Question: What are the specific values of corrugation and inter-layer distances for TBG at a twist angle of 1.0501° and 29.9886°?

Answer: For a twist angle of 1.0501°, the magnitude of corrugation is 0.0683 Å, the maximum inter-layer distance is 3.5183 Å, and the minimum inter-layer distance is 3.3817 Å. For a twist angle of 29.9886°, the magnitude of corrugation is 0.0008 Å, the maximum inter-layer distance is 3.4508 Å, the minimum inter-layer distance is 3.4492 Å, and the average inter-layer distance is 3.45 Å.


Question: How does the maximum rate of flow in a non-uniform network relate to the minimum node-cut-set and the rest time τ?

Answer: The maximum rate of flow in any stage is given by (maximum flow f) / (minimum value of τ for obtaining the max flow f). Therefore, the maximum rate of flow is |Mk|ω / △Tu, where |Mk| is the cardinality of the minimum node-cut-set, ω is a constant, and △Tu is the time required for the network to dissipate heat and become functional again.


Question: How does the transformation of K × K⟨−⟩A-coalgebras into K × (−)A-coalgebras work, and what is the resulting class of power series?

Answer: The transformation of K × K⟨−⟩A-coalgebras X → K × K⟨X⟩A into K × (−)A-coalgebras K⟨X⟩ → K × K⟨X⟩A is done using an inductive scheme. For example, for a polynomial σ, the transformation is defined as follows: σ ˆo(σ) σa (a ∈ A) 1 1 0, xw (x ∈ X, w ∈ X*) o(x) · ˆo({w}), xaw + i(o(x))wa, and for a sum of polynomials Σ kiwi (n ∈ N, wi ∈ X*, ki ∈ K) Σ kiˆo(wi), Σ (wi)a. The class of power series that can be characterized by systems of this type (for finite X) corresponds precisely to the constructively algebraic power series, which are the usual generalization of the context-free languages.


Question: What are the axioms that define a Courant algebroid, and how do they relate to the multi-Courant bracket introduced in the main text?

Answer: A Courant algebroid over a manifold M is a vector bundle E → M equipped with a nondegenerate symmetric bilinear form ⟨·, ·⟩, a bundle map ρE : E → TM (called the anchor), and a bilinear bracket [[·, ·]] on Γ(E), satisfying the following axioms: C1) [[e1, [[e2, e3]]]] = [[[[e1, e2]], e3]] + [[e2, [[e1, e3]]]], ∀e1, e2, e3 ∈ Γ(E); C2) [[e, e]] = 1/2D⟨e, e⟩, ∀e ∈ Γ(E), where D : C∞(M) → Γ(E) is defined by ⟨Df, e⟩ = LρE(e)f; C3) LρE(e)⟨e1, e2⟩ = ⟨[[e, e1]], e2⟩ + ⟨e1, [[e, e2]]⟩, ∀e, e1, e2 ∈ Γ(E); C4) ρE([[e1, e2]]) = [ρE(e1), ρE(e2)], ∀e1, e2 ∈ Γ(E); C5) [[e1, fe2]] = f[[e1, e2]] + (LρE(e1)f)e2, ∀e1, e2 ∈ Γ(E), f ∈ C∞(M). The multi-Courant bracket introduced in the main text is a natural extension of the Courant bracket to the case of field theories, and it satisfies similar properties but is defined on the space of sections of Pr × Ps.


Question: What is the significance of the constant c(k, d) in the context of intrinsic Diophantine approximation on manifolds, and how is it used in the proof of Theorem 4.5?

Answer: The constant c(k, d) is significant because it represents an upper bound on the rate of decay of a Dirichlet function for intrinsic approximation on any k-dimensional nondegenerate submanifold M ⊆ Rd. In the proof of Theorem 4.5, c(k, d) is used to show that if ν-a.e. point of M is D-nondegenerate, then VWAM(ψc(k,d)) is a ν-nullset. Specifically, the proof involves constructing a sequence of hyperplanes Ln,s that contain rational points p/q with q ≤ Tn, and using the properties of the absolutely friendly measure µ to show that the set of points in M that are very well approximable by intrinsic rationals has measure zero.


Question: How do the scaling factors of the Apollonian packings in the candidate text relate to the conformal transformations discussed in the query text?

Answer: The scaling factors of the Apollonian packings are determined by the eigenvalues of the generators of the Apollonian group. Specifically, the string of operators that describe self-similar configurations have one of the eigenvalues of the form ζ2 where ζ = [n + 1; 1, n]. The blue box in Figure 10 shows Apollonians where the scaling factor is unity, while the red circles at the bottom have a scaling ratio of (2 + √3)2 = [3; 1, 2], and the green circles at the top have a scaling ratio of ((3 + √5)/2)2 = [2; 1].


Question: What is the relationship between the phase operator ˆQ derived from the dynamical study and the phase of the condensate ˆθ0, and how do their variances compare?

Answer: The phase operator ˆQ derived from the dynamical study is not the same as the phase of the condensate ˆθ0. The variance of ˆθ0 (at time t = 0) has the property Var ˆθ0(0)Var ˆN = 1, while the variance of ˆQ is larger than one, as shown in Fig. 4. The variances of the two phases coincide only in the BEC limit.


Question: What are the specific initial conditions and ranges used to define Earth analogs in the simulations?

Answer: Earth analogs were defined as the body at the end of each simulation closest to 1 Earth mass and 1 AU within the ranges 0.8–1.25 Earth masses and 0.8–1.2 AU. Initial embryo masses ranged from 0.005 to 0.48 Earth masses, with most simulations using a bimodal distribution of larger embryos and smaller planetesimals.


Question: What is the relationship between the support of the limiting distribution and the support of the empirical distribution in terms of the distance between their corresponding function values?

Answer: With probability at least 1 −δ, for any θ ∈ supp(ρ∞T), we have ∥φ(θ) − φN(θ)∥ ≤ Cr0 ≤ ϵ, where C is a constant from Lemma 8 and r0 is the radius of the ball containing the support of the empirical distribution.


Question: How does the trend of new cases in Italy compare to China, and what is the predicted impact of the home quarantine strategy in Italy?

Answer: Using the home quarantine strategy in Italy should fit the trend of China, with the decrease of new cases, at latest predicted to start on March 15.


Question: What specific structural measures are used to define gateway-hub ports in the GLSN, and how do they contribute to the network's integration and function?

Answer: The gateway-hub ports in the GLSN are defined using three respective structural measures: outside-module degree, inside-module degree, and participation coefficient. These measures help in understanding the modular gateway-ness connectivity and the structural core organization of the GLSN, which is crucial for network integration and the function of realizing cargo transportation in international trade.


Question: What are the transit time distributions for a larger spot diameter at various magnetic fields?

Answer: The transit time distributions for a spot diameter of 60 µm were measured at magnetic fields ranging from 1.72 mT to 2.8 mT (Figure S6).


Question: What specific numerical methods are proposed in the candidate text to solve the Riesz space fractional advection-dispersion equations, and how do they improve the accuracy of the solution?

Answer: The candidate text proposes the use of the weighted and shifted Grünwald difference operators to approximate the Riesz fractional derivative, presenting the finite difference method for the RSFADE. Specifically, the Crank-Nicolson scheme is discussed and solved in matrix form, which is proven to be unconditionally stable and convergent with an accuracy of O(τ^2 + h^2). Additionally, the Richardson extrapolation method (REM) is used to improve the convergence order to O(τ^4 + h^4).


Question: What is the significance of the maximal quantum CHSH value of 2√2 in the context of non-locality and how does it relate to the Popescu-Rohrlich correlations?

Answer: The maximal quantum CHSH value of 2√2 is significant because it sits between the classical value of 2 and the value of 4 attainable by non-signalling correlations. This value is puzzling in isolation, but when considering the full set of CHSH violations given by the singlet correlation function −cos(θ), they are not dominated by any other correlations. The Popescu-Rohrlich correlations, while extreme points in the space of non-signalling correlations, are not distinguished from quantum singlet correlations when looking at the full set of correlations for all angles θ. This suggests that there is no fundamental puzzle about why Popescu-Rohrlich or other purportedly 'super-quantum' correlations are not used by nature.


Question: What are the necessary and sufficient conditions for the sequence w j to be asymptotically a sequence of gradients in the context of the second-order lamination described in Example 6.3?

Answer: The necessary and sufficient conditions for w j to be asymptotically a sequence of gradients are A−B = a⊗n1 and M−C = b⊗n2 for some a, b ∈ Rm, where M = θ1A + (1−θ1)B and X = θ2M + (1−θ2)C.


Question: How does the Dirichlet map D behave when applied to boundary data in L2(Ω) and what is the resulting function space for Du and Du_x?

Answer: When the Dirichlet map D is applied to boundary data in L2(Ω), we have Du = ϕ + ux ∈ H1(Ω) and Du_x ∈ L2(Ω), where ϕ ∈ H1_0(Ω) and ϕ_x ∈ L2(Ω).


Question: What are the definitions of the maximal and minimal Pucci extremal operators and how do they relate to the viscosity solutions discussed in the paper?

Answer: The maximal Pucci extremal operator P+λ,Λ(M) is defined as Λ∑ei>0 ei + λ∑ei<0 ei, and the minimal Pucci extremal operator P−λ,Λ(M) is defined as λ∑ei>0 ei + Λ∑ei<0 ei, where ei are the eigenvalues of the matrix M. These operators are uniformly elliptic and fully nonlinear. A viscosity solution u to the equation satisfies the differential inequalities P+λ,Λ(D2u) + a|Du| − ut ≥ 0 ≥ P−λ,Λ(D2u) − a|Du| − ut in the viscosity sense, where a ≥ 0 is a constant.


Question: What is the approach proposed by Mizuno and Jiménez (2013) for simulating the logarithmic region without considering near-wall dynamics?

Answer: Mizuno and Jiménez (2013) proposed an off-wall boundary condition by taking advantage of the self-similarity hypothesis, where the eddy sizes scale linearly in the logarithmic region at high Reynolds number. The velocity field at the off-wall boundary location is substituted by a re-scaled and shifted copy of an interior reference plane farther from the wall, allowing the logarithmic region to be simulated without considering the near-wall dynamics.


Question: What are the specific deep learning models and techniques used in previous studies for cervical cell classification, and how do they compare to the approach proposed in this paper?

Answer: Previous studies have used various deep learning models and techniques for cervical cell classification, including AlexNet, GoogleNet, ResNet, DenseNet, and VGG-like networks. Some studies, such as [32], required pre-segmentation of cytoplasm and nucleus, while others, like [31], demonstrated that segmentation is not necessary for classification. A comparative study in [34] found that DenseNet161 provided the highest accuracy among ResNet101, Densenet161, AlexNet, VGG19, and SqueezeNet. However, most of these studies focused on binary classification, and the transferred models often failed to capture the characteristics of medical images. This paper addresses these issues by investigating methods for multiclass classification and optimizing the system's performance through hybrid deep feature fusion techniques.


Question: What are the analytic properties of the usual central trinomial polynomials Tn(x) discussed in the candidate text?

Answer: The usual central trinomial polynomials Tn(x) have only real roots, and the roots of Tn(x) interlace those of Tn+1(x) and Tn+2(x).


Question: What is the definition of a quasi-definite moment functional and how does it relate to the existence of an orthogonal polynomial system (OPS)?

Answer: A moment functional u is quasi-definite if its moments an := ⟨u, xn⟩, n ≥0, satisfy the Hamburger condition ∆n(u) := det[ai+j]n i,j=0 ≠ 0, n ≥0. A linear functional u is quasi-definite if and only if there is an OPS {Pn(x)}∞ n=0 relative to u. In this case, each Pn(x) is uniquely determined up to a non-zero constant factor.


