Question: What is the relationship between the empirical bootstrap and the true transition distribution in the context of phased Q-learning, and how does it affect the value output by the algorithm?

Answer: The empirical bootstrap of the estimated function is η′-concentrated around the bootstrap on the true transition distribution, meaning that for all states s, actions a, and iterations i ≤ ℓ, the difference between the empirical average of the estimated value function and the true value function is bounded by η′. This concentration ensures that the value output by phased Q-learning after ℓ steps is close to the optimal value, with the difference given by ζ(s, a) = |ˆQℓ(s, a) − Q∗(s, a)| ≤ η′γ + γℓ / (1 − γ) for all s, a.


Question: What are the projected sensitivities of the Einstein Telescope to the Yukawa interaction coupling α and the dipole emission parameter γ, and under what conditions are these constraints optimal?

Answer: The Einstein Telescope is found to be the most sensitive to such dark sector modifications, with sensitivity as good as O(10−5) and O(10−7) for α and γ respectively. These constraints are optimal when the Yukawa length scale λ is in the range of 102 − 103 km for α and ∼104 km for γ.


Question: How do the final fluxes of νe and ¯νe differ between normal hierarchy (NH) and inverted hierarchy (IH) in the context of Earth matter effects?

Answer: For NH, the final fluxes are Fνe ≈ F 0 νX and F¯νe ≈ cos2 θ12F 0 ¯νe + sin2 θ12F 0 ¯νX + ¯freg(F 0 ¯νX − F 0 ¯νe). For IH, the final fluxes are Fνe ≈ cos2 θ12F 0 νX + sin2 θ12F 0 νe + freg(F 0 νe − F 0 νX) and F¯νe ≈ F 0 ¯νX. The Earth matter effect introduces a regeneration factor freg and ¯freg, which modifies the fluxes differently for NH and IH.


Question: How do the ASR values of the NNI and PRL methods compare to the baseline IBA when different numbers of channels are pruned in the last convolutional layer of the DeepLabV3 model?

Answer: The NNI and PRL methods maintain higher ASR values compared to the baseline IBA across different numbers of pruned channels. Specifically, when 0 channels are pruned, NNI has an ASR of 95.46 and PRL has an ASR of 96.75, while the baseline IBA has an ASR of 93.04. When 30 channels are pruned, NNI has an ASR of 93.52 and PRL has an ASR of 92.72, while the baseline IBA has an ASR of 84.26.


Question: What factors contribute to the discrepancy between the theoretical and experimental neutron multiplicities in the region A ≈ 116 to 130 for nth + 235U fission?

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. The fragments of this mode experience a substantial shape relaxation after scission, which increases the excitation energy available for evaporation.


Question: What is the domain of activity of the jth ReLU neuron and how does it affect the differentiability of the function Nn,o,∆,a?

Answer: The domain of activity of the jth ReLU neuron is defined as Aj = {x ∈ Rdin : nj · x > oj}. Outside the breaklines, the function Nn,o,∆,a is differentiable with DNn,o,∆,a(x) = a′(x) + ∑(j=1 to d) ∆jnj1lAj(x). Along the breakline, the difference of the differential on Aj and Acj equals ∆jnj.


Question: What specific features of the DFT MD and SPC/E spectra are compared with experimental data in Figure 1a, and how do they align?

Answer: Figure 1a shows a direct comparison of the DFT MD and SPC/E spectra with a compiled experimental spectrum of water at 300 ± 2 K. The SPC/E spectrum extends down to 1 MHz, agreeing well with the experimental data from the lowest experimental data point at 100 MHz up until ∼1 THz, reproducing the Debye peak magnitude and position, albeit with a small red-shift. The SPC/E spectrum does not capture the translation band at 5 THz but shows a strong signature in the libration band at ∼20 THz, which is expected for a model where polarization only changes due to molecular reorientations.


Question: What conditions are equivalent for an OX-module FX of finite type in a coherent adic formal scheme X of finite ideal type, and how do they relate to admissible blow-ups?

Answer: The following conditions for an OX-module FX of finite type are equivalent: (a) F rig X = 0; (b) there exists an admissible blow-up π: X′ →X such that π∗FX is an IX′-torsion module, where IX′ is an ideal of definition of finite type of X′; (c) there exists an admissible blow-up π: X′ →X such that the strict transform π′FX is zero.


Question: How is the ordinary boundary triple {G, Υ0, Υ1} constructed from the quasi boundary triple {G, Γ0, Γ1} when G1 is dense in G?

Answer: The ordinary boundary triple {G, Υ0, Υ1} is constructed by defining the boundary mappings Υ0 and Υ1 as follows: Υ0f := ι−eΓ0f, and Υ1f := ι+Γ1f0, where f = f0 + fη ∈ dom A0 ∔ Nη(A∗). Here, ι+ and ι− are isometric isomorphisms from G1 to G and from G′1 to G, respectively, such that (ι−x′, ι+x)G = ⟨x′, x⟩G′1×G1 for all x ∈ G1 and x′ ∈ G′1.


Question: What is the algebraic meaning of the qq-characters in the context of A-type gauge theories, and how are they related to the quiver structure?

Answer: The qq-characters have an algebraic meaning related to the quiver structure of A-type gauge theories. At the classical level, this relation between the Seiberg-Witten curve and the fundamental character of the quiver was first noticed in [42] and was extended to the full Ω-deformed region in [23, 24]. For example, the character of the fundamental representation of su(2) is χ = y + y−1, and the corresponding qq-character can be understood as a doubly-quantized version of the Seiberg-Witten curve. The character for w = 2 corresponds to that of the adjoint representation χw=2 = y2 + 1 + y−2, and is lifted to a more complex expression involving the operators Y A(z1) and Y A(z2).


Question: What is the relationship between the thresholding constants a∗_i,n and the tuning parameter ηi,n in the case of consistent variable selection?

Answer: In the case of consistent variable selection, when n1/2ηi,n →∞, the relationship is given by n1/2a∗_i,n/ξi,n = n1/2ηi,n + Φ−1(1 −α) + o(1). This implies that the thresholding constants a∗_i,n are larger than the standard thresholding constants by an order of magnitude.


Question: What is the structure of the rule ρ in the system 'folding_an_edge' and how does it contribute to proving the system's termination?

Answer: The rule ρ in the system 'folding_an_edge' consists of a single PBPO+ rule over unlabeled graphs. The left-hand side (L) and the left-hand side after the rule application (L') are defined as follows: L { x:0 -P:0-> y:0 }, L' { x:0 -P:0-> y:0, x:0 -XX:0-> x:0, y:0 -YY:0-> y:0, x:0 -XY:0-> <-YX:0- y:0, x:0 -XC:0-> <-CX:0- c:0, c:0 -CY:0-> <-YC:0- y:0, c:0 -CC:0-> c:0 }. The right-hand side (R) is defined as { x.y:0 -P:0-> x.y:0 }. This rule is provably decreasing, as indicated by the weight analysis and successful sliding, which contributes to proving the system's termination.


Question: How are Mpc-structures on a symplectic manifold (M, ω) related to complex line bundles with a Hermitean structure?

Answer: Every symplectic manifold (M, ω) admits an Mpc-structure, and the isomorphism classes of Mpc-structures are parametrised by equivalence classes of complex line bundles with a Hermitean structure over M. Specifically, given a positive compatible almost complex structure J on (M, ω), one can construct a principal U(V, Ω, j)-bundle U(M, ω, J). For any complex line bundle L over M with a Hermitean structure h, the associated Mpc-structure (P(L, J), φ(L, J)) can be defined, and the line bundle associated to this Mpc-structure is L itself.


Question: How is the local benefit of training rule questions computed, and what factors are considered in this computation?

Answer: The local benefit of training rule questions is computed based on three factors: coverage, accuracy, and utility. Coverage is the number of record pairs that satisfy the rule, accuracy is the proportion of correct pairs among those that satisfy the rule, and utility is the extent to which the pairs improve the EM model. The local ranking score for a training rule is given by the formula S(qT) = accuracy(qT) * (sum of utility(p) for all p in qT) / (sum of 1 for all p in qT), and the rules are ranked by the score/cost ratio in descending order.


Question: How does the distribution of YSO candidates in the central dark cloud (Lynds 935) compare to the background contaminants, and what does this imply about the survey's sensitivity to high extinction regions?

Answer: The YSO candidates are primarily located on the central dark cloud (Lynds 935) and are highly clustered. In contrast, background contaminants (galaxies or AGN) are relatively evenly distributed across the field of view, with a notable lack of contaminants in the high extinction regions. This lack of contaminants in high extinction regions is due to the survey being magnitude-limited, causing the observed fraction of galaxies to drop out of the sample because of the extra extinction.


Question: What is the significance of the condition q < q∗{y} in the context of the admissibility of the measure kδy and the inequality u ≥ ukδy?

Answer: If q < q∗{y}, then for every k > 0, the measure kδy is admissible and the inequality u ≥ ukδy holds. However, if q > q∗{y}, the inequality u ≥ ukδy may not hold, as demonstrated by the example of a smooth, obtuse cone where y is the vertex of the cone.


Question: How does the formal rate of change of M become O(λ4) for initial states or ensembles prepared using a finite depth unitary circuit?

Answer: For initial states or ensembles prepared using a finite depth unitary circuit, the formal rate of change of M is O(λ4) at any time t. This is because the O(λ2) and O(λ3) terms in the rate of change of M vanish, as shown by the equations d/dt⟨Ψ(t)|M|Ψ(t)⟩ = λ2 ⟨Ψ0| (R −iλ[X, R]) |Ψ0⟩ + O(λ4) = O(λ4), where the O(λ2) and O(λ3) terms vanish by Eqs. (F10) and (F24).


Question: What are the specific parameters used for the X(π/2) gate on qubit Q2, including drive frequencies, phases, amplitudes, and Gaussian widths?

Answer: For the X(π/2) gate on qubit Q2, the parameters are: Drive 1 frequency ωd = 4.859 GHz, Drive 1 phase ϕd = -1.11, Drive 1 amplitude A = 638.170 mV, Drive 1 Gaussian width σ = 7.4 ns, Drive 1 Gaussian center t0 = 20 ns, Drive 2 frequency ωd = 4.543 GHz, Drive 2 phase ϕd = 0.841, Drive 2 amplitude A = 511.418 mV, Drive 2 Gaussian width σ = 6.82 ns, and Drive 2 Gaussian center t0 = 20 ns.


Question: What method was proposed to improve decision-making performance based on the continuity characteristic of agitation, and how did it perform compared to other methods?

Answer: The CCR method was proposed to promote decision-making performance based on the scores and historic information in decision windows, considering the continuity characteristic of agitation. It slightly improved the decision performance of agitation detection on the basis of RUS under a reasonable threshold range, while requiring less normal data and less time.


Question: What are the contributions of the effective operators a4 and a5 to the WW scattering amplitude, and how do they compare to other operators?

Answer: The effective operators a4 [Tr (VµVν)]2 and a5 [Tr (VµV µ)]2 generate the quadrilinear vertex involving four Goldstone boson derivatives, resulting in a WW scattering amplitude proportional to g2∗(E/Λ)4. These contributions are parametrically enhanced compared to other operators, such as a2 and a3, which contribute amplitudes proportional to g2(E/Λ)2 and g′2(E/Λ)2, respectively. The operators a4 and a5 dominate if the condition g∗(E/Λ) > g is satisfied, which is typically the case at LHC energies.


Question: What are the specific settings used in the CASCADE and HVQDIS programs for predicting b and c jet cross sections?

Answer: CASCADE uses the A0 PDF set with mc = 1.5 GeV and mb = 4.75 GeV, and µr = √(Q² + p²T + 4m²), where pT is the transverse momentum of the heavy quark in the virtual photon-proton centre of mass frame. HVQDIS uses the fixed flavour numbering scheme (FFNS) with mc = 1.5 GeV and mb = 4.75 GeV, and two scale choices: µr = µf = √(Q² + p²T + m²)/2 and µr = µf = √(Q² + 4m²).


Question: How is the string length ℓ estimated in the context of the string network, and what is the significance of the winding length ℓw?

