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In this paper, I investigate the effect of decoherence on the tripartite
entanglement of Dirac field in accelerated frames by using a phase damping
channel, a phase flip channel and a bit flip channel. The effect of amplitude
damping channel and depolarizing channel on tripartite entanglement of Dirac
field in a noninertial frames is recently studied in Ref. {{cite:b3da383d3cc058cbe506627cfa997d09bbf85e4d}}. I
consider three observers Alice, Bob and Charlie that initially share a
GHZ tripartite state. One of the observers, say Alice, stays
stationary and the other two observers move with a constant acceleration. I
work out the effect of acceleration and of decoherence by using the three
channels on the initial entanglement of the shared state between the
observers. I consider different kinds of coupling of each channel with the
system. For example, in one case each qubit interacts locally with the noisy
environment. In the second case, all the three qubits are influenced
collectively by the same environment. I show that the entanglement sudden
death (ESD) {{cite:3d483697e2220ee3432176fbb0fc479295eb0d77}}, {{cite:864b0f0cdb334d2b4164f89b729967b56df68325}} can either be completely avoided or can be slowed
down depending on the various coupling of the system and a particular channel.
Furthermore, I also show that the ESD can happen faster and becomes
independent of the acceleration when the system interacts with a phase flip channel.
{{table:a3cd9101-7656-47a5-adcd-ec4084f7ba0a}} | i | 41c7024bb74d99ea5e29483dbdb3a071 |
The Floquet-Magnus expansion is probably the most widely used
perturbative method in the literature. This technique treats inverse
of the drive frequency (in units of {{formula:bb0dc125-dd85-4975-86d4-c14b7e299a37}} , where
{{formula:4ca98835-591c-4a00-91e0-05455f492130}} is a typical energy scale of the system) as the
perturbation parameter and is therefore expected to be accurate in
the high drive frequency regime. Moreover, it provides a
perturbative expansion which maintains unitarity of {{formula:89a49ad7-b829-4163-b5e7-2e16b48e0915}} at each
order in perturbation theory. Also, the method has been quite
successful in providing a qualitatively accurate picture of
drive-induced generation of topologically non-trivial Floquet states
and the presence of a long prethermal timescale in interacting
many-body systems at high drive frequencies. The main weakness of
this method is two-fold. First, the radius of convergence of the
perturbation expansion and its regime of validity is difficult to
determine. Second, the method may lead to qualitatively wrong
pictures at intermediate and low drive frequencies
{{cite:92da36c3f173f191818e09ec6c5c35ba0f9eee7e}}. The resolution of these issues which might
provide one with a more complete picture of the Floquet-Magnus
expansion method is a long standing challenge; some progress in this
direction has been made recently {{cite:4b372aaa4cb09530b1eeb782f272cf1c01ba9c55}}, {{cite:106a1099fa0e259f68133dca6082fea19a9819a8}}.
| d | b6ac3ba6a207e0183e0b0cb94051ae76 |
As a field of study, information security integrates techniques from a variety of disciplines:
Malware detection is largely a function of information theory as developed by Shannon {{cite:60ff6a147e8a549c907baa58da69e55bfad44b6d}}; incident response borrows techniques from forensic sciences up to and including its need for a chain of custody to ensure evidence has not been tampered with; modern cryptography is built on the foundations of number theory and computational complexity {{cite:c1235bde8523e995ca3b8d905a5155774888ce78}}, {{cite:7086a954d63085cca58e48b7b1525a3495645c15}}.
In addition to this sampling of applications from other fields, information security borrows extensive theory from reliability engineering {{cite:7e94cab7a1bc746450ee8182b992ad635aafe64c}} and operations research {{cite:5bf47049db753929c11ef9b72b29682a8ade675b}}.
Given the applicability of disparate techniques and theories, it stands to reason that other techniques and theories could bear fruit on the rich field of information security.
| i | 31ce9947e4a9f9e0b5454eb25e45e47f |
Results on MORPH2.
Here we provide additional results on MORPH2, despite the performance on it were long saturated. We use the same setting following {{cite:8206ccec8feacb278237822d940393faa1735376}}. The following baselines are tested:
1) DEX w/o IMDB-WIKI {{cite:8206ccec8feacb278237822d940393faa1735376}}: A VGG-16 Net pretrained on ImageNet
2) DEX w/ IMDB-WIKI {{cite:8206ccec8feacb278237822d940393faa1735376}}: The same model as 1) but the ImageNet pre-trained model is further pretrained on a large-scale age dataset, named as IMDB-WIKI, which contains 523,051 images crawled from IMDb and Wikipedia.
3) Ours w/o IMDB-WIKI: our full model with mixed loss, i.e., cost sensitive loss and KL divergence loss. The model is pre-trained on a face verification task.
4) Ours w/ IMDB-WIKI: a VGG-16 network which is the same as 2) but it employs the proposed mixed loss. The model was pre-trained on IMDB-WIKI.
{{table:b38410f8-ab71-42c8-8904-37edc44b3dc3}} | m | d0dc47d7abfcad93ed0865346cfb844e |
Bayesian non-parametric machinery is applied to federated deep learning by matching and combining neurons for model fusion.
Yurochkin et al. yurochkin2019bayesian proposed probabilistic federated neural matching (PFNM) using a Beta Bernoulli Process to model the multi-layer perceptron (MLP) weight parameters.
Observing the permutation invariance of fully connected layers, the proposed FGNM algorithm first matches the neurons of neural models of clients to the global neurons. It then aggregates via maximum a posteriori estimation of global neurons.
However, the authors only considered simple MLP architectures.
FedMA {{cite:10a9d675fe1de4ff120c8a097d48c46b4cb5ac0f}} extends PFNM to convolutional and recurrent neural networks by matching and averaging hidden elements, specifically, channels for CNNs and hidden units for RNNs.
It solves the matched averaging objective by iterative optimization.
| m | 18b0e672f0669bada0a01a11c003708b |
Figure REF shows the clustering accuracy for different variants of the Triplet Loss.
First, we observe that Triplet Loss_2 (REF ) and Triplet Loss_3 (REF ) outperform (REF ) on multicut clustering (left).
However, K-Means (right) performs better on all our experiments given the fact that {{formula:90ce38c2-1719-4110-97ea-1ddbc48365c0}} is known.
The highest performance is achieved when we train the CNN-model with the Triplet Loss_3 (REF ), where the average accuracy is 80.5% (red). For k-means, the Triplet Loss_2 shows worse performance than the regular one {{cite:1ed8392e558debb3d84d0f1c9463c243f8839796}}, while the proposed, simpler version, Triplet Loss_3, performs best in both scenarios.
| r | e134b1e4b683b325fe5abfdc4c9eac50 |
In this section, we compare VHBS with Ncut {{cite:dafc7242a84e05b638d96deeca99cd7643bfa49b}} and KMST {{cite:08016bcfd77dee6bc8151d16e9f918cdafb6d39c}}, {{cite:9b6f835320ab6672c8a42fd22d7936a62c6e196b}} over the entire test set of Berkeley Segmentation Dataset and Benchmark {{cite:79358fa890824939d578551cc391cdd3313b00a8}}. The source codes of Ncut and KMST are got from the author's websites. To fairly compare these three algorithms, we tune the parameters to output the same number of segments of each image in the test set. Figure REF provides the evaluations of {{formula:e7401daa-fe4f-444a-8f56-15d49a933c09}} , {{formula:c92e33bd-4671-4b34-9e97-df5ceb23caff}} , {{formula:d2d42663-dd7e-4666-8ba4-dbf2e8012ebd}} and {{formula:fb9f4d5d-4020-4c39-a8fb-7a2179a1eccc}} based on number of segments.
{{figure:6bfe23a8-b44b-4efc-89f9-af729c583d47}} | m | 0030f66b94c09f90bd436368836baec6 |
We validate Algorithm 1 in CIFAR10 and CIFAR100 with ResNet ({{cite:b646c776c7f7b6ae82720c8ee563a6c14c53a82e}}). The results are shown in Table REF and Table REF . As the tables show, the GL penalty and the shrinkage operator can significantly improve the weight sparsity and the channel sparsity with minor reduction on accuracy. The splitting step before the shrinkage operator can greatly improve the sparsity. Of course, the model performance would be somewhat affected.
{{figure:0b4a5caf-133f-4875-acf9-2e7cf55b0409}}{{table:2e3cdf11-e132-4e9a-8024-4ab51fa0de8f}}{{table:f493c768-8f8a-4858-b956-4c30ff4cdc2d}}{{figure:fdfe26c1-d840-490e-82a9-3919a8affb38}} | r | afa9b286cb9c910b417a7cc90820445c |
In this work, we assume that the pentaquark states are generated by the {{formula:c0720c81-c161-47dd-8fab-3349ba1de338}} , {{formula:c8866b4b-8b22-4374-8b6b-21be26840730}} and {{formula:a32308fe-e0b3-4736-affc-9acd04cd327a}} coupled channels, and neglect the {{formula:10e43bdd-39c7-4e74-8ba1-88c0025efbd6}} contribution, which is shown to be rather small with respect to the other three channels in the chiral unitary approach {{cite:b7da8b8715fb7c357734c8df44bed2d8f139e10a}}, {{cite:e38bf1fc67edb82dd7cc5820bd263695fab58ce0}}. As a result, there are three unknown parameters in the contact-range potentials. Following our previous work {{cite:d28fbf00e94f65b2db478710778de6089c313aeb}}, we study two scenarios A and B. In Scenario A, the spins of {{formula:325085eb-bdc3-4342-866c-c40473dc2984}} and {{formula:7ec20707-b71e-4fe6-b34a-cc1d7a9bcd0a}} are {{formula:9ade1aff-b5e1-49c6-a4e3-abcc79d14393}} and {{formula:174bdec0-dd3f-4fd8-ad15-5a46b91a7afb}} , respectively, while in Scenario B they are {{formula:2fda7047-19de-4be8-8f9d-e62418ffa3af}} and {{formula:ca097f7f-e463-4d7d-99a5-3ab6da760aa7}} . To estimate the uncertainties induced by the cutoff appearing in the Gaussian regulator, we choose two cutoff values, i.e., {{formula:513cbdad-a470-44aa-9511-6cf07f8898a0}} GeV and {{formula:88fd0e7e-44cc-4843-9ba6-add7353cb574}} GeV.
| r | d717298a23d36e8308b4dcec88b485e8 |
which satisfies the Karush–Kuhn–Tucker (KKT) conditions {{cite:0a17acc629e335532b68f8665eb581b5ead5bcc9}}.
| m | 120a106008a2417825423d0f8be556c5 |
Finally, let us briefly discuss the possible realization. This protocol mainly exploits the common linear optics, such as PBS, BS, BD, HWP. Meanwhile, this protocol require the multi-partite hyperentanglement. Such hyperentanglement was also realized in experiment {{cite:427b4ff7bb957f0092b6cd66b592341b871b33c7}}, which show that this protocol is feasible in current experiment condition.
| d | 9df0c1c279a1f6f99f58d2c1c5e7afc0 |
Remark 3.12 The distance {{formula:6dadcf1d-c745-4ec4-acaf-5b15c2ece79a}} in Theorem REF is consistent with the result in {{cite:c0f12746b712cc1179f4908cc048222d4f852a00}}, where it is shown that the slow manifold {{formula:c30f6ec1-6ef5-46a7-aa98-8e63929efefe}} leaves a neighbourhood of a regular fold point at a Hausdorff distance which is {{formula:1b1fa015-f8b8-43e5-9a8c-eb0894e20a57}} from the critical fiber. The only difference between Theorem REF and the result in {{cite:c0f12746b712cc1179f4908cc048222d4f852a00}} is that the distance {{formula:aecf1457-6683-4ba3-8f0e-47140f9011f6}} in Theorem REF is stated for general systems (REF ) in terms of simple transversals in the original coordinates.
| r | c1f31b7529eb9c64abffceae973fc123 |
The inclusion of electroweak radiation in parton showers opens up a rich field of showering phenomena, in particular at energies well above the electroweak scale.
Rather than considering specific sets of observables for specific processes, in this section we first consider radiation spectra of several particles at energies compatible with future colliders,
showing overal branching rates of the electroweak shower, the relative importance of the large number of emission modes and the effects of bosonic interference.
We conclude by showing results for hadron collision processes at LHC energies and comparing with the Pythia electroweak shower {{cite:25df039cc6b25fb4e44dccec7f8fb51827d8ec53}}.
| r | 8bc9ae7116b5bcbfca82a946480f1bf4 |
From our experiments and data collection, We have seen that Gatys et al.{{cite:98242ec94ba44a7593609c12bad8fd39bc2f2de8}} and Johnson et al.{{cite:3c7c2bea073fda2e8e6c1cbc80397d8bb21acf33}} both produce the best quality stylized images, but participants prefer Johnson et al.{{cite:3c7c2bea073fda2e8e6c1cbc80397d8bb21acf33}} stylized images on Bangladeshi paintings. In the future, paintings can be used to neural style transfer in different materials and objects in our day to day use{{cite:d47f7f8670163e2344801d0736aea84e6ec6c820}}. As a result, an artist's value can be preserved and can regenerate their works easily using deep learning technology.
| d | 078143a778a405fd5ca33498d4252378 |
Emerging non-volatile memory (NVM) technologies have been extensively studied, which is due to two reasons.
First, there is a gap of orders of magnitude between access latencies of SSD (e.g., 0.1-1 ms) and DRAM (e.g., 10-100 ns).
Thus, there has been continuous research on finding a new layer of memory hierarchy, e.g., storage class memory (SCM) {{cite:01ac2142f3be3f0836e8b6feea871787ff8f8e82}}, to fill the large performance gap.
Second, conventional dynamic random access memory (DRAM) technology is experiencing significant scalability issues {{cite:5bbe93204f6c9cb1093a509aa99ef08351e9a5ee}}. Therefore, a promising NVM technology should be scalable and provide near-DRAM access latency in order to supplant or supplement DRAM for a high-performance and high-capacity main memory system.
| i | 22a7601aabd8b2acd3c99d182dfc27a0 |
In the present estimations, there is only one model parameter {{formula:35a48c59-893d-4508-a6b1-7c161c951a0c}} introduced by the form factor. As indicated in Ref. {{cite:24ef7cf7207167989dba48ff9b1474c85e237691}} the cutoff {{formula:24d8b14d-78c4-4890-bec5-38ff3dcc49ab}} in the form factor should not be far away from the mass of the exchanged meson. The model parameter {{formula:25416c93-b2f8-4463-ad22-adaa3e540602}} is expected to be of order unity and usually depends on particular processes {{cite:24ef7cf7207167989dba48ff9b1474c85e237691}}. Actually, the model parameter {{formula:a524f4e0-a097-4416-9428-a0be8355a4ae}} can not be estimated from the first principle. It is usually determined by comparing the theoretical estimations with the corresponding experimental data, and check the reasonability of the determined parameter range. However, as for the processes involved in the present work, there are no direct measurements for {{formula:fa51686c-b694-49c4-88bf-90a991e9c230}} . As indicated in the introduction, only the branching ratios of {{formula:749e6e34-a4b7-47a1-b8c7-1cb7f840c7f7}} can be obtained from the experimental data. According to our analysis, we find that the cascade decays {{formula:56faffdd-a703-4f81-bd4d-e59000b7dd87}} play essential roles in understanding the large branching ratios of {{formula:09724005-a8a3-4edd-88d6-197a77104c14}} . Together with the decay processes {{formula:b99ff3de-4877-4ab5-9474-59baab9ccd28}} , we can roughly estimate the branching ratios of {{formula:5e4a7d3c-6d88-4c8c-aa51-2ed3bff28f82}} , which can be compared with the corresponding experimental data.
| r | cd8ef6cfd5dae4704536cdf45dd60a0c |
In fact, the calculation of the effective action using a chiral rotation involving the corresponding Jacobian, which is related to the anomaly,
is rather subtle—as clearly shown in Ref. {{cite:1c550e5fed5480c9741e2b78bef54d6fedfe0cbe}} for the case of a WSM. We find it illuminating to briefly describe the main steps of the procedure. The starting point
is the Lagragian density:
{{formula:7c2cc4fe-f954-4b63-b2d1-9cef7f3f51a1}}
| d | 5a7ef71b29bde6bcca432437e41a5132 |
All energies and matrix elements are calculated on coarse {{formula:53bfd882-9ac5-4b4a-9ac8-676c97e45d18}} and
{{formula:604e48f5-9d52-4009-a69a-e1901a431faf}} meshes using the DFT software JDFTx,{{cite:e010e49893315da47825c0c716087512fd1882e0}}
and are then interpolated to extremely fine meshes in a basis of maximally
localized Wannier functions.{{cite:20cf38c44bb25b63f35387bb92232f3aa21d2f30}}, {{cite:5ce8fb43c467c541a17d2918d5305426176025da}}, {{cite:24ffb2bc05bf12738de94225d4a9c1c084cd1571}}
Starting from at an initial state with a net spin, we evolve the density
matrix {{formula:ce62cc4d-9588-4518-b43f-b2aa05f75ecf}} through the master equation Eq. REF
for a long enough simulation time, typically from ns to {{formula:7f391eec-ebc6-47fa-b523-82a892861143}} s. We
then obtain the evolution of spin observable {{formula:3ccf822d-cc2c-4187-9103-1d63df5f6c89}}
({{formula:992190ea-1c01-48f7-8003-ede43066f12c}} ) from {{formula:c720df22-23f9-4318-9ebc-2ccc99d1a810}} (Eq. S1). At the end, spin
lifetime {{formula:fa535348-9dda-4fa0-94c0-3c8a5bd21549}} is obtained by fitting {{formula:2244881b-33db-4274-96fc-6c0d902253c3}}
to an exponential decay curve with decay constant {{formula:10764b3c-e1d5-461e-ad37-718bdc75d44a}} .
| m | f3dbe25b7c71326f1d661d7e13797f74 |
Inspired by Zhang et al. {{cite:824332804ea9e3446d319720406dae0c33acae7b}}, we propose FloodTransformer to solve segmentation for the flood data domain. It is a fusion architecture of Visual Transformer {{cite:29144ed73b05b782387962742d3a8bf09f772d85}} and Convolution Neural Networks (CNNs) and its model architecture is displayed in Figure REF .
| m | d4f3ff003fd2efad70cd325a17c057b3 |
The quantification of fault slip is achieved using the Rate- and State-dependent Friction (RSF) model for friction evolution, which is considered the gold standard for modeling earthquake cycles on faults {{cite:6383044aa4e82a412ceb236cddff2ba94f0c5d6e}}, {{cite:02e68f9058805c56e8b82d43108b2b55ed020835}}, {{cite:e15229b479f53f41780948112d1de619b9296e47}}, {{cite:6cb4b882dcf8e76f65bbd32997d4a61273226ca3}}, {{cite:fa706c44df9e505b81ec0f2eea28299d628352bd}}, {{cite:d41a3d961b27a05a4c7bf521d43a25ce8bc990ac}}. It is given by
{{formula:e4c9b4b4-b548-4a03-b9c6-22841c4b6185}}
| i | 3e5181aa0774ad47f1db5058f3991835 |
At large {{formula:2290db52-aa34-43e4-81ad-0503be175378}} , the generating function (REF ) localizes close to {{formula:04a8b120-d985-4722-8053-813812a7c808}} , and is therefore dominated by the bulk-channel OPE limit of the discontinuity. This fact puts on solid ground the result first obtained in section via the lightcone bootstrap: in every defect CFT, a subset of the trajectories asymptote the spectrum of the trivial defect at large {{formula:e74a660f-c3fd-474b-8fc9-783e01917acd}} – see eq. (REF ). We call transverse derivatives the operators on these trajectories. Since {{formula:235ed9a2-c4f0-40da-9713-ae7a5b9b42ed}} is a global charge for the defect spectrum, it is interesting to contrast this result with the large charge sector of an ordinary CFT {{cite:7cadd85de1593d1fde7f05a67242fc77425721c6}}, {{cite:e3827d7bbb6c69002fc66991ea332a321a1ada90}}, {{cite:146044eb073d28307fd6c14264f3e45631212e39}}. There, an operator with a large charge creates in radial quantization a state with homogeneous charge and energy density. The scaling of the low-lying spectrum with the charge is not linear, and therefore incompatible with the presence of the transverse derivatives. In fact, as we saw in section , the semiclassical picture in this case is very different from a condensed matter phase. Since a defect CFT is non local, there is no notion of energy density on the defect, and once seen from the bulk, the transverse derivative breaks a spacetime rotational symmetry rather than a global one.
| d | 547675e060121e9f5e818cd3187413fc |
In practice, Bayesian uncertainty estimates often
fail to capture the true data distribution due to intractability of Bayesian optimization {{cite:61f31fdebe4c37a60bbbd73ecd8dafa1dcfd9d90}}, {{cite:5f053a4b9cf5f27ef32b75b3c70a5e4a057e9bff}}. There exists a line of research exploring re-calibration of neural network models - {{cite:61f31fdebe4c37a60bbbd73ecd8dafa1dcfd9d90}} proposed using ensembles of several networks for enhanced calibration, and incorporated an adversarial loss function to be used when possible as well. {{cite:4153a74dfd681c51664b5d5f635bab676e138acb}} proposed temperature scaling, a procedure which uses a validation set to rescale the logits of deep neural network outputs for enhanced calibration. {{cite:9b429709dc7a78c4af6d56eac524ab12a7b61f91}} and {{cite:d3735c2cf38b3fa35da554a822dc198a5cb4e9fc}} propose calibrated regression using similar rescaling techniques, the latter through stochastic weight averaging. {{cite:1d62f03e186c3140e00f4e48288d2a2ef4b8962b}}
| m | 5a44e15c18a859d2cb26cdd95d293f41 |
Line-search methods are particularly appealing as they reduce the possibly high-dimensional optimization problem to a sequence of one-dimensional problems of finding good values for {{formula:dbae6e6b-4db8-4f14-a258-ab6b6aced95e}} . To ensure convergence, {{formula:5e81f8c2-b47a-40c8-97b5-d8c7fddc0713}} has to meet certain conditions {{cite:8591480221e2ed17354b892a033579ffa2fecc50}}, {{cite:985a696c568b097e97cd6ee8510dabf7adee3769}}. Methods to determine the step size that satisfy these conditions are sometimes referred to as globalization techniques. Note that enforcing these conditions only guarantees convergence to a local stationary point, i.e., a local minimum, local maximum or saddle point, but not to a global minimum {{cite:52d25b8eb245f9752313b91a28b831e4e3eb57fa}}.
| m | 5134210c729014c29a77a7f065d19e8d |
One of the major results of the present paper is the analysis of vanishing separation between the particle and the oscillating plate. We showed that the standard lubrication theory developed for modeling near-contact particle-particle or particle-wall interactions in steady Stokes flows {{cite:5e9816b3abb53cc0878fd94c7e710b07f5191d7e}}, can be extended to unsteady Stokes equations. Besides the standard requirement of the lubrication theory, stating that the clearance between the particle and the plate must be much smaller than the particle's radius, {{formula:623f654e-41d4-47bd-ae1a-683d3bacfc84}} , there is an extra requirement, {{formula:b3ffbf6c-5e66-44c9-9c65-64cc38e13ca1}} , that comes from unsteadiness and which imposes strict limitations on {{formula:79cb06cd-a372-4869-bd39-24a5279aed07}} for large particles. Applying the lubrication theory, we demonstrated that in accord with the ad hoc assumption in Ref. {{cite:ef826a0f2666256a1e62218e4be864928e22f69c}}, in the limit {{formula:710a8165-4fca-41bb-8ff1-dbb51944ed1d}} the freely-suspended particle hydrodynamically “adheres" to the oscillating plate, as its translation velocity tends to that of the plate and its angular velocity vanishes. However, such adherence does not warrant the equality of the excess shear force due to a freely suspended and the adsorbed particle, as was suggested in Ref. {{cite:ef826a0f2666256a1e62218e4be864928e22f69c}}, as the difference between the limiting values of {{formula:5d402af3-8fef-4fb2-824b-123a7a1a46b3}} and {{formula:01fe84c8-5547-4b19-b695-ec504e729d3b}} at contact depends on {{formula:0368c8d5-b03d-4915-ae42-6292d6d71dd1}} . The accurate limiting behavior of the excess stress at contact would require more accurate numerical modeling and will be conducted elsewhere. Notice also that our analysis also does not take into account the non-hydrodynamic short-range forces acting on the particle at close proximity.
| d | e1d32d22d06598ba2cbcc60e6adde9e5 |
For online learning problems with general convex losses, the optimal estimation rate depends on both the hypothesis space and loss function (e.g. whether it is Lipschitz or strongly convex). In this paper we did not establish theoretical guarantees for Sieve-SGD when applied to general convex loss, however, we gave some empirical evidence that it can perform well there. We believe our proof techniques might be extended beyond squared-error loss, perhaps using ideas in {{cite:7fcb7c0a65b14fe5f1ca3cf5b786e300aeda68c1}}, {{cite:ba0ab2156c12625f7ba0aade7982e7913506f0b3}}, {{cite:9923c14689532d1eaf969de04f727f09f141697b}}, {{cite:8d8ff3ff2a38e3bfc665924e804683eb18c75905}}.
| d | ab8c5684d55f6d9977755df0038c5d3e |
where {{formula:f4514947-990b-42e8-9ba7-c6071c0f3d67}} is the classic {{cite:dd877726d79b38406004c9623ecefad7f8ed35c1}} stress–energy tensor for electromagnetic field and where {{formula:2d4f540e-f94a-438b-9e51-51b195d011e4}} represents Minkowski metric tensor.
| r | bc3cba4e3da453a5bfe1f5e2cb3c51f1 |
Analogous to the electronic SSH model {{cite:107b5f92fa92bf7c8685698ad81d4c1b0d061819}}, as shown in Fig. REF (a), we construct a 1D FM SSH model including the intracellular and intercellular interactions, i.e. {{formula:17b6315f-edc6-4cf4-9530-dbcb88331026}} and {{formula:9626832b-c2c1-4658-bcc8-9c3e7fbd8395}} , which presents a pure Heisenberg-type model. In general, there are two types of FM ground states, that is, the spin orientation is perpendicular to or along the direction of the 1D chain, as shown in Fig. REF (c), and thus these two quantization axes are considered, respectively.
