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Figure 1: The Unitarity Triangle.

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Table 1: Values of %̄ for different values of β and γ .

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Table 2: Values of η̄ for different values of β and γ .

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Figure 10. Phase-space distribution near a moving resonance predicted by linear perturbation theory (equation 58). Dotted lines mark the current position of the resonance. Linear theory qualitatively describes the near-resonant dynamics (Fig. 9) only in the fast regime (right figure) where libration is absent.

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Figure 11. Density wakes at the equatorial plane (𝑧 = 0) caused by (a) the corotation resonance 𝑵 = (0, 2, 2) , (b) the outer Lindblad resonance 𝑵 = (1, 2, 2) , and (c,d) the direct radial resonances 𝑵 = (1, 0, 2) and 𝑵 = (2, 0, 2) . The density is calculated from the distribution function evolved with the fast-angle averaged Hamiltonian. The bar lies on the 𝑥-axis and is rotating anti-clockwise. The black circles mark the resonant radius for circular orbits (𝐽𝑟 = 0).

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Figure 13. Left: model for the phase-mixed distribution function (equation 77). Middle: original distribution function (equation 52). Right: residual between the mixed model and full distribution. For all panels, the bar slowed from Ωp = 72 to 36 Gyr−1 with slowing rate 𝜂 = 0.001.

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Figure 14. Time evolution of the dynamical feedback (75) by the corotation resonance. The slowing rate 𝜂 increases from 0.001 (blue) to 0.01 (red) with equal spacing 0.00025.

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Figure 15. Dynamical feedback ?̂? from orbits with a specific 𝑱f as a function of the speed of the resonance 𝑠.

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Figure 16. Comparison between dynamical feedback (smooth line) and total torque (jagged line) by the corotation resonance for 𝜂 = 0.001.

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Figure 17. Dynamical feedback by the four strongest resonances for 𝜂 = 0.001.

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Figure 18. Comparison between the total dynamical feedback (smooth lines) and the total torque (jagged lines) computed from 3D test-particle simulations.

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Figure 3. Level curves of the averaged Hamiltonian (bottom) and its unperturbed component (top). Left column: Hamiltonian ?̄? (equation 16) in the static action coordinate 𝐽s. Right column: Hamiltonian 𝐻 (equation 25) in the comoving action coordinate Δ = 𝐽s − 𝐽s,res (𝑡 ) when 𝜂 = 0.002. 𝐽0 is the characteristic width of the resonance defined in equation (34).

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Figure 4. Time variation of the libration action of an orbit trapped and dragged by a moving resonance. The bar slowed from Ωp = 60 to 30 Gyr−1 in approximately 7.5 Gyr (slowing rate 𝜂 = 0.002) and the orbit was initially placed at 𝐽s = 550 kpc2 Gyr−1. The libration action calculated along the contours of the time-frozen Hamiltonian in the comoving frame 𝐻 (blue) is better preserved than that in the static frame ?̄? (black).

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Figure 6. Schematic diagram of the effective potential of the resonance.

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Figure 7. Black curve: The resonant-volume factor 𝐶 (equation 47) as a function of the dimensionless speed 𝑠 (equation 36). Blue and red curves: The local minimum 𝜃res and maximum 𝜃+sep of the resonant potential 𝑉 in the range 0 < 𝜃res < 𝜋/2 < 𝜃+sep < 𝜋. Green curve: The root 𝜃− sep of the equation 𝑉 (𝜃 ) = 𝑉 (𝜃+sep ) closest to 𝜃+sep (see Fig. 6). The difference Δ𝜃 = 𝜃+sep − 𝜃− sep is the angular width of the trapped phase-space.

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Figure 8. Time evolution of the phase-space distribution around a moving resonance in the slow regime (𝜂 = 0.002, 𝑠 ∼ 0.16). Left column: evolution with the 1D averaged Hamiltonian ?̄? using the CBE (equation 52). Right column: evolution with the 3D full Hamiltonian 𝐻 using a standard testparticle simulation.

