Question: What is the estimated accretion rate required for low-mass stars to accrete their own mass in the early disc accretion scenario, and how does this relate to the assumed disc lifetime?

Answer: The accretion rate averaged over 20 Myr has to be about 10−8 M⊙/yr. This is based on the assumption that low-mass stars, such as a 0.25 M⊙ star, need to accrete about 0.25 M⊙ of enriched material to account for the observed abundances in the enriched population. The assumed disc lifetime in this scenario is up to 20 Myr, which is longer than the currently observed disc lifetimes of 5-15 Myr.


Question: What are the B(M1) and B(E2) values for the transitions in the 19/2- to 33/2- band of Ag-105, and how do they compare to the B(M1)/B(E2) ratios?

Answer: The B(M1) values for the transitions from 515.6 keV to 697.0 keV are 1.02(+30-20), 0.986(+10-20), and 0.51(6) μN2, respectively. The B(E2) values for the transitions from 515.6 keV to 697.0 keV are 0.09(+2-3), 0.12(1), and 0.05(5) (eb)2, respectively. The B(M1)/B(E2) ratios for levels of spin 29/2ħ to 33/2ħ are deduced using these B(M1) and B(E2) values.


Question: What are the specific GI and GPL values for Magnus Carlsen and how do they compare to other top grandmasters?

Answer: Magnus Carlsen has a GI of 157, a GIW of 157, a GIB of 157, a GPL of 0.66, a GPLW of 0.70, and a GPLB of 0.63. He has played 1,187 games with 57,422 moves. These values are comparable to other top grandmasters like Fischer and Kasparov, but Carlsen's lower GPL suggests he makes fewer mistakes on average.


Question: What are the specific types of merging phenomena described in the candidate text, and how do they differ from branching phenomena?

Answer: There are two kinds of merging phenomena: (1) Two left contacts merge into a genuine shock (L+L→G), and (2) A right contact merged with a double-sided contact produces a left contact (R+D→L). Merging is not the opposite of branching; in merging, two shocks of different monotonicity produce a single smaller shock, while in branching, two shocks of the same monotonicity are produced. Additionally, the shock curves are not smooth after a merging process.


Question: What does the detailed comparison of STM topography and dI/dV images reveal about the surface defects and inhomogeneities in relation to the observed skyrmion bubble sizes and pinning behaviors?

Answer: The detailed comparison of STM topography and dI/dV images shows that there are no surface defects or inhomogeneities at length scales that could account for the range in skyrmion bubble sizes or variations in pinning.


Question: What are the complexities of solving the eDLP in finite abelian p-groups, and how do they vary with different values of m and p?

Answer: The eDLP in a finite abelian p-group G can be solved using O(log(e + 1) / (log log(e + 2) log |G| + logp |G|) * (m / pm/2)) group operations on a classical computer. When m is polynomial in log log |G| and p is a fixed small prime, the eDLP has complexity polynomial in log |G|. For p = 2 and n = logp |G|, the counts are dominated by the first term of the formula when m = 1, 2, 4, 8, explaining the initial cost decrease with increasing m for a fixed n. Using Shank’s algorithm, the factor n/m can be improved by √(n/m), though this is not relevant for applications requiring n/m close to 1.


Question: What are the multi-wavelength photometric properties and derived quantities for the 450 µm sources with robust counterparts and photometric redshifts?

Answer: The multi-wavelength photometric properties and derived quantities for the 450 µm sources with robust counterparts and photometric redshifts are detailed in Table A3. For example, source c01 has S250 = 42.3 mJy, S350 = 38.5 mJy, S450 = 24.5 mJy, S850 = 3.2 mJy, LIR = 12.8 ± 0.2 L⊙, TK = 40.6+3.7 K, βD = -3.2, M∗ = 2.51+0.2 M⊙, and Re = 11.3 kpc. Similar data is provided for other sources in the table.


Question: What is the Hilbert series of the standard graded k-algebra Fm(q) when q2 = Jq and J is generated by a superficial regular sequence for q?

Answer: PFm(q)(z) = 1 / (1 − z)^r * Σ_{i≥0} dim_k(qi / Jqi−1 + mqi) * z^i


Question: How does the t-layer construction handle the sequence {5(3), p(4), 5(5)} for p = 6 and p = 7?

Answer: For p = 6, a ∇-triangle is attached to the remaining free vertex, and for p = 7, a ∆-triangle is attached to the remaining two free vertices. The sequence {5(3), p(4), 5(5)} can arise only from the sequence {p, 5(4), p} in the layer X−k−1 shown in Figure 2.22-(b). After Step 1, there are two remaining free vertices of the p(4)-gon, and we place a ∇-triangle covering them as shown in Figure 2.24.


Question: How does the pose-aware equivariant point convolution strategy ensure part-level SE(3) equivariance in the feature extraction process?

Answer: The pose-aware equivariant point convolution strategy ensures part-level SE(3) equivariance by aligning point poses to the pose of the group center. This alignment cancels out the influence of varying articulation states on the geometric description of each point. If a point comes from the same part as the group center, information is aggregated as in a normal convolution. If a point comes from a different part, pose alignment canonicalizes the articulation state, ensuring the convolution outcome remains the same regardless of the articulation state change. This strategy allows aggregating context information from different parts while keeping the feature invariant to the transformation of other parts.


Question: How is the word-student matrix V factorized in the context of MOOC forums, and what do the resulting matrices represent?

Answer: In the context of MOOC forums, the word-student matrix V is factorized into two non-negative matrices: the word-topic matrix W and the topic-student matrix H. Each column vector of W can be regarded as a topic, and the memberships of students in these topics are encoded by H.


Question: What is the relationship between the zonal damping rate and the perpendicular wave number kxρe, and how does it vary with collisionality?

Answer: The zonal damping rate normalised to collisionality, γZ/ν, is proportional to kx^2 in the long-wavelength limit kxρe ≪ 1. This relationship is derived analytically and is shown in Figure 4, where the solid line represents the ∝kx^2 scaling. The damping rate is proportional to νei (electron-ion collision frequency) and is due to the Ohmic resistive damping of electron currents, both parallel and perpendicular, arising from electron-ion friction.


Question: What specific human rights are mentioned as providing implicit legal protection against neurotechnology misuse?

Answer: The right to privacy, the right to freedom of thought, and the right to freedom of expression are mentioned as providing implicit legal protection against neurotechnology misuse.


Question: What is the observed behavior of the He I λ5876 and He I λ7065 lines in the spectra of AT 2016jbu, and how does this relate to the interaction with the circumstellar medium (CSM)?

Answer: The He I λ5876 and He I λ7065 lines in the spectra of AT 2016jbu show a clear P Cygni profile with an absorption trough at ∼−500 km s−1, similar to Hα. After the seasonal gap, He I λ5876 is not clearly seen, but at +345 d, a Gaussian emission profile centered at 5897 Å with a FWHM ∼1800 km s−1 is measured, likely dominated by Na I D with minor contamination from He I λ5876. The pEW of the He I lines remains roughly constant for the first ∼120 days, then increases dramatically after the seasonal gap until ∼+300 d, after which it declines. This behavior is consistent with the interaction of the shock front with clumpy dense material far from the progenitor site, reflecting a stratified CSM profile possibly produced by historic eruptions or a variable wind.


Question: What is the charge exchange cross-section between a neutral planetary hydrogen atom and a stellar wind proton, and what does this process produce?

Answer: The charge exchange cross-section is 2×10−19 m2, and this process produces an energetic neutral atom (ENA) and an initially cold planetary ion.


Question: What is the form of the intensity matrix Λ for a Markov jump process with transient and absorbing states, and how is the vector t defined in this context?

Answer: The intensity matrix Λ for a Markov jump process with transient and absorbing states has the form Λ = (T t; 0 0), where T is a sub-intensity matrix of dimension p × p, consisting of jump rates between the transient states, and t is a column vector whose elements are the intensities of jumping to the absorbing state. The vector t is defined as t = −T e, where e is the p-dimensional column vector of 1’s.


Question: What is the proof that the set Ts,a has only finitely many non-loop edges, and what contradiction arises if this is not the case?

Answer: The proof of Lemma 5.7 shows that if Ts,a had infinitely many non-loop edges, there would exist vertices v and w with infinitely many edges connecting them. By choosing a small circle C centered at v, each edge connecting v and w would intersect C at some point zk. Assuming there are infinitely many quasi-fixed edges and no fixed points for Ra on C, the hyperbolic metrics dUs,n at zk can be arbitrarily large, which contradicts the quasi post-critical finiteness of fn.


Question: What are the specific properties of A2TiX6 compounds that make them suitable for photovoltaic applications, and how do they compare to perovskite materials discussed in the introduction?

Answer: The A2TiX6 compounds, specifically CTB and NHTB, have direct band gaps of 1.77 eV and 1.59 eV, respectively, making them potential candidates for multi-junction concentrator solar cells. They possess large absorption coefficients in the order of 10^5 cm-1 in the visible region, suitable band gaps, and relatively high optical absorption. These properties are comparable to the high absorption coefficient and band gap tunability of perovskite materials, which are also key factors in their suitability for photovoltaic applications.


Question: What condition ensures that the asset price process S is a martingale for negatively correlated models (ρ < 0)?

Answer: For negatively correlated models (ρ < 0), the asset price process S is a martingale under the additional condition that for every a ∈ R, the function σ is bounded on (−∞, a]. This condition is denoted as Condition (G).


Question: How do the bolometric light curves and photospheric velocities of the binary-merger model M15-7b-3 compare to the observed data for SN 1987A?

Answer: The binary-merger model M15-7b-3 matches the overall observed bolometric light curve much better than any of the other models. The narrow initial luminosity peak during the first ∼7 days results from a relatively small radius of the corresponding pre-SN model of 37.0 R⊙ and a suitable structure of its outer layers. A combination of the maximum velocity of the bulk mass of 56Ni around 3000 km s−1 and the mass of hydrogen-rich matter of 3.10 M⊙ mixed inward to a velocity of ≲2000 km s−1 produces the smooth and monotonic rising phase.


Question: What are the specific numerical results presented in Section V that demonstrate the efficacy of the model for Si–9 %As and Ni–5 %Cu alloys?

Answer: Numerical results in Section V provide insights on tailoring a desired level for solute trapping and drag. The model parameters can be tuned from available experimental datasets, atomistic simulations, or sharp-interface theories to capture the correct level of partitioning. The model is applied to Si–9 %As and Ni–5 %Cu, and the efficacy of the model is demonstrated by numerical studies on the partition coefficient and interface temperature. A comparative study with the previously developed phase field model suggests that the interface width can be enlarged by a factor of three to twenty-five times in the velocity regime of 0.01 m s−1 to 1 m s−1. For a finite-difference algorithm on a uniform grid spacing, this upscaling expedites the computation by a factor W (d+2), where d is the dimension. Preliminary results on an oscillatory instability leading to the formation of solute bands are also presented.


Question: What are the experimental results when the controllers use accurate and inaccurate system parameters, and how do they differ in terms of pitch and yaw angle errors?

Answer: In the experiments without uncertainty, the yaw angle converges to the desired value, and the pitch angle converges to an arbitrary value with an error around 18 deg when tL = 0. When a positive internal force is applied, the pitch angle error is reduced to about 2.5 deg. In the experiments with 10% uncertainty, the yaw angle still converges to the desired value, but the pitch angle increases for tL = 0 as the load becomes more vertical. For tL > 0, the pitch angle approaches the desired value.