{{figure:b8c7231e-41a0-4d67-8240-29038c1fa155}} | m | 681fb328034b16f646fdd211e740c228 |
Although we considered networks with all-to-all connectivity within and between
populations, the governing equations were derived in the continuum limit. Thus the dynamics of large
but finite networks whose graphs have the same graphon (or graph limit) {{cite:8a9d99a2f008d2c32adc5185d993331674a1b628}} as
the networks studied here, of the form
{{formula:4a9efbe8-1e80-41ec-adb5-42203efdff3c}}
| d | 441cd19aa227ad1ae1c21be33efaf2a6 |
In our daily life, recommender systems {{cite:a89e430053bba3ec9954f2a53df295bf048968e0}} play a vital role to suggest right products, right social media, preferable news or travel places to the target users in different sectors.
Using the knowledge of demographic information, past view history and previous purchasing history, recommender systems understand the users' preferences and suggest suitable products or preferable news or close friends to target users {{cite:a89e430053bba3ec9954f2a53df295bf048968e0}}, {{cite:d02df069740154f95a53300d6b196a6109c944e0}}.
An efficient recommendation model can not only understand users' preferable zones but also increases both contentment for users and earnings for companies.
The Content-Based model {{cite:31db2de428a019382c60f96cbff40bc8c6343388}}, {{cite:0eab229f8ae6c66843aa536a888bcff5e07970f5}} and Collaborative Filtering ({{formula:837e5b01-9847-4ccb-8429-175e78ec6a84}} ) model {{cite:5f2be625df0e4776f7f5dbcf97b7751568114bcf}}, {{cite:e48ace2299216f3211844d6aa794e67f734c60b6}} are the two important categories of recommender systems.
The Memory-Based approach {{cite:ad1b3c4a4dea3a55502b86282248c8ff9d67ea0f}} and Model-Based approach {{cite:7b16de6b55745d07d15d13e82d93ddb21c2da153}} are two types of {{formula:608eae0b-a422-4e9a-8009-c14a0a3aa49e}} .
Memory-based models generally investigate the customer's preferable items from rating values given by users on items.
Each entry of a rating matrix is formed based on users' rating values for their purchased items.
But a model-based approach investigates the way users give ratings on items.
It claims more adequate efficiency than the memory-based approach.
| i | 2b7737ab113a3be8de85d745bd571764 |
In the present paper, the problem has been considered from this last viewpoint, assuming that the ultimate decay stages were amenable to the most abstract level of implementation in terms of probabilistic cellular automata {{cite:32376d83c1f4f43ea6bfeb9393ba90c60beb47cd}}, following {{cite:2c7d426152464f49bc920a942f056a08df1c7e1d}}, {{cite:45116873465d1d94148e38671701ebbd34617d60}}.
We focussed on the specific case of channel flow that offers a particularly rich transitional range.
Its upper part displays regular non-intermittent laminar–turbulent patterns that can better be described using the tools of pattern-forming theory {{cite:d68967cde109c88b247bfda7a8cb3c2b8aa1bd6a}}, {{cite:51cb5ea3ea8b0bd8b17349fbeb5211d9120f318a}}, {{cite:af1e58247708ea6015cc73bd8b8be742b54c0e4e}}.
The lower transitional range is characterised by their spatiotemporally intermittent disaggregation, to which the considered type of modelling is particularly relevant.
The analogy alluded to above has however been severely constrained to fit the empirical observations.
The main assumptions were the introduction of two types of active agents attached to each kind of localised turbulent bands propagating in one of the two possible orientations with respect to the stream-wise direction.
Interactions were assumed local so that the probabilistic cellular automata evolved simple nearest neighbours von Neumann neighbourhoods (Figs. REF –REF ).
Scrutiny of simulation results lead to the introduction of a certain number of probabilities governing the fate of single-occupancy neighbourhoods.
Multiple-occupancy was treated as a combination of single-occupancy configurations supposedly independent, reducing the number of parameters to be introduced and drastically simplifying the interactions (at any rate impractical to estimate in detail).
A clear-cut physical interpretation was however given to each parameter in the set reduced to four, accounting for every possible stochastic event affecting the agents, namely propagation, decay, splitting, either longitudinal or transversal.
A mean-field study of the model, neglecting the nontrivial effects of stochastic fluctuations, reproduced the empirical bifurcation diagram of channel flow at a qualitative level (Fig. REF ).
Transitions have been studied quantitatively by numerical simulation of the stochastic model considering variations of these parameters as putative functions of the Reynolds number {{formula:3a72cfa8-a78e-401e-bd8c-6d722b77124c}} , highlighting three situations:
| d | 885bec24710b218c318a19aa2a0bc42f |
{{formula:725511e7-fe49-4117-918a-fd823f949393}}Normalization: We normalize the base dataset into a unit sphere using the following steps {{cite:6ae9bff07ab653609eb906e918d13c1f0d2902c0}}. First, we calculate the center by computing the mean of point clouds for each model. Then, we find the point which has the maximum distance from this center. Finally, we divide all points over this calculated maximum distance. By normalizing each model, the data is located between -1 and +1.
| r | 461569e9e354d22f3b7219a6de6518d5 |
{{cite:086c37cb7238c89dfd3d3c364c8f4b975bb25eb5}} fit X-ray observations with a single temperature model for the CGM and found that a small selection of observed fields fit to higher CGM temperatures around 0.4 or 0.7 keV. That work included a component at kT = 0.1 keV with free normalization for local SWCX and LHB emission that likely contributed to that fit result, partially masking the warm CGM component. This would have resulted in some fields fitting the CGM partially or totally to the hot component instead of the stronger warm component. This is a result that we observed in our early fitting, prior to making improvements to our background models and model parameters. For those early fits, a small number of fields ended up fitting with a low temperature APEC similar to the LHB component and a higher temperature component at roughly 0.4 (combining part of the warm component emission with the hot component) or 0.7 keV (just fitting the hot component peak). The higher kT fields from {{cite:086c37cb7238c89dfd3d3c364c8f4b975bb25eb5}} appear to be a random subset and exhibit no notable properties in our data.
| d | c62094a87c76f0536035462dad833403 |
As described in Section , the response modes are of
particular interest because they sustain themselves by self gravity
and can affect or dress the response of our system at any
imposed nearby real frequency. For stable systems, the frequency of
these modes have a negative imaginary parts (damped). Most work on
stellar spheres considered stability, the most well-known results
being Antonov's stability criteria. However, {{cite:21124ba0e816e0b8d84ff307cc99f4784852bdd6}}
showed that the {{formula:22787d99-20ca-4596-9e30-c38e8853b672}} modes are often weakly damped while stable.
Therefore, these may be excited and entrained by minor mergers in a
typical galaxian environment and noise in a quiescent environment.
Indeed, we have noticed in prior work that the {{formula:748a5e95-0511-4c74-86e2-114aca6ab8a9}} power in the
halo component is enhanced relative to {{formula:317a4365-f040-4f03-a71c-6d641ee63c93}} in exp simulations
and attributed this to dressed noise as described in
{{cite:21124ba0e816e0b8d84ff307cc99f4784852bdd6}} and more recently by
{{cite:5be0cac29f7444c3a723bc0b068396cfa9f24d96}}. Tracking down this details of this
apparently excess power led to the identification of the instability
described in this paper.
| d | f6ef6a2c9a400ffaec4669eb9b4b9822 |
In the following parts of the paper, we will build on Theorem REF for decoupling Rademacher chaos.
We observe that the book {{cite:dfef9e87606066371f178a16c5b0af60c5793276}} provides some bounds on Rademacher chaos but only for the special case of {{formula:256ef8f2-b084-4b4c-b9d1-8a1fce6454d3}} .
| r | 01e12b9eef67ed6e1774348d215f50b8 |
Many classification approaches in machine learning explicitly or implicitly rely on some measure
of distance {{cite:46d329327960bde9d60129d6ccd0bf20b68093f4}}, {{cite:e2d2a8fe08ed395a80153baab089e3fe167f7fe7}}, {{cite:e08959b38c09d87d1bb971eebb233d19e6f989d3}}. This is particularly
apparent in case of the {{formula:3cc98df2-245e-40ea-a298-1da3b794c411}} -nearest neighbor classifier which classifies data points by
assigning the label of the majority of the {{formula:2cc8d566-14c5-4009-93ea-9343aa1f88fd}} nearest neighbors according to a
given distance {{cite:6ed11626574bd13629e1de9a8e9436fc0c1801e4}}; or in case of learning vector quantization
approaches which classify data points by assigning the label of the closest prototype according
to a given distance {{cite:2c1c3eae13b17514f56f3f40ab7917241869f148}}. The success of such machine learning approaches
hinges on the distance being discriminative, that is, data points from the same
class being generally closer compared to data points from different classes.
If the distance does not fulfill this criterion,
one has to adapt or learn the distance measure with respect to the data, which is the
topic of metric learning {{cite:46d329327960bde9d60129d6ccd0bf20b68093f4}}, {{cite:e2d2a8fe08ed395a80153baab089e3fe167f7fe7}}.
| i | 4d270af5a92367787888c6f1cec005b4 |
Ablation study: We compare the performance of our approach using the benchmarked base CNN architectures such as ResNet-50 {{cite:4d652c687247cf16a22de6299f8b252ad038ec26}}, Inception-V3 {{cite:0ed027114b3332a81d77601bb66cf6441a727a0d}}, Xception {{cite:bb878f7b899295b75670f5097a6b5d6b0e85e037}} and DenseNet121 {{cite:df4c18cd294c86e0f5caca205cfb786f086e81ce}}, as well as SotA lightweight architectures such as NASNetMobile {{cite:c15202cb502b03638013671f2ab75b5befbbb641}} and MobileNetV2 {{cite:edc784c4a39276ebbd353170d8e9c38e29b00341}}. The performance is shown in Table REF . In all datasets, both standard and lightweight architectures have performed exceptionally well when our proposed CAP and classification modules are incorporated. Even our model outperforms the previous best (primary data) for both standard and lightweight base CNNs except in Cars and CUB-200 datasets in which our model with standard base CNNs exceed the previous best. Our results in Table REF & REF are the best accuracy among these backbones. Nevertheless, the accuracy of our model using any standard backbones (ResNet50 / Inception V3 / Xception; Table REF ) is better than the SotA. In Flowers and Pets datasets, the lightweight NASNetMobile is the best performer, and the MobileNetV2 is not far behind (Table REF ). This could be linked to the dataset size since these two are of smallest in comparison to the rest (Table REF ). However, in other datasets (e.g. Aircraft, Cars and Dogs), there is a little gap in performance between standard and lightweight CNNs. These lightweight CNNs involve significantly less computational costs, and by adding our modules, the performance can be as competitive as the standard CNNs. This proves the importance of our modules in enhancing performance and its broader applicability.
| d | f7c1f81bf82df9aed7c626095ca98d64 |
where {{formula:0efe4df3-c7fe-4bd8-a1d5-74a26845bef6}} is the set of rotation, reflection, translation, and uniform scaling maps and their compositions {{cite:be0cc960ea9d276c53901c6bb697823560865d9c}}.
| i | 4201d634f1ac615b0ca9468fe7397618 |
Non-Volatile Memory (NVM) technologies such as Filamentary Oxide-based Resistive RAM (RRAM), Phase-Change Memory (PCM), and Spin-based Magnetic RAM (MRAM) enable low-voltage multilevel operations, making them suitable for implementing analog synaptic weight storage in neuromorphic systems {{cite:470c80d0206fffad0b68d57d664445cff3fb71d4}}, {{cite:e2093a913ca63087b89759f137a74f3147145af6}}, {{cite:d088bd0506e811899fa2c828205bcac038b88beb}}.
Of these emerging new memory technologies, hafnia (HfO{{formula:9c1b2435-5aaf-4813-ab95-3e5a9f431543}})-based RRAM has shown a significant promise due to its CMOS compatibility at scaled nodes, allowing the fabrication of high-density synaptic storage for neuromorphic systems. The synaptic weights are programmed on RRAM cells as conductance. An RRAM cell can be programmed to a high-resistance state (HRS) or one of the low-resistance states (LRS).
Unfortunately, RRAM cells have limited read endurance, i.e., an RRAM cell can switch its state after performing a certain number of reads {{cite:20e0a6c98d326a67ed0f2f846d2cd0bf2116cf82}}.
To give an example, a single quasi-static read for 5000 ms or 5000 reads with 1-ms read time can lead to an abrupt change from HRS to LRS state in an RRAM cell {{cite:20e0a6c98d326a67ed0f2f846d2cd0bf2116cf82}}. To put in the context of neuromorphic computing, an RRAM cell can reliably propagate 5000 1-ms spikes before it becomes absolutely necessary to reprogram the state of the cell.
| i | 664ff5a48f76e62ed4a9c2bdd7be2140 |
The study of quantum thermodynamic machines assists us to interpret the behavior of the thermodynamic quantities in the quantum realm like work, efficiency, heat due to the non-classical features that occur in the quantum regime such as entanglement, quantum superposition, squeezing {{cite:b3549bcca3f614d8fe05083a4e2c1a2fa67e2b85}}, {{cite:65db4fca60993afb89a30469338c594a0acd6660}}, {{cite:6762d52d69ccbd85ee3d709ff3a1d86526bb301e}}. This type of thermodynamic machine has practical importance in the field of quantum computation and refrigeration in micro regimes {{cite:f4dc20fad7ea124bbf00974429602b4027344484}}. Coupled quantum systems, as the working medium for heat engines, have been studied widely in previous works {{cite:36d97632754604be72f3b5eb0f823f54b36f8f4a}}, {{cite:1de41b76ab417de5c7e6f0bb533a4a711598f8af}}. It is shown in the work {{cite:36d97632754604be72f3b5eb0f823f54b36f8f4a}}, {{cite:1de41b76ab417de5c7e6f0bb533a4a711598f8af}} that with appropriate coupling the efficiency of the system can be increased compared with the uncoupled one.
| i | afd4dea0671876bd1e16faf8abfe1bea |
We demonstrate the effectiveness of our proposed method on two benchmark datasets (CIFAR-10 and ImageNet) for the image classification task. Unlike previous NAS works {{cite:5ea6db0b1335645c37aa40151358779610df673a}}, {{cite:5996e918b6c4d9db2df621c1d3927e0636e932c8}} that first learn CNN blocks on CIFAR-10 under small-scale setting (e.g. fewer blocks), then transfer the learned block to ImageNet or CIFAR-10 under large-scale setting by repeatedly stacking it, we directly learn the architectures on the target task (either CIFAR-10 or ImageNet) and target hardware (GPU, CPU and mobile phone) while allowing each block to be specified.
{{table:eaefbbb9-48fe-45db-95ee-b72330445838}} | r | 6e83d71965fdf5baa4297d04481e5916 |
By Proposition REF , it suffices to consider the {{formula:e5b4bc96-bc92-4ec9-8d55-5088128d4b7b}} distribution. Let
{{formula:9f2937d8-205d-4c3e-9c9f-6dd94f61400a}} be its density function, i.e., {{formula:622117ef-703c-42cb-9f85-e7dccf3ae1d7}} as in (REF ).
Fix {{formula:49da6c01-d249-4e17-83e3-f151be03c627}} . To show that {{formula:334651a2-663f-4903-aa4b-f8a4260f787e}} is increasing at {{formula:96971229-886f-44de-8f48-fad4516f9ed1}} , it suffices to
show that the log density ratio of the left and right parts
{{formula:1bb2c5c9-5457-41f8-8e03-1c2a9be2dc12}}
for {{formula:b765aeb1-ff0c-46fc-b530-2afcb5704923}} , has exactly one positive critical point.
Indeed, observe that {{formula:f9d82caf-364c-40b5-9bf4-199aa1798fb2}} as {{formula:6cad8dbe-eb4b-4a2f-bf73-bf657eb0605f}} and that {{formula:fa0e3a65-8f13-4815-8eb5-a2305f28579e}} . Moreover,
(REF ) implies that {{formula:e2fb6a7b-1e41-4cd1-aa2b-229be565477b}} cannot be non-positive for all {{formula:02d21edd-f4d0-44e3-ab4c-b145039601af}} .
Therefore, if {{formula:27885c7d-c61a-492e-b095-4df5fbe796e1}} has a single positive critical point, then it must be a maximum,
and it follows that the conditions of Lemma REF are satisfied and so
{{formula:ed7dcb10-1030-4706-9b21-428289074148}} is increasing at {{formula:c89c88d8-633b-4c0b-8ac4-9fd4ba8aadc4}} .
To identify the positive critical points, observe that
{{formula:2952d5ea-8aa2-457b-adaf-5912adf920a7}}
so the critical points are solutions to the equation
{{formula:f761cff5-6edf-468f-8c7e-b9e3344974c9}}
Further simplification yields a quadratic equation in {{formula:7a531c61-66af-4374-a29d-025e841d7896}} ,
{{formula:bb7e97c8-9e14-4768-909e-d20139f14597}}
which has exactly one root in {{formula:54474339-8404-48c4-9b1f-50a1af4445a3}} if {{formula:2dd2f74e-1ffd-4cc5-ae21-24774a5f39d6}} .
This holds for arbitrary {{formula:a2dc191f-b401-4424-b510-f84c335eeb50}} 1>2/3{{formula:77bb0b61-37d4-4156-abfb-08802d2329ab}} 1>0{{formula:5059c560-978f-42ef-96b3-150a24589a2a}} 0=1/3{{formula:900033a7-ee02-4d4d-b31e-e3fa58f1a32e}} 1=12(1-erfc(12))-2
2.17{{formula:780684e5-fa6b-4f53-b67a-c134948823d8}}
Fix {{formula:ac37bf91-b4b9-43e8-a585-4ca090042124}} and let {{formula:4386f0a5-e34f-4fa0-8abd-9361a74256dd}} be the density function of {{formula:7c1ac1b2-83ff-45dc-bcf9-4c7dee67658b}} , i.e., {{formula:3d910f70-0b7a-4895-83e6-771eca400385}} as in (REF ).
Fix {{formula:792b9c8c-d735-4f52-bc0a-2ac0607471d4}} . There are two cases.
Case 1: {{formula:e6833596-7a9d-4599-9ea7-8dc4a104932a}} . Here, {{formula:b66094d2-eda3-4aa0-86db-6f950cca9c45}} is decreasing on its domain and so true positive skewness follows from
Proposition REF and Theorem REF .
Case 2: {{formula:68457426-407d-41a1-ac83-3b35304aa6ce}} . As in the proof of Theorem REF , to show that {{formula:1280f98e-e2a6-40cf-931e-e74372f9ae22}} is increasing at {{formula:4cc3b221-e9ee-4617-ab6e-c27d1d525d2b}} , it
suffices to show that the ratio of the left and right parts
{{formula:cb29f7cd-0c64-4e25-ab6e-76fb3151c321}}
for {{formula:708ec697-53ce-4937-9bfa-9d7592bb585c}} , has exactly one positive critical point, since {{formula:0a60d78a-b004-4025-9c5f-1c440dc4e361}} as {{formula:e6250ae7-5a10-49f3-b2b5-451be1cf9dd0}} and {{formula:efc494a6-da25-4565-9ae6-609e6030597e}} .
Observe that
{{formula:a20d3a47-82ab-45db-807c-ce4fa1d00498}}
so the critical points are solutions to the equation {{formula:43fc3662-ed0f-4620-81ca-3c2f777476e5}} , i.e.,
{{formula:cc466c25-91c5-4a4f-bc58-37d00dfe475e}}
There is exactly one positive critical point when {{formula:d6cf3820-f7ef-4f63-b15e-19297bf96c94}} . Since {{formula:3ea655ef-5767-4de0-8415-2fe30838e6d5}} has support on the positive half
line, then {{formula:23942906-024b-492a-b75f-338d20be45ad}} is always non-negative by Proposition REF . Thus {{formula:c7a96c66-a092-45d3-b57e-ace4de1712a5}} is increasing at {{formula:cae66dd8-228c-4475-9b9c-a27019cc7323}} if
{{formula:13e5c6fe-7ca2-46d5-9b78-153d8bc55ad7}} . Now by Lemma REF , true positive skewness of {{formula:90066a6b-eb8a-4caf-a7d8-083ddee95f64}} follows if the median satisfies
{{formula:2fa25fc6-9924-4aaa-8b64-87d3747b7e95}} , but this inequality is well-known (see, e.g., Sen {{cite:fb31f29b96a056320611cd5396c92dcfa6dcd992}} and the references therein).
By Proposition REF , it suffices to consider the {{formula:466d6aac-5ce5-433b-b44e-72b85f359a8f}} distribution.
Let {{formula:f0626393-25b3-4dd4-b8b0-2049d27deb9f}} be its density function, i.e., {{formula:43f66c7d-ea37-4e48-8b1e-2278ba80dfcf}} as in (REF ).
It is well-known (see, e.g., {{cite:8fca04b3ff6379a63b6fa9876e7cd7ab5aefddef}}) that the {{formula:649a2e31-2897-4cde-af61-73ea6e966f88}} distribution
has finite moments of all orders and is unimodal with median and mode given by
{{formula:48f89fcd-0ec5-49fd-a075-b6b87db91b4b}}
Notice that {{formula:f084ee17-33aa-48bc-b948-36a132152e7b}} holds if and only if {{formula:ab76ffac-2a3a-4b9f-9cb4-aee0475636c4}} .
Since the Weibull distribution has support on the positive half-line, then the second
part of the theorem follows from Proposition REF . For the first part
of the theorem, the “only if” part follows from (REF ).
It remains to show the “if” part. For {{formula:57b8f0f9-d8f4-4480-bf78-3aa4b1f8ac1a}} , {{formula:206b8541-e182-453e-875d-001746e62323}} is strictly decreasing, so we are
done by Proposition REF .
