Question: What are the asymptotic properties of the ancient ovals as described in Theorem 1.1, and how do they differ from the conditions in Theorem 1.6?

Answer: The ancient ovals described in Theorem 1.1 have the following asymptotic properties: for t → 0, they converge to a round point, and for t → −∞, they have the following asymptotics: (1) asymptotic shrinker: for λ → ∞, the parabolically rescaled flows λ−1Mλ2t converge to the round shrinking cylinder Sj(√(2j|t|)) × Rn−j; (2) asymptotic translators: given any direction v ∈ Rj+1 × Sn−j−1, the blow-downs at the tip in direction v converge to Bowl(v)j+1t × En−j−1, where Bowl(v)j+1t ⊂ Rj+1 × ⟨v⟩ translates in direction −v and En−j−1 is the orthogonal complement of ⟨v⟩ in Rn−j. Additionally, these solutions are α-Andrews noncollapsed and have the following properties: (a) uniformly (n − j + 1)-convex, (b) unbounded rescaled diameter, and (c) type II curvature decay. These properties differ from those in Theorem 1.6, where the conditions (a) uniformly convex, (b) bounded rescaled diameter, and (c) type I curvature decay force the solution to be a family of round shrinking spheres.


Question: What contradiction arises if an outcome W that satisfies µ-PJR-x is assumed to have a W ∗-cohesive group N ′ with a project p that can be added to W without violating the budget constraints?

Answer: This contradicts the assumption that W satisfies µ-PJR-x because there would be a project p that can be added to W to form a new outcome W ∗, such that µ((W ∩ [i∈N ′ Ai) ∪{p}) ≤µ(W ∗), which is not possible if W already satisfies µ-PJR-x.


Question: What does the candidate text reveal about the effectiveness of the metrics used to measure semantic diversity in the patches generated by the repair techniques?

Answer: The candidate text reveals that the metrics for semantic diversity were unable to distinguish between patches for almost every bug, regardless of the repair technique used, suggesting either a lack of patch diversity or weakness in the metrics of semantic distance.


Question: How does the model account for the line intensity contributions from the Local Bubble (LB) and the hot gas halo in the calculation of the total line intensity?

Answer: The model accounts for the line intensity contributions from the Local Bubble (LB) and the hot gas halo by defining the intensity contribution from the LB as ILB(l, b) = n2LBL(l, b)ϵ(TLB) / 4π, where L(l, b) defines the LB path length. The total line intensity is then defined as Itotal(l, b) = ILB + e−σNHI(Ihalo + IF B + Ishell), where the exponential term accounts for attenuation due to neutral hydrogen in the disk, and Ihalo, IF B, and Ishell represent the line intensities from the hot gas halo, Fermi bubbles, and shell, respectively.


Question: How does the QSNR of the MX formats compare to other quantization formats in terms of hardware efficiency and numerical fidelity?

Answer: The QSNR of the MX formats (MX4, MX6, MX9) is higher compared to other quantization formats like FP8, MSFP, and VSQ, while maintaining a good balance in hardware efficiency. In Figure 7, the MX formats are shown to have a higher QSNR (in decibels) on the y-axis, indicating better numerical fidelity, while also having a lower area-memory efficiency product on the x-axis, indicating better hardware efficiency.


Question: What are the implications of the entanglement properties of Gaussian cluster states for scalable measurement-based quantum computation (MBQC)?

Answer: The entanglement properties of Gaussian cluster states, particularly their behavior under finite squeezing and photonic loss, suggest that they may face challenges in being used as universal state preparators (USPs) for scalable MBQC. While the entanglement (quantified using the entanglement width and logarithmic negativity) increases linearly with the grid size, the presence of losses significantly reduces the available entanglement. Additionally, the entanglement width must scale faster than logarithmically in the system size for a state to be an efficient USP, and the criteria derived by Van den Nest et al. for qubit states can be applied to Gaussian cluster states, but the choice of a suitable CV state for comparison is problematic due to the lack of generally-accepted USPs in the CV case.


Question: How does the circular velocity profile of the galaxy change when considering the contributions of all components, and what is the dominant component?

Answer: The circular velocity profile of the galaxy is dominated by dark matter (DM) at all radii. The contributions from the stellar and gaseous disc, bulge, and halo are combined to estimate the total circular velocity profile, Vc, using the equation Vc^2(x) = V_halo^2(x) + V_*,disc^2(x) + V_gas^2(x) + V_bulge^2(x).


Question: How do the posterior mean estimates of the treatment effect compare across the exponential, general Bayesian, and Weibull models when data are generated from the super model with different values of β?

Answer: When data are generated from the super model, the exponential model tends to under-estimate the treatment effect, while the Weibull model tends to over-estimate it. The general Bayesian model provides relatively unbiased estimates of the treatment effect. This is particularly noticeable for larger treatment effects, and when there is no treatment effect, the differences between the distributions are minimal.


Question: What are the challenges and methods used in studying two-temperature plasmas, and how do they relate to the systems discussed in the abstract?

Answer: The study of hot strongly-coupled dense charged fluids, especially near the regime of molecular and atomic species or excitons in electron-hole plasmas, is challenging. Methods based on density-functional theory (DFT) have been used with molecular-dynamics based approaches and multi-component integral-equation approaches to study equilibrium properties and linear transport properties of plasmas. Two-temperature plasmas, where electrons and ions have different temperatures, occur in laser-produced plasmas, shock-wave generated plasmas, and astrophysical settings. These plasmas are influenced by the extreme mass ratio between ions and electrons, and similar but less well-defined situations can arise in electron-hole plasmas, such as in GaAs and Si. Quantum Monte-Carlo methods are not yet available for simulating such systems, so analytical methods based on integral-equation techniques are sought after for their computational simplicity and physical insight.


Question: What is the general form of the symmetry algebra for diffeomorphism-invariant theories, and how does it reduce in the case of general relativity?

Answer: For generic diffeomorphism-invariant theories, the symmetry algebra is Diﬀ(∂Σ) ⋉ SL(2, R) ⋉ R2·(d−2)∂Σ. In the case of general relativity, this algebra reduces to Diﬀ(∂Σ) ⋉ SL(2, R)∂Σ, consistent with the results of [130].


Question: How does the performance of the proposed algorithm change with varying degrees of data heterogeneity, and what is the impact of increasing the number of digits in the dataset split?

Answer: The proposed algorithm performs the worst in the 1-digit case with the highest data heterogeneity, and the performance improves as the number of digits increases due to reduced data heterogeneity. This demonstrates the negative impact of data heterogeneity on the convergence performance.


Question: How does the choice of the proposal pdf q(x) in importance sampling affect the efficiency of the Monte Carlo integration?

Answer: The proposal pdf, q(x), should have heavier tails than the target pdf, eπ(x), meaning that eπ(x) decays faster than q(x) when x is far from the region where most of the probability mass is concentrated. This ensures that the samples drawn from q(x) are representative of the target distribution, leading to more efficient Monte Carlo integration.


Question: How does the presence of a close companion affect the star formation rate (SFR) of HαGS galaxies, and what does this imply about their mass and luminosity?

Answer: Galaxies with a close companion tend to be more massive and luminous than the median across the whole sample, but their SFRs are not enhanced with respect to the median of the distribution at their absolute magnitude. This suggests that the SFR of galaxies with a close companion is enhanced mainly because they are big (luminous, massive) galaxies, not because of the interaction itself.


Question: How does the proposed approach ensure that the grippers are suitable for the assembly task while considering both grasping and collision avoidance?

Answer: The proposed approach ensures that the grippers are suitable for the assembly task by first using mesh segmentation and primitive fitting to determine suitable gripper types and parameters. Then, the segments and grippers are evaluated under assembly constraints, such as affordance and collision avoidance. This process includes checking for collision-free grasps and excluding segments that are not suitable for grasping due to their affordance. Finally, the number of grippers is optimized to reduce the total cost.


Question: What is the nature of the divergent part of the partition function for the Fisher deformation, and how is it handled in the context of BRST-exactness?

Answer: For the Fisher deformation (k2 = 1), there is a logarithmic divergence in the second term of the divergent part. However, this second term is BRST-exact, and thus can be removed by adding a BRST-exact term to the partition function, effectively eliminating the divergence.


Question: What is the historical context and solution to the Kronecker problem, and how does it relate to the classification of indecomposable representations of the 2-Kronecker quiver?

Answer: The Kronecker problem, which involves classifying indecomposable Kronecker modules, was solved by Weierstrass in 1867 for some particular cases and by Kronecker in 1890 for the complex number field case. This problem is equivalent to finding the canonical Jordan form of pairs of matrices (A, B) with respect to elementary transformations over a field k, assuming k is algebraically closed. The solution allows for the classification of indecomposable representations of the path algebra kQ, where Q is a quiver of the form (13). Specifically, the preprojective component of the Auslander-Reiten quiver of the 2-Kronecker quiver has vertices representing isomorphism classes of indecomposable representations of type III, while the preinjective component has vertices representing isomorphism classes of indecomposable representations of type III*.


Question: How does the proposed method for functional motif discovery differ from existing methods in terms of motif identification and clustering?

Answer: The proposed method, probabilistic K-mean with local alignment (probKMA), identifies motifs locally on portions of misaligned curves and allows each cluster to contain multiple portions of the same curve. It also allows each curve to belong to multiple clusters, capturing multiple functional motifs. Unlike existing methods that consider curves globally or require user-specified motif lengths and radii, probKMA does not require the motifs, their number, lengths, or radii to be known in advance, and these parameters are specific to each motif. Additionally, probKMA incorporates derivatives to capture complex shape characteristics and reduces noise through smoothing.


Question: How is the utility of self-supervision quantified in the context of the number of labeled examples needed to achieve the same accuracy?

Answer: The utility of self-supervision is quantified as the ratio of additional labels needed to match the accuracy of the finetuned model. Specifically, if a(n) is the accuracy of a model trained from scratch given n labeled examples and aft(n) is the accuracy of the finetuned model, the utility at n is defined as U(n) = ˆn/n − 1, where ˆn is the number of labels required such that a(ˆn) = aft(n).


Question: How does the LP relaxation relate to the integer linear program (IP) in terms of feasible sets and optimal solutions?

Answer: The LP relaxation of an integer linear program (IP) is obtained by replacing the integer constraint with a real number constraint. The feasible set of the LP relaxation is denoted as 𝒫, and the convex hull of integer points in 𝒫 is denoted as 𝒫I. Solving the IP is equivalent to solving the LP over the convex hull 𝒫I. If 𝐴′ and 𝑏′ are such that 𝒫I = 𝒫(𝐴′, 𝑏′), then the optimal IP solution is also optimal for the LP relaxation.


Question: What are the key improvements made to the nonlocal contact formulation to better predict contact force and radius in elastic confined granular systems?

Answer: The key improvements include the development of an analytical framework for predicting nonlocal effects in the evolution of inter-particle contact area and the introduction of curvature corrections to the description of contacting surface profiles through higher order terms in the Taylor series expansion of the profile functions. These corrections improve the accuracy of the formulation, especially for moderate to large deformations, and are validated through comparison with finite element simulations and experimental measurements.


Question: What are the specific conditions that the derivatives qij must satisfy for a family of discrete ϑ-conformal maps?