Question: What are the synthesis results for LNS adder circuits compared to fixed-point and floating-point circuits for short word lengths up to Q(4, 4)?

Answer: For short word lengths up to Q(4, 4), the LNS adder requires lower or similar area as compared to fixed-point multipliers. The floating-point multiplier is generally smaller, but the LNS multiplier and fixed-point adder are simpler and smaller than the floating-point multiplier.


Question: What are the specific ion lifetime measurements and improvements observed in the cryogenic testing trap compared to room temperature?

Answer: The ion lifetime at 6 K was measured to be 4.5 ± 1.1 hours, which is improved by two orders of magnitude over the lifetime at room temperature.


Question: What does the stability analysis of the travelling wave solutions in the gKG equations reveal, and how is it visually represented?

Answer: The stability analysis of the travelling wave solutions in the gKG equations reveals that these solutions appear to be stable, as shown in Figures 5(a,b). These figures depict the evolution of 16 wavelengths of computed periodic travelling waves for ε = 0.0954 during about 160 periods, under both long and short wave perturbations. However, a more detailed study is needed to confirm this stability with greater certainty.


Question: What methods are used to prove the existence and concentration of nontrivial solutions for the fractional Schrödinger-Poisson type equation with magnetic fields?

Answer: The existence and concentration of nontrivial solutions are proven using the Mountain Pass Theorem to the functional Jε associated with a modified problem. The main issue involves verifying the Palais-Smale compactness condition, which is complicated by the presence of the fractional magnetic Laplacian and the convolution term. A new approximation argument is used to show that solutions of the modified problem are also solutions of the original equation, and a comparison argument is applied to deduce information on the behavior at infinity of the solutions.


Question: How does the mean-field approximation in the model account for the dynamics of gap junctions and their plasticity?

Answer: In the mean-field approximation, the model looks at the average behavior in an infinite system, ignoring spatial variations, fluctuations, and correlations. Every gap junction is assigned a probability to be either strong (f+) or weak (f−). The rate of change of the probability f+ with time is computed by considering only the nearest-neighbor gap junctional interactions. The homeostatic learning rule reinforces synchronous activity by strengthening conductance and weakens conductance during anti-synchronous activity. The probability f+(t + 1) that a gap junction is in the strong state at time t + 1 is given by the equation f+(t + 1) = f+(t) × Prob(∆F = 1) + f−(t) × Prob(∆F = 1), where Prob(∆F = 1) is the probability of a change in strength of a given magnitude.


Question: What is the specific form of the time slices T = const in the Smarr-York slicing of Minkowski spacetime, and how is the radial velocity ν related to the radius r and the pseudosphere radius A?

Answer: The time slices T = const in the Smarr-York slicing of Minkowski spacetime are given implicitly by (t + A − T)² − r² = A², where A is the radius of the pseudosphere. The corresponding radial velocity ν is given by ν = 1 / √(1 + A²/r²).


Question: How does the noise operator ˆN affect the output of the neurons in the reservoir of the echo state network (ESN)?

Answer: The noise operator ˆN affects the output of the neurons in the reservoir by introducing both additive and multiplicative white Gaussian noise. Specifically, the noisy output yi(t) is given by yi(t) = ˆNxi(t) = xi(t) · (1 + ξM(t, i) + ξA(t, i)), where xi(t) is the noise-free output, and ξM(t, i) and ξA(t, i) are the sources of multiplicative and additive white Gaussian noise, respectively.


Question: What is the proposed method for solving Problem (19) to optimality, and how does it work?

Answer: The proposed method is a cutting plane-type algorithm based on the outer approximation method by Duran and Grossmann (1986). The algorithm iteratively tightens a piecewise linear lower approximation of the objective function. The pseudocode for the algorithm, named svar_cutplane, involves finding a warm start using Algorithm 2, computing corresponding binary variables, and then performing cutting plane iterations until the condition c(z(i)) > η(i) is no longer satisfied. The final step involves estimating the coefficients using Theorem 1.


Question: How does the energy constraint model for the players in the game affect their attack and recovery strategies?

Answer: The energy constraint model for the players in the game is given by the inequality k−1 ∑ l=1 βpmp l δp l + βpmp k(t −τ p k) ≤κp + ρpt, where κp denotes the initial energy, ρp is the recharge rate, and βp is a parameter greater than ρp. This constraint affects the attack and recovery strategies by limiting the total energy used by each player. If a player keeps sending jamming or recovering signals, the maximum interval ∆p k on the time duration δp k when the player completes the attack or recovery is given by ∆p k(mp k) := κp + ρpτ p k −Pk−1 l=1 βpmp l δp l / (βpmp k −ρp). This interval is influenced by the number of attacked or recovered edges and the durations of these actions from previous games, thus affecting the players' strategies in the current game.


Question: How does the effective equation of state parameter \( w_{eff} \) change for the second potential when all fields have the same slope and couplings?

Answer: For the second potential, when all fields have the same slope and couplings, the effective equation of state parameter is given by \( w_{eff} = \frac{C_{11}}{n\lambda_1 - C_{11}} \). This shows a strong dependence on the number of fields, in contrast to the first potential type.


Question: What does the frequency distribution of flux imbalance during merging and splitting reveal about its impact on the frequency distribution of cancellation?

Answer: The frequency distribution of flux imbalance during merging and splitting shows that the flux imbalance is less than the detection limit, indicating well-balanced flux content. Additionally, the magnitude of the frequency distribution above the detection limit is larger for merging and splitting by one to two orders of magnitude compared to cancellation. Therefore, flux imbalances in merging and splitting do not have a significant influence on the frequency distribution of flux content.


Question: How are the upper bounds for the expected total backorders within the replenishment cycle defined, and how do they relate to the service level constraints?

Answer: The upper bounds for the expected total backorders within the replenishment cycle, ˜Bub_t, are defined as ˜Bub_t ≥ -˜It + Σ_{j=1}^t ejt/W Pjt. For service level constraints, ˜Bub_t ≤ (1 - βcyc) Σ_{j=1}^t Pjt ˜djt, where βcyc is the prescribed cycle fill rate, ensuring that the expected fraction of demand satisfied from stock is at least βcyc for each replenishment cycle.


Question: How does the VaR Historical Simulation method differ from the VaR Parametric Approach in terms of implementation and data requirements?

Answer: The VaR Historical Simulation method uses a set of historical data to obtain an empirical probability distribution of losses for the portfolio, from which the appropriate VaR is determined by extracting it from the associated quantile. This approach is convenient to implement and can be improved by statistical methods like bootstrapping. In contrast, the VaR Parametric Approach requires an analytic solution to determine the VaR for any cumulative probability, but not all distributions have such solutions, leading to the use of methods like Extreme Value Theory for estimation.


Question: What is the mass ratio and inclination of the binary system in the study of SDSS 0926, and what does this imply about the presence of superhumps?

Answer: The mass ratio (q = M2/M1) is 0.041 ± 0.002, and the inclination is 82.6 ± 0.3 degrees. The presence of superhumps is observed in the 2006 WHT/ULTRACAM data, and the period of this variation is in agreement with the period excess / mass ratio relationship proposed by Patterson et al. (2005).


Question: What is the largest set of unitals found in projective planes of order 16, and how do they compare to previously known unitals?

Answer: The largest set of unitals found in projective planes of order 16 consists of 553 unitals. These unitals are distributed as follows: 149 unitals in Gordon Royle’s representation, 148 unitals in Eric Moorhouse’s representation, and 256 unitals in dreadnaut representation. All found unitals in a plane are non-isomorphic pairwise, and the number of new-found unitals is greater than the number of unitals in previous studies [6] and [3].


Question: How does the model address the first passage time (FPT) problem for reaction-subdiffusion in a bounded domain with a partially absorbing boundary?

Answer: The model investigates the long-time and short-time asymptotic behavior of the FPT density by considering the small-s and large-s behavior of the corresponding Laplace transform. The FPT for adsorption is referred to as the last passage time for exiting the domain, as the particle may visit the boundary multiple times before being absorbed. The survival probability SΨ(t) is defined and used to derive the probability flux JΨ(t) into the boundary at x = 0, which is equivalent to the FPT density for adsorption.


Question: How do the QFS and PR values for the Si2C bending eigenstates change with the bending quantum number νb when λ = 0?

Answer: The QFS and PR values for the Si2C bending eigenstates are plotted against the bending quantum number νb for λ = 0. Red, blue, green, and orange lines correspond to ℓ= 0, 1, 2, and 3, respectively, with QFS represented by solid lines and PR in the U(2) basis by dashed lines.


Question: What are the specific transformations used to achieve unitarization in the model, and what do they represent?

Answer: The unitarization of the model can be achieved by applying the following special non-linear transformations: t = ts_0 −4(t1s_in −ts_0) / [1/V −V]^2, t = tv_0 −4(t1v_in −tv_0) / [1/W −W]^2, t = ts_0 −4(t2s_in −ts_0) / [1/U −U]^2, t = tv_0 −4(t2v_in −tv_0) / [1/X −X]^2. These transformations incorporate the nonzero values of vector meson widths and are used to describe the space-like and time-like data simultaneously. The parameters ts_0 = 9m^2_π and tv_0 = 4m^2_π represent the lowest branch points of isoscalar and isovector Dirac and Pauli nucleon FFs, respectively.


Question: What conditions must be satisfied for the monitoring procedure to detect changes under H∗A,1 and H∗A,2 according to Theorem 4?

Answer: Under H∗A,1, equation (2.34) holds as m →∞ as long as mδ−ε∆β (m) → ∞ when θ ≤ 2, and mθ θ−2+δ 2(θ−1) ∆β (m) → ∞ when θ > 2, for some ε > 0. Under H∗A,2, equation (2.34) holds as m →∞ as long as mδ−εσυ (m) → ∞ when θ ≤ 2, and mθ θ−2+δ 2(θ−1) συ (m) → ∞ when θ > 2.


Question: How is the attention mechanism implemented in the model to interpret the contribution of each atom in the chemical structure?

Answer: The attention mechanism is implemented by feeding the SMILES vector through a linear layer to obtain vector ⃗m and using the similarity between ⃗m and the i-th row ⃗Ri of the SMILES feature matrix as a measure of the importance of ⃗Ri. The attention weight vector ⃗ai is formulated as ⃗ai = 1 / (1 + exp(⃗Ri · ⃗m)). The output of the attention layer ⃗f is then expressed as ⃗f = Σ ⃗ai × ⃗Ri, which is used to extract the contribution of each atom for interpreting the results.


Question: How are the upper limits on the signal strength modifiers µobs and µexp derived, and what is their significance in the context of the 95% CL upper bounds on resonance cross sections?

Answer: The upper limits on the signal strength modifiers µobs and µexp are derived for different MT′ values using a maximum likelihood fit of the mγγ distributions, with the MH parameter fixed at 125 GeV. These limits are estimated at the 95% CL based on the CLs criterion using the asymptotic approximation for the test statistic. The results are verified with pseudoexperiments and cross-checked with detailed bias studies on the parameter µ. These upper limits are then translated into upper limits on σT′bqBT′→tH, which are used to put strict upper bounds on resonance cross sections, consistent with the interpretation derived from the 95% CL upper bounds on resonance cross sections.


Question: How does the protein half-life time affect the existence of oscillations in the gene regulatory network?

Answer: Robust oscillations are guaranteed if the protein half-life time is shortened. For example, when the mRNA half-life time is tr = 3 [min], there exist oscillations for 0.1 [min] ≤ tp ≤ 22 [min].


Question: What specific techniques does DRAGON use to improve dense retrieval training, and how does it compare to common wisdom in the field?