Suppose {{formula:83592004-769a-400c-81a3-b2f3465ecd94}} and fix {{formula:030bf754-81e5-4d90-9374-7305860732a5}} . As in the proof of Theorem REF , to show that
{{formula:1e90fcdb-32cd-4ce4-b751-bb81f6c437cf}} is increasing at {{formula:55e4ce3e-7d59-46af-81fa-bc4d2686def4}} , it suffices to show that the log density ratio of the
left and right parts
{{formula:983d3de7-f709-454f-934b-7ffb835505bf}}
for {{formula:044f0667-9cb2-40b6-9a51-2a91cd5da515}} , has exactly one positive critical point, since {{formula:24f6afa7-a76e-496d-b144-d35ce1820352}} as
{{formula:bfd9b121-e4b8-4142-94a9-244acdc611ae}} and {{formula:251762c5-0801-40f9-8650-bc3e4107e726}} . Observe that
{{formula:69498b9a-2aac-4029-84e3-1a7c401ec5b2}}
so the critical points are solutions to the equation
{{formula:99f4e3f9-6082-4e28-ad37-114b880e981e}}
Since {{formula:e626e9d2-4571-4a9f-8fe6-5393448ddfb5}} , the binomial series for {{formula:4f0cf5a2-4942-40c4-b7dd-7899b82452d4}} and
{{formula:7574531c-942c-4f92-a864-28a8327372dc}} converge, hence
{{formula:61ec0b4c-5937-48d4-b63d-217b831a3049}}
in the notation of the generalized binomial coefficient. Substituting into (REF )
yields the equation
{{formula:aad1b926-dbc7-4465-80a5-64b5f0fa8e52}}
where
{{formula:d197f033-873a-41b3-9f9e-f330d2b32902}}
We analyze the sign changes of the coefficients in (REF ) to determine the number
of its positive roots by splitting into several cases for the value of {{formula:75f2db90-af4b-48b3-abe7-2996d45a3b84}} .
Case 1: {{formula:85988d99-842d-4dfd-85d7-f8816cf7c07f}} . Then {{formula:144e2413-995f-4929-8b8e-2c8e8f9a26cc}} are positive and {{formula:1fd6916d-dc4b-4f6c-8b7e-66fb0306e962}} are negative.
There are an even number (possibly zero, if {{formula:a106dd5b-fb8d-4baa-b959-a87e9e1ebd2b}} ) of negative factors in {{formula:d5c174b3-8a03-4f2a-8c2c-00c8a466a136}} ,
so it is positive. Also note that {{formula:96a0fa10-6e90-4285-befc-d95702768453}} for all {{formula:16f73b52-d093-4cc8-99ed-f4f04ef34be8}} , hence (REF ) is
negative for all {{formula:6609401a-fd0b-4edb-8dee-514b124773d0}} .
For the series expression {{formula:9560c40c-1fcb-4a59-bf5a-3da1c13011d5}} , we have shown that {{formula:2bf9469a-f40e-41d4-8a04-d9b11dc4ce1f}} holds for
non-zero even {{formula:e0b05235-1b95-4b8a-bc8b-2d9c3994ff4a}} and {{formula:5b80d28c-8738-467a-9610-a9bd3dddbe80}} holds for odd {{formula:94a92b93-4384-4b6d-add7-6bf0f7c2b8f3}} . By Descartes' rule of signs for infinite series,
{{formula:2c4fd6f3-a581-4a0c-85b2-82995b6b7baf}} has no positive real roots if {{formula:351c5d29-41aa-4695-815b-35f85fae609b}} and has at most one positive real root if {{formula:e0ded2db-1b20-435d-afab-df3ddfffe13f}} .
Suppose {{formula:ac9da72c-e2d3-445e-9457-488f14c9d20b}} such that {{formula:7ba3192b-9fd6-4103-b958-773ae31f1619}} has no positive real roots. By extension, {{formula:5251a429-a14c-401a-8b5c-0ed1986a1d43}} has no positive
real roots, so {{formula:8eed81b6-c866-4a20-9fe7-e21c1d24212d}} has no positive extrema and is strictly monotonic on {{formula:56a1e51f-1d2f-40d8-8059-e098dde55795}} . Since
{{formula:35395e69-3bd4-436c-8b8f-73043c701d86}} and {{formula:dbe3978d-66cb-4d41-97c2-8eaecfbfd0e7}} , then {{formula:75ee3cd8-ea65-4b55-90de-e1479230ced2}} is strictly negative on {{formula:e2f19923-5e70-4bf1-addf-e8864c442e8c}} ,
hence {{formula:0d720e25-0ac9-4db1-9ee9-0afd9075a577}} for all {{formula:53d49ddc-bb18-40e7-8a44-fa4fe1ae9c18}} . By monotonicity of the integral, this
contradicts (REF ). Thus {{formula:e87ec5ce-a86f-421b-9fa7-12be71e5f9a9}} holds, and {{formula:58ecaf72-8901-4181-b8f4-4ff43f1e3e88}} has exactly one positive root
in {{formula:9a423a2d-5377-45ba-a955-f90a35f19565}} , which implies that {{formula:491fcee3-475d-4424-8f8c-27260df8e8d2}} is increasing at {{formula:e689fd3f-d78b-414c-8fc2-94db92662e1b}} .
This holds for
arbitrary {{formula:e32039ab-2eb9-4190-bcdb-dc38b3bb9bf5}} , so {{formula:fa9faf15-dd15-4925-8ff7-c1ea5d211fec}} is increasing on {{formula:d431a5ab-3172-41fe-a628-1ca7bf7568f1}} . Moreover, since {{formula:b5bee0a4-2714-457b-8046-9a6a2625443a}} holds, then {{formula:db340c14-0e15-4d9c-997c-55a3b6c953d8}} holds for every
{{formula:99b43c6b-018e-4196-9f48-e4131c9c08a2}} , and we are done.
Case 2: {{formula:ade5bf9c-9ff9-449c-8b9e-ffff72ba3ac5}} . The inequality {{formula:8a39f689-5f46-4cf9-90c1-728e28901c59}} still holds for all {{formula:6477b5a4-c1b1-44db-9f48-8bcd96f4592b}} , but we now have
{{formula:31a9d610-0356-48de-9beb-e827f267215b}} and {{formula:df7923ba-ca0b-470c-a6bd-626c3138dfea}} . If {{formula:6505d5a2-e880-4451-a2ee-a1c35ad96f7d}} , then there are an odd number
of negative factors in {{formula:b9153992-6497-44e8-9ff1-96bf9e565da8}} , so the numerator of (REF ) is positive.
If {{formula:ac205e02-5ec3-4d5b-b49f-3b391c9e93c9}} , then the numerator is simply {{formula:85b52182-c572-4e31-982f-be3889f90c6f}} .
Thus our coefficients in {{formula:cd5da964-0425-488d-a0e5-d8ce7179abf5}} are positive for even {{formula:d1a060fc-d026-4de1-8ed1-afe4bfc4f5c7}} and zero for
odd {{formula:90f85ceb-3a77-434e-b517-60bcd01d3711}} , with {{formula:e29f36d4-fefe-4877-94b3-8976cca85c7b}} . By Descartes' rule of signs, {{formula:fb1051d7-b72c-426d-9006-d026237cdcc9}} has at most two positive real roots if
{{formula:475bc118-4dcb-495b-8886-0e5e621f5e29}} and at most one if {{formula:15bd7fcb-7a30-42ee-a2c6-fcb2a7b89e14}} .
Suppose {{formula:a1dac3fe-6ef6-4226-99b3-01a3e78af15f}} . If {{formula:07a47d5d-0cd8-4799-9093-4ca62f2c9002}} has a single positive root, then {{formula:eca53eb0-4bd4-4e16-950f-0514da9a352a}} . But
by continuity this limit tends to {{formula:483187cd-0e53-4a5f-acd2-10d5c9802ae1}} . If instead {{formula:59a9d2ef-56f5-4200-89ed-b48a1838f342}} has no positive real roots,
then by the same argument above we contradict (REF ). Thus {{formula:fb42f25b-63de-4780-bab5-545a53065d52}} . In this
case, if {{formula:e7a5e57c-89b8-4dca-ba00-2e649c324861}} has zero or two positive roots, then again {{formula:65549a96-9890-47ce-85bc-a771543373af}} and we reach a
contradiction. Hence {{formula:0748f123-c8ca-4f81-aeb6-8f7f22f1198d}} has exactly one positive real root. We may now conclude in the same
fashion as in Case 1.
Case 3: {{formula:0520c626-5e6d-407e-9b94-2ce845a47fea}} . We again look at the numerator of (REF ). If
{{formula:8c1b3f61-b121-4c3e-badc-dff5fafdec9a}} , the numerator is {{formula:1f3abf66-4b95-4321-922f-598ec5be41ee}} . If {{formula:1e9bc7e9-7fd4-429c-8dae-4aae5278cd98}} , the numerator is {{formula:fdfc61e4-7ee0-423c-97c7-4ac75cbf0ec1}} . If {{formula:645fa251-98ef-48dc-ae56-f2b466703933}} , then
{{formula:1d88f784-9280-4055-9250-ae6425af091a}} has positive factors {{formula:d0624625-2fad-4e5b-ac42-6a712d45f266}} , and an odd number of negative factors
{{formula:5ebae578-f288-4f21-a069-9a812e5512d9}} . Additionally, {{formula:d0afadf5-d5f4-43e1-9607-bda2a05914fe}} when {{formula:72876c4d-ca6b-4058-a142-433b1b5cfb1a}} , so the numerator of (REF ) is
positive.
Our coefficients in {{formula:50872f88-34ea-4098-b710-a26e1dfbff1e}} are zero for odd {{formula:ef00ddd4-b46b-44e5-9201-c8fd7bdf3bf8}} , positive for even
{{formula:28f1f308-9abe-4c6e-9b59-efdda163a88f}} , negative for {{formula:352daf49-407c-4fd8-bbf6-43acff248ca1}} , and positive for {{formula:34f87b89-926b-4171-b9de-c900ca26b5d2}} . This yields two sign changes and hence at most
two positive roots of {{formula:82468b5f-9581-4884-a694-a0dd2c5fd76f}} if {{formula:04f4d9f5-45c2-4e5a-bf3a-d727fa4d7e65}} . Recall from Case 1 that {{formula:aa79306e-b2be-471f-b125-799f19c5dc83}} holds if and only if
{{formula:078ae632-3e4b-421d-842d-4bbdea4a12d5}} holds. If the latter holds, then {{formula:2e0dc4fb-61ef-4a91-9f6c-4461c00075c7}} must have exactly one positive root since {{formula:ede93216-0187-4e57-a7c4-506a777a3684}}
is negative at the right limit of its domain, and so {{formula:58a06b2f-4078-48a3-9d9e-42368e6a4f29}} is increasing at {{formula:dc9a2d17-92c6-4148-b0c6-7a0588bacb91}} . Now by
Lemma REF , true positive skewness follows if {{formula:2663a596-3033-4043-8ea2-3a9942e7c301}} , but this holds
immediately from (REF ) and our assumption {{formula:91a172e0-b745-43c8-b52a-c19b1210eeef}} .
Case 4: {{formula:51e45bea-3f04-42d7-86be-5859fad69916}} . We can plug {{formula:c51b0256-2efd-4879-aec0-62ea374dfb10}} directly into (REF ) to obtain the roots
{{formula:ae7b32a5-d8dc-4cce-8db7-35158f026dbd}}
The argument in Case 1 can be adapted to show that {{formula:199d022d-eb96-41aa-9535-b23781abfccb}} holds for every {{formula:03507bdd-3266-4e40-a1eb-9646913870d2}} ,
hence exactly one positive root exists.
Case 5: {{formula:42907b34-8af1-40f3-8f58-c337e3a2024c}} . Similarly, we plug {{formula:d7375850-d52f-4113-95df-258212645e0c}} into (REF ) and obtain the roots
{{formula:dd753b67-1e3e-4045-ba98-0a5d545258c6}}
Again, since {{formula:f9a03069-3f74-4024-97fb-eaddef2677c8}} holds for every {{formula:8e9e2a04-a231-4eae-87b7-94ca845c7e93}} , then exactly one positive root exists.
Recall that {{formula:1139855d-97fe-46b8-8707-6000121e1512}} are the density and distribution functions of the standard Gaussian
distribution. For simplicity, set
{{formula:8fef6950-2023-406d-a9fa-c0fa05a4910e}}
We show true positive skewness for positive shape parameter {{formula:e8e750e4-aa65-4d20-b078-2c7e1c3f7c06}} ; the proof for true
negative skewness for {{formula:bc6ba624-2f28-417c-92e4-269a081370bf}} is analogous, and clearly the skew-normal is simply the normal
distribution, which is symmetric, when {{formula:44fb4bc1-8ecc-42e4-b628-876f5102dce5}} . Thus, fix {{formula:8387e4b2-6888-4b6b-8d28-62008718e20b}} and let {{formula:0fdbfd33-5709-4a2c-af4e-541c344bd167}} be the
density function of {{formula:eadfbd53-f818-4be0-95ed-c4c10961b09e}} , i.e., {{formula:10a451a2-12e5-401a-9dc2-867999bc9e41}} as in
(REF ).
Fix {{formula:daadd09e-9c95-4fe7-a929-18f449b49e82}} . To show that {{formula:ffab1cfe-ef18-4b23-ba09-9cce28a7cef7}} is increasing at {{formula:34b8a749-5cbd-4815-bceb-5d26afc9b324}} , it suffices to show
that the log density ratio of the left and right parts
{{formula:d93fff36-e184-42bf-bb59-610d24e7b1ed}}
has exactly one positive root {{formula:d64f9f99-4ddc-41f7-a5c2-b102d4a0b58e}} , satisfying {{formula:03524982-061c-49fb-97b1-3ba32a551534}} for {{formula:1dfeeab0-1f76-4cd2-8e02-8a6c0f1a8107}} and {{formula:e4a58d39-4974-4600-bf52-717484074494}} for
{{formula:a8d0f1f5-0e39-4e5e-a083-a77948c67c22}} . This condition holds if the following four conditions can be verified:
{{formula:c6b615f9-3f60-45c7-9837-af03180a80e3}} ;
{{formula:1075c360-6852-4de2-88a0-37bf0488a8bd}} for all {{formula:446387c4-a213-4fab-96a0-bbab18b6126b}} .
{{formula:0f5cc5f7-e716-43cf-b672-880c545eaea7}} ; and
{{formula:edb8e832-81b6-43b5-8847-da305aae8ba1}} for some {{formula:df207141-a2e9-4ce2-afba-0113fb0b86ff}} .
Condition (I) holds trivially. To prove condition (II), define
{{formula:bf27c90f-0316-4d54-bb3a-db08205db398}}
hence
{{formula:b7b9389e-9780-40da-8bd0-ec248d12a27e}}
Condition (II) holds if and only if {{formula:cc4082a4-8781-4259-912f-822d25d82f80}} holds
for every {{formula:2ced7f94-ab7d-4814-854c-dd466bff0c7d}} . It suffices then to show that {{formula:f0838742-ff4f-4e3c-8805-458f415cda9d}} is monotonically increasing everywhere,
i.e., that {{formula:910c2471-7bc3-4352-8268-7688626b7dd8}} holds for every {{formula:588431c1-69b4-4cde-8b24-ada06321fc67}} .
The third logarithmic derivative of {{formula:2d804118-429f-4579-872e-8ee1b815e930}} is given by
{{formula:d3ed2dea-4cb8-4df2-a28c-8eb75ce8a6f6}}
One may compute directly
{{formula:862b23e3-d314-4670-980e-35f4df675312}}
so if {{formula:4014d1b8-986f-4717-9ce2-ecf70eea12cd}} has no zeroes, then continuity implies that {{formula:523615cd-c1cc-4970-ab39-2a9b5b10ca88}} is positive everywhere.
Define {{formula:782a6fd0-353f-4111-aec0-3955f04a20fa}} , noticing that {{formula:ef0ff72b-365a-4a27-89f9-067eada02cb5}} is strictly
positive. By (REF ), {{formula:4b8e6c1c-5740-48cf-bc29-dc94632f3bcc}} holds if and only if
{{formula:bc6b85f3-57f8-4f92-b068-463397d88fee}}
which, by the quadratic formula, holds exactly when
{{formula:9c667503-67a1-4436-99cb-89bb04228f25}}
In other words, the zeroes of {{formula:4ea423b9-1249-456c-88b3-45ea9e16a2a5}} are solutions to the equations
{{formula:aa73286c-efd9-4fc2-845c-95fc1a38767a}}
with special care needed for the asymptotic values {{formula:319f1fae-e62c-4a6f-94f5-e5fa805a6827}} for {{formula:129d5bbd-0095-4d7f-8498-ca47a6529d70}} . We handled these
cases in (REF ), so it suffices to consider {{formula:529a0418-f892-4508-bc57-ca91c9e3125d}} .
Observe that {{formula:1321d1ac-a4c1-43bc-a1e0-326ecdd0c8d4}} holds if {{formula:0a07b4dc-4cc5-4cee-bd0c-8d887a68c67e}} , so solutions to (REF ) must fall in
{{formula:fe2bdb15-564e-4ff7-a56c-5af4e7871073}} . If {{formula:603f9a90-0761-4fd8-a074-a3de7f993af5}} , then both {{formula:e73fb866-eb84-48e2-b2a0-0e2c42782deb}} and {{formula:acaa8919-fa30-42a1-82cc-c09958ef3163}} are positive and, one can easily
verify the inequalities {{formula:8d6792a9-bde3-4909-bc30-46679831d6d9}} . On the other hand, if {{formula:83d7f181-1606-40d8-9764-1d1edf5093c3}} , then {{formula:a285a3de-1b22-449b-8ecb-b90346ee3bad}}
whereas {{formula:6cf32480-29f8-449b-ba9f-c75b8ca3eda9}} . Therefore, it suffices to show that {{formula:bbea03c5-1fef-4633-b91c-7db0b7ab6116}} for every {{formula:358df1e6-9ad7-4596-9171-89fd7f02ebfb}} ,
{{formula:7ba104c1-d8cd-42ad-b364-a3ac1dfb4275}} .
The key observation is that {{formula:abb947c1-99e5-46d8-90a5-057814cc098d}} is simply the reflection of the Mills' ratio of the standard
Gaussian, i.e.,
{{formula:8bdfb166-0e00-4e9f-9f99-d02908d9f94d}}
for which very precise bounds in terms of elementary functions exist. In particular, we use
the known bound (see, e.g., {{cite:c3fa0bc01869f192b147d4a6976a995ebc1e5e1d}}, or {{cite:dc9e719fb3259704dadb551ee9b6572d79ea91f0}}, {{cite:5f80b8a01a388e17a2bd73dce4c6be34025f15d8}})
{{formula:1206dbb6-37bf-4792-a1b5-4be2577ee1f3}}
which yields
{{formula:3811ca39-6924-401c-98c5-347112e6ab04}}
We have now shown that (REF ) has no solutions and thus that {{formula:e18a642f-59b8-40a9-a3b3-fa01ff614f41}} has no zeroes,
proving condition (II).
Since {{formula:319474bd-e3e0-4097-9639-ac950e2e6d96}} cannot be non-positive on the entire half-line, as a consequence of
(REF ), then condition (II) implies condition (III). For condition (IV),
observe that {{formula:b7f8754a-aea8-4287-9550-28f748770dcf}} is decreasing and tends to infinity as
{{formula:461084c3-a61c-4789-a966-b15efd303211}} , so (REF ) implies that {{formula:c7a87357-96b6-4e44-92e3-938a80c0035a}} as {{formula:cddbd476-e89a-4965-9917-e243d1004d50}} .
It remains to show that {{formula:557bc0ad-34cc-489c-9b2b-90c68b0896ab}} for every {{formula:ad324055-726a-47da-b095-acdac0709738}} . Notice that
{{formula:aa9afbe7-34a5-484e-ab9a-1045ca82b9ee}} is increasing as a function of {{formula:ff78d74b-703b-4cbb-9e2e-602ebecb9179}} given
the decrease of {{formula:11c5cfc2-eead-41b1-bd80-acb6040bbc54}} . The mode {{formula:543b26af-884c-4527-a71d-f08181571b20}} is the unique value satisfying {{formula:41f804ef-26cd-4855-bebc-d3a3444f2e22}} , which
is equivalent to the equality {{formula:3e7893ef-290c-4eac-a333-f96969d989f1}} . Condition (III)
and the increase of {{formula:278991fa-67bf-4fa8-afc8-2f69aaedeb54}} now yield {{formula:5fb1efb7-b9aa-4b3b-b398-03cbcdf84e44}} for every
{{formula:4d30bdbb-dd9d-4c81-9528-10d04655689b}} .
Proofs of results for Section REF
To prove the first part of the theorem, fix {{formula:ccc593d4-1772-42d6-bc0d-5b39c484d578}} and define
{{formula:c8d2687c-288b-4a8c-8e2a-961f13fef2e3}}
{{formula:91e32dde-5416-4002-b3a7-ed58178c16d5}}
so that {{formula:12eee090-954a-45fa-89e7-e13ff8b5942f}} and {{formula:fd011760-ffe7-4c70-ad81-5e2725c6deed}} are the {{formula:822f916e-2aa8-4676-9609-02d7c215b94f}} -means of {{formula:57af28f5-6416-44da-b8fe-9fcb16ecf288}} and {{formula:2cf7462b-05ef-4d27-bf8b-c601b298bac3}} respectively by Definition
REF .
Let {{formula:a1444363-206a-46df-9220-cd1167824cb0}} be any sequence of numbers converging to {{formula:4a60d149-1980-44a7-940c-e78a5aa56144}} .
Observe that the random variable {{formula:72027e6f-d610-4cb5-a3f6-7be431e758e8}} has distribution function {{formula:0c3d7a1b-e445-4cea-8aff-abf392365e04}} .
Similarly, {{formula:098ab65b-c053-4465-ba79-b3764e0a764a}} has distribution function {{formula:ec1c2c03-3cff-439d-9386-44c46c85a713}} , so the uniform convergence
{{formula:6be773b0-c1d6-4fd1-aecf-fb6412c96aae}} implies the pointwise convergence {{formula:a183e92d-7faa-44dc-8b78-ad9bbdd86ee2}} . Therefore, {{formula:4c8973d7-e787-4e4b-ac84-d14098593db7}}
converges to {{formula:64823538-71d9-4c63-a901-fd7bb96a5ebc}} in distribution.
By Jensen's inequality, we have
{{formula:ab756deb-1eb6-4e9c-bb8f-1f5da9770458}}
where finiteness holds by the convergence of {{formula:2334d46a-7a81-48c4-8db8-76f0144f4cd2}} and by assumption on the {{formula:086c02d3-879a-43d7-a36e-c9fbffb9977c}} 's.
It follows from, e.g., {{cite:cb4c94bf2e0a1f322881e9b7dfe5657853c60faf}}, that
{{formula:3ee59d31-712f-4c52-88bd-7c22dfb4f5b7}}
Suppose that the {{formula:d37a8c4e-5a94-4b4c-8e17-293ee752c482}} 's are contained in a compact interval. Then every subsequence
{{formula:c036f153-53be-4998-8c09-c68c6a1e6d18}} has a limit point {{formula:73098f70-ac43-4f30-b53a-b4e76ec3bea6}} . Since {{formula:8888772c-cc4f-4a54-b5db-74a70221e9c7}} for all {{formula:be345942-dd5f-4513-a201-bec97409904b}} ,
then by taking {{formula:d6d50f0b-f7c4-4874-9137-8d053e13559f}} we obtain {{formula:439ed6e7-2eb5-443b-bb5a-41bac803aab2}} holds for every {{formula:9ae5ae75-c39c-445a-97fb-6ad0e8664642}} via
(REF ). Then {{formula:b2eaa694-3a3a-4233-82ff-5db9484a04f5}} minimizes {{formula:8eaf3521-6a4e-4b88-b1cd-2140d4dc6d12}} , and since the minimizer of {{formula:c001df6f-4109-477b-ad24-8ed3f861f141}} is unique
(see Definition REF ), we obtain {{formula:e2b1c880-d4e5-40e9-b45d-a3d8232f6739}} . Every subsequence of {{formula:b925149a-4b81-4d3c-ba15-a99b684fda98}} converges
to {{formula:8e63c271-ba81-472c-a636-739e1aba8075}} , hence {{formula:01397ad9-48de-4d4d-8ebe-b1f7b0371d46}} .