Answer: The derivatives qij on the interior edges [vi, vj] ∈ E must satisfy the conditions: ∑_{vj:vj adjacent to vi} qij = 0 and ∑_{vj:vj adjacent to vi} qij (vi - vj) = 0. Additionally, for a family g(t) of discrete ϑ-conformal maps, qij ∈ ieiϑR.


Question: How does the process of adding time derivatives of constraints to the equation of motion ensure consistency in the theory, and what is the final form of the equation of motion after the nth stage?

Answer: In order to maintain consistency, the time derivatives of the primary constraint (10) are added to the equation of motion (2). This results in N + A1 number of equations containing accelerations, which can be written as Li1 = w1i1j ¨qj + α1i1 = 0, where i1 = 1...N + A1. This process is repeated for each stage, adding the time derivatives of secondary and higher-order constraints. Finally, in the nth stage, the equations of motion for the system are of the form Ln = wninj ¨qj + αnin, where in = 1...N + A1 + An. If no new null eigen vector is found at this stage, the process terminates.


Question: What is the proof that the codes C(m)1 are the only perfect codes in ∆1_ℓ?

Answer: Assume that r ≤ e. In that case the “leftmost” codeword of C(m)1 is (ℓ−r + m −1, r −m + 1), m = 1, . . . , r + 1. Therefore, we have found r + 1 codes with “leftmost” codewords (ℓ, 0), . . . , (ℓ−r, r). Suppose that we try to construct another perfect code by specifying (ℓ−r −k, r + k), k > 0, as its “leftmost” codeword. Since the end point (ℓ, 0) has to be covered, we can assume that k ≤ e−r. Then its “rightmost” codeword is obtained by shifting for i(2e+1) and is therefore either (2e+1−k, ℓ−2e −1 + k) (for i = q −1) or (−k, ℓ+ k) (for i = q). The second case is clearly impossible, and the first fails to give a perfect code because the point (0, ℓ) does not belong to a decoding region of some codeword (its distance from the “rightmost” codeword is 2e + 1 −k > e). Again, the proof is similar for r > e.


Question: How does the unitary shift operator S act on the quantum states in the model of unitary quantum walks on the half-line?

Answer: The unitary shift operator S acts as follows: when located at any site 0, 1, 2, 3, ..., an upward spin migrates one step to the right. When located at any strictly positive position 1, 2, 3, ..., a downward spin migrates one step to the left. When located at the special site 0, a downward spin reverses orientation and remains at site 0, so that, in the next iteration, it migrates to site 1.


Question: What method was used to fit the [O III] λ5007 and Hα λ6563 emission lines in the spectra, and what were the rest wavelengths used for these lines?

Answer: The [O III] λ5007 and Hα λ6563 emission lines were fit with Gaussians over an average continuum taken from line-free regions throughout the spectrum. The rest wavelengths used for these lines were 5008.2˚A for [O III] and 6564.6˚A for Hα.


Question: How does the dynamic phase factor χ(ϵ, t) change in response to a voltage step, and what is its impact on the tunneling of IQPs?

Answer: The dynamic phase factor χ(ϵ, t) changes to χ′(t) = χ(t) + ϕ(t)/2 due to the voltage step. This alteration causes IQPs tunneling at different times t′ to acquire different phases ϕ, commensurate with the time spent in the biased lead since the rise of the voltage step. The resulting mutual interference is captured by the factor exp[i(ϕ(t) − ϕ(t′))/2] in Jp, generating oscillations with frequency ω′J ≈ ωJ/2.


Question: What are the mean and median Monte Carlo estimates for θ1 and VEs prior to and after 20 weeks in the scenario with no confounding and θ1 = log(7)?

Answer: For θ1, the mean is 1.961 and the median is 1.935. For V E≤20, the mean is 0.950 and the median is 0.953. For V E>20, the mean is 0.634 and the median is 0.663.


Question: What is the fix pattern for addressing randomness-related flaky tests, and how is it applied in the example provided in Listing 3?

Answer: The fix pattern for randomness-related flaky tests is to set a random seed value to a constant. In the example provided in Listing 3, the fix is applied by replacing the default non-constant random seed value with a fixed seed value, as shown in Listing 9. Specifically, the lines are changed from first_circuit = random_circuit(num_qubits[0], depth) to first_circuit = random_circuit(num_qubits[0], depth, seed=4200) and from circuit = random_circuit(num, depth) to circuit = random_circuit(num, depth, seed=4200). This ensures that the random circuits generated are consistent across test runs, thus eliminating the flakiness.


Question: How do the new rate measurements from the IfA Deep survey compare to the original BT06 rates and other literature rates?

Answer: The new IfA rate measurements are plotted as downward-pointing blue triangles and are in much better agreement with the consensus of literature rates, while the original IfA Deep rates (BT06) are shown as upward-pointing green triangles.


Question: How does the type I error rate change with different sample sizes in the further experiments, and what methods are used to estimate the covariance?

Answer: The type I error rates remain stable as the sample size increases, as shown in Table 2. The methods used to estimate the covariance include Enorm, EP C, Ec, Bs, BEc, Rz, and Rzs. For example, with a sample size of 100 and ν = 1/2, the type I error rates are .052 for Enorm, .255 for EP C, .056 for Ec, .058 for Bs, .060 for BEc, .059 for Rz, and .001 for Rzs.


Question: What are the estimated inclination angles for galaxies C and D, and how do they relate to the gas kinematics presented in Table 2?

Answer: The estimated inclination angles for galaxies C and D are i = 24.9 ± 3.2 and i = 75.7 ± 10.1, respectively. These inclination angles are used to interpret the gas kinematics listed in Table 2, which includes the position angles (P.A.), maximal velocity offsets (Vmax), and mean velocity dispersions (σv) for these galaxies.


Question: What are the specific experimental setups and evaluation metrics used to compare the proposed method with other vehicle pose estimation methods?

Answer: The proposed method was compared with classical vehicle pose estimation methods like RANSAC L-shape fitting method and convex hull heuristic (CHH) method on the KITTI dataset and a manually labeled dataset. The experiments were conducted on a desktop with an AMD Ryzen™9 5900X processor, running Ubuntu 20.04, and all algorithms were implemented in C++ without parallelization or GPU acceleration. The evaluation metrics used were orientation error and absolute orientation error, with the latter reflecting the estimation bias without considering its direction. The running time was also measured to verify computational efficiency.


Question: How does the Nambu-Eliashberg theory reformulate the AL vertex correction, and what are the components of the Nambu spinor in this context?

Answer: The Nambu-Eliashberg theory reformulates the AL vertex correction by introducing the Nambu spinor representation, which is analogous to the representation used in superconductivity. In this context, the Nambu spinor is composed of each-patch fermion, represented as Fσ = (f+σ, f−σ)ᵀ, where τ3 acts on the +kF particle and −kF particle space. This reformulation allows for a self-consistent framework for boson and fermion self-energy corrections, where the normal fermion self-energy and the fermion pairing potential are components of the matrix self-energy Σ(k0) = (Σ(k0), Φ(k0); Φ*(k0), Σ(k0)).


Question: What are the key characteristics of SRNO that distinguish it from prior continuous SR works?

Answer: The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to 'hallucinate' high-frequency information from the LR image.


Question: How does the matrix ˆD decompose into submatrices for resonance and nonresonance diffraction orders, and what are the corresponding right-hand sides?

Answer: The matrix ˆD can be decomposed into four submatrices: two containing the resonance ˆR = [Dνν′ rr′] and nonresonance ˆM = [Dνν′ NN′] diffraction orders, and the other two being coupling submatrices, ˆU = [Dνν′ rN], ˆL = [Dνν′ Nr]. The resonant and nonresonant right-hand sides are denoted as ˆu = [V ν r] and ˆm = [V ν N] respectively.


Question: What is the key finding regarding the rate of energy extraction in the Blandford-Znajek process for axially symmetric black holes with slow rotation?

Answer: The rate of energy extraction for the split monopole magnetic field depends only on its flux and the black hole's spin frequency, and can be described by a single parameter of deviation from the Kerr spacetime.


Question: What is the transition matrix for a rigid pseudo-Anosov braid in B3 that admits Γ1 as a train track, and how is it constructed?

Answer: The transition matrix for a rigid pseudo-Anosov braid β = (2s; 1; p1, q1, ..., pr, qr) in B3 that admits Γ1 as a train track is M(β, Γ1) = Lp1U q1 · · · LprU qr, where L = [[1, 0], [1, 1]] and U = [[1, 1], [0, 1]].


Question: How is the emission velocity in the observer's frame derived, and what components does it include?

Answer: In the observer's frame, the emission velocity becomes vobs ∼ v = ˆn u(r, u⊥, b∥) − ˆn u⊥ + u0 = ˆn u⊥(ξ − 1) + u0, where 0 ≤ ξ(r, u⊥, b∥) ≤ 1 is the ratio between the local velocity u(r, u⊥, b∥) to the orthogonal entrance velocity u⊥. The observer only senses the component along the line of sight: vobs.ˆl, where ˆl is a unit vector on the line of sight, pointing towards the observer. The emission velocity in the observer's frame, considering the line of sight, becomes vrad = −vobs.ˆl = 0.5 u0(ξ − 1) cos(ϕ) sin(2α) sin(i) + (1 − ξ)u2⊥/u0 cos(i) − u0 cos(i).


Question: How does the linearizability relation ensure that the order of actions within threads and non-overlapping method invocations is preserved?

Answer: The linearizability relation ⊑ on histories ensures that the order of actions within threads and non-overlapping method invocations is preserved by requiring a bijection ρ such that for all i, j, if H(i) = H′(ρ(i)) and (i < j ∧((∃t. H(i) = (t, ) ∧H(j) = (t, )) ∨ (H(i) = ( , ret ) ∧H(j) = ( , call )))), then ρ(i) < ρ(j). This means that the order of actions within the same thread and the order of non-overlapping method invocations must be maintained in the linearized history.


Question: What is the set of tiles obtained after completing the 2-shift automaton for the given rewrite system R?

Answer: The set of tiles obtained after completing the 2-shift automaton for the given rewrite system R is {⊳a, ⊳b, ab, ac, bb, bc, c⊲}.


Question: What are the specific values of the leptonic mixing angles and neutrino masses for the numerical example with minimal χ2 = 2.643, and how do they compare to experimental measurements?

Answer: The leptonic mixing angles are θ12 = 33.351°, θ23 = 49.162°, and θ13 = 9.189°. The neutrino masses are m1 = 0.00398 eV, m2 = 0.00959 eV, and m3 = −0.0505 eV, corresponding to normal neutrino mass hierarchy. The mass squared splittings are ∆m2_atm = 2.53 × 10−3 eV² and ∆m2_⊙ = 7.62 × 10−5 eV². All mixing angles and mass squared splittings are consistent with the experimental measurements in the 2σ region.


Question: How does the effect of regularization on accuracy and gradient flow for dense and sparse networks differ when using a low learning rate on FMNIST compared to a high learning rate on CIFAR-100?

Answer: Figure 15 shows the effect of regularization on accuracy and gradient flow for dense and sparse networks with four hidden layers on FMNIST, using a low learning rate (0.001). The test accuracy and gradient flow for both dense and sparse networks are depicted in subfigures (a) and (b) respectively.


Question: What are the specific results that demonstrate the violation of the CFR at lower pulling speeds?