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Figure 9. Dependence of the phase-space distribution near a resonance on the bar’s slowing rate 𝜂. Top row: the distribution after the bar has slowed from Ωp = 72 to 36 Gyr−1. Middle row: time evolution of the speed parameter 𝑠 (equation 36). The dotted lines mark the critical value 𝑠 = 1 above which trapping is absent. Bottom row: time evolution of the phase-space area of trapping represented by 𝐽ℓ,sep (equation 44). Decrease in 𝐽ℓ,sep at 𝜂 = 0.01 (third column) implies resonant escape.

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Figure 1. This is a flow chart for our pipeline. Training spectra from SDSS DR7 are used to train a Gaussian process kernel with which to model the quasar continuum (i.e., null model, M𝑁 ). Analytic Voigt profiles are used to construct models for absorption from a C iv doublet (M𝐷) or a generic singlet absorber (M𝑆). Conditioning on DR12 spectra produces a posterior probability estimate for each model that can be used to decide if there is a C iv absorber in the given spectrum or not. Moreover, for the absorber models, M𝐷 and M𝑆 , we have a posterior distribution for each model parameter: absorber redshift, Doppler velocity dispersion for the absorption profile, and the absorber column density.

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Figure 10. Purity (Equation 26) and completeness (Equation 27) of the GP catalogue compared to the PM catalogue for different C iv posterior probability (Equation 14) thresholds. The maximum allowed velocity separation between our catalogue and the PM catalogue absorbers is 350 km s−1. The intersection of the purity (dashed blue curve) and completeness (solid red curve) at a threshold of ∼ 95% gives us a balanced purity/completeness of ∼ 80%.

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Figure 11. The ratio of the difference between rest equivalent width from our pipeline with boxcar flux summation (WGP,flux

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Figure 12. Distribution of W𝑟,1548 for absorbers in four categories described in Section 4.4: detected in both the GP and PM catalogues (thick black line), in the GP uncertain (brown), in GP only (green), in the PM catalogue only (blue). The rest equivalent width distribution is similar for all categories. There are some strong absorbers with (W𝑟,1548 > 1.2Å) classified as “PM only”. Visual inspection of the spectra of these systems indicates that they are part of a triplet/complex absorber or a broad mini-BAL system.

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Figure 13. Example spectrum with two C iv absorbers found by GP with high confidence but not included in the PM catalogue. The QSO-ID is 51994- 0309-592 and 𝑧QSO = 2.76. Posterior probabilities for the two searches are P(M𝐷 ) = [1.00, 0.98]. The maximum a posteriori absorption redshifts are 𝑧C iv = [2.288, 2.650], and the rest equivalent widths are WGP,flux

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Figure 14. Example of an absorber at 𝑧PM C iv = 1.822 detected by the PM catalogue, but assigned a relatively low probability (P(M𝐷 ) = 49%) by the GP catalogue. The QSO-ID for this spectrum is 52367-0332-585, and the quasar redshift is 1.87. The vertical dashed lines show the position of PM absorbers. The posterior absorption probabilities are P(M𝐷 ) = [1.00, 1.00, 0.49, 0.15], with maximum a posterior absorber redshifts of 𝑧C iv = [1.556, 1.827, 1.822, 1.693], and the rest equivalent widths are WGP,flux

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Figure 15. Example spectrum containing a PM only absorber for QSO-ID: 51943-0300-475 and 𝑧QSO = 4.31 where 𝑧PM C iv = [3.5309, 3.5389] (vertical dashed lines). GP assigns P(M𝐷 ) = 1 to 𝑧𝐺𝑃 C iv = 3.540574 which is offset by only 110 km s−1from 𝑧PM C iv = 3.5389. Before the second search, we mask 350 km s−1around the first absorber and thus are unable to detect the second PM catalogue absorber.

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Figure 18. Column density statistics for our GP results (blue histograms) compared to the training C iv catalog from SDSS DR7 (red histograms). Each panel is showing a specific redshift bin. Y-axes show the noralized probability distribusion function (PDF). Please note that column density values should be considered as lower limits ((Cooksey et al. 2013)) because they lines are partially to completely saturated so we can only measure lower limits.

