Question: What are the two calibration factors applied during the calorimeter digitization process, and how are they determined?

Answer: During the calorimeter digitization, two calibration factors are applied to every simulated calorimeter hit: first, the deposited energy is normalized to the mean energy deposited by a minimum ionizing particle (MIP) in this particular calorimeter sub-detector; then, the deposited, i.e., visible, energy in units of MIPs is converted to a total cell energy in GeV, such that the sum of all hit energies corresponds to the incident particle’s energy. The calibration of these factors is an iterative procedure. In the first step, the mean energy deposited by MIPs is computed from single µ−-events. In the second step, single particle events with γ and K0L at 10 GeV are simulated and fully reconstructed with the particle flow algorithm, and the resulting energy is used to compute updated calibration factors. This last step is repeated until the reconstructed energy agrees with the true particle energy to within a given margin.


Question: How does the net radial traction force and the effective friction coefficient vary with substrate stiffness for different modes of migration?

Answer: The effective friction coefficient (µeff) monotonically increases in magnitude with increasing substrate stiffness for all modes of migration. Purse-string motion leads to the highest positive µeff, indicating high resistance and sensitivity to substrate rigidity. Crawling driven motility leads to negative µeff, indicating assistive motion. The mixed mode of migration leads to the lowest magnitude of µeff, i.e., the least drag from the substrate.


Question: How does the algorithm ensure that the average of rollouts is close to the expectation with high probability, and what is the bound on the error?

Answer: The algorithm ensures that the average of rollouts is close to the expectation with high probability by applying Hoeffding’s inequality on the 푛1 independent rollouts. Specifically, the probability that the average of the rollouts deviates from the expectation by more than 훿/4 is bounded by exp(−푛1훿2 / 32퐻2(1 + 2퐵)2), which is less than or equal to 휁 / (퐸푑 + 1) if 푛1 = l 32퐻2(1 + 2퐵)2 / 훿2 log((퐸푑 + 1) / 휁) m. With a union bound, the probability that any of these bounds fail for any 휏 ∈ [퐸푑 + 1] is upper bounded by 휁.


Question: How does the accumulation of Fe-NDs in the liver compare to the accumulation of multi-phosphonic acid PEG coated particles in terms of MRI signal intensity changes over time?

Answer: The accumulation of Fe-NDs in the liver leads to a rapid decrease in MRI signal intensity, creating a strong negative contrast. In contrast, multi-phosphonic acid PEG coated particles show a delayed darkening in the liver, with the negative contrast enhancement becoming apparent only about 2-3 hours post-injection.


Question: How does the machine learning model classify the boundary between delocalized or critical phase and localized phase at zero energy, and what does the magenta line in Fig. S1(a) represent?

Answer: The magenta line in Fig. S1(a) shows the boundary between delocalized or critical phase and localized phase at zero energy, as indicated by the neural net model.


Question: How is the scalar potential transformed into a canonical form in the context of the given Kähler potential and superpotential, and what is the resulting expression for the scalar potential in terms of the canonical field χ?

Answer: To make the kinetic term for the scalar field φ canonical, the field φ is redefined to χE via |∂µχE|2 = 18 / (3 + 2|φ|2)² |∂µφ|², leading to φ = √(3/2) tan(χE / √3). Decomposing φ into its real and imaginary parts and assuming the real part is zero during inflation, we have φ = iφI / √2 and χE = iχ / √2. The solution is φI = √3 tanh(χ / √6). Using this, the scalar potential VF in terms of the canonical field χ is given by VF(χ) = 9/32λ² exp(3 tanh²(χ / √6)) tanh⁶(χ / √6) [1 + tanh²(χ / √6)]³ [112 + 466 tanh²(χ / √6) + 777 tanh⁴(χ / √6) + 369 tanh⁶(χ / √6) + 54 tanh⁸(χ / √6)].


Question: What are the specific relations in the monoid Mcℓ(ρ) and how do they correspond to the indistinguishability relations in Corollary 2.4.3?

Answer: The monoid Mcℓ(ρ) has the relations a2 = 1, p4 = p5 = ap4. The relation a2 = 1 corresponds to ∗ + ∗ ≡ 0 (mod cℓ(ρ)) in Corollary 2.4.3.


Question: What are the key estimates provided in Proposition 2.2 regarding the components of the eigensections of Dr?

Answer: Proposition 2.2 provides the following estimates: (i) For any r ≥ c1, if ψ is an eigensection of Dr such that |λψ|2 < 3/4r, then the integral of |β|2 over Y plus r−1 times the integral of |∇rβ|2 over Y is less than or equal to c1r−1 times the integral of |α|2 over Y, where α is the E component of ψ, and β is the EK−1 component of ψ. (ii) If there is a continuous path of eigenvalues λ(s) of Ds which is smooth at r ≥ c1 and |λ(r)|2 < 3/4r, then 9/20 − c1r−1 ≤ λ′(r) ≤ 5/9. In particular, there are only positive zero crossings for the spectral flow of Ds when s ≥ 3c1.


Question: What are the predicted masses and decay constants for the lightest meson states in the 5D model, and how do they compare to the experimental values in Table 1.1?

Answer: The 5D model predicts the masses of the lightest meson states as mρ = mω ≃ 3π/4L and ma1 = mf1 ≃ 5π/4L, which are compatible with observations. The decay constants Fi and couplings gi for the mesons scale as Fi ∼ √M5 and gi ∼ 1/√M5. The best fit to 14 meson quantities, using the parameters 1/L = 343 MeV, M5L = 0.0165, and α = 0.94, results in a minimum Root Mean Square Error (RMSE) of 11%, with the relative deviation of each single prediction below around 15%.


Question: How does the temperature dependence of the nuclear symmetry energy vary with different interactions and densities?

Answer: The nuclear symmetry energy decreases with increasing temperature for all three interactions (MDI, MID, and eMDYI) with both x = 0 and x = −1. The decrease is more pronounced at higher temperatures. The potential part of the symmetry energy with the MDI interaction is temperature-dependent, while the potential part with the MID and eMDYI interactions does not depend on temperature. The kinetic part of the symmetry energy also contributes to the overall decrease with temperature, but the potential part plays a significant role, especially for the MDI interaction.


Question: What is the main result of the paper regarding the injectivity theorem for pseudo-effective line bundles with transcendental singular hermitian metrics?

Answer: The main result is an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, a Nadel type vanishing theorem is obtained.


Question: What are the key components and benefits of the Randomized Incremental Optimal Technique (RIOT) in the context of parallelizing strong-constrained incremental 4D-Var algorithms?

Answer: The key component of the Randomized Incremental Optimal Technique (RIOT) is the use of randomized singular vector decomposition (RSVD) methods, which enables parallel model integrations in place of the traditional sequential minimization steps of the inner loops. RIOT introduces a new level of parallelization for variational DA that is advantageous when the available computing resources exceed that which can be efficiently used by any single forward model evaluation (e.g., through parallel computation with MPI communications).


Question: How does the renormalized on-site interaction energy parameter UR affect the agreement between the TM model and GP simulation in the context of Josephson oscillations?

Answer: The renormalized on-site interaction energy parameter UR = 5.28 × 10−4 ¯hωr, when used in the TM model, results in an excellent agreement with the GP simulation, as observed in Figs. 3 and 4. This renormalization also brings the period of small Josephson oscillations closer to the GP value (Tso = 51.5 ω−1 r).


Question: How do the MAP values for LSA-RAW and CA compare across different datasets and values of k for dot similarity, and what is the optimal α for each configuration?

Answer: For LSA-RAW, the optimal α values and corresponding MAP scores for different k and datasets are: BBCNews (k=6, α=-0.8, MAP=0.701), BBCSport (k=6, α=-1, MAP=0.687), 20 Newsgroups (k=12, α=0.2, MAP=0.495), and Wilhelmus (k=12, α=0, MAP=0.500). For CA, the optimal α values and corresponding MAP scores are: BBCNews (k=4, α=0.896, MAP=0.826), BBCSport (k=6, α=0.853, MAP=0.853), 20 Newsgroups (k=12, α=0.756, MAP=0.642), and Wilhelmus (k=12, α=0.756, MAP=0.642).


Question: How do the implemented chemistry labs address the challenges of open inquiry and student-directed learning, particularly in terms of handling experimental failures and data variability?

Answer: The implemented chemistry labs focus on teaching standard experimental techniques while introducing students to the realistic uncertainty, imprecision, and failure that accompanies open inquiry. Data from all lab groups is available to students to facilitate discussions about variability and drawing conclusions, helping students accept the data that result from their experiments, even if they appear less successful than their peers.


Question: What is the explicit formula for the zeta function of a 3-dimensional Zp-Lie algebra in terms of a ternary quadratic form?

Answer: For a 3-dimensional Zp-Lie algebra L, the zeta function is given by ζpiL(s) = ζZ3p(s) − Zf(s − 2)ζp(2s − 2)ζp(s − 2)p(2−s)(i+1)(1 − p−1)−1, where Zf(s) is Igusa’s local zeta function associated to the ternary quadratic form f(x) ∈ Zp[x1, x2, x3].


Question: How do style-based models achieve improved visual results in high-resolution image generation, and what are the limitations of these models?

Answer: Style-based models achieve improved visual results by first generating a low-resolution image and then growing it in size in a hierarchical manner, which affords additional regularization and stability to the trained model. However, these models are larger and notoriously expensive to train, with some requiring upwards of 50 years of computation time using a top-of-the-line GPU.


Question: What is the relationship between the number of superfluous 1s in permutation tableaux and the number of generalized patterns 31-2 in permutations?

Answer: The number of superfluous 1s in permutation tableaux is related to the number of generalized patterns 31-2 in permutations through a bijection described in [SW]. Specifically, the number of superfluous 1s in a permutation tableau corresponds to the number of generalized patterns 31-2 in a permutation, where an occurrence of a generalized pattern 31-2 in (σi) is any pair (i, j), 1 < i < j such that σi−1 > σj > σi.


Question: What are the specific values of the tau polarisation determined from each year's data, and what is their average?

Answer: The tau polarisation values determined from each year's data are: -0.61 for 1997, -0.41 for 1998, -0.01 for 1999, and +0.11 for 2000. The average of these values is -0.176.


Question: How do different exchange-correlation functionals predict the magnetic transition in Sr1−xBaxMnO3 as x is varied from 0 to 1?

Answer: PBE+U+J and sPBE+Ueﬀ predict an antiferromagnetic to ferromagnetic transition as x is increased from 0 to 1, while sPBEsol predicts that Sr1−xBaxMnO3 remains G-type antiferromagnetic for the entire range of x (0 ≤ x ≤ 1).


Question: How do the SMARAGD and NON-SMOKER predictions compare for the 83Rb(p,γ)84Sr reaction, and what adjustments are needed to match the experimental data?

Answer: The SMARAGD cross section is lower by about 30% than the NON-SMOKER cross section, even when using the same default optical potential. A proton width approximately 0.3 times as large as the width predicted by SMARAGD would reproduce the experimental cross section inferred to be most likely.


Question: What experimental tests have confirmed the predictions of Tomonaga-Luttinger liquids in spin insulators?

Answer: Experimental tests in spin insulators have fully confirmed the TLL predictions, including the existence of power-laws and fractionalization, as well as the universality of the dependence of the exponents in terms of the TLL parameters. These tests showed remarkable accuracy not only in asymptotic regimes but also at intermediate energies.


Question: What are the specific conditions under which a remnant with a 0.35M⊙ core fails to form a planetary nebula?