Answer: Similar results have been found also at low pulling speeds where, again, W ′ does not satisfy the CFR (Section S7 in the SI).


Question: How does the paper address the issue of intractable likelihood density functions in hidden Markov models?

Answer: The paper addresses the issue of intractable likelihood density functions by using a biased approximation of the likelihood density. Specifically, it considers the approximation of the joint density given in [7] and [14], where the likelihood density function gθ is approximated by gϵθ(yt | kt) = ∫Bϵ(yt) gθ(u | kt) du / ∫Bϵ(yt) du. Under strong assumptions, this approximation is shown to be consistent as ϵ tends to zero.


Question: What evidence supports the conclusion that the observed helicity-dependent dynamics are not due to dichroic absorption but rather an opto-magnetic effect like the IFE?

Answer: The analysis of the wavelength- and fluence-dependence of the helicity-dependent dynamics, in comparison to static XMCD spectroscopy and time-resolved ASD simulations, shows that the observed effect cannot be induced by the XMCD, i.e., thermally due to dichroic absorption. Instead, it has the characteristic fingerprints of an opto-magnetic effect, which is further corroborated by first ab initio theory on the IFE at the Fe M3,2 resonance, showing qualitative agreement between the calculated IFE spectrum and the experimental data.


Question: How does the interference term in the probability formula for νe → νe oscillations relate to the neutrino mass ordering and what experimental setup is sensitive to this term?

Answer: The last oscillatory term in the probability formula for νe → νe oscillations describes the interference effect between the oscillations driven by ∆m2_21 and ∆m2_31 + ∆m2_32, which is sensitive to the neutrino mass ordering. This term will cause a fine structure in the energy spectrum of νe → νe oscillations driven by ∆m2_21, and it will be probed in the medium-baseline JUNO experiment with L ≃ 55 km. The energy resolution of the JUNO detector must be good enough to measure this term.


Question: What is the probability bound for the event that the projection onto the Hilbert space fails for more than q subsystems and at most s of the k values βi satisfy |βi|2 > E?

Answer: The probability bound for the event that the projection onto the Hilbert space fails for more than q subsystems and at most s of the k values βi satisfy |βi|2 > E is given by Pr[F n q ∩ T k ≤s] ≤ Psupport, where Psupport = 8k3/2 exp[−k/9 (q/n − 2s/k)²].


Question: How is the field tensor Fµν modified in curved spacetime, and what is the resulting form of the SU(N) Yang-Mills action?

Answer: In curved spacetime, the field tensor Fµν is modified to account for the spacetime curvature, but it retains a simple form: Fµν = ∇µAν − ∇νAµ − i gs [Aµ, Aν] = ∂µAν − ΓσµνAσ − ∂νAµ + ΓσνµAσ − i gs [Aµ, Aν] = ∂µAν − ∂νAµ − i gs [Aµ, Aν]. The SU(N) Yang-Mills action in curved spacetime is given by: A = ∫ d4x √−g [ψ (i emµ γm ∇µ − m) ψ − 1/4 (F aµν)² + gs ψ (emµγm) ψ Aµ − gµνgαβ gs f abc ∂µAaα Abν Acβ − 1/4 g²s f amn f aij gµνgαβ Amµ Aiν Anα Ajβ].


Question: What is the half-life (t1/2) of the U-band light curve for the flare observed on EQ Peg A, and what is the significance of this measurement?

Answer: The half-life (t1/2) of the U-band light curve for the flare observed on EQ Peg A is 2.46 ± 0.13 minutes. This measurement is the FWHM of the light curve, illustrated in red, and it indicates the time it takes for the flare's intensity to decrease to half its peak value.


Question: What is the performance of the CUSUM sign test and the tail-run test in the very sparse regime of the generalized Gaussian model?

Answer: In the very sparse regime of the generalized Gaussian model, the tail-run test is asymptotically optimal and outperforms the CUSUM sign test in numerical experiments.


Question: How many unique phosphopeptides were identified, and what is the significance of the number of replicates in the phosphoproteomic workflow?

Answer: A total of 4,895 unique phosphopeptides were identified from 2,426 phosphoprotein groups. The number of unique phosphopeptides detected from two replicates was 50% higher than that from a single replicate (3,362 vs. 2,237), while the number increased by only 4% (from 4,712 to 4,895) when comparing the total number of unique phosphopeptides from six replicates to that from five replicates, indicating that increasing the number of replicates beyond five contributed little to the total number of unique phosphopeptides identified.


Question: How is the variance of a discrete random variable defined in terms of its moments, and what is the relationship between the variance and the first and second moments?

Answer: The variance of a discrete random variable X is defined as var(X) = E[(X − E[X])^2]. Using the properties of expectation, it can be shown that var(X) = E[X^2] − E[X]^2.


Question: What models and s-process contributions were used to match the observed neutron-capture element signatures in the r-process-rich stars?

Answer: The authors used a composite pattern consisting of the sum of the solar r-process pattern scaled to the stellar Eu value and a fraction of a low-metallicity (Z = 0.0001 and [Fe/H] = −2.3) s-process pattern. They varied the s-process contribution to match the stellar abundance pattern after scaling it to the observed Eu abundance. For seven stars, the s-process fraction that leads to matching the data is between 5 and 30%, while the eighth star, CS 29513−032, requires 50%. The 0.9 and 3 M⊙ models fit the abundances in the element region between the second and third r-process peaks very well, especially Ba to Sm, while the 5.5 M⊙ model provides the closest match for Sr and Zr.


Question: What is the significance of CP violation that can be achieved with the combination of T2HK, Hyperkamiokande, and DUNE for δCP = ±90°?

Answer: The combination of T2HK, Hyperkamiokande, and DUNE can measure CP violation at 10 σ C.L. for δCP = ±90°.


Question: How can the bounds on the principal stratum probabilities be used to identify the treatment that maximizes expected utility?

Answer: The bounds on the principal stratum probabilities can be used to identify the treatment that maximizes expected utility in two cases: 1) when the upper and lower limits coincide, making the joint distribution of {Y a=1, Y a=0} fully determined by the experimental data, and 2) when specific specifications of γ allow for maximization even with only bounds for P(S = s | L = l).


Question: How does the variability in the mRNA produced by the hunchback promoter in the anterior and boundary regions compare, and what does this imply about the precision of the transcriptional process?

Answer: In the anterior region, the variability in the mRNA produced in steady state by different nuclei is large, with a relative error of about 50%. This variability increases to 150% of the mean mRNA produced at the boundary. These empirical estimates are consistent with a two state promoter model and indicate that the precision at the level of the gene readout is not sufficient to form the precise Hunchback boundary up to half a nuclear width.


Question: What changes in diffraction intensities are observed at ambient temperature conditions compared to cryogenic temperatures, and what does this suggest about beam damage?

Answer: At ambient temperature conditions, there is an increase in diffraction intensities at peak locations for both CNTs (~1.8 Å-1) and aluminum oxide (~4.2 Å-1) compared to diffraction intensities obtained at cryogenic temperatures. This suggests that at ambient temperatures, both the CNTs and the ALD AlOx undergo substantial beam damage, indicating that the atomic structure of these materials changes and is no longer representative of the as-formed samples.


Question: How does the inclusion of dislocation reaction affect the evolution of the grain boundary in three-dimensional simulations?

Answer: Dislocation reaction leads to the removal of dislocations, resulting in the coupling motion of the grain boundary. In simulations with dislocation reaction, the initially spherical grain boundary shrinks, and the cross-section of the grain boundary with the z = 0 plane changes from a circular shape to a hexagonal shape as it shrinks. Each edge in this hexagon is pure tilt, consisting of dislocations of only one Burgers vector. The cross-section with the x = 0 plane changes to an ellipse as it shrinks.


Question: What is the main roadblock preventing the large-scale integration of magnonic chips, and how does the spin-orbit torque phenomenon address this issue?

Answer: The main roadblock is the lack of a reliable scheme to perform signal restoration that allows magnonic logic gates to be cascaded. This is due to the finite lifetime of magnons, which limits the total length of a magnonic device to their attenuation length. The spin-orbit torque in layered systems, where a magnetic film is interfaced with a conductive material with strong spin-orbit coupling, can help overcome this difficulty. The electric current in the conducting layer is converted into a transverse pure spin current, which exerts a torque on the local magnetization, counteracting the natural damping torque and compensating for the losses, potentially amplifying the magnetic excitations.


Question: What conditions ensure that the principal symbol σm(t, y, τ, p) has m distinct real roots when considered as a polynomial in τ?

Answer: The conditions are (H1) the coefficients gβ(t, y) are smooth functions, bounded in t and y along with all their derivatives, and (H2) for |p| ≠ 0, the principal symbol σm(t, y, τ, p) has m distinct real roots when considered as a polynomial in τ.


Question: How does the proposed block selection algorithm compare to the layer selection algorithm in terms of runtime, training time, and accuracy on the target datasets?

Answer: The proposed block selection algorithm has a shorter runtime (43 mins, 39 mins, and 2m 42s for CIFAR-100, Food-101, and MangoLeafBD, respectively) compared to the layer selection algorithm (3 hours, 1h 34m, and 12m 19s). The training time is also reduced (21 mins, 19 mins, and 18 mins for BlockSelect vs. 31 mins, 28 mins, and 19 mins for LayerSelect). In terms of accuracy, BlockSelect achieves 0.79, 0.82, and 0.997 on Food-101, CIFAR-100, and MangoLeafBD, respectively, which is slightly better or similar to LayerSelect's 0.77, 0.81, and 1.0.


Question: What evidence is provided for the transition of TDEs from an early-time soft state to a late-time hard state, and how does this affect the detection rates for eROSITA?

Answer: The power law spectrum of PTF09axc and prior literature findings provide evidence that TDEs transition from an early-time soft state to a late-time hard state many years after disruption. This transition helps explain the differences in observed properties such as Lopt/LX of optically and X-ray selected TDEs. The updated TDE rate predictions for the eROSITA instrument range from 3 yr−1 to 990 yr−1, depending sensitively on the distribution of black hole spins and the time delay between disruption and peak X-ray brightness.


Question: What are the observed trajectories of Au nanoparticles when only Brownian motion and drag forces are present, and how do they change when optical forces are introduced?

Answer: When only Brownian motion and drag forces are present, the Au nanoparticles exhibit random movements due to thermal agitation, independent of their position in the simulation domain. When optical forces are introduced by illuminating the system with circularly polarized light, the Au nanoparticles move in spiral trajectories outwards from the cube.


Question: How does the rotation rate of the F1-ATP synthase motor vary with ATP concentration, and what are the key features of this relationship?

Answer: At low ATP concentrations [ATP] ≪ KM ≃ 17 µM, the motor is essentially waiting for the arrival of new ATP molecules, and the rotation rate is proportional to the ATP concentration. At high ATP concentrations [ATP] ≫ KM, the rotation rate is limited by the release of the hydrolysis products ADP and Pi, and the rotation rate reaches its maximum value of about 130 rev/s in the saturation regime. The crossover between these regimes is clearly seen in the Michaelis-Menten kinetics described by Eq. (42).


Question: What are the specific bosonic terms in the Killing spinor equations that must be satisfied to preserve spacetime supersymmetry to lowest order in α'?