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Figure 2. An example learned quasar emission function (red curve) with the normalised observed smoothed flux (blue curve). The shaded red region shows 1𝜎 uncertainties. The SDSS DR7 quasar has QSO-ID: 51630-0266- 280 and redshift 2.57. Note that we search for absorbers starting 3000 km s−1 red-ward of the quasar’s redshift (shown by the solid red vertical line), so the moderate failure to match the quasar C iv emission line in this case does not lead to an artificial preference for C iv absorption. Prominent emission lines are marked by dashed vertical lines.

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Figure 20. Distribution of the maximum a posteriori Doppler velocity dispersion values for absorbers detected in SDSS DR12 with P(M𝐷 ) ≥ 0.95. Our prior distribution for Doppler velocity dispersion was uniform between 35 km s−1and 115 km s−1but the posterior distribution is bimodal. The larger 𝜎C iv posterior values are mostly associated with C iv absorbers found near low SNR pixels.

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Figure 21. The distribution of the rest equivalent width of the 1548 Å (WGP,Voigt

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Figure 3. Learned covariance matrix K (see Equation 8 and Equation 10) for our null (continuum) model. This matrix is built up by considering the observed flux and noise from our C iv-free training set (see Section 2). Brighter pixels show stronger correlations and darker regions weaker ones. The wavelengths of prominent emission lines are labelled. The bright diagonal implies stronger correlations between pixels at smaller wavelength separation.

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Figure 4. The figure shows the spectrum of QSO-ID: 51608-0267-264 with 𝑧QSO=1.89 (blue) where the singlet model (green) is preferred over the C iv doublet model (red), which is in turn preferred over the null model. If we did not have M𝑆 , our pipeline would have incorrectly detected a C iv absorber at 𝑧C iv = 1.635.

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Figure 5. Prior probability for a spectrum containing 𝑘 C iv absorbers as a function of quasar redshift, for 𝑘 = 1−7. We use the average number of absorbers in the PM spectrum in our wavelength search range. C iv is a priori more likely as 𝑧QSO increases but reaches a plateau at 𝑧QSO ∼ 2.5−3. This is because the C iv wavelength coverage is shorter for low 𝑧QSO as the 1548 Å emission line pushes to the blue-end of the SDSS spectral range. Note that we assume the same prior for the singlet model for 𝑘 = 1−7.

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Figure 6. Example SDSS DR7 spectrum with QSO-ID: 51608-0267-264 and 𝑧QSO = 1.89. Both PM and our pipeline find three absorbers between 𝑧C iv = 1.65–1.85. We also find an absorber at 𝑧C iv = 1.489 (probability 92%) that was not detected by PM, due to noise in this part of the spectrum (specifically, the 1550 line was not automatically detected with their parameters, thus the doublet was not visually inspected). The probabilities that our pipeline provides for the existence of the first, second, third, and forth C iv absorber are 𝑃 (C iv) = [1.00, 1.00, 1.00, 0.92], respectively, our maximum a posteriori absorber redshift values are 𝑧C iv = [1.829, 1.672, 1.775, 1.489], and our rest equivalent widths from Voigt profile integration (see Equation 30) are 𝑊GP 𝑟,1548 = [1.37, 0.87, 0.90, 0.79] Å. In the PM-catalogue the absorber redshifts are 𝑧PM = [1.831, 1.673, 1.777] with corresponding 𝑊PM 𝑟,1548 = [1.21 ± 0.18, 1.40 ± 0.20, 0.94 ± 0.19] Å

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Figure 7. Velocity difference between the detected absorbers in the GP pipeline with P(M𝐷 ) ≥ 0.95 in the validation set and the absorbers in the PM catalogue. Only absorber pairs closer than 350 km s−1are shown. The thick red line shows 𝛿vPM,GP = 0 and the dashed lines are 𝛿vPM,GP = ±150 km s−1(the SDSS spectral resolution). The median offset is 𝛿vmed PM,GP ≈-50 km s−1, which is less than an SDSS pixel (69 km s−1).

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Figure 8. Velocity separation (Equation 25) between GP and PM detected C iv absorption systems is shown versus the reported rest equivalent width values for 1548 Å in the PM catalogue (WPM 𝑟,1548). There is no correlation between the velocity separation and the strength of detected absorbers.