Question: How does the decoupled ANM formulation in (28) differ from the vectorized ANM approach in terms of computational complexity and matrix size?

Answer: The decoupled ANM formulation in (28) results in a smaller SDP matrix size of (N + M) × (N + M), which is significantly smaller than the (NM + 1) × (NM + 1) matrix size in the vectorized ANM approach. This reduction in matrix size drastically reduces the computational complexity and run time, making it more efficient for large-scale antenna systems.


Question: What is the significance of the localization principle in the context of the proof of Theorem 1.1, and how is it applied?

Answer: The localization principle is significant in the proof of Theorem 1.1 because it implies that the existence of a square summable λ-eigenfunction leads to the existence of a finitely supported λ-eigenfunction. This implication is proven using the equivalence between conditions (4) and (5) in Theorem 6.6, which is established with the help of the integrated density of states of ∆ and a Følner sequence. The idea for this proof is based on earlier work by Delyon and Souillard, Bellissard, and others, and it is a well-known fact studied and applied in various contexts, particularly in the presence of large symmetry groups.


Question: What are the two measures used to compute the effective number of parameters in WAIC, and how do they differ?

Answer: The two measures used to compute the effective number of parameters in WAIC are pWAIC 1 and pWAIC 2. pWAIC 1 is computed as 2 times the sum over all data points of the difference between the log of the posterior mean of the predictive density and the posterior mean of the log predictive density. pWAIC 2 is computed as the sum over all data points of the posterior variance of the log predictive density. pWAIC 2 is more stable because it computes the variance separately for each data point and then sums, while pWAIC 1 uses a difference of averages.


Question: What are the potential biases in the observed SB profiles of debris disc halos, and how do they affect the estimated radial slopes?

Answer: The observed SB profiles of debris disc halos can be biased due to the presence of unbound grains and the use of size-dependent scattering phase functions (SPFs). Unbound grains, which account for more than 10% of the scattered-light luminosity, can significantly flatten both the τ and SB profiles, especially in collisionally active discs. Additionally, the size-dependent SPF, which peaks for small grains, further flattens the SB slopes to around -3 instead of the standard -3.5. These effects can lead to underestimation of the σ or n slopes by up to 0.5 in GRaTeR-type fits. For edge-on discs, the non-resolution of the disc in the vertical direction can also flatten the SB radial profile by up to +1, leading to an index of -2.5 instead of -3.5.


Question: How is the photon direction generated in the Monte Carlo simulation to ensure efficient sampling of the cross section?

Answer: The photon direction is generated using importance sampling, where more events are generated where the cross section is higher. This is done by using a suitable approximation of the Bethe-Heitler part of the cross section, which is the sum of individual cross sections for radiation off the incoming or outgoing electron, neglecting the interference. The cumulative distribution function is used to transform a uniformly distributed random number to the angle θe,γ with the correct distribution.


Question: What condition must be satisfied for the operator (T − βI)Rβ to be an isometric isomorphism from Hβ to Hβ, and how does this relate to the symmetry of T?

Answer: (T − βI)Rβ is an isometric isomorphism from Hβ to Hβ if and only if the operator T is symmetric on D. Specifically, if Kβ(β) and Kβ(β) are invertible, this equivalence also holds.


Question: How does the MWU map in the atomic congestion game reduce to the MWU map of the non-atomic game, and what is the significance of this reduction?

Answer: The MWU map in the atomic congestion game can be reduced to the MWU map of the non-atomic game by substituting the values c'(i) = α(N-1)x into the update rule, which is equivalent to the non-atomic model map (30) with ai = (N-1)α ln(1/(1-ϵ)). This reduction is significant because it allows us to apply the results and analysis from the non-atomic setting to the atomic setting, showing that the system can exhibit periodic orbits, positive topological entropy, and Li-Yorke chaos when the number of agents N exceeds a certain threshold N0.


Question: How does the presence or absence of a light ring in Universe I and Universe II affect the behavior of light rays near the wormhole throat?

Answer: Universe I possesses a light ring located at rp = −2m, while Universe II has no light ring and only features the wormhole throat at its inner boundary. This means that light rays from Universe II can be reflected by the throat if their turning point is beyond the throat in Universe I, where br ≥ b > bc. In contrast, light rays with a turning point still in Universe II are only bent by the wormhole.


Question: What is the corrected asymptotic formula for the imaginary part of the resonances Ek(h) when r0(x) = 0 and r1(x) is real on R?

Answer: The imaginary part of the resonances Ek(h) is given by Im Ek(h) = −π²r1(0)² / (A'(0) * (τ₁ + τ₂) * τ₃¹/³) * (Ai'(-τ₃⁻²/³ * λk(h)))² * h⁷/³ + O(h⁸/³), where τ₃⁻¹ = τ₁⁻¹ + τ₂⁻¹, uniformly as h → 0+.


Question: What are the key observational findings for the brown dwarf ρ−Oph 102, and how do they compare to the findings for ISO-Oph 200 in terms of disk properties and dust emission?

Answer: ρ−Oph 102 is surrounded by a disk of dust, has a significant mass accretion rate, and drives a wind and molecular outflow. ALMA observations at 0.89 mm and 3.2 mm detected molecular CO gas, with a lower limit of ≈15 AU for the outer radius of the disk. The continuum dust emission was clearly detected at both ALMA bands, with a spectral index of α0.89−3.2mm = 2.29 ± 0.16, indicating dust growth to sizes of at least ∼1 mm. In comparison, ISO-Oph 200 shows a smaller deconvolved size in Band 7 (∼8.6×8.6 au) and a higher spectral index (α = 3.2±0.3), suggesting different dust properties and disk structures.


Question: What is the relationship between the residues of a Riemann-Hilbert gauge and the parameters ci0, ..., cid1 in the context of logarithmic connections?

Answer: The residues Aij leave the parameters ci0, ..., cid1 unchanged, and these parameters are only transformed under the action of the residual subgroup DN. Additionally, for a Riemann-Hilbert gauge, the residues A1, ..., An−1 over z1, ..., zn−1 ∈ C ⊂ CP1 satisfy the condition ∑_{i=1}^{n-1} Ai = - (Wn + N').


Question: What is the spectral process Z of a multivariate max-stable Smith process and how is it defined?

Answer: The spectral process Z of a multivariate max-stable Smith process is defined as Z(t) = (1/p(W))^(1/α) L(t - W), where W is a T-valued random variable with a positive probability density function p(t) > 0, t ∈ T, and L(t) is a given deterministic [0, ∞)^d-valued function with continuous components Li, i ≤ d, satisfying certain integrability conditions.


Question: How are the abundance ratios of the stellar population and ISM calculated in galaxy simulations, and what are the differences in their weighting methods?

Answer: The abundance ratios of the stellar population and ISM in galaxy simulations can be calculated using several methods: mass-weighted (log(∑N MX/∑N MY), log(∑N M MX/MY/∑N M), ∑N M log(MX/MY)/∑N M) and luminosity-weighted (log(∑N L MX/M/∑N L MY/M), log(∑N L MX/MY/∑N L), ∑N L log(MX/MY)/∑N L). For stellar populations, V-band luminosity is used, while for the ISM, the star formation rate of each gas particle is used for weighting, as it correlates with Hα emissions. These different weighting methods can give slightly different results, and the best definition is not obvious.


Question: What are the key findings from the XRMS investigation of Ba2IrO4, and how do they compare to the magnetic structure of Sr2IrO4?

Answer: The XRMS investigation of Ba2IrO4 revealed that the Ir4+ magnetic moments order in an antiferromagnetic structure with a doubling of the unit cell along the in-plane directions, described by a magnetic propagation vector of k = [1/2 1/2 0]. The magnetic scattering was observed only at the L3 edge, with no response at the L2 edge, which is consistent with the Jeffer=1/2 state. These findings are similar to those of Sr2IrO4, where the magnetic moments form a canted antiferromagnetic structure, and the large XRMS intensity ratio IL3/IL2 supports the Jeffer=1/2 state.


Question: What is the significance of the almost-complex structure J in the context of Heegaard Floer homology, and how does it relate to the independence of the resulting homology?

Answer: The almost-complex structure J is chosen in such a way that the resulting homology is independent of the choice of J, and sometimes even of the underlying symplectic or contact structure. This independence ensures that the homology groups are topological invariants of the manifold Y.


Question: What conditions must be met for the simply connected Lie group G, corresponding to the amalgamated product of two unimodular almost abelian LCP Lie algebras, to admit lattices?

Answer: For the simply connected Lie group G, corresponding to the amalgamated product of two unimodular almost abelian LCP Lie algebras, to admit lattices, the quotient t2|θ2| / t1|θ1| must be a rational number. This ensures the existence of t > 0 such that t|θ2| / (|θ1|^2 + |θ2|^2)^(1/2) = k1t1 and t|θ1| / (|θ1|^2 + |θ2|^2)^(1/2) = k2t2 for some k1, k2 ∈ N. Consequently, the matrix of etC, expressed in the basis B1 ∪ B2 of Rn1 ⊕ Rn2, is in SL(n1 + n2, Z), and G admits lattices by Theorem 8.1.


Question: What specific performance trade-offs were quantified in the experiments, and how do they relate to the scalability of blockchain in supply chain management?

Answer: The experiments quantified the trade-offs between transaction validation speed and operation complexity, confidentiality and performance, and performance and decentralization (which affects security and availability). The results show that Fabric supports up to several hundred or a few thousand transactions per second, but performance decreases with the current implementation of gossip dissemination. These findings have implications for the use of blockchain in supply chain management, where performance and scalability are critical.


Question: How is the map g(0, 1 ⊗ m) defined, and what ensures its well-definedness in the context of the de Rham-Witt complex over perfectoid rings?

Answer: The map g(0, 1 ⊗ m) is defined as g(0, 1 ⊗ m) = xdy − u−1du, where x and y are chosen such that xy = pN and ypN = um for some unit u in A. The well-definedness of this map is ensured by the fact that m(xdy − u−1du) = dm, and if different choices of N, u, x, y are made, the difference (x1dy1 − u−11 du1) − (x2dy2 − u−12 du2) is m-torsion and hence pN-torsion. This is further confirmed by the injectivity of the embedding η : Ω1A[pN] → Ω1R[pN], which shows that the construction of g(0, 1 ⊗ m) depends only on m and not on the choices made.


Question: What is the significance of the strong data processing inequality (SDPI) in the context of quantum channels, and how does it differ from the standard data processing inequality?

Answer: The strong data processing inequality (SDPI) asserts that the relative entropy can contract strictly under the action of a quantum channel, meaning there exists a constant c < 1 such that D(Φ(ρ)∥Φ(σ)) ≤ c D(ρ∥σ). This is a stronger statement than the standard data processing inequality, which only states that the relative entropy is non-increasing (D(Φ(ρ)∥Φ(σ)) ≤ D(ρ∥σ)). The SDPI has applications in quantum state preparation and quantum channel capacities, and it provides a uniform control for all amplifications Φ ⊗ idn, which is referred to as the complete strong data processing inequality (CSDPI).


Question: How does the modification of the metric and the non-inertial acceleration affect the trajectories of massive test particles in this theory compared to General Relativity?