Answer: The string length ℓ can be estimated using the winding length ℓw, which is defined as the number of plaquettes pierced by strings multiplied by the physical lattice spacing aδx, corrected by a factor of 2/3 to compensate for the Manhattan effect. This is an estimate of the length of string measured in the 'universe frame', that is, observers comoving with the expansion of the universe.


Question: What are the expected detection capabilities of the Gaia mission for Earth and Mars Trojans, and how do they compare to ground-based observations?

Answer: The Gaia mission is expected to detect thousands of Solar System Objects, including Earth and Mars Trojans. Simulations predict the existence of about 17 Earth Trojans larger than 100 m and about 50 Mars Trojans larger than 1 km. The first Earth Trojan (2010 TK7) was discovered in 2011, and there are only three known Mars Trojans. Gaia's detection capabilities are not limited by the restrictions of ground-based telescopes, such as low Solar elongations, and it can detect objects with apparent magnitudes ranging from V = 17.9 to V = 19.5 for Earth Trojans and V = 16.2 to V = 20.7 for Mars Trojans.


Question: How do the eigen angular frequencies of the PAA and the attached unit compare to the central angular frequency of the ultrasonic probe used in the study?

Answer: The eigen angular frequency of the PAA (𝜔𝑃𝐴𝐴) is about 1.2 × 10^10 rad/s and the eigen angular frequency of the attached unit (𝜔𝑅) is about 5.2 × 10^9 rad/s, both of which exceed the central angular frequency (about 2.5 × 10^6 rad/s) of the ultrasonic probe used in the study.


Question: What visual evidence is provided to support the effectiveness of the proposed method in recovering the original source from the nonlinear mixture?

Answer: Figure 7 shows the nonlinear mixture X (sampled trace (a) and components (f), (g)) of an IC Ornstein-Uhlenbeck source S ((b) and (d), (e)). The estimate ˆS := g(X) of S is shown in brown ((c) and (h), (i)). The source S and its estimate ˆS coincide up to (a transposition and) a monotone scaling of their components, as quantified by (104). A visual comparison of the original samples S(1), S(2) and their estimates ˆS(1), ˆS(2) confirms these results.


Question: What is the proof strategy for showing that all convex cocompact Kleinian groups with Hausdorff dimension less than 1 are classical Schottky groups?

Answer: The proof strategy involves assuming the Kleinian group is torsion-free by Selberg Lemma. It is shown that if a Kleinian group Γ has Hausdorff dimension DΓ < 1, it must be free. This is proven by contradiction: if Γ is not free, it must be convex-cocompact of second-kind, and an embedded surface R in H3/Γ is considered. If R is incompressible, it leads to a contradiction. If R is compressible, cutting along compression disks either results in an incompressible surface or a topological ball, implying H3/Γ is a handle-body, hence free. For a non-classical Schottky group Γ with DΓ < 1, it is shown that every open set about the limit set ηΓ must contain a non-quasi-circle, leading to a contradiction, thus proving Γ is a classical Schottky group. The sharpness of the bound is established by the existence of Kleinian groups that are not free with Hausdorff dimension equal to one.


Question: How is the S-wave contribution minimized in the analysis of B → K−π+l+l− and Bs → K+K−l+l− decays, and what is the impact on the S-wave fraction?

Answer: To minimize the S-wave contributions and project out the P-wave, the subtracted differential decay width is studied using Eq. (63). This method slices the Kπ invariant mass distribution into 4 bins around mKπ ∼ mK∗(892) and subtracts the events in the outer two bins from the central two bins. As a result, the S-wave contributions are reduced to less than 1% for both B → K−π+l+l− and Bs → K+K−l+l−. The disadvantage is the demand for larger statistics.


Question: What are the two limitations of the Gaussian-based model as demonstrated in Figure 5.2?

Answer: The two limitations of the Gaussian-based model are: (1) the threshold voltage distribution of each state as measured from real flash chips has a fatter tail than that of a Gaussian distribution, and the left and right tails of each state have different sizes; (2) the measured distribution demonstrates large second peaks in the distributions of the ER and P1 states, which are not captured by the Gaussian-based model.


Question: What are the specific forms of the reductions for the hydrodynamic type system, and how do they relate to the nonlocal reductions discussed in the query text?

Answer: The reductions for the hydrodynamic type system are given by ri(x, t) = ρqi(ε1x, ε2t) and ri(x, t) = ρ¯qi(ε1x, ε2t), where ε1, ε2, ρ are real constants such that ρ2 = ε2_1 = ε2_2 = 1. These reductions are consistent with the nonlocal reductions discussed in the query text, where ε1 = −1 or ε2 = −1 leads to nonlocal reductions. For example, if ε1 = −1 or ε2 = −1, the system becomes nonlocal, similar to the nonlocal reductions of the form ri(x, t) = τ1qi(ε1t, ε2x) and ri(x, t) = τ2¯qi(ε1t, ε2x) mentioned in the query text.


Question: What is the condition for a multinomial sequence to be almost mean k-divisible for all k according to Theorem 1?

Answer: If GCD(m1, m2, ..., ms) = 1, then the multinomial sequence is almost mean k-divisible for all k.


Question: How is the optimization of the variational parameters λ performed to minimize the KL divergence in the context of the SDGM approximation?

Answer: Optimizing the value of λ in (3) so that qλ(θ) is closest to the posterior density p(θ|y) in the Kullback-Leibler sense is equivalent to optimizing the ELBO (2). The optimization is done by stochastic gradient ascent, where starting from some initial value λ(0) for the variational parameters, we update by λ(t+1) = λ(t) + δt ⊙ ∇λL(λ(t)), for t ≥ 0 until some stopping rule is satisfied; here δt is a vector of step sizes of the same dimension as λ and ∇λL(λ(t)) is an unbiased estimate of the gradient.


Question: What is the relationship between the Gromov-Witten potential and the ε-quasimap potential for semi-positive triples (W, G, θ)?

Answer: The wall-crossing formula relates the Gromov-Witten potential F ∞ g to the potential F ε g for semi-positive triples (W, G, θ). Specifically, for every ε ≥ 0+, the relationship is given by (Jε 0)2g−2F ε g(t(ψ)) = F ∞ g(t(ψ) + Jε 1 / Jε 0). Additionally, for every ε1 ≠ ε2, the formula (Jε1 0)2g−2F ε1 g(Jε1 0 t(ψ) − Jε1 1) = (Jε2 0)2g−2F ε2 g(Jε2 0 t(ψ) − Jε2 1) holds.


Question: How do the simulations in the candidate text differ from the simulations described in the query text in terms of binary mass ratio and spin alignment?

Answer: The simulations in the candidate text consider binary black holes with mass ratios q = 1, 2, and 4, and the spins are either aligned with the initial orbital plane or 45° above it. In contrast, the query text describes simulations of equal-mass binary black holes with equal spins aligned with the orbital angular momentum.


Question: What is the relationship between the forms ⟨−, −⟩γ≤m and ⟨−, −⟩γ on L BS(xˆi) and L BS((xs)bi) respectively, and how does it relate to the first differential in the complex Fs1 · · · FsmFs?

Answer: The relationship is given by the equation ⟨b, Lζb′⟩BS(xs) = ⟨φ(b), φ(b′)⟩γR for all b, b′ ∈ BS(xs), where φ is the first differential in the complex Fs1 · · · FsmFs and ⟨−, −⟩γR denotes the form on L BS((xs)bi) induced by ⟨−, −⟩γ.


Question: How does the momentum conservation differ between the amplitude and the complex conjugate amplitude in the context of two-gluon production diagrams?

Answer: In the context of two-gluon production diagrams, the momenta of the whole incoming nuclei must be equal in both the amplitude and the complex conjugate amplitude. Therefore, for k+1 ≠ k+2, the diagrams represent different redistributions of the projectile nucleus momentum between the nucleons in the amplitude and the complex conjugate amplitude, without changing the same 'plus' momentum of the whole nucleus on both sides of the cut. Hence, the k+1 = k+2 condition is not necessary for the 'crossed' diagrams.


Question: What is the significance of the bar pattern speed in relation to the corotation radius and bar semi-major axis, and how does this relate to the classification of bars as fast or slow?

Answer: The bar pattern speed is parametrised with the bar rotation rate R ≡ Rcor/abar, which is the distance-independent ratio between the corotation radius Rcor and the bar semi-major axis abar. Bars with 1.0 ≤ R ≤ 1.4 end close to corotation and rotate as fast as they can, whereas bars with R > 1.4 fall short of corotation and are termed slow. The dividing value at 1.4 between long/fast and short/slow bars is given by consensus.


Question: How does the proposed method generate dense occupancy labels to improve the network's performance?

Answer: The proposed method generates dense occupancy labels by stitching multi-frame LiDAR points of dynamic objects and static scenes separately, and then using Poisson Reconstruction to fill the holes and voxelize the obtained mesh. This process leverages existing 3D detection and 3D semantic segmentation labels without requiring additional human annotations, resulting in more realistic and dense occupancy labels.


Question: What are the specific conditions that a function c must satisfy to be considered a cocycle function according to Proposition 2.2?

Answer: A non-zero function c : Λ2 → F is a cocycle function if it satisfies the following conditions: (i) c(x, x) = 1 for every x ∈ Λ; (ii) c(x, y)c(y, x) = 1 for every x, y ∈ Λ; (iii) c(x, y)c(y, z)c(z, x) = 1 for every x, y, z ∈ Λ.


Question: What are the specific equations that need to be solved in Taylor series to ensure the uniqueness of the solution for ϕ in the context of the Ricci tensor equations?

Answer: To ensure the uniqueness of the solution for ϕ, we need to solve the equations (3.2.1a)–(3.2.1f) in Taylor series. These equations are: −2 RicNN = N✚ tr LN ✓h + 1/2| LN ✓h|2 −2(N log Ω)✚ tr LN ✓h, −2 RicLL = L✚ tr LL ✓h + 1/2| LL ✓h|2 −2(L log Ω)✚ tr LL ✓h, and −1/2 RicNL = NL log Ω + 1/4N✚ tr LL ✓h −x/8 tr L[N,L] ✓h −x/2[N, L] log Ω −x^2/2 ∆Ω^2 + 1/8✓h(LL ✓h, LN ✓h) + 1/8Ω^2 |[N, L]|2. Additionally, the integrability conditions (3.2.2a)–(3.2.2c) must also be satisfied.


Question: How does the spectral dispersion of the coupled ENZ and SPP modes change as the two branches approach and move away from the anti-crossing point?

Answer: At the anti-crossing point, the upper branch (UB) and lower branch (LB) are associated with hybrid symmetric and antisymmetric PH-ENZ modes, respectively, exhibiting both SPP and ENZ character. Spectrally removed from the coupling point, the UB and LB asymptotically approach the bare SPP and bare ENZ dispersion curves, respectively, and the mode profiles primarily adopt the behavior of the asymptote to which the branch is closest.


Question: How does the Wigner distribution contribute to the understanding of planetary wave modulational instability in the context of the Vlasov equation?

Answer: The Wigner distribution is used to derive a planetary wave Vlasov equation that includes the contribution of the mean flow to the zonal potential vorticity gradient. This equation is applied to the problem of planetary wave modulational instability, where it is used to predict a fastest growing mode of finite wavenumber. The wave-mean flow numerical model is used to test the analytical predictions, and an intuitive explanation of modulational instability and jet asymmetry is given via the motion of planetary wavepackets in phase space.


Question: How is the saddle point αs related to the weight of the internal hole operator and what does it describe in the semi-classical limit?

Answer: The saddle point αs is related to the weight of the internal hole operator, with 2παs being the length of the s-channel geodesic. In the semi-classical limit, it describes a flat cylinder with a ring domain of height s and circumference 2παs.


Question: How can the variance of the empirically computed success rates be used to detect and locate the source of bias in a system?

Answer: The variance of the empirically computed success rates can be used to detect and locate the source of bias by examining the shape of the log-variance curve. For instance, if bias was injected into the system using combinations of two protected attributes (Level M − 2), we might see a bend in the log-variance curve at around those values of K. This could help answer questions related to the sources of bias and identifying gerrymandering.


Question: How are the derivatives of the trimming operation computed in the context of the implicit representation of the convex hull and Bézier patch?