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Figure 9. Receiver Operator Characteristic (ROC) curve for our DR7 validation. True Positive Rate is plotted versus False Positive Rate. True positives are C iv systems in our catalogue at least 350 km s−1apart from an absorber in the PM catalogue with ranking ≥2 given any P(M𝐷 ) threshold between 0 and 1. False positives are those absorbers in our catalogue that do not have any matching absorber in the PM catalogue; though they may be real C iv absorbers (see Figure 6). Above a relatively small False Positive Rate (∼ 0.2), our algorithm procedure obtains True Positive Rate above 80% and, hence, is a successful way to identify C iv absorbers. The area under the ROC curve (AUC) is a quantitative metric for the equality of the GP algorithm; we get AUC = 0.87, a reasonable value compared to an ideal classification that gives AUC = 1.00.

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Table 1. For each sight-line, identified by Column 1 and 2, we report the absorber’s redshift (Column 3), column density in log(cm−2 ) (Column 4), Doppler velocity dispersion in km s−1(Column 5), rest equivalent width for 1548 Å W𝑟,1548 (Column 6), rest equivalent width for 1550 Å W𝑟,1550 (Column 7), the posterior probability of the C iv absorber P(M𝐷) (Column 8), and the posterior probability of the singlet absorber P(M𝑆) (Column 9). We show only absorbers with P(M𝐷)≠NaN. This table demonstrates a portion of the full table for the first ten rows. Note that those measurements with large errors are uncertain (i.e. low absorption model posterior probability). The full table with 445,765 rows is available at https://doi.org/10.5281/zenodo.7872725.

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Table 2. The number of spectra containing different numbers of C iv absorbers for various doublet model probability thresholds, P(M𝐷). The first column shows the number of C iv absorbers found within each spectrum (see Section 3.6) . The second through fourth columns show probability thresholds of > 65%, 85%, and 95% respectively. Cells show the number of quasar spectra falling in each category, together with the corresponding percentage of the 185,425 spectra in our SDSS DR12 sample.

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Fig. 3. L′line/LIR [K km s−1 pc2 L−1 ] ratios as a function of LIR. Clockwise, from top left: CO (5−4), CO (7−6), [C I](3P1 − 3P0), and [C I](3P2 − 3P1). D49 is shown as a red star, BX610 as a turquoise diamond, NGC 6240 as a purple square, and MD94 as a green triangle. The blue and orange points are respectively from the local and high-redshift galaxy samples in Valentino et al. (2020), green points are the samples from Boogaard et al. (2020), and pink points are from Valentino et al. (2021). BX610 and MD94 are not shown in the CO (5 − 4)-IR plane (top left) due to lack of CO (5 − 4) observations. The black dashed and grey lines indicate the best fit in the log(LIR)-log(L′line) space and random sampling of the posterior distribution. Two galaxies in Valentino et al. (2021) are found to have high L′CO(5−4)/LIR ratios (pink down-pointing triangles in the upper left panel), both of which are also found to have strong AGN signatures ( fAGN ∼ 0.9). Three galaxies in Boogaard et al. (2020) are found to have high L′CO(7−6)/LIR ratios (green down-pointing triangles in the upper right panel, one of which is obscured by purple square). AGN signatures are detected in two of them ( fAGN ∼ 0.08), while the other one, with the highest L′CO(7−6)/LIR ratio, is labelled as a non-AGN.

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Table 1. Best-fit results of CO and [CI] lines of D49, obtained by fixing the redshift and FWHM for CO (5 − 4), CO (7 − 6), and [CI] (2 − 1). The upper limit on [CI] (1− 0) is computed within ±3σv, with the bestfit redshift zCO(5−4) = zCO(7−6) = z[CI](2−1) = 2.847. CO (3 − 2) is from Magdis et al. (2017).

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Table 2. Stardust fitting result.

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Table A.2. Hydrogen mass estimations by different methods.

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Figure 1: The workflow of exploratory analysis.

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Table 1: Top-10 most cited documents in the Cambridge Journal of Economics (1985-2013)

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Table 2: Generalized distance correlation between article similarity matrices.

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Table 3: Clusters of articles obtained through Louvain algorithm applied to article similarity matrices based on cited references, topics and bags of words.

