c-2bfc-4119-9bce-085b88fddfbc}} independent. Consider
the half
{{formula:c7f79b03-7beb-4433-97d2-8917aa0035b4}}
where
{{formula:fe6b9795-cabc-4bb1-bb84-dd7e1bee8138}}
Then
{{formula:6ba6e39d-a170-4fab-a6d8-ebb3af819dd5}}
The bound on {{formula:00df04b1-dce6-4ba4-b824-694f6cc2dd11}} is given by (REF ), and we have {{formula:53f0bbac-7f46-4355-9cc5-b3573b64b5f2}} ; the coefficient 2 is absent for the same reasons as above.
For {{formula:e1ff1a55-dc32-4a7c-ba16-a42a184ba890}} we use the trivial bound {{formula:b8170c03-03d6-4f6e-a6a5-579744f900ad}}
and, finally {{formula:f550e38e-9751-4f27-bb9f-1d55029821f7}} simply because {{formula:4d81806f-5840-4464-8222-9141bf45acb9}} is triangle-free.
Plugging all this into the above bound (we leave {{formula:a85ffa93-1a6b-493f-b19b-2428d37e4f00}} alone for the time
being) we get
{{formula:cd8e120d-d487-4743-8b29-daa528ca32c6}}
In this bound, {{formula:92fb0b75-bd55-48c6-980f-a8282f98e18f}} and {{formula:e3ad8ab0-d43e-499f-94f2-a4f6911aa52f}} are the only quantities that depend on the
choice of {{formula:f128d616-5872-48be-a3b8-69caa7d36dd9}} , and we now randomize over all such choices.
The bound (REF ) is concave in {{formula:59c1428d-2405-460b-ae24-3faf798f1ec4}} hence we may simply
replace {{formula:1c97ba96-66b3-4ce7-8755-db914ae06ac9}} with its expected value {{formula:38a80764-ee22-48fa-bce4-86c8ab6d76be}} .
As for {{formula:6bfded9d-01db-4cc4-8ab6-d2e4a2e04112}} , pick {{formula:0b1e542f-64ae-4c0f-abc0-930bdd1de501}} uniformly at random; then by a standard
double counting we see that
{{formula:f86d45c3-5c03-4e98-8cd5-f0deaabe7f9e}}
But we also know that
{{formula:630d7305-cc0b-42ed-a179-81c7b96b96bf}}
where the first inequality comes from (REF ) while
the second follows from {{formula:8eeacbc7-b405-467f-9478-b577935d2475}} . Moreover,
{{formula:6b60b02c-c707-494f-bdd7-46244248dd38}}
Estimating the second moment in a standard way, we get
{{formula:0b790362-ac92-49ce-a5a3-6d967749fe15}}
Finally, plugging all our findings into (REF ), we get
{{formula:94897078-f445-439d-9b25-4cf718c750ac}}
(see the Maple worksheet).
{{formula:4b3ec59b-3871-493d-8ba0-02acb0a40f84}} is quadratic concave in {{formula:89ffc6c3-5ce7-41d9-8940-5c096694ad3f}} and, as before, {{formula:97dbdfab-57ee-4f01-8cc9-b6964062608b}} since {{formula:cdf3386b-b482-48f2-a641-86a3d137ee14}} . Moreover,
{{formula:c9427464-953b-4c2e-893a-245022aed94b}}
Hence
{{formula:ba871254-c95f-40b8-b062-db94767f74f4}}
Finally, {{formula:3e3eb9ff-5491-4d4b-a40e-d3cccf0274bb}} is quadratic concave in {{formula:148a1ee5-bdb1-48cf-9cb2-1038db2cf788}} and {{formula:19244f3a-d393-435d-b5c8-ef8d050b1696}} (as {{formula:0a320b64-a8ba-4dc7-81d6-6dde86c260c1}} ). Since {{formula:6ac84c17-d808-4727-9b40-36f504a801a5}} ,
we get {{formula:c6a6cb4f-c1b7-4ccc-a96c-a8a7eee421e7}} .
This completes the proof.
Graphs of girth {{formula:cfb7bb43-2a09-47a6-a088-a853fd9e428d}}
In this section we prove Theorem REF , and for this particular proof
we resort to absolute sizes of the sets involved rather than their densities.
The reason is that the girth assumption does not survive blowing up a graph, and
this makes the density-based language unnatural.
So we fix a triangle-free graph {{formula:3edec6ba-4969-4cb7-8538-94f1f9affacc}} with {{formula:597e1460-db79-4060-8ac9-37621514ca5a}} , and let
{{formula:ca3b96b6-4b2a-434a-9b0c-9d8dd76b04f2}} be a vertex of the maximum degree {{formula:4f36311f-21b3-4183-aa8d-fb5400520025}} . We may assume that {{formula:aa6b0875-646b-4abf-9c2c-66ce39fa8db6}} (otherwise the result is trivial) and also that {{formula:1cb9ca8d-2af8-4294-a8dc-7de879952fcb}} is a minimal counterexample to Erdős's conjecture, that is {{formula:c964ad6d-0364-4e88-ade5-a057a0f19f3e}} for any proper induced subgraph {{formula:229f56fc-0780-4175-b8bb-ce8173d3cb0f}} of {{formula:198755c3-bbb5-4287-ad54-d2163c53b701}} . We let
{{formula:1fafff2f-a8f9-4a36-afab-b5e2753e5227}}
Then {{formula:54ef59e7-143b-47fc-a424-e48d6f78124a}} implies
{{formula:1342423b-6b26-4863-b5d1-c3d568110415}}
We now apply the minimality assumption to the induced subgraph {{formula:ed13f66b-6b31-48ae-92b2-93b39c740c8f}} . This
gives us a function {{formula:3a951775-8dec-4ed7-8a1f-71e6d6c90792}} such that
{{formula:80100175-6f94-45d3-b579-1705741c7d66}}
We use it to define a half {{formula:dbe5d09c-bf6a-4adc-94d4-a80d68209c00}} in the whole graph {{formula:c78af54f-85b1-4171-b37d-831bf95a31eb}} as follows:
{{formula:3b6660d7-4080-4cc1-8549-58a6a554c467}}
where
{{formula:8af38ad4-8827-472f-a6e8-99fc92c1faa1}}
Then we have
{{formula:4a7242f5-c46e-4361-a98c-5404ee4c3b17}}
We will employ two different methods of bounding the term {{formula:62385196-722c-4e4d-a4ce-9f52986f3415}} .
Firstly, by (REF ) we have
{{formula:222cf765-57d4-4ed5-bef9-401a8e6b439e}}
and thus
{{formula:25e907fa-dece-4c4b-9bfa-81c7b533c9bc}}
We now start a case analysis.
Case 1. {{formula:8c34e70d-fc91-42d4-baf4-70c7ab8e7ce7}} .
In this case, since {{formula:70ba8b26-9c61-4be9-93bf-53646d3842bd}} , we have {{formula:1dcb8275-6944-49d2-a129-cbb4a44a4dda}} , and we are done by (REF ).
Case 2. {{formula:bb36d400-ba0d-409f-9d85-181b1dbbf6a0}} .
This time, the same condition {{formula:51d960e2-6815-4bf8-a6d7-c26cea49e925}} (that can be assumed w.l.o.g.)
provides a new lower bound on {{formula:9794efdd-e975-4b3a-9f05-594c20d5a01e}}
{{formula:62baba9a-a052-46a1-ac26-95c04446c22f}}
To get an upper bound on {{formula:60986c51-d601-4c22-adf6-5da0e9a41cb5}} , we estimate the term {{formula:02cae9e3-6aca-4a6b-8622-4c420bf6b6d4}}
as {{formula:6bbea4df-cc05-480c-907f-de5f97abb0a9}} , simply because the degree of any vertex in {{formula:90e51589-cb40-4aba-a34a-c3f2e70ba39a}} is {{formula:7fe0f5d0-01e3-4cc3-9de2-496c2bcacba1}} , and all of
them are adjacent to {{formula:37e46840-a0c2-4ee4-89dd-b0078704f387}} . Thus
{{formula:aa9356b4-0894-4ec5-b313-5b8b15f0e367}}
Hence we can also assume that
{{formula:0c458602-aa1d-4b9a-9184-fe91a63b7159}}
which immediately rules out the case {{formula:038bc22e-f1c2-488e-b179-757582333912}} . Also, (REF ) and
(REF ) rule out the case {{formula:a5fb00dd-21e7-4579-aa88-3b0ac71c1710}} as well which leaves us with
the possibilities {{formula:22e9b442-9bb1-42e6-9ba9-785a269f11bc}} and 80 potential values for the pair {{formula:cdfe4142-9e19-434f-966d-d1b817b37fbf}} .
Instead of trying to do the remaining analysis manually, we employ a different
strategy. Namely, we record our argument in the form of “unprocessed” (and
recursive) bounds, without attempting to simplify them, and then we simply feed
the formulas to Maple to finish the job.
To start with, let {{formula:e333224f-1aa7-42a6-a091-4c156a59ff11}} be the set of vertices
at distance {{formula:0ac52e0e-23f6-4f2b-ae21-13e3c0bf1c61}} from {{formula:aa6c3103-01eb-4d58-9031-8832f66a04bb}} ; note that
{{formula:907e747f-2087-4eb4-b75a-dc40286bb315}}
Let {{formula:b1a816f8-8d77-4059-b569-c5c8fad98952}} be the off-diagonal Ramsey number; we will only use the following
well-known small values:
{{formula:0e07c128-bb30-4a7a-bcfc-a554c4c72a20}}
For every {{formula:4f2751b6-b45c-4150-952f-76e785aad1b7}} such that {{formula:52ef2915-1670-48a1-a799-da83c97a65d6}} we are going to derive its own bound {{formula:ec405695-4238-433f-8227-9caf14616e93}} , and then we
will minimize over all choices of {{formula:1854f67c-99db-4ee6-9e9c-bf113c36c9c6}} . So let us fix for the time being
some {{formula:25297fc0-995f-4a49-9e53-1e06ae35cdd7}} with the above properties.
Pick a subset {{formula:d7c74404-033d-4a9f-9b90-7222a0592732}} with the only restriction that it contains
as many vertices in {{formula:ad7904dd-5327-4178-bf19-f54bb1a9ac8d}} as possible. Then we have
{{formula:640e2c83-1936-40c1-b085-6b76fea2d70f}}
where {{formula:89ec3fea-e7ee-4e12-b71e-ff586276b984}} .
Finally, let {{formula:aadc91e2-51f8-4930-b14d-2ec70cf6a337}} be an independent subset of size {{formula:a5bd5626-fd9b-4e0f-bc88-936fa2fad97e}} existing
by the definition of Ramsey numbers. Further analysis splits into two more cases.
Case 2.1. {{formula:3a05e069-f561-4b74-a73b-77f1ad9273d6}} .
If {{formula:7f5e99b6-1ebc-403f-8a74-368495ad9722}} is even, we take the half {{formula:73c47647-07a6-4bb8-953f-a448acabcc96}} . If {{formula:2ed78965-0d89-4248-8e1e-a4a4bbd29837}} is odd, we can assume w.l.o.g.
that {{formula:0c7d9bed-a49b-437e-90c1-0fdb15a0218b}} (as otherwise we are done). Let {{formula:ee947bea-5e8f-4e3b-9be8-9a44d0457f01}} be the half
obtained from {{formula:6486112a-3231-4075-ae54-6478e79ff752}} by removing half a vertex in {{formula:929ba727-e186-419e-bd02-bdb46ab47c5c}} ; this will
give us an extra saving of half-edge.
We have two different estimates on {{formula:d7616fa9-35cf-4a37-920c-8664b3797aa2}} : one follows from (REF )
and, on the other hand we, like before, have the trivial bound {{formula:e1c5fbc0-7ff7-451f-9127-fb863800c84a}} coming from (REF ). Summarizing,
{{formula:25b97c2a-c41b-49ee-826a-8f4cdf3f3ad9}}
Let us stress that this bound is defined only when {{formula:462829aa-14d7-4eff-bd0f-105cf91222d8}} .
Case 2.2. {{formula:f2bb0352-5bf0-41e2-9dbe-c1ac957f0ba5}} .
In this case we haveWe do not need the half-edge saving from the
previous case.
{{formula:a47a2839-0591-412b-b47b-220e52ed5305}}
where the function {{formula:2a85263c-092d-41f3-acf0-f46f6a999081}} abstracts our situation as
follows:
{{formula:4c26c2db-a6a0-4dd8-b7e7-8aff878c93d5}}
Here {{formula:72200447-8a66-4808-9d75-66381b8aa121}} runs over all graphs with {{formula:6f070eca-e9e0-412a-94b3-2fb85137d4d3}} vertices and {{formula:0b0594ad-978f-4b35-8510-4bf00a79c19e}} , {{formula:74ca551d-8427-4c23-8284-5dec66fad6ca}}
runs over all sets of vertices with {{formula:8c13011b-b549-4123-a0f2-59157a5edb97}} and {{formula:35431763-313c-4ea8-bb4e-07d03ae17a58}}
and {{formula:8311f839-372d-4461-b498-1fb8360eaf96}} runs over all halves containing {{formula:e5246a53-e72f-4c98-a88b-e01bb84b6f2d}} .
What remains is to give sufficiently good (for our purposes) recursive bounds on
{{formula:7e8c50c7-e715-4c45-b6fa-d5c76342e289}} .
First of all, when {{formula:3468838f-c686-40a9-96f7-3fa63f87c66f}} is even and {{formula:43c3eceb-74c8-47d4-9aa1-3f23cc15be12}} , we clearly have
{{formula:e758cc67-dd16-4cda-815a-075ad89823ff}}
Next, assume that {{formula:3779acf0-a2f0-4b00-a0c4-c703118faa12}} is odd and {{formula:389d88ff-01ac-4f31-8d19-f858cf0f0956}} . Fix the worst-case {{formula:a3aea7f5-e40e-494e-a677-3c29952e2dab}} ,
and let {{formula:9863d93f-045d-41cf-8001-c838ebf85c81}} be the actual number of edges in {{formula:72056f8b-73fb-40f9-99a1-8fe6b780e727}} . Then {{formula:6c3d473e-6e54-44c5-8ff1-661d38b5d647}} has at most {{formula:2ad16a86-c5e2-4cd1-be4a-4a85f54fbc18}} edges. Hence there exists a
vertex {{formula:5f81ec79-c348-44a7-98a5-d636f5e0c221}} with
{{formula:b15e138a-9504-40ca-bfe5-42247f082c5f}}
Adding to {{formula:2b1309f4-c0bc-47e7-b84d-2388458bb997}} half of that vertex, we conclude
{{formula:966a2ca9-5d0a-4e9c-8324-a66d1ee91e62}}
Similarly, for smaller values of {{formula:07e931d6-3e90-4e96-b0ab-2712420a1635}} we apply recursion by
letting {{formula:fdbf2442-6681-4915-ad41-96c2b6a7b489}} , where {{formula:71e92dae-180c-4e11-ba0e-f0f8bfe45b30}} is he vertex satisfying
(REF ). This gives us
{{formula:cc74e171-ac7e-4543-9009-b322a34998dc}}
This completes our description of {{formula:8e0a8182-1fd3-40ab-8fb4-511db8bfb548}} in the case 2.2.
Finally, the “master formula” now reads as
{{formula:05464ee4-d986-4039-96a3-305f822bfe09}}
The bounds (REF ), (REF ),
(REF ), along with recursive estimates (REF ),
(REF ), (REF ) on the auxiliary function {{formula:ffabb01d-9b9a-4281-ba42-b14cc942e973}}
suffice to complete the analysis of the 80 remaining cases. See the Maple
worksheet for details.
Conclusion
In this paper we have proved several partial results on Erdős's half-graph
conjecture. While they make this conjecture even more plausible, it still
remains wide open. The same is true for the last of Erdős's conjectures
on this subject: prove that any triangle-free graph on {{formula:12ce39a7-4746-42ac-a8fd-b7e52b0d2cde}} vertices can be made
bi-partite by removing at most {{formula:8036e622-8726-4577-a087-80e49cc9b4b3}} edges.
As for intermediate, and probably more accessible, goals we would like to ask to
extend Theorem REF to a neighbourhood of the critical value
{{formula:fa6d0ef6-01d1-4364-bc35-469135730a96}} , i.e. prove the half-graph conjecture for triangle-free graphs {{formula:0e84ca0f-a690-4c90-9b7c-749e44a03083}}
with {{formula:9323eabe-cc62-4354-8fed-cc97ae29ca2f}} for a fixed constant {{formula:a7b94162-13b6-48f0-89a2-385e01767daa}} . As we noted
above, such an improvement if known for the minimum degree and the average degree
{{cite:67ef2f952c482544eea466dd4185e68f6cd9327c}}, {{cite:2c94f1d9a0a343921d8e8abc2d2076e4859de677}}.
We have highlighted the extremal problem of finding the minimal density
of quadriliterals in triangle-free graphs with given edge density and have given
its applications to the sparse half problem. Since this quantity can be viewed
(actually, in a quite precise sense) as the measure of non-randomness in a graph,
perhaps it might be worth studying in its own right.