Consider {{formula:fcd534a1-1c9a-47b7-b7ac-e2e39161183d}} and {{formula:29e9d39b-9bbd-4634-8777-d9d9a63f0589}} as functions of {{formula:3bd4c098-c1f1-451a-8456-ac96a73fc318}} , so {{formula:5c621ab8-6ef7-492f-b08a-bb8aaad412c6}} converges pointwise to {{formula:bf335fc0-ba72-436c-a83d-b1a9ce0c2965}} . True
non-negative skewness of {{formula:e94c3dfb-31e8-4329-891a-23f3da0f6d12}} implies {{formula:c77b7cdd-54bd-4aa2-95fc-a07609e455c0}} is non-decreasing, and the pointwise limit of
monotone functions is monotone, hence {{formula:12b78f5b-d809-4f3b-9c62-0822156be164}} is non-decreasing.
It remains to show that the {{formula:ce2c09fc-d149-47eb-a1cf-d317ae7babcb}} 's are contained in a compact interval. Suppose otherwise,
so we have a subsequence {{formula:71b715ef-8a48-41bb-87d6-c3900a85eb31}} as {{formula:554ad5a6-8b0c-43c6-8398-e954265b6735}} . (The argument is similar if
{{formula:8bc19274-f111-40cd-859a-b0b1f968c3ab}} .) We have from (REF ) the pointwise convergence {{formula:1fe5beac-5525-4dc6-afcf-bdcb4a7a4cc8}} ,
so for fixed {{formula:fc3442b9-0892-4334-a008-84cade2c11ee}} , we may choose large enough {{formula:9bcb0f4f-4028-4647-a45e-a3d3028fe39f}} such that {{formula:afd36fcf-b4b5-47f4-8d14-69e675bd76d8}} and {{formula:40a747a3-d112-412d-a053-d24915394aa3}} for all {{formula:4564a680-6c76-47b6-b4c9-49b778fcbe94}} . Since {{formula:ad4b1d24-c855-4eec-973a-5d5487356fef}} is strictly convex with minimizer
{{formula:3d9afd68-702e-4493-99d3-f99d1c688623}} , then {{formula:d4d18496-1f61-4d9a-b2d4-b0dbd118a33b}} for all {{formula:80a9c291-0a4e-4a58-9a6b-82f4cbfd1fd5}} .
Clearly {{formula:a9c08ae4-f1b6-4c2a-a30c-e944dcbfbda3}} as {{formula:5c94529b-05f0-4c7f-81ab-050a0d0b5c94}} , so choose {{formula:f5067f65-2ce7-4fc8-b7e1-81b2219c7581}} large enough such that
{{formula:5cda9b29-a622-4c72-abbd-7aca1c022b59}} . For any {{formula:578a8e6b-7cea-4256-af60-fb8bf7c2c6ba}} , there exists {{formula:856a29fc-97a1-4da6-be9d-88328480a877}} large enough such that
{{formula:edf209b6-598f-4227-bf95-c1c0d88f7549}} and {{formula:ae837a40-9063-4a4c-a58f-592a8f2f049c}} such that {{formula:a2970361-ed22-45c5-9f94-6383a21ca07e}} as shown above.
This contradicts the pointwise convergence {{formula:55fac69e-896d-4d5d-a4af-60c0c30c4d20}} , and so concludes the proof of
the first part.
The second part of the theorem now easily follows from the fact that {{formula:003544da-aadc-4122-a1f0-78f6ff72d41c}} .
Indeed, for {{formula:e8dc9d9d-199e-4adb-8c35-eff776b2667d}} , we have by assumption that {{formula:82e1fc30-0ed6-4771-8cc3-9fdec4abc8a4}} holds for every
{{formula:1a41a99a-b4f7-4468-a7b1-31eb4cab49dd}} , so taking limits yields {{formula:e4a839cf-a2bf-4563-a0f6-ed0daf468c59}} , hence {{formula:5b1659fc-f2aa-4ae6-9ec0-e6955ee9952a}} is strictly
increasing.
Proofs of results for Section REF
Let {{formula:4f17f7ea-4436-439c-be00-294ea3c11fb0}} be a continuous random variable with density function {{formula:8d4f28d1-6e4c-4577-95b5-32db64a15f41}} decreasing on its support {{formula:8338e36c-9512-4abf-bb25-393966b4140c}} for possibly infinite {{formula:465f2b2f-5576-49ab-aa9f-c6f675ce9cd2}} . Let {{formula:61ad4a5d-5a36-40b8-9d8e-ff461a2b154d}} where {{formula:8415354e-de65-4059-92fe-e710d8e54306}} is convex and strictly increasing on the support of {{formula:f5b36bc2-6156-489d-b528-476beaab7f6f}} . We define {{formula:86a8576b-9e48-48b7-86ed-1805c659795f}} . Note that {{formula:1c62f00d-dda9-41e5-a9e0-f9c267f2f5ec}} . We write {{formula:bc5472e1-0bfa-4d5f-90ba-21596efd8810}} fY{{formula:c4029594-f249-434c-ab5b-ce92f71feef8}} Y{{formula:093d0044-6e21-4459-9f29-55ac7339ae1b}} (0, ){{formula:654dc7a1-1c76-4170-9562-1fe9fca35741}} p. First note that as
{{formula:4f58bd06-252d-473a-b135-61c243d9e2c5}} increases, {{formula:a258d69c-ad2f-405e-be3a-72357d77df04}} approaches 0. Once {{formula:430cc493-4d28-4e75-939b-ae0a33db38dc}} , {{formula:c6f5ce81-b9f8-4fce-8e62-ca541c51f133}} . Thus, on the
interval {{formula:c8184413-7802-416a-a1e6-0c84f21c9ff7}} ,
{{formula:ec38c0a2-2dcf-46cf-89b9-c527dc7fe08e}}
On the interval {{formula:25037ebe-8d82-4138-a95e-0ce87a8508b0}} , we have {{formula:2daa119d-e0b0-4700-b881-19076b0f1eee}} , and {{formula:1207af99-7bb0-43ff-a0f5-c3e088b09b33}} . We expand {{formula:f10a84d4-2483-40f2-bace-43fcfe141c35}} :
{{formula:60fe3538-c754-4fc1-a578-a2c84fd7d740}}
Because {{formula:ebeffde6-df4f-44f1-8e54-e358814ef72c}} is an increasing function and {{formula:98d6ee42-5454-4575-93e5-6a1617708c0b}} is a decreasing function,
{{formula:2bc91763-27cf-48ce-821f-35039a0ccbdb}}
{{formula:cb1d25c5-568f-4118-9240-05bd34c77376}}
The function {{formula:941d3f1a-12ff-4a91-a6b5-218c4a1ca4df}} is convex and increasing, {{formula:19e9fda5-b195-470d-8e1e-6b8de82dfb93}} is concave, and therefore {{formula:8fda6917-749c-4ea7-9eec-9c36c3db0b9a}} is decreasing.
It follows that
{{formula:2a22d4bb-5f8a-48d1-8e41-1bbb53c17589}}
{{formula:c4a5cffd-dd7f-40ea-8372-7f77908a19fd}}
which implies that (REF ) is strictly positive on {{formula:231cdd1e-e6ab-401e-a821-d4cb79f46f3a}} . Since ()
is strictly positive on {{formula:5b3c5013-5448-4b56-b29d-de66d40cd65d}} and strictly negative on {{formula:2df536fe-56c6-4f81-9490-4d5da976d904}} , it changes
sign exactly once on the interval {{formula:1f579800-8f34-47c0-87ad-15675814b24b}} . This satisfies Lemma REF for
all {{formula:2a2c6813-b825-418d-bcc0-464d0b7fdb70}}
Let {{formula:7e6337af-ebaa-41b1-b74e-0a5817d17ff4}} denote the density function of {{formula:34748110-aae7-4ca3-aa20-3f9cf7df3b39}} . For {{formula:959caab4-547e-4a90-87e8-096ac9fde71e}} , define
{{formula:51a3e9af-9aaf-44b5-adb0-f1a8eefa1f95}}
and
{{formula:2956116a-3d6e-4bae-a588-4aae1598321e}}
If {{formula:30cfbd95-7ca0-4a18-88d8-0f330696efdc}} is non-empty, then its infimum
{{formula:2219499a-7f3a-424a-bcb8-67b6432f927f}}
exists and is non-negative. Note that if {{formula:daaf41f8-debb-41ff-9771-47bd565fef91}} is continuous, then so is {{formula:9aa70e3a-ed5f-44c9-9e08-2c5c1e563c47}} . Then {{formula:76c1eba2-e34b-4a8a-9887-0c77c1d846e4}}
is the preimage of an open set under {{formula:59e4a4dd-ce8b-4848-9634-ea2508052697}} , so {{formula:d4a91164-d2ae-4e21-ba79-13b6781a4888}} is also open and {{formula:50b43274-5aed-44fb-bb5a-ea127730b147}} .
Since {{formula:ab19bed5-6411-4078-a86b-3852c9b38855}} , then continuity implies
{{formula:6d9bcd4a-71e9-4da6-b2d5-45c48bd18d2d}}
Similarly, if {{formula:5989d8a0-dcc0-460c-abfc-9f4dc14cb5bc}} is differentiable, then so is {{formula:dff742e5-e233-4c41-9d24-40cc02fee9ae}} . Because {{formula:1a6a86dd-a2fc-49ea-ab4b-864794a877ca}} for all {{formula:89b9ad93-369f-4257-b26c-87d39ca77cb4}} , then
{{formula:0927a676-65b2-47e1-abe6-da731655b2a5}}
To prove the theorem, we will require several lemmas.
Lemma 3.1
The density {{formula:f9e78730-c565-44e6-90e2-64f7e7516997}} is convex on {{formula:7c74de9d-8ac1-4995-9e1e-d8605c842358}} , concave on
{{formula:fb791d62-4a61-42c4-90ca-77527232fbb6}} , strictly increasing on {{formula:90cf6083-827d-4954-bf4d-9e40f6011599}} , and strictly decreasing
on {{formula:3b68018f-98d8-4039-aab3-8937e81a1583}} .
Since {{formula:879e5a57-cb3e-416c-83c2-5673f3a03431}} is positive on its support, then it must be increasing on {{formula:98d406ec-4dbe-4d3a-986c-156d443e7a3e}} . If {{formula:e1149b58-ba44-4e79-855a-9a074eaba86d}}
is concave on {{formula:f3fcbd5c-d113-40f0-af5f-b187ffad5454}} , then it is convex on {{formula:470eb471-53e0-415f-894f-096f8854d5c6}} , contradicting
the fact that {{formula:d288e631-9dbc-4c1a-ba03-a64da2ae60c8}} . The convexity part of the lemma follows, and
since {{formula:cf1a30c8-2b15-4ef7-a645-ee9761444216}} and {{formula:0919f9f4-8758-49d4-9ba8-a4c62e5a436f}} is unimodal, then {{formula:a3bab967-d721-4882-98ec-a2735b0e2c6b}} is decreasing on {{formula:e0ffd3c6-a663-4f6b-8b01-f313045f3bbe}} .
Moreover, since {{formula:1feef65d-142c-4dc5-ba78-632d1f8a7bbe}} near 0 and {{formula:4e1844a8-052a-4164-a3b2-4246ddbfeb4c}} near infinity, then {{formula:62af3ef1-1a0e-4b18-a994-ac2862f5dfa0}} has an odd number of
zeroes. Integrability of {{formula:fd67fb6b-f617-4618-9720-ed31def3c46f}} implies {{formula:00f2f96b-0569-479c-a6a1-a6d68430275b}} , hence {{formula:06b72377-6152-4a2d-a3ed-8e3067685cbd}} is non-zero everywhere on
{{formula:112f783f-f1c4-4d1a-814d-2ed14a15e24f}} except at {{formula:899a053e-f0e6-4731-b4e9-eaca181fdac3}} , otherwise {{formula:c81b2667-774c-42df-a175-1160ab7c22a1}} would have at least three inflection points.
Since {{formula:93f0c110-a759-4625-989a-d132aff92641}} is {{formula:820c0ec5-212f-4ac4-b0ae-bd8c633b1209}} on {{formula:b79abaea-1b4c-4943-8252-12e0825d7b01}} , then (REF ) and (REF ) hold. Moreover,
{{formula:523ae989-b6f2-4850-abc0-5ebd3e7f9415}} is {{formula:5b58099d-c9b5-438d-a915-f895c60d8c81}} . Let
{{formula:b81fbbe9-8864-424a-853f-c804be599897}}
By assumption {{formula:135db01f-e0d0-43c1-ba74-02bb7b9601b8}} . We show that membership in {{formula:b8edc938-fad0-41d9-b9be-1a476d64fabd}} is sufficient to make (REF )
a strict inequality.
Lemma 3.2
{{formula:8ca0bcb8-ebf4-46e2-876e-869b9c472b2b}} for all {{formula:08e6f4bf-378b-4279-8245-96b8a8eb7383}} .
First we locate {{formula:f4d9664d-de30-4205-83b8-92cef8c2ebdf}} and {{formula:01ecbf64-7549-4607-a9d7-4cc195b01e50}} . Note that {{formula:0dc1e9a0-9010-4e7b-8bfe-ec45ee7696e4}} is increasing and {{formula:7b05ae49-fe34-4f23-bd90-a973d0db07f4}}
is decreasing for {{formula:7ee8a078-20ee-4bff-8e80-fc54a5d9f26f}} . The fact that {{formula:d6d17c61-7bee-4074-9edd-be121b1851cb}} is negative near 0 and {{formula:7ee21203-ba40-4952-840e-525c20d4ce56}}
implies {{formula:82998b24-3913-433a-9c3b-b08e9547628d}} . By assumption {{formula:c4c14b60-fdf0-477f-8c68-2ae10f3dff6f}} , so it follows that
{{formula:d652901a-b8dc-4801-b211-96e4a7e79d2e}}
On the other hand, we easily have {{formula:fae15f5c-8909-44d0-bc55-32fce678b44f}} , otherwise {{formula:a68de6e2-139f-4e2e-a96a-7ce0954e6869}} . If
{{formula:d1204ea5-7237-4f2d-9974-4683c5f7b1f0}} , then {{formula:cf695944-fdea-4bcc-879c-3a85e439973b}} by Lemma REF , again contradicting
{{formula:62f2504b-c356-4a2f-ac43-4e95c0786f7b}} . Thus
{{formula:7e100346-e705-47e0-9c0d-5a39019fc963}}
Now suppose for the sake of contradiction that {{formula:516ea0cc-8d73-4091-b2e1-3edac82da890}} . Either {{formula:c35e1f8f-d527-4e9c-8171-cba0b11fe7a6}} or
{{formula:53694a3e-318a-4152-8de8-c0da6a5d32ab}} by (REF ). If the latter holds, then {{formula:e86b5b0e-cf4f-40c2-86dd-67134b8f0a3f}} ,
and (REF ) implies {{formula:c9759eba-544f-4a3c-acc9-b4520d1bd2b7}} . Note that
{{formula:81830f43-021c-4d4d-8359-4388e315eae1}} , hence {{formula:05aaaf6d-ebc5-4048-a991-fdaeeb7a535e}} . By continuity of {{formula:81733838-6e49-431c-ae4c-58f4b22f475f}} and
(REF ), {{formula:b35aac10-b01e-4d3d-8891-e85f67d1c9cd}} is strictly convex and thus positive in a neighborhood of {{formula:084ce190-f8f5-432a-bd7f-f8d4157229c5}} (excluding
{{formula:7682b98c-16d3-4192-82f8-4820b2f5caab}} itself), contradicting the minimality of {{formula:e1efbea3-28e1-4eaf-8faa-166219f3a718}} in {{formula:ebd5a12a-7d1d-4e49-9f9c-a798d5e2f707}} .
Suppose instead that {{formula:4d600a1c-5be1-405c-8190-e8a9e552706b}} . Then from conditions (1) and (2) and equations
(REF ) and (REF ), we have the inequalities {{formula:c0ee9f53-2f5e-4d69-aa25-9823a042fa71}} and {{formula:70a80dee-59ba-4661-8dec-23385afc3eea}} . It follows that {{formula:2d2b8e6d-64b7-4b76-84c7-fa38b6672039}}
and we obtain a contradiction.
Next, we prove some properties of {{formula:12b870a5-3bf2-480a-9c6f-86ac6e7470d1}} .
Lemma 3.3
The map {{formula:412fa43b-407a-4f3b-b617-46b75298c04c}} is continuously differentiable in {{formula:95e5ede5-6991-4e0a-84c0-c2ecfc1c6dac}} .
Define {{formula:c5518d7c-310d-4d25-9970-89587dc924ea}} by {{formula:de1a615c-aed6-4935-8a77-8130c54446d0}} . By Proposition REF , {{formula:3a3d3f7a-a178-4b70-9d24-2afcf17b6e8e}} is
differentiable and has partial derivatives
{{formula:5a7b9dd8-b994-4d17-aa33-4e775a4b7969}}
{{formula:43c1c473-b319-46c0-99bf-3aa17d48677c}}
both of which are jointly continuous in {{formula:1f1ae519-8592-40ae-a5d6-212eea4221af}} and {{formula:cdedcc46-1d4a-46f3-9b5f-3d669d8cb02d}} . Thus {{formula:24e5f083-4062-4c33-8f1b-a0e9971ad9fb}} is continuously differentiable.
By Lemma REF , {{formula:8f8109cd-4b66-4bb3-9c56-6f06377aede9}} for all {{formula:b03d4293-c647-4c34-944e-ed28f4b9140b}} and so the
continuous differentiability of {{formula:32942388-09a9-43b2-8d60-725d3f8ffe86}} follows from (REF ) and the implicit
function theorem.
Lemma 3.4
For any {{formula:4b04f703-ad0a-4728-86c6-5073128d5919}} , {{formula:dd66e7eb-dddf-4e98-bb29-d3a0ab2d3d19}} and {{formula:9df7b2d3-6c17-44c9-acb4-7841c76686f7}} have the same sign, and
{{formula:36db019e-9593-44a1-ad25-e2cc81b49f1b}} if they are non-zero.
By (REF ), we have {{formula:3d8025bf-fb62-4324-9cf3-d737fbcd6556}} . The left side is differentiable in {{formula:fb70f7f4-e703-4b10-9556-c3e1a31fd6a5}}
by Lemma REF , so taking derivatives, we obtain
{{formula:9ff2c95f-e16e-4290-96cb-c5e958ceeea8}}
Rearranging yields
{{formula:f11acc0d-0e34-4fbe-a2b9-e128c511bf46}}
where the fraction is well-defined with positive denominator by Lemma REF . The
numerator is positive by Lemma REF , (REF ), and (REF ), so
{{formula:01285021-741d-415d-9344-71965f9130ad}} has the same sign as {{formula:385b6d07-eb7f-4103-9a0b-6b8edc8dc52e}} . If {{formula:27530310-9eb3-4682-b6a6-6d63c65b7979}} , then one can see immediately from (REF ) that {{formula:c06f0a37-b000-456c-a731-675c88c319d6}} .
The reverse also follows.
Our final lemma concerns a criterion for pointwise increasingness of {{formula:035af941-9b74-4b03-bddf-a323a3de4416}} .
Lemma 3.5
Fix {{formula:a4b21d8a-4b98-43e4-9794-1a46a8938bff}} . If
{{formula:323a2eee-5e72-4742-aa8d-ab5b950095e3}}
then {{formula:de629562-f128-480e-81ec-5213b4682102}} is increasing at {{formula:b85e1e51-3ab2-46fb-a580-85333356eae1}} .
Since {{formula:2498264e-2eda-40ec-a7d2-5e704857e24d}} , we may rearrange to obtain
{{formula:dd602e5f-88f9-4741-a231-35b8bbe396d5}}
Define the line
{{formula:8d19a973-0d1f-4960-9fa7-14434f7cffad}}
and note that
{{formula:3396f9e4-7442-45cc-9ce6-e7c3e9712a15}}
Note that (REF ) implies {{formula:01351044-7212-4064-960e-3d94a4cf108b}} , so by convexity of
{{formula:582ea9c2-81a8-4858-8082-f697687a53f5}} on {{formula:cc5d74db-55fb-4269-ab6b-d32b30f34cef}} , we have {{formula:ae9aa250-4555-40fd-ac5b-a899bc1f2fa0}} on {{formula:77876428-c8ea-4037-b48f-445ce90f0d82}} . Via integration we obtain
{{formula:7b12806e-8c75-464e-97c8-af24a12f0711}}
for {{formula:e5bf059d-6335-4b84-8891-796e54647c49}} . We now split into two cases to show {{formula:a0eec75b-6e1d-420d-b0d3-cecd7eb34248}} for {{formula:6264a8e2-5dd3-45e5-b29b-3d20d29f16fb}} .
Case 1. Suppose {{formula:944f73d0-65a2-47ea-8b40-5edbdff234e7}} . By (REF ) and the convexity of
{{formula:5bbe2ae6-2bb3-4219-906e-2912f70955a4}} on this interval, we have {{formula:8139e2b8-be3b-4100-a57c-3d1b5b907384}} for {{formula:d6305b75-2eb0-4566-86fe-5d78c4fb89cf}} , making use of
the fact that convexity is preserved under reflection and translation. A change of variables
gives the inequality {{formula:00d1a3ce-fd6e-4ac9-a9ea-706a73704919}} for {{formula:e7f4e64d-e3a7-4848-bcbb-46112e8a1f9d}} . Combining with
(REF ) yields {{formula:46770eb7-86b8-44e8-9f91-0235134fdfa0}} for {{formula:887ee497-acb5-4ad2-8bee-a103c0ede28f}} . If {{formula:f6ca17ed-3130-473a-a8c2-4eb494633fc2}} ,
then {{formula:52ee6ac1-57c7-43e5-a9bd-6ef2ca74a929}} and we are done.
Case 2. Suppose instead {{formula:8072e3da-afcf-4594-8589-073436fdae0e}} . If it happens that
{{formula:24a7e626-be8a-4068-8663-a8954b0c6c26}} for all {{formula:2837b44a-7e82-4e43-8a6e-66c0a4f5e3af}} , then the argument in Case 1 applies and we
are done. Otherwise, let
{{formula:707322b3-ec8f-4c91-8825-3d6b8e91c773}}
be the line such that
{{formula:3ca6cf3e-1ab6-402a-bba0-40ed4611ee80}}
and
{{formula:dfcafd6f-5c96-491b-b6ef-c1a9883e29e9}}
Lemma REF and convexity imply {{formula:070af90c-da40-47b3-be65-0aa6a410015d}} and thus
{{formula:5cf42799-97c6-434a-8358-a76976b12339}} for all {{formula:4381e646-8a8e-4a7c-917f-830ea59ee94a}} . By concavity of {{formula:1109b70b-09d1-4d3e-91d3-26cd65bb3176}} near {{formula:55325657-64a6-4e60-910c-8f03a52c6b8b}} , one
can easily see that {{formula:077e23ad-ec8b-4f7c-8794-310bb8c467c9}} for {{formula:81380645-d94d-4990-8a4b-bdea076335d0}} . It
remains to show {{formula:aacc5c02-8c2c-4e4c-ae3f-8dd2be331bb3}} for {{formula:42f2e8e6-dcfc-41bf-bc86-9481cae60e11}} .
We have by assumption that {{formula:b53b4f59-5ea3-4ed5-ae1e-957fe41ca27c}} for {{formula:61bbcb01-666e-4b1e-8888-b63e7de5403f}} in a right neighborhood of {{formula:ff0edafe-416f-4b45-93e7-db5bb236ce90}} .
We also have by (REF ) that {{formula:f6e394a6-414f-422a-bbf6-f563f285c1a8}} . It follows that
{{formula:935177be-d985-413d-b81e-ad4080ea0290}}
Substituting with {{formula:831c0291-0b05-4889-a6e4-bf19f07d0bd0}} with {{formula:e324a268-16f4-4c6e-a5b2-c983c264a8dd}} and rearranging yields
{{formula:cd9d52f8-6758-470e-85aa-87c0c081ef32}}
The left side is precisely the root of {{formula:9ff5e47e-71d0-4f8a-b6eb-fe152688fbb7}} whereas the right side is a root of {{formula:d817a271-ef7f-4a30-9ce1-9016c9d6adf9}} .