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Table 4: Values of Cramer’s V for measuring the association between clusters obtained from article similarity matrices based on cited references, topics and bags of words.

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Table 5: Generalized distance correlation between fused matrices and similarity matrices based on topics and cited references.

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Table 6: Generalized distance correlation between fused matrices.

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Table 7: Partial distance correlation between the seven fused networks and the two corresponding layers. Each row reports the values of partial distance correlation between the fused networks and one of the layers, conditioned on the other layer.

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Table 8: Clusters of articles obtained through Louvain algorithm applied to fused networks.

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Table 9: Values of Cramer’s V for measuring the association between classifications of articles in the fused networks.

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Figure 1. Evolution of cosmological comoving length scales with the scale factor at different epochs of the evolution of the universe. Here “RD” stands for radiation domination, “MD” implies matter domination, and “DE” indicates dark energy domination. In each case corresponding equation of state is also mentioned.

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Figure 2. ϕ → SS scenario. Top Left: Reheating temperature Trh as a function of n. The red part of the curve is discarded from present bound on ∆Neff due to PLANCK [61]. The grey-shaded region is in conflict with the BBN limit on Trh.Top Right: Ω (0) GW as a function of frequency, where different curves correspond to different choices of Λ for a fixed gSϕ and n.The orange-shaded regions are discarded from ∆Neff due to overproduction of GWs. Bottom Left: Same as top right, but for fixed Λ and n, with different choices of gSϕ specified below the curves. Bottom Right: Same as bottom left but for a fixed Λ and gSϕ, different curves correspond to different n values. In the bottom and upper-right panels the parameters Λ, n and gSϕ are chosen so that the BBN limit Trh > 4 MeV is satisfied [cf. Fig. 5]. Moreover, in the bottom panels, we show sensitivities of future GWs experiments.

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Figure 3. Same as Fig. 2, but for ϕ → ψψ scenario. In the top left panel the vertical dashed line corresponds to n = 7/2, for which βψ = (n+ 4)/(n+ 1) (see text for details).

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Figure 5. ϕ → SS (top left), ϕ → ψψ (top right) and ϕ → aa (bottom) scenario. In all cases the red shaded region is disallowed from BBN bound on reheating temperature Trh < 4 MeV, the cyan dashed region is forbidden from PLANCK observed ∆Neff bound on GW overproduction at f = fmax, and the “CMB bound” discards scale of inflation Λ > 1.34 × 1016 GeV from constraint on tensor to scalar ratio.

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Figure 7. Left: Contours satisfying relic abundance for ϕ → SS (black solid), ϕ → ψψ (black dotted) and ϕ→ aa (black dot-dashed) scenarios. Right: GW spectrum for the three cases. We fix Λ = 1015 GeV and n = 10, while choose different coupling strengths (see text).

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Table 1. Numerical values of the summation factors appearing in Eq. (A.21), for different values of n.

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Table 2. Numerical values of the inflaton field amplitude and the corresponding number of e-folds N⋆ for different values of n and fixed α = 1/6.

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Figure 1: Transition Rules of MCSat

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Table 1: Instances solved by FFSat, GB, and GBlex, out of 25 polynomial systems per test set.

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Fig. 1: The maximum number of supported players in Servo.

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Fig. 10: Serverless Terrain Generation Performance.

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Fig. 12: Performance improvement of serverless terrain generation (Servo) compared to local generation (Opencraft) for varying workloads.

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Fig. 13: Terrain retrieval latency for local and cloud storage.

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Fig. 2: An operational model of MVEs. Elements highlighted in blue indicate the components most relevant to this work.

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Fig. 3: Download latency (variability) from Azure Blob Storage for two types of game data.Vertical bars indicate approximate network latency thresholds for FPS (blue, left), RPG (green, middle) and RTS (red, right) games [35].

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Fig. 4: Servo design overview. Simulated constructs (SC) and

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Fig. 5: A visual representation of speculative execution in Servo. Colored areas are simulated constructs. White areas are static parts of the virtual world.

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Fig. 6: Example of speculative execution for simulated constructs. Servo simulates the construct locally, until it receives the results from the remote function.