Answer: The 0.35M⊙ core takes 2.1 × 10^5 yr to reach the critical region, which is longer than the estimated 10^4 yr lifetime of the nebula.


Question: What is the relationship between the MEP-threshold and the percentage of misclassifications in the test set, and how does this relationship manifest in the empirical results?

Answer: The MEP-threshold and the percentage of misclassifications are highly correlated. Data sets with good threshold values, such as dna, satimage, and shuttle, have a very low rate of misclassifications. Conversely, the vowel data set, which had a very high threshold of almost 76%, performed poorly with a high rate of misclassifications. This correlation confirms the MEP-threshold as a viable a priori measure for the quality of the computed soft power diagram as a classifier.


Question: How does the sensitivity of the NTK model change with different activation functions, and what does this imply for real-world datasets like MNIST and CIFAR-10?

Answer: For ϕ(x) = x², the sensitivity keeps decreasing as the number of neurons k (and the number of parameters p) grows. In contrast, for ϕ(x) = tanh(x), the sensitivity plateaus at a value which is an order of magnitude larger than for ϕ(x) = x². This difference is still visible even for real-world datasets (MNIST, CIFAR-10), as displayed in the third and fourth plot of Figure 2.


Question: How do dipolar forces influence the formation and behavior of multi-skyrmion domains near boundaries?

Answer: Multi-skyrmion domains bound by dipolar forces are formed near boundaries. These domains are attracted to boundaries by long-range dipolar interaction with image charges, which modifies Bean-Livingston barrier physics. The dipolar attraction to the image 'anti-skyrmions' has a longer range than the repulsion from the boundary due to surface Meissner current. These domains gradually fill the system until merging to form a (checkerboard) square lattice of fractional vortices.


Question: What is the relationship between the noise behavior near the Dirac point and the spatial charge inhomogeneity in single-layer graphene (SLG)?

Answer: The noise behavior near the Dirac point in SLG correlates with the spatial charge inhomogeneity, where the noise starts to reduce when the carrier density is close to the Dirac point. The noise is dominated by the spatial charge inhomogeneity, with the landscape of SLG breaking into puddles of electrons and holes. The noise mainly comes from type-I regions, which are wide regions with almost uniform low carrier densities, while type-II regions with much higher carrier densities contribute less to the noise due to better screening of active traps.


Question: What is the morphology and structure of the Ti3C2Clx MXenes, and how does it affect their properties?

Answer: The Ti3C2Clx MXenes have an accordion-like morphology with homogeneous distribution of Ti, Cl, and corresponding metal elements. This structure allows for the fabrication of Al-Ti3C2Clx MXene multi-layers with few hundred nanometers in lateral size, where Ti, Al, Cl, and C elements are uniformly dispersed in the atomic layers. The as-prepared Al-Ti3C2Clx nanosheets possess good crystallinity with an average lattice distance of 0.26 nm, indexed to the (100) plane, and Ti atoms are regularly arranged in hexagonal symmetry, confirmed by the FFT pattern with six hexagonal diffraction spots.


Question: How are the eigenfunctions for the scalar field represented in the context of spherical symmetry, and what form does the radial function take in the region between the spherical shells?

Answer: The eigenfunctions for the scalar field can be represented in the factorized form ϕα(x) = Z(r)Ylm(θ, φ)e−iωt, where Ylm(θ, φ) are the spherical harmonics with l = 0, 1, 2, ..., −l ⩽ m ⩽ l. In the region between the spherical shells, the radial function Z(r) is expressed in terms of the associated Legendre function as Z(r) = c1P −l−1/2 iz−1/2 (u) + c2P l+1/2 iz−1/2(u) / √sinh r, with integration constants c1 and c2 and the notations z2 = (ω2 − M2)a2 + 6ξ − 1, u = cosh r.


Question: What does Lemma 2.6 state about the regularity of the disconnected prime graph ∆(G)?

Answer: Lemma 2.6 states that if ∆(G) is a disconnected k-regular graph, then k = 0. This holds for both solvable and nonsolvable groups.


Question: How does the concept of quotient graphs relate to the injectivity of the function B in the context of Anosov graphs?

Answer: The quotient graph of a graph G modulo an equivalence relation R is a graph whose vertices are the equivalence classes of the vertices induced by R, and there is an edge between the vertices if the vertices of the equivalence classes were adjacent in G. Since each case in the query text produces a distinct quotient graph induced by ∼ of an Anosov graph, this ensures that B is injective, and thus D∗n ◦ B is an injective function from Λ(n) to An.


Question: How does the 'back tracking' method account for the random component of the magnetic field when modeling the movement of cosmic rays?

Answer: The 'back tracking' method models the movement of an anti-particle starting from the Earth in the direction opposite to the registered cosmic ray, with the same energy. In the first approximation, only the regular component of the magnetic field is considered. The random component of the magnetic field results in widening the area of the source's possible location, but the direction of deflection remains unidentified. This method is most effective for relatively close objects and cosmic rays with low charge, as the distortion of the reconstructed trajectories by the random component is less significant.


Question: How does the generalization of Fine's theorem to n-time LG inequalities ensure the existence of a joint n-time probability distribution?

Answer: For any n ≥3, the joint n-time probability distribution is guaranteed to exist as long as all 2n−1 n-time LG inequalities Eq.(3.4) are satisfied, together with the 4n two-time LG inequalities consisting of the non-negativity conditions on the fixed pairwise probabilities.


Question: What is the significance of the result that P(2X) is not a complemented subspace of Lip0(BX) for Banach spaces with non-trivial type?

Answer: The significance of this result is that it answers Problem 1.1 in the negative for a large class of Banach spaces, including all spaces with non-trivial type, such as super-reflexive Banach spaces. This implies that for these spaces, P(2X) cannot be complemented in Lip0(BX), which has implications for the approximation properties of these spaces. For example, if P(2ℓ2) were a complemented subspace of Lip0(Bℓ2), then Lip0(Bℓ2) would fail the approximation property, as the approximation property passes to complemented subspaces, while P(2ℓ2) fails the approximation property.


Question: How does the hierarchical LSTM model in the action localization framework differ from a typical hierarchical LSTM model in terms of temporal dependencies?

Answer: In the action localization framework, the higher-level LSTMs are impacted by the output of the lower-level LSTMs at the current time step, as opposed to the previous time step, which is typical in a standard hierarchical LSTM model. This allows the model to capture both spatial and temporal dependencies more effectively.


Question: What are the specific Jacobi forms that generate the ring of Jacobi forms of even weight, and how are they defined?

Answer: The ring of Jacobi forms of even weight is generated by the two Jacobi forms: ϕ−2,1(τ, u) = ϑ1(τ, u)² / η(τ)⁶ and ϕ0,1(τ, u) = 4 ∑ (i=2,3,4) ϑi(τ, u)² / ϑi(τ, 0)².


Question: How is the Chern class of M determined from the Chern class of M0 and the local behavior of B at points on Cα?

Answer: The Chern class of M is given by c1(M) = c1(M0) − ∑_{p,i} λi(B; p)ep,i, where λi(B; p) is the rank of M0 at p after blowing up p i-1 times, and ep,i is the i-th class in the sequence e1, ..., em such that φ∗(ei) ∼= OC(p).


Question: How does the performance of the CLASH model compare to baseline methods in terms of Pearson correlation scores?

Answer: The CLASH model significantly outperforms all baseline methods, with a median Pearson correlation of 0.173 compared to 0.120 for raw EEG, representing a 45% relative increase in median performance (p < 0.001).


Question: How does incorporating label information into contrastive learning affect the robustness of the resulting representations, and what are the specific findings on CIFAR-10 and CIFAR-100 datasets?

Answer: Incorporating label information into contrastive learning enhances the robustness of the resulting representations. On CIFAR-10 and CIFAR-100 datasets, contrastive learning without labels shows lower robustness compared to other learning schemes. Semi-supervised learning schemes like SL+CL or SCL+CL achieve better robust performance than the CL scheme.


Question: What theorem provides the lower bound on the truncated Bloch wave U, and how is it derived?

Answer: Theorem 2.11 provides the lower bound on the truncated Bloch wave U. It states that for some c > 0 independent of the energy E, inf |Im(x)|<c ∥U(x)∥≥e−o(n). This is derived by considering the truncated Bloch wave U = U I, where I := [−cN, cN], and using the polynomial bound obtained in Theorem 2.10 instead of estimate (4) in the proof of Theorem 2.6, which improves the conclusion to inf |Im(x)|<c ∥U J(x)∥≥e−o(n). Since ∥U I −U J∥c ≤e−cn, it follows that inf |Im(x)|<c ∥U(x)∥≥e−o(n).


Question: How are the polynomials pi(y) defined for the i-th row in the determinant expansion of f(A)(x)?

Answer: For i odd, pi(y) = (i−1)/2 ∑ n=0 (−1)ni (2n) y2n(1 − y2)(i−1)/2−n = yi−12i−1(−1)(i−1)/2 + O(yi−3), and for i even: pi(y) = y i/2 ∑ n=1 (−1)n+1i (2n) −1 y2n−2(1−y2)i/2−n = yi−12i−1(−1)i/2−1 + O(yi−3).


Question: How does the symmetry group SL(3, R) contribute to the emergence of macroscopic spacetime in the context of classical physics and general relativity?

Answer: The symmetry group SL(3, R) allows stretching and shearing of space. When combined with the Lorentz group SO(3, 1) acting on the same spacetime, it forms the group SL(4, R) of all unimodular coordinate changes, which describes the general covariance of general relativity. This combination suggests that SL(3, R) describes a 'fluid' gravitational 'space' that contributes to the macroscopic spacetime underlying classical physics.


Question: What are the implications of weak asymptotic tail-equivalence for distributions in the class J, and how does this compare to the class S?

Answer: If F ∈ J and F ≍ G, then G ∈ J, indicating that the class J is closed under weak asymptotic tail-equivalence. In contrast, the class S is not closed under weak tail-equivalence, as demonstrated by the example of a Pareto distribution F and the 'Peter-and-Paul' distribution G, where F ≍ G but G /∈ S.


Question: What are the key components of the Ericksen-Leslie system on S2, and how do they differ from the simplified system discussed in the query text?

Answer: The Ericksen-Leslie system on S2 includes the following components: the fluid velocity field u(x, t) : S2 × (0, ∞) → TxS2, the orientation order parameters d(x, t) : S2 × (0, ∞) → S2, and the pressure function P(x, t) : S2 × (0, ∞) → R. The system is described by the equations: ut + u · ∇u + ∇P = −div(∇d ⊙ ∇d) + div(σL(u, d)), div u = 0, and λ1(dt + u · ∇d − Ωd) + λ2Ad = ∆d + |∇d|2d + λ2(dT Ad)d. The convection term u · ∇u is interpreted as the covariant derivative Duu, and the system includes additional terms such as the Leslie stress tensor σL(u, d) and the rotational term Ωd, which are not present in the simplified system discussed in the query text.


Question: What is the union complexity of α-fat triangles and how does it vary with the size of the triangles?

Answer: The union complexity of α-fat triangles is O(r log∗r), where the constant of proportionality depends on α. When the triangles have roughly the same size, this complexity reduces to O(r). When one side of these triangles is unbounded (making them α-fat wedges), the complexity also reduces to O(r).


Question: What are the specific assumptions made about the processing capabilities of the ONU devices in the Access Fog layer?

Answer: The ONU devices are assumed to have a processing capability similar to a Raspberry Pi 3, which can process 2 instructions per clock frequency (Hz). Additionally, the total CPU capacity (in MIPS) per Access Fog device was estimated similarly to the IoT devices, and the Access Fog devices are assumed to have a higher CPU capacity than the IoT devices to account for the difference in efficiencies.