Acknowledgment
I would like to thank Andrzej Grzesik and Jan Volec for pointing out the
references {{cite:94795675cca5b9e7ed1b5df497cf7de725d11080}}, {{cite:f59ad3a93c96038d401b8be295d5a4c60e617913}} and for sharing with me the alternate argument
sketched at the end of Section REF .
| r | 70f289b5531560d9f90e933bb026fd1b |
Section 4.3 in the main paper reports the average accuracy across different domains for each of the method. In this section, we show the complete breakdown of the accuracies across domains.
Table REF , REF , REF , REF showcases the domain generalization (DG) results and zero-shot domain generalization (ZSDG)
results on Fashion-MNIST {{cite:d64d0d229f9df8bf84c549cb197868cf28da75ea}}, CIFAR-10 {{cite:852b119306bd94447682f30934447f2495a7ad99}}, CIFAR-100 {{cite:852b119306bd94447682f30934447f2495a7ad99}} and PACS {{cite:46fa5ce839a0adbe08f6fe42a9e90afc04d00ddc}} datasets.
All the experiments are run with five different seeds and the reported accuracies are in the form of mean {{formula:dfd161b5-dfdf-4fde-bc81-9305091bbed5}} standard-deviation.
Experiments with MTAE on CIFAR-100 was run using single seed because rotations of CIFAR-100 contains 250,000 images, which made it computationally expensive.
Experiments with large scale datasets like CIFAR 100 provide further insights into the problem.
| r | 84db9de37d3d89292193d35fb8c25793 |
[Obstruction to {{formula:312c7c3b-5b4f-44c5-9744-bda6dfc2e955}} over diluted {{formula:9c8cae99-3af2-4039-8d30-394463e2474b}} -spin glass, informal]
For every even {{formula:560ca8f8-8a5f-491e-8fc6-be691999d061}} , there exists {{formula:429035a6-0a43-44c4-b983-f481f22d1a6a}} and the following holds. There exists {{formula:75638169-a3ff-468a-abaa-de95bd1a730d}} such that if {{formula:b7620ce1-5974-4b93-b03b-69d465768b93}} outputs a solution {{formula:7427861d-5eeb-4f3d-8d30-90697575a355}} with {{formula:991fe102-5617-4563-843e-dc6659f1c3a9}} being {{formula:0602ecda-d5d9-4cc5-b516-e086029bc453}} -close to the {{formula:f7346c9a-9b20-406c-b323-65b39cea005d}} of a random diluted {{formula:b79bfd02-557d-4ad9-9482-4b385a22ac60}} -spin glass of average degree {{formula:b7e9e1e5-b698-4b8c-9b44-d721010e0b73}} with probability at least {{formula:07e0f9ca-fda1-481f-af07-b050fb98baa2}} , then {{formula:e34cd65a-782b-4f1d-a816-3629e7dc1e16}} .
The formal version of the theorem is stated in thm:obstruction-1. This result can be interpreted as a weak obstruction to logarithmic-depth {{formula:1fbdf378-6930-4de8-b727-2f06761c4a5a}} in approximating random diluted {{formula:07b36e04-b700-4eb6-b5e5-77576467e71e}} -spin glass, which is equivalent to the random {{formula:6ede2a99-de4e-4933-a60c-2c05d85de287}} problem. See sec:tech overview for comparisons with previous work such as {{cite:6c156b337e4f421488ea4f5b9707cbb141fdc250}}.
| r | 11fcbf5213a88f1f88158109e517b967 |
The collective spontaneous emissions from the identical two-level atomic ensemble was first introduced by Robert Dicke in 1954 {{cite:20bc7b18c642bd60315b3d3be57f5ab9fe6fbc02}}. Since then, the Dicke superradiance (SR) has brought a tremendous attention to the quantum optics community {{cite:7e71274886197ce58ac8c8101b66b2e368c09a40}}, {{cite:72585b860a266b10cd1049595a1984ea5d0d799b}}, {{cite:7317d7bf01862ffd07924e4eb23b074f284d6f46}}, {{cite:74d4954341073fc66b8fd3f577a34a0fe2a1b85b}}, {{cite:321c71251712414ee141682cface877c1fe54530}}, {{cite:9b276935dc55e4a0ae6ddc6a9a26f2e3ee382a4e}}, {{cite:663890201161c1770fda8916cd642d8f50e91a4d}}, {{cite:4bc5d42e1292ba6d0ebce5d297211feb15006b04}}, {{cite:1ae6e9d34e8d2a6524be6a0888aebe754bdc70c1}}, {{cite:e79d02aca1733ee165b6777d5570eefd75f35cfb}}, {{cite:cd716622dc4fbb1d3e80e82a03fa73ccea8cb27c}}, {{cite:b0715b134a65a4b0c483d0296728991ffd38dfc8}}, {{cite:7540e957c2f767389f939840affb1e0eaec7c316}} . However, the recent experimental realizations of the complex quantum systems of strongly correlated many bodies in gas, liquid and solid-state phases {{cite:6c2de13514b25b29365142205c69f1003c3a51e4}}, {{cite:f165f2146909c4d14200f089e481f27b6b3a6246}}, {{cite:9e8630b9ef0d2757c298d5451a6a8baf78426ae1}}, {{cite:18ceec62bff8ffebe32e88f42a8b489cb4d20da9}}, {{cite:ec4c9209fc023467e7b18ef621821f4b79df81de}}, {{cite:4b88d50526216ef99c3f65cea00e44e2b2963959}}, {{cite:dced2d5c2ca63df85fa54f4ee86c2eb1c575f2d6}}, {{cite:910a34634575d4d1b84a5964e54603324b8e7480}}, {{cite:71987114a7e3481a59808bad9a4ec51908c29d7f}}, {{cite:603c03d9053504cb6a0076a73e1f02a58eb7600b}}, {{cite:c3566b64b2f1ce714875bea4cd31eaa37781be54}}, {{cite:49b006c4d255d611d39861f2f980a750b9812a3f}}, {{cite:14863896210e1ec5175b10421d34a437284422b7}}, {{cite:2dbb1171b9d2568096d315e459602a4e05ce9a17}}, {{cite:d637dc8f3baacadd2baf939bf7f8b1f562230f70}}, {{cite:255942344667747936f036db23674bc06002ddd1}}, {{cite:b820c91476f8c87111d0ff9f6903a6c9ea8ec09e}}, {{cite:3e598aa1bba552047ffb2e929809842367af06f7}} demand the understanding of collective phenomena in much broader sense {{cite:4860efac31384b977f147a083789368e121fc4ff}}, {{cite:2631f2c6d0f4bda730ab4af27ce9eddd26b2cf3c}}, {{cite:8a87d09d8542d17e6a23f3fe491bbe4a23791dda}}, {{cite:119f4c6c3eb02390519579399c77e4ededbeb65d}}. For example, the surprisingly tolerant superfluorescence (SF) has been recently demonstrated in perovskite materials at room temperature {{cite:255942344667747936f036db23674bc06002ddd1}}, {{cite:b820c91476f8c87111d0ff9f6903a6c9ea8ec09e}}, which may be understood by a quantum analog of vibration isolation mechanism {{cite:119f4c6c3eb02390519579399c77e4ededbeb65d}}.
| i | 5d37e9a5e6353654518c1436ec5358bb |
Tensor models have received a great deal of attention in machine learning and statistics literature {{cite:1921db7965da5cdabc38eaec4f885b4ac641ad11}}, {{cite:4a7e95d9d7c66744560dba5623996610ab54f4b3}}, {{cite:c5e0c74cd7295beb4910e891471d5c14c29272de}}. For tensor clustering problems, most existing work {{cite:8b18e59f017a12e735f9fa4a39dcb3fc46a69d33}}, {{cite:4dbec51044f7d9d147ea56fbd4b22f86bd78a9eb}}, {{cite:309e20ab17a6d6c88ed18baf8b16d514242be416}} either works with a single tensor or assumes that the clustering structure is the same across different tensor directions. It is also common that they only consider tensors with entries that takes continuous or binary values. In terms of model estimation, clustering is often based on solving a regularized optimization problem, which requires pre-specifying the number of clusters or choosing the cluster number based on certain ad hoc criteria. There are some recent papers on Bayesian tensor models, e.g., {{cite:c853cffdec72f95f2382314e270feaebf04dc341}}, {{cite:1f4530407d271223761b095e3a8788ec28c89d23}}, {{cite:7b884f8d02a02c159a0b53903e1a59a77f536967}}. However, all of them are in the regression context, hence cannot be directly applied to solve our problem. Our proposed approach differs with aforementioned methods in several ways. First, we consider a flexible multidimensional tensor clustering problem, which allows different clustering structures over tensor directions. We believe this is a meaningful relaxation in many applications, e.g., basketball players' shooting choice may differ significantly in terms of shooting distance, angle and game time depending on players' position, shooting preference and role in the team. Moreover, we focus on count-valued tensors (rather than continuous-valued tensors) for the obvious reason that the number of shot attempts is the main outcome of interest in our application. Thirdly, our model is fully probabilistic, which allows an easier interpretation compared to optimization-based method. In particular, we consider a Bayesian nonparametric model under the mixture of finite mixtures framework
{{cite:5ff6371ae214bedcd2794513b1796e18e087fa6c}}, which allows simultaneous estimation and inference for the number of clusters and the associated clustering
configuration for each direction (e.g. distance, angle, and quarter). We show that the posterior inference can be conducted efficiently for our method and demonstrate its excellent empirical performance via simulation studies. We also establish posterior convergence properties for our method. Finally, our proposed method reveals several interesting data-driven insights of
field goal attempts data which are useful for professional basketball players, coaches, and
managers.
| i | 73d092f4e59d8b083a546f1d53ca83f6 |
The results in Theorems REF , REF , REF and REF are new; they are also new in
the case when entire solutions of inequalities (REF ), (REF )–(REF )
in {{formula:144338b6-ef5c-4d1b-b20c-1dd5bfa7b4d8}} belong to the function space {{formula:78f0bd9e-4c22-43fa-ac34-abcf0df22c9b}} . Theorems REF , REF , REF and REF were proved in {{cite:a1c85723ae9a67995e590318370389ae6a1a985e}};
in the present paper, we show that these theorems can be obtained as
simple corollaries of Theorems REF , REF , REF and REF . The result in Theorem
REF and its proof generalize and correct the result in Theorem REF and
its proof obtained in {{cite:b2672fe6226908821724a8cc351b53838a29b4d7}}. We would like also to note that the
results obtained in this paper were motivated by results established
in {{cite:c366b151569bec7745ee2848c50170b5ea992e8a}} and {{cite:35f666d02f9a72d0ef3a1af0d5ed26b08df9bbd3}}.
| r | c3ed73c13044f0b28102aea5348e8576 |
In {{cite:a8e20c18cc55004a89392765ba57825c6a3ebd0e}}, {{cite:01698512aec109dbd3b69e1443313c71abb00a3d}}, Ambainis first designed a {{formula:a9faf3bc-9503-4ada-90e9-97a1efd12135}} -query quantum algorithm {{formula:06bb8b65-cfa9-41b2-9d6a-e6e90b57bd61}} which solves the promised version in which {{formula:2b00ad74-e5e9-4ca5-b8e4-d869423338a0}} contains at most one colliding pair, by using quantum walk search on Johnson graph.
He then reduced the problem to this promised version, by sampling a sequence of exponentially shrinking subsets to run {{formula:35a4fef0-8ef8-407d-ac1e-59f4ae3c1bbb}} .
By showing the probability that the sequence has a subset containing at most one colliding pair is greater than {{formula:67f80196-74ee-49f8-9b67-7f826798befe}} and the overall query complexity is of the same order, Ambainis successfully solved the element distinctness problem with query complexity {{formula:6319658c-4ac2-432d-94b8-4c6475f3f576}} .
This two-stage process shows the promise problem that the algorithm {{formula:5583b35f-7918-4601-88b6-05477108ba1a}} solves is crucial, and we formalize it in the following:
| i | 95007ed7c5012dbd2cc164899a86eb28 |
Recently, domain adaptation has been applied to improve the performance of the combination of datasets in the training step {{cite:0eaa4f7fd62fdedbb883b814ec2f5fad46ddf999}}, {{cite:6bfc02149ce135f9401f86bd90373859f8b66240}}, {{cite:b704b3549543ef0ef2764862d6636e2ec937a5fa}}. {{cite:6bfc02149ce135f9401f86bd90373859f8b66240}}, for instance, combines one synthetic and one real dataset using domain adaptation to improve the result of training. They claim that directly using synthetic data may not improve the results for realistic data evaluation due to dataset bias. They adapt the domain of a synthetic dataset to a real dataset using Style Transfer and combine them to train their models. {{cite:b704b3549543ef0ef2764862d6636e2ec937a5fa}} also performs domain adaptation, and they claim that due to the lack of paired synthetic and real images, the synthetic-to-realistic image translation adds distortions to the depth estimation. They overcome this difficulty by exploring a more complex training procedure involving synthetic-to-realistic and realistic-to-synthetic translations. In addition to using synthetic data, domain adaptation could also be applied to real-to-real translation {{cite:2920ab5b1f8849f3249d94c809cfeb89696e34f6}}, {{cite:f6ed8a28bc10888b13922fd0dec6927546dd10b1}} since the dataset bias also affects distinct real datasets, especially by variations of scale and capture's position of the scenes {{cite:6e0e7ce40323b09c7f62055c9c836ce860a1e810}}.
| d | 70658ae53dee2a63e1a15f1737d21c39 |
Pre-training segmentation network. We train a segmentation neural network to extract lung fields from CXR images. For this task, we adopt the U-Net encoder-decoder to perform semantic segmentation {{cite:6e78ee21195bb53e2050a27cfc4d43a1513bad34}}. With the pre-trained model on {{cite:53f2470c1fff270e93dc4dd308965b0c6a7b4c5f}}, {{cite:77a9446564084fe3c0c3f9ae1a08887e32a6e296}} datasets, we generate pseudo labels for unlabelled COVIDx dataset {{cite:40d716e7b598452a6af3dd7bfc9440d71dedbb7c}}. Before feeding those pseudo masks to the multi-task neural network, we use custom pre-processing algorithms.
| m | de24a3830fc72e5347c430172fdf4413 |
Fig. REF shows the changes in instantaneous propulsion power of the UAV based on different schemes over N time slots. Specifically, compared with the other two benchmark schemes, the instantaneous propulsion power of the proposed scheme is lowest during the entire flying process, since the optimized trajectory based on the GPECM is smoother, and thus is more energy conservation. However, the instantaneous propulsion power of the UAV with the PECM in {{cite:375ec25715305af2f514d29c9032ac8d171ac608}} is much higher than that of the other two schemes, especially when the UAV keeps turning above each user. The main reason is that the PECM in {{cite:375ec25715305af2f514d29c9032ac8d171ac608}} did not fully consider the generalised {{formula:61df93bf-14fc-4c0e-b61f-e4718f549722}} w.r.t. the acceleration and direction change of the UAV. Therefore, the UAV with the PECM in {{cite:375ec25715305af2f514d29c9032ac8d171ac608}} flies in a manner of frequent and sharp turns above each user, which causes addtional propulsion energy consumption. The instantaneous propulsion power of the UAV with the PECM-HT is relatively stable, due to its constant volecity in the hovering and flying status. We also notice that although PECM-HT is a simpler scheme, it is not the most energy-efficient flight scheme for the UAV.
| r | 1bb74fe2c43ff51ce0ef4e799882866b |
In {{cite:d5abdc9ca0b50fe573a2cc348c6d5d7ebdac2f98}}, the relationship between the second-order moments
{{formula:577a1602-d18b-4e5c-b6e9-cf402232cdf9}}
| r | 3a6e9c67f168f57db6fb0c6dcbeec033 |
where {{formula:538250ad-0ecf-493a-ac99-85eb1cedd598}} is a scalar, {{formula:e894b5e9-2261-4e3b-9316-aa2ba434ccd9}} , {{formula:c19ac043-e7b0-4b4a-a2bc-60baba3e6a8f}} is a unitary realisable by Hadamard operations applied to some of the qubits, and {{formula:584f9bce-3209-4bff-bf30-0532217a58ec}} is an operator such that {{formula:1e0fe1e3-dc32-4b6a-8177-9ee2cfb32a95}} .
In our work, the choice of {{formula:0d998f97-2d7b-4c56-8210-983bbe8ebbfe}} and an initial expansion matrix of only principal rows and zero rows corresponds to {{formula:6280492e-1930-44c0-aa10-400ac3d72ff6}} ; then the choice of Gram matrix and expansion matrix can be described by an operator {{formula:f5d82c7b-edd2-4da6-adbc-f554746799ae}} .
By using a representation for {{formula:9d8fc3fa-59d0-449c-9142-510907f1da19}} using the stabiliser formalism, Ref. {{cite:0516594436d0970c40ccb0535884cd1d4561d58d}} is able to simulate one- and two-qubit Clifford operations in time {{formula:24a7b5cf-917f-4e7e-8f30-564c799e4088}} .
Our techniques are better suited to cases where the expansion matrix {{formula:f90fd340-c477-4a7e-a57b-b6ab95b6cc45}} or Gram matrix {{formula:1b343d07-a2ff-434b-8848-0b438bd67fe1}} are expected to be close to {{formula:1f51887f-1d4d-4711-97a1-923040a3bed4}} -sparse, so that unitary gates may be simulated in time {{formula:f1c045cf-38b7-42ac-8dc1-f8ab8ae3c575}} , and where measurements are frequent enough that the techniques of Sections REF and REF provide an advantage over the {{formula:f8a07258-7905-48e4-bc5d-4a79827fa9e9}} -time simulation time using the techniques of Ref. {{cite:0516594436d0970c40ccb0535884cd1d4561d58d}}.
| d | 53bbb9de7f543c4500e9431c1d7977c8 |
Setting 2: For fair comparisons on the leadboard, we use the groundtruth of text bounding-box and transcript provided officially.