Answer: The derivatives of the trimming operation are computed as follows: ∇g = 1/2 * (2h^3∇h + f∇f) / sqrt(h^4 + f^2) - ∇f, and ∇²g = 1/2 * (2h^2 (h∇²h + 3∇h * ∇h) + f∇²f + ∇f * ∇f) / sqrt(h^4 + f^2) - ∇(sqrt(h^4 + f^2)) * ∇(sqrt(h^4 + f^2)) / sqrt(h^4 + f^2) - ∇²f, where the term ∇(sqrt(h^4 + f^2)) can be computed from Equation (19) for the functions f and h^2.


Question: How does the performance of the bivar-MRCCSD(2,2)FOIS method compare to the CASCCSD method in the H8 model system, particularly in the multi-reference case?

Answer: In the multi-reference case, the bivar-MRCCSD(2,2)FOIS method is slightly better with respect to both energy and state projector compared to CASCCSD.


Question: What are the advantages of placing a top gate close to the SiGe nanocrystals for hole-based spin qubits?

Answer: Placing a top gate close to the SiGe nanocrystals allows for strong coupling to the quantum dot, enabling the driving of spin states with Rabi frequencies of the order of 100MHz. Additionally, a narrow gate ensures local action and minimizes the dissipation of high-frequency power.


Question: What is the objective function and the constraints used in the traffic signal timing optimization problem for the incident-affected area?

Answer: The objective function is to minimize the total travel time of the network, expressed as: minimize ∑∫ta(va, λa)dx from 0 to va for all a ∈ A. The constraints include: ∑∑frw for all r ∈ Rw and w ∈ W multiplied by σarw equals va for all a ∈ A, ∑frw for all r ∈ Rw equals Dw for all w ∈ W, va is less than or equal to λaSa for all a ∈ A, and frw is greater than or equal to 0 for all r ∈ Rw and w ∈ W.


Question: How does the linear independence of the family of integrators affect the form of the Riccati equation in the context of the given equations?

Answer: If the family of integrators of scalar type appearing in (3.1) and (3.2) is linearly independent, then the Riccati equation (3.7) can be put in the form dΠ(t) = dτ(t) A(t, Π(t)) + ∑_{a∈I} dMa(t) Ba(t, Π(t)), where A(t) = A(t, Π(t)) and Ba(t) = Ba(t, Π(t)) can be described as the solutions of the operator equations.


Question: What are the measured surface potential differences between the gold substrate and 2L and 3L phosphorene flakes, and how do they compare?

Answer: The measured surface potential difference between the gold substrate and 2L phosphorene is ~536 meV, while for 3L phosphorene, it is ~555 meV. This indicates that the surface potential difference increases with the number of layers.


Question: What is the fitted parameter for the Hubble profile used to fit the observed PDF of the projected interdistances between pair members in the IGPC?

Answer: The fitted parameter is ap = (0.17 ± 0.01) Mpc.


Question: How does the mutation step in the SMC algorithm maintain the diversity of the particle population?

Answer: The mutation step in the SMC algorithm maintains the diversity of the particle population by performing a small number of µn-invariant MCMC iterations. This 'jittering' of the population helps to improve diversity, which is essential because carrying out only selection steps will eventually lead to degeneracy in the diversity of the particle population. During each successive resampling step, only a few parent particles will survive and copy themselves, so the mutation step ensures that the statistical properties of µn are maintained while improving diversity.


Question: How are the submatrices of Ω∗3 and Ω∗4 generated in the two clusters, block diagonal Erdos-Renyi graphs setting?

Answer: To generate Ω∗3, p/2 variables are randomly selected and defined as s1, and s2 is defined as the remaining variables. The submatrix of Ω∗3 corresponding to the indices in s1 is generated such that G = E(A5, p/2), and the submatrix corresponding to the indices in s2 is generated such that H = E(A6, p/2), where A5 and A6 are independent Erdos-Renyi graphs with p/2 edges. The submatrices of Ω∗4 corresponding to the indices in s1 and s2 are generated using R (A7, G, (−.01, .01)) and R (A8, H, (−.01, .01)), respectively, where A7 and A8 are the same as A5 and A6, respectively, with 4 randomly selected edges removed in each.


Question: What are the key findings regarding the stability of phase-locked states in rings of time-delayed Kuramoto oscillators?

Answer: The stability of phase-locked states with a constant phase shift between neighboring oscillators is studied, and it is found that the stability maps for the dimensionless equations show a high level of symmetry. The size of the attraction basins changes periodically over several orders of magnitude as the parameters of the model are varied. Simple heuristic arguments are formulated to understand these changes and predict the most probable states when the system is randomly initialized.


Question: How are the blue straggler stars (BSS) identified and marked in the Gaia DR2 CMD of NGC 7789?

Answer: BSS that fall to the blue of the dashed line are highlighted. BSS non-velocity-variable RV members with Pµ ≥50% are indicated by blue dots, and binary BSS, all of which have orbital solutions, are marked with an additional circle.


Question: What are the specific physical parameters of the shell in the gravastar model under Rastall gravity, and how do they vary with the radial distance?

Answer: The physical parameters of the shell in the gravastar model under Rastall gravity include: (i) Pressure and matter density: The matter density or pressure of the shell is positive and monotonically increasing towards the exterior surface, making the shell more dense at the exterior boundary than the interior boundary. (ii) Proper thickness: The proper thickness or proper length ℓ of the shell is monotonically increasing from the interior junction to the exterior junction. (iii) Energy: The energy of the shell increases with the increase in the radial distance. (iv) Entropy: The entropy S within the shell is gradually increasing with respect to the radial distance r, indicating a maximum value of entropy on the surface of the gravastar.


Question: What does the star formation efficiency (SFE) indicate about the depletion of cold gas in the simulations?

Answer: The SFE measures the efficiency with which the cold and dense gas is being incorporated into stars. Since all the curves increase with time, the sink particles are depleting the cold gas supply faster than gas is able to cool, implying that eventually, the star-forming cluster becomes gas-starved.


Question: What are the conditions under which Eq. (9) and Eq. (10) hold with high probability for the choice of bx = x∗?

Answer: With probability at least 1 − 2δ, Eq. (9) holds for bx = x∗ if t ≥ 900C2 fT C1 and √fT C1t ≥ 30C. Eq. (10) holds for all x ≠ x∗ if t∆x ≥ 30√fT C1t, which is satisfied by the condition t ≥ 900fT C1 / ∆2_min.


Question: What is the observed size evolution of passively evolving galaxies (PEGs) at high redshift, and how does it compare to the local relation?

Answer: At high redshift (1.5 < z < 2.5), PEGs are on average 3-4 times smaller than their local counterparts with similar stellar masses. The size evolution with redshift is described by the relation re ∝ (1 + z)−1.13, which is in agreement with previous studies. The sizes of PEGs need to increase by a factor of about 3 to match the local relation by z = 0. The large scatter in sizes at high redshift (σlogre = 0.22 at z ∼1.6 and 0.34 at z ∼2.1) also needs to evolve significantly to match the smaller scatter observed in local SDSS early-type galaxies (σlogre = 0.16 at z ∼0.06).


Question: What are the rolling velocities of HCT-15 and MDA-MB-231 cells at 50 nL/min under different TNF-α stimulation conditions?

Answer: At 50 nL/min, the rolling velocities of HCT-15 cells were 113.9 ± 4.132 µm/sec for unstimulated HUVECs (- TNF-α), 103.4 ± 2.880 µm/sec for 6h-stimulated HUVECs (+TNF-α 6h), and 98.00 ± 4.552 µm/sec for 12h-stimulated HUVECs (+TNF-α 12h). For MDA-MB-231 cells, the rolling velocities were 118.6 ± 1.349 µm/sec for unstimulated HUVECs (- TNF-α), 105.68 ± 3.340 µm/sec for 6h-stimulated HUVECs (+TNF-α 6h), and 102.1 ± 5.288 µm/sec for 12h-stimulated HUVECs (+TNF-α 12h).


Question: How is the invariance factor K determined for the divergence JI (p∥q) and what is the resulting invariant divergence formula?

Answer: The invariance factor K is determined as K = (Σi pi) / (Σi qi). The resulting invariant divergence formula is JI (p∥q) = (Σj pj)^λ / (λ (λ - 1)) * (Σi [αpλi + (1 - α) qλi] - Σi [αpi + (1 - α) qi]λ), with ¯pi = pi / (Σj pj) and ¯qi = qi / (Σj qj).


Question: How does the phase noise from the LO laser affect the performance of long-haul high-speed optical fiber networks, and what is the formula for the variance of the additional noise due to EEPN?

Answer: The phase noise from the LO laser only goes through the EDC module, which is heavily dispersed in the transmission system without using any optical dispersion compensation. Therefore, the LO phase noise will interplay with the EDP module, and the induced EEPN will significantly affect the performance of the long-haul high-speed optical fiber networks. The variance of the additional noise due to EEPN can be expressed as: \( \sigma_{EEPN}^2 = rac{2 \pi^2 \Delta f_{LO} T_S D L}{c \lambda} \), where \( \lambda \) is the central wavelength of the optical carrier, \( c \) is the light speed in vacuum, \( D \) is the CD coefficient of fiber, \( L \) is the fiber length, \( \Delta f_{LO} \) is the 3-dB linewidth of LO laser, and \( T_S \) is the symbol period of the coherent transmission system.


Question: How does the helicity preference k affect the global structure of the solar corona, particularly in terms of the formation of filament channels and the distribution of magnetic field lines?

Answer: The helicity preference k plays a key role in determining the global structure of the solar corona. For k = 1.0 (100% preference), the twist flux condenses into two primary bands with opposite senses of twist at the inner and outer boundaries of the hexagonal region, with very little twist in the interior. The magnetic field lines are very long near the boundaries and minimally short in the interior. For k = 0.5 (50% preference), the twist flux does not condense into any recognizable global-scale structure, and the field lines have a homogeneous distribution of intermediate lengths. The k = 0.75 (75% preference) case exhibits features of both, with bands of condensed twist flux forming more slowly and with significant intrusions of twisted structures between them, and field-line lengths that are longer at the boundaries but with less contrast.


Question: What is the depth of the Column carry circuit block in terms of unit-gate steps, and how is it composed?

Answer: The Column carry circuit block has a depth of 18 unit-gate steps, composed of 1 CCNOT and 3 SWAPs.


Question: How does the wavefunction asymmetry in four-layer CrI3 depend on the magnetic configuration, and what are the specific wave amplitudes for different spin configurations?

Answer: The wavefunction asymmetry in four-layer CrI3 depends on the magnetic configuration, as characterized by the nonzero value of |𝑎1|² − |𝑎4|². For the spin configurations {+,−, +, −}, {+,+, −, +}, {+,−, +, +}, and {+,+, +, +}, the wave amplitudes are as follows: {0.0504, 0.0419, 0.8749, 0.0328, 0.0176}, {0.3190, 0.5262, 0.0344, 0.1204, 0.1986}, {0.0047, 0.0136, 0.4713, 0.5104, -0.5057}, and {0.0607, 0.2531, 0.4291, 0.2571, -0.1964} respectively.


Question: How do the RMSE values of the CS-LSTM and CS-LSTM(m) models compare to S-LSTM on the HighD dataset, and what does this indicate about the effectiveness of convolutional layers in modeling vehicle movement?

Answer: CS-LSTM and CS-LSTM(m) obtained better results than S-LSTM, showing the advantage of using convolution layers for modeling the dependency of vehicles' movement. Moreover, CS-LSTM(m) has a smaller error because it considers the future path's multi-state nature.


Question: How does the relativistic propagator G1 contribute to the impulse approximation for pion production, and what is the resulting expression for G1Oπ near threshold?

Answer: The relativistic propagator G1 is decomposed into three terms: 1, γ0, and γi. Between two-component nucleon spinors, the product G1Oπ simplifies to G1Oπ = i(E(p1) − mπ + mN) / ((p1 − q)² − mN² + iǫ) Oπ. Near threshold, where E(p1) = mN + mπ/2, this expression simplifies to G1Oπ = i(1 − mπ / (4mN)) Oπ / (−mπ).