Question: What are the specific energy and anisotropy values for the variational wave functions with different patterns and weights of diagonal bonds?

Answer: The energy and anisotropy values for the variational wave functions are as follows: For fx with wd = 0, the energy per site is −0.489281(1) and the anisotropy is 0. For fx with wd = 0.005, the energy per site is −0.489280(1) and the anisotropy is 0.000045(10). For fx with wd = 0.01, the energy per site is −0.489284(3) and the anisotropy is 0.000184(26). For gx with wd = 0.005, the energy per site is −0.489282(1) and the anisotropy is 0.000017(10). For gx with wd = 0.01, the energy per site is −0.489281(3) and the anisotropy is 0.000023(26).


Question: How are the domain-decomposed deﬂation subspaces generated and what is their dimension?

Answer: The domain-decomposed deﬂation subspaces are generated by first decomposing the lattice into non-overlapping blocks Λ of fixed size. A set of quark fields ψl(x), l = 1, ..., Ns, is then generated through a smoothing procedure. These fields are projected to the blocks Λ, and the linear space spanned by the set of all block fields ψΛl(x) forms the deﬂation subspace. The dimension of the deﬂation subspace is equal to Ns times the number of blocks, with typical values of Ns ranging from 12 to 24.


Question: How does the radius of the nth caustic ring in the Milky Way relate to the galactic rotation velocity and the maximum angular momentum, and what is the predicted radius for the n = 1 caustic ring?

Answer: The radius of the nth caustic ring in the Milky Way is given by the formula an ≃ 40 kpc / n * (vrot / 220 km/s) * (jmax / 0.18), where vrot is the galactic rotation velocity and jmax is the maximum angular momentum. For the n = 1 caustic ring, the predicted radius is approximately 40 kpc.


Question: How does the proposed method in the candidate text address the challenge of filming dynamic scenes with multiple UAVs, and what is the role of the leader-follower scheme?

Answer: The proposed method uses a leader-follower scheme where the formation leader carries an onboard camera to film a moving target, and the followers generate trajectories that enable distributed lighting of the target while maintaining desired lighting angles. The leader computes visually pleasant trajectories for filming, and the followers ensure optimal and safe trajectories are computed within convex sets that model free space, ensuring obstacle avoidance and maintaining the formation.


Question: What is the condition for the Cayley transform S(k; A, B) to be uniformly bounded outside a compact set when dim Ker A = 0 and dim Ker B = 1?

Answer: The Cayley transform S(k; A, B) is uniformly bounded outside a compact set containing its only possible pole if and only if P ⊥BP ⊥ ≠ 0. If P ⊥BP ⊥ = 0, and hence PBP ⊥ ≠ 0, then ∥S(k; A, B)∥ = O(|k|) for |k| → ∞.


Question: How does the presence of an odd number of topologically protected gapless states at the surface of a system with reflection symmetry determine its STI phase?

Answer: If the number of topologically protected gapless states at the surface is odd, the system cannot be insulating, and it is concluded that the system is a non-trivial STI. This is because the ensemble symmetry does not allow the insulating phase to appear if the number of gapless states is odd, as defined by the Z2 topological invariant Q = (−1)∆Qd−1.


Question: What additional figures show the histories and mean thrust coefficients for the flow around the pitching NACA0012 airfoil at different conditions?

Answer: Fig. 14 shows the histories of drag coefficient for the flow around the pitching NACA0012 airfoil at different reduced frequencies k, d = 2°. Fig. 15 shows the mean thrust coefficients ¯CT for the flow around the pitching NACA0012 airfoil at AoA=2° and AoA=4°. Fig. 16 shows the mean thrust coefficients ¯CT for the flow around the pitching NACA0012 airfoil at AoA=2° and AoA=4° at different Mach numbers, Ma.


Question: How does the Tree Block Coordinate Ascent (TBCA) method differ from the regular dynamic programming (DP) in terms of message computation?

Answer: The TBCA method differs from regular DP in two ways: i) it depends on the right messages, which are incorporated into the unary costs in ˜gi(s), and ii) there are redistribution coefficients ri in the recursion. To handle the latter, the update equation is modified to mL_i+1(t) := max_s(˜gi(s) + ri mL_i(s) + fi,i+1(s, t)), where ˜gi(s) = gi(s) + (1 − ri)mR_i(s) and ri ∈ [0, 1] are the redistribution coefficients. For r = 1, this recovers the regular dynamic programming.


Question: What is the primary goal of collaborative filtering in the context of recommender systems, and how does it differ from content-based filtering?

Answer: The primary goal of collaborative filtering in recommender systems is to suggest new content to users based on past user behavior (ratings, views, purchases), avoiding the need for additional data collection. In contrast, content-based filtering constructs user and/or item profiles from demographic data or item information, respectively, and uses this information to compute similarity between users or items for making recommendations.


Question: What does the phase diagram in Fig. 6 show regarding the structural and antiferromagnetic transitions and the onset of superconductivity in SrFe2As2?

Answer: The phase diagram in Fig. 6 shows the temperature-pressure (T(P)) relationship for SrFe2As2. Blue and green data points correspond to cell 1 and cell 2, respectively, with low pressure data up to 20 kbar from reference [12] added in pink. Circles and crosses represent the structural/antiferromagnetic transition and the onset of superconductivity, respectively. Triangles indicate the offset temperature of the full superconducting transition. The hatched area denotes the zero-resistance superconducting region, and the very large horizontal error bars are due to pressure uncertainties caused by a small remanent field.


Question: How does the proof of the equivalence between Π0 and ˜Π utilize the properties of the approximate Bergman projection and Corollary 4.7.1?

Answer: Using properties 3 and 4 for ˜Π, we see that ˜Π∗ satisfies 4. Since Π0 satisfies 1, we get (˜Π∗−1)Π0 ≡Φ 0, i.e., ˜Π∗Π0 ≡Φ Π0. By Corollary 4.7.1, we get Π0 ≡Φ Π∗0 ≡Φ Π0 ˜Π. By (4.24) with Π0 and ˜Π exchanged, we get Π∗0 ˜Π ≡Φ ˜Π and hence Π0 ≡Φ ˜Π as claimed.


Question: What is the physical mechanism that leads to the formation of a Mott insulator according to the analysis of the Hamiltonian ˆH (1)?

Answer: The conduction bands, which arise in the traditional electronic theory of metals at half-filling, split into two sub-bands (Hubbard sub-bands). The lower sub-band becomes completely filled with conduction electrons, and the upper sub-band becomes completely empty, creating an energy gap between them. This occurs when the Coulomb interaction is sufficiently strong, leading to the system becoming an insulator, which is the physical reason for the Mott insulator.


Question: How do the success rate, false alarm rate, miss rate, and channel estimation error vary with the number of active devices in the 500 × 500 m² area?

Answer: The success rate, false alarm rate, miss rate, and channel estimation error are plotted against the number of active devices (K) in the 500 × 500 m² area. Specifically, the success rate, false alarm rate, miss rate, and channel estimation error are shown in Figures 5(a), 5(b), 5(c), and 5(d) respectively.


Question: How does the secure communication protocol ensure that the cloud cannot distinguish between isomorphic systems?

Answer: The protocol renders two quadruples Ω and ˜Ω indistinguishable by the cloud by ensuring that the exchanged messages, when using the protocol between the cloud and the plant Ω, and the exchanged messages, when using the protocol between the cloud and the plant ˜Ω, can be made the same. This is achieved through the use of isomorphisms and encoding the plant's parameters and measurements into transformed versions that the cloud processes without knowing the original system.


Question: How are general reflection and uniform scaling versors constructed for a plane and a point, respectively, in the context of general Euclidean transforms?

Answer: For a general reflection in a plane with a normal vector \( \mathbf{E}_n \) passing through the point \( \mathbf{E}_p \), the general reflection versor is constructed as \( V_{	ext{GRP}} = V_p^{-1} V_n V_p \). For a general reflection in a point \( \mathbf{E}_p \), the general reflection versor is \( V_{	ext{GRO}} = V_p^{-1} V_o V_p \). For a general uniform scaling versor in a point \( \mathbf{E}_p \), it is constructed as \( V_{	ext{GS}} = V_p^{-1} V_s V_p \).


Question: How do the eigenfrequency spectra of phononic crystals with different topological phases differ, and what does this indicate about the presence of topological edge states?

Answer: The eigenfrequency spectra of phononic crystals with 29 units-A (PC1) and 29 units-B (PC2) show different behaviors. PC1, with a topologically trivial geometric Zak phase (0), does not exhibit any edge states in the first band gap. In contrast, PC2, with a topologically non-trivial geometric Zak phase (π), shows topological edge states at 0.516 kHz in the first band gap, as indicated by the red line in the spectra. The simulated acoustic pressure fields of these topological edge states are localized at either end of PC2.


Question: What is the background for the µ−µ−E/ final state and how does it compare to the e−e−E/ final state in terms of the effect of kinematic cuts?

Answer: The background for the µ−µ−E/ final state is a six-body final state e−+ e−→µ−+ µ−+ ¯νµ¯νµ + νeνe and is quite small. The effect of cuts is not severe for the µ−µ−E/ final states as compared to e−e−E/ final states, where the cross section is reduced by more than ∼80 −90% of the original one, while the cross section in this case reduces by about 30% for the more stringent cuts listed in Table 2.


Question: What are the boundary conditions for the radial Teukolsky equation that ensure the physical oscillations of the Kerr black hole?

Answer: The boundary conditions require only ingoing waves at the (future) event horizon and only outgoing waves at (future) null infinity. For the perturbations to decay in time, the imaginary part of the frequency ω must be negative.


Question: How do the dynamics of the local and remote task queues relate to the computation offloading policy in the MEC system?

Answer: The dynamics of the local task queue at the MT is given by Ql(n + 1) = [Ql(n) − vt(Pt(n), H(n))τ − vl(Pl(n))τ] + λ(n)τ, and the dynamics of the remote task queue at the MEC server is given by Qr(n + 1) = [Qr(n) − vout(n)τ] + vt(Pt(n), H(n))τ. These dynamics are coupled together by a cascade control, where the departure from the local queue is the arrival to the remote queue, creating complex interdependence and making the computation offloading problem a stochastic optimization problem.


Question: What are the two different dual problems that can be constructed, and how are the solutions of the primal problem recovered from the solutions of these dual problems?

Answer: The two different dual problems are: 1) \( \min_{y \in \mathbb{R}^m} \left( \sum_{i=1}^n \ln \left( \exp \left( (A^T y)_i \right) \right) - y^T b \right) \), and 2) \( \min_{y \in \mathbb{R}^m} \left( \sum_{i=1}^n \left( \exp \left( (A^T y)_i \right) - 1 \right) - y^T b \right) \). The solutions of the primal problem can be recovered as follows: 1) \( x_i = \frac{\exp \left( (A^T y)_i \right)}{\sum_{k=1}^n \exp \left( (A^T y)_k \right)} \) for \( i = 1, \ldots, n \), and 2) \( x_i = \exp \left( (A^T y)_i \right) - 1 \) for \( i = 1, \ldots, n \).


Question: How does the insertion of an invertible gapped domain wall on the trivial 2d SPT order Yid relate to 1d SPT orders, and what does this imply for the higher group of 2+1D SPT orders G3?