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Fig. 7: Comparing the scalability of Servo with alternative systems. Scalability is expressed in maximum number of players while maintaining stable performance. Missing bars indicate the game could not support any players.

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Fig. 8: Efficiency of offloaded simulation for varying tick leads (left), and simulation lengths (right).

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Fig. 9: Latency (left) and number of invocations per minute (right) for varying simulations lengths in ticks.

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TABLE I: Overview of Experiments. Experiments focus on Simulated Constructs (SC), Terrain Generation (TG), and Remote Storage (RS). Components either run locally (L), use serverless computing (S), or combine the two (L+S). For a detailed description of parameters, workload, environment, and explanations on the notation, see Section IV-A.

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Fig. 1. In this figure, leaves are represented by open circles, tree vertices as filled circles, reticulations as filled squares, and the root of the network as a filled, inverted triangle. Network N1 is a level-1 network with |R(N )| = 2. N1 is a reticulation-trimmed subnetwork of N1 with respect to F = ∅. Network N2 = N1〈(d, e)〉, where (d, e) is a simple cherry/reduction. Network N3 = N2〈(e, f)〉 where (e, f) is a reticulated cherry/reduction. N3 is reticulation-trimmed subnetwork of N1 and of N2 with respect to F = {(x, r2)}. Network N4 = N3〈(c, e) · (f, e) · (e, b)〉 and is a reticulationtrimmed subnetwork of N1 and of N2 with respect to F = {x, r2), (v, r1)} or to F = {(w, r2), (v, r1)}. Network N5 ≃ N4, in fact, there are CSs that may head lead to leaf e being any of leaves c, d, e, or f . Each of these networks would have the same label set on that leaf, and all are weakly isomorphic with N5.

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Fig. 2. In this figure, leaves are represented by open circles, tree vertices as filled circles, reticulations as filled squares. A subnetwork without reticulations is represented by a large open triangle, a subnetwork that may be reticulated is represented by a large open blob. This Figure shows an example of the operation of Algorithm 3, note how R(u) ∪ R(v) \ {v} = ∅ in this example. Subnetwork under label (1) is an example network at line 7, the dotted line represents the removed reticulation edge (u, v) by line 5 and both leaves u′ and v′ have been constructed (leaf labels are not shown). The network under label (2) shows the state of network (1) at line 8 when edges (p(u), u′) and (p(v), v′) have been added.The network under label (3) shows the state of the network under (1) at line 9 when vertices in reach(u,N ′)∪ reach(v,N ′) are removed.

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Fig. 3. In this figure, tree vertices as filled circles and reticulations as filled squares. A subnetwork is represented by a large open blob. vertices in red are in the same nontrivial biconnected component. Yellow edges are path π1 1 and green edges are path π1 r . Tree vertex v is a trivial biconnected component itself such that R(v) 6= ∅.

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Figure 2: a) End-to-end and b) iterative Blueprint models. The end-to-end model generates the entire blueprint plan before generating the output text, while the iterative model plans and generates one proposition at a time, conditioning on the input and the sentences generated so far. Each portion of the output is color-coded with its corresponding question-answer pair.

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Figure 3: Schematic representation of the different components of the web browser-based demonstration.

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Figure 4: Example snapshot of the results obtained with the end-to-end Blueprint model for the user query "Why is the sky blue?". Depending on which question-answer pairs the user selects, different summaries can be generated.

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Figure 5: Snapshot of the results obtained with the interactive Blueprint model for the query “What is the Titanic known for?”. Questions highlighted in red were manually added by the user, leading to a different output.

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Table 1: Examples of machine-generated and manually-edited plans and their corresponding summaries. We highlight in bold changes made by the user and the resulting changes to the summary.

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Table 2: Examples of machine-generated and manually-edited plans and their corresponding summaries (Continued). We highlight in bold changes made by the user and the resulting changes to the summary.

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Figure 1. Schematic diagram of the interface separating two fluids in relative motion. The interface (grey color) is located in the 𝑥 − 𝑧 plane. Above and below the interface are fluids I and II with the same Alfvén speed 𝑣A and sound speed 𝑐s. q∥ is the projection of wavevector q onto the interface. We consider the system in the laboratory frame where fluids I and II have shear velocities +𝑣?̂? and −𝑣?̂?, respectively. B0 is the magnetic field vector in the fluid rest frame. 𝜃 is the angle between q∥ and ?̂? (in the laboratory frame) while Ω is the angle between B0 and ?̂? (in the fluid frame).