Answer: DRAGON uses a diverse data augmentation recipe, including query augmentation with a mixture of sentence cropping and generative queries, and progressive relevance label augmentation with multiple teachers. This approach contradicts common wisdom that cross encoders are the most effective teachers and human-like queries are the most suitable training data for dense retrieval. With this recipe, a single BERT-base-sized dense retriever can achieve state-of-the-art effectiveness in both supervised and zero-shot retrieval tasks.


Question: How does the potential function evolve over iterations in the strongly-convex outer objective case, and what is the final bound on the expected value of the potential function?

Answer: The potential function evolves according to the inequality E[bVt+1] ≤ (1 − µfα)E[bVt] + 2α/T + [(2ˆc2β + 8ˆc2βL2K)σ2g + 2(µf + 1)2σ2f]α2. Applying this inequality recursively, we get E[bVt] ≤ (1 − µfα)t[(ℓ(x0) − ℓ∗) + σ2f + ∥y0 − y∗(x0)∥2] + 2/µfT + [(2ˆc2β + 8ˆc2βL2K)σ2g + 2(µf + 1)2σ2f]/µfα.


Question: How do the per-unit prices and total user investments vary with the centrality of nodes in a 50-node PA graph with β = 3 under different pricing schemes?

Answer: In a 50-node PA graph with β = 3, the per-unit prices decrease with the increase in the centrality of nodes under the heterogenous pricing scheme. The total user investments are higher for users with lower centrality under the heterogenous pricing scheme compared to uniform and binary pricing schemes. This is illustrated in Figure 5, where the left plot shows the per-unit prices and the right plot shows the total user investments for heterogenous, uniform, and binary pricing schemes.


Question: What are the main aspects covered in the survey for understanding, visualizing, and explaining DNNs?

Answer: The survey covers three main aspects: (i) Visualizing and Understanding DNNs, which involves visualizing features in pre-trained networks to understand learned kernels and data structures; (ii) Explaining DNNs, which describes the overall and internal behaviors of DNNs, including input-output relations and mathematical derivatives; and (iii) Explainability by Design, which explores DNN architectures that are inherently explainable.


Question: What are the estimated normalised power-law weights and how do they compare to the unconstrained weights?

Answer: The estimated normalised power law has a decay of ˆd = 1.80 (95% CI: 1.61 to 2.01). The unconstrained weights decrease monotonically and closely resemble the estimated power law, with the exception of the fifth order of neighbourhood, which has a slightly higher weight. The results with raw weights are very similar.


Question: What is the impact of the geometry of elongated spheroids on the spontaneous emission rate of CdSe quantum dots, and how does this compare to more symmetric nanoparticles?

Answer: The spontaneous emission rate of CdSe quantum dots increases by 10-15 times when near a silver nanowire, which is effectively an elongated spheroid. This enhancement is due to the presence of plasmon resonances with different eigenfrequencies, an effect that is not observed in more symmetric nanoparticles.


Question: How does the transformation of the minimum point affect the event horizon in extremal black holes, and what does this imply for the Weak Cosmic Censorship Conjecture (WCCC)?

Answer: The transformation of the minimum point, rmin → rmin + drmin, and the event horizon, rh → rh + drh, results in a small shift in h(r), denoted by dhmin. At the new minimum point, h(rmin + drmin) = hmin + dhmin, and since dhmin = 0, the event horizon always exists. This implies that the WCCC is valid in the EPS, and the extremal black hole remains extremal after the absorption, meaning that a fermion with sufficient momentum and charge would not overcharge the extremal black hole in Rastall gravity.


Question: How does the family of Nesterov Lagrangians form an orbit under the action of polynomial time dilations?

Answer: The family of Nesterov Lagrangians (4.11) forms an orbit under the action of polynomial time dilations Tpol. Specifically, if we start from α⋆ = −log t + log 2 (for L⋆(4.11)), then for any p > 0, the time dilation τ(t) = t^p/2 sends us to (α⋆)(τ) = −log t + log p. This means that the Nesterov flows (4.1) can be viewed as the result of speeding up the Lagrangian flow of L⋆(p = 2) or any starting curve.


Question: How does the dynamical Bethe-Salpeter equation (BSE) differ from the static BSE in terms of the screened Coulomb interaction?

Answer: In the dynamical BSE, the frequency dependence of the screened Coulomb interaction W is preserved, and it is replaced by an effective, frequency-dependent ˜W(zm) through Shindo’s approximation. This leads to an eigenvalue problem for the frequency-dependent BSE Hamiltonian ˜Hvck,v′c′k′(ω) = (Eck − Evk)δvv′δcc′δkk′ + 2¯vv′c′k′vck − ˜W v′c′k′vck(ω). In contrast, the static BSE assumes W(0), the zero-frequency, static limit, and the Hamiltonian is frequency-independent.


Question: What are the ratios of the forward-backward and polarization asymmetries for the decays involving muons and tau leptons?

Answer: The ratios of the forward-backward asymmetries are A(K)µ / A(Ds)µ = 0.470(41) and A(K)τ / A(Ds)τ = 0.804(15). The ratios of the polarization asymmetries are P(K)µ / P(Ds)µ = 1.0193(17) and P(K)τ / P(Ds)τ = −0.25(11).


Question: How does the re-emergence timescale of the magnetic field depend on the total amount of accreted matter, and what are the implications for young neutron stars in supernova remnants?

Answer: The re-emergence timescale of the magnetic field depends on the total amount of accreted matter, ranging from ∼103 to 104 years for total accreted masses of ∼10−4 to 10−3 M⊙. Relatively weak magnetic fields (∼1010 to 1011 G) have been measured in a few young neutron stars in supernova remnants (SNRs) named central compact objects (CCOs), which might have experienced the field burial and re-emergence process.


Question: How does the rotating Simpson-Visser metric differ from the Kerr metric, and what are the conditions for it to represent a black hole or a wormhole?

Answer: The rotating Simpson-Visser metric differs from the Kerr metric by the presence of the term ˆ∆ in the metric components. The metric is given by ds2 = −(1 − 2Mr/Σ)dt2 − 4Mar sin²θ/Σ dtdφ + Σ/∆ˆ∆ dr² + Σ dθ² + (r² + a² + 2Ma²r sin²θ/Σ) sin²θ dφ², where Σ = r² + a² cos²θ, ∆ = r² − 2Mr + a², and ˆ∆ = 1 − r²₀/r². For r₀ = 0, ˆ∆ = 1, and the metric reduces to the Kerr black hole. The geometry represents a black hole when r₀ < r+ and a wormhole when r₀ ≥ r+, provided 0 ≤ a/M ≤ 1. For a/M > 1, it always represents a wormhole as r± does not exist in this case.


Question: What does the non-zero correlation in the anisotropic coefficients of D mesons suggest about their behavior in the medium?

Answer: The non-zero correlation in the anisotropic coefficients is consistent with charm quarks sharing with light quarks the collective motion of the medium.


Question: How does the RePtA VPP manage power shortages and surpluses on both monthly and daily timescales, and what are the associated costs?

Answer: On the monthly timescale, a minor wind period in Inner Mongolia from June to August results in a significant drop in renewable generation, with July's output being approximately 74,500 MWh, much lower than the average 102,600 MWh. Despite this, energy demand for ammonia production continues to increase. On the daily timescale, the load is maintained at 110 MW, but renewables have inherent intermittency, with power peaking at 13:00 and falling to 0 MW in the evening, leading to power shortages. The conventional operating strategy can result in the RePtA VPP purchasing expensive electricity during shortages and selling it at a low price during surpluses. For example, in July, the electricity price is $9.98/MWh higher than the annual average, and on a typical day, the price of selling electricity at 14:00 is $15.12/MWh lower than the average daily price.


Question: What does the recent analysis of M dwarf planetary systems reveal about the occurrence rates of planets with minimum masses below 13 M⊕?

Answer: The recent analysis reveals that there is a rapid increase in the occurrence rate of planets with minimum masses below 13 M⊕ for periods below roughly 100 days, which corresponds to a similar increase for planets with radii below 3 R⊕. This increased occurrence rate of smaller planets also applies to longer orbital periods but is shifted towards higher minimum masses of roughly 13-30 M⊕.


Question: How does the time-to-solution for analog and digital computers differ when solving a problem M times as complex as a simple ODE?

Answer: For an analog computer, the time-to-solution remains constant, i.e., T M A = T 1 A, or T M A = O(1). In contrast, a single-core digital computer's time-to-solution scales linearly with the problem size, i.e., T M D = M · T 1 D, or T M D = O(M).


Question: How is the equivariant first Chern class of the equivariant line bundle O(1) related to the Schubert class σT ∨ and the alternative equivariant Schubert class ˜σT ∨?

Answer: The equivariant first Chern class of the equivariant line bundle O(1) is denoted by ζ. The Schubert class σT ∨ is given by σT ∨ = ζ − (xk+1 + ... + xn) = ζ − x∅. The alternative equivariant Schubert class ˜σT ∨ is given by ˜σT ∨ = ζ − (x1 + ... + xn−k) = ζ − xλmax = σT ∨ + x∅ − xλmax.


Question: What are the specific dataset splits and statistics used in the few-shot learning experiments for the birds, cars, aircrafts, dogs, and flowers datasets?

Answer: For few-shot learning experiments, the datasets are split into base, validation, and novel sets. The statistics are as follows: Birds - Base: 100 classes, 5885 images; Val: 50 classes, 2950 images; Novel: 50 classes, 2953 images. Cars - Base: 98 classes, 8023 images; Val: 49 classes, 4059 images; Novel: 49 classes, 4103 images. Aircrafts - Base: 50 classes, 5000 images; Val: 25 classes, 2500 images; Novel: 25 classes, 2500 images. Dogs - Base: 60 classes, 10337 images; Val: 30 classes, 5128 images; Novel: 30 classes, 5115 images. Flowers - Base: 51 classes, 4129 images; Val: 26 classes, 2113 images; Novel: 25 classes, 1947 images.


Question: What are the characteristics of the high-strain regions in Figure 3(c) and (d) and how do they differ from the ideal case shown in (a)?

Answer: In Figure 3(c), the dominant feature has a pore-like structure with intermixed low and high-strain material regions. In Figure 3(d), the high-strain region features a large vertical component. These characteristics differ from the ideal case shown in (a), where a single, horizontal shear band forms.


Question: What are the flux and velocity measurements for the 29SiO(3 −2)v0 transition in RX Boo, and how do they compare to the 30SiO(3 −2)v0 transition?

Answer: For the 29SiO(3 −2)v0 transition in RX Boo, the flux is 45.34 Jy km s−1 with an error of 0.05, and the VLSR is 1.35 km s−1 with an error of 0.08. The Vmin is -5.8 km s−1 and Vmax is 8.3 km s−1. For the 30SiO(3 −2)v0 transition, the flux is 32.73 Jy km s−1 with an error of 0.06, and the VLSR is 1.09 km s−1 with an error of 0.09. The Vmin is -4.8 km s−1 and Vmax is 7.1 km s−1.


Question: How do the skill ratings and other metrics compare between the fully-trained generative models and real data samples in the experiments?

Answer: The skill rating (SR) of real data samples is 1532, while the skill ratings of the experiments are as follows: 5-cond (1530), 4-cond (1528), 1 (1528), 6-auto (1519), 3 (1508), and 2 (1487). The SVHN classifier score (CS) and SVHN Fréchet distance (FD) for real data are 8.09 and 3.03, respectively. The skill rating system ranks experiment 5-cond as slightly worse than real data and slightly better than 4-cond and 1, while the classifier score ranks 5-cond better than real data, and Fréchet distance ranks 5-cond worse than both 4-cond and 1. The system’s rankings agree with Fréchet distance in all other cases.