Answer: The bosonic terms in the Killing spinor equations that must be satisfied to preserve spacetime supersymmetry to lowest order in α' are: δΨM = (∂M + 1/4Ω(−)MABΓAB)ϵ = 0, δλ = (∂MϕΓM − 1/12HMNP ΓMNP)ϵ = 0, and δχ = −1/2FMNΓMNϵ = 0.


Question: How is the term involving f estimated in the proof of the boundedness of weak solutions to nonautonomous parabolic problems?

Answer: The term involving f is estimated using H¨older’s inequality with exponent r/2, 2 + 4/d, and s with 1/s = 1/2 − 2/r + 1/(d+2). This yields the inequality ∫Q(a0,b0) ϑ2n|f|(u −ℓ)+ dt dx ≤ cS (∥(u −ℓ)+∥∞,2 + ∥∇(u −ℓ)+∥2) |Aℓ|^(1/2 − 2/r + 1/(d+2)).


Question: What is the estimated X-ray luminosity per unit comoving volume from Pop III binaries, and how does it compare to other X-ray sources at high redshifts?

Answer: The Pop III binary X-ray luminosity per unit comoving volume is estimated to be logIX ≃(ϵbin/0.5)[40.87−0.12(1+z)] in units of erg s−1 Mpc−3, which is likely higher than those from other X-ray sources of AGNs and normal XRBs at redshifts higher than 6 (Fragos et al. 2013).


Question: How do the wave functions change when a proximitized region is introduced within the quantum dot, and what is the impact on the formation of low-energy ABSs?

Answer: In the presence of a proximitized region within the quantum dot (δx > 0), the wave functions are divided into three regions: a pure quantum dot (∆= 0, V ≠ 0), a proximitized region within the QD (∆≠ 0, V ≠ 0), and a finite length Kitaev chain (∆≠ 0, V = 0). The effective chemical potential within the proximitized region satisfies ˜µ ≡ (µ − V (x)) ⪆ 0, leading to the formation of low-energy partially separated Andreev bound states (ps-ABSs). These ps-ABSs are localized in the quantum dot and are spatially separated by the width of the proximitized region, as shown in Fig. 5(b)-(d).


Question: How does the inclusion of the ℓ2-norm in the elastic net objective simplify the algorithm for sparsity-promoting least-squares reverse-time migration (SPLS-RTM)?

Answer: The inclusion of the ℓ2-norm in the elastic net objective results in a greatly simplified algorithm involving linearized Bregman iterations, which corresponds to gradient descent on the dual variable supplemented by a simple soft thresholding operation with a fixed threshold.


Question: What is the relationship between the preimage morphism f* and the reachable dependency poset rdp(E) in the context of preDSCs?

Answer: The preimage morphism f* can be restricted to the subset of reachable subsets, f*: rdp(E′) →P (E). We wish to consider the class of functions f such that f* factors through rdp(E), which is equivalent to asking that f* preserves reachable subsets.


Question: What functions are available for computing MFCC and PSCC coefficients from audio files, and what parameters do they accept?

Answer: The functions available for computing MFCC and PSCC coefficients from audio files are computeMFCC, computePSCC, computeStandardizedMFCC, and computeStandardizedPSCC. computeMFCC and computePSCC accept parameters such as input_path, input_file, barplot, and zero. computeStandardizedMFCC and computeStandardizedPSCC accept additional parameters like nmel, nmfcc, lmax, and nbins.


Question: What evidence from the neutron scattering and 7Li-NMR measurements supports the existence of two magnetic transitions in LiCu2O2?

Answer: Two anomalies have been observed in the T-dependence of the specific heat Cs and neutron magnetic scattering intensity, which ensures the existence of two magnetic transitions at TN1~24.5 K and TN2~22.8 K. At these temperatures, the dielectric susceptibility ε measured with the electric field E along c has small but clear anomalies.


Question: How does the dynamic partner selection mechanism in the simulations affect the overall cooperation and system efficiency?

Answer: In the dynamic conditions, trustful and trustworthy agents were no longer bound to reciprocate defection but could find new partners with whom they started cooperative relations. This led to an increase in overall cooperation, bringing higher fairness and efficiency in the system. The average investment increased by one third and the average return almost doubled compared to both the experiment and static networks.


Question: How does the proposed method address the non-stationary framing lengths and learning burden issues in DEMUCS?

Answer: The proposed method addresses the non-stationary framing lengths and learning burden issues in DEMUCS by incorporating multi-resolution stationary frequency information into the encoder layer by layer. Specifically, it uses three different window-size spectrograms with {8ms, 16ms, and 32ms} STFT points, {4ms, 8ms, 16ms} hop size, and {8ms, 16ms, 32ms} window length to provide stationary multi-resolution frequency information. This approach ensures short-term stability of the speech signal and reduces the learning burden by avoiding a single output for multiple resolution STFT information.


Question: How does the non-normalisable nature of the reference state |φ0) affect the symmetry transformations of the time-evolved states |t)?

Answer: The symmetry transformations of the time-evolved states |t) are unitary actions on the state, parametrising time and changing the normalisation of the state. Specifically, Ug(α)|t) = [fg(t)]−1|g(t)), where |φ0) is the non-normalisable reference state.


Question: What is the larger algebra formed by combining the conformal symmetries, shift symmetries, and the new symmetries in the DBI and special galileon theories?

Answer: For DBI, the larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. For the special galileon, it is a real form of the special linear algebra.


Question: How do the training and test success rates of the kernel method differ for the wine dataset compared to the iris dataset, particularly for the linear and poly-2 kernels with and without normalization?

Answer: For the wine dataset, the training and test success rates for the kernel method are as follows: Linear, w/o normalization, λ = 10−2: 0.9987 (Training), 0.9955 (Test); Linear, w/o normalization, λ = 10−1: 0.9987 (Training), 0.9883 (Test); Linear, w/o normalization, λ = 1: 0.9987 (Training), 0.9883 (Test); Linear, w/ normalization, λ = 10−2: 0.9357 (Training), 0.9474 (Test); Linear, w/ normalization, λ = 10−1: 0.9230 (Training), 0.9183 (Test); Linear, w/ normalization, λ = 1: 0.6807 (Training), 0.6711 (Test); Poly-2, w/o normalization, λ = 10−2: 1.0000 (Training), 0.9481 (Test); Poly-2, w/o normalization, λ = 10−1: 1.0000 (Training), 0.9558 (Test); Poly-2, w/o normalization, λ = 1: 1.0000 (Training), 0.9683 (Test); Poly-2, w/ normalization, λ = 10−2: 0.9455 (Training), 0.9520 (Test); Poly-2, w/ normalization, λ = 10−1: 0.9259 (Training), 0.9293 (Test); Poly-2, w/ normalization, λ = 1: 0.8413 (Training), 0.8258 (Test).


Question: How are the D4-brane divisors identified in the context of corner lattice points of the toric diagram?

Answer: In the case of D4-branes associated with a corner lattice point of the toric diagram, there is a unique perfect matching p corresponding to the lattice point, and the D4-brane divisor can be identified with the locus {˜Φp = 0} associated with that perfect matching p. In this locus, all the bifundamental fields belonging to the perfect matching are set to zero.


Question: What are the specific constraints on the effective couplings of the Z′ in the muon sector, particularly for the values of CV µ and CAµ, that satisfy the aµ anomaly?

Answer: In Fig. 3 (a), Cνµ is set to 10−5 and CAµ is fixed such that the aµ anomaly is exactly satisfied. In Fig. 3 (b), contours of the values of |CAµ| that satisfy the aµ anomaly are shown as a function of |CV µ| and mZ′.


Question: What modifications to the genetic algorithm improve its robustness in the structural optimization of atomic clusters?

Answer: Admixing twinning mutation moves leads to an improved robustness of the algorithm efficiency with respect to the population size, resulting in a very stable performance over a wide range of population sizes.


Question: What is the definition of a relatively ergodic extension in the context of measure-preserving systems?

Answer: An extension J : (Y, S) → (X, T) is relatively ergodic if J(ﬁx(S)) = ﬁx(T).


Question: How does the construction of twisted modules for the tensor product vertex operator algebra V ⊗k relate to the V -modules in the context of permutation orbifolds?

Answer: For any V -module (W, YW (·, z)), a canonical σ-twisted V ⊗k-module structure on W was obtained using the ∆k-operator, where σ is a k-cycle viewed as an automorphism of V ⊗k.


Question: What are the conditions under which the best linear unbiased estimator (BLUE) of k′β is derived in the context of a linear mixed model with fixed model matrices?

Answer: If k′β is estimable under the model and V ≡ (1/σ2)var(Y) = ZGZ′ + R is known, then the best linear unbiased estimator (BLUE) of k′β is k′ˆβ where ˆβ = (X′V−1X)−X′V−1Y.


Question: How does the BV-BFV formalism handle Wilson lines ending on the boundary in 1D Chern-Simons theory?

Answer: In the 1D Chern-Simons theory, the quantized BFV boundary action coincides with the Kostant cubic Dirac operator, which plays an important role in representation theory. This formalism allows for Wilson lines to end on the boundary of the space-time manifold.


Question: What are the key challenges in designing a distributed algorithm for the decentralized dictionary learning problem over time-varying and directed graphs?

Answer: The key challenges include: (i) Problem P is non-convex and non-separable in the optimization variables; (ii) Each agent i owns exclusively Si and can only compute its own function fi; (iii) Each fi depends on a common set of variables (the dictionary D) shared among all agents, as well as private variables Xi, which need to be treated differently to avoid communication overhead; (iv) The gradient of each fi is generally neither bounded nor globally Lipschitz on the feasible region; (v) G and gi's are nonsmooth; (vi) The graph is directed and time-varying, with no specific structure assumed but some long-term connectivity properties.


Question: How is a local 1+3 splitting of T(M) defined using a Lorentzian metric, and what conditions must be met for a frame field to be time-orthogonal?

Answer: A local 1+3 splitting of T(M) is defined using a Lorentzian metric g, which is a symmetric, second-rank, covariant tensor field that makes each tangent space isometric to four-dimensional Minkowski space M4 with the sign convention η = diag[+1, −1, −1, −1]. A local frame field {eµ, µ = 0, 1, 2, 3} is Lorentzian if it is orthonormal for g in the Minkowski sense, i.e., gµν = g(eµ, eν) = ηµν. A frame field eµ is time-orthogonal if g0i = g(e0, ei) = 0 for i = 1, 2, 3, meaning that the time-like vector e0 is orthogonal to the spatial vectors ei.


Question: What is the key parameter that drives the density inhomogeneities in the proposed model, and how does it differ from other models?

Answer: The key parameter driving the density inhomogeneities in the proposed model is the divergence of the free-falling flow velocity. This differs from other models, such as those involving polarisable media, where the density inhomogeneities might be driven by the gravitational field itself.


Question: What specific evidence supports the presence of very small carbonaceous grains in the clumps of NGC 650, and how does this relate to the low RV value found?

Answer: The low RV value of 2.35, which is indicative of dust with a large amount of carbonaceous material and/or a smaller than usual grain size distribution, supports the presence of very small carbonaceous grains (possibly PAHs) in the clumps, in addition to the larger grains.


Question: What are some recent object-level UVAD methods and their key contributions?