Answer: Inserting an invertible gapped domain wall P on the trivial 2d SPT order Yid results in a 1d SPT order. Since an invertible domain wall defines a higher isomorphism of Yid, by adding invertible domain walls of all higher codimensions on Yφ for all φ ∈ Autbr(Z1(ΣE), ι0), we obtain a higher group of 2+1D SPT orders G3. If E = Rep(G), the formula πi(G3) ≃ H3−i(G, U(1)), i = 0, 1, 2, is obtained.


Question: How do the actual moiré lattice points in the simulated moiré patterns of TBG differ between normal and anomalous commensurate moiré patterns?

Answer: In normal moiré patterns of TBG, each AA-type looking region has one actual moiré lattice point in the center, depicted by green dots. In anomalous moiré patterns of TBG, the apparent moiré period appears quite different from the value of the moiré period supplied in the code, and the pattern of actual moiré lattice points is not specified in the text.


Question: What is the specific form of the plug-in classifier used to match the lower bound in the supervised label shift problem, and how is it constructed?

Answer: The plug-in classifier used to match the lower bound in the supervised label shift problem is defined as: ˆf(x) = 1 if ˆηQ(x) ≥ 1/2, and 0 otherwise, where ˆηQ is an estimator of the regression function constructed by plugging in an estimator of πQ and kernel-based estimators of g0 and g1: ˆηQ(x) = ˆπQˆg1(x) / (ˆπQˆg1(x) + (1 − ˆπQ)ˆg0(x)). Here, ˆπQ is the fraction of samples from the target domain with label 1, and ˆg0 and ˆg1 are kernel-based density estimators: ˆgy(x) = (1 / ny) * Σ (1 / hdy) * K((x - x′) / hy), where Xy = {x : (x, y′) ∈ DL, y′ = y}, ny = |Xy|, and hy > 0 is a bandwidth parameter.


Question: How is the matrix Mik[t] defined for a faulty node i ∈ Fv[t] when (hk[t −1], k, t −1) ∈ Veriﬁed c i[t]?

Answer: If (hk[t −1], k, t −1) ∈ Veriﬁed c i[t], then Mik[t] = 1 / |Veriﬁed c i[t]|.


Question: How does the fixed-vertex positioning in the contrapositioned proof tree differ from fixed-edge positioning in terms of illustrating the relationship between the source tree and its contrapositioned counterpart?

Answer: Fixed-vertex positioning illustrates the relationship between the source tree and its contrapositioned counterpart in a far more understandable manner, while fixed-edge positioning may result in a tree that looks more natural.


Question: What are the specific characteristics of the pulse profiles of SGR 0501+4516 during its burst active phase and immediately after?

Answer: During the burst active phase, the ∼4.5 keV pulse profiles are smooth while those in the lower energy bands show significant subpulses. There are at least four subpulses between 0.3−2 keV, with the most significant subpulse peaking near φ∼0.34 and lasting about 0.72 s, which corresponds to ∼1/8 of the spin period. The significant peak in the 14−40 keV band near φ∼1.0 is weaker in the 8.5−14 keV range and not seen in the lower energy bands. This hard component fades away within a few weeks after the burst activity onset.


Question: What specific reward shaping techniques were used to help the policy escape plateaus during training?

Answer: Two reward shaping techniques were used: 1) Landing Bonus: Increase the sparse success reward. 2) Distance to Table (DTR): A dense version of the success reward given by max{1−d, −2}, where d is the minimum distance between the ball and the opponent’s table surface during the return trajectory.


Question: How does the application of a square-wave voltage affect the sorting of particle populations in the ratchet field, and what is the time scale for this separation?

Answer: Applying a square-wave voltage of 4 V @ 30 Hz causes the particle populations to be fully separated into the respective compartments after 2 s, except for a single 100 nm particle stuck in the shallow ratchet. After 6.3 s, this particle diffused into the deep channel and was transported into the LC.


Question: What are the specific conditions and results of the Iliev Test 3, and how do the GLF and HLL Riemann solvers compare in terms of shadow formation?

Answer: Iliev Test 3 examines the self-shielding of a dense gas clump and the formation of a shadow. The simulation box is 6.6 kpc in size, with a spherical cloud of gas (radius 0.8 kpc) centered at (5, 3.3, 3.3) kpc. The surrounding hydrogen gas has a number density of 2 × 10−4 cm−3 and a temperature of 8000K, while the cloud has a number density of 200nout and a temperature of 40K. A constant flux of 106 photons/s/cm2 from a blackbody spectrum with a temperature of 105K is injected from the x = 0 plane. The HLL solver improves shadow formation compared to the GLF solver, as seen in the temperature profile remaining at the initial 8000K behind the clump at x/L ≥0.9. However, the GLF solver shows the ionization front penetrating much further into the dense clump than the reference solutions.


Question: How does the optimal expected slot degree β and the probability of resolution PR vary with the ratio of the number of slots to the number of users M/N for different packet loss scenarios?

Answer: For a single user class with no packet loss (e(1) = 0), the maximal expected throughput T(1)max ≈ 0.87, with an optimal contention period length M(1)opt ≈ 1.05N slots, PR ≈ 0.93, and β(1)opt ≈ 3.1. For a single user class with a packet loss of 0.375 (e(2) = 0.375), the maximal expected throughput T(2)max ≈ 0.55, with an optimal contention period length M(2)opt ≈ 1.7N slots, PR ≈ 0.93, and β(2)opt ≈ 3.1. For two user classes with equal fractions and packet losses of 0.25 and 0.5, the maximal expected throughput T(3)max ≈ 0.65, with an optimal contention period length M(3)opt ≈ 0.7N, α(3)1 ≈ 3.1, and α(3)2 = 0, indicating that the best strategy is to allow contention only among users with lower packet-loss probabilities.


Question: What is the finiteness property of the kernel of the natural projection PΣAutn → PΣAutn−1 for n ≥ 3, and how does it compare to the pure braid group case?

Answer: The kernel of the natural projection PΣAutn → PΣAutn−1 is finitely generated but not finitely presentable when n ≥ 3. In contrast, for the pure braid group, the kernel of the 'forget a strand' map PBn → PBn−1 is of type F∞ (in fact, it is the free group Fn−1).


Question: How does the paper relate the combinatorial properties of monomial ideals to the algebraic setting of inverse systems and general artinian ideals?

Answer: The paper demonstrates the interplay between the discrete and combinatorial setting (monomial ideals) and the algebraic setting (inverse systems and general artinian ideals). It provides a combinatorial and elementary proof that the docle ∂oc(I) of a monomial ideal I is in bijective correspondence with the generators of the inverse system (IS) for the ideal. Additionally, it explores when these combinatorial properties can be extended to ideals of polynomial rings in general and when it makes sense to view them as inverse ideals of elements in their socles that are not in the ideals.


Question: How is the two-particle density derived from the N-particle density under the assumption of transverse momentum conservation?

Answer: The two-particle density is derived by integrating out N-2 momenta from the N-particle density. Specifically, f2(⃗p1, ⃗p2) = f(⃗p1)f(⃗p2) * [∫F δ2(∑Ni=1 ⃗pi,t) ∏Ni=3 [f(⃗pi)d3⃗pi] / ∫F δ2(∑Ni=1 ⃗pi,t) ∏Ni=1 [f(⃗pi)d3⃗pi]].


Question: What is the estimated Knight shift for the major (slow) component at 300 K, and what does this indicate about the magnetic correlations in LaNiO3?

Answer: The estimated Knight shift K for the major (slow) component at 300 K is approximately 71(10) ppm. This value, combined with the Korringa slope, forms the normalized Korringa product K = 0.40(10), which is significantly less than 1, indicating substantial antiferromagnetic correlations.


Question: How does the mixing term λNvs influence the HS contribution in the ISS-NMSSM, and what is the preferred value of λ for maximizing aHS_ℓ?

Answer: A large mixing term λNvs can render a sizable mixing between the x-field and the right-handed sneutrino field, alleviating the cancellation between contributions from the left-handed and right-handed dominated states. A light µ is favored for large aHS_ℓ, so a small λ at O(0.01) often has a larger aHS_ℓ. This small λ is also preferred by the leptonic unitary condition.


Question: What specific changes does the theory propose in the context of state reduction and where does it specify this reduction occurs?

Answer: The theory replaces the collapse by a more radical transformation, a change in microsystem kinematic description. The state reduction occurs in the sensitive volume of the detector, and the time is that of the detector signal.


Question: How do the results of O/H derivation using the C34He2 and N5He2 parameters from Dors et al. (2019) compare with those from HCm-UV, and what is the mean difference for the sample of 26 galaxies with a prior determination of C/O?

Answer: The mean difference between the O/H values obtained from HCm-UV and the calibrations using the C34He2 and N5He2 parameters from Dors et al. (2019) is -0.2 dex and -0.9 dex, respectively.


Question: What is the expression for the mean-square displacement in the time domain for the fractional diffusion model, and how does it differ from the original expression?

Answer: The mean-square displacement in the time domain for the fractional diffusion model is given by r2(t) = r2_A(t) + r2_B(t), where r2_A(t) = 2v^2 / (αγ(α + γ)) [α(1 − e^−γt) − γe^−γt(1 − e^−αt)] and r2_B(t) = 2γD / Γ(2 + ν) t^(1+ν) e^−γt Φ(ν + 1, ν + 2; γt). The second term, r2_B(t), accounts for the anomalous diffusion of trapped particles, which is not present in the original expression.


Question: What evidence does the study provide regarding the gender effects on self-citations?

Answer: The study shows that men and older researchers are more likely to self-cite, and that self-citations have a small to moderate effect on authors' citation rates, highly inflating citations for a subset of researchers.


Question: What is the significance of KiWi's ability to identify smaller (twig) clusters in large expression datasets?

Answer: KiWi's ability to identify smaller (twig) clusters is significant because it allows the identification of tight co-regulation of small groups of genes across a large number of experiments. This is a novel contribution to the subspace clustering field, as previous studies have often focused on larger clusters. Many important biological processes may involve relatively small numbers of genes, and KiWi's performance in identifying these small clusters has been validated through biological assessments, showing that they are of biological interest.


Question: What method is used to find a set R1 of size k that is disjoint from every rogue set of G0, and how does it differ for different values of n?

Answer: If n ≥ k3(k−1)+k, the existence of an appropriate set R1 is guaranteed, and the application of subroutine Rooted(R1) results in a rogue-free graph G1. For k3(k −1) + k ≤ n < k4(k −1) + k, a brute-force method based on a stronger version of Theorem 2.1 is used. For n ≥ k4(k −1) + k, a more efficient algorithm based on submodular function minimization is applied.


Question: How is the probability P(v ∈I+) related to the volume of a spherical simplex in the context of 3-regular transitive graphs?

Answer: The probability P(v ∈I+) is equal to the relative volume of the (d−1)-dimensional spherical simplex corresponding to the outer normal vectors −ui. This relative volume is determined by the pairwise angles ϕi,j, which can be expressed using the inner products yi · yj = ci,j.


Question: What is the form of the outer solution at O(ϵn) and how does it motivate the factorial-over-power ansatz?

Answer: The outer solution at O(ϵn) is given by ϵn [1/2 log (n) + γ - log (2)/2 - 1/4 log (1 + Y)] Γ(n/2 - 1) [2(1 + Y)]^(n/2 - 1) for n even, and ϵn/2 Γ((n-1)/2) [2(1 + Y)]^((n-1)/2) for n odd. This form motivates the slightly unusual form of the factorial-over-power ansatz previously introduced in (4.2).