Answer: The modification of the metric and the non-inertial acceleration caused by the scalar coupling lead to an exact cancellation between the non-inertial acceleration and the part of the inertial acceleration coming from the modification of the metric. As a result, the trajectories of massive test particles are the same as in General Relativity at the 1.5 post-Newtonian order, even though they are not following geodesics anymore.


Question: What does the structural relaxation of a two-domain structure in (PbTiO3)n/(SrTiO3)n superlattices reveal about the formation of polar vortices?

Answer: The structural relaxation of a two-domain structure in (PbTiO3)n/(SrTiO3)n superlattices reveals an alternating clockwise/counterclockwise polar vortex configuration along the x-axis, with the domain wall thickness extending over one unit cell. The local dipoles form a pattern where the axial component of the polarization is perpendicular to the plane defined by the vortices.


Question: How does the time-rescaling algorithm re-scale the trading times, and what is the range of the re-scaled times?

Answer: The time-rescaling algorithm re-scales the trading times from [0, T] to [0, 2π] by setting ˜ti_h = 2π(ti_h − tmin) / (tmax − tmin) for each asset i and each sample time h.


Question: What are the key features of the electron spectrum and density of states (DOS) in the two-dimensional extended Hubbard model at half-filling for different values of on-site repulsion U?

Answer: For U ≲ 5t and nonzero temperature, the electron spectrum is metallic, and the density of states (DOS) near the Fermi level decreases monotonously as v approaches vc. For larger on-site repulsions, U ≳ 6t, when the DOS contains the Mott gap, the transition is not observed in the considered ranges of intersite repulsions and temperatures. In this case, the growth of v leads to a monotonous increase in the width of the Mott gap.


Question: How does the inconsistency of finite sample models under the canonical sampling operation affect statistical inference in network models?

Answer: When the finite sample models are not consistent under the canonical sampling operation, parameter estimates obtained from a sub-network may not provide reliable estimates of, or may not even be relatable to, the parameters of the whole network, rendering the task of statistical inference based on a sub-network ill-posed.


Question: What experimental results support the theoretical bounds on input precision BX for the Breast Cancer dataset?

Answer: For the Breast Cancer dataset, fixed point simulations with BF = 6 show that BX = 5 is sufficient to achieve a target classification probability of error pe ≤ 0.06. This value of BX is consistent and close to the analytically determined values of BX = 4 using the geometric bound (11) and BX = 5 using the probabilistic bound (13).


Question: What is the significance of NLO corrections in the spin asymmetry for heavy flavor photoproduction, and how do they impact the analysis of helicity PDFs?

Answer: NLO corrections are indispensable for a quantitative analysis and do not cancel in the spin asymmetry as one might naively expect. They are sizable and strongly dependent on the chosen set of helicity PDFs, which is also true for the experimentally relevant spin asymmetry. Theoretical uncertainties due to the choice of the factorization scale are sizable even at next-to-leading order accuracy, while ambiguities from the exact form of the charm quark fragmentation function are less important. These results allow one to include data on charm photoproduction consistently into future global analyses of helicity PDFs.


Question: How does the construction of potential systems change when the dimension of the space of local conservation laws is greater than one, and what are the implications for the basis conservation laws?

Answer: If the dimension of the space of local conservation laws is greater than one, the situation becomes more complicated. Previously, only basis conservation laws were used for constructing simplest potential systems. However, the basis conservation laws may be equivalent with respect to group of symmetry transformations, or the number of G∼-independent linear combinations may be greater than the dimension of the space of conservation laws. This can lead to an unnecessary investigation of equivalent systems or missing a great number of inequivalent potential systems. Class (2) provides examples of all these three cases.


Question: What is the distribution of the number of records in the Bernoulli model, and how does it relate to the Poisson distribution for large n?

Answer: The distribution of the number of records in the Bernoulli model is given by the double generating function ∑∞n=0 ∑∞R=0 P(R, n)λR zn = (1 −qz)λ−1 (1 −z)λ. For large n, with fixed p = 1 −q and R ≥1, P(R, n) converges to the Poisson distribution P(R, n) ≈ 1/pn (ln(pn))R−1 / (R −1)!. This indicates that the record occurrence events become a Poisson process in 'time' ln(pn) for large n.


Question: What evidence supports the idea of a dual halo structure, and what are the key properties of the inner and outer halo components?

Answer: The evidence for a dual halo structure includes studies of the spatial profiles and kinematic studies indicating net retrograde rotation in the outer parts. Carollo et al. (2007) claimed a dual halo structure consisting of a weakly prograde rotating inner halo and a retrograde rotating outer halo of lower metallicity ([Fe/H] < −2). The inner halo exhibits a flattened spatial density distribution with an inferred axial ratio on the order of qH ∼0.6 and n = 3.2 ± 0.2, no rotation at the level of ∼10 km s−1, and a metallicity distribution peaked at [Fe/H] ∼−1.6. The outer halo comprises stars that exhibit a more spherical spatial density distribution, with an axial ratio qH ∼0.9 and n = 1.8 ± 0.3, a clear retrograde net rotation (⟨vΦ⟩∼−80 km s−1), and a metallicity distribution peaked at [Fe/H] ∼−2.2.


Question: What are the limitations of the methods described in [10, 9, 8] for estimating the parameters of the motion model of a moving object, and how does the proposed framework address these limitations?

Answer: The methods described in [10, 9, 8] do not consider the randomness of the time points at which the measurements occur, and their proposed framework does not allow for non-Gaussian measurements. The proposed framework in this paper addresses these limitations by considering a more general dynamical system with arbitrary distributed measurements that occur at Poisson distributed time points, allowing for more general motion models for an object of interest.


Question: How is the skill score for LightGBM compared to QRF at different quantile levels defined and what does it indicate?

Answer: The skill score for LightGBM compared to QRF at a quantile level τ is defined as Sτ,skill := 1 − 𝑆̅𝜏,LightGBM/𝑆̅𝜏,QRF. This score indicates the performance of LightGBM relative to QRF, where Sτ,skill ≤ 1. A value of Sτ,skill = 1 indicates an excellent forecast by LightGBM, and Sτ,skill > 0 indicates that LightGBM outperforms QRF, with higher values indicating better performance.


Question: What is the key implication of the proof of Theorem 2, specifically in the direction (1) ⇒ (2)?

Answer: The key implication is that if G ∈ DX and G ∉ X, then G ∈ Dπ. This is proven through a series of steps, including showing that there exists a non-solvable π-subgroup in G, |π ∩ π(G)| > 2, 2 ∈ π ∩ π(G), 3 ∈ π ∩ π(G), and G has no solvable π-Hall subgroups.


Question: What does the growth of the CmagT -1 peak at 8 K with an increase in magnetic field indicate about the magnetic entropy in volborthite?

Answer: The growth of the CmagT -1 peak at 8 K with an increase in magnetic field indicates that the magnetic entropy at low temperatures shifts to the higher temperature side in the presence of a magnetic field.


Question: How is the ImageNet dataset split for training, validation, and testing in the context of evaluating calibration performance?

Answer: For ImageNet, the original validation set, which contains 50,000 image-label pairs, is split into 12,500 image-label pairs for the validation set and 37,500 image-label pairs for the test set.


Question: What is the relationship between super weakly compact sets and super weakly compact operators, and how does this relationship help in characterizing super weakly compact sets?

Answer: An operator T : X → Y is said to be super weakly compact if T U is weakly compact for any ultrafilter U (equivalently, a free ultrafilter on N). A super weakly compact set can be characterized as a subset of the image of a unit ball of a reflexive Banach space by a super weakly compact operator (see Theorem 2.3). This relationship helps in understanding the properties of super weakly compact sets and their connections to other concepts in functional analysis.


Question: What are the specific values of the monopole 3PCFs, ζ000 and ζ110, for different values of r1 and r2, and how do they decompose into various components?

Answer: The monopole 3PCFs, ζ000 (left) and ζ110 (right), are shown as a function of r2 after fixing r1 to 50, 80, 90, 100, and 130 h−1 Mpc. The 'FG', 'FS', and 'FT' terms arise from the growth, shift, and tidal effects of the non-linear density fluctuation; the 'GG', 'GS', and 'GT' terms arise from those of the non-linear velocity field; the 'BF' term consists only of linear density and linear velocity fields; the 'total' term is the sum of all the decomposed components. For these calculations, the ΛCDM model at z = 0.61, the linear bias b1 = 2.0, and no non-linear bias are assumed.


Question: What conditions must the regularization parameter λ satisfy to ensure exact recovery of the support from noisy measurements?

Answer: The regularization parameter λ must lie in the interval (5.1) such that it is large enough to reject noise but small enough to recover the entire subspace Tx0. The constraint (5.1) is non-empty if the noise-to-signal level ||w||/νx0 is small enough, i.e., ||w||/νx0 ⩽ 1 − IC(x0)/AT min(BT, C(x0)).


Question: How does the Bayesian inversion framework using InSAR data reduce uncertainty in the permeability of a groundwater aquifer?

Answer: The Bayesian inversion framework uses InSAR surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined groundwater aquifer. This framework significantly reduces the pointwise standard deviation (SD) of the permeability posterior distributions compared to the prior pointwise SD in most of the domain, as demonstrated by both the low-rank based Laplace approximation and the MCMC-based posterior.


Question: How does the paper explain the decomposition of the total sum of squares (SS) for a block when orthogonality is used between the common and distinct parts?

Answer: When orthogonality is used in the decomposition, the total sum of squares (SS) for a block can be split into one contribution from the common space (R(X12C)) and one for the orthogonal distinct contribution (R(X1D)). This is expressed as ∥X1∥2 = ∥X12C∥2 + ∥X1D∥2, where ∥.∥2 indicates the squared Frobenius norm of a matrix.


Question: What is the overall effectiveness of the GEVMCMC method compared to other global adversarial attack methods, and why is it considered superior?

Answer: GEVMCMC demonstrates the overall best performance among all proposed global attack methods due to its probabilistic nature, which allows it to generate diverse and intriguing global adversarial examples that are very different from typical local attacks.


Question: What is the nonlinear optical response of the colloidal samples, and which sample exhibits the highest and lowest response?

Answer: The nonlinear optical response of the colloidal samples is negative, with Sample A (HAE) exhibiting the highest response and Sample C (Coumarins) exhibiting the lowest response. The nonlinear coefficient n2 for all samples is of the order of 10-8 cm2/W.


Question: How do the migration energies of vacancy-atom exchanges in fcc Fe and Ni vary with different magnetic states, and what is the significance of these variations?

Answer: In fcc Fe, the migration energies are particularly sensitive to the magnetic state. For example, in the FM state, the migration energy is 0.155 eV, which is significantly lower than in other magnetic states. In the AFS and AFD structures, the intra-plane jumps have lower migration energies compared to inter-plane jumps. In contrast, in fcc Ni, the migration energies are less sensitive to the magnetic state, with the FM state having a migration energy of 1.096 eV and the NM state having 0.967 eV. The mSQS values for Fe and Ni are 0.922 eV and 1.011 eV, respectively. This sensitivity in Fe suggests that the magnetic state plays a crucial role in the diffusion mechanism, while in Ni, the effect is less pronounced.


Question: What are the key findings regarding the electronic band gaps of monolayer and bulk hBN from the DMC and GW calculations?