Answer: Some recent object-level UVAD methods and their key contributions include: LDGK [54] (2017) integrated a generic CNN and environment-related anomaly detector to detect video anomalies and record the cause. DCF [68] (2018) used object detection models and Kalman filtering to extract foreground objects and discriminated anomalies by pose and motion classification. OC-AE [62] (2019) proposed an object-centric convolutional autoencoder to encode motion and appearance and used a one-versus-rest classifier for anomaly detection. Background-Agnostic [44] (2021) used a set of autoencoders to extract appearance and motion features and binary classifiers to detect anomalies. Multi-task [43] (2021) trained a 3D CNN to generate discriminative representations by performing multiple self-supervised learning tasks. OAD [37] (2021) provided an online VAD method with asymptotic bounds on the false alarm rate and a procedure for selecting a proper decision threshold. HF2-VAD [105] (2021) integrated sequence reconstruction and frame prediction for video anomaly detection. VEC [201] (2020) proposed a video event completion framework to exploit advanced semantic and temporal contextual information. BiP [15] (2022) used a bi-directional architecture with three consistency constraints to regularize the prediction task. HSNBM [5] (2022) designed a hierarchical scene normative binding modeling framework to detect global and local anomalies.


Question: What is the significance of the 'distance' d(A) in the context of the ellipticity of the operator A, and how is it related to the ellipticity constants?

Answer: The 'distance' d(A) is a well-known measure of ellipticity and can be bounded from above and below in terms of the usual ellipticity constants. Specifically, d(A) is defined as the minimum over t ≥ 0 of the L∞ norm of 1(CN)d − tA. When A = A*, there is an easy formula for d(A) given in Lemma 2.1.


Question: What does the average neutral domain size and average individual domain size indicate in the context of the line composed of N = 243 sites?

Answer: The average neutral domain size and average individual domain size for the line composed of N = 243 sites are shown in Figure 5(a) and (b) respectively, with data for different predation rates µ, where the swapping rates are given by σ = σn = 1 − µ. The dots in (a) indicate the domination times where one passes from one exponentially decaying regime to the next, as seen in the change of slope in (c).


Question: What is the significance of testing the universality of T-odd TMD fragmentation functions in the context of hyperon polarization in SIDIS and e+e− annihilation processes?

Answer: Testing the universality of T-odd TMD fragmentation functions by studying hyperon polarization in SIDIS and e+e− annihilation processes is significant because it allows for the unambiguous measurement of the sign of these fragmentation functions. This is possible due to the coupling of the T-odd FFs with chiral-even functions, such as the spin-averaged quark PDFs in SIDIS and the spin-averaged FFs in e+e− processes, which are well-determined and positive. This test provides a unique opportunity to study initial/final state interaction effects in the fragmentation process and to verify the associated factorization.


Question: How does Proximity Forest's performance and scalability compare to the Elastic Ensemble on the satellite dataset?

Answer: Proximity Forest scales linearly with the training set size while the Elastic Ensemble (EE) scales quadratically. Additionally, Proximity Forest’s classification accuracy on the satellite dataset is substantially better than EE, even when they train on the same quantity of data. Proximity Forest can learn from 1M time series in 17 hours (on 1 CPU), while it would take over 200 years for EE, resulting in a 103,000x speedup.


Question: What is the theoretical basis for the narrowest synchrotron emission spectrum used in the model comparison, and how is it derived?

Answer: The narrowest synchrotron emission spectrum is derived from a power-law distribution of electron energies with a power-law index of p = 3.8, which makes the synchrotron spectrum narrower. This value is motivated by the typical value found when slow-cooling synchrotron spectra are fitted to the data. The emissivity function of synchrotron radiation in a random magnetic field for a power-law distribution of electron energies is used, implemented in the Python package Naima. The emission spectrum used is referred to as the slow cooling synchrotron model (SCS).


Question: What are the specific product-level impacts of the tariffs on US imports from Spain, as shown in the regression analysis?

Answer: The regression analysis in Table A1 shows that the tariffs had a significant negative impact on the value and quantity of US imports from Spain, with coefficients of -0.578 and -0.558, respectively, at the 1% significance level. The impact on price was less significant, with a coefficient of -0.020.


Question: How does the two-coloring sheaf SimpFinGr(−,K2) behave when applied to a cyclic decomposition of a 5-cycle graph, and what does this imply for the sheaf decision problem on decompositions of other shapes?

Answer: When the two-coloring sheaf SimpFinGr(−,K2) is applied to a cyclic decomposition of a 5-cycle graph, the resulting structured co-decomposition δ has bags that are all two-element sets corresponding to the two proper colorings of any edge. However, since odd cycles are not two-colorable, at least one of the bags of Aδ will be empty. Despite this, the output of Algorithm 1 on δ is isomorphic to δ itself. This implies that while the sheaf decision problem can be efficiently solved on tree-shaped decompositions, it can also be solved on decompositions of other shapes, albeit with slower running times that are still FPT-time under a double parameterization.


Question: What is the expression for ⟨σ2_end⟩ when the condition Eq. (4.41) is satisfied and p < 2?

Answer: ⟨σ2_end⟩ ≃ (3/2) (Hend^2 / (2π)) (Hend / f)^(2+p)/(2-p).


Question: What is the mass distribution of the leading pT C/A R = 1.5 trimmed jet in the multijet and γ + jet selections, and how do the data and MC predictions compare?

Answer: The mass distribution of the leading pT C/A R = 1.5 trimmed jet in the multijet selection shows data and MC predictions with a data-driven normalisation correction, and the systematic uncertainties are indicated as a band in the lower panel. For the γ + jet selection, the data and MC predictions are also compared, with the same data-driven normalisation correction and systematic uncertainties.


Question: How does the choice of initial vortex profiles affect the annihilation process of a vortex dipole in the Gross-Pitaevskii equation?

Answer: The choice of initial vortex profiles strongly affects the vortex trajectories and the time scale of the annihilation process. Specifically, annihilation proceeds more rapidly when low-energy (or 'relaxed') initial profiles are imposed, as these profiles possess less excess energy that needs to be radiated away from the cores before annihilation can occur.


Question: How does the non-local version of the cost functionals affect the distribution of players in the population, and what is the impact of different happiness thresholds on this distribution?

Answer: In the non-local version of the cost functionals, the cost at position x paid by a player depends on an entire neighborhood of x. If the happiness thresholds a1 and a2 are high (e.g., 0.8), players prefer regions of Ω with a prevalent presence of their own population, leading to an interval with non-empty interior where both distributions vanish as ν → 0. However, if the happiness thresholds are sufficiently low (e.g., less than 0.5), the free boundary becomes a point, similar to the local case.


Question: How does the distribution of forward-propagating particles change as they cross the cloud, and what does this reveal about the CR flux over time?

Answer: As particles cross the cloud, a peak in the distribution function of forward-propagating particles near µw = 0 emerges. This peak is due to particles with lower pitch angles (µw = 0.1) having entered the cloud earlier when the CR flux was higher, while particles with higher pitch angles (|µw| = 1) entered more recently when the CR flux had dropped substantially. This overlap of wave-frame distributions at different times explains the positive gradient in the cloud.


Question: What evidence is provided to show that the protocol was effective in influencing subjects' trust in the VA's skills?

Answer: The protocol effectively succeeded in influencing subjects into the desired mental state of either trusting or distrusting the agent’s skills, as evidenced by the significant positive correlation between the difference in scores for each question and the difference in the ratios of equivalent answers for both conditions (r(31) = 0.498, p-value = 0.003). Additionally, a perceptual study by expert listeners found that the subject’s speech could be used to detect which type of VA they were using, with an accuracy up to 76%, compared to a random baseline of 50%.


Question: How does the performance of the RE model vary with different types of supporting evidence, and what does this indicate about the challenges in document-level RE?

Answer: The recall of the RE model is 51.1% for single, 49.4% for mix, and 46.6% for multiple. This indicates that while multiple supporting sentences in mix may provide complementary information, it is challenging to effectively synthesize the rich global information. Moreover, the poor performance on multiple suggests that RE models still struggle in extracting inter-sentence relations.


Question: How do the mean STEM GPAs of URM students, particularly first-generation URM students, change over time compared to non-URM students?

Answer: While non-URM students consistently rise in mean STEM GPA over time, first-generation URM students who major in chemistry, computer science, engineering, mathematics, or physics experience a steady decline in mean STEM GPA from year one to two and year two to three. This decline could be due to a lack of representation and stereotype threat.


Question: How does the PSF vary across the field of view for different f/d values in the parabolic dish design, and what is the impact of using spherical mirror facets?

Answer: The PSF varies across the field of view for different f/d values in the parabolic dish design as shown in Fig. 38. For f/d values of 0.98, 1.22, 1.46, and 1.71, the PSF (RMS) increases with the off-axis angle. The use of spherical mirror facets in the parabolic reflector introduces additional effects on the PSF, including the alignment inaccuracy and non-perfect surfaces.


Question: How does the FF-anti-BRST transformation change the gauge parameter λ in the Lorentz gauge effective action?

Answer: The FF-anti-BRST transformation with the parameter given in Eq. (3.57) changes the gauge parameter λ to λ′ = λ/e−2γ, connecting two effective theories that differ only by the gauge parameter.


Question: What is the relationship between the multiplicity function mH_H(λ) and the stability property in the context of holomorphic discrete series representations and holomorphic symmetric pairs?

Answer: The multiplicity function mH_H(λ) is periodic for sufficiently large parameter λ, and the multiplicities in sufficiently large parameters can be described by the decomposition of Hp+ with respect to ZK∩H(a). This periodicity is referred to as the stability property.


Question: How does the experimental pattern on a square lattice deviate from the nominal four-fold symmetry, and what factors contribute to this deviation?

Answer: The experimental pattern on a square lattice does not perfectly preserve the nominal four-fold symmetry due to small fabrication errors and contamination by dust, grease, etc., which are very difficult to avoid.


Question: How is the unit separability criterion verified for the stabilizer rebit scenario, and what does this imply about the existence of an operationally noncontextual ontological model?

Answer: In the stabilizer rebit scenario, the unit separability criterion is verified by showing that the Choi state of the identity, J(1R), can be decomposed as J(1R) = ∑_{i=1}^4 (1/4) yi ⊗ yi, where yi are elements of the set ¯Y. This decomposition shows that J(1R) ∈ Sep(s, e), proving that the scenario admits an operationally noncontextual ontological model, in agreement with the existing literature [9].


Question: What are the key differences between GPU and Intel Xeon Phi architectures in terms of programming and performance optimization?

Answer: GPUs require explicitly coding in the single instruction, multiple data (SIMD) programming paradigm, where the parallel nature of tasks must be exposed by the programmer. In contrast, the Intel Xeon Phi can be used in a more conventional programming model where the compiler uses heuristics to uncover possible gains from parallel tasks. While coding in this platform is likely to be easier and require less specialized knowledge, the possibilities for gain when the parallelism inherent in the problems is exposed by the programmer are greater for the appropriate class of problems. Additionally, achieving the full potential of these resources will likely require very different approaches, despite both architectures having a similar number of vector processors.


Question: How does the atom-cavity coupling affect the non-Markovianity and quantum speedup in the dissipative Jaynes-Cumming model with an Ohmic reservoir?

Answer: The atom-cavity coupling is the main physical reason for the transition from Markovian to non-Markovian dynamics and from no speedup to speedup process. The critical value of this sudden transition only depends on the Ohmicity parameter. The atom-cavity coupling and appropriate reservoir parameters can effectively improve the non-Markovianity in the dynamics process and speed up the evolution of the atom.