Question: How does the Crank-Nicolson (CN) scheme compare to the Exponential Integrator (EI) scheme in terms of stability and error for different values of α?

Answer: The Crank-Nicolson (CN) scheme exhibits instability, particularly for smaller values of α, as shown in Figure 7. In contrast, the Exponential Integrator (EI) scheme does not show oscillatory effects and maintains stability. The relative error for the EI scheme is also illustrated in Figure 8, showing a convergence rate of h1+α for different values of α.


Question: How is the converse inequality for the large time asymptotics of CT(ω) proven in Theorem 2?

Answer: The converse inequality is proven by considering a sequence (Tk)k∈N tending to +∞ and a minimizing sequence (Yk)k∈N = (y+ k, y− k)k∈N ∈ (L2(Ω) × L2(Ω))N for lim infk→+∞ CTk(ω)/Tk. The sequence Yk is decomposed into Yk = Y∞ + Zk, where Zk converges weakly to 0 in L2(Ω) × L2(Ω). Using the weak convergence and properties from Lemma 4, it is shown that limk→+∞ Jω Tk(Y∞) ≥ g1(ω)∥Y∞∥2. Additionally, the integral Jω Tk(Zk) is analyzed by breaking it into intervals and applying inequalities, leading to the conclusion that the converse inequality holds.


Question: How does the Yang-Baxter deformation with the classical r-matrix (3.1) relate to the twisted periodic boundary condition in the undeformed AdS5×S5?

Answer: The Yang-Baxter deformation with the classical r-matrix (3.1) can be reinterpreted as a twisted periodic boundary condition with the usual AdS5×S5, where the deformed background with the usual periodic boundary condition is equivalent to the undeformed AdS5×S5 with a twisted periodic boundary condition: ˜x2(σ = 2π) = ˜x2(σ = 0) + 2c1η√λc P3, ˜x3(σ = 2π) = ˜x3(σ = 0) −2c1η√λc P2. Here Pi are Noether charges for translation invariance in the xi directions.


Question: How does the coupling between lattice sites differ between the square pulse and continuous drive protocols, and what is the impact on the structure of CAT states?

Answer: For the continuous drive protocol, H1F only induces next-nearest neighbor couplings, while the coupling between odd and higher neighboring sites is induced by H2F. The difference in coupling strength between sites differing by odd and even numbers of lattice sites explains the structure of the CAT states, which are distributed between a site and its next-nearest neighbor rather than the nearest one. In contrast, the square pulse protocol leads to a real-space Floquet Hamiltonian whose range increases with decreasing frequency, but the specific details of the coupling strengths and their impact on CAT states are not explicitly provided.


Question: What is the form of the matrices F2, G2, H2, and K2, and how do they relate to the positivity of the state ρTBCa?

Answer: The matrices F2, G2, H2, and K2 are 4n2 × 4n2 matrices defined as follows: F2 is a diagonal matrix with a on the diagonal, G2 is a matrix with a on the superdiagonal and zeros elsewhere, H2 is a matrix with a in the bottom-left corner and zeros elsewhere, and K2 is a matrix with (1+a)/2 on the diagonal, √(1-a^2)/2 on the superdiagonal and subdiagonal, and a on the diagonal of the lower right block. These matrices are used to show that the state ρTBCa is not positive, indicating that ρa is a NPT state with respect to the bipartite split A −(BC).


Question: What lemma in the candidate text supports the stability of viscosity solutions under uniform convergence, and what does it state?

Answer: Lemma 3.6 supports the stability of viscosity solutions under uniform convergence. It states that if un is a sequence of viscosity solutions of the problems −∆un(x, t) = λn(t) in Ωt(un) and ∂tun = Fn(|Dun|)|Dun| on Γt(un), and if un → u, Fn → F, and λn → λ uniformly on compact sets, then u is a viscosity solution of −∆u(x, t) = λ(t) in Ωt(u) and ∂tu = F(|Du|)|Du| on Γt(u).


Question: What is the estimated number of 7x7 associative magic squares, and how was this estimate obtained?

Answer: The number of 7x7 associative magic squares was estimated to be within the range of (1.125151±0.000051)×10^18 with a probability of 99% by Walter Trump, who used a method combining Monte Carlo and backtracking.


Question: How is the matrix W2 calculated and shown to be zero using the elliptic function identities?

Answer: The calculation of W2, with the help of (A9), gives: (U−BU+A −U+AU−B)ij = Φ(xij −3η) ∑k [U−i U+k (ζ(xik −2η) + ζ(xkj −η) −ζ(xij −3η + λ) + ζ(λ)) − U+i U−k (ζ(xik −η) + ζ(xkj −2η) −ζ(xij −3η + λ) + ζ(λ))]. Using the fact that the sum of residues of the elliptic function (ζ(x−xj−η) −ζ(x−xi+2η)) ∏k σ(x −xk + 2η)σ(x −xk −η) / σ(x −xk + η)σ(x −xk) is equal to zero, we see that (U−BU+A −U+AU−B)ij = Φ(xij−3η)(U−i −U+i) ∑k [U−k (ζ(xik−η) + ζ(xkj−2η) −ζ(xij−3η+λ) + ζ(λ))], so W2 = 0.


Question: How does the rationality of L(k, f) relate to the linear dependence of 1, ζ(k, a/q) over the rationals?

Answer: If L(k, f) is rational, then 1, ζ(k, a/q) for 1 ≤a < q, (a, q) = 1 are linearly dependent over the rationals. Conversely, if 1, ζ(k, a/q) with 1 ≤a < q, (a, q) = 1 are linearly dependent over the rationals, then there are rational numbers c0, ca, 1 ≤a < q, (a, q) = 1, not all zero, such that c0 + ∑(a=1 to q-1, (a,q)=1) caζ(k, a/q) = 0. This implies that L(k, f) is rational.


Question: What are the specific conditions and solutions for the CKV when a(u) = un, and how do they relate to the conformal symmetry classes?

Answer: If n = 2, 1, or 0, then a′′′(u) = 0, and the CKV given by equation (36) is, respectively, a proper SCKV, a proper HKV, and a KV. For all values of n other than n = 1, equation (26) leads to τ(u) = −1/8n(n − 2)u−2 + ωu−2n, where ω is constant. The three solutions for a(u) are ai(u) = un, un sin(qu1−n), un cos(qu1−n) for ω = q2(1 − n)2/8 > 0, and ai(u) = un, un sinh(qu1−n), un cosh(qu1−n) for ω = −q2(1 − n)2/8 < 0. When n = 2, we have an example of case (i), i.e., the SCKV S = u2∂u + (r2/2)∂v + ur∂r, together with two non-special CKV given by equations (36) and (38) or (39) with n = 2.


Question: How does the high-dimensional quantum holography using OAM entanglement improve the security of holographic imaging encryption compared to classical methods?

Answer: The high-dimensional quantum holography using OAM entanglement improves the security of holographic imaging encryption by allowing the use of a superposition state as the key, which makes searching for the key in the corresponding state space (Hilbert space) much harder than for a key encoded in a classical bit. Additionally, the high-dimensional nature of OAM entanglement provides a larger state space, enhancing the security further.


Question: What does the optimal sensor activation pattern look like in the 2D experiment, and how is it represented in the results?

Answer: The optimal sensor activation pattern in the 2D experiment is shown in Figure 5.6, with wk,ℓ= 1 represented in red and wk,ℓ∈(0, 1) shown in green. The measurement domains Ωk obs are numbered as in Figure 3.2.


Question: What are the specific error rates of CLDL-GoogLeNet on the ImageNet dataset compared to other models?

Answer: CLDL-GoogLeNet achieves a top-5 error rate of 10.21%, which is 0.89 points lower than the original GoogLeNet model with a top-5 error rate of 11.1%.


Question: How does the functional BKS2 relate to BKS for functions in the manifold M, and what does this imply for critical points of gBKS?

Answer: The functional BKS2 is defined for functions in H1(0, 2; H) and is related to BKS by the equation BKS2(z) = 2BKS(z|(0,1)) for every z ∈ M. This implies that critical points of gBKS on M correspond to generalized solutions of the perturbed Kepler problem, as shown in Lemma 6.13.


Question: How does the algorithm ensure that the paths listed are simple and do not reuse edges?

Answer: The algorithm maintains the invariant that πs is a simple path s ⇝ u by removing vertex u from the graph G in the call to left_update(C, e) in line 11, ensuring that no path can include the edge e more than once. Additionally, the function choose(C, u) returns an edge e incident to u, and the paths with prefix πs that do not use e are listed by the recursive call in line 8, where e is removed from the graph.


Question: How does the CitedTCR model address the issue of reflecting expertise in the curation process?

Answer: CitedTCR addresses the issue of reflecting expertise in the curation process by using the Personalized PageRank (PPR) algorithm for curator assignment and a multi-task peer-prediction mechanism for reward computation. This ensures that the curation process incorporates the expertise of anonymous curators, particularly in technical content such as academic papers and patents.


Question: What mechanism is responsible for the saturation of the f-mode instability in neutron stars, and how does it affect the gravitational wave signal?

Answer: The f-mode instability in neutron stars is driven unstable by the emission of gravitational waves due to the Chandrasekhar-Friedman-Schutz mechanism. The instability grows exponentially until it is halted by nonlinear effects, such as the coupling of the unstable f-mode to other stable modes, leading to a parametric resonance and the subsequent saturation of its amplitude. The saturation point determines the amplitude of the gravitational-wave signal obtained from an individual source.


Question: What methods were used to generate and process the synthetic training data for the datasets?

Answer: Two large text corpora, namely the English and German Wikipedia dump files, were used as training sources for generating sentences. For validation and test purposes, a corpus from the Leipzig Corpora Collection was used. The texts were rendered using a set of over 2,000 serif, sans serif, and monospace fonts. The generation process first selects a piece of text (up to 40 characters) from the corpus and renders it on an image with a randomly chosen font. The associated attributes (i.e., bounding boxes, baseline positions, and x-height values) used for rendering are stored in the corresponding document layout structure. A counter for the number of occurrences of every individual character in the generated dataset is maintained and used to guide the text extraction mechanism to choose text pieces containing the less frequently represented symbols. The procedure is repeated until the number of occurrences of each symbol reaches a required minimum level (5,000, 100, and 100 in our synthetic training, validation, and test set, respectively), which guarantees that even rare characters are sufficiently well represented in each of the generated datasets.


Question: What evidence from WMAP data supports the Poincaré dodecahedral space model over the infinite, simply connected flat model?

Answer: Several analyses of the WMAP data favour a Poincaré dodecahedral space (S3/I∗) model over the infinite, simply connected flat model (Luminet et al. 2003; Aurich et al. 2005a, 2005b; Gundermann 2005; Caillerie et al. 2007; Roukema et al. 2008a, 2008b).


Question: How does the proposed method's Dice score and pixel accuracy compare to other methods for the IS/OS tissue segmentation?

Answer: For IS/OS tissue segmentation, the proposed method achieves a Dice score of 86.2±0.2%, which is the best performance, and a pixel accuracy of 87.0±0.4%, also the best performance, with an improvement of 0.4% over the second-best method.


Question: What are the reasons for the Drude model's failure in describing the experimental data of the most precise experiments on measuring the Casimir force?

Answer: The Drude model, which has been successfully tested in thousands of experiments and technological applications, fails to describe the experimental data of the most precise experiments on measuring the Casimir force because it leads to violation of the Nernst heat theorem in the Lifshitz theory of dispersion forces and to counterintuitive results in the limiting case of ideal metals. This failure is fundamentally related to the quantum nature of electromagnetic fluctuations giving rise to dispersion forces, as the Drude model is valid for classical electromagnetic fields.