Answer: The DMC and GW calculations reveal that the monolayer band gap of hBN is indirect. GW predicts much smaller quasiparticle gaps at both the single-shot G0W0 and the partially self-consistent GW0 levels. Solving the Bethe-Salpeter equation on top of the GW0 calculation yields an exciton binding energy for the direct exciton at the K point in close agreement with the DMC value. Vibrational renormalization of the electronic band gap is significant in both the monolayer and the bulk. Taking vibrational effects into account, DMC overestimates the band gap of bulk hBN, while GW theory underestimates it.


Question: How is the ground truth translation vector and rotation matrix chosen in the synthetic experiments, and how are the accuracy metrics for translation and rotation defined?

Answer: The ground truth translation vector tgt is chosen as the centroid of the 3D points, and the ground truth rotation Rgt is randomly sampled. The rotation accuracy in degrees between Rgt and R is measured as max3 k=1(arccos(rT k,gt · rk) × 180/π), where rk,gt and r are the k-th column of the respective rotation matrix. The translation accuracy is computed in % as ∥tgt −t∥/∥t∥× 100.


Question: What do the second derivative of the momentum distribution curves (MDCs) at EF reveal about the Fermi surface (FS) of CeCoGe1.2Si0.8 compared to CeCoSi2?

Answer: The second derivative of the MDCs at EF reveals that CeCoGe1.2Si0.8 has double extreme values at kx(z) ∼±0.15 Å−1, while CeCoSi2 has only a single extreme value at kx(z) ∼0 Å−1. This indicates that the peak width of CeCoGe1.2Si0.8 is larger and not a single peak, suggesting a more complex Fermi surface topology in CeCoGe1.2Si0.8 compared to CeCoSi2.


Question: What are the best algorithms identified for the trust-region method with constant and non-constant initialization strategies, and how do they compare?

Answer: The best algorithm with a constant initialization strategy (C Init.) is TR:L2 with m = 14. The best algorithm with a non-constant initialization strategy (Init. 2) is TR:SC-INF with m = 5 and q = 7. TR:SC-INF with non-constant initialization outperforms the best algorithm with a constant initialization (TR:L2).


Question: What is the weak nonlocal gradient of the product of a function in Hs,p,δ(Ω) and a smooth compactly supported function χ, and how is it derived?

Answer: The weak nonlocal gradient of the product χu is given by Dsδ(χu) = χDsδu + Kχ(u) on Ω′. This is derived by calculating the integral involving the nonlocal divergence and using the Leibniz rule for the nonlocal divergence, the formula defining the weak nonlocal gradient, and Fubini's theorem.


Question: What statistical test was used to compare the mean values of the numerical immune score (NIS) between healthy and various disease states, and what were the results?

Answer: A one-way ANOVA was used to compare the mean values of the numerical immune score (NIS) between healthy and various disease states. The results showed statistically significant differences with p-values less than 0.0001 for all comparisons, including Healthy vs Unhealthy, Healthy vs Anaemia, Healthy vs HIV, Healthy vs Hodgkin’s Disease, Healthy vs Leukaemia, and Healthy vs Blood Cancer.


Question: What is the main focus of the paper regarding the use of deep neural networks for ψ-weakly dependent processes?

Answer: The paper focuses on the nonparametric regression and classification problems for ψ-weakly dependent processes, using a penalized estimation method for sparse deep neural networks. It establishes oracle inequalities for the excess risk of the sparse-penalized deep neural networks estimators and derives their convergence rates.


Question: What does the proof of Proposition 9.4 reveal about the symplectic Lie algebra (g10,ωo) in relation to Lagrangian ideals?

Answer: The symplectic Lie algebra (g10,ωo) has symplectic rank four and does not have a Lagrangian ideal. This is shown by observing that the ideal j = ⟨y⟩⊕W ⊕⟨z1⟩ is isotropic and of dimension four, and by considering the symplectic reduction of (g10,ωo) with respect to the isotropic ideal C2g10, which results in an abelian symplectic Lie algebra (a6,ωo).


Question: What modifications were made to the ten orbital tight-binding model to better match the ARPES bandstructure of LiFeAs at kz = 0?

Answer: To better match the ARPES bandstructure of LiFeAs at kz = 0, a handful of the hopping parameters were adjusted. Specifically, ϵ1 = -0.235, ϵ3 = ϵ4 = 0.23, t10_18 = 0.211i, t10_27 = -0.258, t11_33 = 0.267, t11_34 = 0.0225, t10_49 = 0.377, t11_001 = 0.0714, and t101_18 = t101_19 = 0 (in units of eV). The remaining parameters remain unchanged from those specified in the original model. Finally, the resulting bands were renormalized by a factor of 2.17, which is typical for the iron pnictides.


Question: How does the resource allocation game in the context of the butterfly network with non-zero-cost side links differ from the basic resource allocation game with routing flows?

Answer: In the basic resource allocation game with routing flows, multiple end-to-end users compete to send their packets through a single shared link without performing inter-session network coding. The shared link has a cost function C, which depends on the total rate. In contrast, in the resource allocation game for the butterfly network with non-zero-cost side links, users can jointly perform inter-session network coding, and the cost function includes the cost of side links (s1, tN) and (sN, t1).


Question: What is the significance of the non-alternating orientations in the context of v-link diagrams and their resolutions?

Answer: The non-alternating orientations of the resolutions are significant because they are the only ones that survive a closing with an even number of v-crossings. This means that any such saddle is either a multiplication (one bottom component) or a comultiplication (one top component), and these morphisms will always be killed at the bottom or top, respectively. The only saddle that survives the closure of type 0 is the Möbius cobordism, which is a 0-morphism if 2 is invertible.


Question: How does the genetic algorithm ensure that the circle covering solution covers the entire region, and what is the role of the repair function in this process?

Answer: The genetic algorithm ensures that the circle covering solution covers the entire region by incorporating a penalty for the amount of uncovered area into the fitness function. However, this approach can throw away too many good agents. Therefore, a repair function is introduced to fix agents to cover the region as best as possible. This function helps to adjust agents that are near-optimal but miss a small part of the region, allowing them to be retained and potentially improved.


Question: What is the accuracy of the K Star classifier when using the saccade feature set for classifying ADHD?

Answer: The K Star classifier yielded the lowest percent accuracy at 77.21%.


Question: What is the key improvement of the new time-dependent Hamiltonian simulation algorithms over previous methods?

Answer: The new algorithms have gate complexity that scales with the integral of the norm of the Hamiltonian over time, ∫_0^t dτ ∥H(τ)∥, in contrast to the best previous scaling of t maxτ∈[0,t] ∥H(τ)∥. This provides a strict speedup over existing algorithms.


Question: How does the mutual information between the eavesdropper's and legitimate user's channels change with κR, and what does this imply for the secrecy rate?

Answer: The mutual information I(g; ˆhk) can grow large and is limited only by AWGN and quantization noise when κR ≫ 1. When κR ≫ 1, σ2_ˆhk becomes very small, making the channel estimate unreliable, and vice versa. However, the product κRσ2_ˆhk is always less than unity. Since the nullspace noise is a function of ˆhk, which is correlated with g, part of this noise lives in the nullspace of the eavesdropper’s channel, leading to an increase in the eavesdropper's information rate and a significant reduction in the secrecy rate.


Question: How does the signal-to-noise ratio (S/N) for DM annihilation and decay vary with the angular distance from the Galactic Center (GC) in the FIRE-2 simulation compared to the NFW model?

Answer: The S/N for DM annihilation and decay in the FIRE-2 simulation is enhanced compared to the NFW model, particularly at small angular distances from the GC. For annihilation, the S/N is significantly higher in the 10 MeV - 10 GeV and 10 GeV - 10 TeV energy ranges, with the enhancement due to substructure boost. For decay, the S/N also shows a notable increase, especially as the angular distance ψ approaches the GC region (ψ → 0°).


Question: How is the property (iv) of a condition p fulfilled in the context of the forcing poset Qct, κct?

Answer: Property (iv) is fulfilled by choosing, for each α ∈ supp(p, ¯L), a single L* ∈ ¯L such that p(α, L*) has norm at least k, then shrinking p(α, L*) such that its cardinality is minimal with still the same norm, and shrinking all other p(α, L′) to arbitrary singletons; by definition, all of this leaves the ct norms of such segments at least k and the resulting q(ct) is still a valid (part of a) condition.


Question: How does the non-Hermitian skin effect (NHSE) affect the bulk-boundary correspondence (BBC) in non-Hermitian systems?

Answer: The non-Hermitian skin effect (NHSE) can break the conventional bulk-boundary correspondence (BBC) for topological phases, evidenced by an inconsistency between the spectra of a system under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). This phenomenon is eventually solved by the formulation of the non-Bloch band theory, which recovers a generalized BBC for non-Hermitian Hamiltonians.


Question: How do the low-lying excitations in systems with non-spherical ground states, such as molecules and atomic nuclei, manifest according to the effective field theory approach?

Answer: In systems with non-spherical ground states, such as molecules and atomic nuclei, the low-lying excitations are vibrations that are the heads of rotational bands. The moment of inertia is a fit parameter and will, in general, be different for different physical systems (for instance, superfluid and normal systems). Nuclei and molecules differ in that the ground states of even-even nuclei are paired, which accounts for the absence in nuclei of low-lying band heads with odd spin and positive parity.


Question: What are the key differences between Grad-CAM and Grad-CAM++ in terms of their localization maps?

Answer: Grad-CAM produces a coarse localization map using the gradients of the target class with respect to the final convolutional layer, while Grad-CAM++ produces a more fine-grained and accurate localization map by using gradients from both the lower convolutional layers and the final convolutional layer. Grad-CAM++ also introduces a new way of weighting the upsampled map using a weighted combination of gradients and activations.


Question: What does Figure 1c show regarding the effective specific electrical resistance, ρEl, of Pb as the temperature approaches TCrit from above?

Answer: Figure 1c shows that the effective specific electrical resistance, ρEl, of Pb decays as the temperature, TPh, approaches TCrit (7.2 K) from values above. At T = 7.3 K, the porosity of the normal charge carriers is set to π = 1, meaning the contribution of the superconducting charge carriers to current transport is zero.


Question: What are the key milestones in the theoretical and experimental progress of spin polarization in heavy-ion collisions?

Answer: In 2005, Liang and Wang predicted that spin-orbit coupling would polarize strange quarks in non-central heavy ion collisions, resulting in emitted Λ hyperons globally polarized along the collision angular momentum. In 2008, Becattini and collaborators noted that local thermodynamic equilibrium in a hydrodynamic picture implies a relation between spin polarization and rotational flow structure (vorticity). In 2013, an equation relating Λ hyperon polarization and thermal vorticity was derived, predicting polarization at a few percent. In 2016, the first systematic dependence of this effect on collision beam energy was reported, relevant to the beam energy scan program at RHIC. In 2017, the STAR Collaboration published the first observation of global Λ polarization from non-central heavy ion collisions.


Question: What is the specific formula for the sum involving the cosine integral function when x = 1/2 in the context of the given integral representation of the logarithm of the Gamma function?

Answer:
The formula for the sum involving the cosine integral function when x = 1/2 is given by:
\[ \sum_{n=1}^{\infty} rac{(-1)^{n-1} \cos(2n\pi a)}{n} = \log(2) \left[ \sin(2\pi a) + \sum_{n=1}^{\infty} rac{(-1)^{n-1} \cos(2n\pi a)}{n}
ight] \]


Question: What is the mathematical expression for the NLIN power and how does it depend on the mean power spectrum of the interacting channels?