Question: How does the spin detection efficiency (Pdet) vary with applied voltage (Vapp) in the presence of different spin accumulations (µs)?

Answer: The spin detection efficiency (Pdet) strongly depends on the junction bias. At 0 V, Pdet bears the value predicted by the linear model. For negative bias (spin extraction regime), Pdet increases, and for positive bias (spin injection regime), Pdet decreases. The non-linear effect does not show perfect odd symmetry, and Pdet exhibits a non-monotonic dependence, leading to a maximal detection efficiency for a specific voltage Vmax. The maximal value of Pdet is not limited to 2PG and is sensitive to the intensity of the spin accumulation in the vicinity of the barrier.


Question: How does the function φ behave on the set G and its complement in R², and what does this imply about the relationship between φ and the set G?

Answer: The function φ is 0 on the set G and less than 0 on R² \ G. This implies that φ < 0 on R² \ G and φ = 0 on G. If φc(x) = 0, then φ(x) = 0 and thus, x ∈ G. Conversely, if x ∈ G, then c(x, y) − φ(y) ≥ 0 for all y ∈ R², and therefore, φc(x) = 0.


Question: What is the relative contribution of 8Be decays in the fragmentation of 16O, 22Ne, and 28Si nuclei with different multiplicities of α-particles and protons?

Answer: For 16O, the relative contribution of 8Be decays is 35 ± 14% for 2α + (1-4)p, 39 ± 13% for 3α + (1,2)p, 25 ± 12% for 4α + (1,2)p, 13 ± 8% for 5α + mp, and 0/5 for 6α + mp. For 22Ne, it is 32 ± 13% for 2α + (1-6)p, 21 ± 8% for 3α + (1-4)p, 16 ± 6% for 4α + mp. For 28Si, it is 25 ± 20% for 2α + mp, 17 ± 7% for 3α + mp.


Question: What mechanism allows the cell to maintain the M state after the removal of exogenous TGF-β?

Answer: With sufficiently long and strong exogenous stimulus to turn on the second switch, TGF-β can be produced endogenously, allowing the switch to remain at the 'on' state even after removing exogenous TGF-β.


Question: What are the specific results and observations regarding the Q-factor and directivity for meander line antennas in the studied planar region?

Answer: The lower bound on Q-factor with radiation restricted to TM modes only is closely approached by a meander line antenna for a broad range of electrical sizes. The radiation efficiency of the studied meander line antennas is far from an upper bound of a rectangular patch. The performance of maximum efficiency current distributions drops much faster than that of a minimum Q-factor distribution when the initial structure is perforated.


Question: How do the experimental results of jointly trained DDNs compare to DRFs on multi-label classification tasks?

Answer: Jointly trained DDNs are almost always superior to the best-performing DRFs on all datasets. Pipeline DDN models outperform DRF models when performance is measured using the mAP and LRAP metrics. However, when the SA and JI metrics are used, there is no significant difference in performance between pipeline DDNs and DRFs.


Question: How does the final Lagrangian for the Goldstonic Yang-Mills theory incorporate the massless vector Goldstone and pseudo-Goldstone boson multiplet, and what is the role of the tensor hµν in this context?

Answer: The final Lagrangian for the Goldstonic Yang-Mills theory, to first order in (a2/M2), is given by L(aαµ, ψ) = −1/4 Tr(fµνf µν) − 1/2 δ(n · aα)2 + 1/4 Tr(f µνhµν)n2a2/M + ψ(iγ∂−m)ψ + gψaµγµψ − gn2a2/2M ψ(γ · n)ψ. The tensor hµν is defined as hµν = nµ∂ν − nν∂µ + ig([nµ, aν] − [nν, aµ]), and it acts on the infinite series in a2 coming from the expansion of the effective 'Higgs' mode. The massless vector Goldstone and pseudo-Goldstone boson multiplet aαµ gauges the starting global symmetry G, and in the limit M → ∞, the theory is indistinguishable from conventional Yang-Mills theory in a general axial gauge.


Question: How does the inter-layer electrostatic potential affect the energy levels of the t2g orbitals in the (111)-oriented STO interface?

Answer: The inter-layer electrostatic potential increases the distance between the bonding and the anti-bonding orbitals without changing their character, which helps distinguish between the two layers in terms of their distance from the surface.


Question: What is the relationship between the cohomology of the symmetric product of a complex algebraic variety and the cohomology of the variety itself, and how does this relationship extend to mixed Hodge structures?

Answer: There is a canonical isomorphism of sheaves of Q-vector spaces QSnX = (π∗QXn)Sn, which implies canonical isomorphisms H•(SnX, Q) = H•(Xn, Q)Sn = ⨁NnH•(X, Q)Sn. This leads to an isomorphism of bigraded vector spaces H•(S•X, Q) = Sym•H•even(X, Q) ⊗V•H•odd(X, Q). These isomorphisms are compatible with mixed Hodge structures and extend to the case of intersection cohomology.


Question: What is the classification accuracy of the test data from the second quadrotor after applying transfer learning with 20 training images?

Answer: The classification accuracy of the test data from the second quadrotor after applying transfer learning with 20 training images is 91.82%.


Question: How is the mean curvature vector defined for a submanifold in a Riemannian manifold, and what does it mean for a submanifold to be minimal?

Answer: The mean curvature vector −→H is defined by −→H = 1/n trace σ = 1/n ∑ σ(ei, ei), where {e1, ..., en} is a local orthonormal frame of the tangent bundle T M of M. A submanifold M is called minimal in ˜M m if its mean curvature vector vanishes identically.


Question: What is the behavior of the amplitude (4.25) in terms of time evolution and unitarity, and what conditions must be met for probability conservation?

Answer: For λ ≥ 0, (4.25) will evolve non-unitarily. If λ < 0, there are states that could potentially evolve unitarily. For probability conservation, the condition C − 2|λ|ϕ^2 < 0 must be met for all values of C ∈ [0, ∞]. This happens only when ϕ|B(R,L) → ∞, or ϕ|B(R,L) = 1/ϵ.


Question: How does the proof show that the permutations σ and σ′ give the same input, and why is the resulting sum multiplied by 2^(k/2)?

Answer: The proof shows that the permutations σ and σ′ give the same input by demonstrating that the inputs corresponding to σ and σ′ are the same. Specifically, for the permutation σ of type (σ(i1), r2, σ(i2), r3..., σ(ik−1), rk), the input is (−1)σF−σ(i1)σ(i2)er2 ⊗... ⊗F−σ(ik−1)σ(ik)erk. When considering only permutations of this type, the resulting sum must be multiplied by 2^(k/2) to account for the fact that each pair (σ(i2t−1), σ(i2t)) can be swapped without changing the input, thus each valid permutation is counted twice.


Question: How do the coefficients U1, U2, and U3 in the linearized equation of the Riccati equation (6.8) differ in the candidate text compared to the query text?

Answer: In the candidate text, U1 is given by (v1 + v8 −κ2)(1 + v1 + v8 −κ2) / ((κ1 −κ2)(1 + κ1 −κ2)) ∏(i=3,5,7) (v1 + vi −κ1) ∏(j=2,4,6) (vj + v8 −κ1), U2 is given by - ∏(i=2,4,6) (v1 −vi) ∏(j=3,5,7) (vj −v8), and U3 is given by (v1 + v8 −κ1)(v1 + v8 −κ1 −1) / ((1 + κ1 −κ2)(2 + κ1 −κ2)) ∏(i=3,5,7) (1 + v1 + vi −κ2) ∏(j=2,4,6) (1 + vj + v8 −κ2).


Question: What method is used in the study to model the daily maximum temperature extremes in Ireland, and what is the significance of this approach?

Answer: The study employs extreme value theory (EVT) to model the data using the generalised Pareto distribution (GPD) as part of a three-level Bayesian hierarchical model. This approach is significant because it combines predictive processes and EVT, which is the first study to do so in this manner, to solve the computationally difficult problem of modelling data over a very dense spatial field.


Question: What does Figure 3 illustrate about the RABBITT phase of sideband electrons in co-rotating and counter-rotating geometries?

Answer: Figure 3 illustrates the circular dichroism on the RABBITT phase of sideband electrons. Panels a and b show the measured θ-resolved RABBIT traces of SB18 in co-rotating and counter-rotating geometries, respectively. Panel c shows the extracted θ-resolved RABBIT phases in the two geometries, with the uncertainty estimated by the background-over-amplitude approach.


Question: What is the lower bound for D2 when t(D,F) = −∞ and how does it relate to the degree of F and the geometry of the Miyaoka divisors?

Answer: When t(D,F) = −∞, the lower bound for D2 is given by C(D,F) · deg F + 2ǫ∗D.Zk − Z2k, where C(D,F) is defined as 2α(D,F)dk / (−α(D,F) + 2α(D,F)rk − dk) if α(D,F) − 2α(D,F)rk + 2dk ≤ 0, and 3α(D,F) + 2dk − 2α(D,F)rk otherwise. Additionally, if dk = α(D,F)(rk − 1), then D2 ≥ 2dk / rk · deg F + 2ǫ∗D.Zk − Z2k.


Question: How does the kinetic-theory temperature θ[f] relate to the thermodynamic temperature T(ρ, s) for a monatomic ideal gas, and what does this imply for the reduction of noncanonical Hamiltonian systems?

Answer: The kinetic-theory temperature θ[f] and the thermodynamic temperature T(ρ, s) are related by the equations θ[f] = ρ[f]^2/n * (2πe)^(-1) * exp(-2/n * s[fm]/ρ[f]) and T(ρ[f], s[f]) = ρ[f]^2/n * (2πe)^(-1) * exp(-2/n * s[f]/ρ[f]). This implies that the kinetic-theory temperature θ[f] can be described in terms of (m[f], ρ[f], s[f]) through the equation of state T(ρ, s), but the exact reduction of noncanonical Hamiltonian systems is not possible due to the deviation of the distribution function f from its associated local Maxwellian fm.


Question: What are the specific statistics of the MUTAG, DD, PTC-MR, NCI-1, PROTEINS, and ENZYMES datasets used in the experiments?

Answer: The statistics of the datasets are as follows: MUTAG has 188 graphs, 2 classes, an average of 17.93 nodes, and 19.79 edges; DD has 1178 graphs, 2 classes, an average of 284.32 nodes, and 715.66 edges; PTC-MR has 344 graphs, 2 classes, an average of 14.29 nodes, and 14.69 edges; NCI-1 has 4110 graphs, 2 classes, an average of 29.87 nodes, and 32.30 edges; PROTEINS has 1113 graphs, 2 classes, an average of 39.06 nodes, and 72.82 edges; ENZYMES has 600 graphs, 6 classes, an average of 32.63 nodes, and 64.14 edges.


Question: What key lessons were learned during the development of the biochar molecular models, particularly regarding the influence of molecular properties on the structure of condensed-phase systems?

Answer: Key lessons learned include the strong relationship between the aromatic domain size of building blocks and the true density of condensed-phase systems, with larger aromatic domains leading to higher densities. Additionally, curvature introduced by non-hexagonal rings is crucial for developing amorphous morphologies, but it also results in lower true densities due to disrupted packing.


Question: What are the major changes in the electronic structure of ScV6Sn6 below the CDW transition temperature, as revealed by DFT calculations?

Answer: Below the transition temperature, the separations between the bands increase by approximately 30 meV at the Γ point within the energy window [-0.75, -1] eV. The CDW has its major effect along the A-H and A-L directions, where it opens several energy gaps along the entire k-space path. However, the spectral weight immediately fades away from the gap openings, making the gaps challenging to detect experimentally due to their vicinity to the poles, kz broadening, and the experimental resolution (∼15 meV).


Question: What are the main sources of uncertainty that limit the constraints on the Yukawa potential in the MICROSCOPE experiment?

Answer: The main sources of uncertainty are the gold wires used to control the electric charge of the test masses, and a non-zero unaccounted for stiffness that depends on the instrument’s electric configuration, hinting at the presence of patch-field effects. Added to significant uncertainties on the electrostatic model, they only allow for poor constraints on the Yukawa potential.


Question: What is the process for finding equilibrium solutions and performing linearisation in the context of the two-species Vlasov-Poisson system?