Question: How do the scaling relations for fermion and boson fields differ in the context of the tree-level scaling procedure described in the candidate text?

Answer: For fermion states, the scaling relations are k′_0 = etk0, k′_F = kF, and ℓ′ = etℓ, where kF remains unchanged. For boson fields, the scaling relations are k′_0 = etk0 and k′ = etk. The boson fields have a dynamical critical exponent z = 1 at tree-level, and the fields are rescaled to keep the kinetic energies invariant, leading to ψ′ = e−3t/2ψ and φ′ = e−(d+3)/2tφ. This results in the boson interactions being rescaled as λ′_φ = e(3−d)tλφ, setting d = 3 as the upper-critical dimension for the boson fields.


Question: What issues were observed with the min-entropy and the proportion of 1s in the experimental results compared to the simulation?

Answer: A sudden drop in min-entropy was observed in qubit [17] at around 60 hours, which did not occur in simulation. Additionally, the proportion of 1s in the experimental results for each qubit did not fall within the acceptable range under the level of significance α = 0.01, as seen in Fig. 3.


Question: How does the FBDC policy achieve throughput-optimality in systems with server switchover times and random connectivity?

Answer: The FBDC policy achieves throughput-optimality asymptotically in the frame length by solving a Linear Program (LP) corresponding to the MDP solution for the saturated system and applying this solution over a frame in the actual system. It is applicable to many general systems and provides a new framework for developing throughput-optimal policies for network control.


Question: What is the key difference in the convergence rate of the proposed d-CG compared to traditional decentralized algorithms?

Answer: The proposed d-CG exhibits a practical superlinear convergence rate and comes with a priori computable finite-step convergence guarantees, in contrast to traditional decentralized algorithms which typically exhibit asymptotic convergence at a linear rate.


Question: What are the key observations from the stretched period échelle diagrams for different coupling intensities between the g-mode cavities?

Answer: For q1 ≳10−2, additional modes are clearly identified outside of the vertical ridge, and as q1 increases, the vertical ridge widens due to avoided crossings between g1 and g2 modes. If the coupling is weaker (q1 < 10−2), the oscillation spectrum during the subflash is expected to be very similar to that of regular clump stars, except for the value of the period spacing, which could differ.


Question: How is the relative volume change (∆V/V) used to determine if attention toward a topic precedes the creation of a hoax article on Wikipedia?

Answer: The relative volume change (∆V/V) is defined as (V(b) - V(a)) / (V(b) + V(a)), where V(b) and V(a) are the median traffic to neighbors in the 7 days before and after the creation of the article, respectively. When ∆V/V > 0, the majority of traffic occurs before the article is created, indicating that attention toward the topic precedes its creation. When ∆V/V < 0, attention tends to follow the creation of the hoax.


Question: How is the Fisher Information Matrix (FIM) used to assess practical identifiability in ODE models, and what does its rank indicate?

Answer: The FIM is computed by F = XT X, where X is the sensitivity matrix defined by X(i,j) = ∂y/∂pi (tj) for parameters p1, ..., pn and time points t1, ..., tm. The rank of F indicates the number of identifiable parameters and parameter combinations. If F is singular, the model is structurally unidentifiable; if F is close to singular, the model may be practically unidentifiable.


Question: What is the upper bound on the redundancy of optimal binary AIFV codes, and how is it derived?

Answer: The upper bound on the redundancy of optimal binary AIFV codes is given by the expression: rAIFV ≤ 2(1 − q1) + (q1^2 / (1 + q1)) − h(q1) + (q1 / (1 + q1)) * (K−1) * (q2k−1 − q2k). This bound is derived by applying Lemmas 3 and 4 to the terms in the expression for rAIFV, and using the fact that q1 + q2 = 1 and q1 = pmax. The sequence {qk} is non-increasing, which helps in simplifying the bound.


Question: What are the explicit results on the behavior of the Ulam-Kac adder sequence obtained by Kac using generating functions and the method of steepest descent?

Answer: Kac showed that log E[Xn] ∼ 2√n, and log E[X2_n] ∼ √(2n) * (5 + √17).


Question: How does the integration of streaming data and dynamical models contribute to the analysis of complex systems, particularly in the context of neuronal recordings in the brain?

Answer: The integration of streaming data and dynamical models has a nearly seven-decade history. In the modern era, it is increasingly common to build dynamical models directly from data using machine learning. This is especially important in complex systems where first principles models are not available, or where it is not even known what the correct state-space variable should be. Biological systems, such as those that arise in neuronal recordings in the brain, are well suited for such data-driven model discovery techniques, as whole-brain imaging provides insight into how the microscale dynamics of individual neurons produce large-scale patterns of spatio-temporal brain activity. Such systems are ideally suited for leveraging the modern data-driven modeling tools of machine learning to produce dynamical models characterizing the observed data.


Question: What is the relationship between the entropy density and the over-saturated pion gas, and how does this relate to the hadronization process?

Answer: The maximum entropy density S/V occurs for an over-saturated pion gas, where γq ≃ emπ/2T ≃ 1.6. In this state, the entropy density of such a saturated Bose gas is twice as large as that of a chemically equilibrated Bose gas. This large value of γq is necessary and sufficient to allow for the smooth transformation of a QGP into hadrons, as it ensures the absorption of the high entropy content of the QGP by producing abundant pions and other hadrons.


Question: What is the condition for the difference of two composition operators Cϕ and Cψ to be compact on H2(BN) or A2_s(BN) (s > −1)?

Answer: Cϕ − Cψ is compact on H2(BN) or A2_s(BN) (s > −1) if and only if both Cϕ and Cψ are compact or ϕ = ψ.


Question: What is the key feature of Krupchyk and Uhlmann's construction of CGOs, and how does it relate to the approximation of magnetic potentials?

Answer: Krupchyk and Uhlmann's construction of CGOs is based on the use of Carleman estimates, and its main feature is that they only need to make an approximation of the magnetic potentials by smooth vector fields in the L2 sense. This suggests examining magnetic potentials in the Besov spaces B2,r ε, where the modulus of continuity is of Hölder type, say of order ε.


Question: How does the hinge region of the protein differ between the specifically bound and non-specifically bound states, and what does this imply about the role of NMR restraints?

Answer: When the protein is specifically bound, the hinge region forms two tightly bound alpha-helices, and the contacts involving residues Arginine 51, Valine 52, Alanine 53, Glutamine 54, Glutamine 55, Leucine 56, and Alanine 57 are present. In contrast, when the protein is non-specifically bound, these contacts are broken. However, in simulations with NMR restraints, these contacts remain intact, and the helices in the hinge region remain formed, suggesting that NMR restraints do little to disrupt the structure of the protein in this region.


Question: How does the introduction of walls affect the period and average kinetic energy of a harmonic oscillator, and what is the expression for the average kinetic energy in terms of Qk?

Answer: Introducing hard walls at x−= −L/2 and x+ = L/2, with L < ℓ(E, k), the period of the cycle for a harmonic oscillator is given by τ(E; L, k) = 4ω0 arcsin(L/2 √(k/2E)). The average kinetic energy is now expressed by K = E / (2(1 + Qk)) = K0 (1 + Qk) = E − Φk, where Qk = (sin(2 arcsin(L/2 √(k/2E)))) / (2 arcsin(L/2 √(k/2E))) = 1 − (1 / (2K0)) x ∂Φk / ∂x and Φk = (1/2) x ∂Φk / ∂x < Φ0k.


Question: What are the limitations of using trace bases for color space calculations, and what is the proposed alternative?

Answer: Trace bases have advantages in simplicity, recursion relations, and handling gluon exchange and emission. However, they are overcomplete and not orthogonal, which becomes problematic for processes with many partons due to the rapid growth of spanning vectors. The proposed alternative is to construct minimal orthogonal bases using group theory based multiplet bases for any QCD process, which can significantly speed up exact calculations in the color space of SU(Nc).


Question: What was the observation schedule and exposure strategy used by HST to monitor the ejecta from the DART impact on Dimorphos?

Answer: HST used a total of 19 orbits over about 19 days to observe the Dimorphos ejecta. The first orbit was before impact, and the second through the 7th orbit started about T+15 minutes, continuously observing the ejecta except during Earth occultations. In the next 5 orbits, observations were made roughly once every 12 hours, and then once every day in the following three orbits. In the final phase, observations were executed once every 3 days. In each orbit, images were collected at multiple exposure levels, with short exposures keeping the central core of Didymos unsaturated and long exposures saturating Didymos to image the faint ejecta and tail. All images were collected through the F350LP filter.


Question: How does the introduction of capitalist consumption affect the validity of the Marxist law of the tendency of the rate of profit to fall?

Answer: The validity of the Marxist law of the tendency of the rate of profit to fall when the organic composition of capital increases depends on the share of capitalist consumption in total consumption. If this share exceeds a threshold linked to the lifespan of fixed capital, the Marxist law is valid. If this share is lower than this threshold, the Marxist law is not verified, and the rate of profit increases as a function of the organic composition of capital.


Question: How can entanglement gain in system AB be achieved when initial entanglements coincide, and what role does discord play in this process?

Answer: Entanglement gain in system AB is only possible due to nonzero discord DAB|C when initial entanglements coincide, i.e., EA:BC(0) = EA:B(0). To observe entanglement gain, an entanglement-breaking channel can be applied to one of the principal systems at time t = 0, such as a von Neumann measurement, which sets EA:B(0) = EA:BC(0) = 0. This allows for entanglement gain in system AB at a later time τ.


Question: How is the joint probability density function defined for the dual Gaussian sources, and what does it ensure about the relationship between the sources?

Answer: The joint probability density function for the dual Gaussian sources is defined as f(u1, u2) = 1 / (2π √(1 - ρ²)) * exp[-(u₁² - 2ρu₁u₂ + u₂²) / (2(1 - ρ²))], for -∞ < u₁ < ∞ and -∞ < u₂ < ∞. This definition ensures that the covariance between the sources equals the correlation coefficient ρ.


Question: How does the proposed improvement address the issue of frequent recompilations and deoptimizations for methods that can be invoked with both columnar and non-columnar arrays?

Answer: The proposed improvement uses code duplication to preserve both the fast path, which assumes the array is columnar and applies aggressive optimizations, and the slow path, which handles non-columnar arrays and checks the array state upon each access. This approach ensures a stable compilation state and prevents deoptimizations caused by array state changes, even for methods that can be invoked with both columnar and non-columnar arrays.


Question: What is the expression for the kink solution φK(x) in terms of the hypergeometric function, and what is the classical kink energy E[φk]?

Answer: The kink solution φK(x) is expressed in terms of the hypergeometric function as φK(x) = Re{(8/5 - 4i/5)e^(-2(1+i) arctan(tanh(x/2))) √sech(x) [−1 + 2F1(−1/4 + i/2, 1, 1/4 + i/2, (e^x−i)^2 / (e^x+i)^2)]}. The classical kink energy E[φk] is given by E[φk] = ∫_{-∞}^{∞} exp[−2gd(x)]sech(x) dx = sinh(π).


Question: How is the graph product of monoids defined in terms of a presentation, and what are the components of this presentation?