Answer: The NLIN power is given by the expression ⟨|∆u(t)|2⟩= 4 ∑lmn,m̸=n |ρlmn|2⟨|νl|2⟩⟨|ξm|2⟩⟨|ξn|2⟩, which only depends on the mean power spectrum of the interacting channels and is totally independent of modulation format.


Question: How does the proposed method in Co-Mixup ensure diversity among the generated mixup examples?

Answer: The proposed method in Co-Mixup ensures diversity among the generated mixup examples by modeling a similarity penalty between outputs. This is done by representing the input source information of each output and measuring the similarity between these outputs using the inner-product on a matrix A, which is a combination of the identity matrix and a compatibility matrix Ac. The compatibility matrix Ac quantifies the degree to which input pairs are suitable to be mixed together, and by minimizing the inner-product between different outputs, the method penalizes mixup examples with similar input sources and encourages high compatibility within each individual mixup example.


Question: What are the key findings regarding the method's ability to differentiate between benign and malignant inclusions and its sensitivity to noise?

Answer: The method can differentiate between benign and malignant close inclusions, and it is also able to differentiate between one and two inclusions, even when they are quite close. Additionally, the method is fairly insensitive to noise in the data, as is typical in time-reversal methods.


Question: What are the specific dimensions and configurations of the convolutional and pooling layers in the proposed baseline CNN for X-Ray images?

Answer: The proposed baseline CNN for X-Ray images has an input dimension of 244x244. It includes 3 convolutional layers with a filter size of 3x3 and zero padding. The pooling layers use max pooling with a 2x2 filter and a stride of 2.


Question: What is the form of the multidimensional product kernels of averaged type used in the study of approximation in variation?

Answer: The multidimensional product kernels of averaged type are of the form ¯χm(t) := ∏_{i=1}^N ¯χi,m(ti), where ¯χi,m(t) := 1/m ∫_{-m/2}^{m/2} χi(t + v) dv, for some m ∈ N, and χi : R → R are suitable one-dimensional kernels.


Question: What were the results of the forward-backward asymmetry (AF B) measurements in the electron channel during Run 2 of the LHC, and how did they compare to the expected models?

Answer: During Run 2 of the LHC, attempts were made to measure the forward-backward asymmetry (AF B) of events in the electron channel to differentiate between various new physics models predicting a Z′ resonance decaying to leptons. However, the number of observed events in the current dataset made this difficult, and only the most widely varying models could be discriminated. It was estimated that around an order of magnitude more data would be needed to make strong statements about the physics model.


Question: What are the experimental results that demonstrate the effectiveness of the proposed self-supervised learning-based method for manipulators?

Answer: The models trained on the SSG dataset were able to effectively learn, proving the effectiveness of the proposed self-supervised learning-based method for manipulators. Additionally, the accuracy of each model trained on the SSG dataset tended to be higher than that of its counterpart trained on the YCB dataset.


Question: What are the limitations of using CLIP for character grounding in the VIST stories?

Answer: CLIP cannot handle pronouns and generic words. CLIP-based grounding depends on the semantics of phrases, but pronouns (e.g., he, him, and me) and generic words (e.g., men, women) can only express basic characteristics, such as gender, age, and singular-plural. This makes it difficult for CLIP to predict the alignment relation when most characters are similar in these basic characteristics. Additionally, CLIP prefers specific and detailed phrases, and it is pre-trained on image-caption pairs that are usually specific and detailed. This means the model can make more accurate predictions if there is more information in the phrase, such as 'that kid in the orange shirt'.


Question: What are the specific values of the first and second multiplicative Zagreb indices for the starlike tree Tn,k,s?

Answer: For the starlike tree Tn,k,s, the first multiplicative Zagreb index Π1(Tn,k,s) = 4ks−1(n −ks)² and the second multiplicative Zagreb index Π2(Tn,k,s) = 4ks−1(n −ks)n−ks.


Question: How does the conditional expectation in the candidate text relate to the convergence of terms in the query text?

Answer: The conditional expectation in the candidate text, E[ℓ′′_1(ψ(ξ))|V0 = v], is used to derive the limit of the second derivative, k2(ψ), which is shown to converge to -Iξ. This convergence is analogous to the convergence of terms in the query text, where the conditional expectation E∗_θ [f(Xδ_t, Uδ_t) − ¯fθ(Uδ_t) | FUδ, ¯U_t−ϵ ∨ F_˜Uδ, ˜Xδ_t] is shown to converge to zero as δ → 0, due to the ergodicity implied by Condition 2.1.


Question: What are the additional energy edges and critical regions identified in the study of the Aubry-Andr'e model with small wave vector α?

Answer: Besides the almost mobility edges Ec±, there exist additional energy edges Ec′±, at which abrupt changes of the inverse participation ratio occur. The critical regions are identified in the range |Ec±| ≤ |E| ≤ |Ec′±|, with the almost mobility edges Ec± and Ec′± separating the critical region from the extended and localized regions, respectively.


Question: What characterizes the initial data that give rise to fast solutions according to Theorem 3.2?

Answer: The initial data that give rise to fast solutions are those in the graph of a continuous function Φ defined in the subspace N⊥ orthogonal to constant functions. Specifically, for every w0 ∈ N⊥, the solution u(t, x) to equation (1.1) with initial datum u(0, x) = w0(x) + k is fast if k = Φ(w0).


Question: What is the adaptive Newton-Galerkin paradigm and how does it differ from the standard adaptive finite element approach in solving nonlinear PDEs?

Answer: The adaptive Newton-Galerkin paradigm is an alternative approach for the numerical approximation of nonlinear PDEs. It involves applying local Newton-type linearizations on the continuous level to approximate the semilinear PDE by a sequence of linearized problems. These linear PDEs are then discretized using an adaptive finite element procedure based on a posteriori residual analysis. The key difference is the interplay between the Newton method and the adaptive finite element approach, where either a Newton step is performed (if the Newton linearization effect dominates) or the finite element space is enriched based on a posteriori residual indicators (if the finite element discretization is the main source of error).


Question: How does the speed of the zeros of Re A(t, x) evolve over time, and what is the self-similar profile that describes this motion?

Answer: The speed v(t, x) of the zeros located at x at time t follows a self-similar profile, namely v(t, x) = (1+t)^(-1/2) V(x/(1+t)^(1/2)), where V(y) = y^2 - ψ(y)^2 / 2ψ'(y).


Question: How does the sequence Φk evolve over iterations in the proof of Theorem 2.1, and what is the final bound on the expected value of f(xK) − f(x*)?

Answer: The sequence Φk = f(xk) − f(x*) + γ/p ∥gk − ∇f(xk)∥² satisfies E[Φk+1] ≤ (1 − γµ)E[Φk]. Unrolling the recurrence and using g0 = ∇f(x0), we obtain E[f(xK) − f(x*)] ≤ (1 − γµ)K(f(x0) − f(x*)).


Question: How does the learning speed and accuracy of a new agent compare when learning a discrete communication protocol compared to the continuous case?

Answer: When learning a discrete communication protocol, the new agent converges to a lower accuracy at a slower pace compared to the continuous case. Specifically, the population trained from scratch would require 23 epochs to reach its best accuracy, whereas 5 epochs are sufficient to teach a new agent an already accurate protocol.


Question: What is the formula for the initial growth rate of the integral width for a multimode perturbation, and how is the weighted average wavenumber calculated?

Answer: The initial growth rate of the integral width for a multimode perturbation is given by ˙W0 = 0.564σ+A+∆u¯k, where σ+ = (1 −∆u/Us)σ and ¯k is the weighted average wavenumber of the perturbation. The weighted average wavenumber ¯k is calculated as ¯k = sqrt(∫kmin^kmax k^2P(k) dk / ∫kmin^kmax P(k) dk).


Question: How do the background fits and their uncertainties differ between the inclusive and b-tagged selections in the resolved search?

Answer: In the resolved search, the background fits and their uncertainties are displayed for both the inclusive and b-tagged selections. For the inclusive selection, the background fit (blue solid line) is compared to the data (black points), and the fit uncertainty is shown. Similarly, for the b-tagged selection, the background fit and fit uncertainty are displayed. The expected signals from simulated et →qq′ and et →bq′ samples at met = 500 GeV are also shown for both selections. The lower panel of Figure 9 displays the bin-by-bin difference between the data and the fit divided by the statistical uncertainty for both selections.


Question: What is the estimated reflection coefficient of Alfvén waves in the chromosphere, and how does this affect the wave energy flux at the top of the chromosphere?

Answer: The reflection coefficient of Alfvén waves in the chromosphere is 0.99, which means that only about 1% of the wave flux will leak out to the chromosphere. This, combined with the wave dissipation that reduces the wave flux by 4 orders of magnitude, suggests that the Alfvén wave energy flux at the top of the chromosphere should be reduced by 6 orders of magnitude with respect to the wave flux generated in the photosphere.


Question: What mechanism is believed to be robust and relevant for transport by generic turbulent flows, and how does it relate to the zero modes in the Kraichnan model?

Answer: The mechanism of zero modes, which are conserved quantities that scale as a power law with respect to the mean radius of the cloud of particles, is believed to be robust and relevant for transport by generic turbulent flows. In the Kraichnan model, the scaling exponent of these zero modes depends nonlinearly on the number N of particles, and the N-th order structure functions of the advected scalar are dominated by the irreducible zero mode of corresponding order and lowest positive scaling dimension.


Question: What are the key features of ReDas that enable it to support dynamic fine-grained reshaping and multiple dataflows?

Answer: ReDas integrates lightweight and reconfigurable roundabout data paths, which achieve fine-grained reshaping using only short connections between adjacent PEs. It also redesigns the PE microarchitecture and integrates a set of multi-mode data buffers around the array. The PE structure enables additional data bypassing and flexible data switching, while the multi-mode buffers facilitate fine-grained reallocation of on-chip memory resources, adapting to various dataflow requirements.


Question: What are the probabilities of non-detection and detecting 6 or more blobs in one field and none in the other three fields for the bright/large blob sample according to the N-body simulation and Poisson distribution?

Answer: According to the N-body simulation, the probability of non-detection (P(N = 0)) is 41.5%, and the probability of detecting 6 or more blobs in one field and none in the other three fields (P(N ⩾ 6)) is 1.25%. For the Poisson distribution, the probability of non-detection is 23.3%, and the probability of detecting 6 or more blobs in one field and none in the other three fields is 0.022%.


Question: What is the estimate for |B1,1(x)| and how is it derived from the given conditions?

Answer: |B1,1(x)| ≲ |x|^2 / r^2. This estimate is derived by integrating against ϕ′, which is supported on the annulus A(0, 1/2, 2). The integral is bounded by 1/r^2 times the integral of |K(y)||SO|ϕ′(|y|/r)^2 dµ(y). From (4.43), |SO| ≲ |x|^2 / r^2, and since the support of ϕ′ is similar to the support of φ, the estimate follows.


Question: What does the (i′, r′ −i′) CMD reveal about the distribution of RSGs and AGB stars in the peripheral regions of M81?

Answer: The (i′, r′ −i′) CMDs of sources with RGC < 14 kpc show a concentration of objects with i′ > 22.5 and r′ −i′ between 0 and 1.5, which are a combination of RSGs with ages in excess of ∼50−100 Myr and the brightest AGB stars. These objects form an amorphous cloud-like structure in the lower portions of the CMDs.