Question: How is the Lexis optimization problem formulated, and what is the cost function used?

Answer: The Lexis optimization problem is formulated as constructing a minimum-cost Lexis-DAG for the given alphabet S and target strings T. The cost function C(D) is defined as the number of edges in the Lexis-DAG, which is the sum of the in-degrees of all nodes in the graph: C(D) = E(D) = ∑_{v∈V} din(v) = |E|. This problem is NP-Hard, as shown by a reduction from the Smallest Grammar Problem (SGP).


Question: How is the BFKL impact factor calculated from the light-front wave function overlap, and what is the expression for ΦV,T (γ, z, M)?

Answer: The BFKL impact factor is calculated from the light-front wave function overlap using the expression: ℑmAγ∗p→V pT(W, 0) = ∫d²r ∫dz (Ψ∗V Ψ)T(r) · σ0N(x, r) = αs(M · Q0) ∫dγ (1/2 + i∞ to 1/2 - i∞) (1/2πi) ∫dz (1/4π) ˆg(x, M²/Q²0, M², Q0, γ) · ΦV,T(γ, z, M) · (M²/Q²0)γ. The expression for ΦV,T(γ, z, M) is given by: ΦV,T(γ, z, M) = 4π/Nc ∫d²r ∫d²k (k²)² (1 - eik·r) (k²/M²)γ (Ψ∗V Ψ)T(r) = eˆef8π²NT Γ(γ)Γ(1 - γ) (m²f) (m²fR²/8z(1 - z))² e^(-m²fR²/8z(1-z)) e^(mfR²/2) (8z(1 - z)/M²R²)γ [U(2 - γ, 1, ϵ²R²/8z(1 - z)) + (z² + (1 - z)²)ϵ²(2 - γ)/2 · m²f U(3 - γ, 2, ϵ²R²/8z(1 - z))], where U(a, b, z) is a hypergeometric function of the second kind or Kummer’s function.


Question: How does the initial variation of the Higgs VEV (δvini) influence the axion density (ωa) and the axion mass (ma) in the axi-Higgs model?

Answer: The initial variation of the Higgs VEV (δvini) determines the value of Ca2_ini, while the axion density (ωa) determines the value of aini. The axion mass (ma) is determined by the requirements that δvini stays unchanged (or mildly changed) until near or after the recombination and oscillates with a highly-suppressed amplitude at low redshift and today.


Question: What does Contract 3 state about the intentional declassification of secret values?

Answer: Contract 3 states that secret values written to public memory are intentionally declassified by the program.


Question: What are the specific results of the MIRC observations regarding the presence of a companion around zet Aql?

Answer: The MIRC observations found a 3σ upper limit ranging between 1% and 2% (at 90% completeness) on the contrast of any companion located between 1 mas and 200 mas (∼5 AU). The observations mostly rule out the presence of a faint companion as the source of the near-infrared excess around zet Aql up to a distance of about 5 AU, although it cannot be excluded that a faint companion hides in the parts of the search region where the MIRC sensitivity is larger than 1.69%. Companions beyond 8 AU can also be ruled out, thanks to high-contrast adaptive optics observations obtained with CFHT/PUEO.


Question: How does the Nash bargaining solution differ from the Kalai-Smorodinsky bargaining solution in terms of the core of the induced coalition formation problem?

Answer: The Nash bargaining solution guarantees a non-empty core of the induced coalition formation problem, while the Kalai-Smorodinsky bargaining solution can result in an empty core for some coalitional endowments.


Question: What are the key contributions of the APOGEE survey in the context of Milky Way studies, and how does it compare to other spectroscopic surveys in terms of resolution and sample size?

Answer: The APOGEE survey has obtained infrared (H-band) spectra for upwards of 400,000 stars at a resolution of R∼21,000. This high resolution allows for the derivation of detailed stellar atmospheric parameters (Teﬀ, log g, [Fe/H]) and chemical abundance ratios for many elements. Compared to other surveys, APOGEE offers a balance between high resolution and a large sample size, making it particularly useful for studying the chemical and kinematic properties of the main Galactic components, including the bulge.


Question: What are the falloff rates of the nontrivial modes of the constrained variables for near-Kerr initial data, and how do they align with the conditions of weak asymptotic flatness?

Answer: The falloff rates of the nontrivial modes of the constrained variables, as shown in Figure 6, are consistent with the conditions of weak asymptotic flatness. Specifically, the modes |(∆)Kℓ0|, |Re[(∆)kℓ0]|, |Im[(∆)kℓ0]|, and |κℓ0| exhibit falloff rates of r−1.7, r−1, r−2, and r−2, respectively.


Question: What is the significance of PT symmetric and η pseudo Hermitian Hamiltonians in the context of the nonlinear Schrödinger equation (NLSE)?

Answer: PT symmetric and η pseudo Hermitian Hamiltonians admit real eigenvalues despite being non Hermitian. The study of NLSE with complex PT symmetric potentials has drawn a lot of attention due to intrinsic interest and possible applications, particularly in optics. Soliton solutions of the NLSE with constant nonlinearity and complex potentials have been obtained, and schemes for the experimental observation of PT symmetry have been reported.


Question: How is the covariant derivative of a vector field and a differential 1-form expressed in local coordinates according to the provided formulas?

Answer:
The covariant derivative of a vector field \( V^j \partial_j \) and a differential 1-form \( eta_j dx^j \) in local coordinates is given by the formulas:
∇_i (V^j ∂_j) = (∂_i V^j + Γ^j_{ik} V^k) ∂_j,
∇_i (β_j dx^j) = (∂_i β_j - Γ^k_{ij} β_k) dx^j,
where ∂_i = ∂/∂x^i.


Question: What are the specific motion families of 4-cycles identified in the analysis of the graph Q1?

Answer: The analysis of the graph Q1 identifies six groups of motion types of 4-cycles, resulting in eight motion families — irreducible algebraic sets of proper flexible labelings. These families are detailed in [9] and illustrated in Figure 12.


Question: What is the optimality property of the Root stopping time in the context of the Skorokhod embedding problem?

Answer: The Root stopping time minimizes E[τ2] among all solutions to the Skorokhod embedding problem for a fixed terminal distribution µ.


Question: What are the specific results for the Rossiter-McLaughlin effect observed on WASP-17b, including the values of V sin I and β?

Answer: For WASP-17b, the Rossiter-McLaughlin effect is well fitted. The residuals show some dispersion about the model during the HARPS sequence. V sin I is 9.92 km s−1 and β is 148.5°+5.1−4.2, indicating a severely misaligned orbit.


Question: What are the four non-isomorphic rings of order 4, and what are their characteristics and associated geometric structures?

Answer: The four non-isomorphic rings of order 4 are: 1) The Galois ring GF(4), which gives rise to the projective plane PG(2, 4). 2) The chain ring Z4, which is isomorphic to GR(4, 4) and W2(2), with characteristic 4, and is the coordinate ring of a projective Hjelmslev plane. 3) The chain ring D(2), which is isomorphic to A(2, 2), has characteristic 2, and is associated with another projective Hjelmslev plane. 4) The non-local ring B(2), which is isomorphic to GF(2)[t]/(t^2 - t), has characteristic 2, and is associated with a Veldkamp plane with a non-transitive neighbor relation.


Question: How does the presence of the brane effective cosmological constant term (Λ) affect the quasi-homologous symmetry in the vacuum brane system?

Answer: The presence of the brane effective cosmological constant term (Λ) breaks the usual quasi-homologous symmetry. This is because Λ is a fixed parameter, and the system of equations taking into account the cosmological constant term is not invariant under rescaling transformations.


Question: What evidence supports the 27 Myr periodicity of extinction in both the Sepkoski and Paleobiology Database data sets?

Answer: The 27 Myr periodicity of extinction is found in both the Sepkoski (SEP) and Paleobiology Database (PBDB) data sets, as reported in Melott & Bambach (2010) for the 2004 version of the time scale and in this paper.


Question: What is the impact of including BAO data on the H0 tension when using the w(z)CDM dark energy model with Planck + SNIa data?

Answer: Including BAO data in the analysis drags the H0 value towards values that are again in tension with SH0eS. This effect seems to arise from the fact that while in the Planck + SNIa combination the third reconstruction bin yields w3 < −1, when BAO data are included the reconstruction favors w3 > −1, while in all other redshift bins the results obtained in the Planck + SNIa and Planck + SNIa + BAO combinations are compatible with each other.


Question: What is the relationship between finding metric balls with maximum sum of radii and finding a minimum-length set of cycles in the complete geometric graph?

Answer: Finding metric balls with maximum sum of radii is equivalent under linear programming duality to finding a minimum-length set of cycles (allowing 2-cycles) that cover all vertices of the complete geometric graph on the given centers. The maximum sum of radii equals half of the minimum total cycle length.


Question: What are the limitations of using (1 + 1)-dimensional configurations to explore holographic inequalities, and how do they differ from higher-dimensional cases?

Answer: In (1 + 1)-dimensional configurations, the mutual information vanishes easily when the distance between intervals increases, which limits the ability to realize arbitrary patterns of correlations. This is in contrast to higher-dimensional cases where the shape of regions can be deformed to achieve any desired pattern of mutual information. Additionally, in (1 + 1)-dimensions, configurations are constrained, and more complicated setups with multiple intervals of different colors are required, which can obfuscate the pattern of constraints. This limitation implies that generically, all holographic inequalities in (1 + 1)-dimensions collapse down to SSA, and MMI does not provide new information beyond SSA.


Question: What are the specific harmonic field characteristics for Ne atoms exposed to ω–2ω and ω–3ω bicircular fields, and how do they vary with different intensity ratios?

Answer: For Ne atoms exposed to ω–2ω and ω–3ω bicircular fields, the harmonic field characteristics are presented in Fig. 5. The fundamental laser wavelength is 1300 nm, and the sum of the component peak intensities is fixed to I1 + Ir = 8×10^14 W/cm², with r = 2, 3. The results show the real parts of the harmonic field components (Re Σn En,x and Re Σn En,y) for different intensity ratios I2/I1 and I3/I1, as denoted in the upper left corner of each subpanel. The harmonic orders (n1 to n2) are also indicated in the upper right corner of each subpanel.


Question: How is the probability of the first error event (E1) bounded in the context of the modified joint typicality decoder?

Answer: The probability of the first error event (E1) is bounded by Pr {Ec1|m = 1} < ϵ if log p (x) − Hp (Y | X = x) = c, ∀x ∈X, for sufficiently large N. This is derived from the condition that the probability of (X (1), Y) not being in M (N) ϵ,p(x,y) is less than ϵ, as shown in Lemma 13.


Question: What specific observations in the mesospheric meridional wind support the semi-annual variation discussed in the query text?

Answer: A period of westward propagating wave activity can clearly be seen around fall equinox in the S1 mode, as shown in the Hovmöller diagrams of zonal wave numbers 1 and 2 (S1, S2) of the mesospheric meridional wind for the year 2000.


Question: How does the Skip-Gram with Negative Sampling (SGNS) model differ from the typical neural network language model (NNLM) in terms of its architecture and training process?

Answer: The SGNS model, as implemented in the word2vec package, learns word embeddings without a non-linear hidden layer, unlike the typical neural network language model (NNLM). In the SGNS model, each word type w ∈ V is associated with a pivot word representation vector ⃗w ∈ R^d and a context embedding vector ⃗wc ∈ R^d. The training objective is to maximize the probability of predicting context words for each pivot word in the corpus, using the softmax function. The global training objective J is to maximize the probabilities that all pairs from the training set D are indeed observed in the corpus.