Question: How does the planning approach handle the discovery of obstacles and ensure that the robots avoid collisions with each other?

Answer: When a robot discovers an obstacle, it re-synthesizes a plan to reach its remaining goals after updating its representation of the workspace. For inter-robot collision avoidance, the involved robots locally collaborate.


Question: What types of vortices are produced by rotation at different values of magnetic field and stiffness, and how is the density of these vortices determined?

Answer: The types of vortices produced by rotation are: 1) (1, 0) vortices for Ks > Kc or B > Bc, 2) an equal number of (2/3, 1/3) and (1/3, −1/3) vortices for 1/4 < Ks/Kc < 1 and B < Bc, and 3) twice as many (1/3, −1/3) vortices as (1/3, 2/3) for 0 < Ks/Kc < 1/4 and B < Bc. In each case, the density of vortices is determined by the angular velocity of the rotation.


Question: What are the specific challenges in determining the effective temperatures of L and T dwarfs, and how do these challenges impact the isochrone test?

Answer: Atmospheric models for L and T dwarfs have not been confirmed by direct measurement of brown dwarf radii. The molecular opacity line list for CH4 is incomplete or non-existent below 1.6 µm, the FeH line list is incomplete in the 1.2–1.3 µm region, NH3 is important for very late-type T dwarfs but there are no opacities calculated below 1.4 µm, and the K I resonance doublet (0.78 µm) is difficult to calculate. These challenges impact the isochrone test by making it a test of the joint accuracy of evolutionary and atmospheric models, not of either specific class of models.


Question: What are the specific inclinations and position angles of the circumstellar disks of the northern and southern sources in L1551 IRS 5, and how do they compare to each other?

Answer: The circumstellar disk of the northern (N) source has an inclination of 48°.1 ± 0°.5 and a position angle for its major axis of 148°.2 ± 0°.7. The circumstellar disk of the southern (S) source has an inclination of 44°.3 ± 0°.8 and a position angle for its major axis of 147°.8 ± 3°.6. Both disks have nearly the same inclinations and essentially identical position angles for their major axes, indicating that they are closely parallel.


Question: What is the strategy for reducing the number of equivalence classes of short and secondary short elements and long bases in the pseudogroup ˆUt?

Answer: The strategy is to gradually reduce the number of these equivalence classes (of short and secondary short elements) and the number of long bases (i.e., bases that contain long elements), while creating a bounded number of new secondary short and short elements, such that the lengths of the new short elements are bounded below by a fixed fraction of the lengths of the previous short elements, where the fraction and the numbers of new short and secondary short elements are bounded by a constant that depends only on the number of elements in the pseudogroup ˆUt.


Question: What characteristics of individuals with low thresholds are identified in the experimental data, and how do these findings relate to the viability of political mobilisations without traditional leaders?

Answer: The experimental data identified a relationship between extraversion (and, to a less clear extent, internal locus of control) and low thresholds. These findings suggest that contemporary political mobilisations can become viable without leading individuals and organisations to undertake organisation and co-ordination costs, proceeding to critical mass and even achieving the policy or political change at which they are aimed.


Question: What is the definition of a function having 0-Gevrey growth of type (s) at a point p ∈ M \ U, and how does it differ from having Gevrey growth of type {s}?

Answer: A function f has 0-Gevrey growth of type (s) at p ∈ M \ U if for a local coordinate system (x1, ..., xn) around p, there exists a sufficiently small compact neighborhood K of p such that sup_x∈K∩U exp(−h · d(x, K \ U)^(1−s)) |f(x)| < ∞ for all h ∈ (0, +∞). A function f has Gevrey growth of type {s} if the same condition holds for some h ∈ (0, +∞).


Question: What are some recent examples of systemic failures in power grids that highlight the impact of couplings among systems, particularly in developing countries?

Answer: Recent systemic failures in power grids that highlight the impact of couplings among systems in developing countries include the blackout in India in July 2012, which was partly due to imprudent policies for upgrading transmission lines, reluctance to disconnect overdrawing regions, and increased water pumping by farmers to alleviate drought. Another example is the blackout in Brazil in 2012. Additional recent examples include the southwestern USA in 2011, northeastern North America in 2003, Italy in 2003, and Europe in 2006.


Question: What is the condition for the onset of instability in the context of the AT stability criterion, and how does it relate to the divergence of the long-range fluctuations?

Answer: The condition for the onset of instability is αR′(1 −q) = 1, which agrees with the well-known AT stability criterion. This condition is related to the divergence of the long-range fluctuations, as the divergence is expected to occur first for the low frequencies, given by lim ω→0 ˆc(ω) = α / (1 − αR′(1 −q)).


Question: What are the specific limits that contribute to the posterior mean and variance in the context of the Pitman-Yor process?

Answer: For the posterior mean, the limit is given by E[P(f)|X1, ..., Xn] = (1 − λ)P d_0(f) + (1 − σ)λP c_0(f) + λσG(f) P0-a.s. For the posterior variance, the limits are: ∑_{j=1}^{K_n} (N_{j,n} − σ) f(˜X_j) / (n − K_n σ) → (1 − λ)P d_0(f) + (1 − σ)λP c_0(f) / (1 − σλ), and n(n − σK_n)(M + σK_n) / (n + M)^2(n + M + 1) → (1 − σλ)σλ, and −n (∑_{j=1}^{K_n} (N_{j,n} − σ) f(˜X_j))^2 / (n − σK_n)(n + M)(n + M + 1) → −((1 − λ)P d_0(f) + (1 − σ)λP c_0(f))^2 / (1 − σλ).


Question: What evidence from high-redshift Type Ia supernovae led to the discovery of cosmic acceleration in 1998?

Answer: In 1998, two independent teams (Riess et al. 1998; Perlmutter et al. 1999) presented convincing evidence for an accelerated expansion of the universe. They used high-redshift Type Ia supernovae as standard candles and found them fainter than expected. After ruling out possible systemic obscuration by dust or evolutionary effects, they interpreted the dimmer luminosity as a consequence of being farther away, implying an acceleration in the expansion.


Question: What is the estimated contamination from the inner NSD in the outer NSD sample, and how does it affect the age bin 2-7 Gyr?

Answer: The estimated contamination from the inner NSD in the outer NSD sample is ∼5%, which accounts for the ∼1% uncertainty in the age bin 2-7 Gyr.


Question: What evidence is provided to refute the hypothesis that the anomalous Raman modes (ARMs) in tellurides are due to the formation of TeO2 layers?

Answer: The Raman modes of the TeO2 polymorphs, which show narrow and intense bands with several strong peaks below 250 cm-1, near 400 cm-1, and above 600 cm-1, are not consistent with the observed and reported ARMs in tellurides. Additionally, the 62 cm-1 mode characteristic of paratellurite, the most stable phase of TeO2, has not been observed on a regular basis in the Raman spectra of tellurides, including GaTe and GaGeTe.


Question: What is the probability bound for the Langevin dynamic to remain within a certain interval for a long time, and how is it derived?

Answer: There exist two constants C := C(d, V ) < ∞ and c := c(d, V ) > 0 such that, for any T ≥ 1 and any edge e ∈ TL, the probability that |∇φL(t, e)| ≤ RV for all t ∈ [0, T] is bounded by C exp(−c (ln T)^(r/(r−2))). This bound is derived by decomposing the Brownian motion into mutually independent Brownian bridges and increments, and using the properties of these components to establish the probability estimate.


Question: What are the uncertainties in the mass, momentum, and energy calculations for the outflow components listed in Table 3?

Answer: The uncertainties calculated from adopting a different integrated velocity range (± ∼0.4 km −1 at the upper and the lower velocity boundary) are ∼36, 26, and 23 percent for mass, momentum, and energy respectively. Note, errors in source distance, and more importantly outflow inclination angle, could have a much more significant effect. Cabrit & Bertout (1990) suggest uncertainties up to an order of magnitude for energy values if the outflows have large inclination angles (> 70◦).


Question: How does the QCS for a single-mode squeezed vacuum state change with different levels of loss, and what is the significance of the transmission parameter η in this context?

Answer: The QCS for a single-mode squeezed vacuum state with squeezing parameter r (energy sinh²r) that has been subject to loss (overall transmission η) is expected to diminish proportionally to the loss parameter. The measured QCS is always slightly smaller than the predicted one. QCS values less than or equal to unity cannot be used to certify quantumness, even if quantumness is present in some other form. The transmission parameter η > 1/2 is required to certify quantumness for the squeezed-vacuum states.


Question: What are the key differences between the Jeans and hydrodynamic regimes of atmospheric escape, and how do they affect the escape of gas molecules?

Answer: In the Jeans regime, the atmosphere is retained by the gravitational pull on individual molecules and the effective force of collision with other atmospheric molecules, making it 'collisionless.' In the hydrodynamic regime, molecules are so energetic that collisions are insufficient to restrict escape, and the escaping flow of lighter gases can drag heavier gas molecules, making it 'collisional.' Hydrodynamic escape can also occur far below the exobase, similar to boiling, while Jeans escape is more like evaporation below the boiling point.


Question: How does the cell decomposition handle the situation where a 1-cell adjacent to an isolated point is removed, and what is the impact on the local finiteness property?

Answer: If a 1-cell adjacent to an isolated point is removed, the endpoints of the 1-cell must be included as 0-cells, which can cause the local finiteness property to fail. However, the finiteness property can still be checked to hold in the overall cell decomposition.


Question: How does the proof of the parabolic part of Corollary 1.14 demonstrate the necessity of the condition ∑ 1/hn < ∞?

Answer: The proof shows that if the series ∑ 1/hn diverges, then the infinite product ∏ (1 + αn) also diverges, leading to the inequality dncn ≫ ∏ hk for sufficiently large n. This implies that the graph G is CP hyperbolic, contradicting the parabolic part of the corollary. Therefore, the condition ∑ 1/hn < ∞ is necessary for the parabolic part to hold.


Question: What is the relationship between τ-mutation sequences and the Coxeter lattice in the context of the dP3 quiver?

Answer: The τ-mutation sequences can be visualized as walks in a triangular lattice associated with the affine symmetric group ˜A2. This visualization is possible because the relations among the τ′j’s coincide with the relations among the generators of ˜A2. Each mutation sequence (word in the τj’s) corresponds to a gallery of alcoves (faces in the triangular lattice) associated with ˜A2, establishing a bijection between clusters produced by τ-mutation sequences and alcoves in the lattice.


Question: How does the L1 estimate for the derivatives of φ0 in Proposition 4.2 lead to the pointwise estimate for φ0 in Proposition 4.3?

Answer: By applying the fundamental theorem of calculus in the u-direction and using the first estimate of Proposition 4.2 with p = 0, we obtain the required pointwise estimate: |φ0|(u, v) ≤ |φ0|(−∞, v) + C [∫_{-∞}^{u} |∂uφ0|(u′, v0) du′ + ∫_{v0}^{∞} v′−q|∂vφ0|(−∞, v′) dv′], where C > 0 is a uniform constant.


Question: What factors contribute to the discrepancy between the theoretical and experimental neutron multiplicities in the region A ≈ 116 to 130?

Answer: The too large multiplicity predicted in the range between A ≈ 116 and 130 is partly due to the too large amount of very elongated scission shapes originating from the LD fission mode in this region. These fragments experience a substantial shape relaxation after scission, which increases the excitation energy available for evaporation.


Question: What is the closed-form representation of the inverse of the Jacobian in the context of the linearized dynamics discussed in the query text?