Character-Word LSTM is similar to NER {{cite:eb00ba3093d6869c03ce9a96c29863312161c652}}, which applies LSTM on character and word level sequentially.
LayoutLM {{cite:0a29b7384fce56d23d054422299bfdc0a99d8828}} makes use of large pre-training data and fine-tunes on SROIE.
Similar to LayoutLM, PICK {{cite:c8814c642a2a8a089a343a546752c321fb26fc47}} extracts rich semantic representation containing the textual and visual features as well as global layout.
Compared with these methods, our model shows competitive performance.
{{table:d9740f0d-e480-4216-a281-6c738fe207d4}} | r | f258f527017c06890bb49f0fa46f8f4b |
From Corollary REF , we have
{{formula:b1c43b0b-442f-470e-933c-22aec1da2981}}
Since the result depends solely on the behaviour of the characteristic function {{formula:a6666138-0c4a-4ff9-a835-77e5d279941c}} around 0 (see e.g. {{cite:d870b91c0b1a37fb5b3e0bbcaf0ed7f6b5d789fc}}), or equivalently on the tail distributions of {{formula:da5c677b-adbf-4dfd-8e35-775554ee14da}} , the tail asymptotic of {{formula:72438dc0-9726-48cd-afbb-22bf30fd95e8}} can be readily determined.
| r | 3b421099e30d9f0a2ab63ef6da8c4839 |
We will rely on the martingale used in the article by Pemantle and Peres.
Let {{formula:106665c8-afde-4cec-9eab-282c7da220ac}} be a random vector with law {{formula:2f3697c9-fb60-4078-893d-76cce70513b6}} and define the random variables {{formula:e496e0ab-f7bf-40da-b3f3-9592b257e193}} for {{formula:676d5d92-5f10-48d6-9e5f-3985b6e42d1d}} as
{{formula:5a58d78b-2860-4148-ac15-5ee9f2e2f4c0}}
i.e., {{formula:9af2a026-db21-4457-bb06-70385691305f}} reveal in a uniformly random order the elements of {{formula:14a598d4-d8cc-4ed7-9778-ae00b76ac2e9}} .
Let {{formula:3f579e70-0a8e-4fa9-91fd-0f6f344dad8d}} and {{formula:05c52cc1-f39e-427c-8820-fbaaca4191b9}} for {{formula:7f83e85d-493b-4eb7-8bcb-946713db878a}} .
Then {{formula:a03c5375-4789-4ce1-b9a6-500746689fd8}} . We will use the matrix version of Freedman's inequality due to Tropp {{cite:ecc2ddf51dc88029a324d17355d0a2c050658a9e}}, which asserts (in a version specialized for our application) that if {{formula:deda0364-2cd3-4f9f-93d6-58b04261d6fe}} a.s. for all {{formula:c949700a-988c-41d7-a63c-794528c6ffcf}} , and {{formula:5d2924c3-da92-4551-8964-e0708e20b1c9}} a.s., then for any {{formula:2f2a8d81-8ebd-4583-9f2d-d8b6da148ac6}} ,
{{formula:5acc6c2e-8660-4859-9c4c-d028b2889724}}
Consider thus a sequence of pairwise distinct {{formula:cb1e1936-701b-45b9-afcb-672f73d9cbe6}} and denote {{formula:96e55345-502b-42fc-bdd0-a12707aedde0}} . Similarly as in the proof of Lemma REF , if {{formula:76782eec-d66d-4731-9c74-08c665e9a712}} , then we have
{{formula:408a853b-bd28-4a1a-938f-fcbe028f9339}}
where {{formula:ef2b72b1-6fb0-4e8f-a121-af6c49cf2e2c}} . Therefore we have
{{formula:bf77291a-f9ce-416d-a37d-d27d44d7de37}}
Since the SRP implies the SCP, there exists a coupling {{formula:79a45824-303a-4481-9d6a-2b53a542afba}} between the distributions {{formula:5daaddfd-5691-420e-b476-415ea5d1e354}} and {{formula:3e79c23f-f315-4ff2-9798-30ffe53c7c13}} such that {{formula:6c0bd885-fdc5-49f5-8e2b-c7b2c4cf80e0}} and {{formula:db3e8cc0-4ac0-4b57-a432-813ef7560ae9}} differ just at coordinate {{formula:f3ab82a9-ce82-49fa-8084-4b270c0e3832}} and one additional coordinate (at which by {{formula:438f935e-51f4-4fd6-a0af-9f6b50918e8a}} -homogeneity {{formula:46980249-b006-4599-9156-30adeb7bd5a0}} necessarily takes the value one). Let {{formula:18fcfaa4-7b90-44ec-a1c0-6de277ac6575}} be this coordinate.
We have
{{formula:57bfc0a8-b4d1-40d5-9a09-4d6b5c68b9db}}
Since {{formula:1db9a27e-19fa-4733-a392-c459fda00fb1}} , we have
{{formula:68363c60-8a65-4dbf-b5e7-808a29b5789f}}
where in the first and second inequality we used the operator convexity of the function {{formula:99856b91-cc26-490d-aaaf-399ef245bde7}} (see {{cite:ab36d3aff46343f77e2740fe7bfc8e256035ba26}}), and in the last inequality the assumption (REF ).
In particular, using (REF ), we obtain {{formula:92b05024-6632-4321-a12d-b090b6f96595}} , so {{formula:73604c96-3b06-416f-b30c-798834682f50}} .
Moreover as on {{formula:1fec59f6-1641-43e0-baaa-e0237f9b9ba8}} we have {{formula:a21cecbf-d5f2-4027-a53b-387be2a77a80}} , by (REF ) and (REF ) we get that
{{formula:8bb1c671-d341-4213-8f9b-90d855df2892}}
Let us now slightly change our notation and think of {{formula:e5c80bbe-ac64-4553-9d14-e9f141d2a84a}} as of random variable defined on the same probability space as {{formula:b411a7e3-4281-433f-bfaf-cf53cb5180ad}} , with conditional distribution with respect to the {{formula:32cb072a-2a4c-4900-9d58-4e3dd7782e5a}} -field {{formula:df12b366-0db2-43a4-86f1-0dcf2e21b899}} given on each of its atoms {{formula:ca26e7d9-c004-46b1-9aee-539b741fe545}} by the above construction, using the corresponding coupling (which depends on {{formula:d53ffaeb-62df-4e95-9bb3-4efd749ddac4}} ). Then the above inequality can be written as
{{formula:bd13d4e6-ff6b-4e07-aa3b-b7b25dba8f72}}
Let us now go back to the equations (REF ) and (REF ) and let us apply them in the special case of the function {{formula:cabf2afb-8034-4af5-b02d-50749cd02d87}} , denoting the corresponding martingale increment by {{formula:a8b0b8e3-3d0f-42f1-b4b6-d92dd7229e2d}} . We obtain that
{{formula:43d2e882-5269-49c9-9b67-b3a0148a74a2}}
Thus we get that
{{formula:2ea382cb-e141-4def-a28d-2bcb75379c2f}}
i.e.,
{{formula:b756dd42-a102-4e1f-b541-fe95c87555d3}}
which combined with the estimate (REF ) on {{formula:f5d0a6f9-0ebc-47b3-b46d-54f3c9cc6d33}} (replacing {{formula:ad60f303-21d8-474e-9727-e49e06e98f2a}} by {{formula:036612b5-5152-43c5-8898-1d16aafb64f8}} ) gives
{{formula:3228ee67-1121-45c0-b588-1afe506e743d}}
where in the last inequality we used {{cite:f12bc16dc269e799e38da2f362377568ed8c973c}}, which asserts that {{formula:3316fe77-534f-4138-aa2b-6ddf68f30e02}} (we remark that this is the only place in the proof in which we use the full strength of the strong Rayleigh property).
Extending the summation to {{formula:684a3658-9b9f-4a19-97bc-b4299a96d6ef}} , we thus obtain
{{formula:d56a15c0-e332-44f4-9222-372f96a57f9e}}
whence
{{formula:53502b19-79ea-4b64-81a1-27f34395037d}}
Combining this with the already obtained bound on {{formula:647ee432-73da-4182-8f86-0e7d779062dd}} allows us to apply (REF ) with {{formula:9cbafb96-8e39-4eb2-9763-492cde009e8a}} and {{formula:42776b87-79f5-4eb5-995d-224b7af3b198}} , which ends the proof of the theorem.
| r | 5c20e07225766d0d83ebed7e1892d124 |
As discussed in section we choose to focus on six representative models among the many possible scenarios with a colored X. The selected models span a wide range of possible spins and color representation of X so that they can be used to estimate the allowed parameter space in any other scenario. The relic density constraints from Planck {{cite:9c90988a0e277c1eda05ba4c70fe5f9a8510d5f6}} as well as the bounds on the lifetime of X derived in section are shown in figures REF and REF in the {{formula:d8b8a64e-30e7-4b23-96af-bb4158b56a4e}} versus {{formula:d1f420b4-b30e-4ead-9013-2701e4a22b53}} plane. The constraints from the coannihilation partner being long-lived have been calculated assuming LHC13 with an integrated luminosity of {{formula:3353b18a-c469-440b-b77f-17990662f10e}} as detailed in section REF . For each model, we show the annihilation cross sections with and without the Sommerfeld corrections and bound state formation effects.
{{figure:36260e8a-8033-41b9-9b17-9fa6f3ddc8a3}}{{figure:09793b6f-1c97-4164-b714-e8d78d79876a}} | r | dacb8d1c264630bdf81c238f3b7792bc |
Our final result refers to a phenomenon identified by DeMarzo et al {{cite:4756daf2e26bc01d596368b710b853b48e8fa7ca}},
who study the evolution of multidimensional opinions à la
DeGroot {{cite:e300ae9d7c4385342d0bdc0b0a0dcb0656ff1fbf}}. Their main result is that, at some point in the convergence process towards consensus, all the agents' opinions align and keep aligned thereafter. Thus, they claim that the individuals' opinions
over a multidimensional set of issues converge to a single “left-right” spectrum (a phenomenon that they label as unidimensional opinions). We find that this result extends partially to the bi-dimensional version of our BC-PF model. In particular, we obtain that the opinions of
all the agents within each cluster do align at some point of the convergence process. However, this alignment only holds at the level of opinion clusters, and not at the population level. Hence, our results suggest that for dynamics not driving to global consensus, the main result of DeMarzo et al would be expected to hold partially.
| d | 91db21fce862de99dc8336ef099404a9 |
The statistical properties of a system that follows Langevin dynamics can be studied by recasting the theory in Martin Siggia Rose (MSR) formalism {{cite:6d70687fefdc09a9f79629591d6dadfbbd3edcea}}. In this formalism, the statistical weight associated to an order parameter configuration can be written as the exponential of an action, which is referred to as the MSR action. Within this frame work, we can use the standard field theory and RG techniques to extract relevant physical information about the system {{cite:3611504fa488f0c1a6d45d744565952919349d74}}, {{cite:a345e660451e53d59879d8d3140443a9a327304c}}. These techniques are particularly useful in analyzing the critical scaling properties of various models and have been extensively used for the same over the years {{cite:3611504fa488f0c1a6d45d744565952919349d74}}, {{cite:cb95730b0d8660487155fc4f973571b659aaa16c}}, {{cite:ff0663484215feb6b3680200693cf557b2248ada}}, {{cite:19009ae50671a154048b6d4a47fce3b720691e7b}}, {{cite:f09ebc4602d930389355ef18f1e8e9215ed70c41}}. However, the RG analysis can be a formidable task when the calculations involve numerous multi-loop Feynman diagrams. Computational methods are more desirable in such scenarios.
| i | 94038a4564aad7c16c0623e2cb004ac7 |
Classical Harmonic Analysis is strongly related to the two basic geometric operations of translation and dilation. A paradigmatic analytic problem involving these basic operators is the pointwise approximation of the identity of convolution. The basic question is the almost everywhere convergence of {{formula:cfac5862-b25e-47d1-9433-02c2d30d8e75}} to {{formula:a9f55d19-84e2-4ee8-8026-1c34da8aa1ad}} when {{formula:57c5c2e4-5ce3-4dfa-994b-549816c28f77}} , with {{formula:e66a2e5d-78ad-4202-86e6-b7322f734451}} , {{formula:6d73f4d9-e625-4605-818a-20a28d4f86b5}} and {{formula:b9ddc2ff-9746-42ad-9d4d-2df91a0a3a0b}} belongs to {{formula:34b5b64d-8797-48db-947f-c7f93e9d9088}} , {{formula:04d5d921-0ce4-49fa-b410-be26e0ba96a3}} . The most general conditions on {{formula:faccaf87-6d66-4260-bf68-3a9c1e9c923f}} in order to solve the almost everywhere convergence problem goes back to the 1976 results due to Felipe Zó (see {{cite:7417eeed9177bb8fb549b3158c051bb7d2cfcbd2}} or {{cite:93fc76dcc1a01d0e5b8eabcdd246642190f2e897}}).
It is worthy noticing at this point that the main result in {{cite:7417eeed9177bb8fb549b3158c051bb7d2cfcbd2}} can also be obtained from the vector valued approach to the Calderón-Zygmund theory of singular integrals. The classical results (see {{cite:ca76034fee1efdc2579697cc03fa132bd9cb6b28}}) of decreasing kernels are covered by this general approach.
| i | 98f2d056a0b69434a622106c58f05911 |
For the spin-orbit sector, the leading order (LO) effective potential was first computed in {{cite:94b3fee5d932c3affefa29df79b191a46add291d}}. The next-to-leading order (NLO) potential in {{cite:b47ea85701d8cf5ca3ab39ab7fe3e28f0a7a77a2}}, {{cite:54f082afd1a0f5a65ae84ad21ebf7ec956c977c7}}, and N{{formula:724bead1-6e64-461c-b503-026cbe067b09}} LO in {{cite:4481da4fc91877839d86fa4b0513eb52a967c5b8}}. Partial results for the N{{formula:113a0168-b83f-45db-bb20-7d1940ec9759}} LO correction were recently presented in {{cite:875be302c0e09d7adfd8dcc888a3e44a9ddf8cfc}}.
Similarly, for the quadratic in spin sector, the LO effective potential was also computed in {{cite:94b3fee5d932c3affefa29df79b191a46add291d}}. The NLO in {{cite:867602e2016cea5711714989fbddb0502bda4ba6}}, {{cite:1c2e24980cc4db4a4590334b8498264a0e5879c3}}, {{cite:3b641cba3929169003d15a08d3c38b51db91f415}}, {{cite:7869688d9e88525e9886b362ee3f5af3066b414a}},
N{{formula:4a47d6de-72ab-4ba4-86da-7bbc46f592da}} LO in {{cite:6ab49bfef4abf4b5f03e1191e07b88ed3bb908cb}}, {{cite:69d96c2993373d3b0f309c13954354bfb256f6ee}}, {{cite:d09154e245548ba4b44e8e8f11ad831b99ed88e5}} and N{{formula:072d0842-e646-42e4-92be-98df5b0173a6}} LO was computed in {{cite:99429aef58efc86b7d05bfa837bf6b4eb7b873a3}}, {{cite:423de0fe5e2a5eee0f4d15b45f85ccc7c242a08a}}.
The computation for cubic and higher orders in the spin variables can be found in {{cite:55434026cb313add259685ebd49ffbd9c4c717e1}}, {{cite:d82e40e943ff065a7e5103d253f092bb30391d47}}, {{cite:fb7ecf0dfcb3330f61d86062a45d6ad50468956c}}, whereas the finite size effects are described in detail in {{cite:7869688d9e88525e9886b362ee3f5af3066b414a}}, {{cite:d82e40e943ff065a7e5103d253f092bb30391d47}}, {{cite:fb7ecf0dfcb3330f61d86062a45d6ad50468956c}}.
The spin-orbit N{{formula:76ee6bfd-0198-44d2-be0e-23ec67593419}} LO was then completed in {{cite:9377bd3b845ccd111c2efa1acd60ab8fb87dd674}} using the techniques of first-order self-force.
| i | 7642388d18f06ce084c0ef68d5a6bc4f |
Relative to more sophisticated covariate representations (e.g. {{cite:c2703a91ce4b80f4c7e2300f2a5833cb6d3d7809}}, {{cite:348c23c65eb1aeafe7100078d7941ecd2ec50891}}), the PPL model is conceptually simple to understand, and facilitates computationally-simple inference, both of which we consider to be attractive features for the metocean practitioner. Relative to the simpler penalised piecewise-constant (PPC) approach of {{cite:0172a3fe90f0da5ba97447089785f519d75508bd}}, the additional flexibility of the PPL model allows a physically more plausible representation of continuous parameter variation on the covariate domain. Better use of the sample is made, with parameter values at each PPL node informed by observations in all bins adjacent to that node. In contrast, the PPC model is only penalised for the total variance of parameter values over the bins, ignoring bin proximity.
| d | 9c1eafca68b616d3c9c30c8dabe1b9d5 |
Our best performing model, selected on the basis of a systematic analysis of the relationship between cost weights and recall, places us second amongst the 25 teams that submitted their results on this task. We present our score on the test set alongside those of comparable teams in Table REF . We note that the task description paper {{cite:9b9a1819db085bb1340ac93426bbf50d7abe4d00}} describes a method of achieving an F1 score of 60.98% on a similar task although this reported score is not directly comparable to the results on this task because of the differences in testing sets.