Question: How is the continuous family of α|N-cocycles {uh(y, t)}h∈N, (y,t)∈I2 defined in the proof of Theorem 1.9?

Answer: For h ∈N, and (y, t) ∈I2, the continuous family of α|N-cocycles {uh(y, t)}h∈N, (y,t)∈I2 is defined as follows: uh(y, t) = r((1 −t)y)−1ch(t)αh(r((1 −t)y)), for (y, t) ≠ (1, 0), and uh(y, t) = 1, for (y, t) = (1, 0).


Question: What are the masses and widths of X(2120) and X(2370) as reported by the BESIII collaboration?

Answer: The masses and widths of X(2120) and X(2370) are determined to be MX(2120) = 2124.8 ± 5.6 MeV, ΓX(2120) = 101 ± 14 MeV, MX(2370) = 2371.0 ± 6.4 MeV, and ΓX(2370) = 108 ± 15 MeV.


Question: What does Figure C.1 show in terms of temperature maps and core positions?

Answer: Figure C.1 shows the H2CO and CH3CN temperature maps derived with XCLASS for each region. Each core position is marked in red and the derived outer radii from the radial temperature profiles are indicated by dashed red circles.


Question: What spectroscopic evidence supports the rotational motion in coronal jets, and how does this relate to the magnetic setup of larger eruptions?

Answer: Spectroscopy in the UV from IRIS has provided valuable insight into coronal jets, including studies finding rotational (spinning) motion in coronal jets (e.g., Cheung et al., 2015; Liu et al., 2018; Schmieder et al., 2022; Ruan et al., 2019), confirming indications of such rotation from earlier observations in EUV (Pike and Mason, 1998). This rotational motion is consistent with the magnetic setup of larger eruptions, where the same type of magnetic setup appears to be capable of generating similar solar expulsions both on the coronal-jet size scale and on the size scale of typical solar eruptions.


Question: What do the azimuthal intensity distributions in Figure 6 show, and how do they relate to the images in Figure 3?

Answer: The azimuthal intensity distributions I(Φ) in Figure 6 are extracted from the Fourier transform diagrams in Figure 3, delimited by the green circles. The graphs are grouped in pairs, obtained with the same laser power (0, 8.8, 9.5, 10.7 mJ/cm2). The lines are guides to the eyes, with dots and their corresponding full lines coming from the images taken at t = -250 ps, and squares and dashed lines coming from the images taken at t = +190 ps. The vertical dashed arrows show the intensity reduction, accounting for the reduction of the XMCD contrast in the XMCD-PEEM images, and the horizontal full arrows show the width at half maximum of the peaks.


Question: How does the Max Pooling layer in the EC2 network contribute to the reduction of feature map dimensions and what is its specific role in the network architecture?

Answer: The Max Pooling layer in the EC2 network, denoted as MaxPooling_3, reduces the dimension of the input feature maps to a size of h/4 × w/4 × 64. This is achieved by combining nearby inputs to be invariant to small translations and to reduce the dimensionality of feature maps, which is a useful property if the presence of a feature is more important than its exact location. The Max Pooling operation is performed with a kernel size and stride that result in the maximum value being carried over from the receptive field, effectively reducing the dimensions of the feature maps.


Question: What are the details of the MNIST-USPS dataset used in the experiments?

Answer: The MNIST-USPS dataset contains 5000 handwritten digital image samples from numbers 0 to 9, with 10 clusters and 2 views, each with a dimensionality of 1*28*28.


Question: What is the relationship between the time scale of black brane formation and its inverse temperature, and how does this relationship affect the computation of the event horizon?

Answer: The time scale of formation of the brane is much smaller than its inverse temperature. This fact allows us to compute the event horizon of the spacetime in a simple and explicit fashion in a power series in δtT ∼ǫ^2/d. To leading order in ǫ, the event horizon manifold is given by rH(v) = M^(1/d) for v > 0 and rH(v) = M^(1/d) (1 − M^(1/d) v) for v < 0.


Question: What is the probability generating function Gν(u,t) of Nν(t) and how is its Laplace transform derived?

Answer: The probability generating function Gν(u,t) of Nν(t), t > 0, has the Laplace transform given by the integral Z ∞ 0 e−µtGν(u,t)dt = Z ∞ 0 ue−λt 1 −u(1 −e−λt)µν−1e−µνt dt. This is derived by evaluating the Laplace transform as a series expansion and simplifying it using properties of the Mittag-Leffler function and the Beta function.


Question: How do the parameter estimates and standard errors for the ST model change as the degrees of freedom (ν) increase from 2 to 45, and what is the impact on the number of influential observations?

Answer: As the degrees of freedom (ν) increase from 2 to 45, the parameter estimates for L∞, K, and t0 in the ST model become more stable, with L∞ increasing from 35.025 to 35.134, K decreasing from 0.086 to 0.083, and t0 increasing from -2.896 to -3.071. The standard errors for L∞, K, and t0 generally decrease, with SE(L∞) decreasing from 0.059 to 0.068, SE(K) decreasing from 0.001 to 0.001, and SE(t0) decreasing from 0.199 to 0.225. The number of influential observations decreases from 54 (2.01%) for the N model to 10 (0.37%) for the ST model with ν = 51.


Question: What are the properties of the inner solution arcs found in Theorem 4.25, and what uniform bound is established on their time intervals?

Answer: The inner solution arcs found in Theorem 4.25 are classical (collision-less) solutions of the inner problems, re-parametrizations of minimizers of the Maupertuis' functional, and free of self-intersections. There exists a uniform bound on the time intervals of these solutions, such that for any ε ∈ (0, εint), there exist constants c, C > 0 such that c ≤ TPj(p1, p2; ε) ≤ C, where TPj(p1, p2; ε) is the time interval of the solution yPj(·; p1, p2; ε). These constants do not depend on the choice of p1, p2, and Pj.


Question: What does the vibrational IPR (VIPR) indicate about the localization of vibrational modes in Cu-doped Ta2O5, and how does it vary with frequency?

Answer: A small value of VIPR indicates evenly distributed vibration among the atoms, while higher values imply only a few atoms contributing at that particular eigenfrequency. Below ∼300cm−1, the vibrational modes are completely delocalized/extended. Above 300cm−1, higher VIPR values are observed, indicating that high frequency modes are localized on a few O atoms in the network, while low frequency modes are spread among a larger number of Ta and Cu atoms.


Question: How does the mediator ensure that the equilibrium distribution µd is implemented in the mediated game M∗(Γ)?

Answer: The mediator ensures that the equilibrium distribution µd is implemented in the mediated game M∗(Γ) by recommending actions and ensuring that players are truthful and obedient provided that they have been in the past. At truthful histories, the mediator is omniscient and players have an incentive to be obedient provided they have been in the past, which results in the Bayes correlated equilibrium.


Question: How does the gradient-based local search algorithm handle the selection of blocks in the weighted version of the set packing problem?

Answer: In the weighted version of the set packing problem, Loop 1 of the local search algorithm can take the same form as in the LocalSearch, where selected blocks A∗(t) ∈F, t = 1, ..., |P| of the generated partition P are any of those feasible subsets where the sum, rather than the minimum, over elements i ∈A∗(t) of (i, A∗(t))-derivatives ∂ˆW(q(t −1))/∂qA∗(t)i(t −1) is maximal.


Question: How are the switching losses in the IGBT calculated, and what factors influence these losses?

Answer: The switching losses in the IGBT are calculated using the formula 𝑃𝑃sI = 𝑓𝑓s ∙(𝐸𝐸on + 𝐸𝐸off) ∙(1 + 𝑇𝑇𝑇𝑇(𝑇𝑇J − 𝑇𝑇Ref)) ∙ (𝑉𝑉CE / 𝑉𝑉ref)^𝐾𝐾𝑉𝑉. The factors that influence these losses include the switching frequency (fs), the turn-on and turn-off energies (𝐸𝐸on and 𝐸𝐸off), the junction temperature (TJ), the reference junction temperature (TRef), the blocking voltage (VCE), the reference blocking voltage (Vref), and the temperature coefficient (𝑇𝑇𝑇𝑇) and voltage exponent (𝐾𝐾𝑉𝑉). Additionally, the switching losses are highly dependent on the gate resistance, and increasing the gate resistor can significantly increase the switching losses.


Question: What are the specific conditions under which MSO2-model checking is not fixed-parameter tractable according to the main result of the paper?

Answer: If the tree-width of a class C of graphs is not bounded by log16 n, then MSO2-model checking is not fixed-parameter tractable unless SAT can be solved in sub-exponential time. If the tree-width of C is not poly-logarithmically bounded, then MSO2-model checking is not fixed-parameter tractable unless all problems in the polynomial-time hierarchy, and hence in particular all problems in NP, can be solved in sub-exponential time.


Question: How does the initial rotation of the gas halo affect the star formation activity and gas accretion in the disk?

Answer: The initial rotation of the gas halo leads to more gas accretion from the halo to the disk but delays the star formation activity in the disk. The SFR in run DHi (without rotation) is a bit higher than that in run DHir (with rotation) in the middle of the evolution, and the mass of the gas disk is slightly greater in run DHir.


Question: What explains the unpolarized emission of the phonon replicas in both linear and circular basis measurements, despite the parent peaks being linearly polarized?

Answer: The phonons involved in the scattering process giving rise to the replicas are doubly-degenerate, orthogonal circularly polarized (CP) phonons with lphonon = ± 1. The linearly polarized localized X0 can be thought of as a superposition of σ+ and σ−photons, leading to a maximally entangled state of the emitted phonon and replica photon system. When we measure the polarization of the replica photon without access to the phonon angular momentum (AM) information, the state of the photon subsystem becomes a completely mixed state of polarization, independent of the basis, resulting in unpolarized emission.


Question: What does the temperature stratification inside the light bridge suggest about its origin and structure?

Answer: The temperature stratification inside the light bridge is similar to the temperature stratification inside penumbral filaments, indicating that the filaments in the light bridge have a similar origin to penumbral filaments, where magneto-convection is supplying the filaments with hot material from deeper layers.


Question: How does the double-sided algorithm compare to the single-sided algorithm in terms of recovering the long wavelength parts of the model and the vertical boundaries of the perturbation in the smooth background velocity case?

Answer: In the smooth background velocity case, the double-sided algorithm can recover the long wavelength parts of the model and the vertical boundaries of the perturbation, while the single-sided algorithm cannot. The double-sided approach predicts the reflected event coming from the lower boundary more accurately, resulting in a faint image of the vertical sides of the rectangular perturbation, which is more comparable to the true perturbation.


Question: How does the implementation of the MBAR reweighting scheme with waste-recycling in SUNDAE compare to previous applications of MBAR in nonequilibrium path-ensemble averages and Langevin dynamics?

Answer: This study is the first to implement the MBAR reweighting scheme for waste-recycling. However, MBAR has already been applied to nonequilibrium path-ensemble averages for computing free energy differences and to Langevin dynamics for estimating time-correlation functions.


Question: How does the assumption that p is greater than the index of H in ⟨A⟩ affect the Sylow p-subgroups of H and ⟨A⟩?

Answer: The assumption that p is greater than the index of H in ⟨A⟩ implies that a Sylow p-subgroup of H is a Sylow p-subgroup of ⟨A⟩. Since H lies in a Borel subgroup of GLr(K), a Sylow p-subgroup of H is normal in H and is equal to Op(H). All Sylow p-subgroups of ⟨A⟩ lie in H, hence they all coincide with Op(H); we conclude that Op(H) is normal in ⟨A⟩ and is equal to Op(⟨A⟩).