We showed previously that {{formula:aa74d631-6c33-4824-a2a0-aadba9d4b19b}} . Convexity of {{formula:45b416f3-8ce8-4b63-81c7-77b897b12e08}} on
{{formula:928cafd5-d7e6-4e8e-8578-f8a437fd4150}} implies {{formula:6582fe43-7bf0-431b-a739-c1840c44d453}} for {{formula:2b4167eb-d076-4baf-bbd6-9364f7af1259}} ,
and we are done.
We have now shown that, in general, {{formula:3dd90a95-7491-4a8f-8323-d1839bfadb29}} for {{formula:c157c8a4-1360-47be-8b05-2fdce7c02fd3}} . We also know by
definition of {{formula:fa3fc17e-7d89-46d4-997a-e0f22122e71b}} that {{formula:2ae83758-cf4e-4823-a160-4bb9456c1c53}} for {{formula:bdc22eff-84fb-4237-afb8-373aa1d4bfb0}} . Thus the conditions of Lemma
REF are satisfied and so {{formula:928e439e-36dd-4fcd-ba0f-1ed10fdeea0b}} is increasing at the point {{formula:4c440c3a-b07a-476a-801f-c299c3e70ee1}} .
To prove the main theorem, it suffices to show (REF ) for all {{formula:5e75240d-6d98-407a-849d-8667acd5e87e}}
and that {{formula:b1caf70c-d606-45d9-8156-f01c8323d4ff}} . If we have conditions (1) and (2) as they are written, then (REF )
holds for all {{formula:82133222-7056-436d-8af1-9839280d7f5f}} immediately by (REF ) and (REF ). Suppose instead we only
have the weaker condition as stated in Remark REF (b):
{{formula:86bff035-d65f-4d87-9b95-460d7de54370}}
Let {{formula:9a5c0310-bda7-400d-a8f8-cd4a02e8f0de}} on {{formula:c3749007-c651-484c-ac1e-9e28ae4ee243}} , so {{formula:0c2db9d5-7758-492e-8e3d-8662ce04f201}} is differentiable on its domain. Also note that {{formula:f4ea32de-5598-410c-b653-a8d704a7a13b}} by
(REF ). Suppose for some {{formula:0d528240-9168-4124-98ae-65e82e014743}} that {{formula:5b91adf4-f2f2-40b4-a351-c6ff111819ae}} . Then
{{formula:d4723fd5-d9ca-452e-b76e-d6c688db8ff7}}
i.e., (REF ) holds for {{formula:20713bdf-bdc7-4b72-a732-13b2437e7375}} and so {{formula:bcae4b0e-2448-49b3-b3e2-dd7fa75c1f26}} is increasing at {{formula:a42a2ab9-b9b0-4c5a-ba29-094db45aee6b}} . Lemma
REF implies {{formula:84041f01-390a-4330-8f32-663587c96f07}} is decreasing at {{formula:ee15582e-984c-4017-9c80-054a08ea7f37}} . It follows that for all {{formula:a92dbd27-78d4-427a-bda3-674c98429975}} , {{formula:3c20b9b0-50fc-4392-b1f3-2746c97e004b}} is
decreasing and so {{formula:a60ab94f-57e5-422d-b855-f220598e1b0b}} . In particular, {{formula:f6e9f1a4-a454-48b3-8195-6ae5bc102af2}} is increasing for all {{formula:9d9accb1-5ebc-41e4-b3dd-725b6c234377}} .
By assumption, {{formula:27e8ebee-1821-495e-a095-0e2667601c04}} and trivially {{formula:d5427f25-56fd-4c1b-98c1-94c9d5effa70}} . Since {{formula:db9a3c09-27b3-40cb-ae1e-9e03071213d8}} , then
{{formula:c9350971-8045-4d3f-9dd3-9fdf9b50546a}} is increasing for all {{formula:785ea413-53f5-4537-ac4d-56f14972bd15}} . Recall that {{formula:33eadc8f-9f2f-41fa-9419-86d15fefbe02}} if {{formula:886cad5e-79be-4b4e-854d-f3b372bf1e10}} .
Since {{formula:b59767f4-d606-472f-a493-1418b9333a0e}} is increasing at {{formula:d42698b8-644a-4b4b-861b-fd36a4122210}} and for all {{formula:d8792808-518a-497f-a3e7-db2c46005ea1}} , then {{formula:78c13306-7ae2-4673-b1b4-053880fd33a7}}
for all {{formula:1d22d45b-e42b-42c4-9fa0-14f53189c457}} , hence {{formula:621c9bbb-a735-41c0-b92c-131fb0c57b50}} . Clearly the argument
for {{formula:322fa77a-29d8-4dee-9428-2fdfca6aac4d}} still applies if we use the alternative definition for {{formula:98b4460f-bf95-4ddc-883d-16e0fc5b6633}} .
The proof directly extends to the case where {{formula:b3fb0a32-659f-4261-9345-a1ec9a106a5b}} only has a single root {{formula:e3939516-cb7e-4b2a-bbd7-f25efa348c0a}} by
setting {{formula:89aa15e4-4a06-4b4c-8282-37b70470c5af}} . In fact, conditions (1) and (2) are not necessary. Indeed, we use them
once in the proof of Lemma REF in the case {{formula:78412473-26e0-4da5-8f14-ebd69297d35c}} , but this
is no longer relevant if {{formula:37cad00c-62cf-4d41-b92a-0bac095a5240}} has only one root. The only other time we use the conditions
is to prove that (REF ) holds for all {{formula:eed94e60-c5f5-4398-8d78-a8524048a8a5}} . However, if {{formula:b264dc76-584b-4bf2-ae94-5b96f94d0a64}} has only
one root then, (REF ) holds automatically. Indeed, note that {{formula:8802cb27-fcef-4736-bc55-2b73bef466eb}} is concave
and increasing near {{formula:0d90515c-61e6-4b63-9614-8173b604463c}} for all {{formula:a45ba3f9-06e2-4c96-9fc7-97922f4e2cb9}} , hence
{{formula:7be77440-374e-45c8-8e64-9a638481e369}}
Since {{formula:188d93e1-8430-4957-9d74-8312e3808610}} , then by substituting {{formula:870759ff-0707-4308-9b7d-50a91207ebe3}} , we have
{{formula:c4c93598-69af-4e3d-9bcf-a3f5bbe23ef7}}
The right side dominates {{formula:0c348c37-fddb-474d-a961-f7d09714e844}} as a consequence of Lemma REF , so we arrive at
(REF ). This proves the corollary.
Discussion
This work attempts to broaden our understanding of true skewness by demonstrating the true skewness of several additional distributions and by establishing simpler criteria for which one can conclude a distribution is truly skewed. Theorems REF –REF , which establish the parameter regions for which the Lévy, chi-squared, Weibull, and skew-normal distributions are truly positively skewed, all rely crucially on showing that the conditions of Lemma REF hold for every relevant value of {{formula:34f45147-0027-4e21-a782-f596c7c46081}} . This is done by analyzing what we have called the “log density ratio of the left and right parts,” which for given {{formula:943fd088-1c5c-4e8b-84c8-21d17b59bec0}} is
{{formula:c6acc048-1da7-4dd4-9273-091f5b9d4c07}}
where {{formula:5f6286a3-ab76-464a-9010-0561ab252848}} is the density function of the distribution in question. The conditions of Lemma REF hold if one can show that {{formula:733cefd8-b765-476e-aa0f-5e7776f863af}} has exactly one positive root {{formula:e7f36ecb-a852-483f-bedd-f576c6d8d5e2}} , satisfying {{formula:ac140be5-3cce-4f93-8264-9421a98c993a}} for {{formula:bf9da058-7ba8-4f42-a999-e32d073f2147}} and {{formula:35b6a0a1-3c49-45d5-aa93-6040edfc0ce2}} for {{formula:72c5507f-b41e-4ec1-96f8-0fa184300cc9}} . In Theorems REF , REF , and REF , this can be shown in quite a straightforward manner simply by finding the critical points of {{formula:1f476ed2-630a-4cf9-80b0-2790283e0ff5}} . Such an approach may likely be applied to other visibly skewed distributions for which a closed-form expression for its density function exists.
However, for the skew-normal distribution, Theorem REF , the absence of a closed-form expression for its density function makes the computation of the critical points of {{formula:fc29bba4-865e-4e93-b0ed-51e357ea12bf}} intractible. Instead, we exploit the fact that the log derivative of a distribution function {{formula:09fb0c2f-0931-4963-8cc0-b635e582e79a}} is simply {{formula:c20b71c1-f1f2-491a-b208-dbd0ecdc6ec1}} , where {{formula:159ab2d9-e840-4784-b6a3-c8ba296bf1b8}} is the corresponding density function. The ratio {{formula:0dfc3334-7123-4171-8a5c-60cbaccc7351}} is the reciprocal of what is commonly referred to as the hazard rate of the distribution {{formula:68ae4607-9ce0-4025-9bdc-386deaf0e4ae}} , which has been studied with some detail for most well-known distributions. In the proof of Theorem REF , we encounter the reciprocal of the hazard rate of the standard Gaussian, which goes by the special name of the Mills' ratio. The Mills' ratio, as one may expect, has been very well-studied, allowing us to employ a closed-form expression, which very tightly bounds the quantities we care about, to prove the required properties of the skew-normal distribution's log density ratio of the left and right parts.
Indeed, the proof of Theorem REF identifies a key yet unsurprising connection between the notion of true skewness, which is fundamentally a comparison of the rate of decay of the two sides of a distribution, and the hazard rate. It suggests that the notion of true skewness is an accurate reflection, and in a much more rigorous fashion than Pearson's coefficients of skewness, of the essence of what we imagine skewness to mean. More practically, the method by which we prove Theorem REF should work for a larger variety of distributions; a reasonable first direction would be skewed versions of other symmetric distributions, as introduced in, for example, Chapter 1 of {{cite:45863800d488a970b8acd4673bd4020c11ae50c5}}.
The remainder of this section discusses sums and products of truly skewed random variables; examines whether true skewness extends to the discrete case; and provides an interpretation of true skewness for multivariate distributions.
Sums and products of truly skewed random variables
The goal of this paper and this discussion is to present tools for establishing a larger class of distributions which are truly skewed. A natural question is whether or not true skewness is preserved under certain “transformations” of truly skewed random variables.
Theorem REF gives an affirmative answer to this question, but with limitations: transforming a random variable with a strictly decaying distribution by an increasing convex function preserves its true positive skewness. One might also expect that sums of truly positively skewed random variables preserve true positive skewness, but the rather strong requirement that the {{formula:0def1d0a-7bae-4299-867d-26feaf0e147e}} -means of a truly positively skewed random variable be strictly increasing allows one to construct rather straightforward counterexamples. Indeed, we may even take the two summands to be identically distributed. For {{formula:97cd4489-57dd-420c-a197-0ec914be3b80}} , let {{formula:876c09cf-e311-49a4-83c4-51c09c9b3af4}} and {{formula:8b78a3b2-40de-458b-b571-f96bff95d7b1}} be i.i.d. with density
{{formula:e4e36630-5dea-4cea-a8e7-e1371c03ae46}}
Since {{formula:07bb49b9-796b-4276-a0c0-5a4e4aa46781}} is decreasing on its support, {{formula:74ade599-8a79-4676-a525-3f27648b6bcb}} and {{formula:b3c645bf-f6ea-4957-9988-8f0359f7ff80}} are truly positively skewed by Proposition REF . One can easily compute the density function {{formula:1adb330d-0466-4a25-9217-625c7d925f7a}} of {{formula:2c88419b-e115-419a-97be-bedd8e30df5f}} , the convolution of {{formula:65c9eb50-d163-416c-abb2-ec10f6a2c9f9}} with itself and see that, depending on the value of {{formula:90e6bc6f-40c0-4661-8505-c1125e162b34}} , the {{formula:0951e440-b62c-4040-ac1c-58d47b3a60d9}} -means of {{formula:12fece59-eafd-430c-aa90-2f204bb1187a}} are not always increasing. For instance, for {{formula:715d27dd-cfb9-4661-960f-99a551e4e336}} , computing numerically the median {{formula:b5799c2c-8162-441c-8f7d-f06bc39b76c7}} and the integrals in (REF ) yields {{formula:23ab4531-6d10-4a5b-a712-40dc863dfe2a}} and
{{formula:68dcdf51-68a6-4e3b-8c77-e5e42dc03318}}
which implies that {{formula:b54e4dd5-c22f-4989-b21b-f3b538fba866}} is decreasing at {{formula:f7fea0bb-94aa-41ec-a9e3-7eff374c5eed}} , and so {{formula:2784924d-b431-4b66-a3d7-6b18f4007504}} is not truly positively skewed. Numerical computations were carried out using Wolfram Mathematica, version 12.3.0.0.
On the other hand, it may be more fruitful to start with random variables with monotone density functions. One class of truly positively skewed densities that is closed under summation is the class of “decreasing linear densities,” of the form
{{formula:3759d5a5-3b1a-4f4c-874d-106c0acd828b}}
for arbitrary {{formula:759387c7-57c0-48f8-976f-9b8fef585521}} .
True skewness of discrete distributions
In the discrete case, even the simplest sums of truly skewed random variables fail to remain truly skewed. Take {{formula:b4766dcc-75f5-459b-806c-62294eae92ee}} and {{formula:e1245393-15ec-478a-b06c-a16f1d11f823}} to be independent {{formula:0ecf86ea-daff-428e-a1d5-0ab686e7de9b}} random variables; one can show that these are truly positively skewed quite easily. Their sum is {{formula:4733e9a7-a600-47cd-8726-f419fe041432}} , which has median 1 and mean 2/3, hence {{formula:d9bc04ad-4793-4af1-b94b-77c5f10279bc}} , and true positive skewness fails to hold. Indeed, the notion of true skewness is finnicky for discrete distributions, at least in the way it has been formulated here and in {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}}: this could be due to the interpretation of discrete distributions as limits of successfully sharper (continuous) bump functions, in which case distributions like the {{formula:8ffb4a2a-8579-4be5-bbfc-542e43487310}} are no longer unimodal but in fact have {{formula:6ae4cb25-3e02-4f97-9738-1cb396d89e09}} modes.
The proof techniques used in this paper often involve taking the log of a ratio of the density function
of a distribution. In particular, the logarithm of a product of two entities is equal to the sum of the individual
logarithms of those entities. For this reason, it seems natural to examine how true skewness behaves under
the products of random variables and whether it is preserved. In particular, it seems that taking the logarithm
of the density of product of two random variables would yield a sum which could then imply true skewness
depending on how one defines the two random variables.
True skewness of continuous multivariate distributions
The definition of Fréchet {{formula:36b4586d-6770-4c02-9110-760ac763993d}} -mean (Def. REF ) extends naturally to multivariate distributions as follows.
Let {{formula:565d5d6a-eb88-427d-be31-f22d3ff2c102}} be a random vector. The {{formula:517b3c18-08b5-4ee9-b54b-d27120c823de}} -mean {{formula:59d909b5-8e93-4630-9470-381570511729}} of {{formula:20038ec5-8390-4378-84a9-d70b0e80fa83}} is defined
{{formula:95a32e54-8c52-4848-921a-9a090d6edec3}}
where {{formula:7a7326e8-e07f-475a-a4d1-850caa376a74}} is the usual Euclidean norm.
In the univariate setting, true skewness corresponds to the sign of {{formula:5762f539-59c8-4624-bb83-f8a7d09450b7}} representing the direction of trajectory {{formula:97ee8ebd-f7af-45e3-863c-d630732595f3}} .
Following {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}}, we adjust true skewness accordingly. We let
{{formula:43a128af-1f85-41a2-b99a-5a1c8692b8f9}}
denote the unit tangent vector for trajectory of {{formula:fce05911-4ee1-4045-8954-f8429946cecc}} in {{formula:e07a9358-5d57-44f5-942f-67ca3b0797a8}} . We will illustrate true skewness in a multivariate setting by means of an example.
It was conjectured in {{cite:d844093569c5dbd2b703c89a51e0f3ae87da50f5}} that the limiting direction vector {{formula:714af971-17e7-4173-b0f8-3852152a52f1}} with {{formula:5258183b-836b-4049-a894-1ea9e2e9e6c5}} increasing to the rightmost bound in its domain {{formula:96845527-7a1e-4fed-ac0e-9f6d2b373026}} may be interpreted as the “direction” of true skewness.
As an example, considered a multivariate skew normal distribution defined in {{cite:4aa5f944ba4e9c4316a97e9303d539c0705b00fc}}, the probability density function of a multivariate skew-normal random vector {{formula:189549c5-b6e2-409e-877e-c786252f9ffa}} is
{{formula:654e4a46-343e-421e-a974-6dcdf10cb93e}}
where {{formula:093f19ad-0fc2-40fe-a922-de581b5041b2}} is the density function of the {{formula:c764792f-47a6-49ff-ae80-33a4535a3ba2}} -variate normal distribution with mean {{formula:b24bc658-a11d-48fb-9dc7-26e050e04828}} and covariance matrix {{formula:a0a608b3-1411-4e3a-8ea4-3f782fa9c6cf}} , and {{formula:e99c25f4-6c05-48e1-a2c2-73134c6ace15}} is the cumulative distribution function of the univariate standard normal distribution. Taking {{formula:264b1e18-2f33-4067-861a-b9f0cab31894}} recovers the standard multivariate normal distribution with density {{formula:473cdbc4-946e-43ff-abec-b3ab8da6201a}} . We refer to {{formula:5bb263eb-9d61-4947-91bb-1b4af5f0cfb9}} as the skewness parameter vector.
{{figure:a7fab1da-4f5d-453b-a446-03f6f99067d9}}The arrows plotted in Figure REF motivate the interpretation of multivariate true skewness via direction vectors {{formula:5cd4c9e1-ab30-4d73-8cbc-d53dfaadefa2}} , which are naturally co-linear with the skewness parameter vector {{formula:8e77e376-fd04-4695-8426-b6cdb48e8cc5}} .
Acknowledgments
The majority of this research was done as a part of a Research Experience for Undergraduates (REU) program at the Oregon State University Department of Mathematics funded by NSF grant DMS-1757995. We would like to thank Javier Rojo of Indiana University for his helpful feedback on the concepts considered in this paper. We would also like to acknowledge professor Holly Swisher of Oregon State University who as the director of the REU program provided invaluable perspective throughout the summer.
| r | 4e61e2521a99c534183512f486a55b04 |
A key feature of the mutual invasibility criterion is that coexistence is determined by the signs of invasion growth rates. For many classes of multispecies models, positive invasion growth rates of at least one missing species from each subcommunity is a necessary condition for permanence to persist under small structural perturbations, i.e., robust permanence {{cite:7efc4ac453109e469e77299dd1ff489205d7e49e}}, {{cite:3ac14b4d7bb610f53303ec335a8ec9e5d47f2223}}. However, it need not be sufficient as in the case of three competing species exhibiting a rock-paper-scissor dynamic {{cite:e471e38a74c7af1610f68e97a25f81b9dc4f6dca}}. In this case, all single species equilibria can be invaded by a missing species but coexistence depends on quantitative information about the invasion growth rates at these equilibria {{cite:d903064dd0721494f00228dfed12a700666b66fb}}, {{cite:4bf23b5e4605ce4bf4932cbf10c0958bcd1bdd13}}, {{cite:3ac14b4d7bb610f53303ec335a8ec9e5d47f2223}}, {{cite:2e6cef5eda6305951593341e188f7aca682e03ff}}. This raises the question, when is it sufficient to know the sign structure of the invasion growth rates? Are rock-paper-scissor type dynamics the main barrier to qualitative conditions for permanence?
| i | d91fa279e735d0b04608cea80a004fb7 |
In our previous work, we have showed that ResUNet++ outperforms the SOTA UNet {{cite:bc6ae5747ab9bcb8247ed445e488cae0ac167687}} and ResUNet {{cite:84204fbe3bb617e03386072b93e77312fa80135a}} models trained on Kvasir-SEG and CVC-ClinicDB dataset{{cite:ba6765f6ec181067da79f60c43b4e6f324c9c6bb}}. In this work, we aim to improve the results of ResUNet++ by utilizing further hyperparameter optimization, CRF and TTA. In this section, we present and compare the results of ResUNet++ with CRF, TTA, and both approaches combined on the same dataset, mixed dataset, and cross-dataset. Although a direct comparison of approaches from the literature is difficult due to different testing mechanisms used by various authors, we nonetheless compare the results with the recent work for the evaluation.
{{table:9f167ce6-9def-4f58-8780-7b3152a6bc11}} | r | e3df6d49624c3c927902c42357724371 |
The advantage of all these representations lies in the fact that multiple Feynman parameter integrals are reduced
to much lower dimensional infinite sum representations, which are one–fold in the case of generalized
hypergeometric
functions, two–fold e.g. for Appell and Horn functions
{{cite:a172f8bd50be6ab893548b79097c16359bc1223e}}, {{cite:c9e78f9de2910281d55075d39a7959ebc120176a}}, {{cite:118fd396269c311e5e14f18d49aec35e8e9f5563}}, {{cite:d46b82084a4647fc2073e6c6eb23f1265c650c75}}, {{cite:0dd8ca06013c3054d299b1ca5152b95f0ed50459}},
three–fold for the Srivastava functions
{{cite:4a922e96738f5a10f8bf95811c80760fb6c35a2b}} and further given by multi–sum Lauricella-type functions {{cite:e1b21fb180a19267ecdf3c3091f0cfe8c7a6b5d0}} in more
involved cases, with an
early application in {{cite:66cdc712cd6afecd233d3876000d4e18cb92b432}}.
| i | d772a115f2e5a1fa464ca357d4a1d2ec |
In existing literature, discussion on the choice of good landmark points that work for indefinite kernels has been scarce.
In {{cite:ab59a1f80b2edc8af8df167f51eddc487d97a7e2}}, {{cite:bf2880820b4b40b3a872eb2ac52eeecbf59c9cb7}}, uniform sampling {{cite:a3ac8532f1b355c1fd0a3e2e778cccb6167bdab4}} is used to select landmark points for indefinite kernels.
{{cite:ca1be216767610558716e76a78a6177176242af1}} proposed to first use uniform sampling to obtain a Nyström approximation and then apply leverage score method to the Nyström approximation to select landmark points.
However, it is not clear how well the original Nyström method {{cite:a3ac8532f1b355c1fd0a3e2e778cccb6167bdab4}} performs for indefinite kernels in general and how different choices of landmark points affect the approximation accuracy for indefinite kernels.
Moreover, a theoretical study of landmark selection for indefinite kernels is lacking.
We will show that, though Nyström approximation can be used for indefinite kernels, there are a lot more numerical challenges such as numerical instability, as compared to SPSD kernels.
For indefinite kernels, landmark points must be chosen judiciously to prevent a poor Nyström approximation.
| m | fe2318e33b2363042d183c19d303be85 |
Existing literature has been focusing on extending network capacity to achieve state-of-the-art (SOTA) transcription accuracy.
This includes fully convolutional neural networks {{cite:11cb3e59c2d910926663ce3f491b4dc405b01af8}}, hybrid convolutional and recurrent neural networks {{cite:63a93cb530b57f793376bbae142921cf5e678097}}, and convolutional sequence-to-sequence models {{cite:63a93cb530b57f793376bbae142921cf5e678097}}.
In parallel to increasing model complexity, incorporating onset {{cite:9c8ec59b36eca1d84cbbd0e3ed3335ff24f66341}} and offset {{cite:11eca01cc9cf809c6e08684a4c760545171cf771}}, {{cite:8dd94a24eac0a09890b25ab1f5f2647ed9757770}} detection, and leveraging a large dataset {{cite:2e46e9c3363c6450e7852717d9661bd75122f178}} for model training are also shown to improve the performance.
Despite the development, the components responsible for the superior performance remain unclear.