{{table:15874449-7e2f-4b17-a400-e76a56d3d859}} | r | 35c41d1da291b16e1fbf289dbac4e89a |
For all three events studied here, we find that conduction is the dominant cooling mechanism in corona in the early phase of the transients. About 80 s after the heat pulse, radiation looses begin to dominate. We observe that conduction dominated cooling in corona during the early phase of the evolution has also been reported for flares {{cite:58199eba4c56a96e253a9e5d82246a52f7ccdfa5}}, microflares {{cite:b799fc0d1b5ddd662125a2bf944d931a51942aa9}}, and are also expected for nanoflares {{cite:9940a116af0594a050fa3014121020863059888c}}. Our results show that transient events such as the ones observed in Hi-C are similar in character to microflares and nanoflares, and are likely produced through the same underlying physical processes. We also note that our simulated plasma temperatures are lower than the observed values. And since conduction loss increases as {{formula:d0eb2f94-7379-4a28-8112-0c3ba5813197}} while radiative loss decreases in this temperature regime, our assessment of the relative magnitude of conduction loss in the energetics of coronal plasma is an underestimate.
| d | 89a5d709e6b10cb146a9ae14b23a3e23 |
The entanglement entropy has also been computed on the CFT side. The entanglement entropy was obtained by a conformal transformation (REF ) of correlation functions of twisted operators in the analogue of {{cite:c7aa250c6c97a7b9c49185b7790148ba0991db13}}. The entanglement entropy factorized into two modes, which would decouple like the one in the gravity dual. Because a cylinder can be considered as the infinite limit of a torus periodicity, our result should be realized from entanglement entropy on the torus {{cite:f7f623113013b28ad765efc1aec601d3e36e8133}}, {{cite:aa86f7ef21734fc194f0bd9f9d1edc9435f9047f}}. The modular transformation of entanglement entropy on the torus will also be interesting. When the chemical potential conjugate to momentum is zero, {{formula:c592e10b-35dc-4e66-969b-9240cc8033c5}} transforms from zero temperature to high temperature. The modular transformation analysis with {{formula:fe271df8-7322-4709-968c-7dc836ad07e8}} should be applicable for ours.
| d | 92561b3bdb349ae5215aa032efdcc5c4 |
Image synthesis {{cite:606ef4ae0b4405dc93ce6b8e646ab1b7f6eb113a}}, {{cite:ec264483f099a033abdb3b9abb093b21a11d06db}}, {{cite:f454086b4c10fb43d2cc0a226fc7d8e74d24bb98}} and domain adaptation {{cite:97c0f6247a3488daa9eba126b1d002fbf15d003c}}, {{cite:00c79217dabf560d510027895f3b0731f7552faf}}, {{cite:183930c243742b63f84e4bdceff476a4ff2ca8ef}}, where large domain gap and variations exist, recently find normalizations benefit the networks in encoding rich content. Motivated by these findings, our method investigates normalization inside vision tokenizer and supplements tokenization process with capability of context modeling. The clear boost over multiple structures brought by this lightweight design further demonstrates the importance of a good tokenizer.
| d | b9e00168df99207f876169a547ab4400 |
where {{formula:597a306a-ea78-41bd-9d68-23376ac09c8e}} is the distance between AP {{formula:859bc500-5cab-4828-92ab-004ade9d13c3}} and UE {{formula:68e28b5e-700d-48dc-ac69-c319c2e5c1c7}} and {{formula:11fffa3e-66a5-42ba-ad29-f82fcabdf8cb}} is the shadow fading. The shadowing terms from an AP to different UEs are correlated as {{formula:42b35264-d5b9-42d4-a7ad-1f9553043489}} , where {{formula:29bafaf3-793b-4100-98b0-b6f8b6ee05cf}} is the distance between UE {{formula:1ab159fe-0179-4270-b405-255e4178ecdd}} and UE {{formula:edc49a41-7989-4af0-b60d-7d2781cdf9b5}} . The pilots are randomly indexed and assigned to all UEs. We employ 4-QAM modulation scheme and set the signal-to-noise ratio (SNR) 25 dB. We evaluate the performance depending on the pilot-to-user ratio defined as {{formula:56961ffe-8955-4bad-b2d9-df0adb7f00da}} . Finally, we compare the performance of our proposed detector with the CF-MRC {{cite:3b21c707b3bb4370ecf2eccaf86b0989ad94d302}}, CF-MMSE {{cite:999ca2b0cba11673afef0e873ed616d41e890b40}}, and CF-MMSE-SIC {{cite:86eb47107d4c32730a119174055f25af166f2c8a}} detectors.
| r | 67245ff420168b7f25969a0d3f77cd19 |
For the first bias term in (REF ), under the Markovian setting, {{formula:39a756e8-99e8-473b-b532-bad08e62cc37}} is a function of {{formula:34803424-ce16-496f-bf72-dd46caff3382}} , and thus is dependent on {{formula:b7e814b0-1eda-41d1-8594-573b6d4278a5}} , which makes the estimator biased. Our analysis employs a novel information theoretic technique {{cite:ad5a51d2a8a3bb10bd8e98637474f818424160f6}} to bound the bias caused by this coupling.
| d | 3019dea8868486457ead92336b0b5ad4 |
We may list Entropy Rate Superpixels (ERS) {{cite:f7438f3f3fcffc80770565275f4ac85545f468c4}} and the Superpixel Hierarchy (SH) {{cite:f6c9bea22c47128526668e730b7c7d7b7eb44c36}} as the most popular and the most effective graph-based methods, respectively. In ERS, the superpixel segmentation is modeled as an optimization problem of the entropy of a random walk in the graph's topology. While ERS presents high computational time, even when using greedy strategies, SH uses the Borůvka algorithm for generating superpixels with high boundary adherence in linear time. Given it is a hierarchical method, errors in the lower levels are propagated to the upper ones, compromising its delineation performance.
| m | c010fc0a32f4103a524ef18da7e91da3 |
In our theoretical results, the condition (REF ) concerning the asymptotic (for the number of points going to infinity) behavior of the Lebesgue constant plays a crucial role. It is known that for Chebyshev zeros, cf. (REF ), the condition is satisfied, see {{cite:54924148990ef4c888fea5ac8094a3599e235ab2}}. For other meshes the condition is not guaranteed. For instance, for Chebyshev extremal nodes without one or both endpoints it is only known that the quantity at the left-hand side of (REF ) is bounded for {{formula:3f8b3304-0e87-4ef5-86a1-21eb7fba2a02}} tending to infinity {{cite:54924148990ef4c888fea5ac8094a3599e235ab2}}.
However, Chebyshev extrema are widely used for pseudospectral differentiation and integration in the bounded interval, and efficient numerical routines exist for the associated differentiation matrix, and interpolation and quadrature weights {{cite:e542b80398b1c4b0c210180d749a86e81050308e}}, {{cite:deb421365da98d3d531e95d75eb480fd1f3e594d}}.
Our experimental results (not included here), show comparable results when using Chebyshev zeros or extrema, both in terms of accuracy and computational times. It would be interesting if either (REF ) could be verified for a large class of meshes, or the proof of convergence could be restructured such that only a weaker variant of (REF ), known to hold for a large class of meshes, is needed.
| d | 3998aaa0b527b04ed1ee998ab2b8088b |
k-clique {{cite:996be36641f546d3a1d07a4871e1ab9e0707eb6c}} Find k-clique communities in graph using the percolation method.
A k-clique community is the union of all cliques of size k that can be reached through adjacent (sharing k-1 nodes) k-cliques.
| m | 6e38391fa29c02160ebdf9ddcf250634 |
In Fig. REF we show the dependence of the photon energy on
the Lorentz factors of the beam electrons. The set of parameters is
the same as for Fig. REF , except {{formula:5c67c7d4-8155-41eb-87df-cda92933f6b1}} erg/s
and {{formula:30d494d6-c1f2-475c-b4f3-f57602bf4022}} . As we see from the figures, the
QLD may provide the synchrotron emission from multi MeV to several
GeV, which is in a good agreement with the observations {{cite:a50300f487d587721ea6840d971976acdd151fed}}, {{cite:d3a935cc90b9c34b228498b48daa8651209c09de}}. On the other hand, it is worth noting
that according to the multifrequency campaigns, the spectral energy
distribution (SED) of the blazar PKS 1510-089 shows two clear peaks,
one in the GeV energy domain and another peak near {{formula:a7b8c0fd-4e81-4a06-97d7-1178de15178d}} Hz. This
particular peak is formed by mildly relativistic electrons in a
different location, where magnetic induction is supposed to be of
{{figure:663e7398-2abf-4223-a4c8-1d77ca80fca4}} | d | 933bbf5bdec74d99098296af53fdc721 |
[Theorem 6.1, {{cite:da15eeb19ecc8b1528995ad059567f57c2f92665}}]
Consider a bounded domain with a {{formula:f8d40187-d6de-4fa7-9731-0e9f4e0acf97}} boundary, an exponent {{formula:0ef2615b-0776-4808-bded-93fce3d7356c}} . Then for any {{formula:41fb2ee4-a351-4726-85f8-61af5f23dcfd}} and
{{formula:80ca6adb-4138-45d9-a82e-3269176c7e27}} , there exists a unique weak solution {{formula:be8b3b87-a0d0-484f-a618-66fad7e4a190}} of
{{formula:f21544f5-1da9-43b5-807a-8dd98102c8b9}}
| r | b5037e73ae09d6a39bd2164c3e6495b6 |
for any measurable {{formula:ef03458a-ffc1-4fed-985c-3f36258be91f}} .
See Proposition 1 of Méléard & Villemonais {{cite:4a6978b8d5d768f572957fab14c2c37b8ab04e88}}.
Moreover, given the existence of a quasi-stationary measure {{formula:59e457d3-5860-4347-8bdf-a1d855278385}} of {{formula:6878eeca-7a17-410f-a974-22ddfa6dfc0e}} in {{formula:a7a9ad11-2023-4253-8f00-eacb9c3de8c5}} , it is known that the exit time {{formula:bed799ed-e49d-4b24-9af7-7b6a0a321f5e}} is exponentially distributed with respect to {{formula:6bb51d6a-b298-4686-b1d9-0f2e9678ea83}} as an initial condition.
Specifically, there exists {{formula:7f6fa267-bb22-45e3-9f96-b12c3c0c8ed0}} such that
{{formula:bb426e32-1466-4a58-816a-969744ef4362}}
| r | 4fd831d7f8f1df61dd9b3ec217042e6f |
With the extraordinary NICER dataset of MAXI J1820+070 in the bright hard-to-soft transition, we are able to find that the reverberation lag frequency decreases in the transition, with an increasing amplitude, which indicates a larger emitting region that causes reverberation, likely due to an expanding corona. With the reverberation model reltransDCp, we are able to quantitatively measure the coronal height, by fitting both the lag-energy spectrum and the time-averaged energy spectrum. In the time lag fits, the X-ray coronal height increases from several tens to several hundred {{formula:c70ba974-4c72-4847-b208-f0ba46cf72c7}} several days before the radio flare, consistent with the qualitative trend seen in reverberation lags. An expanding or ejected corona during a state transition may explain earlier RXTE observations of the high-mass X-ray binary Cyg X-1, where the low-frequency hard lags were seen to increase during the transition {{cite:42f55545b1630ed1cb89acd73ae6b00487a42411}}, {{cite:a5acd860d2b435bd6d5b478ce86e51a98a59a729}}. We want to stress that to our knowledge, this is the first BHB in which the increase of reverberation/soft lags is clearly observed.
| d | 80661319903d71cdf2eb638c77459a33 |
Once an orbit was classified as a regular one, then we performed an additional categorization, thus classifying regular initial conditions into regular families. Our analysis indicates that several types of regular orbits appear in our disk galaxy model, while the most important ones are: (i) box orbits; (ii) 1:1 orbits; (iii) 2:1 orbits; (iv) 2:3 orbits; (v) 4:3 orbits; and (vi) higher resonant orbits. Specifically, in all studied cases, the relative percentage of higher resonant orbits is always less than 0.1% and therefore we may argue that these orbits do not have any significant contribution to the overall orbital dynamics of the galaxy. The notation of orbits (box orbits and {{formula:f4404acf-79f8-48e1-90fa-608e5ee24a83}} resonant orbits) in the axially symmetric galaxy model is according to the initial works of {{cite:2641515606abe9d4450e489254628c40d43a75e1}} and {{cite:ccf3a550cf70b09e7582eb72317379bf55fa9aaa}}. In panels (a)-(e) of Fig. REF we provide the shapes of the main regular orbits of the system, while in panel (f) of the same figure we give an example of a typical chaotic orbit when {{formula:4766f76e-abce-40d3-9102-d397efcb783c}} . In all cases, the ZVC is the black thick curve circumscribing the orbits. For all regular types of trajectories, we tried to present regular orbits with starting conditions very close to the respective parent periodic orbits, to be able to clearly recognize their shapes.
| m | 048847565155b0a3c4cb176ca3c7ee14 |
We also observe bottleneck behaviors {{cite:a8ebdc24d0f7c967c3a0e54277e0ca56abe3668d}}, {{cite:5af61140400cdeaaf446df938ba8a880bc255221}} that occur when fixing one scaling dimension while increasing others. For instance, OpenCLIP ViT-B/32 and ViT-B/16 are bottlenecked by the number of samples seen at the 13B scale. Increasing the number of samples seen to 34B reveals that LAION-2B brings clear improvement over LAION-400M, which would remain hidden when fixing the number of samples seen scale to a lower value. Similar observations may occur along other scaling dimensions. OpenCLIP ViT L/14 shows an example of data scale bottleneck on LAION-400M scale, as increasing the number of samples seen from 13B to 34B does not lead to improvements. The benefit of using a larger number of samples seen is then revealed when going to the larger LAION-2B dataset.
| d | 43ebbdcbd8d27df34bf549c5aab6bc7f |
Language modelling is a foundational natural language task. In recent times, there has been a surge of research interest in neural language modelling (NLM) which has led to many improvements. This has included a smorgasbord of novel architecture designs {{cite:bac752a10908f731c97597c990bc2a568164ca47}}, {{cite:a32f395f63c3d87d37a78ad5fc3cdcbbdf7eaafe}}, regularization and optimization improvements {{cite:ce83ddf0d81477f7af7b28b03cd5553218b6348d}}, {{cite:f44655f65d590ded637bff1a90c8905afe64ad49}} including an evaluation of (1) activation and weight dropping in embeddings, hidden gates in RNNs and output projection layers, (2) the effects of weight tying between input and output layers to reduce parameters in NLMs while improving performance {{cite:c978f69a417623d6db1c48e61b6b80b89b5b8bfa}}, (3) the effect of varying hidden dimensions of RNNs and embeddings dimensions, (4) optimization techniques such as scheduled cosine annealing of the learning rate with warm restarts {{cite:c520b155903c2a754dcd5f799ba7fbea1008efe8}}, cyclical learning rates where the learning is increased and decreased periodically throughout training {{cite:43f8c03a215d789e328e0b08aac1edcdb5b3bd88}} and separate learning rates for each layer {{cite:4f9ba13b526f18173de87ba6a2d8f0bdd52bbcbc}} and (5) methods for large scale hyperparameter search for optimal performance {{cite:8e404a80e2bc9b7df588e92db60b099dcd828bf1}}. Tangentially, there has been methods for addressing salient problems in structured prediction, such as exposure bias {{cite:3a4613a28114e4777df038d815a82ee61b3a69f1}}, {{cite:287c0424882392ff0274718a02e328b49dbf174d}}, {{cite:6a4c71c898750628e82254be31a658b6c4e40994}}, {{cite:82580fbcdd0a850f01f3ff3b4fb478e8fe92e6f1}} which leads to the compounding of errors is an important concern in language models.
| i | b26c5fb465eab40c4732eabc4ac0320a |
For a given baseline language model ({{formula:9ebe52df-61c1-48e1-ba5c-be90a195017a}} GPT2-Large or DeBERTa-v3-Large), we fine-tune separate copies on the respective SuperGLUE task. We choose a hyperparameter search space based on generally recommended search spaces such as the one given by {{cite:12d422f67874831b4c01ea548a7809a95acb832b}} and then choose the hyperparameter set that performs best on the validation set based on an accepted metric for that dataset. For RTE and BoolQ, this is accuracy and for CB, this is Macro-F1.
| m | edef6921ec77958c1dda4dfffc0a9581 |
In table REF , we report the results for the task of False Friends' detection. We observe that the approach proposed by castro2018high does not perform as well as it does for Spanish and Portuguese, as reported previously. We believe that this approach inherently lacks the linguistic intuition which is needed for the false friends' detection task. Please recall that False friends are word pairs that spell the same but do not mean the same. But for the approach to perform well, monolingual embeddings may not be an appropriate feature. Cross-lingual word embeddings project monolingual word embeddings into a common space and thus should be able to decipher the `meaning' or the `sense' of two different words better, when they belong to different languages. Given the recent advancements in word representation models, cross-lingual word embedding based models should be employed for such a task. Please also note that we do not propose a new approach for the task of False friends' detection and hence do not perform any experimentation with cross-lingual word embeddings. However, merlo2019cross show that cross-lingual word embeddings obtained using the VecMap {{cite:e8c477b64ce15de58485b53e817c6bed0c213d00}} approach have shown promise and can be used to obtain a semantic comparison between two words from different languages.
| r | 9293434374d9277f8865ce952b97a765 |
Competitive games have long been a focal point for AI research {{cite:ef9635c5f82465b86d466c16557a0d2d74cac555}}, {{cite:7e7765fd69e425fb56139bff6c4318a32df35749}}, {{cite:f91ed3686b2bce5db906a6275e99c3b608ca5586}}, {{cite:be19fddb740bd39d202750c127c59fca8be0111d}}. We follow recent calls to move AI research beyond competition and toward cooperation {{cite:c4582ac516a62738356d46d8ec4cc56cbc0d867a}}. Most interaction research on deep reinforcement learning focuses on pure common-interest games such as Overcooked {{cite:70dd1ded09844adf3afd00b2c453b8405155924d}}, {{cite:a746ed478640658724ad265625847f24a9c1d6aa}} and Hanabi {{cite:524b90b6e98cfb96081e4203fc8d9001b22174e0}}, where coordination remains the predominant challenge. Expanding into mixed-motive games like Coins opens up new challenges related to motive alignment and exploitability. For example, participants who played with (and exploited) altruistic agents expressed guilt and contrition. This echos findings that—in human-human interactions—exploiting high-warmth individuals prompts self-reproach {{cite:f7f6b189503ee487fc1b16e2c731784cd7011df6}}. At the same time, it conflicts with recent work arguing that humans are “keen to exploit benevolent AI” {{cite:4b20f526f646ed391d8f3f6312a8d98ff7d6a274}}. Future research should continue to explore these issues.
| d | f3b622b1ac6493d0577c637fdb2af23e |
In the second inequality,
the first term {{formula:3163ddac-4f29-476c-b153-c7f4aca059d3}}
follows from that {{formula:5842f8da-7070-444b-b5d6-8c4c8085a1f3}} is a continuously-distributed random variable with a bounded density due to the
Picard iteration used to establish existence and uniqueness {{cite:a7af337ceecf82da72c5f1faef2db87e4cca1295}}
under the global Lipschitz continuous assumption.