Answer: To find equilibrium solutions for the two-species Vlasov-Poisson system, a pair of functions (푓푒푞푢_푖, 푓푒푞푢_푒) is considered an equilibrium solution if they satisfy the system and are independent of time (휕푡푓푒푞푢_푖 = 휕푡푓푒푞푢_푒 = 0). The existence of such solutions is proven by showing that any pair of distributions (푓푖(푥, 푣), 푓푒(푥, 푣)) = (푓푒푞푢_푖(푣), 푓푒푞푢_푒(푣)) that are independent of 푡 and 푥 defines an equilibrium solution. For linearisation, the system is assumed to be near an equilibrium, and the linearised equations are derived to analyse stability and asymptotic behavior.


Question: What is the role of the birational map eρ in the construction of the log smooth weakly birational model ef′ : (eX′, eA′ + eB′) → eS?

Answer: The birational map eρ : eX′ 99K eX extends naturally to a birational map whose image contains a dense Zariski open subset of supp(eA + eB). This allows the definition of the birational transforms of eA and eB on eX′, denoted by eA′ and eB′ respectively. By replacing eX′ with a suitable blow-up, it is ensured that eX′ is a smooth projective variety, ef′−1σ−1(D) + eA′ + eB′ is a simple normal crossing Q-divisor, and ef′ : eX′ → eS is smooth over eSo.


Question: What condition must be met for the monolayer to remain stable with respect to perturbations in PM_m, and how is this condition related to the relative U-V interface penalisation µ?

Answer: The monolayer of optimal width is stable with respect to perturbations in PM_m if and only if µ ≥ ˜µ(ν), where µ is the relative U-V interface penalisation and ˜µ(ν) is defined as 1 - 2˜ς, with ˜ς being the infimum of ςj over all j ≥ 1.


Question: What is the total number of LCD cyclic codes with length n over Fq when q = 2sf + 1, and how is this number derived?

Answer: The total number of LCD cyclic codes with length n over Fq is equal to 2(t+2)2s−2+1. This is derived from the fact that an LCD cyclic code C is equivalent to the dimension of Hull(C) being zero, i.e., ℓ= dim(Hull(C)) = 0. The defining set S of C must satisfy S = −S, and from Lemma 5.3, the total number of Sij is (t + 2)2s−2 + 1.


Question: How does the resource utilization and performance of NG-Large compare to NG-Medium for the Canny and Harris kernels?

Answer: For the Canny kernel, NG-Large uses 2% LUT, 2% DFF, 4% CY, 1% DSP, and 93% RAMB, while NG-Medium uses 7% LUT, 8% DFF, 13% CY, 5% DSP, and 90% RAMB. For the Harris kernel, NG-Large uses 5% LUT, 13% DFF, 23% CY, 22% DSP, and 36% RAMB, while NG-Medium uses 19% LUT, 38% DFF, 73% CY, 100% DSP, and 63% RAMB. NG-Large has significantly better resource utilization, leaving room for implementing complementary HDL components and increasing parallelization. In terms of performance, Harris in NG-Large is 4.6× better due to a higher clock frequency and processing fewer image stripes, while Canny in NG-Large has a 1.4× performance improvement despite a higher clock frequency in NG-Medium, due to the overhead of processing smaller image partitions.


Question: How does the virtual valuation for timeline C differ from that of timeline A, and what additional constraints are introduced in the monopolist's problem for timeline C?

Answer: The virtual valuation for timeline C is given by ΨC(s, f) = ΓC(s) + M G(ˆλ)−G(s) / g(s) + f for s ≤ ˆλ, which includes an additional term for the lump-sum subsidy f. The monopolist's problem for timeline C includes the constraint that (ΓB(λ) − ΓC(λ)) v(q(λ)) − M ∫_1^λ v(q(s)) ds ≥ f, ensuring individual rationality, and the condition that f may be negative to account for the externality exerted by the self who chooses the bundle on the participation self.


Question: What are the bounds on the Sobolev norms of the canonical quasiderivation ∆(S,σ) in terms of the Sobolev norms of the elements of lCΓ?

Answer: There exist constants C1(n) = C1(n, S, δ) > 0, C2 = C2(S, δ) > 0, and d = d(S, δ) ∈ lN+, such that the canonical quasiderivation ∆(S,σ) satisfies ∥∆(S,σ)(α) ∥n,k ≤ C1(n) · Ck2 · ∥α ∥n,k+d for all α ∈ lCΓ, and all n, k ∈ lN.


Question: What are the specific methods and theorems used to determine the relationship between the vertical and oblique probing frequencies, and how do they need to be corrected?

Answer: The relationship between the vertical and oblique probing frequencies is determined using three theorems: the Law of Secants, the Breit and Tuve Theorem, and the Martyn Theorem. These theorems assume a flat Earth and ionosphere without a magnetic field, so the expressions derived from them need to be corrected to account for the Earth's curvature and the presence of a magnetic field.


Question: What are the specific energy flux requirements and additional criteria for different jet models to stably quench a galaxy in a 10^14 M⊙ halo?

Answer: For kinetic jets, a non-precessing narrow jet requires a wider cocoon to quench efficiently, while a non-precessing wide jet with an opening angle much greater than 45° is not tested but is expected to be inefficient. Precessing narrow jets with a precession angle of at least 45° and a period of 10 Myr are not tested, but extremely large precession angles of at least 30° with a period of 100 Myr are required for quenching. For thermal jets, a constant energy flux of ∼6e43 erg s−1 is required, and the jet material temperature should be at least 10^9 K. A 10% duty cycle with an average energy flux of 6e44 erg s−1 and a duration of at least 100 Myr is also considered. For cosmic ray jets, a constant energy flux of ∼6e42-6e43+ erg s−1 is required, with no additional criteria. Magnetic jets with a constant initial magnetic field are inefficient by themselves but can quench with a wider cocoon.


Question: What is the relationship between the generalized Kronecker function and the quantum group U+ₙ in the context of the subcategory N C₂?

Answer: We have ¯δ = δ on the subcategory N C₂ ⊂ D, and if D = N C₂, the associated quantum group is U+ₙ.


Question: How do different β decay rates and spallation cross sections affect the spallation effects in the third r-process peak for the baseline hot disk wind simulations?

Answer: For the baseline hot disk wind simulations with an initial ejecta velocity of 0.3c, using β decay rates from [Möller et al.(2003)] results in a spallation effect that increases significantly with an increased spallation cross section (σTALYS vs 10 × σTALYS). Using β decay rates from [Marketin et al.(2016)] broadens the third peak and moves it towards solar, resulting in a flatter abundance shape and smaller abundance changes due to spallation (Baseline: ∼30% in average; MKT: ∼8% in average).


Question: What are the key differences between the Erdős-Rényi model and models that aim to reproduce power-law degree distributions in real-world networks?

Answer: The Erdős-Rényi model produces graphs with a Poisson degree distribution, which is highly unrealistic for real-world networks as it has a thin tail. In contrast, models like the Barabási-Albert model and the configuration model (Molloy–Reed model) aim to reproduce power-law degree distributions, which are more characteristic of real-world networks. The Barabási-Albert model uses preferential attachment to grow graphs, while the configuration model and the Chung-Lu model generate graphs with exact or expected degree distributions matching the input graph.


Question: What are the specific conditions and components of the 5D Weyl connection vector in the neutral particle model in the Einstein gauge?

Answer: In the neutral particle model in the Einstein gauge, the 5D Weyl connection vector \( 	ilde{w}_A \) has only one non-zero component, \( 	ilde{w}_4 = 	ilde{w}(l, r) \), while the other components are zero: \( 	ilde{w}_3 = 	ilde{w}_2 = 	ilde{w}_1 = 0 \).


Question: How does the accuracy of the total hydrodynamic force/torque in IBM simulations depend on the flow patterns, and what specific examples are given?

Answer: In IBM simulations, the accuracy of the total hydrodynamic force/torque depends on the specific flow patterns. In a Taylor-Couette flow, where the flow is azimuthally independent, the first-order local errors cannot be canceled out, resulting in a first-order accuracy for the total hydrodynamic force. In contrast, in cases of a uniform flow passing a fixed cylinder or sphere, symmetric flow patterns may form around the cylinder/sphere, allowing the first-order local errors to cancel out, yielding a second-order accuracy for the total force.


Question: How does the proposed IlluRes-SfSNet-Align model enable photo relighting and albedo editing, and what are the specific steps involved in these processes?

Answer: To relight a photo, the source and target images are sent into the well-trained IlluRes-SfSNet-Align, decomposed into corresponding albedo, normal, and lighting information. The lighting of the source image is replaced by the target one, and the novel combination of components is used to generate the relighted image. In albedo editing, the source photo is fed into the full model to produce the decomposition components. After modifying the albedo map, a new photo is generated. The result shows that editing on albedo can generate more realistic results than directly editing on the face photo, as demonstrated in Fig.8 where the cheek in Fig.8(e) contains obvious artifacts, while the result of our model in Fig.8(d) is much more visually pleasing.


Question: How does the magnetic field of a neutron star influence the formation of the plasma and the structure of the magnetosphere?

Answer: The strong magnetic field of a neutron star (Bsurf ∼108−1014 Gauss) induces a huge electric quadrupole field and an electromagnetic force that exceeds gravity by ten to twelve orders of magnitude. Charges are pulled out easily from the surface, resulting in a dense, magnetized plasma that surrounds the pulsar. The strong magnetic field forces the plasma to co-rotate with the pulsar like a rigid body, and the co-rotating magnetosphere extends up to the light cylinder, which separates the magnetic field lines into open and closed field lines. Closed field lines close within the light cylinder, while open field lines would close outside, allowing plasma on these lines to reach highly relativistic velocities and leave the magnetosphere, creating the observed radio beam.


Question: What is the significance of the quadratic functional in the context of the functionals considered in the paper?

Answer: The quadratic functional, corresponding to p = 2, is of particular significance because it is related to other inferential problems in density models such as goodness-of-fit testing and construction of confidence balls.


Question: How does the proof of Lemma 3.3 utilize the decoupling inequality for curves with non-vanishing curvature?

Answer: The proof of Lemma 3.3 uses the decoupling inequality for curves with non-vanishing curvature by applying it to the Fourier extension operator associated with the non-degenerate curve γ(u) = (u, 6u^2 + 4u^3 + u^4), u ∈ [0, 1]. Specifically, the inequality is used to bound the Lp norm of the Fourier extension operator EParp[0,1] ˜h over the region L(B2σ−4) by a sum of Lp norms of the decoupled pieces EParp˜I ˜h over smaller regions B2λ4σ−4. This results in the desired estimate (3.10).


Question: How does the size of the magnetic defect influence the strength of the skyrmion-defect interaction, and what are the potential well and barrier configurations observed?

Answer: The strength of the skyrmion-defect interaction is influenced by the size of the magnetic defect. For small defect areas, the interaction is extremely weak, and the potential can present a double-well potential (or double-barrier potential) instead of a single potential well (or a single potential barrier). As the defect area increases, potential wells and potential barriers arise, indicating stronger interactions.


Question: How do the TENO schemes compare to the WENO-Z scheme in resolving structures in the Rayleigh-Taylor instability problem on both coarse and fine meshes?

Answer: On the coarse mesh ([Nx × Ny] = [128 × 256]), the TENO schemes resolve more structures than the WENO-Z scheme, but TENO5 and TENO5-A produce obvious oscillations near the interface. On the fine mesh ([Nx × Ny] = [256 × 512]), small structures are also better resolved by the TENO schemes, and TENO5 and TENO5-A still produce oscillations near the interface.


Question: How is the graph G \ X constructed, and what is the relationship between its arcs and the original graph G?

Answer: The graph G \ X is constructed by keeping the same set of vertices as G and forming a new set of arcs A′, which is the i-closed subset of A consisting of arcs that do not appear in any edge e in the set X. In symbols, A′ = {a ∈ A | ∀e ∈ X, a /∈ e} ⊆ A. The involution on A′ is the restriction of that on A, and D′ = D ∩ A′.


Question: How does the presence of a boundary affect the entanglement formula in the context of a CFT2 in flat space with a large central charge c?