Answer: The inverse of the Jacobian in (37) has the following closed-form representation: ∂Fimplicit/∂x(0)⁻¹ = -[I, 0, ..., 0; A(0),−1(1,1), A(0),−1(1,1), 0, ..., 0; 0, A(0),−1(N,1), A(0),−1(N,1), ..., A(0),−1(N,N)], where A(0)(i,j) represents the linear sensitivity of the state at time step i with respect to the state at time step j. The sensitivity can be solved by applying the chain rule: A(0),−1(i,j) = ∏(t=j to i) [I - h · ∂f/∂x(x(0)t, u(0)t, w(0)t)]⁻¹.


Question: What are the current upper limits on the branching fractions for B+ →µ+νµ and B+ →e+νe, and how do they compare to the SM predictions?

Answer: The upper limit on B+ →µ+νµ is < 1 × 10−6 at 90% CL, and the upper limit on B+ →e+νe is < 0.98 × 10−6 at 90% CL. These upper limits are close to their Standard Model (SM) predictions, and the decay B+ →e+νe can only be observed if new physics greatly enhances its decay rate.


Question: What is the computational advantage of the prior distribution derived from thin-plate spline radial basis functions in the context of spatial data analysis?

Answer: The prior distribution derived from thin-plate spline radial basis functions allows for direct sampling from the joint posterior via Monte Carlo integration, avoiding the use of a time-consuming MCMC scheme while drawing samples directly from the actual posterior distribution in roughly the same time required to draw samples from the INLA approximation of the posterior distribution.


Question: What are the results of the MiniBooNE experiment in the energy region EQE_ν > 200 MeV, and how do they compare to the LSND data?

Answer: MiniBooNE observes no significant excess of ¯νe events in the energy region EQE_ν > 200 MeV, for a data sample corresponding to 3.39 × 10^20 POT. The result is inconclusive with respect to small amplitude mixing suggested by the LSND data, but more antineutrino data, which are currently being collected, will provide additional information.


Question: How does the concept of the up-down distance relate to the zigzag distance in a discrete poset, and what condition ensures they are identical?

Answer: In any discrete poset, the up-down distance is always greater than or equal to the zigzag distance. If the up-down distance satisfies the triangle inequality, then it must be identical to the zigzag distance.


Question: What is the Bismut curvature tensor formula for the metric g(α, β), and how does it relate to the tensors Uα and Uβ?

Answer: The Bismut curvature tensor for the metric g(α, β) is given by ΩB_ijkl = α[Uα] + 2β[Uβ], where Uα and Uβ are specific tensors. Uα is the curvature tensor of gH with respect to the Bismut connection, and Uβ is defined as Uβ = −δijδkl/|z|^4 + δijzkzl/|z|^6 + −zizjzkzl/|z|^8.


Question: What is the relationship between the 4.5 µm emission and H2 outflow in the context of EGOs, and how does this relationship vary among different EGOs?

Answer: The 4.5 µm emission is often similar in morphology to H2 outflow, suggesting that 4.5 µm emission likely comes from H2 emission. However, this relationship is observed in less than one third of EGOs with H2 outflows. The H2 outflow detection rates for 'likely' MYSO outflow candidates are higher (42%) compared to 'possible' candidates (19%). For EGOs within 4 kpc, these rates increase to 62% and 44%, respectively, suggesting that 'possible' candidates may be affected by nearby point sources or image artifacts, or they may be in different evolutionary stages.


Question: What are the conditions under which the eigenvalues of A(t) are distinct, and how are they derived from the characteristic polynomial?

Answer: The eigenvalues of A(t) are distinct if and only if A(t) ≠ λ+(t)I. The eigenvalues are derived from the characteristic polynomial pA(t)(x) = det[xI − A(t)] = x^2 − (f(t) + g(t))x + f(t)g(t) − |h(t)|^2, resulting in the two eigenfunctions λ+(t) = (f(t) + g(t) + sqrt((f(t) − g(t))^2 + 4|h(t)|^2)) / 2 and λ−(t) = (f(t) + g(t) − sqrt((f(t) − g(t))^2 + 4|h(t)|^2)) / 2. These eigenvalues are distinct if and only if h(t) ≠ 0.


Question: What are the specific parameters and configurations used in the FDTD simulations for the CROW array with bending angles?

Answer: The simulations used 15 monodisperse microdroplets, each represented by an infinite cylinder with a diameter of 18 µm, arranged in a non-touching periodic fashion with a surface-to-surface separation of 6 µm. The bending was implemented by shifting the centers of three central cylinders on a circular locus, with the curvature defining the bending angle. The spatial and temporal resolution were dx = dy = 1/15 µm and dt = 0.11 fs, respectively. The computational cell size was 420 µm × 23 µm for the straight waveguide and 390 µm × 125 µm for the largest bend angle. The carrier mode was excited at λ = 561.4 nm, and eight configurations with bending angles of 5°, 7.5°, 10°, 12.5°, 15°, 20°, 25°, and 30° were simulated.


Question: What are the critical current characteristics of a symmetric 0-π junction and a conventional 0 junction in the short junction limit, and how do they differ?

Answer: In the short junction limit (W/λJ << 1), the critical current for a symmetric 0-π junction is given by Ic(Φ) = I0| sin²(πΦ/2Φ0)/(πΦ/2Φ0)|, while for a conventional 0 junction, it is Ic(Φ) = I0| sin(πΦ/Φ0)/(πΦ/Φ0)|. The 0-π junction shows a squared sine dependence, leading to a different modulation of the critical current with magnetic flux compared to the conventional 0 junction.


Question: What does Lobachevsky's view on the relationship between synthetic and analytic methods in geometry reveal about his stance on the role of spatial intuition in mathematics?

Answer: Lobachevsky stresses the importance of synthetic approaches in science and geometry, believing that spatial intuition and spatial experience are ultimate sources of geometrical truths, and that analytic methods serve only for the derivation of new data from first data.


Question: What evidence supports the conclusion that phonons are responsible for the thermal Hall effect in La2CuO4?

Answer: The thermal Hall conductivity of La2CuO4 is roughly isotropic, being nearly the same for heat transport parallel and normal to the CuO2 planes, i.e., κzy(T) ≈ κxy(T). This shows that the Hall response must come from phonons, these being the only heat carriers able to move as easily normal and parallel to the planes.


Question: What specific experimental setup was used to investigate the differences in muscle velocity imaging between senior and young volunteers using NMES-synchronized MRI?

Answer: The study used an InTENSity Twin Stim III TENS and NMES Combo device with 5.1 x 8.9 cm² rectangular self-adhesive gel-based NMES electrodes placed on the vastus lateralis (VL) muscle. The electrodes were positioned 15 cm apart and 12 cm from the center of the knee joint. Young volunteers were stimulated at 18 mA, while senior volunteers were initially stimulated at a minimum level to achieve visible contraction, with a maximum limit of 22 mA, and an additional acquisition was performed at 18 mA for comparison. The stimulation waveform was a monopolar square wave with a frequency of 150 pulses/s and a pulse duration of 0.3 ms, with a contraction plateau of 1 s and a 2 s relaxation period.


Question: What is the formula for the power corrections in the tail region of the 1-jettiness distribution, and how is the nonperturbative parameter Ω1 defined?

Answer: In the tail region (2ΛQCD/Q ≪ τ1 ≪ 1), the dominant power corrections are given by the formula dσ/dτ1 = dσpert/dτ1 − 2Ω1/QR * d²σpert/dτ²1 + ..., where Ω1 is rigorously defined as a matrix element of a product of soft Wilson lines.


Question: What is the main new aspect of the algorithm described in the candidate text for finding the farthest edge from each vertex in a polygon?

Answer: The main new aspect is the construction of the upper envelope of the coarse cover of each transition edge in linear time, which is achieved by using shortest path trees inside the hourglass of the edge.


Question: What is the proposed mechanism for the formation of blue nuggets and their transition to red nuggets according to Dekel & Burkert (2014)?

Answer: Dekel & Burkert (2014) proposed that blue nuggets form via 'fast-track growth' channels driven by extreme gas inflow from smooth cosmic streams or gas-rich mergers. The gas inflow episodes ignite a chain of events that culminate in violent disk instability, contraction, and rapid star formation. Once blue nuggets run out of gas, they undergo inside-out quenching while maintaining their extreme compactness, making them suitable red nugget progenitors.


Question: What does the N-body simulation in Figure 9 show about the orbital separation of protoplanets during oligarchic growth?

Answer: The N-body simulation in Figure 9 shows that about 10 protoplanets form with mass ∼10^27 g on nearly circular non-inclined orbits with the orbital separation of ≃10rH.


Question: How does the collapse rate in the hydrodynamical model compare to the one-zone model, and what are the implications for temperature and density profiles?

Answer: In the hydrodynamical model, the collapse rate exceeds the free-fall timescale (tff/tdyn > 1) soon after the beginning of collapse, indicating a highly dynamical collapse. This faster collapse rate leads to higher temperatures and densities compared to the one-zone model, where the dynamical timescale is assumed to be equal to the free-fall time (tff/tdyn ≃ 1). For the lowest metallicity cases (Z = 0 and [M/H] = -6), the temperatures remain higher than in other cases, and the temperature dip by dust cooling is less pronounced. Additionally, H2 formation heating results in a large temperature jump at ∼10^8 cm^-3 in the hydrodynamical model, while in the one-zone model, it leads to a gradual increase in temperature.


Question: What is the equivalent condition for a normed algebra to be stable, as stated in the candidate text?

Answer: The normed algebra A is stable if and only if ∥a∥= ||La∥, for all a ∈A.


Question: What are the circular velocity curves for NGC6186, NGC6278, NGC6301, and NGC6310 at R/Re = 0.05?

Answer: At R/Re = 0.05, the circular velocity curves for NGC6186, NGC6278, NGC6301, and NGC6310 are as follows: NGC6186 has Vc = 108.50, ∆V 75th = 0.32, and ∆V 25th = 0.28; NGC6278 has Vc = 88.67, ∆V 75th = 0.12, and ∆V 25th = 0.14; NGC6301 has Vc = 69.73, ∆V 75th = 0.10, and ∆V 25th = 0.08; NGC6310 has Vc = 80.50, ∆V 75th = 0.14, and ∆V 25th = 0.15.


Question: What are the exponents of the upper and conditional lower bounds for Klee’s measure problem and the depth problem in Rd for d ≤ 10, and how do they compare to the previously known lower bounds?

Answer: The exponents of the upper and conditional lower bounds for Klee’s measure problem and the depth problem in Rd for d ≤ 10 are as follows: For d=3, the upper bound is 1.5, the previously known lower bound is 1.5, the SAT-solver lower bound is 1.5, and the covering designs lower bound is 1.5. For d=4, the upper bound is 2, the previously known lower bound is 1.7777, the SAT-solver lower bound is 1.9047, and the covering designs lower bound is 1.8461. For d=5, the upper bound is 2.5, the previously known lower bound is 2.0833, the SAT-solver lower bound is 2.2222, and the covering designs lower bound is 2.1929. For d=6, the upper bound is 3, the previously known lower bound is 2.4, and the SAT-solver lower bound is 2.5714. For d=7, the upper bound is 3.5, the previously known lower bound is 2.7222, and the SAT-solver lower bound is 3. For d=8, the upper bound is 4, the previously known lower bound is 3.0476, and the SAT-solver lower bound is 3.3333. For d=9, the upper bound is 4.5, the previously known lower bound is 3.375, and the SAT-solver lower bound is 3.6818. For d=10, the upper bound is 5, the previously known lower bound is 3.7037, and the SAT-solver lower bound is 4.0540.