The second term follows from the Markov inequality.
Therefore we have
{{formula:17b6a91d-3ef5-42ee-8142-b7d154068b7c}}
| m | 01f7240b2c3be2b840ff95922f385d75 |
Motivated by the excellent performance of deep learning in image processing, people begin to study learning-based point cloud registration methods. PointNetLK {{cite:61be1abe9a2c68217e83108c8e11663af6edb26f}} utilizes PointNet {{cite:c5a062ac33ee3903cb7cb07e9bca802b649dcf1f}} to extract global features for point clouds and then uses the modified Lucas-Kanade {{cite:a1a5603eb063bc253441f84a393f289174d5ed34}} algorithm to optimizes the transformation iteratively. PCRNet {{cite:a07490211680236430d6bcb611ecb72cc08733f6}} compares the features extracted from PointNet to find the transformation between point clouds. DCP {{cite:93f3f75a4ef637c647293b132bda2074f97ea500}} improves the initial features extracted from DGCNN {{cite:2c770d7903291be57758e8b1ea0213b40caee90c}} by incorporating the local and global information, enhancing the ability to approximate combinatorial matching. PRNet {{cite:7c6c5eb71fe12d64929e8c55787d836685a4ac4b}} detects keypoints by comparing the L2 norms of features and uses Gumbel-Softmax {{cite:d0cba2ece9bda4de4a7eefebb8a31526b2c004f1}} with a straight-through gradient estimator to sample keypoints correspondence, which tackles a general partial-to-partial registration problem. IDAM {{cite:102241f0794e922c2ff942c8619f0059fbfd3ffc}} utilizes hybrid features to search correspondences, which incorporates the shape features information and the spatial position information. Furthermore, a mutual-supervision loss is proposed to train the two-stage point elimination without extra annotations. Our work is similar to IDAM, but we independently use shape features and spatial coordinates to guide the correspondences search and fuse them finally. Besides, we calculate the credibility of matching points based on the matching results. RPM-Net {{cite:b7a9ae7bf0c99a2035e4d7ac9f7083421e01187c}} uses the differentiable Sinkhorn {{cite:8281dbe363078f72fab084b5e3662c3507b5cd21}} layer and annealing to get soft assignment matrix from hybrid features, which solves the problem of big difference in dimensions between spatial coordinates and local geometry to some extent, but it requires additional normal information. CorsNet {{cite:e9acd98d60bcbdeb138db95bde427bce666d6554}} concatenates the local features with the global features and regresses correspondences between point clouds, followed by SVD to estimate the rigid transformation.
| m | ba0506071712a4fb716c98241e014775 |
If the pairing strength is not infinitesimal the point or line nodes can be inflated into BFSs. Like for local pairing {{cite:b9cdf2604f1a6046c6b80e9027a969986371fb24}}, {{cite:c5d7789639c12491ac4f31771864f4a0b33503aa}}, the BFSs are given by the zeros of the Pfaffian {{formula:79ddecde-64c4-48d0-8fea-116b1583d08b}} of the BdG Hamiltonian, unitarily transformed into antisymmetric form. There are three possible cases for the inflated nodes: (1) they are forced to contain the original node and thus remain attached to the normal-state Fermi surface, (2) they do not remain attached to the original node and thus generically shift away from the normal-state Fermi surface with increasing pairing strength, or (3) inflated line nodes remain attached to the original line node only in high-symmetry directions (we have only observed the situation that the inflation also vanishes there). For increasing pairing, the BFSs typically grow and eventually merge. In case (2), the merged pockets can eventually shrink and finally disappear if this does not violate any remaining nonzero topological invariants they possess {{cite:63833a0fb40868ad617c234b1858f94dc7623f52}}, {{cite:c5d7789639c12491ac4f31771864f4a0b33503aa}}. In real materials, this only seems likely if the BFSs are located inside small normal-state Fermi pocket(s) of a poor metal. It is an intriguing open question whether the resulting fully gapped superconducting state is topologically nontrivial.
| d | 875ee2ae20ca8aea6c6130cd865a4b72 |
where {{formula:5044b603-8d0d-41ca-b95e-e77f9e0b90d7}} is the number of finite samples and {{formula:d7820993-9c97-4bbe-aa4b-6fc63c3482c6}} represents the digamma function. The parameter {{formula:0da19178-33f6-4b04-9835-0681af3167d1}} represents the dimension of {{formula:ea7b5d04-37da-4ff2-929f-ad7f4b8b140f}} and {{formula:78ce3014-ccfd-48d4-8dcb-97d2256cb417}} is an expression that depends on the type of norm used to calculate the distances, which represents the volume of a {{formula:77f48759-9701-4952-a598-6ac1af345bd0}} -dimensional unit-ball (for {{formula:82c35563-0226-4bce-b4c8-b7eea9cc79c9}} -norm, {{formula:3a7969dd-003a-4479-97e9-6c06f3219a6a}} ). Finally, {{formula:c3d2a11f-6619-45c8-8955-a267fde36f7e}} is the distance of the {{formula:783aad17-8db0-4e8a-a943-a256eebaca5d}} sample to its {{formula:480ebb1e-a2ae-4d79-83d4-9401399b46f0}} neighbour. The reader is referred to {{cite:eb8c14dab04c877f2a7abd9a60dca2ca31e69512}} and {{cite:24b3482f2de434f05de3d2f111ccc17b496b8e57}} for a more detailed discussion of the previous entropy estimator.
| m | 1bb76b172eb24f4406518a6ff409acfc |
We evaluated the proposed methods on image classification tasks with 10 classes, using
MNIST {{cite:c356cdfb6a440048a9ff4321a58a7d771b12530d}} and CIFAR-10 {{cite:8406811008b5f4fa3afcbf5866469b5a71e243c5}} dataset. Our baseline models are Convolutional Neural Networks with ReLU activation functions. We achieved {{formula:365f6ab4-e413-4fe5-bf92-cc047ba1f975}} accuracy on MNIST and {{formula:b1e3e28c-a4b1-4e1e-b2e4-6cf7af136aaf}} on CIFAR-10.
The adversarial attacks in our tests are Fast Gradient Sign Method (FGSM) {{cite:87e2b439cc4bb8bb3b218787652b7ff5cc14fe57}}, Projected Gradient Descent (PGD) {{cite:2badc484a33d8566cb5aafcbbb976e8ec6bfdc40}}, DeepFool {{cite:91b68dc0f4ab09bf407d94c8781b4ac038eb0da2}}, and Carlini and Wagner (C&W) in the {{formula:89604a02-8a46-4b2c-87ed-21e490602519}} norm {{cite:3fc93dc0243cef5e13e8a8c0e16480a3dd3f90ec}}. The attacks just mentioned are described in Section REF of the Appendix.
In all such cases, the maximum distance between an image and its adversarial perturbation is set to {{formula:2677de81-d86c-4396-8f19-da661050f394}} . These methods fall in the white-box category, i.e. they have complete knowledge of their target network. However, due to the transferability property of the attacks, they could also be effective on unknown models.
| r | 67b80060dd974cdd65b1532cca119e58 |
PU learning refers to the classification of unlabeled samples given data from one class during training. A recent survey of PU learning methods can be found in {{cite:aeae008137c95c69560944e3b93e23f3574833b6}}. The class that is usually available in this setup is the novel/ positive class, but PU learning can also be applied to novelty detection in an unlabeled dataset with access to only the negative data during training, and some studies {{cite:40fe283d3c81c76f6ba0e820d3d912a721d9081d}} consider transductive novelty detection as defined in {{cite:37745d239fec82baee5274b3a9e83f0317657d3c}} to be a special case of PU learning.
| m | 1059ab2384df7504164c8dff58432e54 |
An alternate approach to weak supervision for segmentation is to utilize image-level
class labels in conjunction with class attention maps {{cite:4cffc18ef87f1c7d1fb4f573502fc4e5c3204bc9}}, {{cite:267acc97c62c91886adc38ed252237c6ac6a461d}}.
Besides the low labeling cost, the benefit of this approach is that the class
boundary for the object of interest is informed by contextual information in the rest of the image, in contrast
to pixel-level supervision, where the model is encouraged to learn shape and
color cues in the object of interest.
P2P incorporates image-level information by learning characteristics of an object's context via
a second contextual discriminator, {{formula:d8e55727-b528-4140-9cee-4829230f353b}} , as described in Section REF .
| m | 46b4bf020edc675eea29ab779eb3d130 |
We have shown that by {{formula:92adc4ba-e0e7-4113-9cc5-47d5e0809116}} uniaxial pressure tuning Sr2RuO4 to a Lifshitz transition and associated Van Hove singularity, the orbitally-limited {{formula:33cf122f-76b3-4b62-ba4d-43baec1f6bda}} exhibits a sharp rise over an already strong enhancement, pointing to a large gap coinciding with a small Fermi velocity at {{formula:73316faf-bd76-4163-9c2b-06aba426a8e0}} .
In a similar range of strains as the strong enhancement, {{formula:c9c82b45-b967-4a9c-9637-f94e44d7b1b2}} also changes its form from convex to concave.
These sudden changes indicate that the out-of-plane upper critical field is highly sensitive to something occurring around the Van Hove strain.
On the other hand, the in-plane upper critical field is Pauli limited and exhibits a linear dependence in {{formula:1e2031e3-8280-4e18-8b55-2a6c3915015e}} .
In the simplest single-band situation ({{formula:2ab885d5-1354-47f1-98ff-7561db0dc24a}} ), the Pauli limited field is given by {{formula:2900abbd-1e85-4e0e-b5a0-effbb74fae9c}} , where {{formula:e6724674-a31d-41b7-bb38-e9799ccef1a3}} is the renormalization due to the Stoner factor {{cite:c51f2ae5b427206fffee863b581f954874604cca}}.
For materials with {{formula:6c679647-8418-4747-a1e4-97f9ad7b9014}} K and an isotropic gap, a Pauli-limited critical field of {{formula:1867fd1a-0a4a-4d0c-b4a5-7e673755d7b9}} T is expected.
This is about twice the value observed for unstressed Sr2RuO4, suggesting that the Stoner factor is substantial.
On the approach to the Lifshitz transition, Knight shift measurements {{cite:793d3db214f60100e899a7a6d87de4a47d5fb712}}, {{cite:8c3d1ad06b42feac57d7a70af5467f009f6eda1f}} and DFT calculations found a continuously and strongly increasing density of states {{cite:dc514fd264c31e89f2dabf7b348ef81d575b20b9}}, which would naturally result in a gradually increasing Stoner factor.
Indeed, at the Van Hove strain the Stoner factor is enhanced by {{formula:431bc542-c9d1-4694-bdc6-d6798d0b1edb}} over the zero strain value {{cite:793d3db214f60100e899a7a6d87de4a47d5fb712}}.
As a consequence the Pauli limited field should be gradually suppressed, resulting in a sub-linear dependence of the in-plane upper critical field on strain.
Since this sub-linear behavior is not observed, the superconducting gap must increase faster than linearly in {{formula:bd94737f-3f23-437c-be5c-1f810f1a92ad}} to compensate the suppression due to the Stoner factor, resulting in an overall quasi-linear dependence of the Pauli limited field in {{formula:7eec8f80-6445-49ab-b145-37dca12459f0}} .
The strong enhancement of both the density of states and the gap magnitude close to the Van Hove singularity could also explain the strong departure between the experimental values and the weak-coupling calculation of {{formula:5d8eaf71-b4c3-416c-8559-d9dd9f6cc5f8}} close to the Van Hove strain, shown in Figure REF (c).
| d | de25018b455198dd5b82f3aa781e3c92 |
About SDR, despite its relevance in this work, also because it is a usual objective metric, which tends to reduce human-related failures, it may be a less robust metric for some scenarios of speech enhancement or source separation, mainly for monoaural signals, which are of the type discussed in this paper. In order to address the problems associated with this metric, work {{cite:80a66ed34c23c411d60e3f2a13e5814483c0d634}} proposed an alternative metric called SI-SDR. Thus, in a future continuation of this work, this new metric proposal can be explored.
| d | 7e0182b51c842e83521d9b7f82a2d8e5 |
where efficient projections onto {{formula:daf9a6ed-e0ab-44a6-aa27-20ef4fe5adf8}} and {{formula:5e4a7924-8d57-4212-a09a-82ba48d88a63}} are available. One early algorithm for solving (REF ) is the Arrow-Hurwicz method {{cite:b42e7ed9311f355dbc8149b4f23d4156265d48b7}}, which is also known as the primal-dual hybrid gradient (PDHG) method {{cite:bef370991001be5d21506638b048fb589047042a}}. The convergence of PDHG has been established when one of {{formula:281447b9-8b41-425e-8ddb-fdbb069daed1}} and {{formula:23287db4-00f0-4836-91f6-85939c93ae26}} is strongly convex, while a non-convergent example exists if strong convexity is not available {{cite:f639e187a649700078449b9b283d0e1add620389}}. The Chambolle-Pock method (CP) {{cite:30cf0afb2359768457fc82a7aa57c3a1e2f136c1}} is another famous primal-dual algorithm. In terms of the duality gap, the CP method preserves an {{formula:dd123a6f-ded5-46ba-9b1c-e5ae079162f8}} ergodic convergence rate, and is shown to be optimal {{cite:9f7823c70ea68207fe01a257bc426c507adf7111}}.
It is worth mentioning that both PDHG and CP focus on the case where {{formula:562c208d-10b2-4186-9a67-b7cbacb5d3d8}} is bilinear, that is, {{formula:bf912933-8929-4636-b677-01dc0514edfc}} .
Recently, the optimistic gradient descent ascent (OGDA) method has gained popularity due to its superior empirical performance in GANs training {{cite:3c4d1e195652040cacca3ae1ef04ea12e3ecc769}}.
For bilinear objective function, the convergence of OGDA was studied in {{cite:3c4d1e195652040cacca3ae1ef04ea12e3ecc769}}, {{cite:dd6a4e9c7cd6075b4f8e62e41d31a16c2340ee6a}}. By viewing OGDA as an approximate proximal point method, the {{formula:3645cd75-9b83-450a-ae2f-b8aa0414a85b}} convergence was established in {{cite:68de5a95a299cd504e54fc8e1e46fd87aed432d2}} for smooth convex-concave problem, as well as linear convergence for smooth strongly convex strongly concave problem. Further attempt was made in {{cite:c4c032e7636ca3ed9c1794da1c73703d2e7fbcf2}} to prove the last-iterate linear convergence under metric subregularity. To our best knowledge, the OGDA methods has yet been studied for general non-smooth convex-concave problem.
| i | 0d13c5b726b5b232c06179e7b0f534e9 |
Moment-based estimators of a given population are obtained by equating the population moments to their
sample counterparts and solving the resulting equations. The moment-based estimators for the Weibull distribution suffers from numerical computations, see {{cite:29616fd967094687334d351795f212700a2c7eef}}. Also, these estimators are not efficient. The {{formula:e9acd5b7-8b51-460c-bdcb-150350d6b75b}} -th non-central moment for the Weibull distribution is ({{cite:f33b3b679d3f2a336f5700261e6ac33b0be34c6c}}; {{cite:9fbb6c79bdd519d142e5f27430ea5b09698e1cce}}; {{cite:a6741573b0d1056dc46907180191fc90c3203b7a}}):
{{formula:a06330bc-7547-442c-8ffa-42a57d55a9d4}}
| m | 8b007a93d8801c93994bb15700ecaac2 |
We show qualitative results of point cloud segmentation and compare the segmentation quality. For S3DIS dataset, we visualize selected segmentation samples in Fig. REF . From left to right, the RGB view, ground-truth, fully supervised segmentation, WeakSup{{cite:4738bf6871961f19e9535e767279d64039b15f95}} segmentation and our MulPro result are visualized. In these visualization results, both MulPro and WeakSup leverages 1pt labelled points in the training stage.