To the best of our knowledge, only Kelz et al. have attempted to explain AMT models using invertible neural networks {{cite:9769e493d1ce1158a4928932ac965169c9e8902f}}.
Although the work hints towards how the model possibly captures the notion of musical notes, it does not provide further insights on the relevant features for transcription.
| i | 8e4fba5839b0044611f9965960f7c132 |
A RFS is a random variable whose possible outcomes are sets with a finite number of unique elements. That means both the number of elements in the set and the elements themselves are random. Therefore, unlike random vectors, where both the number and the order of the elements are pre-fixed, RFS are invariant to the order of elements and can easily add or remove elements {{cite:8e9c577e5935cf75362c56524f94656aec9f89e9}}, {{cite:0d8dffb04278fb42ac87f5a22e196063ceebc6be}}, {{cite:cda0caa259474200f927719cd97acd801aba183b}}. These merits make the RFS particularly attractive to be used to model the unknown environment and the detected measurements in SLAM problems, since uncertainty in both the number of landmarks/measurements as well as their individual states can be inherently modeled. In RFS-based SLAM methods, different RFS, for example, the LMB RFS in {{cite:466ef25d04a3c94ed85a65092dccce46d7250047}}, {{cite:e9db89217678fc47ea33ffeff7644ffbdd894004}}, the GLMB RFS in {{cite:3a506deb0b7e62f4237dac5a1e02c3da06010d7b}}, {{cite:aea6aa8cd94c7d9b828a88508e245374d7fe7506}}, the PPP RFS in {{cite:06b8a74ecb4106f4d7375119e847281c009d5451}}, {{cite:0d8dffb04278fb42ac87f5a22e196063ceebc6be}} and the PMBM RFS in {{cite:5ba3221b26a1ee094e6a960312a15d964ee25b35}}, {{cite:fde10a769ef7f36f20d70d9706e0accfc39de422}}, are utilized to model the unknown environment. To solve the SLAM problem, the joint posterior density of the ue and landmarks can be approximated by using particles {{cite:5ba3221b26a1ee094e6a960312a15d964ee25b35}}, {{cite:947d66119e6ce47e201102a7164eae705030d13a}}, or sigma-points {{cite:7a3d524b385a99c052b7eb4fa7b21971d3dcf11e}}, {{cite:a6f95b759da15de2c1f0aab9dfa234811215969b}}, or relying on EKF {{cite:fde10a769ef7f36f20d70d9706e0accfc39de422}}, {{cite:6fbf2eae8dd9eb479444cd73f5d5f42c091db004}}, or IPLF {{cite:3dc88505d1ba128e5c84dde9f4cac506392a7387}}.
| m | f77da90ead96427fbe9acf9e4a6b2158 |
We have treated the problem of table structure extraction as an object detection problem by employing the well-known Mask R-CNN model.
We have implemented a novel anchor optimization technique in a region-based convolutional neural network that produces faster network convergence.
We have introduced a simple and effective post-processing method to remove the extra white spaces from the predicted rows. This method can be exploited to recognize tabular structures in realistic scenarios.
After extensive cross-dataset evaluations, our proposed approach has beaten the state-of-the-art results on the ICDAR-13 dataset {{cite:5625c2d281b87eab50ea6c9406caf6765c308a23}} by using same evaluation metrics proposed by Schreiber et al. {{cite:c7682abd8b7ef7850bf783194206c9d6659f3780}}. Furthermore, we have also surpassed the baseline results on the TabStructDB dataset {{cite:ccdd72268afbb599d360b45142c36e10af8ea7cb}}.
| i | 0fdd044795fb11e901055d0557bf0846 |
NSLSs are still slower than iterative methods, but they enjoy a better utilization of GPU resources and a higher degree of parallelization, as can be seen from the comparison with the pure PyTorch implementation of conjugate gradient.
Moreover, the implementation of NSLS in deep learning frameworks like PyTorch {{cite:65b59fceb97deb8430bfdbbebaf7501a7758b8b6}} or TensorFlow {{cite:671c36f1505785ccb688a89f89b7f96a53506cc7}} makes these models hardware-agnostic and highly optimized without much effort.
We are sure that the inference time of neural solvers will be further reduced by the development of technology.
Many research papers already propose more optimized implementations of the sparse matrix multiplication on GPU {{cite:a54550e5c22206292a909177a1ceb01e6a43c978}}, and deep learning frameworks are actively adding support for advanced sparse formats and operations. Anyway, many burdens persist: for example, many neural compilers, like Apache TVM {{cite:d816cd67af7f7279f543d0f0ed9e856078a4e40a}}, do not support variable-size inputs, and thus the optimization of graph neural networks is challenging. Furthermore, the ONNX {{cite:5a45463a3430bc0b701b41dea48a90f32da0bd61}} support for sparse operation is very recent (see ONNX opset 16), and accelerators like ONNX Runtime {{cite:7be89de00d05a94618940b44a66cb0a21e87a480}} still have no support for such operations.
| m | 87f18169bd1207748f8df64a37f51f70 |
{{formula:830142d2-54e2-4eed-a1d1-7a360484f498}} for metric spaces with doubling dimension {{formula:da5b341a-3a63-4a79-b5ab-060a85f8b65a}} . This improves over the {{formula:bbf08c94-ce28-4626-afe5-86c5bc1adfae}} from {{cite:263e18c86833d703ab4216a79c01206f5c81e18f}}. See cor:coreset-doubling.
Since general discrete metric spaces have doubling dimension {{formula:8bbdd077-2e44-43da-bc99-0ae0f19732b3}} , this yields coreset of size {{formula:d7e8752b-aaf0-4c91-bfe3-b32f48bd2489}} . This improves on the bound from Feldman and Langberg {{cite:920efef6af6fd41e549f8f0627cef77599ce994e}} {{formula:6ceee5bf-d51d-4492-9c0c-b09a99f86d09}} , and has an optimal dependency in {{formula:2ce4bdab-0c26-4766-b6bf-a7500db1bf87}} and {{formula:5d2f6a90-7dc3-4e25-82b4-a7e3c5e88fef}} .
{{formula:4e5f1cf8-0279-49e4-8986-20c594630408}} for Euclidean Spaces, see cor:coreset-euclidean. This improves on the recent result from {{cite:4ac169c87ef60cc07e4f056af3d1c2be43f77483}}, who achieve {{formula:bf7dae5b-483b-409b-a5bb-023a2406d6c3}} .
{{formula:628fab4f-e130-4d64-8611-cd93be1ad37d}} for a family of graphs excluding a fixed minor, see cor:coreset-minor. This improves on {{cite:b8709cf9481181ff7f93386b71ddab74ba61ef1e}}, whose coreset has size {{formula:8aed58ee-d2c4-4743-8ee2-366bfd4f7360}} .
{{formula:f0086f08-7996-43b4-a869-b9ddd0258f0d}} in graphs with treewidth {{formula:0e2f9c07-a223-4fc6-af66-ccc51a59c47d}} , see cor:coreset-treewidth.
This improves upon the work of {{cite:d98f6484f51b1884f4c90af7d433af75ba72e53f}} in two ways: their coreset is only for {{formula:edaaae0f-0a85-4738-8aef-8fe17473e0f7}} -Median and has size {{formula:e4396d89-2e3d-44c6-a153-e3a4005c0d8d}} .
{{formula:21a2d78c-654d-43b3-a535-ca6fed59e1d5}} in {{formula:812d9d27-9661-4b23-bf07-86b0b8c0b168}} with {{formula:d533b694-a873-4286-b01e-b9126d19a467}} distance, for {{formula:de0195c8-fe59-4d9a-ac42-fb118a8b19b8}} , see cor:coreset-lp. This improves on
{{cite:4ac169c87ef60cc07e4f056af3d1c2be43f77483}}, who presented a coreset of size {{formula:c8f0d572-d9ee-4706-88e2-035428959761}} .
| r | b65a81b508ea23f71b453762603f91d1 |
paragraph4
.5em plus1ex minus.2ex-.5emDomain Shift. Sec. REF briefly discusses the role of population statistics
in domain shift.
The idea of recomputing population statistics on target domain
is first proposed as “Adaptive BatchNorm” {{cite:39bc2cdede5ae9f1a056f534b3a7c1710991e0e8}}.
This is followed by other works that modify BatchNorm for domain adaptation
such as {{cite:b8fa0253d2914ef37f332eb0c47817843a5fa5a3}}.
Defense against image corruptions or adversarial examples can be seen as
special cases of domain adaptation,
and {{cite:aecbd921de3e1cdb49e8df824a44b06bc9c8bcdd}}, {{cite:f81f8449889bd9d26b8fb0f7222a8092905b46b1}} show that
adaptive population statistics help in these tasks as well.
Moreover, when test-time inputs are given as a sufficiently large mini-batch,
the mini-batch itself can be used as the “population” of target domain.
{{cite:b65ded5ab6af93a5ae822f2e4fea38ed673918cf}}, {{cite:5e2d541cf66a1ae758f7156c7eb29ddc2f51c9a7}} show that
recomputing statistics directly using the test-time mini-batch
also improves robustness against image corruptions or adversarial attacks.
| d | 89fd8501f741b364f566f9e3e7000f3c |
Furthermore, there are approaches to use the closure principle to control other error rates than FWER, such as false discovery proportion tail probabilities {{cite:6c7b4a4172a6d9a155dbe61f30abb8db35becf22}}. There also exist connections between FDR and closed testing. In {{cite:b85732ea909fe379be354c1b7e2c4f0904e98b5b}}, they introduced an approach to use graphical procedures for FDR control and, in {{cite:56cea2ed80cf409b6abab2dde52a62542b27a601}}, a connection between Simes-based closed testing and the Benjamini-Hochberg procedure {{cite:bddfd1dc0fefeccedbd5380f55ad9295dbb2c863}} was shown. In addition, every FDR controlling procedure provides weak FWER control and thereby defines an {{formula:aa6ecd9c-2c58-4987-ae74-8220fc3ea5fa}} -level intersection test. Hence, all procedures that were constructed for FDR control can be used to derive new closed testing procedures with FWER control. It would be interesting to examine the extension of these results in the online multiple testing setting.
| d | a0bb153cd0f711a3688ba01131f3a73e |
However, when using MC simulation, in some cases, it requires to consume extensive computing resource to generate converged set of equilibrium data. For examples, near the phase transition where critical slowing down occurs and thermal fluctuation diverges, or when we do quantum MC simulation, a longer time is also required for the simulation to reach equilibrium due to the computational complexity of the algorithm, which sometimes accompanied with the thorny sign problem {{cite:d3f8492b026e0c94477681d7903a4d5f5e754ed8}}. To obtain the result in the thermodynamic limit, large system sizes are needed to locate the phase transition point accurately {{cite:c403fd69768b98bbed8a7ac56eb9458b4b2c14e9}}. However, when the system size increases, the time required for simulation to reach equilibrium will also increase sharply, which results in a great increase in the time cost and computing resources for generating the training data. Our method being discuss here provides a novel approach to locate the phase transition points using the input data before equilibrium, thus saving the lengthy computational time.
| i | 6cc363c72f21c8d567c7c41c3328ec8c |
Recent advances in machine learning, such as self-supervised learning approaches, have provided powerful techniques to extract semantic information from complex datasets {{cite:ed90c9e6691922abe340e8301aba87ed5bef13a8}}.
Here, we mainly took inspiration from self-supervised generative models combining autoencoder and adversarial learning approaches {{cite:33bb33a22f2ad1e56c3a668869a36610d2cf91a7}}, {{cite:b8a618d9168f2037d5b6b52ac09e55a25537a1ab}}, {{cite:acb009ff1b88128d5c485d4c5f0a404390f0ad66}}, {{cite:ed90c9e6691922abe340e8301aba87ed5bef13a8}}.
In contrast to these GANs variants, our model removes many optimization tricks which are challenging to implement in biological substrates, while maintaining a high quality of latent representations.
As our model is relatively simple, it is amenable to implementations within frameworks approximating backpropagation in the brain {{cite:faa462d2f364a7e1b4bcd2e3e3f04eaa79e2bcb8}}, {{cite:de52756c566ef02b234f22079e421621128b17c2}}.
However, some components remain challenging for implementations in biological substrates, for example convolutional layers {{cite:5c497af4031bd281ed4c42c346abdd6b0aee747f}} and batched training {{cite:673196f1149852c33d8f205d97ca7bef27ec0b2b}}.
| d | 38c1ffcd1dcb8839d67ee424b54cdc45 |
All of the Tait conjectures hold.
In {{cite:34ef81863f08a10678bb956ca3c784349a748776}}, W. Menasco and M. B. Thistlethwaite prove the third Tait conjecture, called the Tait flyping conjecture.
The second Tait conjecture follows from the third one.
The first Tait conjecture
was proved
by L. Kauffman, M. B. Thistlethwaite, and K. Murasugi independently by using the Jones polynomial (see {{cite:c13f54b26fc4cc9ee91763a12edb9e6aa3b654ca}}, {{cite:83f9ec2feee5fd1b771367ac7d6019f5998436ea}}, and {{cite:05e0e4ff124a93925342d1131ef12daf53844e70}}).
Namely, they showed that the number of crossings of any diagram of a link {{formula:35bec3ba-3c71-4407-a9b2-a44c4d0f3237}} is bounded from below by
the breadth of
the Jones polynomial
of {{formula:ebda9c08-3065-461f-a21d-3123c45003fe}} . Furthermore, for any reduced alternating link diagram, this inequality is sharp.
| d | e29125f5e3aab766d2e1836af3865e6f |
To cope with this, one can leverage the social information of users {{cite:e6e9618aaab676d9b8c24b5f7f67f647dbd39653}}, {{cite:8fec2943b235c6f89eae935c3477ea8d5c1b919c}}, {{cite:cfe9797e276f63850f5b5476354a5c3b9c5e595a}}, {{cite:c5c19c107a65a09596c2c1ddc20bf9b6ee7127fe}}.
In this way, we assume that users with social links also share similar item interests.
Therefore, we could simultaneously aggregate social information and user-item interactions {{cite:55bca01f784c4ef9ed439183257331d97376d7f6}}, {{cite:e6e9618aaab676d9b8c24b5f7f67f647dbd39653}}, {{cite:bd1c625feef72dcb39632826168467164e5bdc94}}, {{cite:7185ed116eb930806f6e25bb30a3cb8665089200}} to alleviate the cold-start issue. SocialGCN {{cite:7185ed116eb930806f6e25bb30a3cb8665089200}}, {{cite:95afbbe135a0ec537fe74b1205efcb94d6cbb4e5}} employs the Graph Convolutional Network (GCN) to enhance user embedding by simulating how the recursive social diffusion process influences users.
GraphRec {{cite:e6e9618aaab676d9b8c24b5f7f67f647dbd39653}} and GraphRec+ {{cite:bd1c625feef72dcb39632826168467164e5bdc94}} propose to model three types of aggregations upon social graph, user-item graph and item-item graph.
Thus, it can comprehensively fuse the social links and item transactions. ConsisRec {{cite:55bca01f784c4ef9ed439183257331d97376d7f6}} introduces the social inconsistency problem from context-level and relation-level. It solves this problem by using a sampling-based attention mechanism.
{{figure:d1231106-775f-4eb6-b015-8437cc53b8dd}} | i | 726bce4a110ca0863cb2478737a83743 |
Small systems coupled with heat baths evolve under stochastic dynamics. Their probabilistic description at all time can be understood using the master equation or the Fokker-Planck equation {{cite:abc1a5c00acb1e2b8249e5798f636574502d84b0}}. In this kind of systems, an increasing interest has been devoted in studying thermodynamic observables, such as heat dissipated in the environment, work done on the system, entropy production, probability currents, and so on. The estimation of these fluctuating quantities requires a description at the level of a single stochastic trajectory, which can be obtained within the celebrated framework of stochastic thermodynamics {{cite:2221d34dd14900ff8e0423001905c89922860b42}}, {{cite:1b85c4e59491f6d72903912895e7a52366182eb2}}. Interestingly, these stochastic quantities follow some universal results in the non-equilibrium physics; namely, fluctuation theorem {{cite:92d62c59c22a6452c011c9851177e96634c168d1}}, Jarzynski equality {{cite:dd854a458dace99449ecaa229e4483324cc3c314}}, Crooks fluctuation relation {{cite:114e3ac86fd8fbe6e6b5804e8eb65219ecd7d6af}}, and thermodynamic uncertainty relations (TURs) {{cite:12e9fc146c5f47c581b19e3747e867b0b1291050}}.
| i | 9624ade108a0a3887d54e1f005161d89 |
The subjective nature of the perceptual aspects evaluated must be taken into account for the evaluation and the mean opinion scores are indicative of preferences rather than absolute measures of quantity.
It can be observed that the perceived adherence to melody for the prototypical a cappella voice synthesized by the UTS model has higher preference than the STU models, although a high variance is observed in the ratings for the same. The unison perception evaluation shows that the variations in either timing or pitch alone are not as preferred as variations in both aspects together. Timbre variations do not have as significant an effect on perception of unison as variances in timing and pitch.
The evaluation of audio quality shows room for improvement in the synthesis of the voice signals. This can partly be attributed to the use of the WORLD vocoder {{cite:4e5c2b10599c76c76abf7aebceda2fa9b2ece764}} and we believe that this can be improved on in the future using recently proposed neural synthesis techniques.
| r | 8d49f1ef570fe43ce7464396a9b1a08b |
Once CO is released, we find that its dynamics is different from that modelled in extrasolar systems so far (see Appendix ), i.e., the gas does not evolve viscously inwards as expected in massive disks {{cite:fc60aec3b488955633775ea99d50773255ecb828}}. It is due to two reasons. First the gas quantity we find in the KB is very small and not in the fluid regime in contrast to systems detected up to now. Second, the majority of gas detections has been around A-type stars {{cite:c5d9b9cc889571d3a6f7d69abb3a4d2bda6cebad}}, where stellar winds are not important (only stars cooler than about F5 possess significant convective envelopes and then magnetic fields that can produce strong stellar winds). In contrast, in the Solar System, the Solar wind (SW) drives the dynamics of the gas. We find that once released, CO gets pushed outwards by SW protons on timescales of a few years (at a rate between {{formula:75ca009d-7fe9-4fe6-9f70-7f03fcd08868}} to 10 au/yr depending on the location, see Appendix ). Some of this CO gets dissociated and ionized (when interacting with SW protons, photons from the Sun and/or the interstellar medium, see Appendix ) on its way out. However, the ionization and dissociation timescales are of order 100 yr so that CO remains the dominant species up to {{formula:500dac6c-6b93-473b-b0b8-2f667839ffad}} au (see Appendix ), i.e. well beyond the heliopause (which is the boundary of the heliosphere where the Solar wind is stopped by the interaction with the local interstellar medium) at {{formula:609d328b-8706-49e7-bbb8-a6d06261eb9c}} 150 au {{cite:916493dac97fedb9715462804cf785b9c1a1fa61}}. The daughter products of the CO dissociation, namely C and O, are ionized in {{formula:01e32eb3-1694-4ad0-b947-9e2ce997581f}} 100 yr, leading to an ionized atomic component beyond {{formula:c3a59bfe-bedb-475b-ad76-07bc9cdd748a}} 500 au. These ions will then follow the interstellar magnetic lines and get ejected further in the interstellar medium (see Appendix ). The model predictions for CO, C, O (neutral and ionized) as a function of distance to the Sun are shown in Fig. REF and a summary of the model is given in Appendix .
{{figure:349d42dc-be8f-4df5-af99-69fffd131bbe}} | r | 6761fd89c0e76fad828df910b28535f2 |
Despite the dominance of FM double-exchange interaction, intra-chain AFM coupling is established along the {{formula:708c86ce-2d94-4bfe-8306-b994c13d6a2f}} axis double rutile chains. It has been suggested that the AFM coupling is due to a weak super-exchange coupling through the bond {{formula:cb0b7732-3efc-4ece-960f-51e5c7c06ed0}} {{cite:1a3c97b755ce4094c39ff83c7967365059a00939}}. However, it can naively be argued that an AFM super-exchange interaction across a {{formula:c888a1ba-d3fe-4501-ad34-fae01c100793}} bond requires a bond angle of 180{{formula:49a83f89-3e16-4b76-ac73-61291c035e2c}} {{cite:5195884caf45643cac2547a4ccfbbf45c82e6af1}}. Cases in which the AFM super-exchange mechanism occurs across bond angles that deviate from 180{{formula:0c018cb8-d075-4f46-a2bf-d2b1778e5b13}} can of course be realized in the presence of complex crystal-field symmetries of the magnetic ions (like in our case, with the distorted octahedral coordination of Cr ions). However, in NaCr{{formula:0f6fb8b6-9479-4139-b03b-3cf3dc89a24a}} O{{formula:0dbeab6a-b6fb-4045-8315-fac85d415ea3}} the value of the bond angle {{formula:aab454de-ae3d-41a7-8935-2aaf40ada6a4}} = 125.58(5){{formula:10ab122f-87b7-415c-87bc-c348b356893f}} , is very far from 180{{formula:1b6d7fb9-6857-4a81-b29d-733c3cc9ab98}} (is actually closer to 90{{formula:1d7e50c9-189d-4114-9122-ab1b5097eac2}} , which would favor FM coupling not AFM). Therefore we suggest the mutual hopping of 2 itinerant electrons between the two Cr sites, aided by the O1 ligand holes, as a possible alternative mechanism to explain the intra-chain AFM coupling in NaCr{{formula:fad2cc7b-ef25-4870-8bc6-f89ee1f551b9}} O{{formula:43c6cb82-acaf-42f9-ae24-5f637d866673}} . The hopping of correlated itinerant electrons was suggested as the possible mechanisms to explain the magnetic ordering in CrO{{formula:0082e502-8b87-491a-8ebd-3df5c9bf040e}} {{cite:1a79688eb92746757d9b3e06da9a2edbe021ae7b}}. Such suggestion was though discarded since it would have lead to AFM ordering, which is not observed in CrO{{formula:87991f2b-29de-4035-94d5-021d8263ed29}} . Since the {{formula:8c7db0be-e49c-4a8b-aa49-52a17952738f}} and {{formula:cefa31b1-d8bd-4a22-b1fa-83791ac53ea8}} bonds are structurally equivalent and the O1 and O3 sites are both preferred for the localization of the ligand holes, a similar hopping coupling mechanism would be expected. The only difference between the two bonds would be the mutual orientation of the electronic orbitals in the non-equivalent Cr sites. Here, the different distortions between Cr1O{{formula:cb4182a6-154e-4141-9650-5a3ce0af4aca}} and the Cr2O{{formula:2beef1d6-20ad-4e0e-a0f2-7279bda44f7b}} octahedra, remove the degeneracy of the Cr-{{formula:de35c86f-48d2-4e7d-8916-651a0b5ba249}} orbitals, leading to one localized and two itinerant orbitals for the Cr1 site, and two localized and one itinerant orbital for the Cr2 site {{cite:99d53b05a12b6c5f5a0094d7e265160e18212dbf}}. Due to the lack of inversion symmetry between Cr ions, the canting of the moments in this spin configuration is consistent with an anisotropic AFM exchange including an antisymmetric Dzyaloshinsky-Moriya (DM) exchange term {{cite:32d297ad4bf83bba3a70ec119bb3e660686fa220}}, {{cite:c161f4f8b0a1978aa836d29ed65d9214f6897130}}.
| d | e005136ca11881f526116fe73f602587 |
The model architecture we study is an LSTM-based multi-task model for IC and NER tasks, where we use 300-dimension fastText word embeddings {{cite:b0f70c4792d483a662ad260d63c46bc281aff4ae}}, trained on a large voice assistant corpus.The text corpus contains data transcribed by an automatic speech recognition system. A shared 256-dimension Bi-LSTM encoder and two separate task-specific Bi-LSTM encoders (256-dimension) are applied to encode the sentences. A softmax layer and a conditional random field (CRF) layer are used to produce predictions for IC and NER, respectively.
| m | 566959ae1238b296e66f42a3c1306776 |
In this section, we introduce our scheme of Floquet engineering to
control the phase transitions in non-Hermitian quasicrystals, which is based
on the idea of realizing dynamical localization in Bose-Einstein condensates {{cite:f3880f5520ce1a9c4cfe12a50e5433837fde9b0f}}.