Answer: The graph product G P = G P(Γ, M) of a set of mutually disjoint monoids M = {Mα : α ∈ V} with respect to a graph Γ = Γ(V, E) is defined by the presentation G P = ⟨X | R⟩, where X = ∪α∈V Mα and R = Rid ∪ Rv ∪ Re. The components of R are: Rid = {1α = ǫ : α ∈ V}, Rv = {x ◦ y = xy : x, y ∈ Mα, α ∈ V}, and Re = {x ◦ y = y ◦ x : x ∈ Mα, y ∈ Mβ, (α, β) ∈ E}.


Question: How does the achievable rate for the sum of two binary symmetric channels with different crossover probabilities compare to the matched capacity, and what is the specific rate formula?

Answer: The achievable rate for the sum of two binary symmetric channels with different crossover probabilities δ1 and δ2 is given by R* = log(e^(1 - H2(δ1)) + e^(1 - H2(δ2))), where H2(δu) is the binary entropy function. This rate is identical to the matched capacity, which is given by C* = log(e^C1 + e^C2), where C1 and C2 are the capacities of the individual subchannels. In contrast, the LM rate is strictly less than the matched capacity whenever δ1 ≠ δ2.


Question: What is the one-loop beta function for the Higgs self-coupling λ, and how does it depend on the gauge coupling constants and Yukawa couplings?

Answer: The one-loop beta function for the Higgs self-coupling λ is given by β(1)λ = 24λ^2 − (9/5g1^2 + 9g2^2)λ + 27/200g1^4 + 9/20g1^2g2^2 + 9/8g2^4 + 4T − 2Y, where T = Tr[3Yu†Yu + 3Yd†Yd + Yl†Yl] and Y = Tr[3(Yu†Yu)^2 + 3(Yd†Yd)^2 + (Yl†Yl)^2]. Here, gi denote the gauge coupling constants for U(1), SU(2), and SU(3) groups, and Yf with f = u, d, l represent the Yukawa coupling matrices for up-type, down-type quarks, and charged leptons, respectively.


Question: How do the synaptic time delay, rewiring frequency of the HSP rule, and adjusting rate of the STDP rule jointly affect the degree of CR in small-world and random neural networks?

Answer: The degree of CR depends on the synaptic time delay (τc), rewiring frequency of the HSP rule (F), and adjusting rate of the STDP rule (P). Decreasing the STDP and HSP parameters (P and F) always leads to higher degrees of CR in small-world and random networks, provided that the synaptic time delay parameter τc is fixed to suitable values. Increasing the synaptic time delay τc induces multiple CR (MCR) in small-world and random networks, with MCR becoming more pronounced as the STDP and HSP parameters decrease. The degree of CR varies non-monotonically when the network parameters, including the rewiring probability, average degree, and synaptic time delay, vary.


Question:
How is the term \( \pi^{-1} \left| p^{-1} \sum_{j=1}^p 1\{|W_j| > 	au\} - 2ar{\Phi}(	au)
ight| \) decomposed in the proof of Lemma 5.1?

Answer:
The term is decomposed as \( \pi^{-1} \left| p^{-1} \sum_{j=1}^p 1\{|W_j| > 	au\} - 2ar{\Phi}(	au)
ight| \geq \pi^{-1} \left| p^{-1} \sum_{j \in I_0} 1\{|W_j| > 	au\} - 2ar{\Phi}(	au)
ight| - \pi^{-1} \left| p^{-1} \sum_{j=1}^p 1\{|W_j| > 	au\} - p^{-1} \sum_{j \in I_0} 1\{|W_j| > 	au\}
ight| \).


Question: What are the key characteristics of the UCF-101 dataset and how does it differ from the HMDB-51 and Kinetics-400 datasets?

Answer: The UCF-101 dataset, introduced in 2012, is one of the largest annotated video datasets available and an expansion of the UCF-50 dataset. It comprises 13,320 realistic video clips collected from YouTube and covers 101 categories of human actions, such as punching, boxing, and walking. The dataset has three distinct official splits. In contrast, the HMDB-51 dataset contains roughly 5,000 videos belonging to 51 distinct action classes, with each class holding at least 100 videos. The videos are collected from multiple sources, including movies and online videos. The Kinetics-400 dataset, released in 2017, consists of 400 human action classes with at least 400 video clips for each action, totaling about 240,000 training videos and 20,000 validation videos, making it one of the largest well-labeled video datasets utilized for action recognition.


Question: How do the color factors CV (A) and CV (B) for the vacuum graphs V (A) and V (B) relate to the tree-level color factors, and what does this imply for the absence of double-trace terms in the leading divergence?

Answer: The color factors CV (A) and CV (B) are one-particle reducible and proportional to the tree-level color factors, given by CV (A) ˜f a1a2b ˜f ba3a4 and CV (B) ˜f a1a2b ˜f ba3a4, respectively. This proportionality implies the manifest absence of double-trace terms in the leading divergence.


Question: What is the magnetization direction of the bottom Fe/GaAs(001) electrode in the Fe/Au/Fe/GaAs(001) spin-valve, and how is it used in BEMM experiments?

Answer: The magnetization of the bottom Fe/GaAs(001) electrode is parallel to the [110] direction, which defines the analysis direction for the BEMM experiments.


Question: What are the definitions of ˜v0,0(n, k) and ˜Pn,0,0(x) when a = b = 0, and how do they relate to the Eulerian numbers and polynomials?

Answer: When a = b = 0, the definitions are ˜v0,0(n, k) := v1,1(n −2, k −1) and ˜Pn,0,0(x) := ∑_{k=0}^n ˜v0,0(n, k)x^k = xPn−2,1,1(x). These definitions serve as substitutes for v0,0(n, k) and Pn,0,0(x), which are trivially zero for all n ⩾1, and they generalize the Eulerian numbers and polynomials.


Question: What method is used to estimate the critical temperature Tc and what are the results of the fits?

Answer: A precise determination of the critical temperature Tc is necessary for evaluating the specific heat exponent α and the shift exponent λ. The pseudo-critical temperature Tm(N) is analyzed and fitted to the shift scaling form Tm(N) = Tc(1 − AλN −λ). A linear fit gives Tc = 0.0532 ± 0.0001 and Aλ = 1.5 ± 0.1, assuming λ = 1.0. A three-parameter fit yields Tc = 0.0531 ± 0.0003, Aλ = 1.8 ± 1.5, and λ = 1.1 ± 0.3. Constraining Tc to the theoretical prediction of (3/8)^3 ≃ 0.0527344 and performing a two-parameter fit gives Aλ = 5.2 ± 1.6 and λ = 1.41 ± 0.08. Fixing both Tc and λ = 4/3 = 1.333 results in Aλ = 3.9 ± 0.1, suggesting Aλ = 4.


Question: What ethical considerations were taken into account for the data used in Task 2 of the study?

Answer: For Task 2, the data is publicly available via the Perverted Justice website, which contains conversations between adults and adult decoys pretending to be minors. No actual minors were involved in the conversations, and the website only publishes chats that have led to convictions. While it is highly unlikely that the offenders consented to publish these conversations, this data is the only publicly available source in the domain and has been widely used in the relevant literature.


Question: How does the spectrum of a quaternionic operator T relate to its complex spectrum, and what is the relationship between σH(T) and σC(T)?

Answer: The spectrum of a quaternionic operator T, denoted σH(T), is related to its complex spectrum σC(T) as follows: σH(T) = {q ∈ H; σC(T) ∩ σ(q) ≠ ∅} and σC(T) = {λ ∈ σ(q); q ∈ σH(T)}. This means that σC(T) can be seen as a 'complex border' of σH(T).


Question: How does the mean value of the sampled velocity relate to the true mean velocity, and what assumptions are made in this derivation?

Answer: The mean value of the sampled velocity is equal to the true mean velocity, as we have normalized the sampling function. This is derived under the assumption of statistical independence between the measurement process and the sampled process or the use of RTW weighting.


Question: What are the specific findings regarding the critical exponents and their dependence on chain length N in the context of the loop-length probability distribution?

Answer: The analysis of the data suggests the asymptotic condition cr < 1.5, confirmed by the present evaluation cr = 1.35±0.05. The effective critical exponents cr,1(N) and cr,2(N) show a clear crossover between a short chain length regime and a long chain length regime. The crossover chain length N* is defined as the length beyond which cr,1(N) ∼ cr,2(N) ∼ cr ≃ 1.35 and the correlation lengths ξ1(N, T) and ξ2(N, T) follow the expected scaling behavior.


Question: How does the batch adsorption study of lead at 24 hours HTC time compare to the pseudo second order kinetic model for Pb2+ adsorption at 12 hours HTC time in terms of % removal over time?

Answer: The batch adsorption study of lead at 24 hours HTC time shows a higher % removal over time compared to the pseudo second order kinetic model for Pb2+ adsorption at 12 hours HTC time. The % removal reaches up to 100% at 24 hours, while the pseudo second order kinetic model shows a lower % removal, with the curve not reaching 100% even at 14 minutes.


Question: What are the specific conditions under which the Chebyshev coefficients decay faster than the Legendre coefficients, and how does this relate to the convergence of the quadrature errors?

Answer: The Chebyshev coefficient aj of the Chebyshev expansion of f decays a factor of √j faster than the corresponding coefficient of the Legendre expansion. This faster decay is mentioned in Fox and Parker [18], Hesthaven et al. [25], Petras [30], and Xiang [44]. Additionally, the quadrature errors of the Gauss and Clenshaw-Curtis can be represented using the Chebyshev expansion, respectively, if the series ∑∞j=1 aj is absolutely convergent, as EGn[f] = I[f] − IGn[f] = ∑∞j=2n ajEGn[Tj] and EC-Cn[f] = I[f] − IC-Cn[f] = ∑∞j=n ajEC-Cn[Tj].


Question: How does the self-similar evolution of the MHD plasma affect the magnetic field components, and what are the specific expressions for these components?

Answer: The self-similar evolution of the MHD plasma distorts the dipole-like magnetic field by generating an azimuthal component of the magnetic field with finite a2. The magnetic field components are given by: Br = −1/y^2 * (1/η^2) * R(η) * dΘ(x)/dx, Bθ = −1/y^2 * (1/η) * dR(η)/dη * (1/(1 − x^2)^(1/2)) * Θ(x), and Bφ = +1/y^2 * (a/η) * R(η) * (1/(1 − x^2)^(1/2)) * Θ(x).


Question: How does the Grassmannian formalism represent on-shell diagrams in theories with N < 4 supersymmetries, and what is the significance of the orientation in these diagrams?

Answer: In theories with N < 4 supersymmetries, the Grassmannian formalism is used to integrate over the phase space of internal particles, and the diagrams are fundamentally oriented. This orientation encodes the helicities of the internal particles, which is crucial for representing the physics of ultraviolet singularities present in theories with less supersymmetry. For N = 4, the orientation merely encodes a convenient translation of the on-shell diagram into a particular gauge-fixed matrix-representative C ∈ G(k, n).


Question: What is the relationship between the classifying space of the cobordism category and the Thom spectrum in the context of the Galatius-Madsen-Tillmann-Weiss theorem?

Answer: The Galatius-Madsen-Tillmann-Weiss theorem states that there is a weak homotopy equivalence α : BCd ≃ Ω∞−1MT (d) for all d ⩾ 0, where BCd is the classifying space of the cobordism category and MT (d) is the Thom spectrum associated with the collection of bundles {γ⊥d,n}.