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Figure 2. Dependence of the instability growth rate Im(𝜙) on 𝑀𝑟 andΩwhen 𝜃 = 0 (i.e., wavevector along the shear flow), for three choices of 𝑣A and 𝑐s. Panels in the first, second, and third columns represent 𝑣A = 0.1, 0.5 and 0.9. Panels in the first, second, and third rows represent 𝑐s = 0, 1/2 √ 3, 1/ √ 3. In all the panels, Im(𝜙) is then normalized to its maximum value, which is quoted in the panels themselves. The dashed cyan vertical lines represent the common upper bound 𝑀re = √ 2 across all panels. The dashed white lines in the top three panels represent the exact boundaries of the instability region when 𝑐s = 0 and cos 𝜃 = 1, see Equation 28. The dotted white lines in the middle and bottom rows represent the lower bound of the fast-mode unstable region imposed by magnetic tension, as in Equation 25.

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Figure 3. Dependence of the instability growth rate Im(𝜙) on 𝑀re and 𝜃 when Ω = 0 (i.e., magnetic field along the flow direction), for three choices of 𝑣A and 𝑐s. The dashed cyan vertical lines represent the common upper bound 𝑀re = √ 2 across all panels. The dashed white lines in the top three panels represent the exact boundaries of the instability region when 𝑐s = 0 and cosΩ = 1, see Equation 31. See the caption of Figure 3 for further details.

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Figure 4. Dependence of the instability growth rate Im(𝜙) on 𝑀re and 𝜃 when Ω = 𝜋/2 (i.e., magnetic field perpendicular to the flow direction), for three choices of 𝑣A and 𝑐s. The dashed cyan vertical lines represent the common upper bound 𝑀re = √ 2 across all panels. The dashed white lines in the top three panels represent the exact boundaries of the instability region when 𝑐s = 0 and cosΩ = 0, see Equation 34. See the caption of Figure 3 for further details.

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Figure 1: Left: The time evolution of the upper bound on Nielsen’s complexity for the Hamiltonian evolution of the transverse Ising model at (hx, hz) = (−1.05, 0) (blue) and the Ising model evaluated at a chaotic parameter point (hx, hz) = (−1.05, 0.5) (red). The length of the chain is L = 12. To avoid using candidate minima that are evidently sub-optimal in estimating the complexity (2.24), at every time step we compare the output of the minimization procedure to the complexity associated to the early time solution kn = 0 and the minimum of the two is chosen. Right: A zoom on the time window inside the plateau region used in the numerics.

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Figure 10: Histograms of the eigenvalues of the Q-matrix for Left: the chaotic Ising model (4.4) with (hx, hz) = (−1.05, 0.5) for L = 9, 10, 11 and fixed Nloc/D 2 = 0.15. For the purpose of comparing the concentration properties of the distribution at varying L, we normalize the distributions so that the area below all the staircase curves is unity. The peak of the distribution can be seen to be close to 0.85, as expected from (3.8). Right: The same histograms for the transverse Ising at (hx, hz) = (−1.05, 0), for comparison.

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Figure 13: Left: Saturation values of the complexity as a function of kint, with ksp = kop = ⌈kint/2⌉, for the non-integrable (4.25) and integrable (4.29) spin 1 Hamiltonians. The length of the chain is set to L = 7. Right: In order to highlight the complexity reduction of the integrable model, we display the ratio between the integrable and non-integrable curves on the left plot.

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Figure 2: Histograms of the eigenvalues of the Q-matrix for Left column: an integrable Ising spin chain (4.4) with (hx, hz) = (−1.05, 0) and Right column: a chaotic Ising spin chain with (hx, hz) = (−1.05, 0.5) for varying locality thresholds as Top row: ksp = kop or Bottom row: only varying kop while keeping ksp = 6 fixed. The length of the chain is set to L = 12. In the bottom left inset of each plot we zoom in on the size of the kernel of the Q-matrix. It can be seen that in the integrable case, the number of zero eigenvalues increases in steps of 2, in accordance with the form of the conserved charges (4.10)-(4.13).