Question: How does the localized ETKF (LETKF) approach the problem of localization in the context of the ETKF, and what is the role of the decorrelation factors in this process?

Answer: The localized ETKF (LETKF) is an efficient implementation of localization for ETKF. In the state-space approach to the LETKF, each state space variable is assimilated independently, with the observation space error covariance inverse replaced by R−1i ←−ρi,[j] ◦R−1i, where ρi,[j] is a diagonal matrix with diagonal entries representing the decorrelation factors between all observation space variables and the j-th state space variable. Each diagonal element represents a tapering factor, often chosen as a function of the distance from the state-space variable being assimilated and the corresponding observation-space variable.


Question: What are the specific quantitative results of the proposed CMS-LSTM model on the Typhoon dataset, and how do they compare to other SOTA methods?

Answer: The proposed CMS-LSTM model achieves a PSNR of 28.891, SSIM of 0.907, MSE of 6.24, and MAE of 86.4 on the Typhoon dataset. When using 7×7 convolution layers (CMS-LSTM#), the performance further improves to a PSNR of 30.207, SSIM of 0.921, MSE of 5.21, and MAE of 76.0. These results outperform other SOTA methods, demonstrating the model's superior spatiotemporal expression and prediction capabilities.


Question: What are the two sub-Planckian scales introduced by the large coupling ξ in the Higgs inflation model, and what issue do they cause?

Answer: The two sub-Planckian scales introduced by the large coupling ξ in the Higgs inflation model are MPl/ξ and MPl/√ξ. These scales appear to violate unitarity, thus spoiling the self-consistency of the inflationary approach.


Question: What is the impact of receiver antenna diameter on the capacity for distances from 500Mkm to 10Bkm, and how does the choice of modulation order affect this capacity?

Answer: For distances from 500Mkm to 10Bkm, the capacity ranges from 125 to 3.5 Mbps for a receiver’s diameter of 10m, and from 100 to 0.6 Mbps for a receiver’s diameter of 4m. There is a loss of 0.5 Mbps if M = 16 is chosen along the whole distance.


Question: What are the specific conditions on the spinor that ensure the preservation of 1/4 of the supersymmetry in the D4 brane solution, and how do these conditions relate to the Killing spinor on S5?

Answer: The conditions on the spinor that ensure the preservation of 1/4 of the supersymmetry in the D4 brane solution are given by Γ+ǫ = 0 and Γ+−x2x3x4ǫ = −iǫ. These conditions are equivalent to iγ0123ǫ = ǫ, which is derived from the gravitino variation. The Killing spinor on S5 is given by ǫ = h(θ, ϕa)H−1/2_7 ρ1/2ǫ0, where both ǫ and ǫ0 satisfy the conditions Γ+ǫ = 0 and Γ+−x2x3x4ǫ = −iǫ.


Question: What is the general approach for integrating out heavy fields in a supersymmetric theory to define a low-energy effective theory?

Answer: In a supersymmetric theory, heavy fields are typically stabilized at values that imply a spontaneous breakdown of supersymmetry. The low-energy effective theory is obtained by determining the heavy fields in terms of the light ones by requiring stationarity of the potential with respect to the heavy fields. If the heavy fields are stabilized in an approximately supersymmetric way, the low-energy effective theory for the light fields should be approximately supersymmetric, and one can use an exactly supersymmetric effective theory that differs from the actual effective theory only by small effects related to the contribution of the heavy fields to supersymmetry breaking.


Question: What does the normalized probability distribution, fpairs(θ), indicate about the alignment of satellite galaxies with the major axis of their central galaxy?

Answer: fpairs(θ) > 1 for small θ, indicating that satellite galaxies are distributed preferentially along the major axis of their central galaxy. This is also evident from the fact that ⟨θ⟩= 42.46°± 0.12°, which deviates from the case of no alignment (i.e., ⟨θ⟩= 45°) by almost 21σ.


Question: What are the conditions for a rule to be applicable or discarded in a defeasible theory?

Answer: A rule is applicable for a literal q if q occurs in the head of the rule, all non-modal literals in the antecedent are given as facts, and all the modal literals have been defeasibly proved (with the appropriate modalities). On the other hand, a rule is discarded if at least one of the modal literals in the antecedent has not been proved (or is not a fact in the case of non-modal literals).


Question: How does the extended promotion time model account for time-varying covariates in the context of the cure rate model?

Answer: The extended promotion time model allows for the inclusion of time-varying covariates (TVCs) by incorporating them into both the conditional survival model for non-cured subjects and the regression model for the cure probability. This more general formulation was proposed by Lambert and Bremhorst (2020), and it requires that the follow-up duration after each change of covariate be long enough for identification purposes in the cure probability submodel. In the current paper, a reformulation of the extended promotion time model is proposed, allowing an unlimited number of changes in categorical or continuous TVCs, and additive terms for nonlinear effects of (constant or time-varying) quantitative covariates can also be considered jointly in the long- and short-term survival submodels.


Question: How does the compositional thickness of the transition layer between the γ(f.c.c)- and γ’(L12)-phases change with aging time, and what does this imply about the diffusion mechanism?

Answer: The compositional thickness of the transition layer between the γ(f.c.c)- and γ’(L12)-phases decreases with increasing aging time, varying as \( \delta(t) \propto t^{-0.08 \pm 0.01} \). This implies that the diffusion mechanism is influenced by the range of monovacancy-solute interactions, with the transition layer becoming thinner as the alloy ages, indicating a decrease in the coupling between the solute and solvent fluxes.


Question: What are the main challenges associated with using body-conduction microphones (BCMs) and how does the configurable EBEN model address these challenges?

Answer: The main challenges associated with BCMs include reduced bandwidth due to the intrinsic low-pass characteristics of the biological pathway, and the capture of physiological sounds such as swallowing and blood flow, which add new forms of noise. The configurable EBEN model addresses these challenges by using a deep neural network to extend the bandwidth of the captured speech, specifically by regenerating missing mid and high frequencies. It employs a multiband decomposition with Pseudo-Quadrature Mirror Filters (PQMF) to reduce the dimension of input features, decreasing latency and computational load. Additionally, the model is designed to be lightweight and suitable for real-time execution, making it appropriate for edge computations and frugal AI requirements.


Question: What are the specific results regarding the recovery of artificial signals with different noise models, and why were the MA models considered more reliable?

Answer: The pure white noise model and both first order AR and MA models could not reliably determine the parameter values of the artificial signals, even with amplitudes as high as 5 ms−1. Higher order AR models also yielded severe biases, interpreting signals as noise-related correlations. The MA models, however, were reasonably reliable, though they slightly overestimated the amplitudes of the 20, 50, and 100 day signals. The MA models of order seven and ten yielded the best results for periods and eccentricities of the injected signals. The 50 day signal could not be extracted correctly, and the 20 and 100 day signals had biased estimates, possibly due to existing signals in the data or limitations in the noise models.


Question: How does the paper extend the fitting approach for the linear combination of K ≥ 2 Student’s t RVs using the results from the linear combination of K = 2 Student’s t RVs?

Answer: The paper extends the fitting approach by induction. It shows that if a scaled Student’s t distribution fits the distribution of the linear combination of K = 2 Student’s t RVs, then this approach can be applied to the linear combination of any K ≥ 2 Student’s t RVs. This is demonstrated by the equation σ1T1 + σ2T2 ∼ σz2T(νz2), and by assuming PK i=1 σiTi ∼ σzK T(νzK), it is shown that the linear combination of K + 1 Student’s t RVs can be approximated as σzK+1 T(νzK+1), where σzK+1 and νzK+1 are obtained iteratively using the transformation functions g1 and g2.


Question: How does the multiscale scheme for posterior estimation address the computational challenges of high-dimensional inversion tasks?

Answer: The multiscale scheme for posterior estimation addresses the computational challenges of high-dimensional inversion tasks by introducing a hierarchy of generative models from coarse- to fine-scales. This allows MCMC to explore the posterior in coarse-scales with much reduced cost due to low-dimensional latent spaces and inexpensive forward model evaluations. The computational savings can be even higher if the latent representation on a given scale utilizes the latent information inferred in the immediately coarser-scale.


Question: How does the weak reflection principle solve the static hedging problem for a process that does not possess strong symmetries?

Answer: The weak reflection principle allows us to solve the static hedging problem in a model where S = Xx is the underlying, with Xx_0 = x ≤ 0. For any admissible function h with support in (−∞,0), at any time up until and including the time when S hits 0, the price of a European-type option with maturity T and the payoff h(ST) − W+h(ST) is given by E(h(ST)1{supt∈[0,T] St<0}|Ft∧T0), which coincides with the price of an up-and-out barrier option with the terminal payoff function h. To offset the risks associated with holding an up-and-out barrier option with the terminal payoff function h, one needs to sell the European-type option with the payoff function h−W+h, and buy it back (at a zero price) when, and if, the underlying hits 0.


Question: How are the delay-Doppler information plane and the time-frequency grid related in the OTFS signal model, and what are the specific equations used for the transformation?

Answer: The delay-Doppler information plane is discretized to a grid as Γ = {(k/NT, l/M∆f) | k = 0, ..., N-1; l = 0, ..., M-1}, where 1/M∆f and 1/NT represent the quantization resolution of the time delay and Doppler shift at the carrier frequency. The information symbols x[k, l] on the delay-Doppler grid Γ are mapped to X[n, m] on the time-frequency grid Λ using the inverse symplectic finite Fourier transform (ISFFT) as follows: X[n, m] = (1/√(NM)) ∑(k=0 to N-1) ∑(l=0 to M-1) x[k, l] ej2π(nk/N - ml/M).


Question: How does the prediction likelihood change with respect to fgh and fij, and what optimization method is used to estimate [?

Answer: Changes in fgh have a much larger effect on the prediction likelihood than changes in fij, presumably since the synaptic weight only influences the likelihood in the short interval following each presynaptic spike. Values of fij have negligible influence on prediction likelihood under fixed fgh but not vice versa. Full 2D optimization of the prediction likelihood (bounded gradient descent in this example) accurately recovers the true process noise, as does a simple 1D optimization scheme where we first optimize fgh with fij = 0 then optimize fij using the fgh optimized in the first step.


Question: How does MCSSL use epipolar constraints to reduce the search space for potential bounding boxes in cross-camera pseudo-label sharing?

Answer: MCSSL uses epipolar constraints to map a bounding box in the image from the 'teacher camera' to four epipolar lines in another camera's image, which significantly reduces the search space of potential bounding boxes. For example, it reduces the search space by 12x on the WildTrack dataset. This is achieved by leveraging the epipolar plane, which contains the physical 3D position P and epipoles O1 and O2 of both cameras, and the epipolar lines, which are the intersections of the epipolar plane with the image plane. The epipolar constraints help filter out a large number of bounding boxes that are less likely to be confirmed by reID, thereby reducing computational overhead.


Question: What are the key multi-lepton final states used in the CMS b-tagged multi-lepton search to measure t¯tH associated production?

Answer: The multi-lepton t¯tH signal arises predominantly from H →τ +τ − and H →WW ∗ alongside the semi-leptonic or fully leptonic decay of the t¯t pair.


Question: How does the hybrid chiral bag (HCB) model address the issue of chiral symmetry in the MIT bag model?