Answer: The presence of the boundary has the effect that the entanglement formula now counts the entanglement through the interval from both right- and left-moving degrees of freedom: S(x+, x−) = c/12 log (X(x+) − X(x−))^2 / (δx^2 X′(x+)X′(x−)).


Question: What is the onset temperature of the superconducting transition observed in the resistivity data for Sr4Cr0.8Ti1.2O6Fe2As2 under zero field?

Answer: The onset temperature of the superconducting transition observed in the resistivity data for Sr4Cr0.8Ti1.2O6Fe2As2 under zero field is 29.2 K.


Question: What is the explicit description of the stack H2,1(2/3) and how does it admit a good moduli space?

Answer: The stack H2,1(2/3) is described by a family of Weierstrass tails over A4 given by the equation y^2 = x^5z + a3x^3z^3 + a2x^2z^4 + a1xz5 + a0z6, with the Weierstrass section [1, 0, 0]. Since Gm acts on the base and total space of this family, the family descends to [A4/Gm]. The induced map [A4/Gm] → H2,1(2/3) is an isomorphism, thus H2,1(2/3) admits a good moduli space.


Question: What is the strategy for proving the fixed-parameter tractability of the permissive backdoor detection problem SATpsrbdC0q, and how does it handle the case of a formula with a k-clause?

Answer: The strategy for proving the fixed-parameter tractability of the permissive backdoor detection problem SATpsrbdC0q is based on two main observations: formulas with strong recursive backdoor depth k to C0 have a maximal clause degree k and a diameter bounded by λk in each connected component. For a formula with a k-clause, the existence of this k-clause implies that the strong recursive backdoor depth is at least k. The degree of the k-clause can only be reduced by assigning a variable in its neighborhood.


Question: What specific techniques does DrugGEN use to improve the generation of de novo molecules for a target protein, and how is its performance evaluated?

Answer: DrugGEN incorporates graph transformers into the generator and discriminator modules of the GAN to handle the complexity of target-centric generation, effectively learning long-range dependencies between atoms of small molecules. Its performance is evaluated using benchmarking metrics such as validity, uniqueness, novelty, internal diversity, and quantitative estimate of drug-likeness (QED). Additionally, target-specific properties are evaluated via in silico experiments like molecular docking and deep learning-based drug-target interaction prediction. DrugGEN outperformed other models in an overall ranking based on these metrics.


Question: How does the uniform boundedness of Ka(Xn) contribute to the proof of Theorem 2?

Answer: The uniform boundedness of Ka(Xn) allows the application of the Dominated Convergence Theorem, which is used to deduce that Φ(X) ≥ lim n→∞ ∫ [−N,N] Ka(Xn) dµ(a) = ∫ [−N,N] Ka(X) dµ(a). This step is crucial in establishing the equality Φ(X) = ∫ [−N,N] Ka(X) dµ(a) for all X ∈ LM, completing the proof of Theorem 2.


Question: What are the explicit sets and cubic forms used in the expression for Zreﬂ_c1,c2(q) according to Theorem 1.1?

Answer: For any c1, c2, there are explicit sets D1(c1, c2), D2(c1, c2) ⊂ Z4, and D3(c1, c2) ⊂ Z3 defined in Section 3.3, such that Zreﬂ_c1,c2(q) = − ∑_{v∈D1(c1,c2)} q^C1(v) + ∑_{v∈D2(c1,c2)} 6q^C2(v) + ∑_{v∈D3(c1,c2)} 4q^C3(v), where C1(v) = ∑_{1≤i<j<k≤4} v_i v_j v_k, C2(v) = (v1 + v2)v3v4, and C3(v) = v1v2v3.


Question: What is the key condition for the growth of the sample paths when f is regularly varying at zero, and how does it relate to the proof of the final main result, Theorem 2.10?

Answer: If f is regularly varying at zero, i.e., there exists β > 0 such that for all λ > 0, lim t→0 f(λt)/f(t) = λβ, then the key condition for the growth of the sample paths is that (45) for R = 1 implies (45) for all R ≥ 1. This condition is crucial for the proof of Theorem 2.10, as it ensures the desired lower bound for the growth of the sample paths.


Question: How are the dynamics of the time-1 map Fε and the return map fε,c combined to follow the prescribed invariant objects?

Answer: The dynamics of the time-1 map Fε restricted to Λε,c and the dynamics of the return map fε,c on Dε,c are combined by associating each point of an orbit f n ε,c(z) of fε,c with a point of an orbit F n1 ε(z) of Fε. This is done by finding a unique n1(z, n) such that n1(z, n) ≤ t(z, n) < n1(z, n) + 1, where t(z, n) is the time for which f n ε,c(z) = φ t(z,n) ε(z). This association allows us to construct a sequence of 2-dimensional windows in Dε,c and 3-dimensional windows in Λε,c, which are correctly aligned under some power of fε,c or Fε, or under one of the scattering maps Si ε,c. This construction ensures that there exist orbits of Fε in R6 that follow these windows, and consequently, orbits of the flow φ t ε that follow these windows.


Question: What specific actions were taken to modernize the H1 software and data analysis capabilities after 2012?

Answer: Several software migrations were performed, including the migration to 64-bit operating systems, up to Scientific Linux 6 and CentOS7. The entire software and environment were updated, and previous stable software releases are kept in a central repository. A comprehensive update of the software and analysis frameworks was performed, and the H1 software stack and environment were modernized to allow for easy transition to newer platforms like AlmaLinux 8 or CentOS8 or 9. All core software packages were successfully migrated to a modern computing platform based on amd64 (x86-64) CentOS7, using the GNU 9.2 compiler. External dependencies were updated to the latest releases and are now obtained from the LCG/AA package repository. The common object-oriented data analysis framework H1oo is now based on ROOT 6 and supports the C++17 standard. Interactive analysis in C++ through CLING and Python (v3) is now possible, and several benchmark analyses codes were migrated, such as those for jet production and inclusive DIS cross section measurements.


Question: How does the Selective Memorization (SeMem) policy determine which training cases to memorize, and how does this decision-making process evolve over time?

Answer: SeMem determines which training cases to memorize based on a memorization threshold δ. If the log probability of a case is below δ, the case is memorized; otherwise, it is skipped. This decision-making process dynamically changes over time as the model learns more, making it less likely to memorize similar cases that were previously memorized, leading to a slower growth of the non-parametric memory M.


Question: What are the key differences in the synthetic transmission spectra for carbon-rich and carbon-poor atmospheres of 55 Cnc e?

Answer: Carbon-rich atmospheres (Purple, blue) are defined in shape by HCN and CN, while C/O = 1.0 (Green) is defined by H2O. Low C/O ratio spectra (Yellow) are dominated by the absorption of H2O, and by CO2 for the lowest N/O ratios.


Question: How is the evolution of the concentration of ∇(2)A in L2 estimated in Lemma 10, and what terms are involved in the inequality?

Answer: The evolution of the concentration of ∇(2)A in L2 is estimated by the inequality: d/dt ∫_Σ |∇(2)A|^2 γs dµ + (3/2) ∫_Σ |∇(4)A|^2 γs dµ ≤ c ∫_Σ ∇(2)A ∗ (P 4_3 + P 2_5) (A) γs dµ + c ∫_Σ |A|^6 γs dµ + c ∫_Σ |∇(5)Ric|^2 γs dµ + c ∫_Σ |A|^2 γs−8 dµ.


Question: How does the behavior of the Markovian system compare to the finite-size moment equations in the region where the WC system presents periodic orbits?

Answer: In the region where the WC system presents periodic orbits, the behavior of the Markovian system reflects the properties of the finite-size moment equations, and quasicycles appear in the neighborhood of WC's Hopf bifurcation. The moment equations show that the Hopf bifurcation of the deterministic WC system expands into two branches of generic Hopf bifurcations, leading to stable and unstable cycles with periods close to or around half of the WC cycles' period.


Question: What are the conditions under which the global sensitivity of the squared error loss functions is bounded, and what is the bound?

Answer: When β has ℓ1-norm at most R and the data (Xi, Yi) are restricted to [−1, 1]d × [−r, r], the global sensitivity of the squared error loss functions −2ℓ(β; ·) and −2ℓR(M; ·) is at most (r + R)².


Question: How does the normal form of an element in the amalgamated product relate to the transversals of the subgroup H in the groups Gj?

Answer: Each element α ∈ G can be written in a unique way in the form α = θ1θ2 · · · θℓβ such that β ∈ H and, for each i ∈ {1, ..., ℓ}, there exists j = j(i) ∈ {1, ..., p} such that θi ∈ T∗j = Tj \ {1}, and j(i) ≠ j(i + 1) for all i ∈ {1, ..., ℓ−1}. In particular, α ∈ H if and only if ℓ = 0 and α = β.


Question: What is the effective Lagrangian for the axion-photon coupling, and how is the coupling constant gaγ derived?

Answer: The effective Lagrangian for the axion-photon coupling is given by Laγ = −1/4 gaγ F µν ˜Fµν a = gaγ E · B a, where a is the axion field, F µν is the electromagnetic field strength tensor, and ˜Fµν is its dual. The effective axion-photon coupling constant gaγ is given by gaγ = α / (2πfa) * [(E/N - 2) * (4 + z + w) / (3 * (1 + z + w))] = α / (2πfa) * (E/N - 1.95 ± 0.08), where fa is the Peccei-Quinn symmetry breaking scale, z = mu/md = 0.56, and w = mu/ms = 0.028 are the quark mass ratios, and E and N are the model-dependent coefficients of the electromagnetic and color anomaly of the axial current associated with the axion.


Question: What is the smallest positive integer fp(G) such that fp(G)|G|˜π0 is a generalized Brauer character of G, and how does it relate to e(G)?

Answer: The smallest positive integer fp(G) such that fp(G)|G|˜π0 is a generalized Brauer character of G divides e(G). Specifically, fp(G) divides e(G)p′, where p′ denotes the prime-to-p part of |G|.


Question: What are the specific properties that a (ρ, ϵ, δ)-AWtII code must satisfy to ensure both secrecy and robustness?

Answer: A (ρ, ϵ, δ)-AWtII code must satisfy the following properties: 1) Secrecy: For any pair of messages m0 and m1, and any subset S of size |S| ≤ nρ, the statistical distance SD(Enc(m0)|S; Enc(m1)|S) ≤ ϵ. 2) Robustness: If the adversary is passive, Dec always outputs the correct message. If the adversary is active, the probability that the decoder outputs a wrong message is bounded by δ, i.e., Pr[Dec(A(Enc(m))) /∈ {m, ⊥}] ≤ δ.


Question: What test is used to examine the weak stationarity of the log returns, and what does the null hypothesis of this test assume?

Answer: The Dickey-Fuller test is used to examine the weak stationarity of the log returns. The null hypothesis of this test assumes that the time series under study has a unit root.


Question: What normalization techniques were applied to mitigate bias in the computation of user weights based on semantic matches?

Answer: To mitigate bias, different ad hoc normalizations were applied in the computation of user weights. These included normalizing 𝑤(𝑢, 𝑣) by the number of sentences |𝑆𝑣| written by 𝑣, normalizing by the number of sentences |𝑆𝑖| written on 𝑖, and normalizing by the number of sentences |𝑆𝑢𝑖| written by 𝑢 on 𝑖 when per-item graphs were considered. Additionally, for the One-to-One matching, normalization by 𝑁(𝑢) was tried. For Many-to-One matching, each term in Equation (4) was divided by 1 or 𝑁(𝜎1). For Many-to-Many matching, the terms in the Equation’s (5) sum were divided by 1, 𝛿− = Í 𝜎∈𝑉|𝑠(𝜎, 𝜎2)|, or 𝛿+ = Í 𝜎∈𝑉|𝑠(𝜎1, 𝜎)|, which correspond to extensions of in and out-degrees adapted to the weighted graph.


Question: What are the Bethe ansatz equations in the homogeneous limit, and how do they relate to the eigenvalues of the Hamiltonian?