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Figure 3: Left: The fraction of local generators to the total number of generators, as a function of kop for the two types of locality thresholds T1 and T2 we impose in the main analysis. Middle and Right: The distance between the mean of the Q-eigenvalue distribution and the right edge of the Q-histograms, for, respectively, the chaotic and integrable Ising model as a function of kop. The shape of the curves in all three plots is very similar. The results for the chaotic model qualitatively agree with the RMT prediction (3.8) up to a factor of order one.

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Figure 4: The saturation value of the complexity of the integrable (hx, hz) = (−1.05, 0) and chaotic (hx, hz) = (−1.05, 0.5) Ising model with L = 12 sites as a function of a locality threshold specified by k. The dashed line corresponds to ksp = 6 fixed and varying kop, while for the solid line we vary both locality degrees ksp = kop.

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Figure 5: Histograms of the eigenvalues of the Q-matrix for the L = 12 Left column: integrable XYZ model (4.14) and for the Right column: chaotic XYZ model with magnetic field (4.19) (right) for varying locality thresholds as Top row: ksp = kop or Bottom row: only varying kop while keeping ksp = 6 fixed. For the coupling constants we used the numbers (Jx, Jy, Jz) = (−0.35, 0.5,−0.1) and on the right plot we chose hz = 0.8. In the bottom left corner of each plot we zoom in on the zero eigenvalues. The number of zero Q-eigenvalues in the integrable case increases in accordance with the form of the conserved charges.

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Figure 6: The saturation values of the complexity of the integrable XYZ model (4.14) and the chaotic Hamiltonian with magnetic field (4.19) for L = 12. For the coupling constants we used the numbers (Jx, Jy, Jz) = (−0.35, 0.5,−0.1) in both cases and hz = 0.8 for the chaotic Hamiltonian.

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Figure 7: Histograms of the Q-eigenvalue distributions as a function of k for a set of local generators T3(k). Left: The integrable transverse Ising model at (hx, hz) = (−1.05, 0). Right: A chaotic representative of the Ising model at (hx, hz) = (−1.05, 0.5). In both cases, the length of the chain is set to L = 12.

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Figure 8: Left: Saturation values of the complexity as a function of k for a set of local generators T1, with and without the operators (4.20). This is displayed for chaotic and integrable instances of the Ising model at L = 12. Right: The mean of the Q-eigenvalue distributions shown in the left plot of Figure 7 and the upper left plot of Figure 2.

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Figure 9: Left: The distribution of eigenvalues of the Q-matrix for the integrable XXZ model at (Jx = Jy, Jz) = (−0.35,−0.1) for increasing T1(k), for L = 12. In the bottom left corner, we zoom in on the number of exact zero eigenvalues. Right: A comparison between the late-time complexity saturation values as a function of T1(k) for the integrable XYZ at (Jx, Jy, Jz) = (−0.35, 0.5,−0.1) and XXZ model at (Jx = Jy, Jz) = (−0.35,−0.1).

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Fig. 1. Overview of the systematic review process

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TABLE I HermesBDD NON-PARALLEL EXECUTION TIME BASED ON THE n-QUEENS PROBLEM COMPLEXITY. VALUES ON THE AVERAGE OF 50 SAMPLES USING THE STATIC MEMORY ALLOCATION ON A 32-CORE MACHINE.

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Fig. 1. Overview of ViP-NeRF architecture. Given the images from primary and secondary views, we estimate a visibility prior map in the primary view and use it to supervise the visibility of pixels as predicted by the NeRF. Specifically, we cast a ray through a randomly selected pixel in the primary view and sample 3D points along the ray. For every point p𝑖 , we use the NeRF MLPs to obtain its visibility in primary and secondary views, along with volume density 𝜎𝑖 and color c𝑖 . Volume rendering outputs visibility 𝑡 ′ of the chosen pixel in the secondary view which is supervised by the visibility prior. L𝑣 constrains the visibilities𝑇𝑖 output by network and𝑇𝑖 computed using volume rendering to be consistent with each other.

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