Answer: In the hybrid chiral bag (HCB) model, chiral symmetry is restored by introducing pions into the bag Lagrangian. This allows a baryon to be thought of as three quarks inside a bag, surrounded by a cloud of pions, which makes the nucleon-nucleon interaction description via the exchange of pions accessible.


Question: How does the computational time for the Sinkhorn algorithm compare to the usual Wasserstein distance, and how does this vary with the complexity of the trees?

Answer: The Sinkhorn algorithm is up to 10 times faster compared with the usual Wasserstein distance, although the speed advantage decreases for larger trees. The Sinkhorn algorithm also leads to small errors which increase marginally for trees with more stages.


Question: What is the relationship between the existence of ℓ-good uniform unstable charts and the polynomial nature of Tℓ_x, and how does this relate to the construction of (ℓ+1)-good uniform unstable charts?

Answer: If Λ admits a family of ℓ-good uniform unstable charts and the template Tℓ_x given by equation (3.17) is a polynomial for every x, then Λ admits (ℓ+1)-good uniform unstable charts. Moreover, the (ℓ+1)-good uniform unstable charts can be chosen to depend continuously on the base point if the ℓ-good uniform unstable charts depend continuously on the base point.


Question: How does the choice of activation function impact the robust generalization gap in adversarial training?

Answer: Using activation functions with low (exact or approximate) curvature values has a regularization effect that significantly reduces both the standard and robust generalization gaps in adversarial training. This effect is observed for both differentiable/smooth activations such as SiLU and non-differentiable/non-smooth activations such as LeakyReLU.


Question: How does the inclusion of the nonlinear term in the momentum equation affect the flow waveforms in the proposed model?

Answer: The nonlinear term in the momentum equation affects the flow waveforms, as shown in Fig. 11, where the linear model assumes the coefficient of the nonlinear term α = 0.


Question: What are the key design considerations for achieving efficient acceleration using high-energy single-cycle THz beams without surpassing the damage threshold?

Answer: To achieve efficient acceleration using high-energy single-cycle THz beams without surpassing the damage threshold, two important points must be considered: (i) The electron may gain relativistic energy, which intensifies the effect of the transverse magnetic field of the THz pulse, causing a push from the THz pulse along its propagation direction and leading to a curved trajectory for the electron motion. (ii) A high-energy THz beam should not be focused to small spot-sizes to avoid dark current excitation. To address these, a configuration is devised where two linearly polarized THz beams are symmetrically coupled into a multilayer structure to cancel out the magnetic field where they overlap, and the phase front of the THz beam is matched with the electron trajectory.


Question: What are the four obstacle avoiding patterns observed in the fish school, and how are they related to the school's cohesiveness?

Answer: The four obstacle avoiding patterns observed in the fish school are Rebound, Pullback, Pass and Reunion, and Separation. These patterns are performed by tuning modeling parameters and are related to the school's cohesiveness, which is defined as the school's ability to form and maintain its structure against white noises affecting the school. The results suggest a correlation between the degree of cohesiveness and these avoidance patterns.


Question: What are the key mathematical subtleties and recent advancements in the coherent-state path integral for spin systems?

Answer: Coherent-state path integrals for spin systems have mathematical subtleties such as paths not needing to be continuous, unlike the Feynman-position space path integral. Recent advancements include the inclusion of the Solari-Kochetov (SK) correction, which resolves j versus j + 1/2 arbitrariness and ensures correct short-time behavior. This correction has been extended to many particles and spins, leading to successful solutions for the Bohr-Sommerfeld quantization rule for spin, an extension to instanton calculus, and a quantitatively correct explanation of the spin tunneling spectrum of the magnetic molecule Fe8(tacn)6.


Question: How is the degree of a polynomial g defined in the context of multivariate polynomials over a commutative ring R?

Answer: The degree of g = ∑α aαxα is the maximal number of the set {|α| ∈ N | aα ≠ 0}.


Question: How does the commutativity of ΓF(g1) and ΓF(g2) follow from the given lemmas and corollaries?

Answer: By Lemma 2.32, ΓF(g1) ◦ ΓF(g2) = ΓF(g1 ◦ g2) = ΓF(g2 ◦ g1) = ΓF(g2) ◦ ΓF(g1).


Question: What parameters are introduced in the model to account for the interaction between Wolbachia and WNv in female mosquitoes?

Answer: The parameters introduced include αf (biting rate of female Wolbachia uninfected mosquitoes), αfw (biting rate of female Wolbachia infected mosquitoes), pbf (transmission probability of WNv from infectious birds to WNv-susceptible female mosquitoes), pfb (transmission probability of WNv from WNv-infectious female mosquitoes to susceptible birds), and νf (per-capita transition rate of WNv-exposed female Wolbachia uninfected mosquitoes to the infectious stage of WNv). Additionally, ε is a small parameter modeling the increased time that Wolbachia infected mosquitoes spend in the latent stage of WNv due to the tendency of Wolbachia infection to hamper the replication of WNv in mosquitoes.


Question: What is the pointwise bound for |g(x)| derived in the proof of Lemma 8.2, and how is it transformed using the substitution (t, y) ↔ (t, u)?

Answer: The pointwise bound for |g(x)| is given by |g(x)| ≤ Cλ^(d/2) ∫∫ e^(Θ(x,y,t)) |x − y| |Sg(tx + (1 − t)y)| dt dm(y), where Θ(x, y, t) = Φ∗(x) + 2Φ∗(y) − Φ∗(tx + (1 − t)y) − γ(y). After the substitution (t, y) ↔ (t, u) where tx + (1 − t)y = x + u, the bound becomes |g(x)| ≤ Cλ^(d/2) ∫ (∫ e^(Φ∗(x) + 2Φ∗(x + su) − Φ∗(x + u)) s^(d-1) ds) |u| |Sg(x + u)| dm(u).


Question: How does the system's adiabatic evolution change when g2 is increased from 0 to g1/2 and beyond, and what are the resulting eigenstates?

Answer: For g2 < g1/2, the system evolves as an equal weight superposition of the symmetrized energy eigenstates, |Ψ⟩ = 1/√2(|Ψ(1)+⟩ + |Ψ(1)−⟩). At g2 = g1/2, a basis transformation to the eigenstates |Ψ(2)±⟩ takes place, with |Ψ(1)+⟩ = |Ψ(2)+⟩ and |Ψ(1)−⟩ = −i |Ψ(2)−⟩. For g2 > g1/2, the system evolves as |Ψ⟩ = 1/√2(|Ψ(2)+⟩ − i |Ψ(2)−⟩).


Question: What are the specific Hilbert spaces G0 and G1, and how do they relate to the trace operators τD and τN on the maximal domain of the Laplacian?

Answer: The Hilbert spaces G0 and G1 are defined such that the Dirichlet and Neumann trace operators admit continuous and surjective extensions from the maximal domain of the Laplacian onto the anti-dual spaces G′1 and G′0, respectively. Specifically, τD : Hs∆(Ω) → Hs−1/2(∂Ω) and τN : Hs∆(Ω) → Hs−3/2(∂Ω) for all s ∈ [1/2, 3/2].


Question: What measures were taken to mitigate contamination issues when using WISE data in the study of Tr 37?

Answer: The study used Spitzer data in preference to WISE data because the larger WISE beam often suffers from strong contamination due to cloud emission, resulting in overestimated fluxes, especially in bands 3 and 4. However, when there was a good agreement between IRAC and WISE bands 1 and 2, MIPS 24µm and WISE band 4 (22µm), it was assumed there was no significant contamination, and the 12µm WISE data was used. The WISE fluxes were obtained applying the Wright et al. (2010) color corrections depending on the Spitzer SED shape.


Question: How does the inclusion of chemistry affect the ionization fraction and temperature profile in the shock model?

Answer: When chemistry is included, the ionization fraction is much lower, leading to weaker ion-neutral coupling and a broader shock region. Consequently, the maximum temperature is lower because the energy is dissipated over a larger region, as shown in the bottom panel of Figure 4.3.


Question: What are the limitations of using only local optimization or gradient optimization in fPEPS, and how do they complement each other?

Answer: Local optimization and gradient optimization have complementary features. Local optimizations of adjacent sites do not seem to lead towards a minimum of the entire fPEPS, whereas gradient-based optimization is, by construction, unable to modify the bases on the local tensor bonds to form an fPEPS that efficiently approximates the many-body ground state. Combining the methods partially overcomes these problems, but the synergy between the two methods is limited, so that convergence is still very slow.


Question: How does the proof of Theorem 2 demonstrate the convergence of the numerator and denominator of the expression for √nknlˆρ(k, l)?

Answer: The proof of Theorem 2 shows that the denominator converges to √(Σ1(k, l)Σ2(k, l)) at the rate of O(1/n), and the numerator converges to γ, which is normally distributed with mean 0 and variance Σ1(k, l)Σ2(k, l). Specifically, the numerator is expressed as √nknl ˆΣ12(k, l) = γ + O(1/n^2), where γ is the sum of A1(i, j)(A2(i, j) - p) over all i and j, and p = Prob(A2 = 1|k, l). The term O(1/n^2) converges to 0 by the law of large numbers, and γ is normally distributed by the central limit theorem.


Question: How does the proof of the fourth claim in the infinite family of almost-positive knots with the second Jones coefficient equal to -2 utilize the induction hypothesis?

Answer: The proof of the fourth claim uses the induction hypothesis to show that the leading coefficient of the Conway polynomial for Dw+1 is equal to the leading coefficient of the Conway polynomial for D0. Specifically, it shows that lead coeﬀ∇(w + 1) = lead coeﬀ∇(w), and by the induction hypothesis, lead coeﬀ∇(w) = lead coeﬀ∇(0), thus lead coeﬀ∇(w + 1) = lead coeﬀ∇(0).


Question: What are the median sizes of star-forming and quiescent galaxies at z=2.25 for stellar masses of 10^10 M⊙, 10^10.5 M⊙, 10^11 M⊙, and >10^11.3 M⊙?

Answer: At z=2.25, the median sizes are: for star-forming galaxies, 2.7±0.1 kpc (10^10 M⊙), 2.8±0.1 kpc (10^10.5 M⊙), 3.0±0.1 kpc (10^11 M⊙), and 4.8±0.2 kpc (>10^11.3 M⊙); for quiescent galaxies, 0.6±1.8 kpc (10^10 M⊙), 1.1±0.2 kpc (10^10.5 M⊙), 1.9±0.2 kpc (10^11 M⊙), and 3.6±0.4 kpc (>10^11.3 M⊙).


Question: How is the reference polynomial fref(x1, x2, x3) defined, and what is its significance in the context of the diced cube representation?

Answer: The reference polynomial fref(x1, x2, x3) is defined as fref(x1, x2, x3) = x4_1 + x4_2 + x4_3, which represents a dice-shaped surface when its isovalue is set to 1. This polynomial is significant because it exhibits the sought octahedral symmetry, which is the lowest polynomial order with this property, and it is invariant under any argument inversion and/or permutation.


Question: How are the maps κ, ι, and ζ defined in the context of an action operad, and what do they represent in the construction of a cloning system?

Answer: In the context of an action operad G = (G•, π, {◦i}i, id), the maps κ, ι, and ζ are defined as follows: κn_j(g) = g ◦j e2, ζn(g) = e2 ◦2 g, and ιn(g) = e2 ◦1 g. These maps are used to construct a quintuple (G•, ι, ζ, κ, π), which forms a restricted operadic cloning system if G is an action operad, and an operadic cloning system if G is a general action operad.