Answer: In the homogeneous limit, the Bethe ansatz equations become c σ2(η)σ(2µj + η)σ(2µj + 2η)σ2N(µj + η) = ∏γ=± ∏l=1^3 σ(µj + α(γ)l + η) / σ(α(γ)l) ∏l=1^(N+1) σ(µj + µl + η)σ(µj + µl + 2η), for j = 1, ..., N + 1. The eigenvalues E of the Hamiltonian are given by E = σ(η) / σ′(0) [∑j=1^(N+1) (ζ(uj) − ζ(uj + η)) + (N − 1)ζ(η) + ∑γ=± ∑l=1^3 ζ(α(γ)l)].


Question: How does the inclusion of self-shielding affect the neutral hydrogen density distribution in the IGM during reionization?

Answer: In the absence of self-shielding, the neutral hydrogen density distribution is more uniformly ionized. However, when self-shielding is included, high-density locations within ionized regions can remain neutral due to their high recombination rates, as shown in Figure 2. This results in a more complex ionization topology, with high-density regions far from sources remaining neutral longer, and reionization proceeding more deeply into underdense regions.


Question: How does the decay rate of the most subradiant eigenstate in the |Fz|max manifold scale with atom number, and what is the field intensity pattern emitted by this state?

Answer: The decay rate of the most subradiant eigenstate in the |Fz|max manifold scales as Γ/Γ0 ∼ 1/N^3. The field intensity pattern emitted by this state is largely evanescent transverse to the bulk of the chain, with most of the energy radiated out through scattering at the ends of the chain, as expected for an optical waveguide.


Question: What are the visual indicators of nucleation observed in the droplets during the cooling process?

Answer: The visual indicators of nucleation observed in the droplets during the cooling process are the formation of a dendritic ice shell across the air-liquid interface and an increase in the temperature within the droplet and at its surface.


Question: How do the BLYP energy components for Cl2 and HCl compare between CCG and ACG, and what are the specific values for ⟨T⟩, ⟨V⟩, and ⟨E⟩?

Answer: For Cl2 (R = 4.2 a.u.), the BLYP energy components are: ⟨T⟩ = 11.21504 (CCG Set A), 11.21577 (CCG Set B), 11.21570 (Ref. [77]); ⟨V⟩ = −40.98344 (CCG Set A), −40.98434 (CCG Set B), −40.98424 (Ref. [77]); ⟨E⟩ = −29.76840 (CCG Set A), −29.76857 (CCG Set B), −29.76855 (Ref. [77]). For HCl (R = 2.4 a.u.), the BLYP energy components are: ⟨T⟩ = 6.25431 (CCG Set A), 6.25464 (CCG Set B), 6.25458 (Ref. [77]); ⟨V⟩ = −21.74382 (CCG Set A), −21.74417 (CCG Set B), −21.74411 (Ref. [77]); ⟨E⟩ = −15.48951 (CCG Set A), −15.48953 (CCG Set B), −15.48953 (Ref. [77]).


Question: What are the overlaps between the PCI wavefunctions for bases A, B, and C, and what do they indicate about the intrinsic states after backbending in 48Cr?

Answer: The overlaps between the PCI wavefunctions for bases A, B, and C are very large, with |⟨ΨA(I)|ΨB(I)⟩| ≈ 1 at I > 10, and |⟨Ψg.s.+A(I)|Ψg.s.+C(I)⟩| at least 92%. This indicates that different intrinsic bases generate almost the same wavefunctions after angular momentum projection, suggesting that the intrinsic states after backbending in 48Cr are not unique and that the apparently contradictory CHFB and PSM explanations are qualitatively equivalent.


Question: What are the specific income definitions used for the child and parent generations in the US sample, and how do they compare to the Scandinavian data?

Answer: In the US sample, child generation income is defined as three-year averages of annual labor income measured at ages 35-37. Parental income is defined as the average of maternal and paternal individual income, measured as three-year averages of annual labor earnings around age 18 of the child. These definitions are chosen for easy comparisons with recent literature and are similar to those used in Scandinavian administrative data, though the PSID has constraints in estimating intergenerational mobility trends.


Question: What are the two main linear trends observed in the closure delays versus closure TEC, and what do they indicate about the data quality?

Answer: The two main linear trends observed in the closure delays versus closure TEC have slopes of 40.5 ± 0.1 ps/TECU and 63.8 ps/TECU, respectively. The slope of 63.8 ps/TECU is variable, ranging from 59.9 ps/TECU to 68.9 ps/TECU, indicating that it varies from source to source and from triangle to triangle. This variability suggests that the data quality and conditions can differ significantly across different observations.


Question: How do the current vertices and diagrams in the time-dependent Wannier basis emerge from the interband transitions in the original Bloch basis?

Answer: The current vertices and diagrams in the time-dependent Wannier basis emerge from neglecting the internal dynamics of the propagator 1/(ω −ϵCBk −∆) for the virtual photoexcited electron in the inert conduction band (CB), which is valid if the detuning to interband transitions is large. In this case, the propagator for inert bands can be contracted, and all virtual interband contributions can be resummed to yield the effective two-particle current vertex that depends solely on the quantum geometry of the partially-filled bands near the Fermi energy.


Question: How is the Kloosterman sum S(m, n; c) defined in the context of Petersson’s trace formula, and what is its role in the trace formula presented in Proposition 5.2?

Answer: The Kloosterman sum S(m, n; c) is defined as S(m, n; c) := ∑_{v∈(Z/cZ)∗} e^{2iπ(mv+nv)/c}, where v is the inverse of v in (Z/cZ)∗. In Proposition 5.2, the Kloosterman sum appears in the trace formula as part of the sum over c, where it is multiplied by c^{-1} and the Bessel function J1(4π√mn/c). Specifically, the formula includes the term -2π ∑_{c>0, N|c} c^{-1} S(m, n; c) J1(4π√mn/c).


Question: What is the upper bound for the integral involving the Hardy potential and the test function βR, and how is it derived?

Answer: The upper bound for the integral involving the Hardy potential and the test function βR is given by CR^(N+µ-1)p-N-1-µ-τ/p-1 (ln R)^p/p-1. This is derived by combining the estimates from (2.24), (2.25), and (2.26), which show that the integral is bounded by CR^-2p/p-1 (ln R)^p/p-1 times the integral of |x|^(p-1-(τ+µ)+µp/p-1) over the region R < |x| < √2R, xN > 0. This integral is further bounded by CR^(N+µ-1)p-N-1-µ-τ/p-1 (ln R)^p/p-1.


Question: How does the addition of pendant vertices to the graph G affect its essential spectrum, and what are the zero eigenstates of the modified graph G′?

Answer: The addition of pendant vertices to the graph G to form G′ results in the essential spectrum of LG′ being a superset of the essential spectrum of LG, specifically σess(LG) = [−1, −1/3] ∪ [1/3, 1] ⊂ σess(LG′). Additionally, LG′ has zero eigenvalues with infinite multiplicity, with zero eigenstates Ψ(n) defined as Ψ(n)(x, s) = 1/√2 for (x, s) = (n, 1), −1/√2 for (x, s) = (n, 2), and 0 otherwise.


Question: What was the functional unit selected for the environmental and energy assessment of the two cropping alternatives in the marginal area of the farm?

Answer: The functional unit selected for the assessment is 1 ha of each of the two alternatives tested in the marginal area of the farm.


Question: What conditions must be satisfied for the Sturm-Liouville equation to have a fundamental system of solutions with specific properties as described in Lemma 2.1?

Answer: The conditions are that the integrals of q(t) from negative infinity to x and from x to positive infinity must both be greater than zero for all x in R. Specifically, ∫_{-∞}^{x} q(t) dt > 0 and ∫_{x}^{∞} q(t) dt > 0 for all x in R.


Question: What are the specific curvature invariants E and F for de Sitter space, and how do they relate to the Hubble parameter H?

Answer: The curvature invariants E and F for de Sitter space are given by E = 24H^4 and F = 0, where H is the constant Hubble parameter.


Question: What properties must a functional ρ satisfy to be considered a maturity-independent convex risk measure according to Definition 3.1?

Answer: A functional ρ : L → R is called a maturity-independent convex risk measure if it satisfies the following properties for all f, g ∈ L, and λ ∈ [0, 1]: (1) ρ(f) ≤ 0, ∀f ≥ 0 (anti-positivity), (2) ρ(λf + (1 − λ)g) ≤ λρ(f) + (1 − λ)ρ(g) (convexity), (3) ρ(f − m) = ρ(f) + m, ∀m ∈ R (cash-translativity), (4) for all t ≥ 0, and π ∈ Abd, ρ(f + ∫₀ᵗ πₛ dSₛ) = ρ(f) (replication and maturity independence).


Question: What conditions must be met for a vertex v with four neighbors to be admissible in a (2, 2, 0)-gain-tight gain graph?

Answer: v is admissible if and only if the following conditions hold: (i) v is not contained in a balanced subgraph isomorphic to K4, and (ii) v is not contained in a balanced subgraph isomorphic to K1,1,3 with the property that v has degree 4 in this subgraph.


Question: What theorem and proof demonstrate that the standard RPF natural evaluation function is defined for all Dyck words in D?

Answer: Theorem 2.3 states that the domain of the standard RPF natural evaluation function αNr is D. The proof shows that for any d ∈ D, αNr(d) is defined: for d = ǫ, αNr(d) is 0; for d ≠ ǫ, αNr(d) is recursively defined as the product of p_i^αNr(ei) for all chunks ei in d, and it is shown that this recursion always terminates due to Lemma 2.2 and the finite nature of recursion chains.


Question: What are the mechanisms of heavy quark production and their behavior in the QGP, and how do these mechanisms affect the measurement of J/ψ suppression?

Answer: In nucleus-nucleus collisions, the dominant production mechanism for heavy quarks is gluon-gluon fusion in the initial hard scattering of nucleons. Thermal production of heavy-quark pairs in the medium is negligible after the initial stages of the collision, and the number of heavy quarks is essentially 'frozen.' The presence of a QGP leads to the dissociation and suppression of heavy quarkonia through Debye screening of color charges. PHENIX measurements of RAA for J/ψ show less suppression than expected based on SPS data, which may be due to the regeneration of J/ψ through the recombination of dissociated charm (anti)quarks into new c¯c pairs.


Question: What are the specific performance improvements of SimMTM over other pre-training methods in the in-domain forecasting task?

Answer: On the average of all benchmarks, SimMTM achieves 8.3% MSE reduction and 4.3% MAE reduction compared to the advanced masked modeling baseline Ti-MAE [21], and 14.7% MSE reduction and 12.0% MAE reduction compared to the contrastive baseline CoST [46].


Question: What are the specific results that demonstrate the better fit of the skew-symmetric links compared to the logit and probit links in the Bliss (1935) data set?

Answer: The Bayes factors of the different links against the skew–logistic link, AIC, and BIC values, shown in Table 6, favour the use of an asymmetric model and slightly favour the skew–logistic link over the other competitors. Table 5 shows the predicted observations with these models obtained by multiplying the number of subjects nj by the predicted probability at the corresponding dose level, suggesting a better fit of the asymmetric models.


Question: What does Theorem 3.4 state about the convergence of optimal controls for the adapted problem (ePα) to the optimal control of the original problem (P0)?

Answer: Theorem 3.4 states that for any sequence {αn} ⊂ (0, 1] such that αn ց 0 as n → ∞, there exists a subsequence {αnk} of {αn} and, for every k ∈ IN, an optimal control uαnk ∈ Uad of the adapted problem (ePαnk) with associated state (µαnk, ραnk) such that, as k → ∞, uαnk → ¯u strongly in L2(Q), and eJ((µαnk, ραnk), uαnk) → J((¯µ, ¯ρ), ¯u).


Question: What are the self-similar solutions for the surface density, angular velocity, radial velocity, and pressure in the context of advection-dominated discs?

Answer: The self-similar solutions for the surface density, angular velocity, radial velocity, and pressure are given by: Σ(r) = ω0Σ0( r/r0 )^(s-1)/2, Ω(r) = ω1 (GM*/r0^3)^(1/2) (r/r0)^(-3/2), Vr(r) = -ω2 (GM*/r0)^(1/2) (r/r0)^(-1/2), and P(r) = ω3Σ0 (GM*/r0) (r/r0)^(s-3)/2, respectively.