A schematic illustration of our system and approach is
shown in Fig. REF . The Hamiltonian of the system describes
a tight-binding superlattice with onsite quasiperiodic potential and
driving field, i.e.,
{{formula:732d522a-0178-4b3c-9020-897c0656de18}}
| m | 9c4cbd3bbd0bc61e0e3c02360fb77eeb |
Using the above as input, the two-photon annihilate corrections can
be calculated directly. We use the package FeynCalc {{cite:5e801ce0101b0a2442194eb0e0a9eed8251931ef}}
and LoopTools {{cite:144bc55988e82009bbdfb66a2a1c1121c4fb06ee}} to carry out the calculation. The IR
divergence in the {{formula:e5aac4cc-1f4e-4915-9d31-5d864d13653d}} intermediate case is treated as
{{cite:06878753e4488d3180c6ef1243c22bd804a948f9}} and there is no divergence in the {{formula:90e61928-bd47-4bb0-b35f-e9e21a61a955}}
case. The numerical results for {{formula:f3764a9d-4ad7-4cac-b420-f44a91ddaf24}} are
showed in Fig.3. The similar calculation can be applied to the
polarized quantities {{formula:527c9aaa-a5c1-434f-8d36-9b2820230c93}} and {{formula:45ef88c8-0537-46e5-91f3-87fbb36939a7}} as {{cite:dcdbd3face91d4db40581f4cdde18fff07fbb9f9}}, {{cite:06878753e4488d3180c6ef1243c22bd804a948f9}}
with the definitions
{{formula:b49f2750-0d5b-4e49-85f6-dabc50898b1d}}
| r | fe1cca22cb8ec14a9bed91d4af16820b |
In this work, we modify SimCLR {{cite:9c7c773109d20b5035f0f7df902e32dd2a780733}}, a state-of-the-art contrastive representation learning method, by explicitly adding information compression using the Conditional Entropy Bottleneck (CEB) {{cite:fee985c7ba4cdb951c5694d1d1973a8464e3cece}}.
Furthermore, we show how BYOL {{cite:d50c7820a7ae7335a72d624ed956a086fa7c401c}} representations can also be compressed using CEB.
By using CEB we are able to both measure and control the amount of information compression in the learned representation {{cite:af60da09b46f78ba90e96646f41275b0610c639d}}, and observe its impact on downstream tasks.
We empirically demonstrate that our compressive variants of SimCLR and BYOL, which we name C-SimCLR and C-BYOL, significantly improve accuracy and robustness to domain shifts across a number of scenarios.
| i | b589faf5a5fa2c16283bf90974c6832b |
As an extension of the MTP, Souza, Wilkens, and Martin {{cite:1e238bea5736555adf32f55f0a0100a77bf7efc7}}
introduced the so called gauge invariant cumulants, which are
essential in the study of
polarization {{cite:34be652b2810fe0909e75d2a3579f506d3369a05}}, {{cite:15f95f280a57d71321e71164e3e64a83fa239455}}, {{cite:697fa53e985e53c82222fb7ffa94397d3ae535e4}}, {{cite:d16e33c8450482b4a978ef0afd05f53cb20f218a}} as well
as charge transport, since they provide access to important related
quantities, such as the variance of the polarization {{cite:12fd9e501243c3aaa4d85b106f0d2d95d42d5dc9}},
or the shift current {{cite:d0b496ff5529e126349e2fb9f2be94b55df83ea2}} (related to the third
cumulant). Patankar et al. {{cite:d0b496ff5529e126349e2fb9f2be94b55df83ea2}} use the ratio of the
third cumulant and the second as a gauge of nonlinearity in the
Su-Schrieffer-Heeger model {{cite:5ba3e3bb1f3272989013f9a251c59fc545346c1d}}. The Zak phase {{cite:01cd55b8ee034dbf0c935b9fce5841b941adfb3a}} is
also the starting point to construct topological invariants, the
quantities which characterize the different phases of topological
insulators {{cite:7b6894767081cd3f7ad271383f0b4e315bde70ff}}, {{cite:9a6b9b2add0e76580c5400836fed86a5432af94b}}, {{cite:04fe277d3dcaef30100c2c612e0cec17894c2957}}, {{cite:4f71e807e5f42716dbea587f8929bb3e9ac1a08d}}, {{cite:89d2f619a22ed411e7a40b81cea28c99474ee674}}. The
MTP formalism has also been applied {{cite:30aa18dbeffd5c51e5609d6c58e32a0a87836c8e}} to study the
topological Haldane model.
| i | 0e0bdbda1188fd857373feadd33ce5ea |
In this section, we compare CIRCLE with 12 multi-view representation learning or clustering baselines, including Deep Canonical Correlation Analysis (DCCA) {{cite:fa763e3aee19993279df3eed2641f92886e141d0}}, Deep Canonical Correlation Analysis Autoencoder (DCCAE) {{cite:a01a1758d294b8896d7067e437a23c952b5311cb}}, Multiview Graph Learning (MVGL) {{cite:3ac36889c7cd8608b9c3c994d3d8878d0f1a06c5}}, Multiview Consensus Graph Clustering (MCGC) {{cite:c77975e5d032a553e786ff3b00a07af9f0adec5b}}, Graph-based Multi-view Clustering (GMC) {{cite:ba88b3400f51c4afd5392a9b67d3ef1e9292bf18}}, Autoencoder in Autoencoder Networks (AE{{formula:263be8ca-cc30-4bff-8731-7fba49ecb2d5}} -NET) {{cite:cbd8b9b44bf9879c866acc128312472be5362099}}, Large-scale Multi-view Subspace Clustering (LMVSC) {{cite:f33e93164b720624a3b3c1a10ed69637fbac892e}}, One-pass Multi-view Clustering (OPMC) {{cite:d764593a885c8fd532bbc1a99078595f9b1b895e}}, Scalable Multi-view Subspace Clustering (SMVSC) {{cite:45d8dd7b92c7fde2ce714700841a558732587d5c}}, Partially View-aligned Clustering (PVC) {{cite:3f0401822413bdbde46dcff8bfb923685909fc9e}}, Multi-view Clustering Method for Unknown Mapping Relationships (MVC-UM) {{cite:329acc2fed7840d98c73b48b478619d6e3c0b77b}}, and Multi-view Contrastive Learning with Noise-robust Loss (MvCLN) {{cite:4aa28b9d61296c24482b993cef3d5f1239f02e96}}.
| m | b072a98da7a32b66e8d2b506393205b8 |
In the near Maxwellian framework, global existence and large-time behavior of solutions to the spatially inhomogeneous equations is proved in {{cite:e347129ba2db78931d64dd7995a932a65c815a1b}}, {{cite:bb79188f18eec2da011cabb498eee6c8c4bb22c2}}, {{cite:99e8e94c56a07481c470b9c6e5db547bfb27aa21}}, {{cite:826cfe3536ee4b9f447a438893274f8a2841873c}} for the cutoff Boltzmann equation and in {{cite:a3729bc04be53ec5aafe41faa15513e258deca48}} for the Landau equation. For the non-cutoff Boltzmann equation it is first proved in {{cite:0de27e49dc42da736a3ef1f97684dc0e9629e61d}}, {{cite:8633343036ede3889b852792d264bb13a7aa3d47}}, {{cite:255614acc5216b10c32d699fa6c2f2ba4e3fbf97}}, {{cite:bfbea219cba04fe29d0fcb652c559a12c22f38ce}}, {{cite:6375802878696da9b5637609d6c38f47f5cd10c5}}, {{cite:316c3c845dabb679229ac517407dc3a7aef8060a}}, see also {{cite:be9b87e4a3d4f53ea9b5a06edd69dc7df77c46a6}} for a recent work on such topic.
We also refer to {{cite:47e9b2fff3ac1bf2253ea3820336c7684b5cd587}}, {{cite:ea8276a4753031530164896fc3569b29c31ce479}}, {{cite:33921f91bf047eacf8e2e3942aa2449714c930f4}}, {{cite:9f87b3639f2ad2fccdb3c3300a7b34be8954329f}}, {{cite:414521d31249656326cf8de21828fb1c7cf434f4}}, {{cite:11ce22015144b3b1cc48eea9775b3dbb49d7c6f0}}, {{cite:7c623fe92c704dd0658d9cc6852369afb4e1655a}} for the former works on the Vlasov-Poisson/Maxwell-Boltzmann/Landau equation near Maxwellian. We remark here all these works above are base on the following decomposition
{{formula:6ac4ff1b-5048-422d-8627-62ee5fabffb9}}
| r | e8c14e87b169b79d8bd1b4176703adec |
The approach of modelling a sentence pair based on neural networks usually consist of two steps. First, a sentence encoder transforms each sentence into a vector representation. Second, a classifier receives two sentence representations as features to make the classification. The sentence encoder can be regarded as a semantic compositional function which maps a sequence of word vectors to a sentence vector. This compositional function takes a range of different forms, including (but not limited to) sequential recurrent neural networks (Seq-RNNs) {{cite:a6a6524534e53d613554f78890c2018005ae6fa6}}, tree-structured recursive neural networks (Tree-RNNs) {{cite:b07cf5e497009227f974c07c611a089333133fa9}}, {{cite:8eb08d04b20b5e29f75b6e7a3d0aff4a39b97ce3}} and convolutional neural networks (CNNs) {{cite:a582b03cb59a8c51d1b44c7218d69e5d7a151c4b}}.
| i | b6f2813315f00925fc65ac47f4e6d24d |
The results in Table REF demonstrate how model complexity may be offloaded from the internal model structure to the input features. In black box models, the relationships between inputs are generally represented by complex data structures (tree ensembles in random forests, hidden layer node interconnections in neural networks), which are difficult to interpret by human domain experts. Using symbolic regression for feature engineering, these relationships can be represented in human-readable form{{cite:e59e6363fc4f0ec1a93e28945eed3adbaa07c599}}. The human-readable features exhibit higher correlations with the target variable than any of the original features in the dataset, which unlocks the potential for using simple linear regression models instead of black box algorithms to achieve the same predictive power. Transfer of the model complexity to the input features significantly enhances model portability, as the resulting linear models can be fully represented by just 2 coefficients: slope and intercept, which makes them easy to deploy in a broad range of settings without the need for specialized machine learning experts.
| r | 4124fc156eeb922d94d4d1d0d0a505ae |
We follow the classical routine {{cite:788094017ddba9ca130caf1d21c6dcd41e26e72c}} to define the generalization error of QGLMs as follows. When either the kernel or the target distribution {{formula:6c1dfd31-154b-4298-9d21-faf00a84fc4b}} is classical, the generalization error of QGLMs yields
{{formula:635e0f77-7f53-4e9f-a3a1-8f8c58d7f389}}
| r | c7ecec048f648dc891bbd58bf62b12bd |
Comparison with Image-Based Methods.
In Table REF , we compare our FastMETRO with the image-based methods for 3D human mesh reconstruction on 3DPW {{cite:fa2a86215ad15975d777cb85c7b313548aa38de8}} and Human3.6M {{cite:3c2bca5dfb4dbdae667924277691f89e7016b403}}.
Note that existing methods are implemented with
ResNet-50 {{cite:255b1e021652c6a782aec784e76b0d02ec2192ae}} (R50) or
HRNet-W32 {{cite:72023e52b116863504f0d3147160476b36612b90}} (H32) or
HRNet-W64 {{cite:72023e52b116863504f0d3147160476b36612b90}} (H64).
When all models employ R50 as their CNN backbones,
FastMETRO-S achieves the best results without
iterative fitting procedures or test-time optimizations.
FastMETRO-L-H64 achieves the state of the art in every evaluation metric on the 3DPW dataset and PA-MPJPE metric on the Human3.6M dataset.
{{figure:0018aede-73e3-4acc-8d1b-035c6862111f}}{{figure:969d997f-c04b-4ed7-b69f-4d7f37bc042c}} | r | b3263cfea885a022c540f0e1a85dadc9 |
As discussed earlier, the theory developed to describe such effects is quite general, as it is applicable to any Kerr nonlinear oscillator coupled to one or more continuua with frequency-dependent couplings. The consideration of Kerr nonlinearity is not so restrictive: many systems in nature with self-interactions are described by a Kerr Hamiltonian, with some value of the {{formula:69261f4c-6de3-46ad-9a5e-4919eba23e1b}} parameter which can be predicted from first-principles, or measured. Such systems include: bulk optical materials (where Kerr comes from {{formula:5707edb2-c486-439c-b4bf-8090f321e507}} {{cite:357cc7dc4507ce24668743430270a8723de65dbd}}), exciton-polaritons (where Kerr comes from Coulomb interactions {{cite:17839d8f20a268cbd39edf347c2b9e70b1f53c9b}}), superconducting circuits (where Kerr comes from nonlinear inductance {{cite:48fb35578256e9df86cdeb94bc6f51b29899ddb2}}), magnons (where Kerr comes from magnon-magnon interaction {{cite:31d708818032fee221c741208bcb75cc413a64b0}}), Rydberg atoms (where single-photon nonlinearities arise from Rydberg blockade {{cite:dce31a0aadcf429b15206e88471ef8a72c9f38c6}}), and cavity-QED systems (where single-photon nonlinearities arise from photon blockade {{cite:336a21abcc3e18367d825158ec47653e55e8a252}}).
| d | a18880359919dc70fe9d2e315df0fcc3 |
We first solve the model by performing calculations in an armchair-edged sample with the width {{formula:027d0563-f063-437b-b74f-fe960141e3b9}} . Two terminals L and R are
assumed to be semi-infinite. A relatively-short scattering region with the length {{formula:11f9b818-0447-4d11-af2d-d8de0164d6e0}}
is fixed, and the main electrically-tunable parameters are the bias voltage {{formula:ae8fa983-ae33-43bc-b293-0334cfa09416}} and the Fermi energy {{formula:2f396f61-a7e1-4cf0-88e7-1842c68e7821}} (see details in Appendix B) and Ref. [GavLaz]). We keep {{formula:a6d96241-1174-430f-883d-1a9f9b0e2a71}} the same for P and AP configurations and fix the parameters {{formula:afc6f661-6028-4748-b23e-97547e91e89c}} eV and {{formula:ebf2cb03-06ce-419a-8313-5f76db8dc8d8}} eV from bare GBL {{cite:3a5a45c8c47a000533b39c2dd72b5cf97d0a64e3}} and {{formula:faff8f46-41d8-4e4d-98b3-c7f450ce13f1}} meV, {{formula:a15c3b38-0524-49f9-a17e-4c30dfd9a657}} meV and {{formula:6253f2f6-4aa2-4f23-bf80-c685f0a4abdd}} meV.
| r | 09e6f9f042998756dbe439c80d859b03 |
As shown in Table REF , we compare all methods appeared in the main paper, including WAGE {{cite:ad38b89c958c5f63949fff71824305fb0b6e4654}}, LQ-Net {{cite:577e5fa933edfc4e211572454e40fbcf65f5d7fe}}, PACT{{cite:0614a961ad197bb3e852f4cfdb023b547b4ca302}}, RQ {{cite:677307753707d5c46b8473f750656283dde3df5e}}, UNIQ {{cite:e8c792919f0408e38f50ce1b1623384596369ee4}}, DQ {{cite:329fcdc55569268669c910cf4ad9c61aa59f59f7}}, BCGD {{cite:c324abd0e93782cceae2d330bffa26f31dcf5012}} {{cite:329fcdc55569268669c910cf4ad9c61aa59f59f7}}, DSQ {{cite:a3502f4467855262aca59b0c7d38cab452cc8a0b}}, QIL {{cite:3119acaedc6c400b1261fcb6c70cc14b87a5eb70}}, HAQ {{cite:d8ca130b4c7bb059758d26d9cbd1df2d32572e08}}, APoT {{cite:e7dbb6846fcbdaec1ae0d9f43597a7f28e82dfd4}}, HMQ {{cite:87cc00da679dd3bd3f81cb9d6453094d004144c7}} DJPQ {{cite:712ab44439038b9dbbd3852a4b32923bab196b0e}}, LSQ {{cite:576bdab91be36e2c53077fff4af084753ad90dee}}.
| m | fbec1d3f1c9558d75ea6b17ec7ff6e86 |
In the next section we first review the construction of the operators {{formula:b996ca37-73ec-4246-be43-fb4a0455d48c}} {{cite:6c0f461a9cb1f18ef83d50f88de1c8573cf09438}}. In Section III we turn to the example of the Ising model, focusing on the massless critical case.
The massive non-critical case with only {{formula:19cbb3fa-5586-40b6-aa79-a91ad5df6c08}} can also be dealt with in rather great detail as described in section VI.
| i | 9f5796229f473acedc4b79a9733a089e |
Significant research efforts have been made on interpretation techniques that explain deep neural network models on image data. While existing techniques are commonly partitioned into instance-level and model-level methods, considerable attention has been paid to instance-level explanations that explain the prediction for a given input instance by discovering salient features in the input through the underlying explainable method. As one of the earliest instance-level explanation methods, sensitivity analysis (SA) was presented in {{cite:6560b0aa7d7bf634f8d9866f284d90d2b214d505}} by providing a gradient heatmap for a given input instance as the explanation. Guided backpropagation (GBP) {{cite:7ad9934b5d1163e7de0ab000ab9746b223a67179}} was also proposed as another gradient-based explanation approach. As one of widely used techniques, layer-wise relevance propagation (LRP) {{cite:225ecc0b15d570fbb9b558926990f6bdf1522ba0}} and its variants {{cite:8cbb805b5514194f9dfa6141fe6da621eab8a1b9}}, {{cite:1485bc08fd62b5aba49973f80fc3dbd6a25ad7d1}} were shown by devising redistribution rules such that the model output values are redistributed proportionally to the activation values in forward propagation. Besides, Grad-CAM {{cite:f2ed181df3db283261f137664d779d21634bc1a8}} was presented by producing explanations based on a course heatmap highlighting salient regions in the given image. Instead of heatmap-based explanations, LIME {{cite:402ffbbab0172feeb5bd1a7b0528ea81bdc6a746}} was designed as an interpretable surrogate model such as linear regression and decision tree models.
| m | b60828415860e7b18aabdde73500ec5e |
The most natural direction for future research is to generalize to the non-realizable setting, where there is no perfect classifier from the hypothesis class. Here we may need to relax our requirement of computational efficiency, as the problem of learning a linear classifier in the non-realizable setting (also called agnostic learning) is known to be computationally intractable even in the single-party setting {{cite:87583e2e0fdc6f75efe00f6d384339ade4d64b57}}. However one may be able to obtain efficient protocols assuming access to an efficient (single-party) learning algorithm.
| d | baa9c8fd46c34dc3168896777815aba2 |
Another source of uncertainty is introduced by parametrizing each momentum distribution by two variables, {{formula:1cb1ca81-d226-41f6-be85-e7fd3c7e0686}} and {{formula:db9ecc88-b0f7-4b0e-9117-bcd4f6dc01e3}} , only. This is of course perfectly justified for the electron sector. For the neutrino sector, the non-thermal contribution is below the percent-level {{cite:38a47bdd03f240ec17edf1a76d1ba5d980b2437e}}, {{cite:935c265dfe53f8133c8e4284681d4b97a0c9fadc}}, and it is expected to be even smaller with flavor-blind dark sector-neutrino interactions. When we present our numerical results below, we will demonstrate that we recover the state-of-the-art prediction for {{formula:0b1b74c0-2ebf-43d0-9ad2-51ff9d726329}} . Finally, a description in terms of temperature and chemical potential in the dark sector is expected to give accurate results for the problem at hand as well. The reason is that kinetic equilibrium through self-scattering is typically maintained until after freeze-out. The interactions of the dark sector with itself and with SM particles require specification of the particle physics model, of which we now present an example and for which these conditions are satisfied.
| d | 83141a1d20b12a1064e7cdeeb14f07c9 |
As already mentioned, the computation of the stochastic reduced-order model (REF ) is costly. The cost of the reduced-order model in the moment-mean has complexity identical to classical deterministic model reduction methods and various state-of-the-art algorithms could be used to decrease the cost further. In fact, note that to determine model (REF ) we just need to compute the vector {{formula:37c68e91-fda8-4c21-a514-b5f6c10cbb21}} , where {{formula:aaffab0c-e912-4ce4-a12f-171863e252bb}} is the solution of the Sylvester equation (REF ). Thus one could proceed in a number of “deterministic” ways by defining an auxiliary deterministic system for which its steady state is described by the solution of the Sylvester equation (REF ). When {{formula:2b01918c-d3fe-4779-adb9-36c6a7eac0d5}} one could directly use the IRKA algorithm {{cite:532db004e6d2f3544b790a4a050b2e565156dc28}}, which uses efficient Krylov projections, and then efficiently extract the matrix {{formula:273768b0-26ab-4ab6-abf6-b9403eed7e2a}} (since the obtained model is low dimension). Otherwise, one could use the auxiliary deterministic system to generate a trajectory and then apply the data-driven method presented in {{cite:91b84e26bc6d1d217b7a7ee1162eb2e71d9b0ae5}}. In particular, this second method computes directly the matrix {{formula:5f49aaf2-ac7c-472f-97f8-549625d148fa}} from data and it has a complexity of {{formula:793e6c72-0d02-42be-900c-cd9a1a900008}} , for some positive {{formula:25aa483b-a2eb-470f-b062-7a2e10dd090c}} , in its most efficient form. When {{formula:873ac7f2-4573-4a2f-a984-9ab415b0672d}} one could determine the solution of (REF ) with an efficient method of choice {{cite:adfa4cc7c826a9acd2039fe0e4f1425c51b29c3c}}.
| d | ea0e661a27d5d535d6797044a09a901d |
By Noether's Normalization Lemma (see {{cite:acdf0acb1960a38d12e25512c4880acc4a6d3aed}}, §13, Theorem {{formula:1cc669b3-bed0-41a8-9dae-f56666e4cb45}} ), {{formula:e28d2ff2-86be-49ca-989a-1e535c7dd6dd}} is isomorphic as a {{formula:ff372e62-0e5e-4803-8639-6a981a0ef5ad}} -algebra to {{formula:aae0a42f-26d2-4c05-a023-6be61f68a546}} where each {{formula:22d18f19-ce95-4976-b37c-5720fd3bc602}} is irreducible and of the form
{{formula:d96ef5f4-b1c5-4e03-8977-f95efdf834d6}}
| r | 3233e0c1bd72aad9b2af05e326309dad |
In terms of analyzing biological data, our results demonstrate that we should not conclude any degree of functional modularity simply by observing a moderate degree of structural modularity. This potentially poses problems for approaches such as connectomics {{cite:74c458899e064034f722da119217b63cbcfcaa42}}, which assume that knowing the structural properties of networks will give us good constraints on their functional properties.
| d | 847ae7ad2b72c7cabf0ccd0a3bc7db66 |
Both player 2's objective in (REF ) and (REF ) are equivalent because the operation performed is monotone, and (REF ) remains convex in {{formula:fccbc7be-92a2-478c-a61b-da21fd1b4167}} {{cite:18774e28901377ef432bcddc431669f86f50f346}}.
Even though we enforced that {{formula:452946e5-6343-4936-a70a-5ef38de2b870}} in the objective, we know that such solution is non-empty (see assm:non-empty-set).
| m | e9071cdbc1c06c92768d0f61290e62cc |
The main difference between the semi-classical and autonomous master equations (Eqs. (REF ) and (REF )) concerns the emergence of a transient character in the semi-classical framework. This is manifested by an explicit time-dependence in the kinetic and eigenoperators.
The different temporal behaviour replaces the fixed point of the map by a time-dependent instantaneous attractor. Moreover, in contrast to the autonomous case, the semi-classical master equation mixes coherence and energy. The generated coherence can be traced back to the initial coherence in the control {{cite:f393d532b91b33e51acb4e4d321d110fd9779df2}}, {{cite:e041dc30b4108e61cf28d69b5ab5098469bea153}}. Meaning that coherence is transferred from the control to the primary-system. As a consequence, if the control is initially in a stationary state, coherence and energy will evolve independently.