Question: What is the form of the perturbed system (6.6)-(6.9) and how are the new unknown functions σ and u defined?

Answer: The perturbed system (6.6)-(6.9) is rewritten in the form Az = Fz, where A is a linear operator and F is a nonlinear perturbation. The new unknown functions are defined as σ(x1, t) = ρ(x1, t) − s(x1, t) and u(x1, x2, t) = v(x1, x2, t) − pε_0(x1, x2) − χ(Λ) ∂pε_0/∂x2 (x1, x2)σ(x1, t), where pε_0 solves problem (6.1) and s is defined in Lemma 6.2.


Question: What are the key findings regarding the impact of non-linear dispersion on the Kondo problem and spin relaxation time?

Answer: The key findings include the derivation of an effective spin-spin interacting model with θ = π/3 symmetry rotation in the Brillouin zone, leading to exceptional points (EP) in a Hermitian model when combined with anisotropic Dzyaloshinskii-Moriya (DM) interaction. The spin relaxation time shows the signature of coalescence in momentum-resolved couplings and an ideal logarithmic divergence in resistivity over a range of nonlinearity (β). The model also exhibits a 'Sign Reversion' (SR) regime for a Hermitian problem with a critical value of nonlinear coupling Jk3, and out-of-phase RKKY oscillations above and below the critical value of the chemical potential.


Question: How is the term K1 bounded in the proof of the global a priori bounds for the 2D Boussinesq equations with critical dissipation?

Answer: To bound K1, we have for any s ∈ (0, 1), |K1| ≤ ∥Λs(G|G|q−2)∥L2 ∥Λ1−s([Rα, u]θ)∥L2. By Proposition 4.5, (5.4) and (5.5), we obtain ∥Λ1−s[Rα, u]θ∥L2 ≤ C (∥G∥L2 + ∥θ0∥Lq1 + ∥u∥L2)∥θ0∥Lq2 + C ∥u∥L2 ∥θ0∥L2 ≡ B(t), where 2 − 2α < s, q1 = 2/α, 1/q1 + 1/q2 = 1/2 and B(t) is a smooth function of t depending on ∥u0∥L2 and ∥θ0∥L2∩Lq1∩Lq2 only. Choosing s satisfying 2 + s − α − 2(2 − α)/q = α/2 and applying Lemma 6.2, we obtain ∥Λs(G|G|q−2)∥L2 ≤ C ∥G∥q−2L2q/(2−α) ∥G∥˚H(α/2). Combining the estimates, we obtain, for any t ≤ T, |K1| ≤ B(t) ∥G∥q−2L2q/(2−α) ∥G∥˚H(α/2) ≤ C0/2 ∥G∥qL2q/(2−α) + C ∥G∥q/2˚H(α/2), where C is a constant depending on T and the norms of the initial data.


Question: What are the periods of the Mira candidates with light curves available in public surveys, and how were these periods determined?

Answer: Periods were recovered for 27 out of 41 Mira candidates. The periods were determined using the CRTS survey when available, as it has a larger number of epochs. When only Gaia or ASAS-SN light curves were available, the corresponding period was adopted. The recovered periods range from 65 to 450 days, except for EE13, which has a period of only 1.13 days according to the SSS catalogue, but this period is considered unreliable due to considerable scatter in the light curve.


Question: What are the specific invariant amplitudes for the gQ → gQ scattering process with massive heavy quarks and massless gluons?

Answer: The invariant amplitudes for the three graphs (shown in Fig.15) for the case of massive heavy quarks and massless gluons is given according to Combridge [13] and Cutler and Sivers [12] and expressed in the appendix by Eq. (B.1). The lowest-order amplitude for the process gQ → gQ is obtained from the Feynman rules of the gauge theory by the sum of the amplitudes (B.1).


Question: What are the key findings regarding the biomechanical constraints of the lips and their impact on speech sounds?

Answer: The study found evidence for potential links between variability in the anatomy of the lips and variability in articulatory and acoustical characteristics of speech sounds. It also demonstrated that the deformation capabilities and muscle activations of the lips allow for a certain amount of freedom in jaw positioning during bilabial stops, which could be used for co-articulation planning in bilabial stops-vowel-coronal stops-vowel sequences at fast speaking rate. Conversely, protrusion and rounding of the lips require high jaw positions.


Question: How is the set of symmetric discrete N-marginals defined, and what are its extreme points in the context of the physical measure of electrons?

Answer: The set of symmetric discrete N-marginals, denoted as ΠN,sym, is defined as the set of symmetric, non-negative measures on (R|X|)N that sum to 1. An extreme point of ΠN,sym is given by the tensor product of basis vectors ec1 ⊗ ... ⊗ ecN, where c1, ..., cN ∈ {1, ..., |X|}. For the physical measure of electrons, the set ˜ΠN,sym is a restricted subset of ΠN,sym where µ(i1, ..., iN) = 0 if any two indices are the same, ensuring that two electrons cannot be in the same state. The convex hull of the set ˜ΓN, which consists of symmetrized tensor products of distinct basis vectors, equals ˜ΠN,sym.


Question: What adjustments were made to the photometric redshifts to ensure consistency between the Northern and Southern hemispheres?

Answer: An average offset between the photometric redshifts in the North and South was calculated and plotted as a function of redshift. For each galaxy located in the North (δ1950 > 2.5°), this offset was added to the photo-z’s derived with ANNz to ensure consistency between the two hemispheres.


Question: How does the proposed Projective Spatial Transformer (ProST) module enable the integration of deep convolutional layers for 2D/3D registration?

Answer: The ProST module enables the integration of deep convolutional layers by providing a projection layer that can be embedded as a PyTorch layer with tensor variables. This projection layer, which does not have learnable parameters, allows for analytical gradient flow from the projection domain back to the spatial domain. By integrating deep convolutional layers, ProST makes end-to-end 2D/3D registration feasible.


Question: What are the properties of the Bass-Serre tree and the fundamental group of a graph of groups?

Answer: The Bass-Serre tree X is constructed from a graph of groups (X; Gv, Ae) such that the group G, called the fundamental group π1(X; Gv, Ae), acts on X. The quotient graph X/G is isomorphic to X. The stabilizer of any vertex in X is isomorphic to the group Gv, and the stabilizer of any edge in X is isomorphic to the group Ae. Additionally, starting from a left action of a group G on a tree X, G is isomorphic to the fundamental group of the graph of groups it defines, and the action on the Bass-Serre tree is conjugate to the action of G on X.


Question: Which figures in the supplementary material show the surface density of star formation for NGC 5194?

Answer: Figure D7 shows the surface density of star formation in NGC 5194.


Question: What is the significance of the decay of (s)neutrinos of the second and third generation during the reheating process?

Answer: The decay of (s)neutrinos of the second and third generation is responsible for an increase in the radiation number and energy densities on two slightly distinct time scales. This decay is efficient as long as the radiation production rate exceeds the Hubble rate, and it contributes to the first local maximum of the radiation energy density, marking the first stage of the reheating process, referred to as the stage of N2,3 reheating.


Question: What are the numerical results for the first two Lyapunov exponents at A = 0.073, and how do they support the chaotic nature of the attractor?

Answer: The first two Lyapunov exponents at A = 0.073 are λL1 = 0.0793 ± 0.0008 and λL2 = −0.1834 ± 0.0007, and λL1 = 0.057 ± 0.0009 and λL2 = −0.1921 ± 0.0007, which justify the chaotic nature of the attractor.


Question: How does the Raman intensity map at 750 nm excitation help in identifying the nature of the resonance in the NPoM nanocavity shown in panel (b) of Figure 4?

Answer: The Raman intensity map at 750 nm excitation indicates a preferential longitudinal input coupling, which allows us to associate the single resonance observed in dark field to the longitudinal L01 mode rather than the S11 mode.


Question: How does the color-type gradation of the principal subspaces WL(kΛ0) and WN(kΛ0) relate to the weight and the number of elements in the generating functions?

Answer: The color-type gradation of the principal subspaces WL(kΛ0) and WN(kΛ0) is determined by the sum of the indices of the generating functions. For integers mpi,i, ..., m1,i, the elements xnp2,2α2(mp2,2) · · · xn1,2α2(m1,2)xnp1,1α1(mp1,1) · · · xn1,1α1(m1,1)vkΛ0 and xnp2,2α2(mp2,2) · · · xn1,2α2(m1,2)xnp1,1α1(mp1,1) · · · xn1,1α1(m1,1)vN(kΛ0) have weight −mp2,2 − ... − m1,2 − mp1,1 − ... − m1,1 = −m. Their color-type is (r2, r1), where ri = npi,i + ... + n1,i.


Question: How is the eavesdropper's channel vector g expressed in terms of the legitimate user's channel estimate ˆhk, and what is the covariance matrix of the estimation error ǫ?

Answer: The eavesdropper's channel vector g can be expressed as g = √κRˆhk + ǫ, where κR is the received power ratio between the eavesdropper and intercepted user k, and ǫ is an uncorrelated Gaussian error vector with covariance matrix Cǫ = (1 − κRσ2ˆhk)IN.


Question: What specific conditions must be met for the first player (PI) to guarantee a win in the described transitive avoidance game?

Answer: PI can guarantee a win if he can ensure that the position after his turn is of the form (A ∪{(x, y)}, ¯A) for some set A and point (x, y). If the position is a direct win, which is of the form (A ∪{(x, y)}, ¯A ∪{(z, w)}) for some A and 1 ≤z −x ≤b′, PI plays (x, y + 1) and ensures he gets one of every other pair set except he never plays the point (z, w + 1) opposite to (z, w). This means that after PI’s last turn, he has one of every pair, except he has both of the xth pair and neither of the zth pair, which is a winning set under Condition 2.


Question: How does the topological influence manifest in the biaxial nematic phase of liquid crystals, and what is the role of the quaternion group Q8 in this context?

Answer: In the biaxial nematic phase of liquid crystals, the topological influence is described by the action of π1(G/H) on πn(G/H), which results in an automorphism map on πn(G/H). The order parameter manifold for the biaxial nematic phase is SO(3)/D2, which is isomorphic to SU(2)/Q8, where Q8 is the eight-element quaternion group. Since Q8 is non-Abelian, vortices in this phase have non-Abelian charges. When a vortex with charge iσy rotates around a vortex with charge iσx, the topological charge iσy changes to −iσy, demonstrating the non-Abelian nature of the charge transfer.


Question: How does the introduction of disorder affect the lattice positions and the Hamiltonian in the Haldane-Shastry model?

Answer: Disorder is introduced by choosing the lattice positions as zj = e2πi(j+αj)/N, where αj is a random number chosen with constant probability density in the interval [−δ/2, δ/2], and δ ∈ [0, N] is the disorder strength. For δ = 0, the uniform model is recovered. When 0 < δ < 1, there is some disorder, but the spins are still ordered from 1 to N on the circle. For δ > 1, the order of the spins can change, and for the maximal disorder strength δ = N, all spins can be anywhere on the circle. The Hamiltonian is given by H = −2 ∑_{i≠j} [zizj / (zi − zj)^2 + (zi + zj)(ci − cj) / 12(zi − zj)] ⃗Si · ⃗Sj, where hij = −2 [zizj / (zi − zj)^2 + (zi + zj)(ci − cj) / 12(zi − zj)].