Question: How does the system described in the candidate text monitor and adjust the current sharing among parallel MOSFETs?

Answer: The system includes two IRFP4688 MOSFETs, two ACS758 hall current sensors, two NTC Thermistor Sensors, and a dummy loader. It monitors the current flowing through each MOSFET and their temperature, and allows the adjustment of the 𝑉𝐺𝑆 applied to each MOSFET separately.


Question: What are the main phases of Algorithm 1 that contribute to the memory consumption, and how do they impact the overall memory usage?

Answer: The main phases of Algorithm 1 that contribute to memory consumption are: (1) the computation of all visibility regions of all vertices V in line 3, which stores O(n^2) x-coordinates in memory, and (2) the determination of the visibility regions of the remaining points in U \ V in line 9, which generates the largest chunk of data, keeping O(n^3) x-coordinates in memory. These phases are necessary because a guard at the far right of the terrain may still see a region at its very left, making it difficult to apply filters like WitnessFilter before knowing all extremal points. The memory bottleneck is further amplified by the use of the EGC paradigm, which ensures a correct and consistent representation of all visibility regions.


Question: What is the dimension of the orbifold that identifies the set of all generalized solutions of the Monge-Ampère equation with k distinct non-removable singular points, and how does it vary with n and k?

Answer: The dimension of the orbifold is given by dn,k, where dn,k = (k−1)(k+2)/2 if k −1 ≤ n, and dn,k = (k −1)(n + 1) − n(n−1)/2 if k −1 > n.


Question: What are the numerical results for the temperature field retrieval using the linear system, and how do they vary with the number of cells?

Answer: The numerical results obtained by solving the linear system are presented in Figs 3a and 3b for N = 5 and 500 cells, respectively. The Planck function is correctly determined for x ≤ nRλ. However, the method fails in the area x > nRλ, even with exact non-noisy data, resulting in oscillatory and inaccurate results with an error increasing rapidly.


Question: What does Figure 4 illustrate regarding the performance of different views on selected classes?

Answer: Figure 4 shows the average precision (AP) of different views on selected classes.


Question: What are the key structural and physical properties of the 122 family of iron-based superconductors that make them particularly interesting for research?

Answer: The 122 family of iron-based superconductors, such as BaFe2As2, have a ThCr2Si2-type structure and are characterized by their relatively high critical temperatures (Tc’s up to 55 K), very high critical fields (at least 70 T), and the richness of their crystal chemical and physical properties. Superconductivity in these materials emerges in the FeAs layers built up by edge-sharing FeAs4/4 tetrahedra, and is assumed to be unconventional due to the proximity to structural and magnetic transitions in their parent compounds. These properties, along with the metallic nature of the parent compounds and the multi-band character of the Fermi surfaces, make them particularly interesting for research.


Question: How does the gauge current flow through the domain walls in the context of anomaly cancellation?

Answer: When a small gauge transformation is performed at the 4-dimensional junction, the gauge current flows through the M domain wall, but never flows into the µ domain wall, since there is no CS term which can absorb the gauge noninvariance. Instead, a large gauge transformation can create exotic instantons on the µ domain wall and flip the sign of the partition function.


Question: What are the two main directions in which pattern mining has evolved beyond the general framework of using a complete lattice as a pattern language?

Answer: First, various mining problems have been investigated that come down to searching for closed patterns which cannot be considered, strictly speaking, as support-closed patterns, such as convex hulls of subsets of a given set of points, or sequential patterns with wild-cards. Second, various mining problems have been addressed in which the pattern language is not a lattice, in particular problems where closed patterns are support-closed patterns with respect to some dataset of objects.


Question: What is the unique critical point of the map FSr: W1(i(S)) → W for a small S > 0, and how is it determined?

Answer: The unique critical point of the map FSr: W1(i(S)) → W for a small S > 0 is cS(Fr). This is determined by the holomorphic motion of the critical point cS(F⋆) under the map FSr, and it is shown that JS(Fr) starts at this unique critical point.


Question: What is the impact of removing noise from the Langevin dynamics on the quality of the generator in the context of training EBMs with maximum likelihood?

Answer: Removing the noise from the Langevin dynamics improves the quality of the generator, as it allows for a more stable and higher-quality sample generation. The reintroduction of noise does not lead to a qualitative change in the behavior but merely reduces the quality of the generator.


Question: What alternative method is proposed to replace Canonical Correlation when the marginal distributions are not Gaussian or the mapping function is not linear?

Answer: One proposal is to look for α and β that maximize Spearman’s rank correlation between α′X and β′Y, which can be formulated as max α∈Rdx,β∈Rdy E [Fα′X (α′X) Fβ′Y (β′Y)] s.t. α′ΣXα = 1 and β′Σβ = 1. This technique, however, does not deal with any kind of unobserved heterogeneity.


Question: What does the candidate text reveal about the existence of amplifiers under the death-Birth (dB) Moran process compared to the Birth-death (Bd) Moran process?

Answer: The candidate text reveals that amplifiers under the death-Birth (dB) Moran process, if they exist, must be both transient and bounded. Transience means that any population structure can amplify selection only in a limited range of relative fitness values r ∈ (1, r⋆). Boundedness means that even when a population structure does amplify selection for a fixed r > 1, it can do so only to a limited extent. In particular, quadratic amplification, which is achieved by Star graphs under Birth-death (Bd) updating, is impossible to achieve under dB updating. As a consequence, there are no super amplifiers under dB updating. These results are in sharp contrast to the Bd Moran process, for which amplifiers and super amplifiers have been constructed repeatedly and can be abundant.


Question: What factors contribute to the variability in the latency times observed during the cork's rise in the experiment?

Answer: The variability in latency times is influenced by multiple factors, including slight variations in the cork's density from one measurement to another, the non-uniform distribution of salt at the bottom of the jug, the perturbation of salt dissolution due to the cork's finite size, and changes in the transport properties of the mixture due to uncontrolled water temperature.


Question: How does the frequency of the drive affect the heating time scale and the lifetime of the time-crystalline order parameter for the polarized state?

Answer: Upon increasing the frequency of the drive, we observe an increase in the heating time scale τ ∗ for the polarized state, which is mirrored by an increase in the lifetime of the time-crystalline order parameter—the system is in the PDTC Floquet phase.


Question: How does the Hall resistivity (ρyx) behavior change with magnetic field (B) for different concentrations of Mn (x) in Ba(Fe1−xMnxAs)2 at 2 K, and what does this indicate about the carrier compensation?

Answer: For x = 0.028, ρyx deviates from its linear behavior at a lower value of B than dRM/dB. This deviation may originate from the strong compensation between holes and electrons at low field strength, resulting in the sign change in ρyx. While both dRM/dB and ρyx are affected by scattering in the low-B regime, dRM/dB is less sensitive to it, leading to a wider B-linear regime in dRM/dB than in ρyx.


Question: What conditions must an affine k-supergroup G satisfy to be considered globally strongly split, and how does this relate to the concept of a fine supergroup?

Answer: An affine k-supergroup G is globally strongly split (gs-split) if there exists a closed subsuperscheme G1 of G such that: (a) 1G ∈ G1, (b) the product in G restricts to an isomorphism G0 × G1 ≅ G of pointed left G0-superschemes, and (c) G1 is isomorphic to a totally odd affine superscheme A0|d−k. If G is globally strongly split and Lie(G) is quasi-representable, then G is also fine, as per Definition 2.3.3.


Question: How does the size of BaTiO3 nanoparticles affect the pyroelectric and electrocaloric properties, and what are the implications for their applications?

Answer: By changing the particle size, one can induce maxima of the pyroelectric coefficient and electrocaloric temperature variation, control their width, height, and sign. This size effect opens the possibility to control pyroelectric and electrocaloric properties of ferroelectric nanocomposites, which can be important for their advanced applications in energy convertors and cooling systems.


Question: What is the purpose of QCDml and how is it being used in the lattice QCD community?

Answer: QCDml is an XML-based markup language designed for sharing QCD configurations and ensembles worldwide via the International Lattice Data Grid (ILDG). It is intended to provide starting points for colleagues in the community to markup valuable configurations and submit them to the ILDG.


Question: How does the effective U(1)weak × U(1)strong gauge field theory apply to composite monopoles/dyons, and what are the implications for Lorentz non-invariant string effects?

Answer: The effective U(1)weak × U(1)strong gauge field theory is applied only to composite monopoles/dyons, particularly those that are slowly moving and heavy compared to the matter-fermion mass. This is necessary because wave-function renormalization for slowly-moving monopoles/dyons violates unitarity bounds set for elementary particle asymptotic states. Lorentz non-invariant string effects, which might appear in the theory, affect only the phases of the respective scattering amplitudes and disappear once the charge quantization condition is imposed.


Question: What is the proposed solution for detecting signals with unknown moments of appearance and disappearance, and how does it compare to the FMA procedure?

Answer: The proposed solution is a strictly optimal modified CUSUM procedure that maximizes the probability of detection in a window of random size, distributed according to a geometric distribution, in the class of detection procedures with a given maximal probability of a false alarm in a prescribed finite window. Monte Carlo simulations for the Gaussian example show that the operating characteristics of the optimal procedure are typically close to those of the more practical FMA procedure.


Question: How does the 1D flow model's performance in predicting volume flow rate during vocal fold vibration compare to the 3D FSI results, especially in Cases 7 and 8?

Answer: The 1D flow model shows significantly better agreement with the 3D FSI result compared to the untrained 1D flow model. In Cases 7 and 8, where the medial thickness is smaller, the vocal fold maintains a significant gap at the closing phase, leading to a high flow rate and low-magnitude oscillations during vibration.


Question: What does the optimal density ρ look like for values of m greater than 4.5, and what is the theoretical implication of this observation?

Answer: For values of m greater than 4.5, the optimal density ρ seems to be a 'homogenized' spherical cap, which is a surprising result. This has an important theoretical implication: even if the ball Bm were optimal for µ1 among shapes Ω such that |Ω| = m ≤ |Sn|/2, it would be impossible to prove it using the standard mass transplantation technique of Weinberger [21].


Question: What is the explicit formula for the sine-Gordon breather, and what are the conditions on its parameters?

Answer: The sine-Gordon breather is given by the explicit formula u∗(x, t) = 4 arctan(m sin(ωt) / ω cosh(mx)), where the parameters m, ω > 0 satisfy m^2 + ω^2 = 1.


Question: How does the excitation spectrum in 1D optical lattices compare to the 2D case when the superfluid flows along the (π, 0) direction, and what role does the CDW fluctuation play in both cases?

Answer: In 1D optical lattices, the excitation spectrum shows rotonlike minima in the AB mode spectrum due to enhanced CDW fluctuation near half filling. The critical velocity vc is smaller at n = 0.9 compared to n = 0.5 due to strong CDW fluctuation. Below a certain value of interaction (U/J ≲−2), vc becomes almost constant, reflecting the enhanced CDW fluctuation. In 2D, the rotonlike structure is more remarkable near half filling due to strong CDW fluctuation, and the critical velocity is smaller near half filling compared to quarter filling, also due to the enhanced CDW fluctuation.


Question: What are the specific classes of algebras that are not detectable in Lω,ω and L∞,ω according to the candidate text?

Answer: The class of completely representable algebras and strongly representable ones of dimension > 2 is not detectable in Lω,ω, while the class Nr_nCA_m for any ordinals 1 < n < m < ω is not detectable even in L∞,ω.