Therefore, the source of the transient character is the non-stationary state of the control. In this context there is a hidden assumption of a timescale separation between the fast direct influence of the environment on the primary-system and a slow indirect effect on the control.
| d | 0331b32560d77cb3962ea00d6ba08b88 |
The results above confirm, that we have experimentally verified the canonical commutation relation by weak measurement of the path-qubit observable in a neutron interferometer.
Accordingly, from the quantum foundational perspective, our experiment thus provides a genuine direct test of one of the fundamental tenets of quantum theory. Our test is as fundamental as the direct measurement of a quantum wave function by Lundeen et al. {{cite:c49a2940f1fb25434614c4e0d5850107faf91075}}. Heisenberg's uncertainty relation is a direct consequence of the canonical commutation relation and several formulations of it have been tested experimentally {{cite:411eb8f152b992d40f2ee08d3208ec8aa5ef88d5}}, {{cite:0903f4acfa49c415b930e1802f9918d9be261dca}}, {{cite:092361cadd4cf401695e97016a6e332c9422dd80}}, {{cite:0073f3ebd8d158cf6dfc0ffd2603849ced762c6a}}, {{cite:bd95ab42ce5ebabc14832ebaf6a6f9e0f7dcafca}}. However, a genuine test of the canonical commutation relation has hitherto not been performed.
| d | efb162ea8a8baea2c2df1636cd770454 |
Although the study on the dynamics in the presence of an electromagnetic field is an old issue but due to the following facts it may be still an important issue in the recent technology. The investigation on the ion conducting electrolytic materials is a key area in physics and chemistry {{cite:e18e8ea96c43ba4dfa5d3eda72a18afa9a98b504}}, {{cite:2d1e1e0d85bddaa2bdabcd934a21f48039c80067}}, {{cite:e1138c1d387e5e9f653e40e6bf0eba98b53ca0a8}}, {{cite:3927868a3cabd90ea12f8668424ebf090a6510b8}}, {{cite:640d8d312da1b989ba78a06ebdd2a0313b756ab0}}, {{cite:d7550e6061e70a2025497bc4b8536e6da312da8c}}, {{cite:35bcb73dc96d0d60501981983b17b3ab4f77ad33}}. The materials have potential applications in a diverse range of all-solid-state devices, such as rechargeable lithium batteries, flexible electrochromic displays and smart windows {{cite:e18e8ea96c43ba4dfa5d3eda72a18afa9a98b504}}. The properties of the electrolytes are tuned by varying chemical composition to a large extent and hence are adapted to specific needs {{cite:d7550e6061e70a2025497bc4b8536e6da312da8c}}, {{cite:35bcb73dc96d0d60501981983b17b3ab4f77ad33}}. High ionic conductivity is needed for optimizing the glassy electrolytes in various applications. Then it would be very interesting if one can tune the ionic conductivity according to specific need by a physical method. In this context, very recent studies {{cite:df82fffcc06f5aed5ff839cacbb1b6b9bbb2410c}}, {{cite:8fad51f4d7c9777cc5ce97a277d24dd0edbe71de}}, {{cite:3aa296c707daa08576675a21efb316a52b152abb}}, {{cite:93e16e3150fe9c9a79cded8d86a39c505d989b3c}}, {{cite:f6d5bff1da40083fbe17a6da2fcd0c075a435096}}, {{cite:50d76c499e74d98cca1162ef90dfebc8bdbd96c8}}, {{cite:c2d9c4c44d7101fb78c00d483db58cd913fdece4}}, {{cite:d60c7b0877f4d096882ed0209873616a9d91b0ae}}, {{cite:d9db68430c302ee65fb8acc9d2f4438d73a118a8}} show that the conductivity of an electrolytic material can be tuned by an applied magnetic field. To tune the conductivity of ions in the solid electrolytes the combination of both magnetic field and time-dependent electric field may be an important choice. Then one may be interested to know the basic dynamics as well as energetics of an analytically solvable model like driven damped harmonic oscillator in the presence of a magnetic field. The driven damped harmonic oscillator is a well studied text book material {{cite:cc4dc34d6819d6b8c3e8ef6996a122d970e56d9c}}, {{cite:b96e6b6f5cf65dffac7a17d6ecb3ba17435dae5f}}. At the same time, the dynamics of a particle in the presence of both constant magnetic and electric fields is also a well studied issue in the text book {{cite:b96e6b6f5cf65dffac7a17d6ecb3ba17435dae5f}}. This study has been extended in different contexts{{cite:dde68453e41a69c2a216ea4749bce4c9f9825a77}}, {{cite:c301078c3736b530324abe45e7b12e85c52a94b4}}.
But to the best of our knowledge, the study on the dynamics of periodically driven damped harmonic oscillator in the presence of a magnetic field was not addressed. It does not mean that this issue is not an important one. In other words, the
model study may be a very relevant one in the context of barrier crossing dynamics as mentioned above. At the same time it may also be an important one to explain the refractive
of a dielectric material in the presence of a magnetic field{{cite:659fda9728c2cc33ae309675c2f95115f2923f74}}.
Thus our objective is to explore distinguishable feature (if any) of this dynamics including the energetics and the resonance at the steady state. Then we start with the transients of the two dimensional damped harmonic oscillator (having same frequency along both {{formula:9211b909-e3d4-4115-bc82-d2761733a493}} and {{formula:859a52a0-081f-442a-96ce-a3a6a04a73db}} -directions) in the presence of a magnetic field. {{formula:0aa6c3ce-af34-4ff2-8454-17d5240e17c3}} and {{formula:02d9017b-1f33-4c38-b333-d807a3cc7f4a}} are superposition of two periodic terms in the absence of damping. Thus the motion may be quasi periodic in nature. Then we determine the condition for simple periodic motion. At the same time we determine how the frequencies of the periodic terms in the damped oscillation may depend on the damping strength. These calculations corroborate to the resonance conditions which are determined based on the steady state dynamics for the driven system. Here we find that the magnetic field induces an asymmetric splitting of the spectrum of the output signal with the finite values of the amplitude at the resonance conditions. But the values of the amplitudes at resonance condition become infinite in the absence of damping. It is to be noted here that for this case, the phase shift between the input and output signals at the resonance condition is similar as that of the driven damped harmonic oscillator. It proofs indirectly that the finite value of the amplitude at the resonance condition in the presence of damping is solely due to the dissipation of energy. In other words, the phase shift has no significant role in this context. Major points like these have been included in the conclusion section.
| i | d0ebf326d9ade6a75d8fa4aef7710312 |
Before concluding this paper, we would like to emphasize the following points: First, the technique of simulating isothermal processes with adiabatic processes
and isochoric processes are important to our proof, which enables us to establish the connection between large time limit and large N limit.
Second, the calculation of exact expression of microscopic work in our paper is non-trivial because
the work contributions in the different steps are not identically distributed. Hence, it is different from the law of large numbers, with
time as the large number {{cite:c81f62488023e7d2e995bea31ea19ecac9d5402b}}. Third, we proved the “minimum work principle" formulation of the second law stands for
even small system, though other formulations may be transiently “violated"
probabilistically {{cite:00120bb0cb702639ec2d94f39ce3623cfba55b59}}. This is not surprising because “minimum work principle" concerns infinite-long-time processes,
which has no contradiction with the transient “violation" of the second law for small systems predicted by the Fluctuation Theorem. Actually, the Fluctuation Theorem
does not constitute real violation of the second law, which is a statistical law and holds when averaged over different realization of the process.
Fourth, the isothermal process is reversible, but the finite {{formula:70e61f10-478a-4b35-a21b-0af30c2e0f05}}
“step path" is irreversible, due to the QIP
(thermolization) is irreversible. We can thus expect that the work
dissipation {{cite:c81f62488023e7d2e995bea31ea19ecac9d5402b}}, {{cite:3728887d996fc866d01a2394e56be8260439ca93}} for the finite {{formula:51417b12-373c-403f-b712-731dc894ce79}} step
path will be finite and will decrease with the increase of {{formula:89dd27da-2321-43d2-a980-ed3271005e86}} , and
finally vanishes when {{formula:7f1c9590-40d3-4801-adf3-f3fb7266af5c}} approaches infinity.
| d | d96e44a46b1429f308d3d8d9a4247389 |
Recalling Theorem 8.2.1 in {{cite:065cdc1ae74d9a619240b9bf4f5a41575dcfa42b}}, and Theorem 1 in {{cite:cdd87764d3b46fb650f92245eeff42ad8a330f18}}, we have that
| r | a21061e836b966f5a442f6bcb02079f5 |
In this section, we report the results of our proposed active learning methods, NDS and NDS+, in comparison to random selection, minimum margin {{cite:9edb7fd1a93dd04c0e50f564d10a84ea9222f7be}}, and Bayesian AL with variation ratio acquisition function {{cite:0b891a262844323c95866385d0b5c3bff5a8a921}}.
| r | 289fa400e134479030f89d93eb619517 |
We report the results of our model and compare these results with the results obtained using the approaches described in {{cite:b6a1a5324e584a589cc1faa690ecb6f91ec5f8ac}}, {{cite:cc6d3249f530e0b16bf725dc90752cd026ecb428}}.
We show that when the input is a very low resolution image, where most of the information is missing, the use of the gradient has a huge impact on the results.
As illustrated in Fig. REF , the reconstructions of our model are closer to the real images than other methods, but also more coherent since the structural properties of the images is being preserved by making the model learn the gradient of the image.
| r | 7db9ea496d84d3ba9a901450c81298ee |
The next lemma is the analogue of Lemma 5.7.22 of {{cite:b27d91d5b59172726f0365ee10b0cb5f597c5017}}.
| r | b83c9c6f71e272eeba89bd4553cc8f7a |
Since introduced in {{cite:817e899c9e1cc74487ac43b710a0ae73da0f9787}}, {{cite:d5717034075ce554fd219bd82e385f65d29b126b}}, {{cite:0845e163d085172b455b40c2cbe4c31e1b494018}}, the idea of plug-and-play has received great attention for its flexibility and effectiveness to solve a wide range of inverse imaging problems. In the area of traditional Gray/RGB image restoration, there have been many attempts {{cite:d5717034075ce554fd219bd82e385f65d29b126b}}, {{cite:17fe0b0fb7092ecd17a511086d45df5cb18f503f}}, {{cite:3b87fa2c9e2cb0def701d7d830f8fbf820481006}}, {{cite:f57353a2eab63671a378e600f428227384d544d2}}, {{cite:928c1a86ee1509298ea5a635b4da6e14df3a84ed}} for injecting denoising priors into the plug-and-play framework. In {{cite:d5717034075ce554fd219bd82e385f65d29b126b}}, Danielyan et al. combined the augmented Lagrangian technique with a BM3D prior {{cite:2a38966687469c50ac8bc39eb0716f20afe5491c}}.
By leveraging DnCNN {{cite:707556e13470b02452c8784b133a418a3202a2b9}} denoiser as the implicit regularizer, Sun et al. {{cite:17fe0b0fb7092ecd17a511086d45df5cb18f503f}} developed a block coordinate regularization-by denoising (RED) algorithm. Inspired by the design of FFDNet {{cite:30f2f72647ee0e64d7de8113b0dec5398ec3581d}}, Zhang et al. {{cite:3b87fa2c9e2cb0def701d7d830f8fbf820481006}} proposed a DRUNet that adopts a noise level map as additional input for plug-and-play denoising. The recent work in {{cite:928c1a86ee1509298ea5a635b4da6e14df3a84ed}}, from a new perspective, proposed to use a deep super-resolver instead of denoiser as the implicit prior for plug-and-play single image super-resolution. Although all the denoisers {{cite:707556e13470b02452c8784b133a418a3202a2b9}}, {{cite:f57353a2eab63671a378e600f428227384d544d2}}, {{cite:30f2f72647ee0e64d7de8113b0dec5398ec3581d}} for plug-and-play Gray/RGB image restoration can be directly extended to the HSI cases, none of them specifically explore the extensive domain knowledge of HSIs.
| m | 39218b3fcebda27839e46b862dc6519f |
where {{formula:33ac5e6b-1b7c-490a-aa9f-89bccbc19793}} is the Hadamard product as described in {{cite:824332804ea9e3446d319720406dae0c33acae7b}}.
| m | 13dfbbc2336bebe4bb66b90ef720ccc8 |
In contrast to some other studies (e.g., {{cite:a854fbf0da8bc8ffb20de156e83619ea2dcc90cc}}, {{cite:64bf78600ccbae57ae241a840f83ad9ce2b9c976}}), we do not find any evidence for a concentration of satellites near the pericentres of their orbits, nor for a strong tangential anisotropy of their velocities. We stress that this assessment is based on the limited selection of galaxies that does not include the LMC and its likely satellites, which are, of course, mostly near their pericentres, but should not count as independent samples. However, there are two further reasons for our different conclusion. One is that we use posterior-weighted uncertainties rather than raw measurements, which downweights the high-velocity tail of the observational error distribution. The other is that since we let the potential vary in the fit, the models choose the MW mass profile that prefers a more uniform distribution of orbital phases by construction. Nevertheless, we highlight a possible caveat in this line of reasoning: if the observed catalogue of satellites is incomplete and there exist yet undiscovered objects at large distances (hence more likely to be at apocentres), our inference on the MW mass would be biased up, as illustrated in the [sec:potentialbias]Appendix, and in the true (lower-mass) potential, the observed distribution of satellites should indeed be concentrated near pericentres. If the selection function of the observed sample could be reliably estimated, this effect may be taken into account in our modelling scheme, as illustrated e.g. by {{cite:47577a62494f22970e6d25be3fbedef7d8230ba7}} in a similar context.
| d | 222685cdd2c3ce6f11e730ef0d0c3f6a |
In this regard, we observe that without modifying {{formula:eabdc408-956e-4043-8be2-473897a43152}} , which is in agreement with realistic performances in such fibre-coupled ridge waveguides, our results could be straightforwardly enhanced by simply replacing our balanced photodetection with one with higher efficiency. By correcting the data by the HD efficiency ({{formula:b28239b2-d8d2-4bff-8000-6d07b73b5973}} ) {{cite:b62c6b01eca3157d95d921cb773d82833468c0b9}}, the non-Gaussian state obtained just after the 95:5 f-BS exhibits a negative Wigner function as expected for Schrödinger cat states, the corresponding value at the phase origin being {{formula:9ab9249c-b86f-4fe7-b4db-316673eb8d56}} (see Fig. REF -(b)). We point out that residual losses in the fiber components and low HD efficiency are not intrinsic to the followed guided-wave approach but only to our specific realization. In particular, the lack of efficient photodiodes, that represent the most critical loss factor in the HD, affects in the same way both guided-wave and bulk realizations.
| r | 806aea4b7981e89a7e262be9036f1f62 |
The NA64 experiment in the "visible mode" configuration, i.e. configured for searches for dark matter particles, such as dark photons A'
or {{formula:6c4127ea-c1b0-4163-b16d-e306d2abc8c5}} particles, decaying visibly, into {{formula:177eeeed-bbeb-43ad-b64e-ef0ac0ad4493}} pairs, is described in Refs. {{cite:5b8800950b14323d8d609d0de986378c465e84a2}}, {{cite:8f3cf7c1d569109820ce9485aeaf126dbb8551fe}} and
shown in Fig. REF .
{{figure:eefee2b6-464b-4165-bf9c-5f120ba94868}} | m | d7f868d1d76d0bcb08f466e03c355cd2 |
Comparisons with supervised pre-training. In the original ViT paper {{cite:aa9a21fdd6868296cede97d8e26669e20b47502b}}, ViT-L degrades when trained in IN1K. Our implementation of supervised training (see REF ) works better, but accuracy saturates. See Figure REF .
| r | c83f0ec2cce7bf7d87a7834d9a711f20 |
We also find the ranges of the individual parameters allowed
at different confidence levels from the analysis of the observational data. To obtain this,
we find the variation of {{formula:6a4d9f59-6854-49bb-acc3-2b884ee5f32f}} with each of the parameters of the set
{{{formula:42d94a3c-91dd-4cfb-ad1c-ec93f5e441e5}} , {{formula:193f33c2-0ed5-4041-94de-f64d9cd2cb66}} , {{formula:404c0615-3425-4910-9ffb-ece7850f87d9}} , {{formula:a78da36a-942f-4cf6-bfc3-51da14df42d6}} , {{formula:39d67128-a779-4b90-a367-fb3a81f47945}} } at a time, keeping values of
all other parameters fixed at their respective best-fit values.
The confidence interval for the single parameter may then
be obtained from the distribution of the function {{formula:86731dda-6e3d-425d-91de-fed0594ac2d1}} {{cite:94722d0e576d005305d0124735e3be3af19a5885}}.
The range of values of the parameter for which {{formula:e9ca1a16-ce1e-4bf4-84d1-2b7f71f8771c}} , {{formula:d674e4a3-6fea-4d90-8158-396091471a1b}} and {{formula:6b3b2207-10ec-47d6-b39b-108b3b2cb195}}
respectively correspond to
{{formula:42be2bad-d69a-4c1a-8e05-990945b61035}} (68.3% Confidence Level (C.L)), {{formula:b092fe60-a1c6-4438-b32a-06e27188c09d}} (95.4% C.L)
and {{formula:1529d732-79ad-48dc-b76d-d175cfc49245}} (99.73% C.L) {{cite:94722d0e576d005305d0124735e3be3af19a5885}} allowed intervals of the parameter.
We have shown in Fig. REF
the nature of dependence of {{formula:7d8c9d53-5dc0-4463-81f5-c0fccdb76543}}
on each of the individual parameters. For demonstrative purpose, we have shown the plot for CPL model only.
However, the obtained 1{{formula:d2e7c119-9b7c-44c5-bc73-80e1bd7becee}} and {{formula:d56df578-f5bc-4954-85dd-8473ff119ff1}} ranges of the individual parameters for different models
of parametrisations of {{formula:e88e1af4-1089-4904-b376-a98de92ced75}} , are given in Tab. REF .
| r | ce0e78c164ae1a130e3cdbc540053316 |
Generative Adversarial Networks GANs {{cite:a4e973807b6817f296507047bb31b9d5fc0cbc7d}} have shown to be extremely powerful at generating images and sequences {{cite:41e8434747b6174397a1012e7ba60aed7f297bf1}}, {{cite:5c75ee544edd48450c5e00cd6f2822f66107494b}}, {{cite:6a37dc6d98fb6d21e50437f786f6f7386c090a19}}, {{cite:af8a70be5781f0a3ab6df7c7dfa53c061d96fc27}}, {{cite:12c2ef1f8bbb5ef6b783e838a9456f0d04a47dfe}}. Applying such techniques to the problem of generating computer code from an input image is so far an unexplored research area. GANs could potentially be used as a standalone method to generate code or could be used in combination with our pix2code model to fine-tune results.
| d | 9268b31ea50ce93b452ed21aea24f454 |
Groups of isometries of finite dimensional hyperbolic spaces have been studied by a number of mathematicians. To name a few Anderson {{cite:94270a1c3e807e1f4cc603819444d397dd5faff7}}, Chen and Greenberg {{cite:cc6c66c43b98e060f0d94ab339856dcb1ad6be31}}, Parker {{cite:f7efc845f9b285a3d97bd8dd72088199af762d95}}. Hyperbolic spaces can largely be classified into four classes. Real, complex, quaternionic hyperbolic spaces and octonionic hyperbolic plane. The respective groups of isometries are {{formula:3b4b45a2-8dc1-4bc5-9994-d3b434f21abf}} and {{formula:0b9d330c-e18b-46b0-839b-887bb85589c3}} . Real and complex cases are standard and have been discussed at various places. For example Anderson {{cite:94270a1c3e807e1f4cc603819444d397dd5faff7}} and Parker {{cite:f7efc845f9b285a3d97bd8dd72088199af762d95}}. Quaternionic spaces have been studied by Cao and Parker {{cite:6751a460bb04ab2adf612dd7201b32ee50a3cce4}} and Kim and Parker {{cite:de59cc37f863182fd94fa6d4f895844e37c552f9}}. For octonionic hyperbolic spaces, one may refer to Baez {{cite:0be96527a3ce59f2866c007cec02004efd7afb14}} and Markham and Parker {{cite:1bee03673777356f1eb4965c3a5aa9ee7441c7ee}}. The most basic model of the hyperbolic space happens to be the Poincar{{formula:f48ff033-fcee-4a1d-97f9-9084c4593063}} disc which is the unit disc in {{formula:dc28cf8f-083c-4890-9b14-44ece0360151}} equipped with the Poincar{{formula:bae9bbf1-cde0-4c78-97e3-dae2d1bdf81f}} metric. One of the crucial properties of this metric is that the holomorphic self maps on the unit ball satisfy Schwarz-Pick lemma. Now in an attempt to generalize this lemma to higher dimensions, Carath{{formula:a1f79bb5-9bba-4834-afef-ab3a12432351}} odory and Kobayashi metrics were discovered which formed one of the ways to discuss hyperbolic structure on domains in {{formula:0f79c12c-e1bd-4c39-96ec-bd7419fabdbf}} . It is therefore natural to look for an infinite dimensional counterpart of the finite dimensional hyperbolic spaces. In late 20th century, Franzoni and Vesentini studied infinite dimensional Hilbert ball equipped with the Carathéodory metric. A description of the group of isometries of the hyperbolic ball has been given in {{cite:56a5349f8bc7563e1469b6227af57f11ce1a7de9}}. Motivational sources behind studies carried out in this article are {{cite:8ad033ad080cd7aaf1edf09f31fd61ffa1568f1e}}, {{cite:559826390ef8b561e69e5e8c94d95191278cdf9d}} and {{cite:dd52617c3e9444773e24e4c9128b9c361346b7f9}}. In this article, we intend to study dynamical aspects of the group of isometries of infinite dimensional hyperbolic ball. To this end, we consider this group, focus on some special subclasses of this group and explore their properties. Sectional detail is as follows. Section 1 is a brief literature survey. Section 2 gives the description of group of holomorphic isometries and its linear representation. In section 3, we will study a class of isometries having a two dimensional reducing subspace, class of normal isometries, self adjoint isometries and involutory isometries. In section 4, we will discuss the condition on an isometry to be unitarily equivalent to its inverse and compute cardinality of the group of isometries, class of self adjoint elements and set of unitary equivalence classes of normal operators on {{formula:8bde7d42-f39d-4a89-9ee3-5fe6759d317c}} .
| i | 14174da503a0cff2cc8ae98cd980b5f1 |
To (ii) integrate the estimated confidence into self-training, inspired by the single-teacher-single-student update {{cite:762c388cc85babd1e482a4b58d7af0afc29136c0}}, {{cite:8e8f9afe9b61130f0258f41760a3c6fdbc7d1c53}}, we develop mutual training with self-training based on consistency training for a training network (student) and distillation-based update for the two networks (teachers).
For training the student network, we build a unified self-training framework that can work favorably for the two tasks.
Motivated by supervised or weakly-supervised learning for jointly estimating both tasks {{cite:1104023535f3c968d27bd5582e47382cb35028c6}}, {{cite:7823262e8bb55cda95030cc42145d3c2d5bb5ad9}}, {{cite:f67626d903b4eb9ac40ca304a9ee857bff85198b}}, {{cite:ab60e3d4355c109947a49e13293c014624e15773}}, {{cite:0c60f5910217d176bce2c345f22408913d83fb50}}, we expect that jointly adapting both tasks will allow one task to provide useful cues to the other task even in the unlabeled target domain.
Specifically, we enforce the student network to generate consistent predictions for both tasks under geometric augmentation.
We weight the loss of the consistency training using the confidence estimated from the divergence of the teachers' predictions.
This can reduce the weight of the noisy predictions during the consistency training.
To learn the two teacher networks differently, we train the teachers independently from different mini-batches by knowledge distillation, which matches the teacher-student predictions in the output level.
This framework enables the teachers to update more carefully than the student and prevent over-fitting to the noisy predictions.
Such stable teachers provide reliable confidence estimation for the student's training.
| i | 39cd2cf3604c45994f1572592f91904e |
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