Question: What is the key strategy for solving differential equations for master integrals in the context of the DE approach, and how does it differ from traditional methods?

Answer: The key strategy is to turn to a much more convenient basis of master integrals consisting of pure functions of uniform weight. This strategy involves using integration-by-parts (IBP) relations to reveal master integrals and then expressing the differential equations in a canonical form, d f = ǫ d ˜A f, where ˜A is a matrix and d ˜A is a logarithmic one-form independent of ǫ. This approach allows the general solution to be written down in terms of Chen iterated integrals, making the singularity structure of the integrals transparent and simplifying the determination of boundary constants of integration.


Question: How does the 'filling-in-value' change with misalignment in the horizontal and vertical configurations, and what does this indicate about anisotropy?

Answer: The 'filling-in-value' is more negative (better filling-in) for the horizontal configuration compared to the vertical one, indicating anisotropy in filling-in tolerance. This means that filling-in is more sensitive to misalignment in the horizontal configuration, showing that it is less tolerant to misalignment compared to the vertical configuration.


Question: What does the EDX analysis reveal about the Fe concentration in sputtered FGT compared to crystallized FGT?

Answer: The EDX analysis shows that the Fe concentration in sputtered FGT is much less than that of crystallized FGT, with measured values of 38.09% and 38.24% at positions P1 and P2, respectively.


Question: How does the coherent wMFPCA model compare to other models in terms of forecast accuracy and desirable properties?

Answer: The coherent wMFPCA model outperforms the first proposed model as well as the current models in terms of forecast accuracy, in addition to several desirable properties.


Question: How does the piecewise smooth medial axis transform differ from the discrete medial axis representation in terms of shape approximation and control points?

Answer: The piecewise smooth medial axis transform provides a compact yet accurate shape representation, requiring far fewer control points to define spline curves to accurately approximate the medial axis transform, compared to the discrete medial axis representation which needs a set of densely sampled medial circles to well approximate the shape.


Question: How does the proposed approach incorporate the nearest target in the LSTM initial memory state, and what are the benefits of this method?

Answer: The proposed approach incorporates the nearest target in the LSTM initial memory state by mapping the retrieved target yn into a fixed-length representation and using an affine transformation to match the dimensionality of the LSTM states. This is achieved through the expression ryn = Wnf(yn), where Wn is a learned parameter that projects the vector representation of the retrieved target to the same dimensionality of the LSTM. The benefits of this method include improving result quality and providing cues for the model’s decisions, making models more interpretable.


Question: How does the number of X-pQTL per gene compare to the number of eQTL and pQTL for the 85 genes assayed in all three data sets?

Answer: For the 85 genes assayed in all three data sets, the mean number of X-pQTL was 7.2, compared to 1.2 eQTL and 0.6 pQTL.


Question: What is the upper bound for the rate gap between the perfectly aligned and misaligned HB-NOMA user, and how does it depend on the system parameters?

Answer: The upper bound for the rate gap between the perfectly aligned and misaligned HB-NOMA user is given by ∆Rn,m ≤ log2(1 + (1 − ρ²n,m)κmax(S) + σ²K−1NBS,mN−1BS N−1U |βn,m|−2 / (ρ²n,mK−1NBS,1κmin(F) ∑m−1k=1 Pn,k)). A low misalignment factor can substantially increase the rate gap.


Question: What are the key assumptions of linear regression that must be fulfilled to ensure the validity of its results?

Answer: The key assumptions of linear regression include: 1) linearity, which requires the target variable to be representable as a linear combination of the features; 2) normality, which assumes the target variable follows a normal distribution given the independent variables; 3) homoscedasticity, meaning the variance of the error term is constant over the entire feature space; 4) independence between all input instances; 5) absence of multicollinearity, meaning different features must not be highly correlated; and 6) fixed features, requiring the independent variables to be fixed and not carrying any error or variation.


Question: How does the feature selection process impact the performance of the ResNet50 model in the AQA pipeline?

Answer: The performance and efficiency of the ResNet50 model, which uses a SVR regressor, drop prominently when the feature dimension grows too large, indicating a need for feature selection. Figure 3 shows how the accuracy of ResNet50 changes with the dimension of selected features, and a two-stage feature selection strategy is implemented to optimize the feature set.


Question: How does the moment map µΓ relate to the standard moment map µ, and what is the level set µΓ−1(Γb)?

Answer: The moment map µΓ is obtained by composing the standard moment map µ with Γ, and the level set µΓ−1(Γb) is exactly the moment-angle manifold ZΓ.


Question: What are the key components of RAINCOAT that enable it to handle both feature and label shifts in time series data?

Answer: RAINCOAT uses time and frequency-based encoders to learn time series representations, motivated by the inductive bias that domain shifts can occur via both time or frequency feature shifts. It uses Sinkhorn divergence for source-target feature alignment and provides both empirical evidence and theoretical justification for its superiority over other popular divergence measures. Finally, it introduces an 'align-then-correct' procedure for universal DA, which first aligns the source and target domains, retrains the encoder on the target domain to correct misalignments, and then measures the difference between the aligned and corrected representations of target samples to detect unknown target classes.


Question: What is the contribution of the PDF uncertainty to the measurement of sin2 θW at different dilepton masses in the HL-LHC scenario?

Answer: The contribution of the PDF uncertainty to δ sin2 θMS W ( ˆmℓℓ) in the HL-LHC scenario is shown in Table 3 for different PDF sets (NNPDF31, NNPDF40, MSHT20, CT18A, ABMP16) at various dilepton masses (133, 175, 250, 400, 1000, 3250 GeV). For example, at 133 GeV, the contributions are 0.5%, 0.3%, 0.6%, 0.9%, and 0.5% for NNPDF31, NNPDF40, MSHT20, CT18A, and ABMP16, respectively.


Question: What are some key contributions of previous works in the context of stochastic partial monitoring and information-directed sampling?

Answer: Bartók et al. (2012) proposed an algorithm for stochastic feedback that adapts to the game structure. Komiyama et al. (2015) studied asymptotically optimal instance-dependent bounds. Vanchinathan et al. (2014b) analyzed an algorithm that exploits prior knowledge on the loss distribution and proposed a computationally more efficient variant based on Thompson sampling. Kirschner and Krause (2018) extended the information-directed sampling framework to the frequentist setting, and Kirschner et al. (2020) further extended instance optimality to other model classes. For the linear bandit setting, Kirschner et al. (2021) showed that IDS achieves the asymptotic lower bound while also being near worst-case optimal.


Question: How is the Diﬀusive Capacity/Productivity Index (PI) defined for gas filtration in porous media, and what do the terms Q(t) and PDD(t) represent in this context?

Answer: The Diﬀusive Capacity/Productivity Index (PI) for gas filtration in porous media is defined as Jg(t) = Q(t) / PDD(t). Here, Q(t) is the total flux through the well-boundary Γi, given by Q(t) = ∫Γi ρu · N ds, and PDD(t) is the pressure drawdown in the domain U, given by PDD(t) = (1/|U|) ∫U p^2 dx − (1/|Γi|) ∫Γi p^2 ds.


Question: How does the construction of the gauge δ ensure that the Riemann-Stieltjes sums are bounded for any δ-fine Perron partition of [a, b]?

Answer: For any δ-fine Perron partition ˙P = {(ξk, [vk−1, vk]) : k = 1, 2, ..., p} of [a, b], the gauge δ is constructed such that for every j = 1, 2, ..., m, we can form non-degenerate compact intervals [cj, dj] with δ-fine Perron partitions ˙P∗j. By (9), the Riemann-Stieltjes sums for each ˙P∗j are bounded by 1, leading to the overall sum being bounded by m.


Question: What are the average flux and optical depth measurements within the Lyβ forest at different redshifts?

Answer: The average flux and optical depth measurements within the Lyβ forest are as follows: at zabs = 5.6, ⟨F Lyβ, obs⟩ = 0.0989, σ⟨F Lyβ, obs⟩ = 0.0193, ⟨τ Lyβ, obs eﬀ⟩ = 2.3140, σ⟨τLyβ, obs eff⟩ = 0.1952; at zabs = 5.8, ⟨F Lyβ, obs⟩ = 0.0466, σ⟨F Lyβ, obs⟩ = 0.0106, ⟨τ Lyβ, obs eﬀ⟩ = 3.0658, σ⟨τLyβ, obs eff⟩ = 0.2279; at zabs = 6.0, ⟨F Lyβ, obs⟩ = 0.0082, σ⟨F Lyβ, obs⟩ = 0.0062, ⟨τ Lyβ, obs eﬀ⟩ > 4.3935.


Question: How does the phase shift difference change when focusing on the lower surface in the semi-reflective mode compared to the regular reflection mode?

Answer:
In the semi-reflective mode, the phase shift difference is given by \( \Delta \phi = rac{2\pi}{\lambda} \left( h_2 n_a - h_1 n_a - t (n_f - n_a)
ight) \), where \( t \) is the thickness of the transparent matter and \( n_f \) is the refractive index of the transparent matter. This differs from the regular reflection mode where the phase shift difference is \( \Delta \phi = rac{2\pi}{\lambda} (h_2 - h_1) n_a \).


Question: What is the groupoid structure associated with the action of a group G on a derived stack X, and what is its quotient?

Answer: The groupoid structure associated with the action of a group G on a derived stack X is given by G × X with source map pr_X and target map ρ. The quotient of this groupoid is denoted as [X⧸G].


Question: How does Rule 2 affect the share of codeword v when there is an Fn−2-father in N(v)?

Answer: Rule 2 shifts (1/6 + 1/(n −2) −1/(n −5))/(n −3) share from codeword v to each other codeword in I(a). Since Rule 2 shifts share to v, there is an Fn−2-father in N(v) of weight three. Let us assume without loss of generality that a = e2 + e3 + en ̸∈C is the Fn−2-father. Observe that the only non-codewords in N[a] are e2+e3, e3+en and a itself. We may assume that we shift the share from word w = a + ei ∈I(a) where i ∈{1} ∪[4, n −1] to v. Hence, there has to be a special father u′ in B2(w) and another codeword in I(w). Observe that w(u′) = 6. Indeed, since w(w) = 4 and d(w, u′) = 2, we have w(u′) ∈{2, 4, 6} and since |I3(0)| = n, we have w(u′) = 6. Thus, there exists a codeword w + ej ∈I(w) ∩I(u′) of weight five. Since d(w + ej, a) = 2, they have two common neighbours and both of them belong to the code C since the non-codewords in N(a) have weight of two. However, both of these words locate in B3(u′) \ N[u′] and thus, |I3(u′)| ≥n + 1. Hence, Rule 2 does not shift share from w to v (a contradiction) and thus, the claim follows.


Question: What are the key findings regarding the leakage rate to a non-coherent eavesdropper in the ANAM system?

Answer: The analysis in the paper reveals the degrees of freedom (DoF) of the leakage rate and identifies upper and lower bounds of the constant terms that are irrespective of the eavesdropper's SNR. As the coherence time increases over the number of BS antennas, the bounds become tight. Notably, when the dimensionality of the signal and AN space is scaled up to the level of coherence time, the leakage rate DoF converges to zero, regardless of the number of eavesdropper antennas and without the aid of wiretap codes.