Question: How is the asymptotic expansion of the composite mean K(N1, N2) derived in the context of symmetric means K, N, and M?

Answer: The asymptotic expansion of the composite mean K(N1, N2) is derived by considering the specific properties of the means N1 and N2. Despite N1 and N2 not being symmetric, their difference is antisymmetric and their sum is symmetric. This allows the composition K(N1, N2) to be symmetric for symmetric means K, N, and M. The symmetric form of the asymptotic expansion of K(N1, N2) is difficult to deduce directly from Theorem 2.1 in the paper by Burić and Elezović, so a similar procedure was conducted from scratch to obtain the desired result.


Question: How is the distance between two expression trees defined, and what are the specific cases that contribute to this distance?

Answer: The distance between two expression trees is defined as the number of nodes assigned different operators/operands. The specific cases that contribute to this distance are: a change of a node assignment from an operator/operand to another operator/operand, an addition of a new node, and a deletion of a node. Changes in constant values are not counted in this distance.


Question: What is the significance of the root in a rooted binary phylogenetic tree, and how does it relate to the concept of speciation events?

Answer: The root in a rooted binary phylogenetic tree corresponds to the common ancestor of all the species in the tree. The directed edges from the root indicate the direction of time. In a binary tree, every vertex other than the root has degree one or three, and the root itself has degree two. The degree three internal vertices represent speciation events, where one species at the time just prior to the vertex splits into two species that emerge from the vertex.


Question: How is the LST of the batch sojourn-time distribution of an arbitrary batch eT (ω) calculated in the context of locally-gated service?

Answer: The LST of the batch sojourn-time distribution of an arbitrary batch eT (ω) is given by eT (ω) = ∑_{k∈K} π (k) eTk (ω), where eTk (ω) is given by (19) or (67). Alternatively, it can be written as eT (ω) = 1/E (C) ∑_{j=1}^N ∑_{i=1}^N (E (Vj−1) fW (Vj−1) i (ω) + E (Sj−1) fW (Sj−1) i (ω)) × π (Kj,i) eK (B∗ j,i|Kj,i).


Question: What are the implications of long-ranged effective interactions in the context of large deviations in non-equilibrium systems, and how do they affect the interpretation of these systems?

Answer: Long-ranged effective interactions in non-equilibrium systems can lead to unusual new behavior, such as phase transitions to states with long-ranged order in one-dimensional systems. This makes interpreting and analyzing results for large deviations more difficult, as it is not clear what effective interactions might be required to stabilize these states, and it is therefore challenging to understand what kinds of equilibrium or non-equilibrium protocols might be used to prepare them in the laboratory.


Question: How does the introduction of a secondary periodic stimulus current affect the firing patterns of an integrate-and-fire neuron, and what are the conditions for the continuity of the return map F?

Answer: The introduction of a secondary periodic stimulus current E cos ωt introduces a competing time scale Tdrv = 2π/ω, leading to a loss of simple periodic behavior. The neuron can exhibit both periodic and aperiodic firing patterns. The return map F is continuous when RI ≥ E + ∆v, and discontinuous otherwise. When condition (2) fails, the model has a solution with ˙v = 0 at threshold, causing a jump discontinuity in the return map F.


Question: What is the reliability of spectroscopic redshifts compared to photometric redshifts, and how are they prioritized in the redshift determination process?

Answer: Spectroscopic redshifts are the most reliable, so if a matching optical source has an associated spectroscopic redshift, we use that value as the redshift of the source. If the source does not have a spectroscopic redshift, but does have an associated photoz2 redshift, we used the specific photoz2 redshift depending on the apparent magnitude of the source. The errors for the photoz2 photometric redshifts are much smaller than the errors associated with the more simply calculated photoz photometric redshifts. In other cases, we used the redshift given by the photoz table as the redshift of the source.


Question: What does the study by Friebe et al. (2022) reveal about the gaze cueing effect in human-robot interaction, and how does it compare to previous findings?

Answer: The study by Friebe et al. (2022) found a gaze cueing effect of 14 ms when the target stimulus was predicted by the gaze of an embodied robot. This effect is smaller than the 25 ms found by other studies but is consistent with findings from more controlled settings. The study also showed that the virtual version of the robot consistently triggered a gaze cueing effect, indicating that gaze following in HRI is not limited to physically present robots. Additionally, there was no significant difference in gaze following behavior between the presence conditions, suggesting that physical presence does not significantly influence the gaze cueing effect.


Question: How does the Goldstone theorem ensure the absence of infrared divergences in the scalar-sector Goldstone mode of the extended-AHM in Lorenz gauge?

Answer: The Goldstone theorem, together with the U(1)Y Ward-Takahashi identities, ensures that the extended-AHM T-matrix vanishes as one of the pion momenta goes to zero (i.e., 1-soft-pion theorems), provided all other physical scalar particles are on mass-shell. This asserts the absence of infrared (IR) divergences in the scalar-sector Goldstone mode (in Lorenz gauge). Although individual Feynman diagrams are IR divergent, those IR divergent parts cancel exactly in each order of perturbation theory.


Question: How does the time at which Alice starts clinching in auction LOW compare to the time at which bidder 1 starts clinching, given that bidder 1 has a higher initial budget than Alice in HIGH?

Answer: Alice starts clinching at least δ later than the time instant at which bidder 1 starts clinching, i.e., q0 ≥ y0 + δ, where δ = B1 − B0∗.


Question: What are the key differences between the abrasion model proposed in this paper and the previous models, and how do these differences affect the morphological outcomes?

Answer: The key differences include (i) the rate of volume loss being associated with radial abrasion speed rather than speed normal to the surface, and (ii) a false interpretation of zenith angle leading to nonconstant abrasion speeds on convex surfaces and inconsistent results. The previous model predicted surface singularities, but otherwise radically different morphological characters compared to the current model.


Question: How is the crystallographic T-duality map defined in terms of the Poincaré line bundle and the homomorphisms involved?

Answer: The crystallographic T-duality map is defined as the composition φTtG := π! ◦([P] ⊗V/Π×ˆΠ ) ◦ˆπ∗, where ˆπ∗ is the pull-back, [P] ⊗V/Π×ˆΠ is the internal tensor product with the Poincaré line bundle [P], and π! is the push-forward. The Poincaré line bundle P is given by (V × ˆΠ × C)/{(v, χ, z) ∼(v + t, χ, χ(t)z)} with the φ-twisted G-action g · [v, χ, z] = [gv, χ, φ(g)z].


Question: What are the conditions for thermodynamical equilibrium in a universe filled with dark energy and bounded by the horizon?

Answer: The conditions for thermodynamical equilibrium in a universe filled with dark energy and bounded by the horizon are: i) ˙Sh + ˙Sfh ≥ 0 (the entropy function cannot decrease, the second law of thermodynamics) and ii) ¨Sh + ¨Sfh < 0 (the entropy function attains a maximum), where Sh and Sfh are the entropies of the horizon and the fluid within it, respectively.


Question: What are the limitations of current alternatives to MdJRQA, and how does MdJRQA address these limitations?

Answer: Current alternatives such as bi-variate analyses (Cross-Correlation, Cross-Recurrence, relative phase analysis, convergent-cross mapping) only capture bivariate relationships among individual observables and do not capture higher-level dynamics. Reducing the dimensionality of the data, for example by averaging across all component time series, leads to problems of normalization or weighting and risks averaging out interesting dynamics. Other techniques like principal component analysis or fractal analysis are also vulnerable to auto-correlation properties. Correlation methods like canonical correlation or multidimensional cross-recurrence quantification analysis require a matching number of data points and variables. MdJRQA, however, generalizes to situations where we seek to quantify correlation between two multivariate time series that do not have a matched number of observables, and it retains the dynamics of the time series, avoiding the loss of information inherent in fluctuations that are otherwise lost in the averaging process.


Question: How is the algebra D0 defined and what properties does it have in the context of a complete n smooth representation M?

Answer: D0 is defined as the algebra consisting of those φM where φ comes from Ln. Assuming M is n square, D0 is a subalgebra of the Crsn with domain ℘(Cn(M)), so D0 ∈ Crsn. The unit Cn(M) of D0 is symmetric and closed under substitutions, so D0 ∈ Gn. Since M is n flat, cylindriﬁers commute by definition, hence D0 ∈ CAn.


Question: How does the use of Entity Generalization reduce the number of features in the Relation TM compared to the Propositional TM?

Answer: Using Entity Generalization, the number of features no longer depends on the cardinality of set En,1≤n≤e in the context of the whole dataset. Instead, it only depends on the context of the single sample. Thus, if each sample contains r Relations, and a Relation R involves e different entities (E1, E2, ..., Ee), and maximum possible cardinality of sets E1, E2, ..., Ee are |E1| = |E2| = ... = |En| = r, the number of features become o = { |r| × |r| × ... × |r| } × r = r(e+1). In most scenarios, this number is much lower than the one obtained without the use of Entity Generalization.


Question: How are the bounding lines for the high-resolution TVD region on non-uniform meshes defined, and what are their non-dimensional slopes?

Answer: The four bounding lines for the high-resolution TVD region on non-uniform meshes are defined as follows: φ− = (2 + a + b) / (1 + a) * f, φleft = (2 + a + b) * f, φ+ = (2 + a + b) / (1 + b) * (1 − f), and φright = (2 + a + b) * (1 − f).


Question: How does the monodromy action relate to the lifting of loops in a topological cover, and what is its functorial property?

Answer: The monodromy action is defined by lifting loops at a point x in the base space X to paths in the cover Y starting at a point y in the fiber p^(-1)({x}). Specifically, for a loop γ at x, the monodromy action γ ↷ y is the endpoint of the unique path eγ in Y such that p ◦ eγ = γ and eγ(0) = y. The monodromy action is functorial in the sense that for two covers p : Y → X and q : Z → X and a homomorphism ϕ : Y → Z, the induced map ϕx : p^(-1)({x}) → q^(-1)({x}) is π1(X; x)-equivariant. This means that for any loop γ at x, γ ↷ ϕ(y) = ϕ(γ ↷ y).


Question: How does the second stage cost function v(x1α, x1γ, ˜ξ) vary based on the realization of parameters, and what is the criterion for partitioning the support of (˜δ1, ˜δ2)?

Answer: The second stage cost function v(x1α, x1γ, ˜ξ) takes different forms depending on the realization of parameters. The support of (˜δ1, ˜δ2) is partitioned based on whether hα ≥˜δ1γhγ or hα < ˜δ1γhγ. This criterion determines the task to allocate a worker cross-trained to work on task γ, when needed. If hα ≥˜δ2γhγ, it is more profitable to use the fully productive capacity of task α to satisfy ˜dα when needed. If hα < ˜δ2γhγ, it may be more profitable to shift some of this capacity to task γ and lose task α demand.


Question: What specific diseases were used in the evaluation of the ConvAE model, and how were the patients for these diseases selected?

Answer: The specific diseases used in the evaluation of the ConvAE model were Crohn's disease (CD), breast cancer (BC), prostate cancer (PC), and attention deficit hyperactivity disorder (ADHD). Patients with these conditions were selected using SNOMED–CT (Systematized nomenclature of medicine – clinical terms) to find all patients in the data warehouse diagnosed with these conditions.


Question: How does the HCRF model compare to other methods in terms of Dice, IoU, and RVD metrics for image segmentation?

Answer: The HCRF model achieves the highest Dice value of 0.4629, an IoU of 0.3259, and an RVD of 1.4135, indicating overall better image segmentation performance compared to other methods.


Question: What is the impact of inaccurately computed inner iterations on the outer iteration in the context of the JBD process?