We observe that our results better respect the ground-truth for classes with large intra-class variation. For example, a “clutter” category (in black color) exists in S3DIS which covers multiple types of objects that do not fall into the other 12 predefined classes. Because of the multi-prototype classifier, our model is able to identify subclasses within this “clutter” category. This is reflected by the more consistent predictions for “clutter” class. In contrast, WeakSup makes more erroneous predictions on the “clutter” class.
For ShapeNet, we show the segmentation results in Fig. REF .
These examples again demonstrate competitive performance by our model when facing categories with large intra-class variation, such as the examples of the car, lamp and plane.
{{figure:a9557adf-b736-40d2-8844-0e56d4e39c76}} | r | 642458423c020235548d9f4212f51cbe |
In this work, the problem of orbit flips caused by the eccentric von Zeipel–Lidov–Kozai effect is systematically investigated under the test particle limit at the octupole-order approximation. By integrating the secular equations of motion, the flipping regions in the whole {{formula:4dd1d3c1-bddc-42db-9a45-9d783c195532}} space are produced by taking the initial angles at {{formula:b2a2192d-b253-430b-859f-386345fbd247}} and {{formula:9981d1e4-3bfa-4612-8919-e4720615639a}} (case I) and {{formula:2554e56c-4ea4-41b0-a9ea-2562bf28a80b}} and {{formula:5ba54043-20be-40cf-9f34-0e360669e262}} (case II). The regions corresponding to case I are distributed in the low-eccentricity and high-eccentricity spaces and the regions corresponding to case II are distributed in the intermediate-eccentricity space. The results cover both the low-eccentricity high-inclination (LeHi) case and the high-eccentricity low-inclination (HeLi) case, which are discussed in {{cite:80009764aea3c7e62f2ceac832cba5aa014170f0}}. Numerical simulations show that (a) there are three distinct regions, denoted by I (low-eccentricity region), II (intermediate-eccentricity region) and III (high-eccentricity region) from the left to right, and (b) the structures of flipping areas remain qualitatively similar under dynamical models with different magnitudes of {{formula:c06063f2-e97e-4dbf-ad1d-2c25aba6a62d}} . In particular, when the eccentricity is higher than a critical value {{formula:23b37a14-ef9a-4364-a89b-4aea7deb26cb}} , orbit flips can take place from nearly-coplanar configurations (corresponding to the HeLi case where flips can occur with nearly coplanar configurations).
| d | 7591e260bab191f9cd6436288a1cd0e1 |
Usually, the computation of the GMB fluctuations has been constrained only to the balanced case i.e., with the “Zeeman magnetic field” {{formula:916d10bf-92e8-4027-a274-3d8c24b8af54}} set to zero, where {{formula:e29c3ae6-306d-4d6f-aa39-4b089880efb5}} and {{formula:c42c071f-d7bf-444f-9bae-a19e491b3b3f}} are the chemical potential of the spin-up and spin-down (non-interacting) fermions, respectively. Spin-imbalanced Fermi gases with a finite {{formula:a7ddafb7-ce65-42fb-b137-ed43c8910a83}} exhibit rich scenarios {{cite:169af6f48c29f11312c1c498774046076217138d}}. Starting with a superfluid (BCS) phase at zero temperature with {{formula:23841cbb-86bc-4fd9-85ce-b12173288e58}} , adding one of the spin species (spin-down atoms for instance) will increase {{formula:46ab7cbb-dd8d-488f-a9b7-330b3aee6b6f}} and for {{formula:0cd5721a-a9ac-474b-ab1c-fb1c7939d8d0}} there will be a first-order quantum phase transition between a pure BCS phase and a system with phase separated superfluid and normal states {{cite:c2ad2300047453ea1e86e2e703ec66631d0d9240}}, {{cite:0e365b33466d56f66923049d9f4e872296af6aca}}. Experiments in 3D have observed phase separation between the superfluid and normal phases in the trapped gas {{cite:7219685b1f089cb1ce813a22a5c51e640f05f9e2}}, {{cite:8f459ad06a8099eec3475ebf3a8d978dbb197d7d}}, {{cite:90697838d2edae92ba997b31e77107e6f1e2bacf}}.
| i | 5f5d16666f9ef8a4c373209bb4ce086c |
subject to the uniform initial condition {{formula:5694e8ee-a949-4683-8a36-9725e56aeb3f}} and mixed
Robin-Neumann boundary conditions {{cite:5f61d9ce7134199e62b28137d0ef442fa3e93e8f}}:
{{formula:4ac56e37-a10c-4c15-ba6c-29fb29b604cd}}
| r | 182cfbded9bf2980afd537f9d81e8950 |
Finally, we apply the Non-Maximum Suppression algorithm {{cite:23945ee1d3c3bfaa2594b1071ea029b32d0d29ee}} to {{formula:71f8bf9b-9b8e-4f4d-9459-bc15f471a442}} , which takes in a set of boxes and confidence scores and returns the final set of detections for the image. We calculate the confidence score {{formula:443b8ef9-b572-4d61-a0cb-5731e43adfba}} for each box {{formula:29f22ae2-8553-4e1a-b0cc-e6ac73ee58a2}} as {{formula:ff02cada-9da1-4183-bb15-d546668168ba}} .
| m | 7b9458af5dce4a5f2e4a3cff2e90474f |
Algebraic hyperstructure theory is a well generalization of classical algebraic theory and it was introduced by Marty in 1934 {{cite:a6e9eeb5c67ed590e19df8e5d6513b864d7755aa}} at the 8th Congress of Scandinavian Mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Since then, several mathematicians studied Algebraic hyperstructures, see {{cite:8437e4302c70d9cbce6f96dd3fc1b1e5b60809e1}}, {{cite:b9dd90c10a66af8ac47b9dd323a0128350dd8afd}}, {{cite:c4761d6946ef28527c3b57c460f0edf8287dda49}}, {{cite:f0d94385fb3272433e90554d72f5d881931935d9}}, {{cite:ce90cc6129fcb0edc3b4a8468468eba3a7812fd9}}, {{cite:59e4593a4b7e378db49de0aa0e3c69d1c02b7ec2}}. Marty introduced the concept of hyperoperations on hypergroups which are generalization of groups, and this led to many concepts of hyperrings. Hyperrings are hyperstructures together with two binary (hyper)operations, addition and multiplication, in which addition or multiplication is a hyperoperation. For examples, {{formula:79f65186-2b25-4478-9146-211f1988486a}} is a general hyperring when the addition and the multiplication are two hyperoperations such that {{formula:1b09b6e3-1089-4fd7-9a1c-4ad05f4a3491}} is a hypergroup and {{formula:694f14ef-8b58-4e5a-a5c8-65db01bac9e2}} is a semihypergroup and the multiplication is distributive over the addition, we have also multiplicative hyperrings. Another type of hyperring is the additive hyperring, if only the addition is a hyperoperation and the multiplication is a usual operation. One of the additive hyperrings is the Krasner hyperring which was introduced by Krasner in 1983 {{cite:b3fbc8aeb988156c98d83ee9a4567c1f230750f7}}. We refer the reader to {{cite:ff34e13b110c1a0890c8552efb6342abd327c065}}, {{cite:c4761d6946ef28527c3b57c460f0edf8287dda49}}, {{cite:835528988262e251008792f029e2aec42cbc351f}}, {{cite:8c261c4657082e67a359ca313288f2397dcc18c3}} for the notions of hyperrings. In this paper, we will take hyperring to mean Krasner hyperring unless otherwise specified.
| i | 55aa43c86be13c5e36ec2d345b588cc6 |
We note that the uniqueness assumption is often required in the analysis of OMWU even for normal-form games {{cite:ef8bbac6ccb21185362c323c1f1cd40abc6419e9}}, {{cite:5cf221b15b50452877056ef69aa07966083887c8}} (although {{cite:5cf221b15b50452877056ef69aa07966083887c8}} provides empirical evidence to show that this may be an artifact of the analysis).
Also note that for normal-form games, {{cite:5cf221b15b50452877056ef69aa07966083887c8}} show a linear convergence rate, whereas here we only show a slower sub-linear rate, due to additional complications introduced by treeplexes (see more discussions in the next section). Whether this can be improved is left as a future direction.
| r | 177926f66d4823b75b1aef328f527855 |
CIFAR-10.
On CIFAR-10, we compare lottery jackpots with many competitors including OBD {{cite:ad6490ee0ab36bdeab6d03f2771f8ef33efecd79}}, SET {{cite:53881532a6044f9aece9bd5c8f12f2d45348f618}}, Deep-R {{cite:fa13766b78b1f8eb0c06a18f346bb27d8e74dcc3}}, Lottery Tickets (LT) {{cite:f29dba5bf1a92eb549817c63ff60beb3bc984d34}}, SNIP {{cite:fbb75035f5567049bc6a77bc1d724bd84ed5458e}}, and GraSP {{cite:4123257717e15a4bef93458bc6fa305c0f4e3e89}}.
We report top-1 classification accuracy for pruning VGGNet-19 and ResNet-32 under two sparsity levels {{formula:80576fe1-0b1f-4ee1-a59b-1aefb345d9f5}} .
The weight tuning or search cost, which refers to epochs expenditure for obtaining the final pruned model, is also listed for comparison of the pruning efficiency.
Quantitative results are shown in Tabs. REF .
| m | e3a6e9139c9d719f36663a3fc7955b64 |
In contrast, projection of the 5-d surface on ({{formula:64e25fe5-52b5-4d85-989c-796d9777e015}} )-plane (Fig REF ) has the two basins along the {{formula:428ebba5-47f0-48cf-afa0-c7c35501584f}} -direction alone, but the transition region aligned along both {{formula:48c2016e-3d0b-48c9-98b0-1eaba9f61ca6}} and {{formula:b46b88c7-2132-4a56-b421-5c1043869855}} . This second feature is consistent with the importance of {{formula:a354f572-5b6e-432f-b74f-4f76ced869a0}} in determining the barrier crossing
dynamics {{cite:e22c9d3eb132516aa4ac5d28922dcc224f0dcc8e}}, {{cite:6bee35e9496693b58f627effbdd2c2035ca56cad}}, {{cite:8190f625f7d863d51e6a3821a2699229d78d539f}}, {{cite:89283035c315ce9bd9bab9e8d406f6e6214fa791}}, {{cite:877e9d4d88cd1266fedb7ac739ac485d43c3d301}}, {{cite:58cbe13defc7e4f2c06504b1e08f8f04d8ab33d0}}, {{cite:48594ca6edb5caefe4ca6677feb17c5d5eaad31e}}.
| d | 7574dcfb75c14e341bf0d7312bd53a39 |
As described in the introduction, in this section we recapitulate the Hilbert space hypocoercivity method from {{cite:429fd0e6155705f05add20a36d2c50e8f1b536aa}}. It will be applied later on to establish hypocoercivity of the Langevin dynamics. The method in {{cite:429fd0e6155705f05add20a36d2c50e8f1b536aa}} is a rigorous extension of the original hypocoercivity method from {{cite:f46b442c07e635be9520bd8e9f54ec86306b00c9}} in which domain issues are not yet included. Moreover, the formulation of the method in {{cite:429fd0e6155705f05add20a36d2c50e8f1b536aa}} is made for studying Kolmogorov (backward) evolution equations. Below, {{formula:cb33c379-b31b-4320-8bd4-4ef7e1c65420}} always denotes a real Hilbert space with scalar product {{formula:4fd2d5ed-490a-42ff-ba22-fef905d90c56}} and induced norm {{formula:80689950-f531-44d2-8994-08f854c4cdb1}} . All considered operators are assumed to be linear, defined on linear subspaces of {{formula:ce607c6a-2ab0-4d35-9fe5-7591714e5a53}} . An operator {{formula:71b5f55a-e6cf-4ac0-810f-0c536008c60f}} with domain {{formula:06125cb4-5d57-443f-a861-94a3ddc22bc6}} is also abbreviated by {{formula:19e01b1d-dcef-4d97-b166-bb9f1a2afb24}} . Basic knowledge from the theory of operator semigroups is assumed, see e.g.{{cite:1b40913b26afc607b5cefa4fd7991c7c23f0230f}} and {{cite:bed3d8266b796c3e82ee8cffa803da75b27ce15f}} for references. The upcoming data conditions (D) are assumed until the end of this section without mentioning this explicitly again.
| m | 6e68c85537abb2931314fafba199bd1c |
The OFU problem (REF ) and the pessimistic optimistic problem (REF ) may appear much more challenging than the CMDP problem (REF ) because they involve a minimization over all models in {{formula:e3ef3673-eda0-4db9-a6e7-badd4a176dcc}} , which is non-trivial. However, finding the optimistic model (and the corresponding optimistic policy) from a given confidence set is a standard step in OFU style algorithms for exploration in RL {{cite:4e5e9bee194e6ddc4f91775a86ac7bb345b2b338}}, {{cite:1396e5b05dcd19b6cf47b6ff9b9ee328173b293e}}. In the case of standard (unconstrained) MDP, this problem is solved using a approach called extended value iteration {{cite:4e5e9bee194e6ddc4f91775a86ac7bb345b2b338}}. In the case of constrained MDP, (REF ) (and similarly (REF ) ) can be solved by an approach called extended linear programming. The details are given in {{cite:1396e5b05dcd19b6cf47b6ff9b9ee328173b293e}}. We give a brief description below for completeness. Note that the description below mainly focus on solving (REF ). Solving (REF ) is identical, just by replacing the constraint cost function {{formula:ad5e7a7f-d166-4e56-8f32-0ed45703ab6e}} with pessimist constraint cost function {{formula:6e37d4de-d412-4ec2-af57-660c1b14e720}} , and is mentioned at the end of this subsection.
| m | bcf51e769048be4f8e76516192fdb064 |
The key ingredient for many clustering algorithms is modularity, which is at the same time a global criterion to define communities, a quality function of community detection algorithms, and a way to measure the presence of community structure in a network. The definition of modularity for graphs was first introduced by Newman and Girvan in {{cite:22b4e49d8ce9b3cb1aa8c78b749f26d520a2b137}}. We present the definition in Subsection REF .
| i | da9315ecb238a09555993a45d39b4ad0 |
We argue that although such a pipeline demonstrates reasonable performance for SR problem, there are two main drawbacks: First of all, it is difficult to accurately estimate blur kernels of HR space directly from LR images due to the ambiguity produced by undersampling step {{cite:43892ca2bc2ebb421853ef452649d283dfd052cf}}, {{cite:90254ffc82707c3c5eba1142c1f3720878aaeda7}}. And the mismatch between the estimated kernel and the real one will cause significant performance drop and even lead to unpleasant artifacts {{cite:b0f80b1a318c77d9f78e33f1a7ecaa6e97a4cc85}}, {{cite:fe77bca8665e80d2452d589d7db075ed84e53179}}, {{cite:0e8b6e45a8816591fe3a9043de58810bac9c26ca}}, {{cite:96b9a23b2ceabecc4b273b00f6295e78a3dcd646}}.