Question: What does the phonon band structure in Fig. 8 reveal about the {10¯12} twinning mode under homogeneous shears?

Answer: The phonon band structures at the homogeneous shears corresponding to the {10¯12} twinning mode show the phonon frequencies at different wave vectors. The structures are drawn for shears s/st = 0, 0.2, 0.4, 0.6, 0.8, and 1, with phonon frequencies below zero indicating imaginary frequencies. The filled circles at the M′ points indicate the soft phonon modes that have the eigenvector shown in Fig. 5 (b).


Question: What does the figure showing the running time of random scan divided by EM-complexity as a function of the logarithm of the problem size indicate about the relationship between the measured running times and the EM model predictions?

Answer: The figure shows that the running time of random scan divided by the EM-complexity is not a constant function, indicating that the measured running times do not align with the predictions of the EM model.


Question: What specific Raman wavenumbers were used to identify the pigments in the study?

Answer: Raman wavenumbers from the present work and from the consulted Raman libraries are reported in Table TS1.


Question: How does the rod-climbing behavior of the 0.3 wt. % PIB Boger fluid compare to the 0.3 wt. % PIB solution and the 3.0 wt. % PIB solution, and what does this indicate about the influence of fluid elasticity?

Answer: The 0.3 wt. % PIB Boger fluid exhibits pronounced rod-climbing due to the dominant effect of elasticity overwhelming inertial effects. In contrast, the weakly elastic 0.3 wt. % PIB solution undergoes rod-descending, and the 3.0 wt. % PIB solution, which is more concentrated, also exhibits rod-climbing. This indicates that increasing the fluid elasticity, as achieved by increasing the solvent viscosity, can dramatically enhance rod-climbing behavior.


Question: What is the relationship between the length of plaque-chains and the domains of the holonomy maps in the context of the induced pseudogroup G*F(W)?

Answer: The length of plaque-chains required to cover X by images of W is bounded by αW, but there is no control over the domains of the holonomy maps defined by the overlaps between the plaques in these chains. By allowing plaque-chains of length βW, which is defined as 2αW δF_U / ϵF_U, uniform estimates on the domains of the holonomy maps can be obtained. Specifically, for any hI|Uh ∈ G*F(W) where I has length at most αW, and for any w ∈ Uh ⊂ W, there exists an admissible sequence I′ with length at most βW such that V ⊂ Dom(hI′) and [h]w = [hI′]w, where V is an open subset of W with diamX(V) ≤ δW.


Question: What is the growth estimate for the Dirichlet integral D(ρ) for ρ ≤ R/2, and how is the constant C defined in this context?

Answer: For ρ ≤ R/2, the growth estimate for the Dirichlet integral D(ρ) is given by D(ρ) ≤ C (ρ/R)^(1/δ), where δ = K(1 + α)/(α + K(1 + α - 2α^2)) and C(K, K′, R, α) is defined as C = 2^(1/δ) (k1 + k2R + m1/(2(δ - 1)R) + m2/(4(2δ - 1)R^2)), with m1 = 18πK^2M^2/ν̄ and m2 = 4πK′[1/3 + 6δν̄/(2K(1/α + 1) - 1)].


Question: How does the use of CNOT gates help in compensating for limited qubit connectivity in quantum devices, and what are the challenges associated with this approach?

Answer: CNOT gates are used to compensate for limited qubit connectivity by allowing a control operation between distant qubits that do not share a direct connection. This is achieved by using a series of CNOT gates through intermediate qubits, known as ancilla qubits. However, the presence of ancilla qubits can introduce new sources of error, and their repeated use in algorithms can compound these errors. Additionally, proper handling of ancilla qubits is crucial, especially when they need to be reset for future use.


Question: How does the typical size of newly formed carbon grains vary with the gas density and onset time of dust formation in Type II–P and Type IIb supernovae?

Answer: For Type II–P SNe, with a gas density of ˜con ≃108–109 cm−3 at ton = 330 days and γ ≃1.5–1.7, the typical grain size is aave,∞=0.03–0.3 µm. For Type IIb SNe, with a gas density 100–300 times lower at the same onset time, the typical grain size is aave,∞∼0.001 µm.


Question: How does the local-equilibrium description of nanoparticles and solvent in a nonisothermal system differ from the global isothermal equilibrium?

Answer: In a nonisothermal system, nanoparticles and solvent can still be described by a local-equilibrium where a (molecular) temperature field T(r, t) can be defined, representing the local molecular temperature at position r and time t almost down to the atomic scale. However, important hydrodynamic degrees of freedom of the system are not locally thermalized at T(r, t) unless this field is everywhere equal to the constant ambient temperature T0, which is the case in global isothermal equilibrium.


Question: What is the Moreau envelope and how does it relate to the proximal mapping in the context of the proximal gradient method?

Answer: The Moreau envelope is a well-defined real-valued function, locally Lipschitz continuous, that lower bounds ψ and shares with ψ infima and minimizers. It is defined as ψV(x) ≔ min { ψ(w) + 1/2∥w − x∥²_V }, where w ∈ ℝ^r. The proximal mapping is available in closed form for many useful functions, including widely used regularizers in machine learning, such as the ℓ0 and ℓ1 regularizers, which correspond to hard and soft thresholding operators, respectively.


Question: How does the average path length ˜s scale with l⋆/R, and what is the exponent of this scaling relationship?

Answer: The average path length ˜s scales as ˜s ∝ 1/l⋆, with an exponent m = −0.975 ± 0.05, as indicated by a power-law fit to the data.


Question: What are the specific results for the B and Bs decay constants and the b-quark mass in the lattice QCD computation using Nf = 2 twisted mass Wilson fermions?

Answer: The results are mb(mb, MS) = 4.29(12) GeV, fBs = 228(8) MeV, and fB = 189(8) MeV.


Question: What clustering technique was found to be most appropriate for UK domestic electricity data, and how does it differ from the approach used in Portuguese data?

Answer: The preferred technique in the Portuguese work, a two-stage process combining Self Organised Maps and Kmeans, was found to be not appropriate for the UK data. The work shows that up to nine clusters of households can be identified with the differences in usage profiles being visually striking, demonstrating the appropriateness of breaking down electricity usage patterns into more detail than the two load profiles currently published by the electricity industry.


Question: How does the frame formulation of gravity in three dimensions facilitate the introduction of interactions in the Chern-Simons formulation?

Answer: In three dimensions, the frame formulation of gravity helps make contact with the gauge formulation of gravity as a Chern–Simons gauge theory, where interactions are rather easy to introduce by means of finite- or infinite-dimensional higher-spin algebras.


Question: What is the computational complexity of the algorithms used for cycle decomposition, and how do they achieve this complexity?

Answer: The computational complexity of the algorithms used for cycle decomposition, including Algorithm 2 and Algorithm 3, is O(n). This is achieved by following the given cycle and checking the optimal decomposition conditions at each vertex, which requires constant computational time. Since each vertex in the cycle is visited exactly once, the overall complexity of the methods equals O(n).


Question: What is the estimated inner radius of the accretion disk in 4U 1705-44, and how does it compare to the neutron star's radius?

Answer: The inner radius of the accretion disk in 4U 1705-44 is estimated to be 10.5+1.0−1.7 Rg, which corresponds to 15.6+1.5−2.5(M/ M⊙) km. This upper limit on the radius of the neutron star is in agreement with previous studies, and there should be a gap of at least ≈3.5 km between the surface of the neutron star and the edge of the inner accretion disk if the mass range is restricted to > 1.4 M⊙.


Question: What is the relationship between the Gauss map ν and the slant angle α in the context of Theorem 2.5?

Answer: The Gauss map ν : N → G(2, 4) ∼= S2+ × S2−, and its projection ν− : N → S2−, maps the slant surface N to a small circle S1α of S2−, where the slant angle α determines the specific small circle. The image ν−(N) lies in this small circle, and the relationship is further characterized by the formula (ν−)∗ω = 1/2Θ, where ω is the arclength element of S1α and Θ is a form related to the slant angle α.


Question: How does the embedding tensor ΘA_MN introduce minimal coupling in the context of six-dimensional N = (2, 2) supergravity, and what is the form of the covariant derivative Dµ in this case?

Answer: The embedding tensor ΘA_MN introduces minimal coupling of various fields via the covariant derivative Dµ = ∂µ − g Aµ ΘA_MN tMN, where g is a gauge coupling constant. The embedding tensor identifies generators XA = ΘA_MN tMN of the gauge group G0 ⊂ SO(5, 5) with particular linear combinations of the SO(5, 5) generators tMN.


Question: What specific observations and data analysis were conducted to study the chromospheric evaporation during the X1 flare on 2014 March 29?

Answer: RHESSI imaging and spectroscopy were used to determine the timing, location, and amount of electron-beam deposited energy. IRIS observations of the Fe XXI λ1354.1 line provided the location, speed, and intensity of evaporating material at a temperature of ∼10 MK. The IRIS data were obtained during a coordinated flare observing campaign from 14:09-17:54 UT, with 180 8-step rasters performed at a cadence of 75 seconds. Each raster had a field-of-view of 14′′ x 174′′, and the Fe XXI spectral window was not overexposed for the entire duration of the observations.


Question: How does the singularity of the Wilson-Fisher fixed point in the complex field plane relate to the convergence of the various expansions, and what form does this singularity take for finite N?

Answer: The convergence of the various expansions is controlled by singularities of the Wilson-Fisher fixed point in the complex field plane, related to poles in the quartic (infinite N) or sextic (finite N) scalar self-interaction. On the level of the RG flow for the field-dependent mass, the singularity relates to a non-analytical dependence on the complexified field of the form \( u'(ρ) ∝ (ρ - ρ_s)^{-n} \) with \( n = 3 \) for finite N, showing that certain n-point functions at the fixed point at vanishing external momenta display singularities for specific points in the complex field plane (outside the physical domain).


Question: What is an example of a Dirichlet form that does not admit energy measures, and how is this demonstrated?

Answer: Consider the classical Dirichlet integral on the unit interval [0, 1], given by E0(g) := ∫_0^1 g′(x)² dx for any function g from F0 := {g ∈ C([0, 1]) : E(g) < ∞}. The form (E0, F0) is a resistance form on [0, 1]. For a function f ∈ F0|X with E(f) > 0 and g ∈ F0 such that f = g|X, we have 2E(ϕf, f) − E(ϕ, f²) = 2 ∫_0^1 ψ(x)g′(x)² dx for all ϕ ∈ F0|X with ϕ = ψ|X, ψ ∈ F0. Approximation by piecewise linear functions shows that F0 is dense in C([0, 1]), and any bounded Borel function on [0, 1] can be approximated pointwise by a uniformly bounded sequence of functions from F0. If (E, F) would admit energy measures as in (11), then for a uniformly bounded sequence (ψn)n ⊂ F0 approximating 1X pointwise, we would obtain ∫_X ψn|X dΓ(f) = ∫_∆ (ψn|X)∧ dˆΓ(f) = ∫_0^1 ψn(x)g′(x)² dx, which leads to a contradiction, showing that (E, F) does not admit energy measures.


Question: How does the paper address the challenge of ensuring the feasibility of QPs in the context of high relative degree constraints?

Answer: The paper proposes a sampling-based learning approach to learn a new feasibility constraint for CBFs, which is then enforced by another HOCBF added to the QPs. The accuracy of the learned feasibility constraint is recursively improved by a recurrent training algorithm.


Question: What are the specific components of the full spatial DLM model and how are the hyperparameters defined?

Answer: The full spatial DLM model is defined as Xti = Ftiθti + νti, where νti ∼ N(0, diag{V 1, ..., V nz}), and θ3,ti = θ3,ti−1 + ktiωti, with ωti ∼ N(0, diag{W 1, ..., W nz} + K3). The hyperparameters η3 = (σ3, φ3)T govern f3(·), while η1 = (σ1, φ1)T and η2 = (σ2, φ2)T govern f1(·) and f2(·) respectively.


Question: How does the event-driven ECMC algorithm handle rejection distances for multiple interaction energies?

Answer: The rejection distances for all possible interaction energies are determined, and the shortest distance triggers the next lifting event. This is analogous to the first-reaction method in a Gillespie algorithm.