Answer: When the inner iteration is inaccurately computed, the outer iteration is not equivalent to the Lanczos bidiagonalization of QA and QL any longer but with a perturbation of order O(κ(C)τ), where κ(C) is the condition number of C and τ describes the solution accuracy of (1.2).


Question: How is the force fB due to the interaction between the vortex and the boundary calculated using the method of images?

Answer: The force fB is calculated by removing the boundary, regarding all the space as the superconductor, placing an image antivortex at (x, y) = (−x0, 0) to satisfy the boundary condition, and evaluating the force due to the image. The force is given by fB(x0) = jimg(x0) × φ0ˆz = −(φ2_0/4πµ0λ2x0)ˆx, where jimg(x0) = −φ0/(2πµ0λ2 · 2x0)ˆy is the current circulating the image antivortex for x0 < λ.


Question: What is the outcome of applying Algorithm 3 to the TCP model under Threat Model 1 with channels, and what does it indicate about the existence of a For-all attack?

Answer: Algorithm 3 returns a non-empty HCNa with 52783 states and 626 marked states, and Lm(HCNa) contains the string SY NBC3.SY NC3N.SY N ACKNC4.SY N ACKC4B.ACKBC3 which steers GTM1sm to its marked states. This indicates that there exists a For-all attacker SP with respect to Ga and Ha in this setup, and the attacker can always eventually win by disabling some transitions in Ga.


Question: What specific conditions are required for the convergence of the neural network with exponential activation according to Theorem 1.1?

Answer: The conditions required for the convergence of the neural network with exponential activation according to Theorem 1.1 are: 1) Let λ > 0 denote the minimum eigenvalue of the neural tangent kernel with respect to exponential activation. 2) Let m = Ω(λ−2 log(n/δ)n2+o(1)) represent the number of neurons. 3) Let wr be random Gaussian weights and ar be the random {−1, +1} weights. 4) Let η = Θ(λ/(mn2+o(1))) denote the learning rate of the gradient descent algorithm. 5) Let T = Ω(λ−2n2+o(1)·log(n/ǫ)) denote the number of iterations of the gradient descent algorithm. With these conditions, after running the algorithm with T iterations, and with probability at least 1 − δ, we obtain a w(T) such that ∥F(T) − y∥2^2 ≤ ǫ.


Question: What is the significance of the Gauss-Bonnet term in the context of the Einstein-Gauss-Bonnet (EGB) gravitational model, and how is it motivated?

Answer: The Gauss-Bonnet term in the EGB gravitational model is significant because it is a quadratic term in the curvature that appears in the action. It is motivated by string theory and is given by L2 = RMNPQRMNPQ − 4RMNRMN + R2. The inclusion of this term is crucial for the model's application in cosmology, particularly in explaining the accelerating expansion of the Universe based on supernovae (type Ia) observational data.


Question: What is the detection threshold used for identifying sources in the CTA GPS, and how does it compare to the H.E.S.S. GPS?

Answer: A threshold of TSDet > 30 is used for the detection of sources in the CTA GPS. This threshold is adopted from the H.E.S.S. Collaboration (2018b) and leads to an estimated false-positive fraction of 3% for the H.E.S.S. GPS. However, the false-positive rate with this detection threshold is likely larger than 3% for the CTA GPS, since the CTA point spread function is narrower than for the H.E.S.S. instrument.


Question: What is the significance of proving Soergel's conjecture on the characters of indecomposable Soergel bimodules, and how does it relate to the Kazhdan-Lusztig polynomials?

Answer: Proving Soergel's conjecture on the characters of indecomposable Soergel bimodules implies that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. This, in turn, provides an algebraic proof of the Kazhdan-Lusztig conjecture.


Question: What is the estimate for the means An in Schur’s problem for the unit disk, and for which values of n does it hold?

Answer: The estimate for the means An in Schur’s problem for the unit disk is |An| ≤ 8√(log n / n), for n ≥ max(M, 55).


Question: What are the numerical results for the unoptimized multilayer shells with different numbers of layers, and how do they indicate the cloaking efficiency?

Answer: The numerical results for unoptimized multilayer shells with different numbers of layers (M = 2, 4, 6, 8, 10, and 12) are presented in Table 1. The values of J(k), Ji(k), and Je(k) decrease as the number of layers increases, with J(k) taking the value 1.6 × 10−2 for M = 12. These results indicate low cloaking efficiency of the designed cloak (Ω, k).


Question: What are the three qualitatively different dynamical regimes observed in the system, and what are the conditions for each regime?

Answer: The three qualitatively different dynamical regimes observed in the system are: (1) Rabi oscillations, which occur when the interaction term is weak compared to the driving strength (g/4π ≪ K), (2) chaotic dynamics, which are expected for intermediate interaction strength, and (3) self-trapping, which occurs when the particle interactions dominate over the driving strength (g/4π ≫ K). The critical interaction strength for the self-trapping transition is given by gc = 8π max(|Γ+|, |Γ−|).


Question: What specific measurements would the EIC perform to create detailed images of the proton's gluonic matter distribution and sea quarks?

Answer: The EIC would measure exclusive processes to create detailed images of the proton's gluonic matter distribution and sea quarks, as shown in Fig. 1.4. These measurements would reveal aspects of proton structure that are intimately connected with QCD dynamics at large distances.


Question: What are the mechanical modes and their characteristics for a Z-cut LN waveguide, and how do they vary with the rotation angle θZ?

Answer: For a Z-cut LN waveguide, the mechanical dispersion analysis reveals a high density of modes with different mechanical polarizations. The bulk acoustic velocities for this orientation of LN are shown in the figure, with the slowest mechanical modes (below the QS1 line) exhibiting SAW speed characteristics. The mechanical frequencies for these modes, fixed at βm = 14.37 µm−1, were calculated as a function of the rotation angle around the Z-axis, θZ. Modes 1 and 2 are localized in the lower corners of the suspended LN structure and are not of interest, while mode 3 shows promising displacement in the central waveguide region. The normalized SBS gains (GB) normalized by the mechanical quality factor (Qm) reveal that the lowest frequency modes may have reasonable normalized SBS gains, with the maximum normalized gain of GB/Qm = 0.113 W−1m−1, which is about 30% of the normalized gain observed for backward SBS in suspended silicon waveguides.


Question: How does an increase in differential rent in Section I affect the rate of surplus-value and the rate of profit?

Answer: In the first case, where the increase in differential rent occurs in Section I, there is an increase in the rate of primary surplus-value k, and an increase in c*’. The parallel increase in variable c*’ has no effect on the rate of surplus value. Therefore, in the first case, one is assured to be in the 'Ricardian' situation where an increase in differential rent and the rate of surplus-value entails a decline in the rate of profit.


Question: How does the probability of a moving aphid stopping (PMS) change with the distance to its nearest neighbor, and what is the functional form used to model this relationship?

Answer: The probability PMS decreases monotonically with the distance to the nearest neighbor d and levels off. It is modeled using the functional form PMS(d) = P ∞_MS + (P 0_MS − P ∞_MS) e^(-d/dMS), where P 0_MS is the probability of an aphid becoming stationary when infinitely close to its nearest neighbor, P ∞_MS is the probability of transitioning when isolated, and dMS characterizes the transition between these two regimes.


Question: How does the concept of virtual menus extend the set of observed menu paths, and what is the relationship between the observed and extended stochastic choice functions?

Answer: The virtual set of menus, denoted by ¯J t, includes all possible nonempty subsets of Xt, extending the observed set of menus J t. The set of all menu paths, including observed and virtual ones, is denoted by ¯J, where the observed menu paths J are a subset of ¯J. The extended stochastic choice function ¯ρ is defined over all menu paths in ¯J, and it agrees with the observed stochastic choice function ρ on the observed menu paths, meaning ¯ρ(xi|j) = ρ(xi|j) for all j ∈ J and i ∈ Ij.


Question: How does the Nusselt number vary with the Reynolds number for the ice ball fixed in the flow produced by two rotating discs, and how does this compare to the one-disc configuration?

Answer: The Nusselt number (Nu) is in the range [100, 400] for the Reynolds number (ReD) in the range [5·10^3, 65·10^3] for both the two-discs and one-disc configurations. Despite the different turbulence levels, the Nusselt number is of the same order of magnitude in both cases, indicating a weak impact of local turbulence on heat transfer in fully turbulent flows. The Nusselt number is well fitted by an empirical power law Nu ∼ α + βReD^γ in both configurations.


Question: How does the Es-TPU membrane's reflectivity and IR emittance change after exposure to soil and subsequent cleaning?

Answer: The reflectivity of the Es-TPU membrane with soil attached to the surface decreased significantly, especially in the UV–vis range due to the absorption of sunlight by the soil, but it is restored upon cleaning. In contrast, soil contamination affected the infrared emissivity negligibly.


Question: Which specific pulsars have been detected by LWA1, and what are some notable observations?

Answer: LWA1 has detected several pulsars, including B0531+21 (the Crab Nebula pulsar) and J2145-0750 (a millisecond pulsar). The giant pulses from the Crab Nebula pulsar are detectable throughout the frequency range of LWA1, and the millisecond pulsar J2145-0750 has been detected using coherent dedispersion at frequencies from 37 MHz to 85 MHz.


Question: What are the specific values of the GUT Higgs VEVs used in the anomalous U(1)A GUT models, and how do they influence the partial decay widths of nucleon?

Answer: In the anomalous U(1)A GUT models, the values of the GUT Higgs VEVs are x = 1 × 10^16 GeV, vc = 5 × 10^14 GeV, and vφ = 5 × 10^15 GeV. These values are fixed by their U(1)A charges and are important for calculating the partial decay widths of nucleon.


Question: How does the average particle velocity in the steady state of a dissipative quantum particle hopping on a one-dimensional lattice relate to the topological winding number?

Answer: The average particle velocity in the steady state is proportional to the topological winding number and to the inverse of the average time between quantum jumps. Within each phase, the average velocity depends smoothly on system parameters, but nonanalytic behavior arises at phase transition points where the winding number changes.


Question: How does SAM compare to LIFE and HRFAE in terms of visual quality and age transformation accuracy?

Answer: SAM is able to more naturally alter facial features that change as we age, such as facial hair and wrinkles, compared to LIFE, which introduces unwanted artifacts. When compared to HRFAE, SAM achieves visually pleasing results while better modeling changes in head shape and texture, despite HRFAE's ability to generate high-resolution images. However, all methods struggle with modeling changes in hair color and capturing receding hair lines.


Question: How does the Euler class of the pull-back bundle relate to the open string product in the context of the given morphisms and inclusions?

Answer: By formula (9) (Naturality of Euler class), e(g◦q)∗(TM) = (q ◦ev1/2)∗(eν), where eν is the Euler class of the normal bundle. This relationship is used to show that µ∗f,g,h ◦µ!f,g,h(x) = x ∪ev∗1/2 ◦q∗(eν).


Question: How are the derivatives of ζ with respect to x, y, and z derived using Proposition 2.2-(iv) and the given equations?

Answer: The derivatives of ζ are derived as follows: ζx = (κ2_3)x − 2κ3((κ3)x + (e−P)x tan ϕ) + 2e−P (Ix,z)z − 2(e−P)y sin^2 ϕ Ix,y − 2((e−P)z + e−P ϕz tan ϕ)Ix,z + 2(ψxzz + ϕxzz cot ϕ − ϕzϕxz sin^−2 ϕ)e−2P. ζy = (κ2_3)y − 2κ3((κ3)y − (e−P)y cot ϕ) + 2e−P (Iy,z)z − 2 cos^−2 ϕ(e−P)x Ix,y − 2[(e−P)z − e−P ϕz cot ϕ]Iy,z + 2(ψyzz − ϕyzz tan ϕ − ϕzϕyz cos^−2 ϕ)e−2P. ζz = (κ2_3)z − 2κ3((κ3)z + e−P ϕz) + 2e−P((Ix,z)x + (Iy,z)y) + 2(2e−Pϕx tan ϕ − (e−P)x cos^−2 ϕ)Ix,z − 2(2e−Pϕy cot ϕ + (e−P)y sin^−2 ϕ)Iy,z + 4 cos 2ϕ sin^−1 2ϕ((ϕxx + ϕyy + ϕzz)z 2 + (ϕxx − ϕyy) − ϕzz cos ϕ sin^−1 2ϕ ϕz − ϕxϕxz cot ϕ + ϕyϕyz tan ϕ)e−2P.