Secondly, it is also challenging to find a suitable way to fully utilize the information of the estimated HR space kernel and LR space image. A common solution is to employ a kernel stretching strategy {{cite:b0f80b1a318c77d9f78e33f1a7ecaa6e97a4cc85}}, {{cite:fe77bca8665e80d2452d589d7db075ed84e53179}}, {{cite:c967de1a9f1ee60050ba0e5927dac7daf5c3af71}}, where the principal components of the vectorized kernel are preserved and stretched into degradation maps with the same size as the LR input. These degradation maps then can be concatenated with the input image or its features to generate a clean HR image. However, the spatial relation of the kernel is destroyed by the process of vectorizing and PCA (Principal Component Analysis), which causes insufficient usage of the kernel. The subsequent reconstruction network requires a huge effort to harmonize the inconsistent information between LR features and HR-specific kernels, limiting its performance in super-resolving images.
| i | 091fe04389bd78793afceed41d5fbbc8 |
Current fusion models suffer from the problem of misalignment and information loss {{cite:eb73601072a7dc1624e899a4a10a030f75ce772d}}, {{cite:278ebda293f92c69cec6a8ce8d2b8a677979f5c8}}, {{cite:cb01795b2ca0036595643d6a20a897f62d189cd5}}. Besides, the flat fusion operations {{cite:2f9b50d1c3570070dafd17fb3c5111eefaa65a2a}}, {{cite:504eb4d2f67700c9127bcdedbfb12641cdd09790}} also prevent the further improvement of perception task performance. We summarize them as two aspects: Misalignment and Information Loss, More Reasonable Fusion Operations.
| m | d18710a740bf437e8965d3ddeab0e7a4 |
Fluency
The simplest method for the generator to fool the discriminator is to output broken sentences whose style cannot be deciphered by the discriminator. To ensure the fluency of generator outputs, we employ a discriminator to distinguish between real and generated sentences. However, as {{cite:43e0387b4d68de5f31baa0c2f8b799fc71422526}} explain, the error signal from using a classifier to evaluate the fluency of the entire sentence is often too weak to force the generator to create fluent outputs. Instead, their approach uses a language model to provide token-level feedback at every step of the generator output. While the task in {{cite:43e0387b4d68de5f31baa0c2f8b799fc71422526}} is text style transfer, we draw inspiration from their method and feed the output of the generator at each time step into a one-layer Gated Recurrent Unit (GRU) RNN discriminator, generating a score for every word. Given generator pre-training (which is explained shortly), the initial outputs of the generator during training are fluent. As a result, the purpose of the fluency loss is not to provide a distribution that the generator has to learn but rather to guide the generator to stay fluent throughout training. As a result, we do not impose the gradient penalty on the fluency discriminator during training. Therefore, the loss function for the training of the fluency discriminator is
{{formula:e1e99668-2762-46d0-9652-c7d99227861d}}
| m | 9d62d080ccc67b8e1dfa09636943b36a |
The main purpose of the paper is to develop a joint stochastic model that characterizes the marginal behavior of electricity prices, demand, and renewable energy sources by capturing the related dependence structure.
To this end, we exploit the advantages of the copula methodology, which
has been used for economic and financial applications in a number of works (e.g., {{cite:97ae9360d83b971b979b5ec17e265f29a7fe012c}}, {{cite:ae9fb445b42d1146a7bc939a2baf43acac9fd451}} and {{cite:e2274d834d1f805341bcf5c86b7ab3c790226943}}, references therein). Specifically, an {{formula:b4299d39-7725-46b1-8699-fc3337b1d932}} dimensional copula is a distribution function supported on the unit cube {{formula:3c5b7fef-7973-4cce-8bb8-9c294eeeb5ce}} with a uniform marginal distribution. As well-known, an {{formula:838de8d6-3010-4344-aab0-893e29ac3314}} dimensional joint distribution function can be decomposed into its {{formula:2faf0bc4-d786-4599-9133-5da69f5102b5}} univariate marginal distributions and an {{formula:25a92faa-704b-4f0a-811d-280af5ea7bf7}} dimensional copula, which is unique when the marginal distributions are continuous. For more details, see also {{cite:bd718a5a3ea4d178050881d5e7ce58032a385d00}} and {{cite:820a0f3deec1e90ca12a3b430396402e236b5fb6}}.
| m | c5ee8bc15c1412bed4b789cc2733ede2 |
WETAS fully exploits the instance-level (or weak) labels for training its model and also for inferring the sequential anomaly labels (i.e., pseudo-labeling) that can be used as rough segmentation masks.
Based on dynamic time warping (DTW) alignment {{cite:00ece1d0414674ad643231db981d51d077b8e480}} between an input instance and its sequential pseudo-label, WETAS effectively finds variable-length anomalous segments.
To be specific, WETAS optimizes the model to accurately classify an input instance as its instance-level label, and simultaneously to best align the instance with its sequential pseudo-label based on the DTW.
As the training progresses, the model generates more accurate pseudo-labels, and this eventually improves the model itself by the guidance for better alignment between inputs and their pseudo-labels.
| i | 8a7fb7f92d2fd0c42e792b038fd13c30 |
Table REF shows overall results in splitGSC. `Acc' stands for FSL accuracy, and we use the threshold-free area under the receiver-operating characteristics (AUROC) as an OSR measure following previous FSOSR approaches {{cite:8affd2820887d51550f039a4f78cb27a1c6ed1c1}}, {{cite:c921669014c4ebc3c8d73a6e016f412e309b89fc}}. By introducing dummy prototypes and additional losses to various backbones, our D-ProtoNet significantly improves vanilla ProtoNet in AUROC and shows better FSL accuracies. Other baselines also improve vanilla ProtoNet, but D-ProtoNets shows clear margins. SnaTHCer is the strong baseline, and they suggest directly using distance metric instead of softmax output to detect open-set, i.e., {{formula:26da3345-0afc-4e6a-baf7-e9695232808e}} is an open-set sample if {{formula:fc69bc06-fdae-4db6-8451-ce7e28e1ecd5}} while other approaches {{cite:8affd2820887d51550f039a4f78cb27a1c6ed1c1}}, {{cite:55754bbb0ea3e6ff3ff275ba2c273c96dfd294a9}}, {{cite:a4ba433366980650d39249e315810e3589dbdc1f}} usually use {{formula:e6334d81-8c3e-41c1-86bd-1b8ef5559847}} . However, we observed that the unnormalized distance metric does not always hold for detecting open-set and shows poor AUROC with Conv4-64 in splitGSC and our training details.
{{table:9a7464ad-1d2b-4279-b52b-f93f6311037b}} | r | 2957ea2485886b9506da9687b981c181 |
In comparison to measurements, self-testing of quantum states is relatively well understood. For example, we know that any (pure) bipartite entangled state in {{formula:c03d1208-8024-46ea-8d1c-276d831af6d3}} can be self-tested from a correlation with 3 inputs and {{formula:df4bd1c4-9b6d-4bd1-bef5-296b2df570f5}} outputs {{cite:5e9cac98ba85223482e89f386daf7e7c3f33c4d7}}. For applications, it would be efficient to have small-sized correlations that robustly self-test states with large dimensions. The only family of constant-sized correlations that self-test states of arbitrarily large dimension {{cite:cf11a0c75664fe3e5567e576afb35700f03eaf66}} are constructed by building upon Slofstra's group embedding procedure {{cite:f87830b1f998ef9963f794c22d035d365ff5b6ca}} for linear binary constraint system games. Specifically {{cite:cf11a0c75664fe3e5567e576afb35700f03eaf66}} shows that for each {{formula:dc96e14e-d892-4eb3-b720-399cd58bd986}} , where {{formula:db18bb93-f124-4241-bd30-22566e99fbbc}} is an infinite subset of the primes given by
{{formula:a548d863-f661-4557-96c6-6b245e0c6b5d}}
| i | 2c1e13cdeaa359f50157f4be88844f19 |
The grains are subject to a uniform gravitational field orthogonal to the substrate line, and to short range binary interactions. Besides the visco-elastic
and coulomb friction interactions used in past works {{cite:8bd0d34622c6048588093461ee27dde38ae69fbb}}, {{cite:75b4fc1c45b5bfd5fd2baef7dd2004c4384a384e}}, {{cite:fbf2ff3c7c7c594560b0485cbeabb0e61bb77730}}, {{cite:1f76c87bcb63447866d11105f09cf689ffa3cb76}}, {{cite:96dcad6a99d388f8b4f184d85bbdd5ce5a20a099}}, {{cite:5f1fa06ff0a00fc2a9e87db69b015a993f14584e}}, {{cite:e66830faf2a22b56c6db3661e458e7df3ed0a027}}, two grains in contact are also subject to
a rolling resistance moment due to the finite contact length {{formula:3fcae669-4e05-4f77-b646-fd6666d642a0}} . The rolling resistance is introduced through a micromechanical model of the contact line
between two grains {{cite:e29079900f40ccda0f2e5649e7b23f7705ff73b9}}. This model treats the contact as an object formed by a set of springs and dashpots connecting the borders of the two grains.
As one grain rolls over the other, the springs in one side of the contact line contract while the springs in the opposite side stretch. This configuration
generates an unbalanced force distribution and a consequent moment with respect to the grain center (see Fig. REF ). This moment grows linearly as
the grain rolls, until the rolling displacement {{formula:7d8f3f61-920e-4134-94d7-5f5dfcbb646a}} reaches some threshold value at which the springs located near one end of the contact line break
up and new ones emerge at the other end. At that time, the moment saturates at some value that depends on the properties of the grain. The rolling
displacement is defined by {{formula:fae9ba42-e32f-498e-b7ef-16bdf73e87f7}} , where {{formula:0f731ff9-a801-42b3-8cde-262278f2f447}} refers to the angular velocity of grain {{formula:85a4f6f5-c2bb-4af1-8af3-1b5034167911}} , {{formula:252bc965-0493-4433-a43d-7adad5630731}} is
molecular dynamics time step, and the sum runs over time during the whole existence of contact. Based on these assumptions, the authors have derived an
analytic expression for the rolling resistance moment as a function of rolling displacement. They have also proposed a simplified version - the one used
in this study - in order to improve numerical computations:
{{formula:92b2f3a6-ea07-4acb-be47-b0bf2d5fe41e}}
| m | e38b68457bfc6496f6a8b5b1f0b55baf |
To put the above numbers into context, we also examined the margin of improvement of successive architectures published on the corresponding datasets, as listed in {{cite:aa089f81c380245404f499c630b28c127740857e}}, {{cite:e6b70b382b378af2132edb9999f4a05a9a2e5793}}. We sorted the results with respect to the test performance and then calculated the differences between successive models. The median difference was for {{formula:b33dc315-b4f5-4986-bc8d-e5a7011e8a53}} for CIFAR-100 and {{formula:d3dd99d1-72b9-43d3-8da4-e9dc1be59e7e}} for cbtcn.
| r | 87409e4e0e9377f6221b7e83662d0ff9 |
Exploiting a three-band Hamiltonian model {{cite:f530755c10cc4ccd3c31b0458fb6170e70d3d4d4}}, {{cite:89e713af1b8e64f8d5db1f0a7ad8d48baed30702}}, {{cite:e92ba3570fb929ed4fb465c3230b177bb516a5bc}}, {{cite:084cb56d0ef416ff43ce3ac1fb51d7ce1353cdc8}}, {{cite:29c1cb0028aeb0e7fe87eb74c1ff3c2197a3a23a}}, {{cite:546e41a476d1f1797b05d7da29d7cf4cb66f0376}} and the NEGF method, besides the spin polarization studies in lateral two- dimensional {{formula:9f77d473-0ba0-4fae-8731-8b38716b9f70}} TMD heterostructures, it has been shown, at the two K and K’ corners of first Brillouin zone (BZ), the Bloch states in monolayer {{formula:66708f70-6f0a-4c6d-b2e3-b9f3a8f7fab0}} (M=Mo, W and X=S, Se) mostly consist of {{formula:7ba6df08-3761-4bf3-b772-7b1d3d102da9}} orbitals for the conduction band minimum (CBM), and {{formula:d644437c-f1f9-4a14-a5fb-105b994c6f9f}} and {{formula:5a04594a-270f-4210-aae6-6602d7e15da6}} for the valence band maximum (VBM) {{cite:85d2e2210114893eecb306f6bc21f150bc5cfd8e}}, {{cite:bdd0e287802e7f6a8e3aa59094d1d7c1161c6cf4}}, {{cite:e9319a3850929a59d57c8d1a63df1810283aa382}}, {{cite:d9bec34624330c4e0b25b8b0691bd6c532995492}}, {{cite:9b534cd0a03982f603e8a193c324f1c266deba96}}, {{cite:286abdf12e1fb43433f37e35ed4d6ea9fd9e29be}}. Additional information including the hopping matrix between metal atoms in each displacement direction and on-site matrix of each metal atom is provided in appendix .
In order to calculate the spin polarization and spin current which are defined as Eq.REF and Eq.REF respectively, one should evaluate the transmission probability, T +(-), of spin up (down) electrons through the NEGF method {{cite:4becde873d5c824a7937ef2dfe0b809886fb61c8}}, {{cite:ec75b69b9f1238ca69bf78ea10fba7f2a4469966}}, {{cite:81aa4314f492315cc0d5a368ae757920ed53312e}}, {{cite:321010443db6863ffcabc665bc3388dbe8188825}}.
{{formula:38f7e438-c1ff-4953-afdf-7ae465de00ab}}
{{formula:6916ad82-5dda-466b-ba85-1dc557a9d0a1}}
{{figure:ce9835a3-5774-456d-8c7d-bbf8d26297e4}} | m | 81307c0bd6507355a3d78bd00a15ff35 |
Observations over several years revealed that GRB have a bimodal distribution in terms of prompt emission time, with an explicit separation around two seconds {{cite:8ed947a59e4be6c5d23be418b933b5e452dd36a7}}. This bimodality suggests the existence of two different types of progenitors: short Gamma-ray bursts (sGRB) and long Gamma-ray bursts. Short timelines (on the order of milliseconds) in the case of sGRB imply a notion of a progenitor based on the merging of two compact objects, such as neutron star - neutron star or black hole - neutron star {{cite:644f61d5c0666fb9113c95d042bd56e4867a2d7c}}, {{cite:076fddb4ac52cb7ca13a57e56fd7551d1a776485}}, {{cite:e0ccefcf698749470edcd7187d0b04410c66b43c}}, {{cite:38a82db514998269561c54489f28aae1b303e14b}}, {{cite:3971f71dcbbb67f67b798881dcf9933231acfe79}}. LGRB progenitors, on the other hand, are connected with the core-collapse (CC) of a massive star, resulting in an Ic-type supernova due to their unique placement in low-metallicity host galaxies with active star formation {{cite:d1c5663796c3c78ddcf37a6637d8046080a7f2bd}}, {{cite:d391e306dd5f06e045844996e60c352c54cdfb16}}, {{cite:dd2b48946ed7514321cfa1cad3bbd37b9f9477f0}}, {{cite:8e8c247f13f147e67842083e050c079b3b2d1146}}, {{cite:660194203b9a5f02fface9e6bd3b38bc2bfaabb8}}, {{cite:7572a2d29dcbf80ffcd94628c1075c56c397e307}}, {{cite:ed62c790762eed89ee6746492167ee0fe2836c1d}}. Massive star collapse can occur in one of two ways. The most common involves the creation of a hyper-accreting stellar-mass black hole {{cite:a37f4aa4eb1c7bf9146e944731decd18f4a3cbb6}}, {{cite:8be007669e087672b50c21372bde07ee196fdcf3}} from which a relativistic jet is launched by {{formula:3d286cbc-8ecb-4d29-be9f-3a8493e74148}} - annihilation processes {{cite:dd955a8c23a6143482480097ce284646eb75158e}}, {{cite:46cb0d814c03e14ec0fa925ec517ebc79c658952}} or the Blanford-Znajek (BZ) mechanism {{cite:2ce2f4052a72ffb1e2040f6877d5654e2b1b31e7}}, whereas in the second scenario, a rapidly spinning, and strongly magnetized neutron star ("magnetar") is formed with enough rotational energy to avoid gravitational collapse {{cite:076fddb4ac52cb7ca13a57e56fd7551d1a776485}}, {{cite:ed6f72965ccd97937c3b01a34e9fa625b8b01ffc}}, {{cite:1db9dded5459eff75b653852660998157e64f752}}, {{cite:2405236272e7c84ef160abab596f94568f02d80d}}, {{cite:237c8e46f35fe5ba27a5c7ab8ed7c3f94c10b3a6}}. In either case, a spinning disk of long-lived debris is left near the compact object. Because the temperature exceeds the rate of {{formula:41e0ee86-050b-475b-b2cb-00bb0860bb6a}} pair production, nuclei are photo-disintegrated, and the plasma formed at the base of the progenitor so-called fireball is predominantly made up of free pairs, gamma-ray photons, and baryons {{cite:83f5a0e5bfde0d2db4dd5a7509fea26d08d21d76}}.
| i | c8c5eb1bd2d61bf3f5994c1546071aef |
Once the estimation theory is established, we can develop the inferential theory. First, we provide the simultaneous confidence regions (SCR) based on our sieve estimators (c.f. (REF )). Then we apply the SCR to conduct structural testings on the smooth functions. For example, we can test whether the functions are time-invariant (c.f. Example REF ) or have multiplicative separability structure (c.f. Example REF ). Second, we propose a {{formula:393642f5-1714-440b-b2ab-f8854d073422}} statistic to test whether the smooth functions are equal to some given functions (c.f. (REF )). The key technical input is to establish a Gaussian approximation result for high dimensional locally stationary time series for the affine forms and quadratic forms. We prove such a result in Theorem REF and it can be of independent interest. Especially, when the temporal relation of the locally stationary time series decays fast enough, the approximate rate matches that of {{cite:580faa0636f7bc3f4a39cb14b236dee5936e1555}} which is claimed to be the best known rate. Moreover, to obtain the critical values for the SCR, we establish the maximum deviation of a Gaussian process using the device of volume of tubes {{cite:f555ff326fefe1000136d6b9ac02a8d4a845f0c7}}, {{cite:09999495caf8472584c94fc6f5b01bed30facb04}}, {{cite:a0af06a047ccbbc4e416294598487ac2a4c536d2}}. For practical implementation, we propose a multiplier bootstrap procedure as in {{cite:c22c60318cc3dfe09f8e74c10c18bb068e711008}}, {{cite:eeb05f8b23573d12a1eede16399f8e132f4d41ce}} which is both theoretically sound and empirically accurate and powerful; see Theorems REF and REF . Numerical simulations and real data analysis are provided to support our results and methodologies.
| r | 8ed00632653c06b3a579a241de2ec56e |
In the context of Narain lattices which have U(1){{formula:ffe3decf-3181-4107-b6fe-4c4122c03adc}} Kac-Moody algebras, {{cite:c176fb18bd018e233e2177eb0efec5d56c3737b2}} proposed that the gravity dual is a topological U(1){{formula:e3f863ef-7e99-48fc-bceb-2dd63a0002af}} Chern-Simons theory. In the same spirit, we have suggested that the gravity dual for WZW theories based on non-abelian SU({{formula:c093bbc2-d89f-41de-848a-5ded3e0e4e67}} ){{formula:406df8ce-0d6c-4adf-b9b8-5627ab74667a}} {{formula:de5112a9-1f08-40c6-9981-81d8f9f78751}} SU({{formula:7d620581-ccbd-44b2-ae1c-273c66bccc63}} ){{formula:7acb52ea-1240-4ebb-98ba-cf69024b3619}} Kac-Moody algebras is a topological SU({{formula:0141f2f9-1bc6-45c8-a816-a397823d4718}} ){{formula:5d7e66e3-073c-4dc9-bc1d-1d99625ab2a0}} {{formula:5ffbc16d-74a7-4fd5-99c0-7f50198f0725}} SU({{formula:bfa6bf6a-2440-48fc-9834-3dc77c9f4fca}} ){{formula:1385c01b-0615-47ef-bbe0-225dcaa6b360}} Chern-Simons theory. However we did not really make use of this in the present work. Perhaps one can obtain more information about the dualities discussed by invoking properties of the “gravity” side.
| d | 0c087038df35f63424b0d2f9b918a91d |
The scientific motivation for the eROSITA design was the desire to detect and spatially resolve a large number (of order {{formula:6bd8caa8-482e-487e-9e48-761b825ac43b}} ) of clusters of galaxies over a wide redshift range (up to at least {{formula:09e404ba-aee4-4b5b-863f-d9ae6242aa88}} ) in order to constrain cosmological parameters {{cite:51eb254e1b0d2d1eaf97f2cacf9f2efd13346238}}, {{cite:cb3d5d46bb33ba48f67a9f43a62f036bf9d31c58}} at the level of a Stage-IV Dark Energy experiment {{cite:2abee3d253e5e641e275f1eb08b8ed6a7a8ce56d}}, via characterization of the growth of structure.
| i | c62b6c3da0e1ea7630d3c4865601285f |
Finally, we apply an optimisation method inspired by statistical physics, namely simulated annealing {{cite:3948e0e98fe9bb1e426c301ea2ece648862cd47f}}. Given {{formula:c0ac005e-5c5d-4903-8387-c8ec982014df}} orthonormal bases {{formula:d024d6f4-f48e-4bd6-b6a8-85f4698b4164}} in a Hilbert space of dimension {{formula:a08f07d8-3138-44b5-9a29-3280ad321f28}} , our goal is to maximise the expression of Eq. (), which we rewrite here for completeness (we keep the {{formula:8dcce6e2-2a78-4994-9a72-509dcd91fd83}} dependence implicit throughout this section, and keep the {{formula:4110c08e-ef57-4a1c-a8ec-c86e56a3933b}} superscript implicit in Eq. ()):
{{formula:ec3b262f-f090-4d4a-aaf1-783a259965ea}}
| m | 51638f7e40f47c41b93d51992280fb7e |
Our experiments showed that a convolutional network
was able to
generalise successfully to unseen digits
without featural overlap between training and test instances.
Thus, one potential solution to the identity learning challenge
{{cite:069da408384d9085a2925c6317cbbb5c70ed68fc}} turns out to be a standard architectural
design that predates it by almost two decades
{{cite:0fcd5cdaaa0de26b810430b46d93dbf3e0d86042}}.
In concrete terms, weight sharing across
digit positions avoids the problem of training
independence identified by {{cite:069da408384d9085a2925c6317cbbb5c70ed68fc}},
allowing what is learned at seen positions
to be transferred to unseen positions.
More abstractly, symmetry provides a notion of sameness
that goes beyond featural overlap,
allowing us to apply the same operation to
features that were unseen during training.
Here, we identified translational symmetry as reflecting
the intrinsic structure of the problem, in terms of
inputs and outputs being sequences of binary digits.
| d | 983298b48da9351af715e8e58b3fc2a1 |
More recently, quantum computers are considered for mathematical optimization {{cite:4612d329d706c0585ae125b1aebb41949c1d285b}}, or simulation of molecules for chemistry {{cite:73620e270ff20f4c3efe8cf2924efb99773e5e92}}.
Current quantum computers are still in their infancy and are restricted by the capacity and fault-tolerance {{cite:5f4dc490fcad356fa8937c9c41e33dd0b5294d95}}, {{cite:651f64e0c94c6e5175fda589593db2e36e51c1bd}}, {{cite:172a44478f5385c7610645872ef3bdd536dc61d6}}, making them more interesting for quantum research than for real-world application.
This motivated the creation of hybrid classical-quantum algorithms like the Quantum Approximate Optimization Algorithm (QAOA) {{cite:4612d329d706c0585ae125b1aebb41949c1d285b}} that employ a quantum device to explore high-dimensional search space and classical routines to optimize the search procedure.
By only relaying parts of the computation to the quantum computers, these hybrid classical-quantum-algorithms are more resistant to error {{cite:4612d329d706c0585ae125b1aebb41949c1d285b}}, making them more suitable to near-term devices.
| i | 483002853e3e4dce06f3392aa8200ecc |
To demonstrate the superiority of our proposed modules, we present a coarse-to-fine
framework termed UniMVSNet (or UnifiedMVSNet), named for
its unification of depth representation and
focal loss, which replaces the traditional representation of recent works
{{cite:26e84c8d4a2d5e9899b0edf2d79503dde0205231}}, {{cite:e37950032cfa84d8b6dccaef4da7d4fe4e777359}}, {{cite:62dc1da9233d39b438cd3f28ea00d7ec3325d4b6}} with Unification and adopts UFL for
optimization. Extensive experiments show that our model surpasses all previous
MVS methods and achieves state-of-the-art performance on both DTU
{{cite:c854e030041ea7a2625610cc1f343478d3aa3611}} and Tanks and Temples {{cite:4bbf1df070d5852ea2ccf071780c03cc61f51687}} benchmarks.
| i | 4f67ee86ff437df8928e30756bd46298 |
Artificial electronic states confined at heterointerfaces are a basis for modern semiconductor electronics and a fundamental theme in solid state physics {{cite:672c6dd2e569992ae62baf7931c75b946c9bf923}}, {{cite:9b77741d069086ec6f92582ee9829eb5366942d7}}, {{cite:5d35be71f0923140fc5a70e59eea08c8d2091c82}}, {{cite:b72761c3a1130a90735d1842c173b58d06fb1465}}, {{cite:cca5ff0fa9b7a693910c9db8218c67be0377add6}}. While most devices thus far have utilized electrical charge modulation, the introduction of other physical degrees of freedom suggests an opportunity to design novel interfacial properties. In that context, strongly correlated electron systems of transition metal oxides provide optimal interfaces where charge, spin, orbital, and lattice degrees of freedom interact with each other, leading to novel properties {{cite:d88dfb79546f0298604126ed71b683ffe3b538da}}, {{cite:2b49a44c255f4bb7b4f200d3e746e24f83d0523c}}. Notably, charge modulation at the correlated oxide interfaces is characterized by a significantly shorter length scale compared to that of semiconductor heterojunctions, as the charge screening in correlated electronic phases is usually expected to be of atomic-scale due to their high carrier densities {{cite:0543527758f1dfdbbadb8d750eaa1a2eda7943a2}}, {{cite:1ab7c6fdf0e00c1dec5b3d2d4bce53934c1eda80}}, {{cite:1dd73e613efe85ebd83714998b862ff7652fa24d}}. Furthermore, previous studies have shown that magnetic {{cite:54637cf60da6bbc5c16bec6e6fb045ab416129f4}}, orbital {{cite:6ff15a2970e800ac49cfc1f07c19c855e88d5008}}, and structural {{cite:8a6706e5d436b0f23f29b8c89a5b9bf4af8e3b47}} reconstructions also occur within several atomic layers from the interface. The short length scale of the interfacial reconstruction indicates the spatially abrupt modification of physical properties, which can translate into a high level of integration of tunable correlated oxide interfaces.
| i | 04d939220908a6590e749fc34f13c99f |
Limitations.
Our work has some limitations.
First, we demonstrate the application of our framework primarily using the Happy Merchant meme as a case study and focus on a single fringe social media platform (i.e., 4chan's /pol/).
Despite this limitation, we anticipate using our framework to generalize to new datasets from other social media platforms due to the great generalizability of large-scale AI models like OpenAI's CLIP model {{cite:d0cf47e956a140f99846a80b50f8e4bb1c66a475}}.
Indeed, by running our framework on other memes (e.g., Pepe the Frog, see appendix: casestudypepe in the Appendix), we show that our framework generalizes beyond the Happy Merchant case study.
Second, our framework for identifying variants and influencers of hateful memes generates false positives that need to be considered carefully (see section: evolution), highlighting the need to keep humans in the loop when moderating content.
Despite this limitation, we argue that our framework can assist in understanding the evolution of hateful memes and help in moderating them.
| d | 9a9e237bbc0b1aa00d13686330e8fa5e |
[sketch]
The upper bound follows from Theorem .
For the lower bound, we reduce from the satisfiability problem
of the basic modal logic {{formula:6a16e363-51f0-48c0-b365-a34610a8d5ce}} extended with the global modality
(which we will denote as {{formula:9286aa71-c16d-4d93-838b-543f00fd2332}} ).
This logic is known to be ExpTime-complete (cf. {{cite:81e47cbfe6c4754408160c8d611259317dcebb0b}}).
By a simple encoding trick due to Halpern and Vardi {{cite:b6c6ed69f74d2622654a9ef7ae8e28d57f6b9870}},
the satisfiability problem for {{formula:81fd7d91-badc-49cb-85a2-a058090167b3}} reduces to the satisfiability
problem for the multi-modal logic {{formula:eb167133-d551-4edb-b6eb-e84a095a5833}}
(where {{formula:c18cfe7a-7829-4d27-a5cf-2bf9172be380}} is the bi-modal “fusion” logic
that has two S5 modality without interaction axioms). The coding trick
in question consists of replacing every occurrence of {{formula:18849579-2079-4e84-83fd-82ae574286c3}} by
{{formula:9cbd0b00-63a1-42cc-b159-aa9e79651dc4}} where {{formula:8f6bb290-88a2-4f9a-9dba-082d57908d74}} and {{formula:f80ee9dd-aebc-4af6-b7f8-47d22a9dbf9f}} are the two {{formula:9e363cb8-c649-4067-b2e4-68059cdd36bc}} -modalities (cf. {{cite:b6c6ed69f74d2622654a9ef7ae8e28d57f6b9870}} for the proof
that this preserves satisfiability).
The satisfiability problem for {{formula:80e00cdc-4589-4e61-869b-225e8b1fbadd}} ,
finally, embeds straightforwardly into the monadic fragment of LFD
(even without using dependence atoms): each proposition letter becomes
a unary predicate, {{formula:4a57305b-37dc-4265-a94d-644c8eea496c}} becomes {{formula:662298d0-d96e-40b9-a132-205f734473c2}} and {{formula:7ed9a5dc-8b4a-40a2-b0c3-0b0fef628c4b}} becomes {{formula:4c05a8ea-e7e8-49a4-8c4c-5eeba3414e6d}}
for {{formula:7e1123bc-41eb-4592-86d6-ed1aa502373d}} two distinct LFD-variables, whereas the global modality
(in its existential form) becomes {{formula:c2709a32-6ac0-4792-a059-ad9f915e4406}} .
| r | 432bf15ea3de4904c1d2cca32a1bd2eb |
The QCD axion is a well motivated dark matter candidate as it can resolve the Strong CP problem of quantum chromodynamics (QCD) {{cite:76d2f8869f449228f2dc85ff96bb3853eea3d8d0}}, {{cite:c9b36dd6d46dc383e261050b63c0d958892927ef}}, {{cite:d1d77c03a8c3bd84ecf594a0596f32507b017bfc}}, {{cite:1a589c9548b4f8dbcf8f0c695d5b74e0bbf16a99}} while simultaneously having a favorable production mechanism which can populate the universe with the currently observed abundance of cold dark matter (DM) {{cite:0389fbd8b06b47736d0875eb10ab01057c9734ed}}, {{cite:517c992308f00674c301b875b9a4bd645207dbce}}, {{cite:61892be7c21cbb40f50555c839b4f07586b3bc04}}. In particular, through interactions with QCD, the axion obtains a mass, {{formula:f6c5013a-7fec-4360-a250-fbf920e18529}} , where {{formula:7ec4dabb-ee6a-4691-a5a8-822af268c652}} is the energy scale associated with the breaking of the Peccei-Quinn symmetry {{cite:5354ee952b1ed9b9bde87ad1b6afb26add5b9f4d}}. Recent theoretical results have shown that the axion can be a well motivated dark matter candidate if this symmetry breaking scale occurs above the energy scale of inflation {{cite:d49c1c5d93628715ade2c21d59c51e444730c434}}, {{cite:0507f19e5cdc239ebe5ffd5ab613691cce0de85e}}. Since {{formula:dc8a9547-a45c-4af0-82e2-69dc3f2380bd}} could be as high as the Planck scale, this has motivated experimental efforts to search for low mass (peV-{{formula:0dbfa24d-447e-4abc-81ac-4d0c0f527796}} eV) axions.
| i | 4a800a9520f6ef92eb5204c809ece22c |
How does self-normalization improve adversarial robustness?
We complement the differential private noise layer with a self-normalizing layer. The self-normalizing layer incorporates a scaled exponential linear unit (SELU) activation function which has been shown to elicit three distinct properties. First, the SELU activation function sets to control the average learning rate (u) of the network using negative and positive values. Secondly, The SELU maintains a fixed point in the neural network with the aid of a continuous curve. Thirdly, the saturation region dampens the variance alpha (learning rate) and the positive slope augments the alpha (learning rate). These three properties help to reduce the invariance of the neural network model {{cite:52c5d90815a5c60eefec979e67f82a17e5e97273}}.
| d | d2befe07eb7bfe2773cad422f7728f72 |
The automated CCE implementation inside AFLOW enables the correction of an extensive library of ionic materials that are made available via the AFLOW APIs {{cite:10517d556402e98b83c6564e8397e238c9e23677}}, {{cite:c9becc15b4b43ff0f671d7044fc77ffa142187e4}} and web interfaces {{cite:7acc7546d5abad9cb855d64b19ad49d353a620d0}}.
The implementation features three ways of user interaction depicted in Fig. REF : (i) a command line tool, (ii) a web application, and (iii) a Python environment.
The command line tool (Fig. REF (a-f)) provides the CCE corrections and formation enthalpies, (automatically determined) oxidation numbers, and cation coordination numbers for the given structure file that can be in any format recognizable by AFLOW, such as VASP POSCAR {{cite:930496da8a3748e4e566058f8b328be24290e066}}, Quantum Espresso {{cite:2f1293e91c0ed4181137d6d762a4d7edbc316933}}, FHI-AIMS {{cite:7039dcd8ab8635fe3fba7a21af997138fc565a32}}, ABINIT {{cite:25bdd109f4a6491b433bf932c0c146e2751d2084}}, ELK {{cite:36755e447022804637f0038bda7d71e855285ac1}} and CIF {{cite:9a9883d7082f2f290f04052c7350aa9a74227ecd}}.
Available options
are described in Section “CCE command line interface”.
The web interface (Fig. REF (g)) prints the cation coordination numbers,
oxidation numbers and CCE corrections for the selected functionals using the given structure.
The output also includes the CCE formation enthalpies when precalculated DFT values are entered in the designated fields.
The Python environment is distributed with the AFLOW source and can be generated with the command aflow –cce
–print=python.
It connects to the command line functionality and imports the results into a CCE class similar to the Python modules of AFLOW-SYM {{cite:dad9d20f2e074f09bb19fdc9276add0431bcdbc7}} and AFLOW-CHULL {{cite:5f82c6e78ebbcca1cf6ff0e50eba210e5cf472c3}}.
An example script leveraging the functionality is depicted in Fig. REF (h).
The CCE object has three built-in methods:
| r | 009e8b349031280376f4674f012b34c5 |
DFM can be cast into the general update scheme (REF ). Moreover, it has a close connection to the weighted gradient method {{cite:257ce3bce15e1a62640b9b9f9b2dae33ae02804d}} when the local constraint sets are absent.
[Connection with weighted gradient method]
Suppose that {{formula:6a146538-1afe-44fe-a3a8-d3dac1ce40c5}} and {{formula:114f563f-939e-44c1-adf2-6f1fa1fd7ac2}} for all {{formula:97569815-0604-4d8b-9092-26e312139b80}} . Then, by letting {{formula:a14b225f-d228-4d15-8d23-7cc910ce66ac}} {{formula:3223f104-b8c1-4d7d-850e-e79c1b2c2fb0}} , DFM (REF )–(REF ) becomes the weighted gradient descent method:
{{formula:25bdf04e-c2ab-40cb-9ef6-7430da997929}}
| m | e510ac855d2965fe1f9d20e3c5b8375b |
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