Question: What specific elements are included in the finite set used to control the lengths or ratios of lengths during the iterative shortening process?

Answer: The finite set includes: (1) the fixed set of generators of dominant QH vertex groups and generators of dominant edge groups in Λ1, ..., Λi, (2) the elements in a ball of radius i in the Cayley graph of L with respect to the generators s1, ..., sr, and (3) the fixed set of generators of some of the QH vertex groups and some of the edge groups in Λi+1.


Question: What are the Hp- and La-numbers for the CeAl simulation with a fixed 8x8x8 FE mesh and varying enrichment support radii?

Answer: For the CeAl simulation with a fixed 8x8x8 FE mesh and varying enrichment support radii, the Hp- and La-numbers are shown in the right plot of Figure 5.7. The Hp-numbers are between 2 and 6, and the La-numbers are between 0 and 4, with no apparent dependence on conditioning.


Question: What specific observations in the simulation suggest the presence of a singularity at t ≈ 0.86?

Answer: The distance between ωmax and |∇θ|max suddenly drops down to almost zero around t ≈ 0.86; for enough fine resolutions, the ωmax(t) curves converge before t ≈ 0.86 and suddenly diverge after t ≈ 0.86; β > 2.


Question: What are the key principles of the OECD Guidelines on the Protection of Personal Privacy and Transborder Flows of Personal Data, and how do they extend the original FIPs?

Answer: The OECD Guidelines on the Protection of Personal Privacy and Transborder Flows of Personal Data, adopted in 1980, extended the original five Fair Information Practices (FIPs) into eight principles: 1. Collection Limitation: Collection should be lawful, fair, and with the knowledge or consent of the data subject. 2. Data Quality: Personal data should be purpose-relevant, accurate, complete, and kept up-to-date. 3. Purpose Specification: Purposes should be specified earlier than collection and complied with. 4. Use Limitation: Personal data should not be disclosed, made available, or used for non-specified purposes. 5. Security Safeguards: Personal data should be protected by reasonable security safeguards. 6. Openness: There should be a general policy of openness about developments, practices, and policies with respect to personal data. 7. Individual Participation: An individual should have the right to access his data, be timely informed on data collection, be given a disputable reason for denied lawful request, and challenge his data to have the data erased, rectified, completed, or amended. 8. Accountability: A data controller should be accountable for complying with measures which give effect to the principles stated above.


Question: What are the specific technical requirements for the laser and timing systems to achieve the mission's ranging accuracy?

Answer: The pulse width is 50 ps, the repetition rate 100 Hz, and the pulse energy is around 10 mJ. For a ranging uncertainty of 0.9 mm in a distance of 2.55 × 10^11 m (1.7 AU), the laser/clock frequency needs to be known to one part in 10^14 by comparison with ground clocks over a period of time. Stability to 6 × 10^−14 in 1,000 s (round-trip time) is required.


Question: What are the relative amplitudes and phases for the D0 →K∗−π+ and D0 →K∗+π− decays in the D0 →KSπ+π− Dalitz plot, and how do they compare to the experimental values?

Answer: For the D0 →K∗−π+ decay, the relative theoretical amplitude is 1.472±0.187 with a phase of 188±7 degrees, while the experimental amplitude is 1.735±0.005 with a phase of 133.5±0.2 degrees. For the D0 →K∗+π− decay, the relative theoretical amplitude is 0.114±0.015 with a phase of 27±7 degrees, while the experimental amplitude is 0.164±0.003 with a phase of –44.0±1.1 degrees.


Question: What does the spectral behavior across the water snowline in V883 Ori indicate about the dust properties and grain growth?

Answer: The spectral behavior across the water snowline confirms recent model predictions: dust fragmentation and the inhibition of grain growth at higher temperatures result in soaring grain number densities and optical depths.


Question: Under what conditions does the FKM model exhibit a third-order GWW phase transition on a general graph?

Answer: The FKM model on a general graph exhibits a third-order GWW phase transition in the scaling limit when the graph consists of plaquettes (fundamental cycles) of the same size, as in the square lattice case.


Question: How does the feedback loop between ML prices and realized sale prices affect the estimation of machine learning error in the housing market?

Answer: The machine learning error can be estimated using the ML price and the realized sale price. However, this is an approximation since the true home value is not observed. The confounded ML error is inflated, and the confounded ML error estimate is deflated if α is large enough. The confounded ML error estimate is decreasing in α if α is large enough, and these conditions are always true at equilibrium α*.


Question: What is the proof for the eviction- and reception-compatibility of the propositional logic satisfaction system Λ(Prop)?

Answer: Since we need only to consider finitely many symbols, there are finitely many possible valuations. If there are n propositional atoms, there are at most 2^n distinct models, meaning that there are at most 2^m distinct sets of valuations where m = 2^n. Consequently, FR(Λ(Prop)) is finite. Additionally, since both the empty set and the set of all valuations are representable in this satisfaction system, we obtain as a consequence of Corollary 17 that Λ(Prop) is both eviction- and reception-compatible.


Question: What are the three conditions that must be satisfied for a weighted graph to be a θ-Harnack graph under the ellipticity condition?

Answer: The three conditions are: 1. The volume doubling condition, where there is a constant CD such that for all x ∈ X and all r > 0, π(B(x, 2r)) ≤ CDπ(B(x, r)). 2. The Poincaré inequality with constant CP rθ holds on the ball B(x, r) for all x ∈ X and all r > 0. 3. The cut-off function existence property CS(θ) is satisfied.


Question: What is the key advantage of using a Slice sampler over a basic Gibbs sampler in the context of the posterior distribution discussed in the query text?

Answer: The Slice sampler does not require sampling from the full conditional distributions, and it only needs to know the target full conditional posterior up to normalization, which is important since solving the normalizing constant in this model is not possible analytically.


Question: What is the conformal pairing between three-point structures and their shadow structures, and how is it computed?

Answer: The conformal pairing between three-point structures and their shadow structures is defined as P a,b 123 = (T a 123, T b ˜1˜2˜3) ≡ ∫ ddx1 ddx2 ddx3 vol(SO(d + 1, 1)) ⟨O1O2O3⟩a ⟨˜O1 ˜O2 ˜O3⟩b. This can be simplified to P a,b 123 = 1 / (2d vol(SO(d − 1))) ∑ ϵ1,ϵ2,ϵ3 T a 123(ϵ1, ϵ2, ϵ3)(1) T b ˜1˜2˜3(ϵ∗1, ϵ∗2, ϵ∗3)(1), where the denominator is the volume form of the 'little group' that keeps the frame (0, 1, ∞) fixed. For the vector-vector-general ⟨V1V2O⟩ case, the pairings between Even or Odd structures are given by P E 11O = 16PsNE 11O (J3+1)(J3+2) / (J3−1)J3 and P O 11O = 16PsNO 11O (J3+1)(J3+2) / (J3−1)J3.


Question: What are the key observations regarding the impact of noise on the reconstruction of heat source locations using amplitude and phase information?

Answer: Using the amplitude information, the location detection is almost unchanged as noise is increased, but from around 5% onwards, the reconstruction breaks down very suddenly. For some simulations using phase information, the reconstructed spectrum amplitude values change radically, which can be explained by a best fitting source location around 10 units towards or away from the true location.


Question: What is the estimated value of the chiral condensate ⟨¯ψψ⟩/e^4 for N = 2, and how does it compare to the numerical solution?

Answer: The estimated value of the chiral condensate ⟨¯ψψ⟩/e^4 for N = 2 is approximately -4.64 × 10^-12, which is very close to the numerical solution of -5.57 × 10^-12.


Question: What does the simulation with semi-random perturbations reveal about the robustness of the resonant run-up mechanism?

Answer: The simulation with semi-random perturbations shows that the resonant run-up mechanism is robust, and the resonant frequency dominates the run-up, as the run-up timeseries obtained with the perturbed signal is the same as that obtained with the unperturbed monochromatic wave.


Question: What are the limitations of existing PML approaches that focus on the smoothness or low-rank assumptions?

Answer: These approaches either mainly focus on the smoothness assumption that (dis-)similar instances should have (dis-)similar ground-truth labels, or the low-rank assumption that the ground-truth label matrix should be low-rank. However, these assumptions cannot handle cases where two instances without any common candidate label but with high feature similarity, or two instances with some overlapped candidate labels but with low feature similarity. As a result, smoothness-based methods ignore the negative information that two instances with high (low) feature similarity but with low (high) semantic similarity from the candidate labels, and do not make good collaborative use of the information from the label and feature space for PML.


Question: What is the relationship between the operators P (k)ab and the Lie algebra structure for N ≥ 3?

Answer: For N ≥ 3, P (0)ab = 0 and P (ei)ab = 0 for i ≠ a, b. The subset of elements P (ei)ab spans an abelian algebra with generators {P (e2)12, P (e1)i1 | i = 2, ..., N}.


Question: What are the specific performance metrics of the proposed system on the BDD100K and local datasets, and how does it compare to other lane detection approaches?

Answer: The proposed system achieves 82.57% F1 score, 74.14% IoU, and 82.21% confidence on the local dataset, and 76.29% F1 score, 75.84% IoU, and 83.55% confidence on the BDD100K dataset. Compared to other lane detection approaches, such as CLRNet and CondLaneNet, the proposed system outperforms them in terms of F1 score and IoU on both datasets.


Question: How does the entropy difference between the non uniform black string and the uniform black string vary with the horizon radius, and what does this imply for the phase transition?

Answer: Above some critical values of the horizon radius, the uniform black string is more entropic than the non uniform one, while for larger values of rh, the non uniform black string becomes more entropic. This implies that the phase transition can be of first order or of higher order for the same value of the dimension, depending on the value of the horizon radius.


Question: What are the key components and transformations involved in the neural network architecture described in the candidate text?

Answer: The key components of the neural network architecture include the input layer, hidden layers, and output layer. Each layer is fully connected to the neighboring layers, and the transformations involve nonlinear functions. The parameters, referred to as weights and biases, are adjusted using gradient descent to better fit the dataset. The activation function, φ, defines the transformation represented by edges, and the sum total of the edges going into a neuron is a composition of the activation function with an affine function evaluated at the previous layer, of the form: φ(w · x + b).


Question: What is the definition of a Fixed-to-Variable Complementary Delivery (FV-CD) code and what are its key components?

Answer: A set (ϕn, bϕn(1), bϕn(2)) of an encoder and two decoders is an FV-CD code for the source (X, Y ) if and only if ϕn: X^n × Y^n → B^*, bϕn(1): ϕn(X^n, Y^n) × Y^n → X^n, bϕn(2): ϕn(X^n, Y^n) × X^n → Y^n, e(X)_n = Pr{X^n ≠ bX^n} = 0, e(Y)_n = Pr{Y^n ≠ bY^n} = 0, where bX^n = bϕn(1)(ϕn(X^n, Y^n), Y^n), bY^n = bϕn(2)(ϕn(X^n, Y^n), X^n), and the image of ϕn is a prefix set.